A51385	----	The poor man's dyal· With an instrument to set it. Made applicable to any place in England, Scotland, Ireland, &c. By Sir Samuel Morland knight and baronet. 1689. Morland, Samuel, Sir, 1625-1695. 1689 Approx. 9 KB of XML-encoded text transcribed from 4 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2006-06 (EEBO-TCP Phase 1). A51385 Wing M2781B ESTC R221912 99833157 99833157 37632 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A51385) Transcribed from: (Early English Books Online ; image set 37632) Images scanned from microfilm: (Early English books, 1641-1700 ; 2191:21) The poor man's dyal· With an instrument to set it. Made applicable to any place in England, Scotland, Ireland, &c. By Sir Samuel Morland knight and baronet. 1689. Morland, Samuel, Sir, 1625-1695. [2], 5, [1] p. ; ill. And are to be sold at all the button-sellers, cutlers and toyshops about the town. And will be shortly publisht in several other ... and dimensions, for the good of the publick, and for the con... of the manufacture of our nation, [S.l.] : [1689] Date of publication from title. Copy stained and torn. Reproduction of the original in the Lambeth Palace Library, London. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org). The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data. Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Dialing -- Early works to 1800. 2005-12 TCP Assigned for keying and markup 2006-02 SPi Global Keyed and coded from ProQuest page images 2006-03 Andrew Kuster Sampled and proofread 2006-03 Andrew Kuster Text and markup reviewed and edited 2006-04 pfs Batch review (QC) and XML conversion THE Poor Mans DYAL . WITH AN INSTRUMENT To Set It. Made applicable to any place in ENGLAND , SCOTLAND , IRELAND , &c. BY Sir SAMVEL MORLAND Knight and Baronet . 1689. 〈◊〉 are to be Sold at all the Button-Sellers , Cutlers , and 〈◊〉 about the Town . And will be shortly publisht in several other 〈◊〉 and Dimensions , for the Good of the publick , and for the 〈◊〉 of the Manufacture of our Nation . The Poor Man's Dyal , With an Instrument to Set it . To Set the Dyal . FIrst , Set the small Instrument upon any Level place , where the Sun comes two or three hours before , and as many after Noon , and by the 4 Marks at the bottom , make the 4 Points ( A , B , C , D ) and by those Marks make two Lines , ( A , C ) and ( B , D ) crossing each other in the Point ( O. ) Then mark the Point ( E ) where the shadow of the Pin terminates in the Forenoon . And having from the Distance ( O E ) described a Circle , watch in the Afternoon , when the shadow of the Pin cuts the Circle in the Point ( F ) for the Line ( E F ) is a true East and West Line ; And the Hours of ( VI ) and ( VI ) upon the Dyal , being placed upon the said Line , the Dyal is truly set . DIRECTIONS . FOR London , or any place within 20 Miles , the Dyal must be placed exactly Level ; But for the following place , the North-side of the Dyal ( where is the Hour of ( XII ) must be elevated higher than the Opposit , or South-side , as is hereafter exprest , which every Carpenter and Joiner knows how to perform . IN ENGLAND . BEDFORD , about 1 twentieth part of an Inch. Berwick , 1 tenth . Buckingham , 3 hundreths . Cambridge , 1 twentieth . Carlisle , 1 fourth , or a quarter . Chester , 1 tenth . Colchester , 1 twentieth . Darby , 1 tenth . Durham , 1 fourth . Glocester , 3 hundreths . Hartford , 1 fiftieth . Hereford , 1 twentieth . Huntington , 1 sixteenth . Ipswich , 1 twentieth . Kendal , 3 sixteenths . Lancaster , 1 fifth . Leicester , 1 sixteenth . Lincoln , 1 tenth . Northampton , 1 sixteenth . Norwich , 1 tenth . Oxford , 1 fiftieth . Stafford , 1 tenth . Shrewsbury , 1 tenth . Stanford , 1 tenth . Warwick , 1 sixteenth . Worcester , 1 sixteenth . York , 1 fifth . WALES . ANglesey , 1 eighth . Bermouth , 1 tenth . Brecnock , 1 twentieth . Cardigan , 1 sixteenth . Caermarthen , 1 twentieth . Carnarvan , 1 twentieth . Denbigh , 1 tenth . Flint , 1 tenth . Landaff , 1 hundreth . Monmouth , 1 fiftieth . Montgomery , 1 twentieth . Pembroke , 1 fiftieth . Radnor , 1 twentieth . St. David , 3 hundreths . The Isle of Man , 3 sixteenths . SCOTLAND . ABerdeen , 43 hundreths . Dunblain , 1 third . Dunkel , 3 eights . Edinburgh , 3 tenths . Glascow , 3 tenths . Kinsale , 45 hundreths . Orkney , 59 hundreths . St. Andrews , 1 third . Skyrassin , 1 half . Sterlin , 3 tenths . IRELAND . ANtrim , 1 fifth . Arglas , 15 hundreths . Armagh , 17 hundreths . Caterbergh , 17 hundreths . Clare , 8 hundreths . Cork , 3 hundreths . Drogheda , 1 eighth . Dublin , 1 tenth . Dundalk , 1 eighth . Galloway , 1 tenth . Kenny , 7 hundreths . Kildare , 1 tenth . Kingstown , 1 eighth . Knockfergus , 1 fifth . Kinsale , 1 hundreth . Limmerick , 7 hundreths . Queenstown , 1 tenth . Waterford , 1 twentieth . Wexford , 1 sixteenth . Youghall , 3 hundreths . For the Places hereafter mentioned , the North-side of the Dyal must be lower than the South-side , viz. In ENGLAND . BRistol , 1 hundreth . Canterbury , 1 tenth . Chichester , 1 sixteenth . Dorchester , 1 twentieth . Exeter , 1 sixteenth . Gilford , 1 fiftieth . Reading , 1 hundreth . Salisbury , 3 hundreths . Truro , 7 hundreths . The ISLANDS . GVernsey , 1 eighth . Jersey , 1 eighth . Lindy , 1 hundreth . Portland , 7 hundreths . Wight , 1 twentieth . If it be required to fix any Dyal in any place of England , Scotland , &c. not mentioned in this Catalogue , it must be set according to the nearest of the places that are mentioned , and it will serve without any sensible Error , and much better than those ordinary Brass Dyals , which are usually made by ignorant Apprentices and Journey-men : Or that Cobbled Pewter Dyal , which was lately made by a Brazen-fac'd Founder , in Imitation of this Porr Man's Dyal , and deserves at least , by way of Transposition , the Name of That Man 's Poor Dyal ; For tho' he had the Wit to make a Circle of the same Diameter , and to set off all the Divisions of the Hours , Halfs , and Quarters from the other ; yet the Style is so False and Defective in all its Parts , that it is not to be mended by one , who knows nothing of a Dyal : Yet , notwithstanding , in one thing he is to be commended , that he has made the Hour-Lines so short , that the Shadow , of the False Gnomon , will not so easily discover his Errors . Besides , he has carefully fill'd up the Vacancies in the middle of he Dyal , with the Points of the Compass , which are only proper for an Upright Style , and clogg'd the Tail of it with five Leaden Bells , within a pair of Rams-horns , to ring aloud the praises of his Sheeps-head , for attempting to Imitate or Counterfeit another Man's Contrivance , without being able to perform it like an Artist : Forasmuch as any person who has the least knowledge of those matters , will soon distinguish between the true Original , ( which was first Calculated , and afterwards Exactly Delineated by Sir Samuel Morland's own Hands , and the Molds of the Style , which is the principal part of the Dyal , carefully Contrived and Corrected ) and that ignorant Founder's Counterfeit , and Ill-contrived Dyal , to which he has put a Date of 1690 , to let the World understand that it was none of his own Contrivance , but that he did Counterfeit one that was made before , viz. in 1689. And of which the Style comes False out of the Mold , and by Cutting and Scraping is made Ten Times worse . If by Accident in sending about , or otherwise , any of these Dyals , or the Stiles of them being but Block Tin , shall happen to be Bent ; Then with the help of a Square , or for want of that , of a quarter of a Sheet of any Paper folded up in a double fold , ( which makes an exact Square ) it may be bended back , and set right again . FINIS . 
A45349	----	A plain declaration of the vulgar new heavens flatform serving not onely fore this age, but also fore the future age of 100 years. Halley, Edmond, 1656-1742. 1679 Approx. 18 KB of XML-encoded text transcribed from 3 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2009-03 (EEBO-TCP Phase 1). A45349 Wing H452 ESTC R39228 18283100 ocm 18283100 107300 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A45349) Transcribed from: (Early English Books Online ; image set 107300) Images scanned from microfilm: (Early English books, 1641-1700 ; 1635:8) A plain declaration of the vulgar new heavens flatform serving not onely fore this age, but also fore the future age of 100 years. Halley, Edmond, 1656-1742. 4 p. s.n., [London? : 1679] Attributed by Wing to Halley. Date of publication suggested by Wing. Reproduction of original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org). The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data. Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Astronomy -- Observations -- Early works to 1800. Sun -- Observations -- Early works to 1800. Dialing -- Early works to 1800. 2008-04 TCP Assigned for keying and markup 2008-06 SPi Global Keyed and coded from ProQuest page images 2008-07 Paul Schaffner Sampled and proofread 2008-07 Paul Schaffner Text and markup reviewed and edited 2008-09 pfs Batch review (QC) and XML conversion A plain DECLARATION of the vulgar new HEAVENS FLATFORM . Serving not onely fore this Age , but also fore the future Age of 100 years . HEre you are at first to knowe , that the motion of the Sun and the time do alwayes concur , and therefore is the one the measure of the other . Fore by the Time is to be knowne the place of the Sun , and again by the Suns place you may knowe the Time : therefore you must either knowe the Time or the place of the Sun , both of them you may easily find in the Heavens-Flatform , do but lay the dial A or B on the desired day of the yeare and see then what degree the dial doth touch upon the Sodiack , and you wil find the thing desired . I. EXEMPLE . How to knowe at anny time of the yeare in what Degree of the Sodiack the Sun is . ●y Exemple on the first day of May , lay the dial A upon the suid day , and see what degree the dial doth cut upon the Sodiack , you wil finde it to be the 12th of Taurus being the place of the Sun. II. EXEMPLE . How at anny time to knowe the rising and going under of the Sun. Suppose it be the 24th . of May , then you must lay the dial on the said day , and see where it cuts the Sodiac , there you must make upon the dial a signe of chalck or anny thing else that may easily be rubbed out , which signe demonstrates the Sun , turn then the dial first so long to the East-side of Heaven , til the said signe comes to touch the crombe Horizon , see then without upon what houre and minute the dial doth lye on the houre cirkle , you wil find the Suns rising at 4 a clock in the morning , and if you turn the said point to the West-side of Heaven upon the Horizon , then you wil see the dial to lye upon 8 a clock in the evening , being the going under of the Sun , then you wil also see that the Sun riseth then 36 degr . 30 minut . from the East to the Northward , and by consequence so manny Degrees and min. lesse from the West to the Northward . III. EXEMPLE . Hou you may see in the Night by the Starrs what time it is , Suppose you doe but see anny acquainted Starre to rise or stand in the S. N. E. or Westward , let it be the three Kings arising the first day of October , in the night , and that by it you would knove how late it is ; then you must lay the dial A on the said day , and the dial B over the 3 Kings , and give then a signe upon the dial B over the 3 Kings , then you must firmly turn both dials alike to the Eastward , til the said signe upon the dial B doth cut or touch the Horizon , and see then upon what houre and minute the dial A lieth , you 'l find it to be 11 a clock in the evening , beingh the right time of the night . I. PROBLEME . How you may upon every Poles higth find the riseing and going under of the Heavens lights after you have taken the Poles higth , and the declination or anny aquainted Starres . You must place one foot of the Compassis in the Centrum of the Heavens-Mirrour , upon the Dial A , and the other downewards , as far as the Degree of the Declination of the Sun or Starres ; with this opening of the Compassis you must place the one foot upon the edge-side of the dial A upon the complement of the Poles higth which you have taken ; and slide the dial A towards 6 a clock , ( or to the dial B which may be laid along by 6 a clock ( or the edge-side of the dial B. ) See then how in anny degrees the dial A lieth upon the houre-circkle from 6 a clock , which wil be the true breadth of the rising and going under of the Sun or Starres . I. EXEMPLE . Desiring to knowe on the 21th . of June , where the Sun riseth and goeth under , being beginning of Cancer , on the Poles heigth of 52 degrees . Then you must place one foot of the Compassis in the Centrum of the dial A , and the other downewards as far as on 23 degr . 30 min. being then the Declination of the Sun ; with this opening of the Compassis you must place one foot on the edge-side of the dial A upon the Complement of the Poles heigth , being 38 degr . then you must turn the dial A towards 6 a clock , til the other foot of the Compassis comes triangularly to touch the lin . of 6 a clock . See then how manny degrees the dial A lieth off from 6 a clock . You 'l find it to be very neare 40 degr . 20 min. And so far doth the Sun then rise from the E. to the N. upon each Northern breadth of 52 degr . and goeth likewise 40 degr . 20 min. under from the W. to the Northward . And if the Sun be in the first degr . of Capricornus , then it is just the same . II. EXEMPLE . Desiring to know on the Northern breadth of 50 degr . how manny degrees the Southern Ey of the Bul called ●●debaran riseth from the E. to the N. Fore the doing of it , you must place one foot of the Compassis in the Centrum , 〈◊〉 the dial A , and the other as far as the Declination of Aldebaran being 16 degrees , with this opening of the Compassis you must place one foot upon the complement of the Poles heigth of 40 degrees , on the dial A , then you mus● turn the said dial til the other foot of the Compassis comes triangularly to touch te line of 6 a clock . See then how manny degrees the dial A lieth off from 6 a clock , you 'l find verry neare 25 deg . 20. min. And so manny d●g . doth Aldebaran then rise from the E. to the N. it doth also go under 25 deg . 20 min. lesse from the W. to the N. III. EXEMPLE . Desiring to knowe on the Southern breadth of 20 degr . how many degrees the Spica Virginis riseth from the E. 〈◊〉 the S. and that , because the declination of Spica Virginis is Sourhly . You must place one foot of the Compassis in the Center of the Dial A , and the other downewards as far as on 9 degr . being the declination of Spica Virginis , with this opening you must place one foot of the Dial A upon the complement of the Poles higth of 70 degrees , then you must turn the Dial A from the E. to the S. till the other foot of the compassis comes triangularly to touch the East line or the line of 6 a clock . See then how manny degr . the Dial A lieth off from 6 a clock , you find verry neare 10 degr . and so far doth S. Virginis from the E. to the S. and goeth like wise so far unter from the W. to the S. II. PROBLEME . How to find the rising and going under of the Sun , or of anny acquainted Starrs , and that upon every Poles higth . Place one foot of the Compassis in the center of the dial B , and the other downeward , along by the Edge-side as far as on the degree of the Declination of the Sun or Starrs , with this opening of the Compassis you must place one foot on the edge-side of the dial B upon the Complement of the Poles heigth , add slide the Dial B from the E. to the N. or S. til the other foot comes triangularly to touch the line of 6 a clock . Then you must see o● what houre and min. the dial B lieth , which is the true time of the Suns rising , which you may also being to the going under . I. EXEMPLE . Desiring to knowe the rising of the Sun , on the N. breadth of 25 degr . being the 21th . of June , when the Suns Declination is Northly 23 degr . 32 min. you must place one foot of the Compassis in the Centrum of the dial B , and open the other foot douwnewards as far as on 23 degr . 32 min. with this opening of the Compassis you must place one foot on the edge-side of the dial B upon the Complement of the Poles hitgh of 38 degrees , sliding the dial B from the E. to the N. til the other foot of the Compassis comes triangulary to touch the line of 6 a clock . See then upon what houre and min. the dial B lieth on the houre circkle , you 'l find it to be in the morning at 5 a clock 15 min. being the right ti●● of the Suns rising , the same is in the evening at 8 a clock 15 min. the Suns going under . II. EXEMPLE . Desiring to knowe on the Southern breadth of 40 degr . being on the 21 of June , at what time the Sun doth there rise . Then you must place one foot of the Compassis in the Centrum of the dial B , and the other foot downewards , as far as on 23 deg . 30 min. being at the said time the Declination of the Sun , with this opening you must place one foot of the Compassis on the dial B upon the complement of the Poles higth of 50 degr . and turn the dial B from the E. to the N. til the other foot of the Compassis comes triangularly to touch the East line of 6 a clock , see then upon what houre and min. the dial B lieth , you 'l find neare enough in the morning 17 houres 30 min. being there the rising of the Sun , the same is its going under in the evening at 4 a clock 35 min. Nota. You must knowe that if you wil , use the Heavens Flatform over the South-side of the Equinoctial Line , then you must take the house contrary to that as they are signed upon the Heavens Mirrour , fore that which is over the North-line 4 a clock in the morning , the same is Southly from the line 8 a clock in the morning , and so is the rest accordingly . III. EXEMPLE . Desiring to knowe on the Northern breadth of 40 degrees , being the first of August , what time the great Dog Syrius shal rise . Lay the Dial B over Syrius and the Dial A upon the first of August , then you must place one foot of the compassis upon the Center of the Dial B , and open the other as far as on 16 degr . 15 min. being the declination of Syrius , with this opening you must place one foot the compassis on the Dial B , upon the complement of the Poles higth of degrees . Then you must firmly turn both Dials alike from the E. to the S. , till the other foot of the compassis comes triangularly to touch the East line of 6 a clock . See then upon what houre and minutes the Dial lieth , you 'l find neare enough 4 a clock 42 minutes . Fore to find its going under , you must firmly turn both dials alike from the West to the Southward til the other foot of the compassis comes triangularly to touch the West line of 6 a clock , see then upon what houre and min. the Dial A lieth , you 'l find 2 a clock 48 minutes . III PROBLEME . How to find at al set times the Declination of the Sun upon the Heavens mirrorr . Which is indeed verry proffitable for al Sea men , fore it serves not onely fore this present Age , but also fore the future Age of 100 years ; when al Books that are made fore that purposse shal be of no worth . Fore to find the Declination of the Sun upon the Heavens-Mirrour , you must knowe that the Suns place is there set according to the two Jears , before and after the Leape-yeare , and that especially upon the future Age , which doth almost differ a whole degree in the Sodiack with this present Age to the yeare 1700 , and in the Suns Declination in March and September about 24 minutes . So that al Tables of the Suns Declination which are reckened out with such a difficult calculation , shal after the yeare of 1700 be of no use or worth to a●ny Seaman ; because the yeare of 1700 must be a common yeare . Desiring then to knowe the Declination of the ●un upon some certain or set day in this present Age , then you must alwayes lay the dial A one daye farther then the set day , and in the second yeare after the Leap-yeare you must lay the dial upon the midst of the day , but being the third yeare after the Leape-yeare , then you must lay the dial A on the first fourth part , being in the Sodiack about 15 min. backward . But being the first yeare after the Leap-yeare , then you must lay the dial upon the third fourth part of the day , being verry neare 20 min. farther in the Sodiack , then in the third yeare . And when it is a Leap-yeare , then you must lay the dial A upon the beginning of the day , til to the 28 of February ; but being after the 28 of February , then you must al the yeare along lay the dial A upon the end of the day . And if you do truely understand and perform this , then you wil at al times knowe the Declination of the Sun so perfect and exactly , as the Navigation requires . This is the Head thing I have to say of my Heavens-Mirrour . I. EXEMPLE . Desiring to knowe the Suns Declination on the 30th . day of April 1691 or 95 being the third yeare after the Leap-yeare , then you must lay the dial A upon the first of May , to wit upon the fourth part of the parck of that day . See then where the Dial dath cut or touch the Sodiack , you find it to be verry neare the 10th . degr . 15 min. ( being at the ●ame time the true place of the Sun ) that is 40 degr . 15 min. of ♈ . Farther you must lay the Dial A over the 40 degr . 15 min. off from ♈ . , in the degrees of the Equinoctial or houre cirkle , then you must place one foot of the Compassis on the dial A upon the greatest Declination of the Sun , being 23 degrees 32 min. then you must open the other foot of the Compassis towards the Line of 6 a clock , to come triangularly with this opening of the compassis . Then you must place one foot of it in the Center of the Dial A , and turn the other downewards , and see where it falls , you find it to be verry neare 14 degrees 55 minutes , being at the said time the Suns Declination . II. EXEMPLE . Desiring to knowe the Suns Declination on the 30th . of April 1688 or 92. being the first yeare after the Leap-yeare . Then you must lay the dial A on the first of May , upon the utmost of the days parck , see then where the dial A doth cut the Sodiack , you find it to be verry neare the eleventh degree of Taurus , being the true place of the set time , that is 41 degrees of ♈ . Then you must further lay the dial A on 41 degrees of ♈ . as before , and place one foot of the Compassis on the dial A upon the greatest Declination of the Sun , being 23 degr . 32 min. then you must open the other foot of the Compassis , til it comes rect-angularly to touch the Line of 6 a clock , with this opening of the Compassis , you must place one foot of it in the Center of the dial A , and the other you must turn downewards , seeing upon what degree and minute it falls , you 'l find it to be verry neare 15 degr . and 10 min. being at the set time the true Declination of the Sun , differing litle or nothing with the wise and artificial calculation , do so at al other times , til to the yeare of 1700 , but after that time you must lay the dial upon the set day , considering that this use is principally ( as I told you ) practised and formed upon the future Age , where upon we shal also give some exemples , which also wil serve fore the better understanding of the former . III. EXEMPLE . Desiring to knowe the Suns declination on the 30th day of April 1706. being the second yeare after the Leape-yeare . Then you must at the same time lay the Dial upon the midle of the parck of the said day , and see where the Dial A doth toutch the Sodiack , you 'l find it to be verry neare the 9th degree and 30th minute of Taurus , being at the said time the true-place of the Sun , differing almost a whole degree with the Yeare of 1686 , fore the Sun is 39 degr . 30 minut . of ♈ Then you must further place the Dial A on 39 degrees 30 minutes of ♈ . upon the Dial , in the same manner as I told you before , and then you must place one foot of the Compassis on the Dial A upon the greatest declination of the Sun , being 23 degrees 32 min. , then you must open the other foot of the Compassis , till it comes triangularly to toutch the Line of 6 a clock : with this opening you must place one foot of Compassis in the Centrum of the Dial A , and the other downewards , see then on what degre and minute the foot stands , you 'l find it to be verry neare 14 degrees 42 minutes , which is on the said 30 day of April about noon , the declination of the Sun , differing also in the Suns declination with the Yeare of 1686 almost 18 min. So that by this exemple you may see that al Boocks and Tables ●●●●●ning this matter which are formerly made , wil altogether be in vain , and of no vallue , as soon as ever the Yeare of 1700 begins . IV. EXEMPLE . Desiring to knowe the Suns Declination in the yeare of 1710. whe the Sun is in the 18th . degr . of ♌ . Then you must lay the dial A upon the 18 degr . you 'l also see that the dial doth then lie on the eleventh day of August , about on the third fourth part of the parck of that day , being in the evening about 6 a clock . Then you must further lay the dial A upon the 42th . degree , from ♎ . to Cancer upon the houre circkle , being on the 18th . degree of ♌ , then you must place one foot of the Compassis on the dial A upon 23 degr . 32 min. as before , en then you must open the other foot , rectangularly unto the Line of 6 a clock , with this opening you must place one foot of the Compassis in the Centrum of the dial A , and then you must turn the other foot downewards , and see on what degree and minute it falls , you 'l find it to be verry neare 15 degr . 30 min. being at the said time the Declination of the Sun. FINIS . 
A29764	----	The triangular quadrant, or, The quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as Davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by John Brown ... Brown, John, philomath. 1662 Approx. 38 KB of XML-encoded text transcribed from 14 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A29764 Wing B5043 ESTC R33264 13117453 ocm 13117453 97765 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29764) Transcribed from: (Early English Books Online ; image set 97765) Images scanned from microfilm: (Early English books, 1641-1700 ; 1545:11) The triangular quadrant, or, The quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as Davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by John Brown ... Brown, John, philomath. [2], 24, [1] p. : ill. To be sold at [his, i.e. Brown's] house, or at Hen. Sutton's ..., [London] : 1662. Added illustrated t.p. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Quadrant. Dialing. Mathematical instruments. 2004-01 TCP Assigned for keying and markup 2004-03 Apex CoVantage Keyed and coded from ProQuest page images 2004-04 Mona Logarbo Sampled and proofread 2004-04 Mona Logarbo Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion THE TRIANGULAR QUADRANT : OR The QUADRANT on a SECTOR . Being a general Instrument For Land or Sea Observations . Performing all the Uses of the ordinary Sea Instruments ; as Davis Quadrant , Forestaff , Crossstaff , Bow , With more ease , profitableness , and conveniency , and as much exactness as any or all of them . Moreover , It may be made a particular , and a general Quadrant for all latitudes , and have the Sector lines also . To which is added a Rectifying Table , to find the Suns true Declination to a minute or two , any day or hour of the 4 years : Whereby to find the latitude of a place by a Meridian , or any two other altitudes of the Sun or Stars . First thus Contrived and made by Iohn Browne at the Sphere and Dial in the Minories , and to be sold at 〈◊〉 house , or at Hen. Sutton's in Thredneedle-street behind the Exchange . 1662. triangular quadrant THE TRIANGULAR QUADRANT : Being a GENERAL INSTRUMENT for Observations at Land or Sea , performing all the Uses of all ordinary Sea Instruments for Observations , with more speed , ease and conveniencie than any of them all will do . Contrived and made by Iohn Browne at the Spheare and Sun-dial in the Minories , and sold there or at Mr H. Suttons behind the Exchange . 1662. THE Description , and some uses of the Triangular Quadrant , or the Sector made a Quadrant , or the use of an excellent Instrument for observations at Land or Sea , performing all the uses of the Forestaff , Davis quadrant , Bow , Gunter's crosstaff , Gunter's quadrant , and sector ; with far more convenience and as much exactness as any of them will do . The Description . First , it is a jointed rule ( or Sector ) made to what Radius you please , but for the present purpose it is best between 24. and 36 inches Radius , and a third peice of the same length , with a tennon at each end , to make it an Equilateral Triangle ; from whence it is properly called a Triangular Quadrant . ●●Secondly , as to the lines graduated thereon , they may be more or lesse as your use and cost will please to command , but to make it compleat for the promised premises , these that follow are necessary thereunto . 1. A Line of degrees , of twice 90. degrees on the moveable leg , and outer edge of the cross piece : for quadrantal and back observations . 2. Such another Line of 60. degrees , for forward observations on the inside of the cross piece . 3. On the moveable leg ▪ a Kallender of Months , and dayes in 2 lines . 4. Next to them , a Line of the Suns place in degrees . 5. Next to that , a Line of the Suns right Assention , in degrees or hours . 6. Next above the Months on the same Leg , an hour and Azimuth Line , fitted to a particular Latitude , as London , or any other place , for all the uses of Gunter's Quadrant as you may find in the former discourse ( called a Joynt-rule . ) 7. On the head leg , and same side , a particular scale of altitudes , for the particular Latitude . 8. Next to that a general scale of Altitudes , for all Latitudes . 9. A Line of 360 degrees , divided so as to serve for 360 degrees of 12 Signs ; and 24 hours , ( and in foot and 2 foot rules for inches also . ) 10. A Line of 29½ laid next to the former , serving to find the Moons coming to South , and her age , and place , and the time of the Night , by the fixed Stars . 11. A perpetual Almanack , and the right Assension , and declination of several fixed Stars . 12. A Line of Lines next the inside , may be put on without Trouble , or incumbering one another , all these on one side . Secondly , on the other side may be put the Lines of natural Signs , Tangents and Secants , to a single , double , and treble Radius , and by this means more then a Gunters Sector , ( the particular Lines being inscribed between the general Lines . ) Thirdly on the other edge there may be Artificial Numbers , Tangents , Signs , and versed Signs , and by this means it is a Gunters Rule or a Crosstaff . Fourthly , on the insides , inches , foot-measure , and a Line of 112. parts , and a large meridian Line , or the like : as you please . Lastly , two sliding nuts with points in them , fitted to the Cross piece , makes it a proper beam Compass to use in working by the Numbers Signs and Tangents on the edge , or flat side , also it must have four or five sights , a Thred , and Plummet , and Compasses , as other instruments have , thus much for the Description , the uses follow . An Advertisement . First , for the better understanding and brevity sake , there are ten things to be named and described , as followeth ; 1. the head leg , in which the brass Revet is fixed , and about which the other turn ; 2. the moveable leg , on which the Months and Days must always be ; 3. the Cross piece , that is fitted to the head , and moveable Leg , by the two Tennons at the end ; 4. the quadrant side where the degrees and Moneths are , for observation . 5. the other or Sector side for operation , 6thly the head center , being Center to the degrees on the inside of the crosse peece , for a forward observation as with the forestaffe . 7 The other Center , near the end of the head leg , being the Center to the moveable leg , for backward observations , ( as the Davis quadrant is used , and the bow ) which you may call the foot center , or leg center for backward observations . 8thly the sights , as first the turning or eye sight , which is alwayes , set on one of the centers , with a screw to make it fast there , which I call the turning sight , 9thly the Horizon sight that cuts the degrees of altitude , and sometimes is next to the eye , and sometimes remote from the eye , yet called the horizon ( slideing ) sight , 10thly . the object or shaddow sight of which there may be 3 for convenience sake , as two fixed and one moveable to slide as the horizon sight doth : the other two do serve also to pin the crosse peece , and the two legs together , through the two tennons , all whose names in short take thus : 1. The head leg : 2. the moveable leg : 3. the crosse peece : 4. the quadrant side : 5. the sector side , 6. the head ( or forward ) center . 7 the leg ( or backward ) center . 8. the turning sight . 9. the ( slideing ) horzon sight . 10. the object ( or shaddow ) sight . of which there be 3. all differing according to your use and occasions : one to slide to any place , the other 2. to be put into certain holes . nigher , or further off : as will afterwards largely appear . THE USES : I. To find the suns declination , true place , right assention , and rising , the day of the moneth being given . First open the rule to an angle of 60. degrees , which is alwayes done when the cross peece is fitted into the Mortesse holes , and the pins of the object sights put in the holes through the tennons , or else by the second Chapter of the Joynt-rule : then extend a thred from the center pin in the head leg , to the day of the month , & on the degrees it cuts the suns declination , in the line of right assention his right assention , in the line of true place his true place , and in the hour line his true rising and setting , in that latitude the line is m●de for : Example , on the first of May I would know the former questions , the rule being set by the crosse peece , and the thred on the leg center pin ; and drawn straight and laid over May 1. it cuts in the degrees 18. 4. north declination , and 20. 58. in ♉ Taurus for his place , and 3 hours 14. minutes right assention in time , or 48. 32. in degrees : and the rising of the sun that day is at 4. 23 , and sets at 5. 37 , in 51. 32 latitude . The finding of hour and azimuth , either particularly , or generally , with other Astronomical propositions , are spoken enough of before in the Joynt-rule , and in all other authors that write of the Sector , or Gunter's rule , so that all I shall speak of now , shall be onely what was forgot in the first part , and what is new as to the using the instrument in sea observations . II. To find the Suns or a Stars Altitude , by a forward Observation . Skrew the turning sight to the head Center , and set that object sight , whose holes answer to the Sliding horizon sight , in the hole at the end of the head leg , and put the horizon sight on the crosse peece next the inside ; Then holding the crosse peece with your right hand , and the turning sight close to your eye , and the moveable leg against your body , with your thumb on the right hand thrust upwards , or pull downwards , the horizon sight : till you see the sun through the object sight , and the horizon through the horizon sight , then the degrees cut by the middle of the horizon sight , on the crosse peece shall be the true altitude required : III. To perform the same another way . If your instrument be parted , that is to say the crosse peece from the other , and an altitude be required to be had quickly , then set the two object sights , in two holes at the end of the line of naturall signs , then set the head of the rule to your eye , so as the sight of the eye may be just over the Center , then open or close the Joynt , till you see the horizon through one sight , and the sun or star through the other , then is the sector set to the angle required , to find which angle do thus , take the parallel sign of 30 and 30 , and measure it from the Center , and it shall reach to the sign of half ●he angle required . Example . Suppose I had observed an altitude , and the distance between 30 and 30 , should reach from the center to 10. degrees on the signs , then is the altitude of the sun 20. degrees for 10 doubled is 20. IIII. To find the suns Altitude by a back observation , Skrew the Turning sight to the leg center , ( or center to the degrees on the moveable leg ) and put one of the object sights , in the hole by 00. on the outer edge of the crosse peece , and set the edge of it just against the stroke of 00 , or you may use the sliding object sight and set the edge or the middle of that , to the stroke of 00 , as you shall Judge most convenient ; and the horizon sight to the moveable leg , then observe in all respects as with a Davis quadrant , till looking through the small hole of the horizon sight , you see the crosse bar and button , in the turning sight , cut the horizon : and at the same instant the shadow of the edge or middle of the object or shadow sight , fall on the middle of the turning sight , by sliding the horizon sight higher or lower , then the middle stroke of the horizon sight , shall cut on the moveable leg , the suns true altitude required . As f 〈…〉 t stay at 50 degrees , then is the sun 50 degrees above the horizon . V. But if the sun be near to the Zenith or 90 degrees high , then it will be convenient to move the object sight , to a hole or two further as suppose at 10 , 20 , 30 degrees more , toward the further end of the crosse peece and then observe as you did before in all respects , as with a Davis quadrant , and then whatsoever degrees the horizon sight cuts , you must ad so much to it , as you set the object sight forwards , as suppose 30 , and the horizon sight stay at 60 , then I say 60 , and 30 , makes 90 : the true altitude required . Note that by this contrivance , let the altitude be what it will , you shall alwayes have a most steady observation : with the instrument leaning against your brest , a considerable thing , in a windy day , when you may have a need of an observation in southern voyages , when the sun is near to the zenith at a Meridian observation . VI. To find the suns distance from the zenith , by observing the other way , the sun being not above 60 degrees high , or 30 from the zenith . Set the turning sight as before on the leg Center , then set an object sight in one of the holes in the line on the head leg , nigher or further of , the turning sight : as the the brightnesse or dimnesse of the sun will allow to see a shadow , then looking through the small hole on the horizon sight , till you see the horizon cut by the crosse bar of the great hole , in the turning sight , turning the foreside of that sight , till it be fit to receive the shadow of the middle of the object sight ; then the degrees cut by the horizon sight , shall be the suns true distance from the zenith , or the complement of the Altitude . VII . Note that by adding of a short peece about 9 inches long on the head leg , whereon to set the slideing shadow sight , you may obtain the former convenience of all angles , this way also , at a most steady and easie manner of observation ; but note whatsoever you set forwards on that peece , must be substracted from that the sight sheweth , and the remainder shall be the suns distance from the zenith required . As suppose you set forward 30 degrees , and the horizon sight should stay at 40 , then 30 from 40 rest 10 , the suns distance from the zenith required ; thus you see , that by one and the same line , at one manner of figuring , is the suns altitude , or coalitude acquired and that at a most certain steady manner of observation . VIII . To find an observation by thred and plummet , without having any respect to the horizon , being of good stead in a misty or cloudy day at land or sea . Set the rule to his angle of 60 degrees by putting in the crosse peece , then skrew the turning sight to the head center , then if the sun or star be under 30 degrees high , set the object sight in the moveable leg , then looking through the small hole in the turning sight , through the object sight , to the middle of the star or sun , as the button in the crosse bar will neatly shew ; then the thread and plummet , hanging on the leg center pin , and playing evenly by the moveable leg , shall shew the true alti●ude of the sun , or star required counting the degrees as they are numbred , for th : north declinations from 60 toward the head with 10 20 , As if the thred shall play upon 70 10 then is the altitude 10 degrees . IX But if the sun or star be above 30 degrees high , then the object sight must be set to the hole in the end of the head leg : then looking as before , and the thred playing evenly by the moveable leg , shall shew the true altitude required , as the degrees are numbred . Note that if the brightnesse of the sun should offend the eye , you may have a peice of green , blew , or red glasse , fixed on the turning sight , or else remove the object sight nearer to the turning sight , and then let the sun beams pierce through both the small holes , according to the usuall manner and the thred shall shew the true altitude required . Note also if the thred be apt to slip away from his observed place , as between 25 and 40 it may : note a dexterious handling thereof will naturally shew you how to prevent it : X. To find a latitude at Sea by forward meridian Observation or Altitude . Set the moving object Sight to the Suns declination , shewed by the day of the Month , and rectifying Table , and skrew the turning sight to the leg center , and the Horizon sight to the moveable leg , or the outside of the Crosse piece , according as the Sun is high or low ( but note all forward observations respecting the Horizon , ought to be under 45 degrees high , for if it be more it is very uncertain , by any Instrument whatsoever , except you have a Plummet and then the Horizon is uselesse ) then observe just as you do in a forward observation , moving the Horizon sight till you see the Sun through the Horizon sight , and the Horizon through the object sight , or the contrary . ( moving not that sight that is set to the day of the Month or Declination , ) then whatsoever the moving sight shall shew , if you add 30 to it , it shall be the latitude of the place required ; observing the difference in North and South Latitudes ; that is , setting the sight to the proper declination , either like , or unlike , to the latitude - Example . Suppose on the 10. of March when the Suns declination is 0 — 10. North , as in the first year after leap year it will be , set the stroke in the middle of the moving object sight to 10 of North Declination , and the Horizon sight on the moveable leg , then move it higher or lower , till you see the Horizon through one , and the Sun through the other , then the degrees between , is the Suns meridian altitude , if it be at Noon , as suppose it stayed at 21 30 ▪ then by counting the degrees between , you shall find them come to 38. 40. then if you add 30. to 21. 30. it makes 51. 30. the Latitude required , for if you do take 0 10′ minutes from 38. 40. there remains 38. 30. the complement of the Latitude . Note , that this way you may take a forward observation , and so save the removing of the ●urning sight . Note also , That when the Horizon sight shall stay about the corner , you may move the object sight 10. or 20. degrees towards the head , and then you must add but 20. or 10 degrees to what the sight stayed at ; or if you shall set the sight the other way 10 or 20. degr . then you must add more then 30 so much . As suppose in this last observation , it had been the latitude of 45 or 50 degrees , then you shall find the sight to play so neer the corner , that it will prove inconvenient , then suppose instead of 0 10. I set it to 20 degrees 10′ North declination , which is 20. degrees added to the declination , then the Suns height being the same as before , the sight will stay at 41. 30. to which if you add 10 degrees , it doth make 51. 30. as before ; here you must add but 10 degrees , because you increased the declination 20. degrees ; but note by the same reason , had you set it to 19. 50. South declination , then it had been diminished 20 degrees , and then instead of 30 you must add 50 ▪ to 1 - 30. the place where the sight would have stayed . Thus you see you may very neatly avoid this inconvenience , and set the sights to proper and steady observations , at all times of observation . XI . To find the latitude by a backward Meridian Observation at Sea. This is but just the converse of the former , for if you set one sight to the declination , either directly , ( or augmented or diminish'd as before , when the moving sight shall stay , about the corner of the Triangular Quadrant ) then the other being slipt to and fro , on the outside of the Crosse peece , till the shadow of the outer edge , shall fall on the middle of the turning sight , then 30 just , or more or lesse added , to that number the moving sight stayed at , ( according as you set the first Horizon sight to the declination ) shall be the true latitude required . Example . Suppose on the same day and year as before , at the same Noon time , I set my Horizon sight to just 10′ of North declination , you shall find the moving sight to stay at 21. 30. neer to the corner , now if the Sun shine bright , and will cast the shadow to the turning sight , then set the Horison sight at the declination , forward 10 or 20. degrees , then the moving sight coming lower you , add but 20. or to that it shall stay at , and the summe shall be the latitude . But it is most likely that it will be better to diminish it 20. degrees , then the moving sight will stay about 2 - 30. on the Crosse piece , and so much the better to cast a shadow ; for if you look through the Horizon and turning sight to the Horizon , you shall find the shadow of the former edge of the moving shadow sight , to stay at 2 - 30. to which if you add 20. the degrees diminished , and 30. it makes 51 30. the latitude required as before . Note also for better convenience of the shadow sight , when you have found the true declination , as before is taught , set the moving object sight to the same , on the Crosse piece , counted from 00. towards the head leg , for like latitudes and declinations ; and the other way for unlike latitudes and declinations , then observing as in a back observation , wheresoever the sight shall stay , shall be the complement of the latitude required . If you add or diminish consider accordingly . Note likewise , when the declination is nere the solstice , and the same way as the latitude is , and by diminishing , or otherwise the moving sight shall fall beyond 00 on the Crosse piece ; Then having added 30. and the degrees diminished together , whatsoever the sight shall stay at beyond 00 must be taken out of the added sum , and the remainder shall be the latitude required . Example . Suppose on the 11. of Iune , in the latitude of 51 30 north , for the better holding sake I diminish the declination 30 degrees , that is in stead of setting it to 23. 32. north declination , I set it to 6 : 28 south then the sun being 62 degrees high will stay at 08. 30. beyond 00. the other way now 30 to be added , and 30 diminished , makes 60 , from which take 8. 30 , rest 51 30 the latitude required . XII . To find a latitude with thred and plummet , or by an observation made without respecting the Horizon . Count the declination on the cross peece ( and let 00 be the equinoctiall and let the declination which is the same with the latitude be counted toward the moveable leg , and the contrary the other way , as with us in north latitude , north declination is toward the moveable leg , and south declination the contrary and contrarily in south latitudes ) and thereunto set the middest of the sliding object or horizon sight , then is the small hole on the turning sight and the small hole on the horizon sight , two holes whereby the sun beams are to pierce to shine one on the other : then shall the thred shew you the true latitude of the place required . Example . Suppose on the 11 of Decem. 1663 , at noon I observe the noon altitudes set the middle of the Horizon sight to 23 ▪ 32 counted from 00. toward the head leg end , then making the sun beams to peirce through the hole of this , and the turning sight , you shall find the plummet to play on 51. 30 , the latitude required , holding the turning sight toward the sun . Note also that here also you may avoid the inconvenience of the corner , or the great distance between the sights , by the remedy before cited , in the back and forward observation . For if you move it toward the head leg , then the thred will fall short of the latitude , if toward the moveable leg then it falls beyond the latitude , as is very easie to conceive of : Thus you see all the uses of the forestaff , and quadrant , and Mr. Gunter's bow are plainly and properly applyed to this Triangular quadrant , that the same will be a Sector is easie to perswade you to believe , and that all the uses of a Gunter's quadrant , are performed by it , is fully shewed in the use of the Joynt-rule , to which this may be annexed , the numbers signs and tangents and versed signs makes it an excellent large Gunter's rule , and the cross peece is a good pair of large compasses to operate therewith ; lastly , being it may lie in so little roome it is much more convenient for them , with whom stowage is very precious , so I shall say no more as to the use of it , all the rest being fully spoke to in other Authors , to whom I refer you : only one usefull proposition to inure you to the use of this most excellent instrument , which I call the Triangular Quadrant . Note that in finding the latitude , it is necessary to have a table of the suns declination for every of the four years , viz. for the leap year and the 1 , 2 and 3d. after , now the table of the suns declination whereby the moneths are laid down , is a table that is calculated as a mean between all the 4 years , and you may very well distinguish a minute on the rule ; now to make it to be exact I have fitted this rectifying table for every Week in the year , and the use is thus : hang the thred on the center pin , and extend the thred to the day of the moneth , and on the degrees is the suns declination , as near as can be for a common year , then if you look in the rectifying table for that moneth , and week you seek for you shall find the number of minutes you must add to or substract from the declination found for that day and year : Example , suppose for April 10. 1662. the second after leap year , the rule sheweth me 11. 45 , from which the rectifying table saith I must substract 3′ then is the true declination 11 42 , the like for any other year . Note further that the space of a day in the suns swiftest motion being so much , you may consider the hour of the day also , in the finding of a latitude , by an observation taken of the Meridian , as anon you shall see that as the instrument is exact , so let your arithmetical calculation be also : by laying a sure foundation to begin to work upon , then will your latitude be very true also . A Rectifying Table for the Suns declination .   D 1 year 2 year 3 year Leap year Ianuary 7 Sub 5 Sub 2 Add 1 Add 4 15 s 6 s 2 a 2 a 5 22 s 7 s 3 a 1 a 6 30 s 7 s 3 a 2 a 7 Februar 7 s 8 s 3 a 2 a 7 15 s 8 s 4 a 2 a 8 22 s 9 s 3 a 2 a 7 March 1 s 4 s 1 a 7 Sub 11 7 s 3 Ad 3 Add 9 Sub 9 15 Add 3 ad 1 Sub 9 Add 9 22 a 2 Sub 4 Sub 9 ad 8 30 a 1 Sub 4 Sub 9 ad 8 April 7 a 2 Sub 3 s 8 a 8 15 a 2 s 3 s 8 a 7 22 a 1 s 3 s 8 a 6 30 a 1 s 3 s 6 a 5 May 7 a 0 s 2 s 6 a 5 15 a 0 s 2 s 5 a 3 22 a 0 s 1 s 3 a 3 30 a 0 s 1 s 2 a 1 Iune 7 a 0 s 1 s 1 ad 0 15 a 0 s 0 s 0 Sub 0 22 s 1 s 0 ad 2 s 2 30 s 1 Ad 1 ad 3 s 3 Iuly 7 Sub 2 Add 1 Add 3 Sub 4 15 s 1 a 1 a 5 s 5 22 s 2 a 1 a 5 s 6 30 s 2 a 2 a 6 s 7 August 7 s 3 a 2 a 7 s 8 15 s 3 a 2 a 7 s 8 22 s 3 a 3 a 8 s 9 30 s 3 a 3 a 9 s 9 September 7 s 3 a 3 Add 9 Sub 9 15 Add 3 Sub 3 Sub 9 Add 9 22 a 2 s 4 s 9 a 8 30 a 2 s 4 s 9 a 8 October 7 a 3 s 3 s 9 a 8 15 a 2 s 3 s 8 a 7 22 a 2 s 2 s 7 a 8 30 a 2 s 2 s 7 a 8 November 7 a 1 s 1 s 6 a 7 15 a 1 s 2 s 5 a 5 22 a 1 s 3 s 4 a 3 30 a 0 s 2 s 3 a 2 December 7 0 s 1 s 1 Add 1 15 Sub 1 s 0 s 0 Sub 1 22 s 1 s 0 Add 1 s 3 30 s 3 s 0 Add 2 s 5 The Declination of the Sun being given , or rather the Suns Distance from the Pole , and the Complement of two Altitudes of the Sun , taken at any time of the day , knowing the time between : to find the Latitude . Suppose on the 11. of Iune , the Sun being 66 degrees , 29′ distant from the North Pole , and the Complement of one altitude be 80. 30. and the Complement of another altitude 44. 13. and the time between the two observations just four hours ; then say , As the Sine of 90 00 To Sine of Suns dist . f. the Pole 66 28 So is the Sine of ½ time betw . 30 00 To the Sine of ½ the 3d. side 27 17½   27 17½ of a Triangle as A B. 54 35 The side A P 66 28 The side P A 66 28 The side A B 54 35 whole sum 187 31 half sum 93 46 The differ . betw ½ sum and AP side op . to inqu . Triangle 27 18 Then say , as S. of 90 90 00 To S. of Suns dist . from Pole 66 28 So is the Sine of A B 54 35 To the Sine of a fourth Sine 48 23 Then as that 4 to Sine of ½ sum 93 46 So is S. of the difference 27 18 To a seventh Sine 37 44½ or the versed Sine of P A B 77 02 Then to find Z A B Z B is 80 30 And Z A is 44 13 The former side A B is 54 35 Sum 179 18 half sum 89 39 difference 09 09 As S. of 90 90 00 To Sine of A B 54 35 So is S. of Z A 44 13 To a 4th . Sine 34 38 As S. 4th . 34 38 To S. of ½ sum 89 39 So is S. diff . 09 39 To a 7th . Sine 16 15¼ or to the vers . sine of Z A B 116 07 Then if you take P A B from Z A B there will remain Z A P 39 05 Then say again by the rule as before , As the sine of 90 00 To Co-sine of Z A P 50 55 So is the Tang of A Z 44 13 To the Tangent of A C 37 04 which taken from A P 66 28 remaineth P C 29 24 Then lastly say ,   As the Co-sine of A C 52 56 To the Co-sine of C P 60 36 So is the Co-sine of Z A 45 47 To the Co-sine of Z P 51 30 The Latitude required to be found . This Question or any other may be wrought by the Sines and Tangents and versed sines on the rule , But if you would know more as concerning this or any other , you may be fully satisfied by Mr. Euclid Spidal at his Chamber at a Virginal Makers house in Thred-needle Street , and at the Kings head neer Broadstreet end . Vale. FINIS . The Triangle explained . S P Z N a Meridian circl . Ae Ae the Equinoctial . ♋ B A ♋ the , tropick of Cancer . P B 5 P the hour circle of Five , ante , mer. P A 9 P the hour circle of Nine , ante , mer. Z B the Suns coaltitude at 5 — 80 — 30 Z A the Suns coaltitude at 9 — 44 — 13 P A & P B the Suns distance from the Pole on the 2 h. li. of 5 & 9 66 — 28 5-9 the equinoctial time between the two Observations — 60 — 00 B A in proper measure is as found by the first measure working — 54 — 35 Z C a perpendicular on A P from Z 29 — 24 P Z the complement of the latitude that was to be found — 38-30 
A69643	----	A nevv quadrant, of more naturall, easie, and manifold performance, than any other heretofore extant framed according to the horizontall projection of the sphere, with the uses thereof. By C.B. maker of mathematic instruments in metall. Brookes, Christopher, fl. 1649-1651. This text is an enriched version of the TCP digital transcription A69643 of text R4412 in the English Short Title Catalog (Wing B4917A). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 26 KB of XML-encoded text transcribed from 12 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2017 A69643 Wing B4917A ESTC R4412 99834885 99834885 39505 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A69643) Transcribed from: (Early English Books Online ; image set 39505) Images scanned from microfilm: (Early English books, 1641-1700 ; 806:10, 1944:10) A nevv quadrant, of more naturall, easie, and manifold performance, than any other heretofore extant framed according to the horizontall projection of the sphere, with the uses thereof. By C.B. maker of mathematic instruments in metall. Brookes, Christopher, fl. 1649-1651. [4], 18 p. [s.n.], London : printed in the yeare 1649. Dedication signed C.B. (Christopher Brookes). Identified on UMI microfilm (Early English books, 1641-1700) reel 806 as Wing B43 (entry cancelled in Wing 2nd ed.). Reproductions of the originals in the Henry E. Huntington Library and Art Gallery (reel 806), and the Bodleian Library (reel 1944). eng Dialing -- Early works to 1800. Quadrant -- Early works to 1800. A69643 R4412 (Wing B4917A). civilwar no A nevv quadrant, of more naturall, easie, and manifold performance, than any other heretofore extant; framed according to the horizontall pr Brookes, Christopher 1649 4680 5 0 0 0 0 0 11 C The rate of 11 defects per 10,000 words puts this text in the C category of texts with between 10 and 35 defects per 10,000 words. 2006-05 TCP Assigned for keying and markup 2006-05 Aptara Keyed and coded from ProQuest page images 2006-09 Judith Siefring Sampled and proofread 2006-09 Judith Siefring Text and markup reviewed and edited 2007-02 pfs Batch review (QC) and XML conversion A NEVV QUADRANT , OF More Naturall , Easie , and Manifold Performance , than any other heretofore Extant ; Framed according to the Horizontall Projection of the Sphere , with the Uses thereof . By C. B. Maker of Mathematic Instruments in METALL . LONDON , Printed in the Yeare 1649. TO My Singular good Friend Mr. WILLIAM BADILEY , Mariner , and a lover of the Mathematics . Worthy Sir , HAving diligently inquired ▪ the reason of the projection of the Sphere into plaine , as the ground of all Mathematic Instruments ( the making whereof in Metall is my Trade and Livelihood ) and compared the severall manners ; I found none so genuine , simple , easie , and manifoldly usefull , as is the Horizontall ; which lively representeth the Globe rectified to some certaine elevation , and naturally performeth the uses thereof . And having likewise compared the severall Quadrants , and pocket Instruments hitherto made , and finding them all pieced up with many unnaturall and forced lines and divisions , presupposing an exact diligence both in the Calculator , and in the workman ; and yet the performance difficult , troublesome , and tedious : I bethought my self whether out of the Horizontall projection I might not by some smal alteration frame a Quadrant , that might remedy the defects of all the former Instruments , and that with greater ease and certainty . And having by the help of God happily attained my desired intent , my many respects represented you under whose Approbation and Patronage I might send out into public view this my new Quadrant , with the many Uses thereof ; as being one to whom I stand obliged for your love and manifold favours to mee both at Sea in divers Voiages , and at land ; and who through your skill in the Mathematicall Sciences are able to judge and patronize the first attempts of Your affectionately devoted Servant , C. B. THE Description of the Quadrant , and the parts thereof . THe limbe of the Quadrant divided into 90 Degr. representeth the Horizon . That side of the Quadrant where the Sights are , is the Meridian , or XII a Clocke line , unto which is joyned the scale of Moneths with every fift Day , untill they grow so little toward the Solsticeas , that they cannot bee distinguished . This scale ▪ hath five Rowes , the midst whereof hath the very same divisions which are on the Meridian line : The two next on both sides are for the parts of the Moneths , which in the two outermost Rowes are noted by their first Letters . The other side of the Quadrant hath on it the scale of Altitudes above the Horizon . The short Arching lines within the Quadrant beside the Meridian , are Houre-lines , noted by their Figures , both for the forenoone , and afternoon ; and halfe Houre lines : each halfe houre containing 30 min. of an hour , or Deg. 7. 30. Of these Horary lines , those which serve in the morning before the Sunne is full East , or in the evening past the West , ( which is onely in Summer halfe yeare ) are reversed . And all the Hour lines are noted with two Figures ; whereof the upper next the Center and scale of Altitudes , serve for the afternoon ; and the lower for the forenoon . The two Arches which crosse the Houre lines , and meet at the beginning of the Horizon and scale of Altitudes , are two Quarters of the Ecliptic , and are divided into 90 Degr. a piece , in which are noted the XII Signes by their proper Characters , namely on the upper next the Center are ♈ ♉ ♊ & ♋ ♌ ♍ , the Summer or Northerne Signes : and on the lower next the Horizon are ♎ ♏ ♐ & ♑ ♒ ♓ , the Winter or Southerne Signes , and contain 30 Degr. a piece . This is the Circle of the Sunnes annuall motion . The long Arches , which beginning at the Scale of moneths in the Meridian betwixt the two Quarters of the Ecliptic , crosse all the Houre lines , are the parallels of Declination , or the Semidiurnall Arches of the Sunne ; the middlemost of which is the Equinoctiall , the outermost above is the Tropick of ♋ , and the outermost below is the Tropick of ♑ : although between the Equinoctiall and each Tropick Innumerable parallels are understood to be contained , yet those which are in the Instrument drawne , at every second Degree of Declination , may be sufficient to direct the eye in tracing out an imaginary parallel from every point given in the Scale of moneths . The Equinoctiall and every tenth parallel are for distinction sake made somewhat grosser than the rest , and all the Summer parallels at the East and West line are continued reversedly back unto the Horizon . Note that upon the right estimation of that imaginary parallel , the manifold use of this Instrument doth especially rely ; because the true place of the Sunne all that day is in some part or point of the same Circle . And note that in this Instrument , the direct Horary lines , and parallels before their reversion , shew the houre of the day like a direct South upright Diall : And the Arches of them reversed serve like a direct North upright Diall . Use I. To finde the Declination of the Sun every day . Seek the day proposed in the Scale of moneths very exactly , & mark upon what point it falleth in the middle Row of that Scale , or ( which is all one ) in the Meridian , for there is the Declination of the Sunne from the Equinoctiall , either North or South : which if it fall not directly upon a parallel , but in the space between two , supposing each halfe of that space to containe 60 minutes , estimate with your eye proportionally what minute the point giveth . Example 1. What is the Sunnes Declinaclition upon Novemb. 13 ? the day will fall in the space after 20 Degrees , from the Equinoctiall Southward , about 30 minutes : Wherefore the Sunnes Declination is 20° . 30′ South . Example 2. What is the Sunnes Declination upon August 19 ? the day wil fal in the space after 8 Degrees , from the Equinoctiall Northward , one Degree and about 40 minutes : Wherefore the Suns Declination is 9° . 40′ . North . Note that the Declination thus found is to be kept in minde all the day . Use II. To finde the Semidinunall Arch , or parallel Circle in which the Sunne moveth every day . Seeke out the true point of the Sunnes Declination upon the Meridian by Vse I : then from that point by the estimation of your eye , trace out an imaginary parallel : which when it commeth to the East and West line ( as in all Northerne parallels it doth ) is to bee reversed unto the Horizon or Limbe at the same proportionable distance as before . This operation requireth exact diligence . Use III. To finde the time of the Sunes Rising and Setting every day . Seek out the imaginary parallel , or Semidiurnal Arch of the Sun for that day by Vse II , and marke where it meeteth with the Horizon ; for that is the very point of the Sunnes rising and setting , and the Hour-lines on both sides of it , ( by proportioning the distance reasonably , according to 30 minutes for halfe an houre ) will shew the time of the Sunnes rising and setting . Thus at London , Novem. 13. the Sun will be found to rise at 9 min. before 8 , and to set at 9 min. after 4. Also August 19 , the Sunne will be found to rise 12 min. after 5. and to set 12 min. before 7. Use IV. To finde the Suns Amplitude , Ortive and Occasive : that is , how many Degrees of the Horizon the Sunne riseth and setteth from the true East and West points every day . The imaginary parallel of the Sunne , together with the time of the Sunnes rising , and setting , sheweth upon the Horizon the Degree of his Amplitude from East and West , which in all the Northerne parallels is on the North side , and in the Southerne on the South side . Thus at London , Novem. 13. the Ampl. Ort. will be found 34 Degreees . Also Aug. 19. the Ampl. Ort. will be found 15° . 10′ . Use V. To find the Length of every day and night . Double the houre of the sunnes-setting , and you shal have the Length of the day : or double the houre of the sunnes-rising , and you shall have the Length of the night . Use 6. To know the reason and manner of the Increasing and Decreasing of the Dayes and Nights throughout the whole yeare . When the Sunne is in the Equinoctial , it riseth and setteth at 6 a Clocke : But if the Sunne be out of the Equinoctial , declining toward the North , the Intersection of the parallel of the Sunne with the Horizon is before 6 in the morning , and after 6 in the evening ; and the Diurnall Arch greater than 12 houres , and so much more great , the greater the Northerne Declination is . Againe , if the Sunne be declining toward the South , the Intersection of the parallel of the Sunne with the Horizon is after 6 in the morning , and before 6 in the evening ; and the Diurnall Arch lesser than 12 hours , and by so much lesser , the greater the Southern Declination is . And in those places of the Ecliptic in which the Sunne most speedily changeth his Declination , the Length also of the day is most altered ; and where the Ecliptic goeth most parallell to the Equinoctiall , changing the Declination slowly , the length of the day is but little altered . As for Example ; When the Sun is neare unto the Equinoctiall on both sides , the dayes Increase and also Decrease suddenly and apace ; because in those places the Ecliptic inclineth to the Equinoctiall in a manner like a straight Line , making sensible Declination . Againe , when the Sunne is neare his greatest Declination , as in the height of Summer , and the depth of Winter , the dayes keep for a good time , as it were , at one stay ; because in these places the Ecliptic is in a manner parallel to the Equinoctiall , the Length of the day differeth but little , the Declination scarce altering ; and because in those two times of the yeare , the Sunne standeth as it were still at one Declination , they are called the Summer Solstice , and Winter Solstice . Wherefore wee may hereby plainely see , that the common received opinion , that in every moneth the dayes doe equally increase , is erroneous . Also wee may see , that in parallels equally distant from the Equinoctiall , the day on the one side is equall to the night on the other side . Use VII . To take the height of the Sunne above the Horizon . Hold the edge of the Quadrant against the Sunne , so that the Sunnes Ray or Beam may at once passe through the hole of both the sights ; then shall the thread with the Plummet shew the Sunnes Altitude . Use VIII . To finde the Houre of the day , or what a clock it is . Having the imaginary parallel or Semidiurnall Arch of the Sunne , already found and conceived in your minde by Vse II , take the Sunnes height above the Horizon , then stretching the thread over the scale of Altitudes , set the Bead to the Altitude found , move your thread untill the Bead exactly falleth upon the imaginary parallel , for there is the houre fought ; and that is the true place of the Sun in the Quadrant at that time ; to bee estimated upon the Horary lines , either direct , or reversed , according as the parallel is . Use IX . To finde the Sunnes Azumith or Horizontall distance from the foure Cardinall points . The Bead being set to the houre of the day , as was shewed in the Vse next before , the thread shall in the Limbe cut the East or West Azumith ; that is , how many Degrees of the Horizon the verticall Circle in which the Sunne is , is distant from the East and West points : The complement of which number giveth the Azumith from the South Meridian , if the Bead fell in the right parallels : But if the Bead fall upon the reversed parts , the Azumith is to be accounted from the North Meridian . Use X. To finde the Meridian Altitude of the Sunne every day . Stretch the thread over the Meridian , and set the Bead to the true Declination of the Sunne therein ; then apply the thread to the scale of Altitudes ; and the Bead shall give the Meridian Altitude sought . Use XI . To finde at what time the Sunne commeth to bee full East or West every day in Summer . This is shewen by observing at what houre the imaginary parallel meeteth with the East and West line , at which it beginneth to reverse . Use XII . To finde how high the Sunne is above the Horizon at any houre , every day . Set the Bead to the point in which the imaginary parallel of that day crosseth the houre given : then applying the thread to the scale of Altitudes , mark upon what Degree the Bead falleth ; the same shall bee the Altitude of the Sun required . Use XIII . To finde how high the Sunne is being in any Azumith assigned every day : and also at what houre . Set the Bead to the point in which the imaginary parallel of that day crosseth the Azumith assigned ; There also shall bee the houre sought : Then applying the thread to the scale of Altitudes , marke upon what Degree the Bead falleth ; The same shall be the Altitude of the Sun required . These two last Uses serve for the Delineation of the ordinary Quadrants , as that of Gemma Frisius , Munster , Clavius , Master Gunter , &c. and also of Rings , Cylinders , and other Topicall Instruments ; and for the finding out of the houre by a mans shadow , or by the shadow of any Gnomon , set either perpendicular , or else parallel to the Horizon . Use XIV . To finde the Sunnes Longitude , or place in the Ecliptic . The imaginary parallel of the day being exactly traced will cut in the Ecliptick the Signe and Degree wherein the Sunne is : and note , that each semicircle of the Ecliptic is doubly noted with Characters of the Signes ; the first and third Quarters goe forward from the Equinoctial point unto the Meridian , containing ♈ ♉ ♊ & ♎ ♏ ♐ : the second and fourth Quarters goe backe from the Meridian unto the Equinoctiall point , containing ♋ ♌ ♍ & ♑ ♒ ♓ . But because neare unto both Tropicks ( namely from May 11 , to July 10 , in the height of Summer , and from November 13 , to Januarie 12 in the depth of Winter ) the Declination altereth so slowly , that the true place of the Sunne in the Ecliptic cannot be distinguished with any certainty , worke according to this foure-fold Rule following . 1. Before June 10 , out of the number of dayes from May 0 , subduct 11 : the remains shall be the Degrees of ♊ : thus for June 3 , ( because there is all May and three dayes of June ) say 34 — 11 = 23 ♊ , the place of the Sunne . 2. After June 10 , out of the Number of dayes from June 0 , subduct 10 : the remains shall bee Degrees of ♋ : thus for July 3 , say 33 — 10 = 23 ♋ , the place of the Sun . 3. Before December 13 , out of the Number of dayes from November 0 , subduct 13 : the reamines shall be Degrees of ♐ : thus for December 3 , say 33 — 13 = 20 ♐ , the place of the Sun . 4. After December 13 , out of the Number of dayes from December o , subduct 13 : the remaines shall be Degrees of ♑ : thus for January 3 , say 34 — 13 = 21 ♑ , the place of the Sunne . Use XV . To find the Suns Right Ascension every day . Having by Use XIV . found the place of the Sunne in the Ecliptic , mark diligently upon what houre , and as neare as you can estimate what minute it falleth , counting the houres in the first and third Quarters of the Ecliptic , from the Equinoctiall point ; but in the second and fourth Quarters , from the Meridian : and adde thereto in the second Quarter six hours , in the third twelve houres , and in the fourth eighteen houres : so shall you have the Sunnes Right Ascension , not in Degrees , but in time , which is more proper for use . Example , in ♌ 6. the Sunnes Right Ascension will bee eight houres , one halfe , and about three minutes ; that is H : 8 : 33. min. reckoning 30′ for halfe an houre . Use XVI . To find the Houre of the Night by the Starres . For this , I have set a little Table of five knowne Stars dispersed round about the Heavens , with their Declination and Right Ascension for Anno Dom. 1650. Namely the left shoulder of Orion , noted O. The heart of the Lion , noted ♌ . Arcturus noted A : the Vulture volant , noted V. The end of the wing of Pegasus , noted P. The Table .   Declinat . Rec. As . O 5° 59′ N H5 6,5′ ♌ 13 39 N 9 50 A 21 4 N 14 00 V 8 1 - N 19 34 P 13 15 N 23 55 , 5 THe Operation is thus ; first by the height of the Starre taken , and the parallel of its Declination exactly traced , seek out the houre of the Starre from the Meridian , as before was taught for the houre of the Day by the Sunne . Secondly , out of the Right Ascension of the Starre , subduct the Right Ascension of the Sun ; the remain● sheweth how long time from the Noone before the same starre commeth into the Meridian . Lastly , if the Starre be not yet come to the Meridian , out of the houre of the Starres comming into the Meridian , subduct the houre of the Starre : but if the Star be past the Meridian , adde both the houres together ; so shall you have the true houre of the Night . Note , that if the hours out of which you are to subduct bee lesser than the other , you must adde unto them 24. Use XVII . To finde out the MeridianLine upon any Horizontall plaine . About the middle of your plaine describe a Circle ; and in the Center thereof erect a straight Piece of Wire perpendicularly . When the Sunne shineth , note the point of the Circle which the shadow of the Wire cutteth , which I therefore call the shadow point ; and instantly by Vse IX . seeke the Sunnes Azumith from the South or North : keepe it in minde . Then from the shadow point , if your observation be in the foore-noon , reckon upon the circle an Arch equall to the Azumith kept in minde , that way the Sunne moveth , if the Azumith bee South : Or the contrary way if it bee North . But if your observation bee in the afternoone , reckon the North Azumith that way the Sunne moveth ; Or the South Azumith the contrary way . Lastly , through the end of the Azumith and the center , protract a Diameter for the Meridian line sought : which you may note with S. at the south end , and with N. at the North end . You may also note the point of the Circle Diametrally opposite to the shadow point with sun ; , because it is the Azumith place of the Sun , at the moment of your observation . Use XVIII . To finde the Declination of any Wall or plaine . The safest way ( because the Magneticall Needle is apt to be drawne awry ) will be by an Instrument made in this manner . Provide a rectangular board about ten Inches long , and five broad : in the midst whereof , crosse the breadth , strike a Line perpendicular to the sides ; and taking upon it a Center , describe a Circle intersecting the same Line , in two opposite points , to be noted with the Letters T. and A : divide each semicircle into two Quadrants , and every Quadrant into 90 Degrees , beginning at the points T and A , both wayes ; the first Quad. beginning on the left hand of T. the second Quadrant on the right hand : the third Quadrant above it toward A : And lastly , the fourth Quadrant . And in the Center erect a Wier at right Angles . The use of this Instrument . Apply the long side of the board next T to the Wall when the Sunne shineth upon it , holding it parallel to the Horizon , that it may represent an Horizontall plaine . Marke what Degree the shadow of the Wyer cutteth in the Circle ; and instantly seek the Sunnes Azumith , either South or North : Reckon it on the Circle from the shadow to the Meridian , as was taught in the Use next before , noting that end with the Letter contrary to that of the Azumith : as if the Azumith bee South , note it N. and the opposite end S ; if the Azumith bee North , note it S , and the opposite end N : whereby also you have the East ▪ and West sides : So shall the Arch S A. or N A. give the Declination of the plaine , and the point A , the coast or quarter into which it is . Example , June 2 in the forenoone , applying the instrument to a wall , I found the shadow in 23 Degr. of Quadr. 2. and the height of the Sunne was 26 Degrees , whereby I found the Azumith to be North 84 Degr. which reckoned from the shadow against the Sunne , fell upon 61 Degr. in Quad. 1. for one end of the Meridian ; and the Opposite end ▪ which is N. upon 61 Degr. in Quad. 3. And A was on the East side of N. Wherefore the Declination of that Wall is 61 Deg. from the North Eastward . Use XIX . To finde the Declination of an upright wall by knowing the time of the Sunnes comming to it , or leaving it . And contrariwise , the Declination of an upright Wall being known to finde at what time the Sunne will come into it . Because the Declination of a plain is an arch of the Equinoctiall intercepted between the Horizontall section of the plaine : and the East or West points : Or else ( which is all one ) between the Meridian , and A , the axis of that Horizontall sexion . Watch till you see the Center of the Sunnejust even with the edge of the Wall : then instantly take the Sunnes Azumith from East or West , by Use IX . the same is the Declination of the wall . Likewise if the Declination be given , reckon it upon the Limbe of your Quadrant from the East and West point ; and the thread being applyed to the end of that Arch , shal in the Suns imaginary parallel for that day , cut the houre and time desired . Use XX . Certaine advertisements necessary for the use of the Quadrant in the night . In which Questions as concerne the night , or the time before Sunne-rising , and after Sunnesetting , the instrument representeth the lower Hemisphere , wherein the Southern Pole is elevated . And therefore the parellels which are above the Equinoctiall toward the Center , shall be for the Southerne or winter parellels : and those beneath the Equinoctiall , for the Northerne or Summer parallels : and the East shall be counted for West , and the West for East ; altogether contrary to that which was before , when the Instrument represented the upper Hemisphere . Use XXI . To finde how many Degrees the Sunne is under the Horizon at any time of the night . Seek the Declination of the Sunne for the day proposed by Use I. and at the same Declination on the contrary side of the Equinoctiall imagine a parallel for the Sunne that night ; and marke what point of it is in the very houre and minute proposed : Set the bead to that poynt ; then applying the thread to the scale of Altitudes , marke upon what Degree the bead falleth : for the same shall shew how many Degrees the Sunne is under the Horizon at that time . Use XXII . To finde out the length of the Crepusculum , or Twi-light . It is commonly held that Twilight is so long as the Sunne is not more then 18 degrees , under the Horizon , the question therefore is , at what time the Sunne cometh to be 18 Degrees under the Horizon any night . Seek the Sunnes declination for the time proposed , and at the same declination , on the contrary side of the Aequinoctiall , imagine a paralsel for the Sunne that night : then set the bead at 18 degrees in the scale of Altitudes ; and carry the thread about till the bead fall upon the imagined parallell : for there shall be the houre or time sought . And in this very manner you may find the time or houre of the night at any other depression of the Sunne under the Horizon . FINIS . 
A15752	----	A short treatise of dialling shewing, the making of all sorts of sun-dials, horizontal, erect, direct, declining, inclining, reclining; vpon any flat or plaine superficies, howsoeuer placed, with ruler and compasse onely, without any arithmeticall calculation. By Edvvard Wright. Arte of dialing Wright, Edward, 1558?-1615. 1614 Approx. 47 KB of XML-encoded text transcribed from 27 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-10 (EEBO-TCP Phase 1). A15752 STC 26023 ESTC S111551 99846866 99846866 11861 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A15752) Transcribed from: (Early English Books Online ; image set 11861) Images scanned from microfilm: (Early English books, 1475-1640 ; 946:04) A short treatise of dialling shewing, the making of all sorts of sun-dials, horizontal, erect, direct, declining, inclining, reclining; vpon any flat or plaine superficies, howsoeuer placed, with ruler and compasse onely, without any arithmeticall calculation. By Edvvard Wright. Arte of dialing Wright, Edward, 1558?-1615. [52] p. : ill. Printed by Iohn Beale for William Welby, London : 1614. Another issue, with 3 figures numbered 16-18 on a bifolium signed G1,G2, and with new prelims., of: Wright, Edward. The arte of dialing. Running title reads: A treatise of dialling. Signatures: A² B-F⁴ G² ² G² . Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Dialing -- Early works to 1800. 2004-10 TCP Assigned for keying and markup 2004-11 Apex CoVantage Keyed and coded from ProQuest page images 2005-01 Emma (Leeson) Huber Sampled and proofread 2005-01 Emma (Leeson) Huber Text and markup reviewed and edited 2005-04 pfs Batch review (QC) and XML conversion A SHORT TREATISE OF DIALLING : SHEWING , THE MAKING OF ALL sorts of Sun-dials , Horizontal , Erect , Direct , Declining , Jnclining , Reclining ; vpon any flat or plaine superficies , how soeuer placed , with ruler and compasse onely , without any Arithmaticall calculation . By EDVVARD WRIGHT . LONDON , Printed by Iohn Beale for WILLIAM WELEY . 1614. The Contents of this Booke . Chap. 1. The making of the Clinatory . Chap. 2. The first diuision of dials into Horizontall , erect inclining , and reclining . Chap. 3. The second diuision of Dials into direct & declining . Chap. 4. The third diuision of Dials , either agreeing with the plaine of the Meridian , or disagreeing from the same . Chap. 5. To find the eleuation of the meridian line aboue the Horizon . Chap. 6. The describing of the figure of the dial first on paper , pasteboard . Chap. 7. The making of Aequinoctial Dials . Chap. 8. The finding of the substilar line , and stile , in grounds not Equinoctial direct , aud Polar . Chap. 9. The finding of the distance of the stile from the Meridian line in dials , that be neither equinoctial nor polar . Chap. 10. The finding of the substilar line & the distance of the stile frō it , when the stile maketh a right angle with the merid ▪ line . Chap. 11. From which end of the Meridian line the eleuation of the stile is to be counted . Chap. 12. The finding of the substilar line & stile in dials that be not Polar nor Equinoctial the stile making oblique angles with the Meridian line . Chap. 13. The drawing of the line of contingence , and of the Equinoctial circle , and how it must be diuided . Chap. 14. The drawing of the houre lines in all dials that be not Aequinoctiall . Chap. 15. What number must be set to the houre lines . Chap. 16. What houre lines are to be expressed in all sorts of dials . Chap. 17. How to translate the dial drawne on paper , or paste-board vnto the dial ground . line . Errat . Correct . B2 b 28 neither nether B3 b 2 cliniatorie clinatory B3 b 2 liue line B3 b 10 made make B4 a 12 alother all other B4 b 14 horozontal horizontal C1 b 21 l●aueth leaneth C3 a 25 ground grounds C3 a 29 drawe drewe D1 a 25 neather nether D3 a 20 contingency contingence D4 b 2 gen●y gence E1 a 8 u●ust must E1 b 23 night from thence : night : from thence E2 a 17 zersection tersection E2 b 4 continue conteine E3 a 8 as the in the E3 a 9 in the as the E3 a 15 nto into E3 b 12 on the one on the E3 b 14 bastbord pastborde . IT is also to be remembred that there should haue been numbers set to euery one of the figures , or pictures of the dials , adioined to the end of this treatise , beginning at E4 b , as there is to the three last , viz. the 16. 17. and 18. figure . As in the 7. Chap. where I shewe the making of Aequinoctial dials sect 4. when I referre you to the first and second figure , I meane the figues or pictures of the Aequinoctial dials in E4 b. To the first of these therfore there should be set [ 1. Figure , to the next [ 2. Figure , and to the first of the polar dials in F1 b , there should be set [ 3. Figure , and to the next in F2 a [ 4 Figure &c. Also in the 18. figure the line F. E. should haue beene continued vpwards so much longer , and it is between R. and E , and at the end thereof being so continued the letter ▪ should be set . A TREATISE OF DIALLING CAP. I. The making of the Clinatory . 1 DIals are diuersly made according as they are diuersly placed . Therefore their situation must first be knowne : which may be done by an instrument not vnfitly called a Clinatory . 2 Let this instrument be made iust foure square , and let the thicknesse bee about halfe a quarter of the breadth of it , vpon one side therof describe a quadrant , whose two semidiameters or sides must be paralel to the side of the quadrate . 3 The quadrant must bee diuided into 90. degrees , with figures set to euery fifth or tenth degree ( as the manner is ) both forward and backward , and without the peripherie thereof , a groofe or furrowe must be made so deepe that a plummet hanging by a thrid from the Center of the quadrant may fall into it , in such sort that the thrid may come close to the degrees of the quadiant . 4 Close within the limb of this quadrant make a great round hole or box , for the placing of a magneticall needle within the same , whose true Meridian line must bee perpendicular to one of the sides of the quadrant , which shall be called the North side , and the other side of the quadrant shall bee called the East-side , to the which the Westside of the Clinatory is opposite , as the South side of the Clinatorie is opposite to the Northside of the quadrant : and the magneticall Meridian , must bee drawne in the bottome of the box according to the variation of the place where you are . The figure of the Clinatory . CAP. II. The first diuision of Dials into Horizontall , erect inclining , and reclining . EVery flat whereupon a Diall is to bee made ( which is also called the Diall ground ) either lieth leuel with the horizon , or els is eleuated the one side higher then the other . 2 The first kinde is thus known : Take the Clinatorie and hold it so that the plummet fall vpon on of the Semidiameters , or sides of the Quadrant : then if the nether side of the Clinatory , which way so euer you turne the instrument , will touch the flat , it lieth leuel with the Horizon , and Dials made vpon such flats , are called Horizontal Dials . 3 Those flats which are eleuated the one side higher then the other , stand either vpright ( which are called erect ) or else , they stand leaning : if they leane to you ward , when you stand right against them , they are called inclining : otherwise reclining , if they leane from you ward . 4 All these flats are thus knowne : Holding the Clinatory as before ; if either the right , or the left side thereof ( whereto the plumbline is aequidistant ) will touch and lie close to the flat , it is erect : but if either of the neither corners onely touch it , it is reclining : if either of the vpper corners onely touch it , it is inclining . 5 And how much the reclination or inclination is , you shall know after this manner . 6 Set one of the sides of the Clinatory to the flat , in such sort , that the plumbline hanging at liberty , may fall vpon the circumference of the quadrant : for then the arke of the quadrant , betwixt the plumbline and that side of the quadrant that is parallell , or aequidistant to the slatte , is the reclination thereof , if the center of the quadrant be from the flat , or else the inclination , if it bee towards the same . CAP. III. The second diuision of Dials into direct and declining . ALl flats are either direct , or declining . 2 All flats lying leuel with the Horizon are direct . 3 But if the flat lie not leuel with the Horizon : you shall thus know whether it be direct or declining . First , draw therein a line parallel to the Horizon , after this manner : holde the Clinatory to the flat in such sort , that the plumbline may fall vpon one of the sides of the quadrant ; then draw a line by the nether side of the Clinatory in recliners ; or by the vpper side in incliners , or by either of those sides in erect flats , for that line shall be parallel or aequidistant to the Horizon , and may be called the Horizontall line . Set the North side of the Cliniatorie to this liue , if the North end of the needle looke towards the flat : then if the Magnetical Meridian be right vnder the needle , it is a direct flat : but if it differ from it , it is declining , and that so much as that difference is , and that way which the North end of the needle declineth from the Northend of the Meridian line in the clinatory . 4 If the Southend of the needle looke towards the flat , made your account contrary wise . CHAP. IV. The third diuision of Dials , either agreeing with the plaine of the Meridian , or disagreeing from the same . ALl flats doe either agree with the plaine of the Meridian circle ( which may therefore bee called meridian flats ) or else they disagree from the same . 2 They are knowne thus : If the flat bee erect and declining 90. degrees , it is a Meridian flat , otherwise it is no Meridian flat ; and then you must first draw therin the meridian line , after this maner . 3 If the flat be Horizontal , take the clinatory and lay it flat downe thereupon ; and turning it about till the needle hang precisely ouer the Magneticall meridian , by that side thereof that is parallel to the true Meridian line of the Clinatory , drawe a right line , for that shal be the Meridian line desired . 4 In erect flats the Meridian line is perpendicular , and therefore laying the Clinatory close to such a flat in such sort that the plumbline hang precisely on either side of the quadrant , a line drawne by the side of the clinatory , parallel to that side of the quadrant , shall be the meridian line . 5 In direct flats , a line perpendicular to the line aequidistant from the Horizon , is the Meridian line we seeke for . 6 In flats reclining or inclining , declining also 90. degrees ( which are commonly called , East , or West reclining or inclining ) the meridian line is parallel to the horizon . 7 For alother inclining or reclining , & withall declining flats , drawe a line vpon some pastbord or paper which shall bee called the horizontal meridian AB . wherein settingone foot of your cōpasses , with the other draw an arch of a circle ; & therin reckon the complement of the declination FC . drawing a right line BC. by the end thereof out of the center B. This right line you shall crosse squire-wise with another as AC . which may be called the base of inclination or reclination , and must also meet with the horizontall meridian at A. and setting one foot of your compasses in the crossing at C. with the other foote draw an arke , counting therein the complement of the reclination or inclination AG. drawing a right line by the end therof , out of the center of the foresaid ark CGD . & from A erect AD. perpendicular to AC . which may meet with CGD . the line of reclination or inclination at D. Also from A. draw the line AF. perpendicular to the horozontal meridian , AB . in the point A. and equal to the former perpendicular AD. and from the end therof draw a line to ( the center of the arke of declination ) B. Then continuing foorth AC . to N. ( that CN . be equall to CD ) from N. you shall draw a line to B. which ( if you haue wrought truely ) must be equall to BE. Now the angle contained betweene the lines NB. and BC. sheweth how much the Meridian line in your Diall ground should be distant from the line which you drew aequidistant to the Horizon heere represented by BC. In this line therefore ( in the Dial ground ) set one foote of your compasses , and extending the other that way which the Diall declineth , drawe an arke of a circle , vpwards in recliners , but downewards in incliners : and therein count the said angle from the line parallel to the Horizon , and drawe by the end thereof a line , which shall bee the true Meridian in the Dial ground . 8 From A draw AH perpendicular to EB . make BI . equal to BH . from I. let IK be drawn perpendicular to BN . make CL. equal to CK and drawe a line from L to A. of these three lines AH . IK . and LA. make the triangle AHM. for then the angle AHM. is the angle which the dial ground maketh with the plain of the meridian . CHAP. V. To finde the eleuation of the Meridian line aboue the Horizon . THe Meridian line is either parallel to the Horizon , or else eleuated the one end higher then the other . 2 If the flat bee either horizontal , or East , or West , and inclining , or reclining , the meridian is parallel to the Horizon . 3 In all other flats that disagree from the plaine of the Meridian circle , the Meridian line is eleuated the one end higher then the other . 4 This eleuation is either vpright , as in all erect Dials not declining 90. degrees , or else leaning , as in all inclining , and reclining flats not declining 90 degrees , which if they be direct , is equall to the complement of reclination , or inclination . 5 But if they decline , then the angle ABE . in the former figure , is the eleuation of the Meridian line . 6 If the meridian line bee not erect , it leaueth either Northwards , when the eleuated end thereof looketh towards the North , or else Southwards when the eleuated end looketh towards the South . 7 All flats are either Polar ( which being continued ▪ would goe by the poles of the world ) as all leaning flats , wherein the eleuation of the meridian line is Northwards , and equall to the poles eleuation ▪ and all erect decliners 90. degrees . Otherwise they are no polar flats . CHAP. VI. The describing of the figure of the Diall first on paper or pasteboard . NOw it shall bee best to take a sheet of paper , or rather a pastebord , that you may therein describe the figure of your Dial , before you draw the Diall it selfe vpon his ground : that is , vpon the Truncke , Stone , wall , &c. 2 This paper , or pasteboord therefore , you shall place , or vnderstand to be placed so as your Dial ground is or must be placed , and therein write the names of the parts of the world , as they lie in respect of your Dial ground , as East , West , North , South , Zenith , Nadir , vpper part , nether part , &c. which you may do by helpe of the magnetical needle : for the North end thereof ( hanging at liberty ) sheweth the North , whereto the South is diametrally opposite ; and your face being turned towards the North , your right hand sheweth the East , your left hand the West , the Zenith , or verticall point is aboue your head , the Nadir vnder your feete . Note also , which end of the Meridian line must be higher , and which lower ; if the Meridian be not parallel to the Horizon . CHAP. VII . The making of Aequinoctiall Dials . ALl Diall grounds are aequinoctiall , or not aequinoctiall . 2 An aequinoctiall ground is that which agreeth euen with the plaine of the aequinoctiall Circle : which is thus knowne . If the Diall ground be direct , and the Meridian line eleuated Southwards , equally to the complement of the poles eleuation , it is an Equinoctiall Diall ground , otherwise not . 3 In an Equinoctial Dial you shall describe the houre lines after this manner . 4 Set one foot of your Compasses in the Meridian line AB . and with the other , drawe a circle DBC . and deuide it into 24. equall parts , as D. E. F. G. &c. beginning at B. the crossing therof , with the Meridian line ; for then right lines , as AD. AE . AF. AG. &c. in the 1. and 2. figure drawne out of the Center , by those diuisions shall bee the houre lines . 5 The stile must stand vpright out of the center of the Diall . 6 Of Equinoctiall Dials there be two sorts , the vpper and the nether . 7 The vpper Equinoctial Diall looketh vpwards to the eleuated Pole of the world : And it sheweth the houre of the day , onely in the Spring and Summer time , as in the first figure . 8 The nether , or lower Equinoctial dial , is that which looketh downewards to that Pole of the world which is beneath the Horizons and sheweth the houres onely in Autumne , and Winter , as in the second figure . CHAP. VIII . The finding of the substilar line , and stile , in grounds not Equinoctiall direct , and Polar . IN all Dial grounds that are not Equinoctiall , the substilar line , and the distance of the stile from the substilar must bee found . 2 The substilar line is that right ouer which the stile must be set . 3 The distance of the stile from the substilar , is the angle , or space contained betweene the stile , and the substilar line . 4 The finding out of these is diuers , in diuers kinds , and therefore must bee specially shewed in each kinde . 5 In direct Dial grounds not Equinoctiall , and Polars not Meridian , the substilar line is the same with the Meridian line , or else parallel thereto , in declining polars . 6 In Polar ground ▪ agreeing with the plaine of the Meridian , the substilar line may thus be found . 7 Set one foot of the compasses in the South-end of the line that you drawe equidistant from the Horizon and extending the other foot towards the North end of the same line , draw an arke of a circle : therein reckon the eleuation of the Pole beginning at the foresaid line : for a right line drawne thereby out of the center , shall be the substilar line AB . figure . 3. 8 In al Polar grounds draw a parallel CD . ( figu . 3. 4. 5. 6. 7. 8. ) to the substilar line at a conuenient distance from the same ; for that shall be the line representing the stile . CHAP. IX . The finding of the distance of the stile from the Meridian line in Dials that be neither equinoctial nor polar . IN all Dial grounds that be not aequinoctiall nor polar , before the substilar line , and distance of the stile from it can be found , first the distance of the stile from the Meridian line must be found after this maner . 2 If the Meridian line be parallel to the Horizon , as BC. the distance of the stile from the Meridian line , is equal to the height of the Pole , as BR . 3 But if the eleuation of the Meridian be either vpright , as AG. or leaning towards the North , and withall greater then the Poles eleuation , as AH . the height of the Pole BR . taken out of the height of the Meridian line BH . or BG . shal leaue the distance of the stile from the Meridian line RH . or RG . 4 If the eleuation of the meridian line be Northwards , and lesse then the height of the Pole , as BI . take the eleuation of the meridian line BI . out of the height of the pole BR . and there shall remaine the distance of the stile from the Meridian line RI. 5 If the eleuation of the Meridian line be Southwards , and either greater , or equal to the complement of the Poles eleuation , as AF. and AE . then the complement of the Meridian lines eleuation , FG. or EG . added to the complement of the Poles eleuation GR. shall make the distance of the stile from the meridian line . 6 If the eleuation of the meridian line be Southward and lesse then the complement of the poles eleuation as CD . the eleuation of the meridian line CD and the height of the Pole C● . put together shall make the distance of the stile from the meridian line . CHAP. X. The finding of the substilar line and the distance of the stile from it , when the stile maketh a right angle with the meridian line . SEcondly , in a ground not Equinoctial nor Polar we must consider whether the stile make a right angle , or an oblique angle with the meridian line . 2 The stile shall make a right angle with the meridian line , if the eleuation of the meridian line be Southwards and equall to the complement of the Poles eleuation , as in the 9. 10. 11. and 12. figure Herein a right line drawne squirewise ouerthwart the meridian line , towards that part of the world , which is opposite to that whereto the dial ground declineth , shall be the substilar line , as BA . in the 9. 10. 11. and 12. figu . and the distance of the stile from the substilar line shal bee equall to the angle which the dial ground maketh with the plaine of the meridian circle as the angle BAD . fig. 9. 10. 11. 12. which angle is found by the third Chap. CHAP. XI . From which end of the Meridian line , the eleuation of the stile is to be counted . IF the stile make an oblique angle with the meridian line , we must first finde out from whether end of the meridian line , the eleuation of the stile must be reckoned , thus : 2 If the meridian line be parallel to the horizon as in the 13. figure , the eleuation of the stile shal be reckoned from the North end of the meridian line in reclining , and horizontal flats looking vpwards , as BR . from B in the former figure , but contrariwise in incliners as PC . from C. in the same figure . 3 If the meridian line be eleuated the one end higher then the other from the horizon , and the dial ground looke towards the South , the eleuation of the meridian being also Northwards , and lesse then the eleuation of the pole : the eleuation of the stile shal be counted from the vpper end of the meridian line : as IR. from I. 4 But if the eleuation of the meridian be greater then the eleuation of the pole , or vpright , or southwards and greater then the complement of the poles eleuation ; the eleuation of the stile shall bee counted from the neather end of the meridian line , as PM , PN , PO , from MNO . 5 If the eleuation of the meridian line be Southwards and lesse then the complement of the poles eleuation , the eleuation of the stile shal be counted from the vpper ende of the meridian line as DP . from D. 6 If the Dial ground looke toward the North , the eleuation of the stile from the meridian line shal be reckoned contrariwise in euery kinde . CHAP. XII . The finding of the substilar line and stile in Dials that be not Polar nor Equinoctiall , The stile making oblique angles with the Meridian line . HAuing thus found out from whether end of the meridian line the eleuation of the stile is to be reckoned , set one foot of your compasses in the meridian line as in A. and stretching foorth the other foot towards that end of the meridian line , from which the eleuation of the stile is to bee reckned as towards L. draw an arch of a circle MDLN. and ( beginning at the Merîdian line ) reckon and marke therein the eleuation of the stile from the Meridian line , LD . figure 13. 14. 15. in the rest LO . either Eastwards or Westwards in direct Dials , as in the 13. 14. 15. fig. but in decliners towards that part of the world which is opposite to the part whereunto the Dial declineth , as in the 16. 17. 18. fig. 2 Then in direct Dials , a right line ACD . fig. 13. 14. 15. drawne out of the center of the said arke by the marke of the stiles eleuation from the meridian line shall be the line representing the stile , and therefore the distance of the stile from the substilar line shall be the distance of the stile , from the meridian line . 3 But in decliners you shall thus finde the substilar line : From O the point of the stiles eleuation from the meridian line in the foresaid arke drawe OP . a perpendicular to the meridian line AL. and taking the length of this perpendicular with your compasses , leaue one foote in P. the concurse therof with the meridian line , and with the other describe a quadrant of a circle QRO. beginning from the Meridian line , and so proceeding vnto O the other end of the perpendicular line : and in that quadrant beginning at the meridian ALQ. reckon and marke QR . the complement of the angle conteined betweene the plaines of the diall ground and of the meridian circle , and take with your compasses RS. the distance of that marke from the meridian line , and setting one foote of the compasses in P. the meeting of that perpendicular with the meridian line , with the other make a prick T , in the same perpendicular line : for then AB . a right line drawn by this prick T. out of the center of the foresaid arke MDLN. shall bee the substilar line . 4 Then take with your compasses TR. the distance of the foresaid marke in the quadrant , QRO. and this pricke , and leauing one foote of your compasses in the same pricke T. with the other make another pricke V. in the arke you first described ; for then a right line AV. drawne thereby out of the arch you first described shall bee the stilar line , or line representing the stile . 5 In Dials not polar nor aequinoctiall , if the distance of the stile from the substilar line be but smal as in the fig. 10. 12. 17. it may bee increased by drawing a paralel CD . to the stile already found , which for distinctions sake may bee called , the stile augmented . CHAP. XIII . The drawing of the line of Contingence , and of the Equinoctiall circle , and how it must be diuided . NOw in all Dials that be not aequinoctiall , draw a right line , EHF . so long as you can , making right angles with the substilar line , which is called the line of contingence , or touchline . 2 Then describe the Equinoctiall circle GHI . after this manner : Take with your compasses the shortest distance betweene H. the intersection of the line of contingence with the substilar line , and the stilar line , and leauing one foot in that intersection , with the other make a pricke B. in the substilar line , whereupon describe a circle GHI . which shall be called the equinoctiall circle . 3 If the distance of the stile from the substilar be augmented , you must draw two touch lines and two aequinoctial circles : as in 10. 12. 17. figures . 4 The halfe of the aequinoctiall circle next the line of contingence must be deuided into 12. equal parts , beginning at H , the intersection thereof with the substilar line in all direct dials , and erect or meridian polars which are commonly called East or West dials erect , as in the 3. 4. 5. 6. 13. 14. 15. figures . 5 In polars not meridian nor direct , let HK , in fig. 7. & 8. ( the complement of the angle which the dial ground maketh with the plaine of the meridian ) be numbred and marked in the aequinoctial circle , beginning at the substilar line , and proceeding that way which the diall ground declineth as from H. to K. for at that marke K you must begin to diuide . 6 In decliners not polars , if the stile make a right angle with the meridian line , as in the 9. 10. 11. 12. figu . a paralel to the line of contingence , drawne by the center of the aequinoctiall , shall shew the beginning of the diuision , as BK in figu . 9. 10. 11. 12. 7 But if the stile make an oblique angle with the meridian line , and the line of contingencye , cut the meridian line , as in the 16. figu . your ruler laid to that cutting at X and the center of the Equinoctial B. shal shew in the peripherie thereof , the beginning of the diuision K if the distance of the stile from the substilar be not augmented . 8 But if it be augmented ( as in the 17. figure ) the shortest distance HX betweene H the intersection of the touch line , with the substilar line , and the stile not augmented AV must bee taken with the compasses , and resting one foot in that intersection H , with the other make a pricke Y in the substilar line , towards B the center of the Equinoctiall ; by which pricke Y & Z the mutuall intersectiō of the next touch line with the meridian line , let a right line YZ be drawne , for BK . and BK . paralels to it drawne out of the cenrers of both the Equinoctials , towards the meridian line , at their crossings with the Equinoctials K & K shall shew the beginnings of their diuisions . 9 But if the touch line cut not the meridian line as in the figure 18. let a paralel thereto XY bee drawne , which may cut the meridian line in Y and take with the compasses the shortest distance ZA betwixt the intersection thereof with the substilar line and the stile not augmented ; and leauing one foote in that intersection Z , with the other make a pricke B in the substilar line towards the center of the Equinoctiall ; from this pricke drawe a right line BY from B to Y the intersection of the said paralel with the meridian line ; for BKA paralel to this line drawne out of the center of the Equinoctiall B. shall shew the beginning of the diuision K. CHAP. XIIII . The drawing of the houre lines in all Dials that bee not Equinoctiall . HAuing thus , deuided the Equinoctial circle , lay your ruler to the center thereof B. and to euery one of those prickes 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 by which it is deuided , and make marks 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 &c. in euery place where it crosseth the line of contingence for then 2 In all polar Dials paralels to the substilar line , drawne by those markes , shall bee the houre lines , as in the 3. 4. 6. 7. 8. figu . 3 In Dials not polar , in which the height of the stile is not augmented , right lines drawne out of the center of the dial by those markes shal be the houre lines as in the 9. 11. 13. 14. 15. 16. 18. figu . And if any of the diuisions of the aequinoctiall circle doe fall in to the substilar line , a paralel to the line of contingence drawne by the center of the said diall , shall shew two opposit houres , distant by the space of six houres from the substilar line ; as for example in direct Dials , six in the forenoone and six in the afternoone , as in the 13. 14. 15. figure . Also if the ruler laid to B. the center of the Equinoctial circle , and some diuision thereof , as V in the 16. and 18. figu . cannot crosse the line of contingence , and yet draweth neerer to it , : draw BY. a right line from the center of the Equinoctial by that diuision , and draw AF a paralel to that line , which may crosse the substilar and line of contingency in F. then let HA. the other part of the substilar that is betweene the line of contingence and the center of the dial A. be cut in such sort that the segments , of the substilar line concurring at the line of contingence AH and HB . may keepe the same proportion which the greater segments BH . and HA. haue , which are contained betweene the center of the Dial and line of contingence , and betwixt the center of the Equinoctiall and the line of contingence . And let aright line BF . bee drawne by that section B and the section of the line of contingence F. For AI a paralel to this right line drawne out of the center of the Diall shall be the houre line that wee seeke for . 4 In those dials wherein the distance of the stile from the substilar is augmented , right lines drawne by those markes in both lines of contingence which are proportionately distant from the substilar line shall be the houre lines . CHAP. XV. What number must be set to the houre lines . IN meridian Dials , the substilar line is the line of the sixth houre : but for the rest , we must consider whether it be an oriental or an occidental dial . 2 An oriental Dial looketh to the East , and the forenoone houres onely must bee set in this Dial , and therefore the substilar line sheweth six of the clocke in the morning ; from which towards the South are the morning houres before sixe , viz. 5. 4. 3. &c. but towards the North after six , 7. 8. 9. 10. 11. as in the 3 figure . 3 An occidental Dial looketh directly Westwards : and onely the houres after noone can bee set into this dial . Therfore the substilar line sheweth the sixt houre after noone : from which toward the North are the houres before six in this order . 5. 4. 3. 2. 1. but towards the South after six thus . 7. 8. 9. &c. as in the 4. figure . 4 In Dials not Meridian , if a ruler laid to the center of the aequinoctial and the beginning of the diuision thereof doe crosse the touchline ; then the houre line drawne by that crossing is the line of twelue a clocke . But if it cannot crosse the touch line , imagine notwithstanding , that crossing and the twelue a clocke line , drawne thereby without the bounds of your Dial , whereabouts you thinke it would bee , if the ruler and touch line were continued foorth long inough . 5 Then in al Dials not meridian , imagine the stile to be fastned in his place , in aequinoctial Dials perpendicularly erected out of the Center . In Dials that be not Equinoctial , conceiue it to be placed exactly ouer the substilar line , so much raised from the same as the stilar line in your paper or pastbord , is distant from the substilar line . 6 After this , place your paper or pastebord ( wheron the figure of your Dial is described ) in the same site or position that the dial ground is , or must be placed ▪ so that the quarters of the world written thereupon , may answer in like position to the quarters of the world as they lie in respect of your dial ground : for then if the 12. a clocke line be towards the North , from the stile it is the line of the 12. houre of the day . From hence therefore towards the West are the forenoone houres , 11. 10. 9. 8. 7. &c. and toward the East , the afternoone houres , 1. 2. 3. 4. 5. 6. &c. 7 But if the 12. a clocke line bee Southward from the stile , it is the line of the twelfth houre in the night from thence : therefore on both sides are the night houres : toward the West , after midnight , 1. 2. 3. 4. &c. towards the East before midnight , 11. 10. 9. 8. &c. CHAP. XVI . What houre lines are to be expressed in all sorts of Dials . IN al Dials , those houre lines onely are to be expressed , vpon which the shadow of the stile shal fall . Therefore the houres of the day onely are to bee expressed . 2 In Dials not Polar , wherein the height of the stile is not augmented , if the stile point vpwards , and the eleuation thereof from the substilar line , bee not lesse then the complement of the sunnes greatest declination , all the houre lines seruing for the longest day , are to be expressed therein . 3 But if the cleuation of the stile from the substilar be lesse then the complement of the Sunnes greatest declination , draw a right line out of the intersection of the line of contingence , and substilar perpendicularly ouerthwart the stilar line : and setting one foot of your compasses in the center of the dial , and extending the other towards the other end of the stilar line , draw an arke there from equal to the complement of the Sunnes greatest declination : and thereby draw a line out of the center of the Dial , and setting one foot of your compasses inthe intersection of this line with the foresaid perpendicular , extend the other foot to the stilar line : Then keeping this distance , set one foote of your compasses in the center of the aequinoctial circle and with the other crosse the line of contingence on both sides the substilar : now if you lay your ruler to these crosses and the center of the Dial : right lines drawne thereby beyond the center of the Dial shal continue betweene them the space wherein no houre lines are to be expressed . 4 This rule holdeth also in meridional Dials inclining , when the eleuation of the stile is counted from the vpper end of the Meridian line , and the eleuation of the stile from the substilar is lesse then the complement of the Sunnes greatest declination . 5 If the stile point downwards , no houre lines are to be expressed aboue a line parallel to the horizon drawne by the center of the Dial. 6 And if the crosse in the line of contingence ( made as before was shewed ) be aboue the line aequidistant to the horizon , drawne by the center of the Dial ; no houre lines are to be expressed aboue a right line drawne from the crosse and continued beyond the center of the Dial. 7 If any part of the Dial whereupon the shadow of the stile may fal , bee void of houre lines : let the houre lines before described bee continued foorth into that part of the Dial , as in the 13 and 15 figure . CHAP. XVII . How to translate the Diall drawne on paper or paste-board vnto the Dial ground . THe figure of your Dial being thus described , you shall translate the same into the Dial ground , after this manner . 2 Place the paper or pastebord whereas the figure of your dial is described in such sort , in the Dial ground is placed , so as the quarters of the world written on the paper or pastebord may answer in like position to the quarters of the world as they lye in respect of the Dial ground . 3 Then as the houre lines and substilar line are described in your pastebord , so in like manner , and in like position , let them be inscribed nto your Dial ground that so little part of the ground as may be , be left voide of houre lines seruing for vse , and that the spaces on both sides from the substilar line drawne on the Dial ground bee proportionable to the number of houre lines that are to bee expressed in the Dial. 4 In Polar dials draw a right line squire-wise ouerthwart the substilar in the Dial ground ; then take with your compasses the distances of the houre lines from the substilar in the pastebord , and set them into that line drawne squire-wise in the Dial ground , setting alwaies on foot in the intersection thereof with the substilar line , and with the other foote making pricks in the said line drawne squire-wise : And let paralels to the substiar line be drawn by those prickes , for they shall bee the houre lines we seeke for , set into the dial ground . 5 The stile must be paralel to the substilar line , and must be placed directly ouer it , so much distant from the same , as the stilar line is distant from the substilar in the figure of your Dial drawne on the pastbord or paper . 6 In Dials that be not polars , wherein the eleuation of the stile from the substilar is not augmented , describe two peripheries of equal bignesse on the Dial ground , the center thereof being placed in the meridian line , the other vpon the center of the Dial in the bastbord : then in this peripherie take the distances of the substilar and the houre lines from the Meridian with your compasses , out of the figure of your Dial in the paper or pastbord , & set those distāces likewise into the dial grounds , and by them draw the houre lines and the substilar from the center of the dial . 7 The stile must bee fastned in the center , and must hang directly ouer the substilar , eleuated so much from the same , as the stilar line in the figure of your dial is distant from the substilar . 8 But in Dials that be not polars , wherein the eleuation of the stile from the substilar is augmented , let the substilar line bee described in the Dial ground so much distant from the Meridian , which you first described therein , as the substilar is distāt from the Meridian in the figure of your dial . And let two lines of contingence be drawne squire-wise ouerth wart that substilar in the Dial ground , so much distant each from other , as the lines of contingence in the paper are . And let the distances of the houre lines from the substilar line bee taken in both lines of contingence in the figure of the dial , and be set in like manner in to the lines of contingence , answering to them in the Diall ground , setting one foot of your compasses alwaies in the substilar line , which is in the Diall ground , and with the other making markes in the lines of contingence drawne therein : for then right lines drawne by those markes , differing alike from the substilar line , shall bee the houre lines . The stile must hang perpendicularly ouer the substilar line , so much distant from the same , and from the sections thereof with the lines of contingence , as the stile augmented in the figure of your Diall is distant from the substilar . Equinoctiall Dials . North direct reclining 51. degrees , ●0 . minutes , or the vpper Equinoctiall Diall . South direct , inclining 51. degrees , 30. minutes . The manner of finding the substilar line in Meridian Polar Dials . This example serueth for the Oriental dial . Meridian Polar dials . A South direct dial reclining 38 degrees 30. minutes , or a South direct polar dial . A North direct dial inclining 38 degrees 30. minutes , or a North direct polar dial . South declining Eastward 27. degrees , reclining 34. degrees 40. minutes . North declining Westward 36 degrees inclining 32. degrees 15 minutes . North declining Eastward 43. degrees reclining 42. degrees 20. minutes wherein the stile and meridian line make right angles . North declining 84. degrees westward , reclining 7 degrees 20 minutes , the stile pendicular to the Meridian line . South declining Eastward 31 degrees , inclining 48 degrees . ●0 min. the stile perpendicular to the meridian . South declining Westward 86 ▪ degrees 40. min. inclining 4. degrees stile perpendicular to the meridian . In all the figures following , the stile maketh a sharpe angle with the meridian : a direct dial lying leuel with the Horizon , commonly called an Horizontal dial . A South dial erect direct . A North Dial erect , direct . A South dial erect declining Eastward 30. degr . A South erect dial declining westward 80. degr . North declining westward 41. de . 40. min. reclin . 4● . de . 30. min. Notes, typically marginal, from the original text Notes for div A15752-e450 Also in the Diagramme placed there , and in the leafe following , let K & ● be chāged each into others place . Notes for div A15752-e1580 The Dial ground . Horizontal flats : and how they are known . Eleuated flats ; Erect , Inclining Reclining : and how they are to be known To know how much the reclination or inclination is . Notes for div A15752-e1780 How to drawe the Meridian line , in Horizontall flats . In erect flats . In direct flats . In reclining or inclining flats , declining 90. degrees . In incl●ning or reclining flats declining lesse then 90. degrees . To finde the angle which the dial ground ( or flat ) maketh with the plaine of the meridian . Notes for div A15752-e1970 How to finde the eleuation of the meridian line . The eleuation of the meridian line in erect dials . In reclining or inclining direct flats ; in reclining or inclining declining flats . The eleuation of the meridian whether North or South . Flats polar . Not polar . Notes for div A15752-e2170 How to make Equinoctial Dials . Placing of the stile . Vpper Equinoctial dial . Neather Equinoctial dial . Notes for div A15752-e2320 Substilar line . Distance of the stile from the substilar . The finding of the substilar line . In direct flats not aquinoctial In Polars not Meridian . In Meridian Polars . The stilar line in all Polar Dials . Notes for div A15752-e2730 The stile augmented . 
A01089	----	The art of dialling by a new, easie, and most speedy way. Shewing, how to describe the houre-lines upon all sorts of plaines, howsoever, or in what latitude soever scituated: as also, to find the suns azimuth, whereby the sight of any plaine is examined. Performed by a quadrant, fitted with lines necessary to the purpose. Invented and published by Samuel Foster, professor of astronomie in Gresham Colledge. Foster, Samuel, d. 1652. 1638 Approx. 60 KB of XML-encoded text transcribed from 30 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-10 (EEBO-TCP Phase 1). A01089 STC 11201 ESTC S102472 99838255 99838255 2628 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A01089) Transcribed from: (Early English Books Online ; image set 2628) Images scanned from microfilm: (Early English books, 1475-1640 ; 1097:07) The art of dialling by a new, easie, and most speedy way. Shewing, how to describe the houre-lines upon all sorts of plaines, howsoever, or in what latitude soever scituated: as also, to find the suns azimuth, whereby the sight of any plaine is examined. Performed by a quadrant, fitted with lines necessary to the purpose. Invented and published by Samuel Foster, professor of astronomie in Gresham Colledge. Foster, Samuel, d. 1652. [6], 39, [1] p., [2] folded plates : ill. Printed by Iohn Dawson for Francis Eglesfield, and are to be sold at the signe of the Marigold in Pauls Church-yard, London : 1638. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Dialing -- Early works to 1800. Quadrant -- Early works to 1800. 2003-06 TCP Assigned for keying and markup 2003-06 Aptara Keyed and coded from ProQuest page images 2005-02 Andrew Kuster Sampled and proofread 2005-02 Andrew Kuster Text and markup reviewed and edited 2005-04 pfs Batch review (QC) and XML conversion THE ART OF DIALLING ; BY A NEW , EASIE , AND MOST SPEEDY WAY . SHEWING , HOW TO DESCRIBE THE Houre-lines upon all sorts of Plaines , Howsoever , or in what Latitude soever Scituated : As also , To find the Suns Azimuth , whereby the sight of any Plaine is examined . Performed by a Quadrant , fitted with lines necessary to the purpose . Invented and Published by SAMVEL FOSTER , Professor of Astronomie in Grosham Colledge . LONDON , Printed by Iohn Dawson for Francis Eglesfield , and are to be sold at the signe of the Marigold in Pauls Church-yard . 1638. To the Reader . READER , HEre is presented to thy view a short and plaine Treatise ; it was written for mine owne use , it may become thine if thou like it ; The subject indeed is old ; but the manner of the Worke is all new . If any be delighted with recreation of this nature , and yet have not much time to spend , they are here fitted , the instrument will dispatch presently . If they feare to lose themselves in a wildernesse of lynes , or to out-runne the limits of a Plaine , by infinite excursions ( two inconveniences unto which the common wayes are subject ) they are here acquitted of both , having nothing to draw but the Diall it selfe , contracted within a limited equicrurall triangle . If want of skill in the Mathematicks should deterre any from this subject , let them know that here is little or none at all required , but what the most ignorant may attaine . If others shall thinke the Canons more exact ; so doe I , but not so easie to bee understood , not so ready for use , not so speedy in performance , nor so well fitting all sorts of men : and withall an instrument in part must bee used , this will doe all , and is accurate enough . If it must needs be disliked , let a better be shewed and I will dislike it too ; It is new , plaine , briefe , exact , of quicke dispatch . Accept it , and use it , till I present thee with some other thing , which will bee shortly . Imprimatur . Decemb. 1. 1637. SA . BAKER . THE DESCRIPTION OF THE QVADRANT , and the manner how the lines are inscribed and divided . CHAP. I. 1. The description of the fore-side . THe limbe is divided into 90 degrees , and subdivided into as many parts as quantity will give leave . The manner of division , and distinction of the subdivided parts is such as is usuall in all other Quadrants . To describe the other Worke in the superficies ; Take from the upper edge of the limbe about 3 degrees , and set off that space from the center R to A. Then divide AE into seven parts , whereof let EB containe two . Or in greater instruments , if AE be 1000. let EB containe 285. Make SC equall to EB , and drawe the line BC. From C , draw CD parallel , and of equall length to AB . Upon AB and CD , and BE also ( as farre as it is capable ) insert the 90 sines , from B towards A and E , and from C towards D , but let them be numbred from A unto B to 90 , and so to E 113 degrees 30 minutes , from D to C unto 90 degrees . Againe : Draw ES cutting CD at F ; so shall BCFEB containe a parallelogram , whose opposite sides , being parallel , are divided alike , and in this manner . BE and CF as whole sines , doe containe the 90 sines , or as many of them as can distinctly bee put in : and from the divisions are drawne parallel lines , having every tenth , or fifth , distinguished from the rest . These serve for the 12 Signes and their degrees , and therefore you see upon every 30th degree , the characters of the 12 Signes inserted , in such manner as the figure sheweth . And these lines may bee called , The Parallels of the Suns place . In like manner , The lines BC , EF , being first bisected at X and Z , shall make 4 lines of equall length . These 4 lines XB , XC , ZE , and ZF , are each of them divided as a scale of Sines , beginning at X and Z , and from each others like parts are parallel lines protracted , having every tenth and fifth distinguished from the rest . They are numbred ; upon BC , from B to X 90 , to C 180 ; upon FE , from F to Z 90 , to E 180. These lines are called , The lines of the Sunnes Azimuth . This done ; Upon the center R describe the two quadrants VT , and BC , let their distance VC bee one sixth part of Rc , or more if you will. Divide them each into 6 equall parts , at e , o , y , n , s ; and a , i , u , m , r , drawing slope-lines from each others parts , as Va , ei , on , ym , nr , sb : and these lines so drawne are to bee accounted as Houres . Then dividing each space into two equall parts , draw other slope-lines standing for halfe houres , which may be distinguished from the other , as they are in the figure . Then from the points V and T draw the right line VT . Lastly , Having a decimall scale equall to TR , you must divide the same TR into such parts as this Table here following alloweth , the numbers beginning at T , and rising upto 90 at R. Vpon your instrument ( for memory and directions sake ) neere to the line AB , write , The summe of the latitude and Sunnes altitude in Summer ; The difference in Winter . Over VT , write , The line of Houres . Neere to CD write , The summe of the latitude and Sunnes altitude in Winter ; The difference in Summer . By TR , write , The line of latitudes for the delineation of Dialls . A Table to divide the line of Latitudes . 90 10000 62 9360 46 8259 30 6325 14 3325 85 9982 61 9311 45 8165 29 6169 13 3104 80 9924 60 9258 44 8068 28 6010 12 2879 78 9888 59 9203 43 7968 27 5846 11 2650 76 9849 58 9147 42 7865 26 5678 10 2419 75 9825 57 9088 41 7738 25 5505 9 2186 74 9801 56 9026 40 7647 24 5328 8 1949 72 9745 55 8962 39 7532 23 5146 7 1711 70 9685 54 8895 38 7414 22 4961 6 1470 69 9651 53 8825 37 7292 21 4772 5 1228 68 9615 52 8753 36 7166 20 4577 4 984 67 9378 51 8678 35 7036 19 4378 3 739 66 9519 50 86●● 34 6902 18 4176 2 493 65 9496 49 8519 33 6764 17 3969 1 247 64 9454 48 8436 32 6622 16 3758 0 ● 63 9408 47 8348 31 6475 15 3543   S●… 2. The description of the backe-side . Upon the backe-side is a circle only described , of as large extent as the Quadrant will give leave , noted with ABCD , divided into two equall parts by the Diameter AC . The semicircle ABC is divided into 90 equall parts or degrees , every fifth and tenth being distinguished from the rest by the longer line ; They are numbred by 10 , 20 , 30 , &c. unto 90. The same parts are also projected upon the diameter AC , by a ruler applyed to them from the point D. These are numbred also from A to C by 10 , 20 , &c. unto 90. The other semicircle ADC , is first divided into two Quadrants at D. And then upon these two quadrants are inscribed 90 such parts as this Table insuing doth allow . The inscription is made by helpe of a Quadrant of a circle equall to AD or CD , being divided into 45 equall degrees , out of which you may take such parts as the Table giveth , and so pricke them downe , as the figure sheweth . Every fifth and tenth of these parts is distinguished from the rest by a longer line ; they are numbred from A and C , by 10 , 20 , &c. unto 90 ending in D. A Table to divide the upper and nether Quadrants of the Circle . 1 1.00 14 13.36 27 24.25 40 32.44 53 38.37 66 42.25 2 2.00 15 14.31 28 25.09 41 33.16 54 38.59 67 42.38 3 3.00 16 15.25 29 25.52 42 33.47 55 39.19 68 42.50 4 3.59 17 16 18 30 26.34 43 34.18 56 39.40 69 43.02 5 4.59 18 17.10 31 27.15 44 34.47 57 39.59 70 43.13 6 5.58 19 18.02 32 27.55 45 35.16 58 40.18 72 43.34 7 6.57 20 18.53 33 28.35 46 35.44 59 40.36 74 43.52 8 7.55 21 19.43 34 29.13 47 36.11 60 4054 75 44.00 9 8.53 22 20.32 35 29.50 48 36.37 61 41.10 76 44.08 10 9.51 23 21.21 36 30.27 49 37.03 62 41.27 78 44.22 11 10.48 24 22.08 37 31.02 50 37.27 63 41.43 80 44.34 12 11.45 25 22.55 38 31.37 51 37.51 64 41.57 85 44.53 13 12.41 26 22 40 39 32 . 1● 52 38.15 65 42.11 90 45.00 Thus have you both sides decribed . Besides all this , there are two sights added , with a threed and plummet like as in other instruments . The threed hath a moovable bead upon it for speciall use . The same threed passeth through the center R. quite behind the Quadrant , and is hung upon a pinne at the bottome of the Quadrant , noted with W. The reason of the threeds length will be seene when wee come to the uses of the instrument . CHAP. II. The use of the Quadrant in generall . FIrst upon the fore-side . The limbe serveth especially for observation of all necessary angles . The lines AE , CD , with the Parallelogram BCEF , are to find out the Suns Azimuth in any latitude whatsoever . The slope-lines within the arkes VT , cb , by helpe of the threed and bead , doe serve artificially to divide the line of Houres TV , into its requisite parts ; which together with TR the line of latitudes , doe serve to protract all plaine Dialls howsoever scituated . Secondly upon the back-side . Note that ABC is called the Semicircle : AC is called the Diameter : AD the Vpper quadrant : CD the Nether quadrant . The uses of these are to find out the necessary arkes and angles , either for preparation to the Dialls description , or serving after for the Dialls scituation upon the Plaine . In all these uses the threed bearing part , and therefore having asufficient extent of length , that being loosed it may with facility reach over either side of the Quadrant . CHAP. III. To find the Azimuth of the Sunne in any Latitude whatsoever . BEfore you can make any draught of your Diall , you must know the scituation of your plaine , both for declination and inclination . The best way to come to the plaines declination is by helpe of the Sunnes Azimuth . By having the Latitude of the place ; The place of the Sunne in the Eclipticke , and the altitude of the Sunne above the Horizon , you may find out the Azimuth thereof in this manner . Adde the Sunnes altitude , and your latitude together , and substract the lesser of them from the greater ; So shall you have the summe of them , and the Difference of them . With this summe and difference , come to your Quadrant , and according to the time of the yeare ( as the lines will direct you ) Count the said Summe and Difference respectively , and applying the threed unto them , find out the Sunnes place in the Parallels serving thereto , and where the threed cuts this Parallel , observe the Azimuth there intersecting , for that is the Azimuth from the South , if you number it from the line whereon the summe was numbred . Example 1. In the North latitude of 52 gr . 30 min. in the Summer-time the Sunne entring into 8 , and the altitude being observed 30 gr : 45 min. I adde the latitude 52 gr . 30 min. and the Sunnes altitude 30 gr . 45 min So I find the summe of them 83 gr . 15 min. and substracting the lesser of them from the greater , I find the difference of them 21 gr . 45 min. The summe I number in the line AE , and the difference in DC ( because it is in Summer ) and to the termes I apply the threed , and where it crosseth the parallel of the beginning of 8 , there I meet with 66 gr . 43 min. which is the Azimuth from the South , being reckoned from the line AE whereon the Summe was counted . Example 2. The latitude and Sunnes place being the same if the altitude had beene 9 gr . 15 min. The summe of the latitude and altitude would bee 61 gr . 45 min. The difference 43 gr . 15 min. and so the threed applyed to these termes would have shewed 96 gr . 52 min. for the Azimuth from the South . A third Example . In the same Latitude of 52 gr . 30 min. in the Winter-time , the Sunne entring the tenth degree of ♏ , and the altitude being 9 gr . 30 min. I would know the Azimuth of the Sun from the South . I adde the Altitude 9 gr . 30 min. to the Latitude 52 gr . 30 min. and so find the summe of them 62 gr . 0 min. And substracting the Altitude out of the Latitude , I find the Difference of them 43 gr . 0 min. The summe ( because it is in Winter ) I count upon the line DC in the Quadrant , and the Difference upon AE . So the threed applyed to these tearmes cutteth the tenth of ♏ , at 49 gr . 50 min. which is the Azimuth numbred from DC the South . The Amplitude . Note here by the way , That the threed applyed to the Latitude of your place numbred upon both lines AE , DC , will shew you , for any place of the Sunne , the due Amplitude of his Rising or Setting , or the Azimuth whereon hee riseth or setteth , if you number the same from the middle line noted with XZ which here representeth the East and West Azimuths . CHAP. IIII. To find out the Declination of a Plaine . THe declination of a Plaine is numbred from the South or North points towards either East or West . And it is the arke of the Horizon comprehended betweene the South-North , and a line infinitely extended upon the Horizon perpendicular to the horizontall line of the Plaine ; which line may be called the Axis , and the extremity of it , the Pole of the Plaines horizontall line . To find out this declination you must make two observations by the Sunne : The first is of the Distance or angle made betweene the Axis of the horizontall line of the Plaine , and the Azimuth wherein the Sunne is at the time of observation . The second is of the Suns Altitude . Both these observations should bee made at one instant , which may bee done by two observers , but if they bee made by one , the lesse distance of time betweene them , will make the worke to agree together the better . 1. For the Distance . Upon your Plaine draw a line parallel to the horizon , to this line apply the side of your Quadrant , holding it parallel to the horizon . Then holding up a threed and plummer , which must hang at full liberty , so as the shadow of the threed may passe through the center of the Quadrant , observe the Angle made upon the Quadrant by the shadow of the threed , and that side that lyeth perpendicular to the horizontall line , for that angle is the Distance required . 2. At the same instant as neere as may be , take the Sunnes Altitude ; These two being heedfully done , will helpe you to the plaines Declination by these rules following . When you have taken the Altitude , you may find the Sunnes Azimuth by the former Chapter . Then observe , whether the Sunne bee betweene the Pole of the horizontall line and the South North point or not . If the Sunne be betweene them , adde the Azimuth and Distance together , and the summe of them is the Declination of the plaine . If the Sunne be not betweene them , subduct the lesser of them from the greater , and the difference shall be the Declination of the plaine . ¶ By your observation you may know to what coast a Plaine declineth . For if the South North point bee in the midst betweene the Sunnes Azimuth and the pole of the Plaines horizontall line , then doth the Plaine decline to the coast contrary to that wherein the ☉ is : If otherwise , the declination is upon the same coast with the Sunne . CHAP. V. To find the Inclination of a Plaine . THe Inclination of a Plaine is the angle that it maketh with the Horizon . When you have described your horizontall line upon a Plaine , as in this figure EF , crosse it with a perpendicular GH , for the Verticall line . And because the inclinations of the Upper and Under faces of the Plaine , are both of one quantitie in themselves , if therefore you apply the side of the Quadrant noted with AB unto the verticall line of the under face , or to the under side of a Ruler applyed to the verticall line of the upper face , as is here shewed in this figure ; Then shall the degrees of the Quadrant give you CAD the angle of inclination required . CHAP. VI. Of upright declining Plaines . THose Plaines are upright , which point up directly into the Zenith or verticall point of the Horizon , and may be tryed by a perpendicular or plumb-line . In these , as in the rest that follow , before the Houres can be drawne , two things must bee found ; 1. The Rectifying arke ; 2. The Elevation of the Pole above the Plaine . 1. To find the Rectifying arke . Extend the threed from your Latitude counted in the upper Quadrant of the circle on the backeside , to the complement of the Plaines declination numbred in the Semi-circle ; so shall the threed shew you on the Diameter the Arke required . 2. To find the Elevation of the Pole above the Plaine . Extend the threed from the Rectifying arke numbred in he upper quadrant , to your Latitudes complement taken in the Semicircle ; so shall the threed shew upon the Diameter , the Elevation of the Pole above the Plaine . According to these rules , in the latitude of 52 gr . 30 min. supposing an upright Plaine to decline 55. gr . 30 min. I find the Rectifying arke to bee 28 gr . 36 min. And the elevation of the Pole above the Plaine to be 20 gr . 10 minutes . CHAP. VII . In East and West incliners . THose plaines are called East and West incliners , whose horizontall line lyeth full North and South , and their inclination is directly towards either East or West . 1. To find the Rectifying Arke . Extend the threed from your Latitudes complement taken in the upper quadrant of the Circle on the backside , to the complement of the Plains inclination counted in the semicircle ; so shall the threed shew upon the Diameter the Arke required . 2. To find the Elevation of the Pole above the Plaine . Extend the threed from the Rectifying-arke counted in the upper quadrant , to your latitude taken in the Semicircle ; so the threed upon the Diameter gives the elevation of the Pole above the Plaine . Thus in the latitude of 52 gr . 30 min. If a Plaine incline Eastward 40 gr . to the horizon , the Rectifying-arke will be 35 gr . 58 min. And the elevation of the Pole 37 gr . 26 min. above the plaine . CHAP. VIII . In North and South incliners . SUch Plaines are called North and South incliners , whose horizontall line lyeth full East and West , and their inclination is directly upon either North or South . 1. For the Rectifying-Arke . There is no use of it in these plaines , because they all lye directly under the Meridian of the place . 2. To find the Elevation of the Pole above the Plaine . If the inclination be toward the South , adde the inclination to your latitude ; for the summe is the Elevation of the pole above the Plaine . If the summe exceed 90 degrees , take it out of 180 , and the supplement gives you the Poles elevation . If the inclination bee Northward , compare the inclination with your latitude , and subduct the lesser out of the greater : the Difference is the elevation of the Pole above the Plaine , If there bee no difference , it is a Direct polar Plaine . CHAP. IX . In declining Incliners . THose Plaines are called Declining incliners , whose horizontall line declineth from the East or West , towards either North or South , and their inclination also deflecteth from the coasts of North and South towards either East or West . The best way to find the Rectifying-arke , and the poles elevation for these Plaines , will be First , to referre them to a New latitude , wherein they may lye as East or West incliners . For which purpose you are first to find out an Arke , which in respect of its use may fitly be called , The Prosthaphaereticall arke , it is found by this rule : ¶ Extend the threed from the complement of the Plaines declination counted in the upper quadrant , to the inclination numbred in the Semicircle ; so the threed shall give you upon the Diameter the Prosthaphaereticall-arke required . This Prosthaphaereticall-arke is to be used as the Inclination was in the precedent Chapter . For , If the Plaine doe incline towards the South , it must be added to your Latitude : and so the summe ( if lesse then 90 degrees ) gives you the New Latitude : but if the summe bee greater than 90 , then the residue , or supplement of it to 180 degrees will be the New Latitude required . If the Plaine incline toward the North , compare this Prosthaphaereticall-arke with your Latitude , and subduct the lesser of them out of the greater ; So the Difference shall give you the New Latitude . If there be no difference , it is a declining Polar plaine . Secondly , it will be required to know what Inclination these Plaines shall have in this their New latitude ; and that is done by this rule : ¶ Extend the threed from the Prosthaphaereticall-arke taken in the upper quadrant to the Plaines declination counted in the Semicircle : so the threed shewes on the Diameter , the New-inclination in their New latitude . Being thus prepared , you may now proceed as in East and West incliners you did before . 1. To find the Rectifying-Arke . Extend the threed from the New latitudes complement taken in the upper quadrant , to the New-inclinations complement numbred in the Semicircle ; so the threed upon the Diameter shewes the Arke required . 2. To find the Elevation of the Pole above the Plaine . Extend the threed from the Rectifying-arke in the Vpper-quadrant to the New latitude in the Semicircle ; so the threed upon the Diameter gives the Elevation of the pole above the plaine . According to these rules , supposing a Plaine to incline towards the North 30 degrees , and to decline from the South towards the West 60 degrees in the latitude of 52 gr . 30 min. First I find the Prosthaphi-arke 60 gr . 6 min. and because the Plaine inclineth toward the North ; I compare this arke with the Latitude of the place , and taking it out of the Latitude there remaineth 36 gr . 24 min. for the New Latitude . Then I find the New inclination to bee 25 gr . 40 min. and so the Rectifying-arke 59 gr . 8 min. and the Elevation of the Pole above the Plaine to be 32 gr . 20 minutes . CHAP. X. To draw the Houre-lines upon the Horizontall , the full North or South plaines , whether standing upright or inclining . IN the foure last Chapters we have seene the uses of the Circle on the backe-side of the Quadrant : in this and the next Chapter we shall shew the use of TR the line of latitudes , and of TV the line of Houres ; which two lines with the helpe of the limbe VCTB , and of the threed and Bead , will serve to pricke downe any Diall , by the Precepts hereafter delivered . And first we begin with those Plaines which have no declination , whose Poles lye directly under the Meridian of the place ; of which sort are the Horizontall , the Erect South and North plaines , with all Incliners looking directly North or South . Having then by the former Precepts found the Elevation of the pole above your Plaine , you may begin your draught in this manner . First , Draw the line RAT of sufficient length , and out of the line of Latitudes in your Quadrant , take off the Elevation of the pole above the plaine , and pricke it downe from the point A , unto R and T both wayes . 2. Take the line of Houres TV also out of the Quadrant , and with that extent of your Compasses upon R and T as upon two centers , draw the arkes BV and CV , crossing each other in V ; and draw the lines RV and TV : then comming to your Quadrant againe ; 3. Apply the threed to every houre point in the limbe VT or CB , as first to s , or r , so shall it cutte the Line of houres TV in 1 ; Then take off with your Compasses T1 , and pricke it downe here from V to 1 , and from T to 7. Again , Apply your threed to the next houre in the limbe at n or m , it will cut the Line of houres TV in 2 take off T2 , and prick it down here from V to 2 , and from T to 8. So againe , the threed applyed to the third noure at y , or u , cuts the line TV , in 3 ; take off T3 , and pricke it downe here from V to 3 , and from T to 9. In like manner , the threed applyed to the fourth houre at o , or i , will cut the line TV in 4 take off T4 , and pricke it downe here from V to 4 , and from T to 10. So also the threed laid upon the fifth houre at e , or a , cutteth TV in 5 ; take off T5 , and prick it downe here from V to 5 , and from T to 11. Thus are all the Houres pricked downe . An horizontull Diall to 52 gr : 30 m : lat : Lastly then , laying your Ruler to the center A , through each of these points , you shall draw the houre-lines A7 , A8 , A9 , A10 , A11 , AV which is 12 , A1 , A2 , A3 , A4 , A5 , RAT is the line of the two sixes . So having 12 houres , which is halfe the Diall , drawne , you may extend the necessarie lines , as many as you will , beyond this center , as 5A5 , 4A4 , 7A7 , 8A8 , &c. In the same manner may the halfe houres bee pricked downe and drawne , by applying the threed to the halfe houres in the limbe , &c. And note also that in these Plaines before mentioned ; As the extent from V to 1 , is the same with that from T to 7 , so likewise is it the same with V11 , R5 ; And as V2 is the same with T8 , so likewise is it the same with V10 , R4 : So likewise V9 and T9 are all one , and both equall to R3 and V3 . So that the three first houres taken from the Quadrant , that is to say , T1 , T2 , T3 , will give all the houres for these Dialls . T1 , gives V1 , V11 , R5 , T7 . T2 , gives V2 , V10 , R4 , T8 . T3 , gives V3 or R3 , V9 or T9 . But in other Plaines it is not so , for which cause I have rather set downe this way before at length , as a direction for what comes after , for that is generall . Here note againe , that if you desire to make your draught greater , you may in your description either double or triple every length which you take in your Compasses . And so I proceed to all declining Plaines . CHAP. XI . To draw the Houres upon all sorts of declining Plaines , whether erect or inclining . BY the former precepts you must first get the Rectifying-arke , with the Elevation of the pole above the Plaine . After they are had , you may pricke downe the Houre points in this manner following , little differing from the former . A Plaine ▪ inclininge Eastward 40 gr : The horizointall line , parallel to the line of 12. 1. Asbefore ; Upon the line RAT , set off the Elevation of the pole above the plaine , being taken out of the line of latitudes in the Quadrant , from A both wayes , to R and T. 2. Take the line of Houres TV out of the Quadrant , and with that extent upon R and T as upon two centers , describe the two arkes BV and CV crossing at V , and draw the lines RV , TV , and AV. Thus farre we goe along with the last Chapter . 3. If we take the example in the seventh chapter , that plaine is the upper face of an East incliner , whose Elevation is 37 gr . 26 min. and so much doth this line TA reach unto in the line of Latitudes : the Rectifying arke is 35 gr . 58 min. This arke I number below in the limbe of the Quadrant ES , and thereto applying the threed I observe in the upper limbe Vcb T which of the Houres and where it cutteth , I find it to cut the slope line o u in the point P ; to this point P I set the Bead , which by this meanes is rectified and fitted to the description of the Diall . Here you see the use of the Bead , and the reason why this arke counted upon the limbe is called the Rectifying arke : and here bee carefull that you stretch not the threed . 4. The threed and Bead being thus placed and rectified , you shall see the threed to cut the line TV at a upon the Quadrant ; take T a in your Compasses , and pricke it downe here from V to 12 , and from R to 6. Here by the way observe , that because this plaine is an Eeast-incliner , the face of it looketh toward the West , and then if you imagine the true scituation of this Diall upon the plaine whereon it must stand , you will easily conceive that the line of 12 is to stand on the right hand from the line AV. and so the line of 6 on the left hand , whereas if this plaine had faced toward the East , the line of 12 must have stood on the left hand , and 6 on the right hand . Your owne conceit , together with the precepts of the chapter following , must helpe in this , and in other things concerning the right scituating of the lineaments of your Diall . To proceed then , In the same manner must you apply the Bead to every houre line , as in the next place I remove it to the line y m in the Quadrant , and then I see it to cut the line TV in b ; I take 1 b in my Compasses , and with it doe pricke downe from V to 1 , and from R to 7. Againe , the Bead being applyed to the lines nr , sb , the threed will cut the line TV upon the Quadrant in c and d ; I take the points TC , Td , in my Compasses , and pricke them downe from V to 2 and 3 , and from R to 8 and 9. Then againe , the Bead applyed to the lines ei , Va , the threed will cut the line TV in the points e and o ; I take then Te and Tf , and pricke them downe from U ●o 11 and 10 , and from R to 5 and 4. 5. Lastly , lay your rule to A , and draw A10 , A11 , A12 , A1 , A2 , A3 , A4 , A5 , A6 , A7 , A8 , A9 . Thus have you twelve houres , and if you extend these beyond the Center , you shall have the whole 24 houres , of which number you may take those that shall bee fit for the Plaine in this scituation . The halfe houres may thus bee pricked on and drawne also , by applying the Bead to the halfe houres pricked downe in Vcb T the upper limbe of the Quadrant , for so the threed will give you the halfe houre points upon the line TV , which may be taken off , and set downe upon the Diall as the houres themselves were . CHAP. XII . How to place the Diall in a right Scituation upon the Plaine . AFter the houre-lines are drawne by the last Chapter , they are to be placed in a right scituation upon their Plaine . Which to doe , upon some Plaines is more difficult than the Description of the Diall it selfe . To give some directions herein , I have added this Chapter , where you have 9 ▪ Questions with their Answers , giving light sufficient to what is here intended and required : but first be admonished of three things . 1. That the inclination mentioned Chap. 8. is the very same in Use with the Prosthaphaereticall arke mentioned Chapter 9. And therefore when I mention the Prosthaphaereticall arke , because it is of most frequent use , you must remember I meane both the Prosthaph : arke , Chap. 9 , and the Inclination , Chap. 8. 2. That these rules , though given primarily for places of North-latitude , lying within the Temperate , Torrid , and Frigid Zones , yet are also as true , and may bee applyed to all places of South-latitude , if we exchange the names of North and South , for South and North. Here by the way note , that the North part of the Torrid Zone extendeth from 0 degrees of latitude to 23 gr . 30 min. the Temperate Zone reacheth from 23 gr . 30 min. to 66 gr . 30 min. the Frigid Zone extendeth from 66 gr . 30 min. to 90 gr . of latitude . And so I come to the 9 Questions . 1. What Pole is elevated above the Plaine . Upon all Upright plaines declining from the North : Upon the upper faces of all East or West incliners : Upon the upper faces of all North-incliners , whose Prosthaph : arke is lesse than the latitude of the place : On the under faces of all North-incliners , whose Prosthaph : arke is greater then the Latitude of the place ; and on the upper faces of all South-incliners , The North pole is elevated . And therefore contrarily , Upon all upright Plaines declining from the South : On the under faces of all East and West , and South incliners : On the under faces of all North-incliners , whose Prosthaphaereticall arke is lesse than the Latitude of the place : On the upper faces of all North-incliners , whose Prosthaph : arke is greater than the Latitude of the place , The South pole is elevated . 2. What part of the Meridian ascendeth or descendeth from the Horizontall line of the Plaine ? In all Upright plaines the Meridian lyeth in the Verticall line , and if they decline from the South it descendeth , if from the North it ascendeth . Upon both faces of East and West Incliners the Meridian lyeth in the Horizontall line . In all North-incliners , the North part of the Meridian ascendeth , the South part descendeth : in all South incliners the South part of the Meridian ascendeth , the North part descendeth : upon both upper and under faces . And if these North and South incliners be direct , then the Meridian lyeth in the Verticall line , and so maketh a right angle with the Horizontall line : but if they decline , then the Meridian on the one side maketh an acute angle with the horizontall line . 3. To which part of the Meridian is the style with the substyle to be referred , as making with it an acute angle ? The style is the cocke of the Diall ; the substyle is the line whereon it standeth , signed out in the former descriptions by the letters AV. In all Plaines whereon the North pole is elevated , it is referred to the North part of the Meridian , and maketh an acute angle therewith . In all Plaines whereon the South pole is elevated , it is referred to the South part of the Meridian , and is to make an acute Angle therewith . Except here only those South-incliners , whose Prosthaph : arke is more than the complement of your Latitude : for on these plaines the substyle standeth on that part of the Meridian , whose denomination is contrary to the Pole elevated above the Plaine . For on the upper faces the North pole is elevated , but the substyle standeth toward the South end of the Meridian : and on the under 〈◊〉 the South pole is elevated , but the substyle lyeth toward the North end of the Meridian . Note here , that in South-incliners whose Prosthaphaereticall arke is equall to the complement of your Latitude , the substyle lyeth square to the Meridian upon the line of 6 a clocke ; which line in such plaines alwayes lyeth perpendicular to the Meridian line . Amongst these falleth the Equinoctiall plaine . 4. On which side of the Meridian lyeth the substyle ? In all direct plaines it lyeth in the Meridian . In all Decliners it goeth from the Meridian toward that coast which is contrary to the coast of the plaines declination . And so doe all houres also goe upon the Plaine to that coast which is contrary to the coast whereon they are ; As all the morning or Easterne houres goe to the Westerne coast of the Plaine , and all the Evening or Westerne houres goe to the Easterne coast of the Plaine . Which being observed will bee a great helpe to place them aright . 5. What plaines have the line of 12 upon them , and which not ? All upright Plaines , in all latitudes whatsoever , declining from the South have the line of 12 ; and decliners from the North in the temperate Zone have it not , but in the other Zones they also have it . The upper faces of East and West incliners in all Latitudes have it , the underfaces have it not . The upper faces of all North incliners whatsoever have it ; their under faces in the Temperate Zone want it , in the Frigid Zone have it , and in the Torrid Zone likewise if the Prosthaph : arke bee greater than the Sunnes least North Meridian altitude , but if it be lesse they want it also . For South incliners , consider the Sunnes greatest and least Meridian altitude upon the South coast . For if the Prosthaphaereticall arke bee such as falleth betweene them , that is , if it be greater than the least , or lesse than the greatest , then have bothsides the line of 12 upon them ; but if it be lesse than the least , then doth the Underface want it universally , and the upper face alone hath it ▪ if greater than the greatest , then doth the Upper face want it , and the under face alone hath it : Except in the Frigid Zone where the upper face hath it also , by reason of the Sunnes not setting there for a time . 6. Whether the North or South part of the Meridian serveth for the line of 12 ? In those Plaines that have the line of 12 , where the North pole is elevated , there the North part of the Meridian serveth for 12. and where the South pole is elevated , there the South part of the Meridian serveth for the line of 12 or mid-day . Except , in all Latitudes , the under faces of those South incliners , whose Prosthaphaereticall arke falleth betweene the Sunnes greatest and least Meridian altitudes , for in them the South pole is elevated , but the North part of the Meridian serveth for the line of 12. Except in speciall those Upright Plaines in the Torrid-zone which looke toward the North , and the Under faces of North-incliners also , whose Prosthaphaereticall arke is greater than the least North-meridian-altitude ; for these have the South or lower part of the Meridian serving for 12 , though the North pole be elevated . 7. Which way the style pointeth , and how it is to bee placed ? In Plaines where the North pole is elevated , it pointeth up towards it ; and where the South pole is elevated , it pointeth downe towards it . The style lyeth perpendicularly over the substyle , noted in the former figures with AV , and is to be elevated above it to such an angle as the Elevation of the pole above the Plaine shall be found to be by the 6 , 7 , 8 , and 9 Chapters . 8. When is it that that part of the Meridian next the substyle , and the line of twelve doe goe contrary wayes ? In all Latitudes , Upon the upper faces of South-incliners , whose Prosthaphaereticall arke is greater than the complement of the Latitude , but lesse than the Sunnes greatest South Meridian altitude : And on the Under faces of those South-incliners also , whos 's Prosthaph : is lesse than the complement of the Latitude , but greater than the Sunnes least South meridian altitude : In the Torrid-Zone alone you must adde hither also , North upright Plaines , and those North-incliners on the Under-face , whose Prosthaphaereticall-arke is greater than the least North-meridian altitude of the Sun ; for these have the line of midday standing on that coast which is contrary to the coast of that part of the Meridian next the substyle , and none else . The line of 12. I call herethe line of midday because in the Frigid-zone , where the Sunne setteth not in many dayes together , there are two twelves , the one answering to our midday , and the other to our midnight : and so all Upper faces of South-incliners , whose Prosthaphaereticall arke falleth betweene the least and greatest South meridian altitudes , have there two 12 a clockelines upon them . 9. How much the Meridian line ascendeth or descendeth from the Horizontall line ? The quantity of the Angle is to be found upon the circle on the back-side of your Quadrant , in this manner ; Extend the threed from the complement of the Plaines inclination taken in the lower Quadrant , to the complement of the Plaines declination counted in the Semicircle , and the threed will shew you upon the Diameter , the degrees and minutes of the Meridians Ascension or Descension . In the example of the 9. Chapt. taking the Upper face of that Plaine , I find the Meridian to ascend above the Horizontall line 33 gr . 41 minutes . ¶ These directions are sufficient for the bestowing of every line into its proper place and coast . As may bee seene in the Example of the ninth Chapter . For , First , upon the upper face of that North incliner , because his Prosthaph : arke 16 gr . 6 min. is lesse than 52 gr . 30 min. the Latitude of the place , therefore the North pole is elevated above it : by the Answer to the first Quest. 2. Because it is a North-incliner , therefore the North part of the Meridian ascendeth above the Horizontall line , by the answer to the second Question . 3. Because the North pole is elevated , therefore the Style with the substyle maketh an acute angle with the North end of the Meridian , by the Answer to the third Question . 4. Because this Plaine declineth toward the West , therefore the substyle lyeth on the East-side of the Meridian , and so doe the houres of the afternoone : by the Answer to the fourth Question . 5. This Plaine , being the Upper face of the North-incliner , will have the line of 12 to bee drawne upon it , by the Answer to the fifth Question . 6. Because the North Pole is elevated , therefore the North part of the Meridian serveth for the line of 12 : by the Answer to the sixt Question . 7. Because the North pole is elevated , therefore the style pointeth upward toward the North pole ; by the Answer to the seventh Question . 8. That part of the Meridian next the Substyle , and the line of 12 are both one , and so therefore goe both one way : by the Answer to the eight Question . 9. By the second the Meridian line ascendeth , and the quantity of the ascent is 33 gr . 41 min. above the Horizontall line : by the Answer of the ninth Question . Thus you see every doubt cleared in this example : the like may be done in all others . CHAP. XIII . The making and placing of Polar Plaines . Place this Diagram betweene folio 32. and 33. The horizontall line of the Plaine . These Plaines may have Dialls described upon them by this Quadrant , but the better way is the common way , to protract them by an equinoctiall circle , for otherwise the style will be alway of one distance from the Plaine , be the Diall greater or lesser . The Polar plaines that decline , before they can be described , must have their New-inclination known , and then their delineation will be easie , the manner of it may be seene in this Example . Suppose the upper face of a North-inclining Plaine , lying in the Latitude of 52 gr . 30 min. to decline from the South toward the East 68 gr . and to incline towards the North 73 gr . 57 min. you shall find by the ninth Chapter , the Prosthaph : arke to be 52 gr . 30 min. the same with the Latitude of the place , and therefore you may conclude this plaine to be Polar . By the same Chapter you shall find the New inclination to be 63 degrees . When you have these you may draw your Semicircle AB4 , and divide it into 12 equall parts for the houres : so signing the new-inclination 63 degrees from A to B , draw CB : and supposing the altitude of your style to be CD , through D draw the perpendicular D 12 ; and where the lines drawne from C through the divisions of the semicircle doe cut the line D 12 , there raise perpendiculars for the houres , and so finish it up as the manner is . The style lyeth directly over and parallel to the substyle CB , & the distance of it from the plain is CD , and in this Example the substyle CB standeth from the line of 12 Westward , because the plaine declineth Eastward , according to the rules in the former Chapter , and so doe the morning houres also . For the placing of the Diall in a true site upon the Plaine , you shall find by the answer to the 9 Quest. in the former Chapter , that the Meridian ascendeth 55 gr . 38 min. for other necessaries , the precepts of the former Chapter will direct you . Onely observe , that in Upright East and West plaine , the line of 6 is alwayes the substyle , and it ascendeth above the North end of the Horizontall line , as much as the Latitude of the place commeth to . FINIS . AN APPENDIX Shewing a ready way to find out the Latitude of any place by the Sunne . BEcause in the third Chapter , and quite through this Treatise , the Latitude of the place is supposed to bee knowne , when as every one perhaps cannot tell which way to find it out ; I thought good therefore to adde this Appendix as a ready helpe to shew how it may bee attained sufficiently for our purpose . Know then that for the finding out of the Latitude of a place by the Sunne , these things are required . 1. To find the Meridian line . The readiest way to find the Meridian line is by the North-starre . This starre is within 2 degr . 37 min. of the North-pole . The North-pole lyes very neere betweene Allioth , or the root of the great Beares tayle , and this starre ; You may therefore imagine where the Pole is , if you conceive a right line drawne from the Pole-starre to Allioth , and by your imagination suppose ⅔ parts of the distance of the next starre of the little Beares taile from the Pole-starre towards Allioth , for there is the very Pole-point . Now then if you set up two poles aslope , and from the tops of them hang two cords with weights at the ends of them , and turne them till you standing on the South-side of them may see them both together with the Pole-point , as it were all in one line , then be sure these two cords doe hang in the Meridian line , or very neere it , yea so neere it , that though you should erre 3 degrees herein ( wherein you need not to erre one degree ) yet will not the Meridian altitude in these Climates ( especially more Northward ) faile you above 3 minutes , which is neere enough to our purpose . I have here given you the chiefe starres of the great and little Beares , that by them you may come to know the starres used in this observation , and so find the very Pole-point it selfe . 2. To find the Sunnes Meridian altitude . Observe diligently about noone when the shadow of the South cord shall fall upon the North cord , for then is the Sun in the Meridian . At that instant observe the Suns altitude stedily and carefully , for that is the Meridian and greatest altitude of the Sun for that day . 3. To find the Sunnes declination . For this purpose the limbe hath the characters of the 12 Signes fixed to each 30 degree , and a scale of declinations under the limbe noted with MN . The Scale is divided by this table ; for looke what degr . and min. of the Eclipt . doe answer to the degr . of declination in the table , the same are to be numbred in the limbe , and by a ruler applyed to them , the degrees of declination are drawne upon the Scale . A Table to make the Scale for the declination of every part of the Eclipticke . Degr. of decl . Deg. of the ecl . Degr. of decl Deg. of the ecl . Degr. declin . Degr. eclipt . Degr. declin . Degr. Eclipt Degr. declin . Degr. Eclipt . Degr. declin . Degr. Eclipt . 0.0 0.00 4.0 10.04 8.0 20.26 12.0 31.26 16.0 43.44 2.0 59.04 0.15 0.38 4.15 10.43 8.15 21.06 12.15 31.09 16.15 44.34 20.15 60.14 0.30 1.15 4.30 11.21 8.30 21.46 12.30 32.52 16.30 45.25 20.30 61.26 0.45 1.53 4.45 11.59 8.45 22.26 12.45 33.36 16.45 46.17 20.45 62.41 1.0 2.31 5.0 12.37 9.0 23.06 13.0 34.21 17.0 47.09 20.0 46.00 1.15 3.08 5.15 13.16 9.15 23.46 13.15 35.05 17.15 48.03 21.15 65.22 1.30 3.46 5.30 13.54 9.30 24.27 13.30 35.50 17.30 48.57 21.30 66.48 1.45 4.24 5.45 14.33 9.45 25.08 13.45 36.35 17.45 49.52 21.45 68. ●● 2.0 5.01 6.0 15.12 10.0 25.49 14.0 37.21 18.0 50.48 22.0 69.58 2.15 5.39 6.15 15.51 10.15 26.30 14.15 38.07 18.15 51.45 22.15 71.44 2.30 6.16 6.30 16.30 10.30 27.12 14.30 38.54 18.30 52.43 22.30 73.41 2.45 6.55 6.45 17.08 10.45 27.53 14.45 39.41 18.45 53.43 22.45 75.53 3.0 7.33 7.0 17.48 11.0 28.36 15.0 40.28 19.0 54.44 23.0 78.30 3.15 8.10 7.15 18.27 11.15 29.17 15.15 41.16 19.15 55.47 23.15 81.52 3.30 8.48 7.30 19.6 11.30 30.00 15.30 42.05 19.30 56.50 23.30 90.00 3.45 9.26 7.45 19.46 11.45 30.43 15.45 42.54 19.45 57.56 Finis . Before you can find the Declination , you must know the Sunnes place , and for such as know not the use of the Astronomicall tables , an Almanacke will serve , where for every day at noone , you shall find the Sunnes place in signes , degrees and minutes . The degr . and min. must bee numbred in their Signes upon the limbe , and the threed applyed thereto will shew the declination answerable . As for example . September 21. 1637 in the Almanack for this yeare , the Sunne is found to be in 8 gr . 23 min. of ♎ . In the Quadrants limbe I looke for the Signe ♎ and number there , 8 gr . 23 min. whereto apply the threed , I find it to cut in the scale of Declinations 3 gr . 20 min. 4. By the Meridian Altitude , and declination of the Sun had ; how to find the Latitude of the place , or the Elevation of the Pole above the Horizon . Compare the Sunnes Meridian altitude and declination together , and if the Sunne be in a North Signe as ♈ ♉ ♊ ♋ ♌ ♍ , then substract the declination out of the Meridian altitude , so shall the difference give you the height of the Equinoctiall . But if the Sun be in the South Signes , as ♎ ♏ ♐ ♑ ♒ ♓ , then adde the declination to the Meridian altitude , so shall the summe give you the height of the Equinoctiall , which being taken out of the Quadrant or 90 degrees , leaveth the Latitude of your place , or the Elevation of the Pole above your Horizen . For Example . Upon the 21 of September 1637. I observed the Sunnes altitude in the Meridian to be 34 gr . 10 min. Upon which day I find the Sunnes place to be ( as before ) 8 gr . 23 min. of ♎ , and the declination 3 gr . 20 min. And because the Sun is in a South signe , I adde this declination and Meridian altitude together ; the summe 37 gr . 30 min. is the altitude of the Aequator , and this taken out of 90 degrees leaveth 52 gr . 30 min. for the Latitude of Coventrie . 
A75737	----	Speculum nauticum A looking-glasse for sea-men. Wherein they may behold, how by a small instrument, called the plain-scale, all nautical questions, and astronomical propositions, are very easily and demonstratively performed. First set down by John Aspley, student in physick, and practitioner of the mathematicks in London. The sixth edition. Whereunto are added, many new propositions in navigation and astronomy, and also a third book, shewing a new way of dialling. By H.P. and W.L. Aspley, John. 1662 Approx. 136 KB of XML-encoded text transcribed from 39 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2008-09 (EEBO-TCP Phase 1). A75737 Wing A4013 ESTC R229501 99899278 99899278 152809 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A75737) Transcribed from: (Early English Books Online ; image set 152809) Images scanned from microfilm: (Early English books, 1641-1700 ; 2323:2) Speculum nauticum A looking-glasse for sea-men. Wherein they may behold, how by a small instrument, called the plain-scale, all nautical questions, and astronomical propositions, are very easily and demonstratively performed. First set down by John Aspley, student in physick, and practitioner of the mathematicks in London. The sixth edition. Whereunto are added, many new propositions in navigation and astronomy, and also a third book, shewing a new way of dialling. By H.P. and W.L. Aspley, John. H. P. W. L., 17th cent. [4], 64 [i.e. 72] p. : ill., charts printed by W. Leybourn, for George Harlock, and are to be sold at his shop at Magnus Church-Corner, in Thames Street, near London-Bridge, London : 1662. Page 72 misnumbered 64. Reproduction of original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Dialing -- Early works to 1800. Navigation -- Early works to 1800. Nautical astronomy -- Early works to 1800. 2007-12 TCP Assigned for keying and markup 2007-12 Apex CoVantage Keyed and coded from ProQuest page images 2008-01 John Pas Sampled and proofread 2008-01 John Pas Text and markup reviewed and edited 2008-02 pfs Batch review (QC) and XML conversion Speculum Nauticum . A Looking-Glasse FOR SEA-MEN . Wherein they may behold , how by a small Instrument , called the PLAIN SCALE , all Nautical Questions , and Astronomical Propositions , are very easily and demonstratively performed . First set forth by John Aspley , Student in Physick , and Practitioner of the Mathematicks in London . The Sixth Edition . Whereunto are added , many new Propositions in Navigation and Astronomy , and also a third Book , shewing a new way of Dialling . By H. P. and W. L. LONDON , Printed by W. Leybourn , for George Hurlock , and are to be sold at his Shop at Magnus Church-Corner , in Thames-Street , near London-Bridge , 1662. TO THE WORSHIPFVLL , THE MASTER , WARDENS , & ASSISTANTS OF THE TRINITY HOVSE ; JOHN ASPLEY , IN TESTIMONY OF THE HONOUR HE BEARS TO THE GOVERNOURS & PRACTISERS OF THE ART OF NAVIGATION , DEDICATES THESE HIS FIRST LABOURS . The Printer to the Reader . THis little book having been well accepted of among Sea-men , being the first fruites of Mr. Aspley's Mathematical Studies , hath passed five Impressions , without any alteration ; and so I doubt not might have done still : But because since that time there have been severall bookes put out of this nature , I have procured this to be revised , and severall alterations and additions to be made therein , So that here you have both the old , and a new booke intermingled all in one , with a third part added thereto , concerning Dialling ▪ by a way not formerly published by any . All which I doubt not you will kindly accept of , and receive much delight and profit thereby . Your . G. H. ERRATA . PAge 34 line 26 read 360. Page . 45. l. 8. r. Distance I M. Page 50. line 13. for 14 &c. 〈…〉 which is just the length of the Gnomon . Page . 50 line . 28. for increase , read decrease , Page 52 line 4. r. H A I. line 18. r. point O. Page 57 line 11 r. point L. Also for some lite●all faults we shall desire your Pardon ▪ Speculum Nauticum , OR THE SEA-MANS GLASSE . The First Book . CHAP. I. The Explanation of certain Terms of Geometry . BEing intended in this Treatise of the plain Scale , to declare the manner of projection of the Sphere , in plano , I have thought fitting first , to shew unto you some tearms of Geometry which are necessary for the unlearned to know , ( for whose sake chiefly I write this Treatise ) before they enter into the definition of the Sphere . First therefore I intend to relate unto you , what a point or prick is , and afterward a Line both right and crooked , and such sorts thereof as are appertinent unto the operations and use of this Scale . Punctum , or a point , is the beginning of things , or a prick supposed indivisible , void of length , breadth , and depth : as in the Figure following is noted by the point , or prick A. Linea , or a Line , is a supposed length , or a thing extending it self in length , not having breadth nor thickness , as is set forth unto you by the Line BAD . Parallela , or a Parallel Line , is a line drawn by the side of another line , in such sort that they may be equidistant in all places . And of such parallels , two only belong unto this work of the plain Scale , that is to say , the right lined Parallel , and the circular Parallel . Right lined Parallels are two right lines equidistant one from another , which being drawn forth infinitely , would never touch or meet one another , as you may see in the Figure , where the line H I is Parallel unto the line CE , and the line GF is Parallel unto them both . A circular Parallel is a circle drawn either within or without another circle upon the same center , as you may plainly see by the two circles BCDE , and XVYW . These circles are both drawn upon the center A , and therefore are parallel the one unto the other . There is another kind of Parallel also , which is called a Serpentine Parallel , but because it is not belonging unto the use of this Scale , I will omit it , and so proceed unto the rest . Perpendiculum , or a Perpendicular is a line raised from , or let fall upon , another line , making equal Angles on both sides , as you may see declared in the figure , where in the line AC is perpendicular unto the line BAD , making equal ●ngles in the point A. Diameter circuli , or the Diameter of a Circle , is a right line drawn thorow the center of any circle , in such sort that it may divide the circle into two equal parts , as you may see the line BAD is the Diameter of the circle BCDE , because it passeth thorow the center A , and the two ends thereof do divide the circle into two equal parts , in the two extreams B and D , making the semicircle BCD equal unto the semicircle DEB . Semidiameter circuli , or the semidiameter of a circle is half of the Diameter , and is contained betwixt the center , and the one side of the circle , as the line AD is the Semidiameter of the circle BCDE . This Semidiameter contains 60 degrees of the line of Chords , which we sometimes call the Radius . Semicirculus , or a Semicircle , is the one half of a circle , drawn upon his Diameter , and is contained upon the Superficies , or Surface , of the Diameter , as the Semicircle BCD which is half of the circle BCDE , and is contained above the Diameter BAD . Quadrans circuli , is the fourth part of a circle , and is contained betwixt the Semidiameter of the circle , and a line drawn Perpendicular , unto the Diameter of the same circle , from the Center thereof , dividing the Semicircle into two equal parts , of the which parts , the one is the Quadrant , or fourth part of the same circle . As for example , the Diameter of the circle BCDE is the line BAD , dividing the circle into two equal parts : then from the center A raise the Perpendicular AC , dividing the Semicircle likewise into two equal parts ; so is ABC , or ACD , the Quadrant of the circle BCDE , which was desired . CHAP. II. The manner how to raise a Perpendicular from the middle of a line given . 〈◊〉 first a ground line whereupon you would have a Perpendicular raised , then open your Compasses unto any distance ( so it exceed not the end of your line , ) placing one foot of the said Compasses in the point from whence the Perpendicular is ●o be raised , and with the other foot make a mark in the line on 〈…〉 removing your Compasses unto any other distance that 〈…〉 set one foot thereof in one of the marks , and with the 〈◊〉 foot make an Arch over the middle point , then with the same distance of your Compasses set one foot in the other mark upon the line , and with the other foot make another Arch of a Circle over the middle Point , so that it may cross the first Arch , and from the meeting of these two Arches , draw a right line unto the middle Point , from which the Perpendicular was to be raised , which line shall be the Perpendicular desired . Example , suppose your Base or ground line whereupon a Perpendicular is to be raised be the line FLK , and from L the Perpendicular is to be raised , set one foot of your Compasses in the Point L , and with the other , make the marks G and M on both sides of the point L , the● opening your Compasses wider , set one foot in the point M , and with the other draw the Arch S over the point L , then with the same distance of your Compasses , set one foot in G , and with the other make the Arch R , crossing the Arch S in the point T , then from T draw the line TL , which line is perpendicular unto the line FLK from the point L , which is the perpendicular desired . CHAP. III. To let a Perpendicular fall from any Point assigned , unto the middle of a line . LET the line whereupon you would have a Perpendicular let fall be the line LFK , and the point assigned to be the point T , from whence you would have a Perpendicular let fall upon the line FLK , first set one foot of your Compasses in T , and open your Compasses unto any distance so that it be more than the distance TL , which here we suppose to be the distance TM ; then make in the line FLK the marks G and M , then with your Compasses take the one half of GM , which is the point L , then from L draw a line unto the point T , so the line TL shall be the Perpendicular , which was desired to be let fall from the assigned point T unto the middle of the line FLK . CHAP. IV. To raise a Perpendicular upon the end of a Line . SUppose the line whereupon you would have a Perpendicular raised , be the line FLK , and from the point F a Perpendicular is to be raised : first open your Compasses unto any distance , which here we put to be the distance FG , and set one foot of your Compasses in the point F , and with the other draw the Arch DEG , then set one foot of your Compasses in the point G , and with the other draw the Arch E ; then placing one point of your Compasses in E , with the other draw the Arch DB ; then place your Compasses in D , and with the same distance draw the Arch A , cutting the Arch DB in C , then draw a line from C unto the end of the line FLK , unto the assigned point F , so shall the line CF be a Perpendicular raised from the end of the line FLK , and from the assigned point F. CHAP. V. To let a Perpendicular fall from any point assigned unto the end of a Line . LET the line FLK be the Base or ground line , and from the point I a Perpendicular is to be let fall upon the end of the line at K , first from the assigned point I , draw a line unto any part of the Base , which shall be the line IHM , then find the middle of the line IM , which is at H ; place therefore one foot of your Compasses in the point H , and extend the other unto I , with which distance draw the Arch INK upon the Center H , cutting the Base or ground-line in the point K , then draw the line KI , which line shall be the Perpendicular desired . CHAP. VI. A right Line being given , how to draw another parallel there unto at any distance required . LEt the line given be AB , unto which it is required to draw another right line CD which shall be parallel to the former line AB , and at the distance AC . First open your Compasses to the distance AC , then set one foot in the point A , with the other describe the Arch C ; again , place one foot in B , and with the other describe the Arch D ; Then draw the line CD , so that it may only touch the two Arches C and D , so shall the line CD , so drawn , be parallel to AB , and at the distance required . CHAP. VII . A right line being given , how to draw another parallel thereunto , which shall also pass through a point assigned . LEt AB be a line given , and the point assigned be C : and let it be required to draw another line parallel thereunto , which shall pass through the given point C. NOw I doubt not but you understand the way to let fal , or to raise any manner of Perpendicular line , either from , or upon any part of a line : as also to draw lines parallel one to another at any distance required , therefore now I intend to proceed unto the main point here aimed at , which is , to declare , and make known unto you the several operations performed by the plain Scale , which though it be in use with very few , yet it is most necessary for Sea-men , because all questions in Navigation are thereby easily and plainly wrought . And also all questions in Astronomy ( belonging unto the expert , and industrious Sea-men ) may both speedily and easily be wrought by the same Scale : in regard whereof I have declared in this little Book , that knowledge ( which God hath been pleased to bestow upon me ) concerning the necessary use and practice thereof ; hoping that you will as kindly accept it , as it is freely offered unto your courteous considerations . CHAP. VIII . Of the description of the Scale . The figure of the plane Scale . The second part of the Scale , is the single Chorde of a circle , or the Chord of 90 , and is divided into 90 unequal divisions , representing the 90 degrees of the Quadrant : and are numbred with 10 , 20 , 30 , 40 , &c. unto 90. This Ghord is in use to measure any part or Arch of a circle , not surmounting 90 degrees : The number of these degrees from 1 unto 60 is called the Radius of the Scale , upon which distance all circles are to be drawn , whereupon 60 of th●se Degrees are the Semidiameter of any Circle that is drawn upon that Radius . The third part of the Scale is divided into eight parts , representing the Points or Rumbes of the Mariners Compass ; which in all are 32 points : but upon the Scale there are only S reckoned , which is but one Quadrant or quarter of them , being to be reckoned from the Meridian of North and South both wayes , as you may see more plainly by this figure , representing the order of the points of the Compasse . It is usuall also to have another line placed upon your Scale , to she● you how many leagues make a degree of longitude in every latitude , concerning which you shall have directions in the 14 Chapter following . CHAP. IX . Knowing the course any ship hath made , and the leagues she hath sailed : to find how much she hath raised or depressed the Pole ; and likewise how much she is departed from her first Meridian . The Course is South-west and by South , the leagues sailed are 100 , the difference of the Latitude , and the distance of the Meridians is required . Now you must heedfully observe this point D , for this represents the place where your Ship is , and doth shew both the differencefo the latitude of the place you are in , and also your distance or departure from your first meridian . First for the latitude , you see the line DF , being paralel to the line AB , cuts the Meridian line AF in the point F : So that if you take the distance FA with your Compasses , and apply it to the scale of equal leagues , you shall find it is just 83 leagues , which counting 20 leagues to a degree , makes 4 degrees 9 min. and so much you have altered your latitude by the said course , which degrees and minutes being added to , or substracted from the latitude of the place you came from , according as your course requires , shews you alwaies the true latitude you are in . Likewise from this point D , take with your Compasses the distance DF , and you shall find it by your scale of equal leagues to be 56 leagues , and so much you are departed from your first Meridian to the West-ward ; which when you are near the Equinoctial , where the degrees of longitude are equal to the degrees of latitude , would shew the longitude , by taking 20 leagues for one degree , &c. so it would be two degrees , and 48 min. for your difference of longitude , from y●ur first Meridian AF. But in other places , you must first 〈◊〉 howmany leagues make a degree of longitude about that latitude where you are , and so turn your leagues of distance from the Meridian , into degrees and minutes of longitude , of which more hereafter , Chap. 14. I have been the larger in these two Propositions , because they are ●he first , for the better understanding of all the rest ; and because they are most necessary , for thereupon depends the knowledge of the true Traverse point , and the keeping of your dead reckoning . Now because this cannot alwaies be kept exactly , it is to be corrected by the observation of the latitude , according to this following proposition . CHAP. X. Knowi g the difference of latitude of two places , and the Rumb you have sailed upon , to find the leagues you have sailed , and the difference of Meridians . The Pole depressed four degrees and the Rumb South-West by South or the third from the Meridian , to find your true Traverse point , viz. how far you have sailed , and how much you are departed from your first Meridian . In the first figure DRaw the lines as in the former Chapter , so that AKF may represent the Meridian line , and ACD may represent the third Rumb from the meridian ; then because you have altered your latitude 4 degrees , which make 80 leagues , take 80 leagues with your Compasses out of your Scale , and set them upon the meridian line AF , from A to L : Then keeping the same distance of your Compasses , draw the line LM parallel to AB , ( or else you may erect LM perpendicular to the line AF , in the point L ) and mark where the said LM crosseth the Rumb line ACD , which is in the point M. This point M is the true Traverse point , the leagues sailed are shewed by the line AM , which being measured in the Scale , will be found to be 96 leagues and an half , and the departure from the Meridian is LM , which is 54 leagues . Now by this Proposition ( as I said ) you may correct your dead reckoning ; for suppose by the former proposition you reckon you had sailed 100 leagues upon the ●hird Rumb , then as you see there , you should have been at the point D , and have altered your latitude 83 leagues , and departed from your Meridian 56 leagues ; but now suppose that by a good observation of the latitude , you find that you have altered the latitude onely 80 leagues , from A to L , by drawing this line LM , which crosseth the Rumb or Ships way in M , you may conclude your true Traverse point to be at M , so that you have sailed only from A to M , which is 96 leagues ½ , and departed from your Meridian 54 leagues . So that as you are short of the latitude you reckoned for 3 leagues or 9 min. you are also short of your way you reckoned 3 leagues ½ , and two leagues less in your departure from the Meridian . And this you must account for your true reckoning , being thus corrected . CHAP. XI . By the difference of the latitudes of two places and the distance between their Meridians , to find the Rumb by which you must sail from the one place to the other , and how far it is from the one place to the other ? The difference of latitude between the two places is 4 deg . 9 min. and the distance between the two Meridians is 56 leagues , and it is required to find the Rumb from the one place to the other . IN the former figure draw the quadrant AKCB , then turn your four degrees 9 min. of latitude into leagues , it maketh 83 leagues , which you must place upon the meridian line from A to F. And from the point F draw the line FD parallel to the line AB . Then open your Compasses to the distance of the meridians which is 56 leagues , and set it on the line FD , from F to D. Then lay your Ruler by this mark D and the Centre A , and draw the line ACD . Then mark where this line cuts the quadrant , which is in the point C , and setting one foot of your Compasses in the point C , open the other to K , and keeping your Compasses at that distance CK , measure it upon your Scale , either in the line of Chords , or in the line of Rumbs , you sh ll find it to be in the one 33 deg . 45 min. and in the other just the third Rumb from the meridian . So that the Rumb from A 〈◊〉 D ●s South-west and by South , and the Rumb from D to A is the Rumb opposite thereunto , which is North-east and by North. Then for the distance between the two places in the Rumb , ●et one foot of your Compasses in the one place at A , and open the other to the other place at D , and the length of the line A D ineasured in the Scale of leagues , shews the distance between them to be just 100 leagues . These three ( or rather these six ) Propositions , ( for they are each of them double ) are the most usefull and necessary in the art of Navigation . By the first of these , knowing the point of the Compass you ●ail upon , and judging howmany leagues you have sailed thereon , you know and are able to give a reasonable account where you are , both in respect of latitude and longitude . By the second having a fair observation of the latitude at any time , you may more perfectly know where you are ; and thereby correct your former account . And by this third you may know how to direct your course from any place to your desired haven . So that in effect you need no more , but yet for your better instruction by variety of cases and examples , I shall proceed . CHAP. XII . The difference of Latitude and the lea●●es sailed being given , to find the distanee from the Meridian , and the Rumb you have sailed upon . Sailing 100 leagues between South and West , untill the Pole be depressed 4 deg . 9 min. the distance from the Meridian is demanded , and what Rumb you have sailed upon ? IN the first figure draw the Quadrant AKCB , as in the former Chapters , and then reduce your degrees of latitude into leagues , so 4 deg . 9 min. make 83 leagues , which you must take with your Compasses out of your Scale of leagues , and set them off in this Meridian line from A to F. Then from the point F draw the line FD , parallel to the line AB , which you may do with the foresaid distance of your Compasses . Then open your Compasses unto your distance failed , which is 100 leagues , and setting one foot of your Compasses in the point A , with the other draw the little Arch HG , cutting the line FD in the point D. So the line FD measured in the Scale of leagues , shall shew you the distance from the M ridian , which is 56 leagues , and if you draw the line ACD , it i● the Rumb line upon which you have sailed , and the Arch KC 〈◊〉 ed in the Scale of Rumbs , shews it to be the third Rumb from the Meridian , or South-west by South . CHAP. XIII . To find the distance of any Island from you , that you may discern at two stati n● , knowing the po●ut of the Compasse , the Island beareth unto each of the stations . Suppose , being at Sea you discover an Island bearing North-east off you , which place let it be your first station , and then sailing seven leagues full North you observe the Island to bear full East off you , which let be the second station ; the aemand is to find the distance of the said Island from both the said stations ? IN the second figure , or demonstration , let A be the first Station , and upon the Center A draw the Quadrant ABDE ; Then in regard you found the Island to bear North-East from you , take 4 of your 8 points of the Compass our of the Scale , and place them upon your Quadrant from B to D , then from the Center A by the point D , draw the line ADF , representing the visuall line passing between your sight and the Island , being at the first station A. Then seeng when y●● had sailed 7 leagues North , you observed the Island to bear full East off you , set off the said 7 leagues from A to C , ( reckoning every 10 leag● s of your Scale to be but on● ) and from this point C , which is the second station , draw the line C F parallel to AE , and it will cut the line ADF in the point F : So shall the point F , be the place of the Island desired , and the distance AF , is the distance of the Island from the first station , viz. 9 leagues 90 parts or almost 10 leagues : Likewise the distance from C , to F , is the distance of the Island from the second station , which is just seven leagues . And by this manner of work , you may find the ▪ distance of any Island or head land from you , or you may take the distances of as many places as you will or can see at any two such stations , and by the crossing of their visuall lines , find their position and distances each from other . CHAP. XIV . To find how many leagues , miles , and parts do make one degree of longitude in every latitude . Note , All this while we have been sailing according to the Rules of the plain Chart , which supposith the degrees of longitude to be equal to the degrees of latitude , in all latitudes , but that is very false and erroneous ; it being true onely in places near the Equinoctiall , where every degree of longitude contains 20 leagues , as the degrees of latitude do ; But in places neer the Poles it alt●rs very much , so that in the latitude of 60 degrees , 10 leagues make a degree of longitude : and in other latitudes the degrees of longitude alter , as in this little Table , which shews at what degree and minute of latitude , any nnmber of leagues make a degree of longitude , by which you may divide a Line upon your Scale for your ready use . Leagues in one Degree . 20 00 d 00 m 19 18 11 18 25 50 17 31 47 16 36 52 15 41 25 14 45 34 13 49 27 12 53 08 11 56 38 10 60 d 00 m 9 63 15 8 66 25 7 69 31 6 72 32 5 75 31 4 78 28 3 81 22 2 84 16 1 87 08 Now to return to the Question , and shew you by demonstration how to find how many leagues , miles , and parts , make a degree of longitude in any degree of latitude ? The larger you make your Quadrant , the more exact will the work be , and shew the leagues and miles more exactly , which you may make into a Table , as this following . A Table shewing how many leagues , miles , and hundred parts of a mile make one degree of longitude in any latitude . Latitude Leagues Miles Parts Difference   Latitude Leagues Miles Parts Diffe ence   Latitude Leagues Miles Parts Diff●rence 0 20 0 0 —   30 17 0 96 —   60 10 0 0 — 1 19 2 99 1   31 17 0 43 53   61 9 2 09 91 2 19 2 96 3   32 16 2 88 55   62 9 1 17 92 3 19 2 92 4   33 16 2 32 56   63 9 0 24 93 4 19 2 85 7   34 16 1 74 58   64 8 2 30 94 5 19 2 77 8   35 16 1 15 59   65 8 1 36 94 6 19 2 67 10   36 16 0 54 61   66 8 0 40 96 7 19 2 55 12   37 15 2 92 62   67 7 2 44 96 8 19 2 42 13   38 15 2 28 64   68 7 1 47 97 9 19 2 26 16   39 15 1 63 65   69 7 0 50 97 10 19 2 09 17   40 15 0 96 67   70 6 2 52 98 11 19 1 90 19   41 15 0 28 68   71 6 1 53 99 12 19 1 69 21   42 14 2 59 69   72 6 0 54 99 13 19 1 46 23   43 14 1 88 71   73 5 2 54 100 14 19 1 22 24   44 14 1 16 72   74 5 1 54 100 15 19 0 96 26   45 14 0 43 73   75 5 0 53 101 16 19 0 68 28   46 13 2 68 75   76 4 2 52 101 17 19 0 38 30   47 13 1 92 76   77 4 1 50 102 18 19 0 06 32   48 13 1 15 77   78 4 0 48 102 19 18 2 73 33   49 13 0 36 79   79 3 2 45 103 20 18 2 38 35   50 12 2 57 79   80 3 1 42 103 21 18 2 1 37   51 12 1 76 81   81 3 0 38 104 22 18 1 63 38   52 12 0 94 82   82 2 2 35 103 23 18 1 23 40   53 12 0 11 83   83 2 1 31 104 24 18 0 81 42   54 11 2 27 84   84 2 0 27 104 25 18 0 38 43   55 11 1 41 86   85 1 2 23 104 26 17 2 93 45   56 11 0 55 86   86 1 1 18 105 27 17 2 46 47   57 10 2 68 87   87 1 0 14 104 28 17 1 98 48   58 10 1 80 88   88 0 2 09 105 29 17 1 48 50   59 10 0 90 90   89 0 1 05 104 30 17 0 96 52   60 10 0 0 90   90 0 0 0 105 CHAP. XV. The difference of latitude , and the Rumb or distance sailed being known , to find the distance of the Meridians , and thereby to find the degrees and minutes of the difference of longitude in any latitude . Sailing from the North parallel of 56 degrees and 5 min. latitude , 100 leagues upon the third Rumb from the Meridian ▪ viz. South-west and by South untill I find the Pole is depressed 4 deg . 9 m. and the Meridional distance 56 leagues ; the longitude is desired thereby ? I● the first figure Now to reduce this 56 leagues into degrees of longitude , you must consider from what latitude you have sailed , and to what latitude you are come , viz. from latitude 56 d. 5 m. to 4 deg . 9 min. lesse , which is 51 d. 56 m. and take the middle latitude ( or somewhat more ) between the two places , which in this example falls out to be 54d . 01 m. Then by the Table in the former Chapter , find out howmany leagues and miles in the said middle latitude make one degree of longitude , and you shall find in that Table , that in the latitude of 54 d. there is but 11 leagues , and 2 miles , and 27 parts in one degree of longitude ; Therefore open your Compasses upon your Scale of leagues , to this 11 leagues , 2 miles , 27 parts , and keeping your Compasses at that distance , set one foot of them at 56 leagues in your Scale of leagues , or in the line DF in the figure , ( or upon the like line in your Chart at any time ) either at F or D , and measure howmany times you find that distance either to the end of your Scale coming backward , or in the line DF , for so many degrees is the difference o● longitude , and if any odde part remain , you may proportion i● by your eye , judging it to be a quarter , a third , an half , or any part more or lesse of a degree , which you may either reckon by parts , or 15 , 20 , 30 &c. minutes , Thus this line DF being 56 leagues , opening your Compasses to 11 leagues 2 miles 27 parts , you will find this distance in it , 4 times and 3 quarters ; so that the difference of longitude is 4 deg . 45 min. Or you may reduce it into miles and work by the rule of proportion , so you shall find As 11 leagues , 2 miles , 27 parts , that is 35 miles 27 parts . 35,27 To one degree of longitude in the latitude of 54 d. 01,00 So is 56 leagues , or 168 miles . 168,00 To 4 degrees , 76 parts . 04 76 But if your Scale be large , the other way with your Compasses will give you the degrees and parts of longitude as exactly as you need for most uses . Also if the latitude fall not out in equal parts , you may find out for your odde minutes by proportion , for which purpose I have set the differences between each degree in the Table . So that as one hundred parts or 60 minutes being one degree , to the difference in the Table between the two next degrees ; So the odde hundred parts or minutes of latitude , to the parts and minutes proportional to be allowed . CHAP. XVI . Sailing from the South latitude of 60 degrees , 51 min. and from longitude 25 degrees , 24 min. 99 leagues , upon a South-west course : the latitude and longitude of the second place is demanded . IN the second demonstration , draw the Quadrant ABCDE , as is formerly taught , then in regard you sail South-west , take 4 points of the Compasse from your Scale , and place them from B unto D , then by the point D draw the line ADF , then place your ninety nine leagues upon the line ADF , from A unto F , so shall F be the place of your Ship. Then from F draw the line FC parallel unto AE , cutting the line ABC in C , so shall the distance CA be the leagues you have run South , which is seventy leagues , or 3 deg . 30 minutes , which being added to the latitude from whence you dearted , makes 64 deg . and 21 minutes for the latitude of the second place : then take the distance CF , and apply it unto the line of equal parts , and you shall find it likewise 70 leagues : Then finding the middle latitude 62 degrees 36 minutes in the Table , Chap. 14. you shall find that 9 leagues and 0 miles , and 61 parts , do alter a degree of longitude in that latitude . Then opening the feet of your Compasses to 9 leagues 0 miles 61 parts , in the Scale of equall leagues , and keeping the Compasses at that distance , see howmany times that distance is in the line CF , which is seven times and somewhat above an half , the true difference of longitude being 7 deg . 36 m. which being substracted from the longitude from whence you departed , leaves 17 degrees and 48 minutes for the longitudeof the second place . CHAP. XVII . A Ship sayling from the North Parallel of fifty degrees , having an hundred leagues to sail South-west , and by West , by the way is enforced by contrary winds to sail upon several points of the Compasse , first sailing thirty leagues upon a direct course , then West North-west twenty leagues , then South sixty leagues , the question is to find the latitude of the second place , how far it is to the place whereuuto you are bound , the distance of the Rumb that is betwixt them , the distance that you are from your first Meridian , and thereby the difference of longitude . IN the third demonstration , draw the line AD , and from the point A , raise the perpendicular AB , then open your Compass unto the Radius of your Scale , and place one foot thereof in the center A. and with the other draw the Quadrant BCD , then take three points of the compasse & place them upon the Quadrant from D. unto C , then from the Center A , by the point C , draw the line ACL , 100 Leagues in length , which is the true course you are to saile , Then in regard you sayled thirtie leagues direct , take thirtie leagues from your Scale of equall parts , and place them upon the line AEC , it extends from A unto E : then in regard you turned your Course , West , Northwest , from the Center E , draw the Line EG parallell unto A. D. and again from the center E draw the line EH perpendicular to EG , and parallell to AB , then witn the distance of the Radius , set one foot of your compasses in the center E , and with the other draw the Quadrant GMH , and in regard you sayled West , Northwest , which is two points from the West Northward , take from your Scale two points of the Compass , and place them upon the Quadrant GMH , from G unto M , then from the center E unto the point M , draw the line EFM , then take 20 Leagues with your Compasses from the Scale of equall parts , and place them upon the line EFM , from E unto F , then is your Ship in the point F. Lastly , in regard you run South 60 Leagues from F , draw a Line Parallell unto the Meridian AB , which is the line FI , then take from your Scale of equall parts sixtie Leagues , and place them from F , unto I , then is your Ship in the point I : then last of all is to be found how far it is to the place where unto you were bound , the distance of the Rumb that is betwixt you , the degrees and minutes you have raised the Pole , the distance of departure from the first Meridian , and thereby t●e difference of Longitude : and that you may so doe , first draw the line OIK , Perpendicular unto the line IF in the point I , and with your Compasses opened unto the distance of the Radius , set one foot of your Compasses in the Center I , and with the other draw the Quadrant KNF , then in regard your ship is in the point I , and the place whereunto you are bound is the point L , therefore from I , thorow the point L draw the line ILN , cutting the Arch KNF , in the point N , therefore let IL , be the Leagues you have unto the place whereunto you are bound , which is fortie one Leagues and a halfe , and the Rumb the distance KN , which is West , and by North , and three degrees unto the Northward , so likewise is the line AO , the number of Leagues you have run due South , which is sixtie eight Leagues and one mile , or three degrees and twenty five minutes , which being taken from fiftie degrees , the parallell from which you departed , leaves fortie six degrees and thirtie five minutes for the Parallel you are in . Last of all , shall the line IO , be the Leagues that you have departed your first Meridian , which are fortie two leagues and one mile , Then take the middle latitude which is fortie eight degrees seventeen minutes and in the Table chap. 14 you shall find that thirteen Leagues 0. mile , 92 parts , do answer unto a degree of Longitude in that Parallell ; then setting one foot of your Compasses in thirteen Leagues , and ninety two parts , extending the other to the beginning of the Scale , keeping the Compasses at that distance , turn them over the line I O , and you shall find it contains that distance three times and almost a quarter , So the difference of longitude is three degrees eleven minutes . CHAP. XVIII . Two Ships departing from one Parallel , and Port , the one in sayling eight Leagues betwixt the North , and the West , hath raised the Pole two degrees , the other in sailing a hundred Leagues betwixt the North , and West , hath raised the Pole four degrees , I demand by what Rhumbs the said Ships have sailed , and the Rhumb and distance that is betwixt them ? IN the fourth Demonstration , draw the Quadrant ABCDE , then in regard the first Ship hath raised the Pole two degrees , which is fortie leagues , take fortie Leagues off your Scale , and applie them unto the Meridian line AGL , from A unto G : then from the point G , draw the line GF , parallel unto AB , then opening your compasses unto 80 Leagues , set one foot in the Center A , with the other make a marke in the line GF , which will be at F , so shall F be the place of the first ship ; the second Ship hath raised the Pole four degrees , which is 80 Leagues , therefore place 80 leagues upon the Meridian line AGL , from A unto L , and from the point L draw the line LM , parallel unto GHF , then open your Compasses unto the distance of a hundred leagues , which are the Leagues the second ship did run , and set the foot of your Compasses in the Center A , and with the other make a mark in the line LM , which will be at M , then draw the line MA , which is the course of the second Ship , and the line FA , is the course of the first ship , then from F let a Perpendicular fall , being Perpendicular to the line GF , which is the line FK , then opening your Compasses unto the Radius of your Scale , set one foot in the Center F , and with the other draw the Quadrant HIK , likewise from F , the place of the first Ship , draw a line by the point M , the place of the second , cutting the Quadrant KHI , in I , so let IK , be the course that is betwixt them , that is , if you will saile from the first ship unto the second , you must saile North and by East , and one and fortie minutes to the Eastward , likewise let F M , be the distance that is betwixt them , which in this Demonstration is fortie Leagues , two miles , so shall BC , be the course of the first ship from the West Northward , wh ch here is found to be thirtie degrees and one minut from the West Northward , or Northwest by West , and three degrees and fortie four minutes to the west ward . Lastly the Arch ED , is in the distance of the course that the second Ship made from the North Westward , which is found by this Demonstration to be Northwest and by North , and three degrees five minutes to the Westward . CHAP. X●X . Two Ships departing from one Parallell and Port in the Parallell of 47 deg . 56 min. the first in sayling 80 leag . betwixt the North and West , hath raised the Pole two degrees , I demand by what course the second ship must runne , and how much she shall alter in her first Meridian or longitude , to bring her selfe 40. leagues and two miles North and by East , and 41. minutes to the Eastward of the first ship ? IN the fourth Demonstration draw the Quadrant ABCDE , then multiplie your two degrees you have altered your latitude by twentie and it maketh fourtie Leagues ; which fourtie Leagues set upon the line AEL , from A unto G , then from the point G draw the line GF , parallel unto AB , then open your Compasses unto the distance of 80 Leagues , which are the Leagues your first ship did runne , and place one foot of your Compasses in the Center A , and with the other make a marke in the line GF , which will be at the point F , then from the Center A unto the point F draw the line AF , representing the distance of the Course of the first Ship 80 leagues : Then from F let fall a Perpendicular FK , and upon the Center F , with the Radius of the Scale draw the Arch HIK , Then in regard you must bring the second ship North and by East , and 41 minutes Eastward of the first ship , take 11 degrees 56 minutes from your Scale of Chords , and place them from Kunto I , upon the quadrant KIH . Then from F draw the line IF , and upon the line , FI , place the distance that you must bring the second ship from the first ( which is fourty leagues and two miles ) from F unto M. So is M the place of your second ship . Then from M draw the line ML parallel unto FG , cutting the line AGL in L , then draw the line MA , cutting the Quadrant BDE in D. So shall the Arch DE be the course that the second ship must run , to bring her self fourty leagues and two miles North and by East , and 41 minutes East of the first ship . Then to know what you have altered the latitude , first take the distance LA and apply it unto the Scale of equall parts , and you shall find it to be 80 leagues , which is just 4 degrees , which you have altered your latitude , or Poles elevation : which 4 degrees added unto the latitude you depar ed from , it makes 51 degrees 56 min. for the latitude that your second Ship is in , then take the distance LM and apply it to the Scale , it gives 60 leagues ; then open your Compasses unto the distance of the middle latitude , which is 40 deg . 5● min. of the Chord , and apply it unto the Table of longitudes , and it gives 12 leagues , and 2 miles , and 62 parts , to alter one degree of longitude in that Parallel : Then set one foot of your Compasses in 12 leagues 2 miles , and 62 parts , and open the other to the beginning of the line , and with that distance measure the line L M , being 60 leagues , and you shall find that it is contained there in four times and two thirds , so the longitude is 4 degrees 40 minutes . CHAP. XXI . Of the Ebbing and Flowing of the Sea , aud of the Tides , and how to find them in all places . A generall Table for the Tides in all places . The Moons age . Hours and minutes to be added . Hours and minutes to be added .   The Moons age . Hours and minutes to be added : Hours and minutes to be added : Daies . Degrees : Minutes :   Daies . Degrees : Minutes : 1 0 48   16 0 48 2 1 36   17 1 36 3 2 24   18 2 24 4 3 21   19 3 12 5 4 0   20 4 0 6 4 48   21 4 48 7 5 36   22 5 36 8 6 24   23 6 24 9 7 12   24 7 12 10 8 0   25 8 0 11 8 48   26 8 48 12 9 36   27 9 36 13 10 24   28 10 24 14 11 12   29 11 12 15 0 0   30 0 0 The use of the Table of the Tides . FIrst it is to be understood , that by the swift motion of the first Mover , the Moon and all the rest of the Stars and Planets , are turned about the World in four and twenty hours , upon which swift motion of the Moon , the daily motions of the Sea , do depend , which motion of the Sea falleth not out alwaies at one hour , the reason thereof is , because of the swift motion of the Moon in regard she goeth almost thirteen degrees in four and twenty hours , and the Sun moveth scarce one degree , which gives every day twelve degrees , that the Moon cometh slower to any point in the Heaven than the Sun : which twelve degrees makes fourty eight minutes of time for the difference of every full Sea , according unto the middle motion of the Moon , which difference is here set down in this Table for every day of the Moons age . Therefore if you would know the full Sea at any place in the World , first you must know at what hour it is full Sea at the new or full Moon ; which hours and minutes keep in mind , then seek the age of the Moon as is before taught , and with the number of her age enter this Table , under the Title of the Moons age , and having found her age in the Table , against it you shall find the hours and minutes which are to be added unto the time that the Moon maketh full Sea in any place , and the whole number of hours and minutes is the time that the Moon maketh full Sea in that place upon the day desired . As for example , I desire to know the full Sea at London Bridge upon the 13 of July 1624. the age of the Moon being found as before , is eight daies , then in the Table I find eight daies , and against it 6 hours , and 24 minutes , which being added unto 3 hours , the full Sea upon the change day gives 9 a clock 24 minutes for the time at the full Sea upon the 13 day of July 1624. THE SEA-MANS GLASSE . The Second Book . VVherein is declared the Definition of the Sphear , a Description of the six great Circles , and also of the four lesser Circles , last of all , certain Questions Astronomicall , performed by the said Scale . CHAP. I. Of a Sphear , and the Circles thereof . The figure of the plaine Scale . A Sphear according to the Description of Theodosius , is a certain solid Sup● ficies , in whose middle is a Point , from which all lines drawn unto the Circumference are equall ; which Poi●● is called the Center of the Sphear , by which C●●●er a right Line being drawn , and excending himself on either side unto that part of the Circumference whereupon the Sphear is turned , is called Axis Spherae , or the Axle-tree of the World. A Sphear accidentally is divided into two parts , that is to say , in Sphaeram rectam & Sphaeram obliquam . Sphaera recta , or a right Sphear , is onely unto those that dwell under the Equinoctiall , Quibus neuter Polorum magis altero elevatur : that is , to whom neither of the Poles of the World are seen , but lie hid in the Horizon . Sphaera obliqua , or an oblique Sphear , is unto those that inhabit on either side of the Equinoctial , unto whom one of the Poles is ever seen , and the other hid under the Horizon . The Circles whereupon the Sphear is composed are divided into two sorts : that is to say , in Circulos majores & minores . Circuli majores , or the greater Circles , are those that divide the Sphear into two equall parts : and they are in number six , viz. the Equinoctial , the middle of the Zodiack , or the Ecliptique line , the two Colures , the Meridian , and the Horizon . Minores vero Circuli , or the lesser Circles , are such as divide the Sphear into two parts , unequally , and they are four in number ; as the Tropick of Cancer , the Tropick of Capricorn , the Circle Artike and the Circle Antartike . CHAP. II. Of the six greater Circles . I. THE Equinoctial is a Circle that crosseth the Poles of the World at right Angles , and divideth the Sphear into two equall parts , and is called the Equinoctial , because when the Sun cometh unto it , ( which is twice in the year , viz. In principio Arietis , & Librae , that is , in March and September ) the daies and nights are equal thoroughout the whole World , whereupon it is called Equator diei & noctis , the equall proportioner of the day and night artificiall : and in the figure is described by the line CAE . II. The Meridian is a great Circle passing thorow the Poles of the World , and the Poles of the Horizon , or Zenith point over our heads ; and is so called , because that in any time of the year , or in any place of the World , when the Sun ( by the motion of the Heavens ) cometh unto that Circle , it is noon , or twelve of the Clock . And it is to be understood , that all Towns and places that lie East and VVest one of another , have every one a severall Meridian : but all places that lie North and South one of another , have one and the same meridian . This Circle is declared in the figure following by the Circle BCDE . IV. The two Colures , Colurus Solstitiorum , or the Summer Colure , is a Circle passing by the Poles of the World , and by the Poles of the Ecliptick , and by the head of Cancer and Capricorn , whereupon , the first scruple of Cancer , where the Colure crosseth the Ecliptick Line , is called Punctus solstitiae aestivalis , or the point of the Summer Solstice : to which place when the Sun cometh , he can approach no nearer unto our Zenith , but returneth unto the Equator again . Arcus vero Coluri , The Ark of the Colure contained betwixt the Summer Solstice and the Equator , is called the greatest declination of the Sun , which Ptolomy found to be 23 degrees , 31 minutes : but by the observation of Copernicus it was found to vary , for ●e found the declination sometimes to be 23 degrees 52 minutes , and in the processe of time to be but 23 degrees 28 minutes . And in these our daies ( by the observation of Ticho de Brahe , and that late famous Mathematician , Mr. Edward Right ) it is found distant from the Equinoctiall 23 degrees , 31 minutes , 30 seconds . V. The other Colure passeth by the Poles of the World , & by the first point of Aries and Libra , whereupon it is called Colurus distinguens Equinoxia . These two Colures do crosse each other at right Angles in the Poles of the world , whereupon these , verses were made . Haec duo Solstitia faciunt Cancer Capricornus , Sed noctes aequant Aries & Libra diebus . CHAP. III. Of the four lesser Circles . THe Sun having ascended unto his highest Solstitial Point doth describe a Circle , which is the nearest that he can approach unto the North Pole , whereupon it is called Circulus Solstitii aestivalis , the Circle of the Summer Solstice , or the Tropick of Cancer , and is noted in the figure before , by the line H Y I. The Sun also approaching unto the first scruple of Capricornus , or the Winter Solstice , describeth another Circle , which is the utmost bounds that the Sun can depart from the Equinoctiall Line towards the Antartike Pole , whereupon it is called Circulus solstitii hyemalis , sive Tropicus hyemalis , vel Capricorni : the Circle of the Winter Solstice , the VVinter Tropick , or the Tropick of Capricorn , and is described in the figure by the line GXF. So much as the Ecliptick declineth from the Equinoctiall , so much doth the Poles of the Ecliptick decline from the Poles of the VVorld , whereupon the Pole of the Ecliptick , which is by the North Pole of the VVorld , describeth a certain Circle as it passeth about the Pole of the VVorld , being just so far from the Pole as the Tropick of Cancer is from the Equator , and it is the third of the lesser Circles , and is called Circulus Arcticus , or the Circle of the North Pole , and is described in the Diagram , in the second Chapter by the line PO. The fourth and last of the lesser Circles is described in like manner , by the other Pole of the Ecliptick , about the South Pole of the world , and therefore called Circulus Antarcticus , the Antarctick Circle , or the Circle of the Antarctick or South Pole , and is demonstrated in the former figure , by the line NM . CHAP. IV. Definitions of some peculiar terms fit to be known by such as intend to practice the Art of Navigation or Astronomy . THe Zenith is an imaginary point in the Heavens over our heads , making right Angles with the Horizon , as the Equinoctiall maketh with the Pole. The Nadir is a prick in the heavens under our feet , making right Angles with the Horizon under the earth , as the Zenith doth above , and therefore is opposite unto the Zenith . The declination of the Sun is the Ark of a Circle contained betwixt the place of the Sun in the Ecliptick , and the Equinoctiall , making right Angles with the Equinoctiall . But the declination of a Star is the Ark of a Circle let fall from the Center of a Star , perpendicularly unto the Equinoctiall . The Latitude is the Ark of a Circle contained betwixt the Center of any Star , and the Ecliptick Line , making right Angles with the Ecliptick , and counted either Northward , or Southward , according to the scituation of the Star , whether it be nearer unto the North or South Pole of the Ecliptick . The Latitude of a Town or Countrey , is the height of the Pole above the Horizon , or the distance betwixt the Zenith and the Equinoctiall . The Longitude of a Star is that part of the Ecliptick which is contained betwixt the Stars place in the Ecliptick , and the beginning of Aries , counting them from Aries according to the succession or order of the signes . The Longitude of a Town or Countrey are the number of degrees , which are contained in the Equinoctiall , betwixt the Meridian that passeth over the Isles of Azores , ( from whence the beginning of longitude is accounted ) East wards , and the Meridian that passeth over the Town or Country desired . The Altitude of the Sun or Star is the Arch of a Circle , contained betwixt the Center of the Sun , or any Star , and the Horizon . The Amplitude is that part of the Horizon which is betwixt the true East or West points , and the point of the Compasse that the Sun or any Star doth rise or set upon . Azimuth's are Circles , which meet together in the Zenith , and crosse the Horizon at right Angles , and serve to find the point of the Compasse , which the Sun is upon at any hour of the day , or the Azimuth of the Sun or Star , is a part of the Horizon contained betwixt the true East or West point , and that Azimuth which passeth by the Center of the same Star to the Horizon . The right ascension of a Star is that part of the Equinoctiall that riseth or setteth with the Star , in a right Sphere : or in an oblique Sphere , it is that portion of the Equinoctiall , contained betwixt the beginning of Aries , and that place of the Equinoctiall , which passeth by the Meridian with the Center of the Star. The oblique ascension is a part of the Equinoctiall , contained betwixt the beginning of Aries , and that part of the Equinoctiall that riseth with the Center of a Star , in an oblique Sphere . The difference ascensionall , is the difference betwixt the right and oblique ascension : or it is the number of degrees contained betwixt that place of the Equinoctiall that riseth with the Center of a Star , and that place of the Equinoctiall that cometh unto the Meridian , with the Center of the same Star. Almicanterahs are Circles drawn parallel unto the Horizon , one over another , untill you come unto the Zenith : these are Circles that do measure the elevation of the Pole , or height of the Sun , Moon , or Stars above the Horizon , which is called the Almicanter of the Sun , Moon , or Star : the Ark of the Sun or Stars Almicanter , is a portion of an Azimuth contained betwixt that Almicanter which passeth thorow the Center of the Star , and the Horizon . QUESTIONS ASTRONOMICAL , performed by the plain Scale . CHAP. V. The true place of the Sun being given , to find his declination . The Sun being in the head of Taurus , his declination is desired . BY the seventh Demonstration , draw the line AD , then upon the Center A raise the Perpendicular AB , then opening your Compasses to the Radius of your Scale , place one foot in the Center A , and with the other draw the Quadrant BCD , then opening your Compasses unto the greatest declination of the Sun , place it upon the Quadrant , from D unto K , then from the point K draw the line KH , parallel to DA , cutting the line AB in H , then with the distance AH draw the small Quadrant GEH , and in regard the Sun is in the head of Taurus , which is 30 degrees from the beginning of Aries , let AD be the Equator , and D the beginning of Aries , DC 30 degrees , or longitude of the Sun , then from the point C draw the line CA , cutting the Quadrant GEH in E , then from E draw the line EI parallel to AD , cutting the Quadrant BCD in I , so shall the Arch ID be the declination of the Sun desired , which in this demonstration is found to be eleven degrees , and thirty one minutes . CHAP. VI. The declination of the Sun , and quarter of the Ecliptick that he possesseth , being given , it is desired to find his true place . The Declination is 10 deg . 31 min. the first quarter that he possesseth , is betwixt the head of Aries and Cancer . FIrst , by the seventh Demonstration , draw the Quadrant ABCD , as is taught in the former Chapter , then set the greatest declination of the Sun upon the Chord from D unto K , which is 23 deg . and 31 min. then from K draw the line KH parallel unto the Equator DA , cutting the line BA in the point H. So shall HA be the sign of the Suns greatest declination , then with the distance AH draw the Quadrant GEH , then from D upon the Quadrant DBC set the declination of the Sun , which is 11 degrees 31 minutes from D unto I , then draw the line IE parallel unto AD , cutting the Quadrant GEH in E. Then from the Center A by the point E , draw the line AEC , cutting the Quadrant BCD in C. So shall the Ark CD be the distance of the sun from the head of Aries , which is here found to be just 30 degrees , which is in the beginning of Taurus . CHAP. VII . By the elevation of the Pole , and declination of the sun , to find the amplitude of the sun , or his distance of rising , or setting from the true East or West point . The elevation of the Pole is 51 deg . 32 min. the declination of the sun is 14 deg . 52 min. North. BY the eight Demonstration , first draw the line BD , then upon the Center A draw the Circle BCDE , then from A raise the Perpendicular CAE , then is your Circle divided into four equall parts : then suppose the elevation of the Pole to be 51 degrees , 32 minutes , which must be placed upon the Circle , from D unto F , then from the point F , by the Center A , draw the line FAG , representing the Pole of the World , F being the North Pole , and G the South Pole , then substract 51 deg . 32 min. from 90 deg . and the remainder is the height of the Equinoctiall , which is 38 deg . 28 min. which must be placed upon the Circle from the Horizon B , unto the point I , then from I , by the Center A , draw the line IAH , representing the Equinoctiall Circle . Then from I unto M set the declination of the Sun , being here supposed 14 deg . 52 minutes North , then from the point M draw the line , or Parallel of declination MTN , parallel unto the Equator I A H , cutting the Horizon BD in T , then from T raise the perpendicular TV , cutting the Circle BCDE in V , so shall the distance CV be the true amplitude of the sun desired , which here is found to be 24 deg , 21 minutes North. CHAP. VIII . By the Amplitude of the Sun , to find the variation of the Compasse . HAving found the Amplitude of the Sun by the last Chapter , first observe with a Compasse , or rather with a Semicircle , upon what degree and minute the Sun riseth or setteth , beginning to reckon from the East or West , and ending at the North or South at 90 degrees : and when you have diligently observed the Magneticall rising or setting , by the Semicircle , or by some other like fitting Instrument : and also the true Amplitude found , as is declared in the last Chapter , the difference of these two Amplitudes , is the variation of the Compasse : But when the Sun riseth upon the same Degree of the Compasse , as is found by the Scale , the variation is nothing , but the Needle pointeth directly unto the Poles of the World , which by M. Mulinux was affirmed to be at the Westernmost part of S. Michaels , one of the Islands of the Azores , from whence he will have the Longitude reckoned . Secondly , when the Sun is in the Equinoctial Circle , where he hat● no Amplitude , look what distance the Compasse maketh the Sun to rise from the East or West of the Compasse , the same distance is the Compasses variation , from the North or South . Thirdly , if the Sun rise more to the South of the Compasse , or setteth more to the North of the Compasse , than is shewed by the Scale , the difference betwixt the Amplitude given by the Scale , and the Amplitude given by the Needle , is the variation of the Compasse from the North Westward . Fourthly , if the Compasse sheweth the Sun to rise more Northward , or set more Southward , than is shewed by the Scale , the difference is the variation of the Compasse , from the North Eastward . Fifthly , if the Scale shew the Amplitude of the Sun rising Southerly , and the Compasse shew it to be Northerly , adde both the Amplitudes together , and they shew you the variation Westernly . CHAP. IX . The place of the sun being given , to find his declination , by a whole Circle . The suns place is the tenth degree of Taurus . ACcording unto the eighth Demonstration , first draw the Circle BCDE , then draw the Horizon BAD , and then the Equinoctial IAH , as is before taught : and then the Tropick of Cancer KL , twenty three degrees and a half from the Equinoctial : then draw the Tropick of Capricorn PO , of like distance from the Equinoctial , and after from K to O draw the Ecliptick line KAO. And when you have thus laid down the Sphere , suppose the Sun to be in the tenth degree of Taurus , at which time his declination is desired . And in regard the Sun is more near unto the Tropical point Cancer , than unto Capricorn ; first find how many degrees he is from the Tropick of Cancer , and you shall find him to be 50 degrees ; therefore take with your Compasses 50 degrees from the Chord , and apply it from the Tropical point Cancer at K , unto V , upon one side , and unto P on the other side : then draw the Line VP , cutting the Ecliptick KO in the point R , then from R draw the Line MRN parallel unto the Equinoctial IAH , and cutting the Quadrant BC in the point M. So shall the arke MI be the declination of the Sun desired , which being applyed unto your Scale , gives you 14 deg . and 52 minutes . CHAP. X. The elevation of the Pole , and declination of the sun given , to find his height in the vertical Circle . The Pole is elevated 51 degrees 32 minutes , the declination of the sun is 14 degrees 52 minutes North , his height in the Verticall Circle is found as followeth . FIrst , according unto the former Chapter , draw the Circle BCDE , then the Horizon BAD , and after the verticall line CAE , then the Axis of the World FG , and likewise the Equator IAH , this being done , place the declination of the Sun 14 degrees 52 minutes , upon the Circle from I unto M , and also from H unto N , then draw the line MN , cutting the line CAE in S , then from S draw the line SVV , parallel unto the Horizon BAD , cutting the Meridian Circle BCDE in VV : so shall the distance DVV be the height of the Sun in the vertical Circle , for the time demanded , which by this proposition is found to be 19 degrees and 8 minutes . CHAP. XI . The elevation of the Pole , and the Amplitude of the sun , being given , to find the declination . The elevation of the Pole is 51 degrees 32 minutes , the suns amplitude is 24 degrees 21 minutes , the declination is found as followeth . FIrst , as in the eight demonstration , upon the Center A , draw the Circle BCDE , then draw the Line BAD , representing the Horizon : dividing the circle into two equall parts then draw the Line CAE , perpendicular to BAD , representing the East and VVest points of the Compasse , then placing the elevation of the Pole 51 degrees and 32 minutes , from D unto F , from F , by the center A ▪ draw the Line FAG , which let be the Pole or Axletree of the world , then from B unto I , and from D unto H , set the complement of the Poles elevation : which shall represent the Equinoctiall , in regard it maketh right Angles with the Pole of the world , in the center A. Then from C unto V place the amplitude of the Sun , which is 24 degrees and 21 minutes : then from V let fall the perpendicular VT , cutting the Horizon BAD in the point T , then from the point T , draw the Line MTN parallel unto the Equinoctiall IAH , and cutting the Circle BCDE in the points , M and N , so shall the distance , M , or HN , be the declination of the Sun , which was desired : which being applied unto your Scale , gives you fourteen degrees and fifty two minutes . CHAP. XII . The elevation of the Pole , the declination of the Sun , and hour of the day being given ▪ to find the Almicanter . The elevation of the Pole is thirty degrees , the declination of the Sun is twenty degrees North , the hour is nine in the morning , at which time the Almicanter is found , as followeth . BY the ninth demonstration , first upon the Center A , draw the Circle BCDE , then draw the line BD for the Horizon , then place your Poles elevation , which is thirty degrees , upon the Circle from D unto R , then from R by the center A , draw the Line RAS , representing the Axis of the World , then from B unto F place the complement of the Poles elevation , which is ●0 degrees , and from the point F , by the Center A , draw the line FAH , representing the Equinoctial line , and then set the declination of the Sun from F unto L ▪ and from L draw the Line LPO parallel unto the Equator FAH , cutting the Axis of the World in the point P , then set one foot of your Compasses in the point P , and extend the other either unto L or unto O , and with the same distance of your Compasses , upon the Center P , draw the circle LNOQ , which is called the hour circle : so shall L be the point of twelve a clock at noon , N the place of six a clock after noon , O the place of twelve a clock or midnight , and Q the place of six a clock in the morning : Every one of the four quarters must be divided into six equall parts , or hours , making the whole Circle to contain twenty four parts , representing the twenty four hours of the day and night , then in regard the hour of the day was nine of the clock , which is three hours before noon , take three of those twenty four hours , and place them upon the circle LNOQ , from the Meridian point L unto K , the nine a clock point in the morning , and unto M the point of three a clock after noon , then draw the line MK , cutting the parallel of the Sun LO in the point I , then from I draw the line IG parallel unto the Horizon BAD , which shall cut the Meridian Circle BCDE in the point G , so shall the distance of G and B be the Almicanter the Sun , which was desired , which in this demonstration is found to be fourty eight degrees and eighteen minutes . CHAP. XIII The elevation of the Pole , the Almicanter , and declination of the Sunne , being given , to finde the houre of the day . The elevation of the Pole is thirty degrees , the declination of the Sun , is twentie degrees , the Almicanter of the Sun , is fortie eight degrees , and eighteene minutes , the houre of the day is found as followeth . FIrst , as in the ninth demonstration , upon the Center A , draw the Circle BCDE , then draw the Diameter BD , representing the Horizon , then from D unto R , set 30 degrees , the elevation of the Pole , then from R unto the point A , draw the line RAS , representing the Pole of the World , then draw the line FAH , crossing the Pole in A , at right Angles , cutting the Meridian circle in F , then from F , set twenty degrees , the declination of ●he Sun unto L , and then from the point L , draw the line LPO , representing the parallell of the Sun , and cutting the Pole of the World in P , then placing one foot of your Compasses in P , extend the other unto L , with which distance of your Compasses , draw the hour Circle LNOQ , then from the Horizon at B , place the Suns Almicanter : ( which is fortie eight degrees , and eighteen minutes ▪ ) upon the Quadrant BGL , from B unto G , then from the point G , draw the line G● parallel unto the Horizon BAD , cutting the Line LO , in I , then from the point I , draw the line KIM , parallell to the Pole of the World QAN , cutting the Circle LNO , in M , then let LN , be divided into six houres , whereof LM , are there : whereupon I conclude , that is is three houres from noon , that is , at nine a clock in the morning , or three in the after noon . CHAP. XIV . The Latitude of the place , the Declination of the Sun , and the Altitude of the Sun being given , to finde the Hour of the day : By a n●w way differing from that in the former Chapter .   deg . min.   deg . m The Suns Altitude is 48 18       The Lat●ude of the place is 30 00 its Comple . 60 00 The Suns declination is 20 00 N. 70 00       Sum 130 00       difference 10 00 The Complement of any arch lesse then 90 degrees , is so much as the arch wants of 90 degrees , as the Complement of 20 degrees is 70 degrees , &c. FIrst , finde the sum and difference of the Complement of the Suns declination , and the Complement of the Latitude , as above is done , where the sum is 130 deg . and the difference 10 deg . Then your Compasses being opened to the Radius of your line of Chords : describe the Semicircle ABC , and divide it into two Quadrants by the perpendic●lar BD , then out of your line of Chords ; take 48 deg . 18 min. the Suns Altitude , and set it from B to E , and draw E F parallel to B D : Then from your line of Chords take 130 deg . the sum , and set it from A to G , ( or its Complement to 180 deg . which is 50 deg . from C to G ) and draw the line GH also parallel to BD. Again , out of your line of Chords , take 10 deg . ( which is the difference ) and set that distance from A to K , and draw K L parallel to EF or BD. This done , take with your Compasses the distance from F to H , and seting one foot in A , with the other describe the Arch MP , likewise take the distance from F to L , and seting one foot in C , with the other describe the arch NQ . Lastly draw the streight line PQ ▪ which only touching the two former arkes will cut the line AC in O , Upon the point O , therefore , erect the perpendicular OR , cutting the Semicircle in R , so will CR being measured upon your line of Chords , give you the degrees of the Sun from the South part of the Meridian , which here you will finde to be 45 degrees , which make 3 hours , allowing 15 degrees for an hour , for 15 degrees make one hour , and one degree makes 4 minuts of an hour , so that it is either 9 of the clock in the morning , or 3 in the afternoon . CHAP. XV. The Almicanter , or height of the Sun being given , to finde the length of the right shadow . The Almicanter is 45 degrees . ACcording unto the tenth Diagram , draw the line AF , and upon the center A , raise the perpendicular AC , then upon the center A , draw the Quadrant CDF , then suppose the height of your Gnomon , or substance yelding shadow be the Line , AB , which is to be divided into 12 equall parts , which Gnomon , I have here made just 12 degrees of the equall Leagues of the Scale , then from B , to the top of the Gnomon draw the Line BE , parallel unto AF , then set the Almicanter which is fortie five degrees from F , unto D , and from the point D , draw the Line DA , cutting the Line BE in the point G , so shall BG , be the length of the right shadow desired , which here is found to be fourteen degrees and eighteen minutes , which is but just the length of your Gnomon , and 2 / 12 and ⅓ of a twelfe over : Note that the right shadow , is the shadow of any poste , staffe , or steeple , that standeth at right Angles with the Horizon , the one end thereof respecting the Zenith of the place , and the other the Naedir . CHAP. XVI . The Almicanter , or height of the Sun being given , to finde the length of the contrary shaddow . The Almicanter given is 70 deg . BY the verse or contrary shadow , is understood the length of any shadow , that is made by a staffe or Gnomon , standing against any perpendicular wall , in such a manner that it may l●e parallel unto the Horizon , the length of the contrary shadow , doth increase as the Sun riseth in height , whereas contrariwise the right shadow doth increase in length , as the Sun doth increase in height : the way to finde the verse shadow is as followeth . First , draw your Quadrant as is taught in the last Chapter , wherein let AB , be the length of the Gnomon , likewise from B , draw the line BE , parallel unto AF , as before , then set your Almicanter from C upon the Quadrant which is given to be seventie degrees and it will extend from C unto H , then from the point H draw the line HA , cutting the line BE , in the point K , so shall KB , be the length of the contrary shadow , which here is found to be thirtie four degrees and eight minutes , or twice so long as your Gnomon , and ●0 / ●2 about ½ part of a twelfth more . CHAP. XVII . The latitude of the place , the Almicanter , and declination of the Sun being given , to find the Azimuth . The latitude of the place is fiftie one degrees , thirtie minutes , the declination of the Sun twenty degrees North , the Almicanter thirtie eight degrees thirtie minutes , the true Azimuth of the Sun is desired . FIrst as in the eleventh Demonstration upon the Center A , draw the Circle BCDE , then draw the Diameter BAD ▪ and from D unto F , set the Elevation of the Pole , which is one and fiftie degrees , and thirtie minutes , whose complement is eight and thirtie degrees and thirtie minutes , which must be placed from B unto H , then from H , draw the line HAL , representing the Equinoctial line , and from F , draw the line FAG , representing the Pole of the World , then from H unto P , and from I unto Q , set the declination of the Sun , which is twentie degrees , and by those two points draw the line PQ , for the Parallel of the Suns declination ; then upon the Circle from B unto H ▪ set the Suns Almicanter , thirtie eight degrees , and thirtie minutes , then from H , draw the line HR ▪ parallel unto the Horizon cutting the Suns parallel POQ in O , then draw the Line TVAE Perpendicular unto the line BAD , in the Center A , and cutting the line HVR , in V , then seting one foot of your Compasses in the point V , extend the other unto R , and with the same distance draw the Semicircle HLR , then draw the Concentricke Circle upon the Radius of the Scale MTN , and where the Line POQ , and the line MON do meet in the point V , raise the Perpendicular OL , cutting the Semicircle HLR in L , then lay the Scale from the Center A to the point L , and draw the line LK , cutting the Semicircle MTN , in K , so shall M K , be the true distance of the Sun from the East , or West point Southward , or the Suns true Azimuth , which is here found to be seventie two degrees , and fortie minutes from the South part of the Meridian . CHAP. XVIII . The Latitde of the place , the Declination of the Sun , and the Altitude of the Sun being given to finde the Azimuth : By a new way differing from that in the former Chapter .   deg . min. S. deg . m. The Suns Declination is 20 00       The Latitude of the place is 51 30 its Comple . 38 30 The Suns Altitude is 12 00   78 00       Sum 116         difference 39 30 HAving found the sum and the difference of the complement of the Suns Altitude , and the complement of the Latitude as above is expressed where you finde the Sum of them to be 116 deg . 30 min. and their difference 39 deg . 30 min. Secondly , take 116 deg . 30 min. the sum out of your line of Chords , and set it from C to G , and draw the line GK parallell D to B , Thirdly take 39 deg . 30 min. the difference , out of your line of Chords , and set it from C to H , and draw the line HL parallell also to BD. Fourthly Take in your compasses the distance from F to K , and setting one foot in A , with the other describe the arch S. Fifthly , Take the distance from F , to L , and setting one foot in C , with the other describe the arch R. Sixthly , Lay a rular , that it may only touch these two arches , S , and R , and by it draw a line as SR , cutting the line AC in N. Lastly , upon the point N , erect the perpendicular NM , then the distance AM , measured upon your Line of Chords , is the Azimuth from the South part of the Meridian , which in this example will be found to be 34 deg . MC the Azimuth from the North 146 deg . And MD , the Azimuth from the East or West , 56 deg . CHAP. XIX , The place of the Sun being given , to find the right ascension , Suppose the Sun be in the twentieth degree of Taurus , his right ascention is found as followeth . FIrst , as in the 12 demostrastion , draw the line BAF , for the Pole of the World , the ● upon the Center A draw the Circle BCDE , then from the Center A , raise the Perpendicular CAE , for the Equator , then place your greatest declination from C unto Q , and from E unto P , then daw the line QAP , which doth represent the Eclipticke line , then in regard the Sunne is in the twentieth degree of Taurus , which is forty degrees , from the head of Cancer , which forty degrees , place from Q unto L , and unto K , then draw the line KL , cuting the Eclipticke in I , then from the point I draw the line HI , parallel unto CAE , cuting the Pole of the World in O then set one foot of your Compasses in O , and extend the other unto G , with which distance draw the Semicircle HDG , then opening your Compasses unto the Radius of the Scale , and upon the Center O , likewise draw the Circle HNFG , then draw the line IM , parallel unto AOD , cutting the Semicircle HMDG , in M , then lay your Scale from the Center O , unto the point M , and draw the Line NM , cutting the Concentricke Circle in N , so shall the distance NF , be the right ascention , which is here found tobe two and fortie degrees , seven and twentieminutes . CHAP. XX. The elevation of the Pole , and declination of the Sunne given , to finde the difference of the ascensions . The Poles elevation is 51 degrees , 32 minutes , the declination of the Sun is 21. degrees . FIrst , as in the 13th . demonstration , draw the Line BAK , representing the Horizon , then upon the Center A , draw the Circle BCDEF , Then from K unto D , set the elevation of the Pole which is 51 degrees , and thirty two minutes : then from the point D , by the Center A , draw the Line DAF , representing the Pole of the World , then from B unto C , set the Complement of the Poles elevation which is thirty eight degrees , and 28 minutes : then from C by the center A , draw the line CAE , representing the Equinoctiall Line ; then from C unto G ▪ and likewise from E unto H , for the declination of the Sunne , which is 21 degrees , then from G unto H , draw the parallel of the Sunnes declination , cutting the Pole of the world in L , and he Horizon in I , then set one foote of your Compasses in the point L , and extend the other unto G , then with that distance of your Compasses draw the Semicircle GMNH , then opening your Compasses unto the Radius of your Scale , upon the same Center draw the Concentricke Circle , GXOH , then from I , where the declination of the Sunne doth cut the Horizon , draw the Line IN , parallell unto the Pole of the World AM , cutting the Circle GMH in N , then lay your Ruler from the point I unto the point N , and so draw the line NO , cutting the Concentricke Circle GXOH , in O , so shall the distance of O and X , be the difference of the ascentions , which is here found to bee eight and twentie degrees , and foure and fiftie minutes . CHAP. XXI . The right ascention of the Sun or of a Star being given , together with the difference of their ascention , to finde the oblique ascention or descention . The Sun is in the 4th . degree of Sagitarius , his right ascention is 242 degrees , or 16 hours 8 minutes , the difference of ascention is 1 houre 53 min. or 28 deg . 28 min. the oblique ascention or desce●tion is required . THe right ascention of any point of the Heavens being known , the difference of the ascention is either to bee added thereunto , or else to bee substracted from it , according as the Starre is situate in the Northern or Southerne Signes : As for example , if the Sunne be in any of these sixe Signes , Aries , Taurus , Gemini , Cancer , Leo , or Virgo , then the difference of the ascentions is to bee substracted from the right ascention , and the remainder is the oblique ascention . Suppose therefore the Sunne to be in the fourth degree of Gemini , where the right ascention is found to be foure houres , and 8 minutes , or 62 degrees , and the difference of ascention where the Pole is elevated 51 degrees , is found to be one houre 53 minutes , otherwise 28 degrees 50 minutes , which being taken from the right ascention , leaves two houres and 16 minutes , or 33 degrees and 42 minutes , which is the oblique ascention of the Sunne in the fourth degree of Gemini . But if the Sun be upon the South side of the Equinoctiall , either in Libra , Scorpio , Sagitarius Capricornus , Aquarius , or Pisces , then the difference of the ascentions is to bee added unto the right ascention , and the Product will be the oblique ascention . Suppose the fourth degree of Sagitarius is given , for which Sign and degree the oblique ascention of the Sun is desired , his right ascension being then found to be 242 degrees , or 16 hours 8. min. the difference of the ascensions is one hour , 53 minutes , or 28 degrees , 18 minutes : which being added unto the right ascension , makes 18 hours , and one minute ; or in degrees 270 degrees , and 18. minutes : which is the oblique ascention of the Sunne , when he is in the fourth degree of Sagitarius . And if you would finde the oblique descention , you must adde the difference of the ascentions unto the right ascention , when the Sunne is in these six Signes . Aries , Taurus , Gemini , Cancer , Leo , Virgo : and contrariwise , when the Su●n is in the other six Signes , you mnst substract the difference from the right ascention , and you shall have the oblike descention of the Sun or any Starre , whose right ascention and difference of ascentions is knowne . But it is to be understood , that this manner of operation , doth serve no longer than you are upon the North side of the Equinoctiall . For if the South Pole be elevated , the worke is contrary : for so long as the Sunne is in any of the Northerne Signes , the difference of the ascentions is to be added unto the right ascention , to find the oblique ascention . And contrariwise , substracted to finde the oblique descention . Likewise if the Sunne or Star be in the South●rn Signes , then is the difference of ascentions , substracted from the right ascention , to finde the oblique ascention , and added , to finde the oblique descention . The end of the Second Book . THE SEA-MANS GLASSE : The Third Book . Shewing how by the Plain-Scale , to delineate Houre-lines upon all kinde of Upright Plains , either Direct or Declining , in any Latitude . The figure of the plaine Scale . CHAP. I How to draw hour lines upon an Horizontal Plain , in any Latitude . VVith the Radius of your line of Chords , upon E as a Center , describe the Circle ABCD , and crosse it with he diameters AB , and CD . This done , out of the line of Chords take the complement of the Latitude of your place ( which we here suppose to be London , whose latitude is 51 deg . 30 m. and its complement 38. deg . 30 m. ) which set from B to G , from G to N , and from D to M ; then lay a ruler from A to G , and it will cut the line CD in H , and from A to N it will cut C D in O , and from A to M it will cut the same line in F. This done , upon O ( as a center ) place one foot of your compasses , and extend the other foot to F , and with this distance describe an arch of a circle , which ( if the rest of your worke be true ) will fall just in the points A and B , and so constitute the arch AFB , representing the Equinoctiall Circle , and so we shall hereafter call it . Having drawn the Equinoctiall AFB , divide the Semicircle ADB , into 12 equall parts in the points *** , &c. Then laying a ruler to the Center E , and every one of these marks *** &c. it will divide the Equinoctiall circle into 12 unequall parts in the points ●●●● &c. Again , Lay a ruler to H , and every of these unequall parts ●●●● , &c. it will cut the semicircle ADB in the points 7 , 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , 4 , 5 and 6. Lastly , If you lay a ruler on the center E , and from thence draw right lines to the severall points 7 , 8 , 9 , 10 , &c. they shall be 12 of the true houre-lines belonging to an horizontall diall for the latitude of 51 degrees , 30 minutes . But for the houres before 6 in the morning , and after 6 at night , do thus ; draw the hour liues of 4 and 5 in the evening , quite through the center E , and they shall be the hours of 4 and 5 in the morning ; also , 7 and 8 in the morning drawn through the center , shall give the hours of 7 and 8 at night , as in the figure . CHAP : II. Concerning direct South Dials . A Direct South diall is no other then an horizontall diall , the makeing whereof is before described , the difference consisting only in the numbring of the houres , and in the placing of it , the one being to be fixed on a poste or the like , and the other to be fixed to a Wall which exactly beholds the South , I say here is no other difference : for   degrees   degrees An Horizontall Diall for the Latitude of 10 Will be a direct South Dial in the Latitude of 80 20 70 30 60 40 50 50 40 60 30 70 20 80 10 And the like in any other Latitude , as 15 , 16 , 33 , &c. CHAP. III. Of driect North dialls . A Direct North diall , is the same with a direct South diall ; for , i● you take a South diall and turn it upside down , causing the Sc●le or cock to point upwards , as the Cock of the South doth down wards ; and leaving out the hours neer the Meridian , in these Northern Latitudes ; as the hours of 9 , 10 , 11 , and 12 at night , and 1 , 2 and 3 in the morning , all which time the Sunne is under the Horizon . I say a South diall so disposed , and fixed against a direct North Wall , shall give you the true houre of the day . CHAP. IV. How to draw the houre lines on a direct East or West plain . This done , upon the point G , with the radius of your Chord , descirbe an occult arch of a Circle H I , and set thereon 15 degrees fr om H to I , then from G , through I , draw the line G K , cutting N Min K , On K , as a center , with the radius of your Chord , describe the quadrant K S T , which divide into 6 equall parts in the points ●●●● , through which points and K , draw the lines , K● , K● , &c. cutting the Equinoctiall EB in **** &c. Through these points *** , &c. draw right lines quiet through your plain perpendicular to the equinoctial , which will be parallel to your lines of VI , and XI , and will be the true hours of VII , VIII , IX , and X , then the like distances of VII and VIII , set above VI , on the other side , and drawn parallel thereto , shall be the true hours of IIII. and V. and thus have you all the hours of an East dial truly drawn , which is from Four in the morning , till Eleven at noon , and is the same with a West diall only naming the hours contrary : for , in the East diall 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , in the morning , are in the West diall 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , h in the evening . The Stile of either of these dials , is a t in plate of brasse , made directly of the breadth of the distance between the hours of VI , and IX , and must be placed directly perpendicular upon the line of VI , and so is your diall finished . CHAP. V. Of upright declining Plains . BEfore we come to draw the houre lines upon a declining plain , two things are first to be discovered , viz. First . The height of the pole above the plain , which is the height of the Cock or Stile . Secondly , The deflexion , or distance of the substile from the Meridian or line of Twelve a Clock . 1. To finde the height of the Pole above a declining Plain . VVIth the tadius of your line of Chords , upon A , as a center , describe the Quadrant AB C , then your Latitude being 51 deg . 30 min. take it out of your line of Chords , and set it from B to F , and draw the line ED parallel to AB , cutting the line AC in D , then with the distance DE , on the center A , describe the Quadrant GHR . Then supposing your plain to decline 30 deg . set 30 deg . from B to F , in the Quadrant BEC , and draw the line FA cutting the Quadrant GHR in H , through which point H , draw the line SHN parallel to CA , and cutting the Quadrant BE C in N , so shall the arch CN be the height of the Pole above the plain , and in this example contains 32 deg . 37 min. 2. To finde the Deflexion , or the distance of the Substile from the Meridian . Out of this figure , take the distance HS , and set it in the line DE , from D to K ; through which point K , draw the line AKL , cutting the Quad ant BC in L ; so shall the arch CL be the distance of the Substile from the Meridian : and in thls Example will be found to be 21 degrees 42 minutes . CHAP. VI. How to draw the Houre-lines upon an upright Plain declining from the Meridian towards the East or West . VVE will here take for Example a South erect plain , declining Eastward 30 deg . Having ( by the Fifth Chapter of this Book ) found the Defl●xion of such a plain to be 21 deg . 42 min. And the height of ●he ●●ile ( by the same Chapter ) to be 32 deg . 37 min. we may proceed to draw the Diall in manner following . With the radius of your line of Chords , on the Center C , describe the Circle XNSW ; and in it , draw SN through the Center C , for the Meridian , or line of 12. Then the deflexion being found to be 21 deg . 42 min. set that from N to E , and draw the line ●C through the center to G ▪ This line representeth the Substilar line of your Diall , upon which line the Stile or Co●k must stand ▪ Also , out from your line of Chords take 32 deg . 37 min. the height of the S●ile , and set that distance from E to H , and draw the line CH for the Stile of your Diall ; so shall the Triangle ECH , be the true pattern for the Cock of your Diall . The Substilar line EG being 〈◊〉 ●●aw the line XW through the center C , and perpendicular to EG . This done , take the distance EH , ( which is equall to the Stiles height ) and set that distance from A to B , and from W to D. Likewise , take the distance from W to B , and set it from B to I. These three points I , B and D , being found in the circumference of the Circle XNSW , lay a ruler from X to I and it will cut the substilar line EC being extended in the point G , which is the center upon which the equinoctiall Circle must be described . Again , a ruler laid from X to B , will cut the substilar line in F , and a ruler laid from X to D , will cut the substilar in O. Now , if you set one foot of your Compasses in G , and extend the other to X or W , you may describe the Equinoctial circle XOW , which ( if you have not erred in your former worke ) will passe exactly through the point O in the substilar line before found . In the next place , if you lay a Ruler from F to N , it will cut the Equinoctiall circle in P , and a ruler laid from C to P , will cut the Diall circle in V. These things being performed , the next thing is to draw the hour lines , which will be easily effected if you 〈◊〉 the former directions . First , from the point V last found , begin to divide your houre circle into 24 equall parts ( or only one halfe of it into 12 parts ) which you may do by taking 15 deg . out of your line of Chords and set that distance on both sides of V at the marks ⚹ ⚹ ⚹ &c. so many times as the plain is capable of hours . This done , If you lay a ruler on the center C , and every of these points **** &c. you shall divide the equinoctiall Circle into 12 unequall parts in the points ●●●● &c. Now a ruler laid from F to every of these unequall points ●●●● , &c. will divide the houre circle into 12 other unequall parts marked with 4. 5. 6. 7. 8. ▪ 9. 10. 11. 12. 1. on the one side of V , and with 2. 3 ▪ ●n the other side of V. Lastly , a ruler laid from C to the severall points 4. 5. 6. 7. 8. 9. 10. 11. 12. 1. 2. 3. and lines drawn by the side thereof they shall be the true houre lines belonging to such a declining plain of 30 deg . in the Latitude of 51 deg . 30 min. But if you desire more hours then 12 , the equinoctiall may be divided into more unequall parts , being continued beyond X and W , and if you will , quite round the whole Circle , but that is needlesse without you would make 4 Dialls in the makeing of one as you may easily do . For , The hours that are on the West side of the Meridian of a South East diall , being drawn through the Center , will make a North West diall of the same declination . And the hours on the east side of the Meridian of a South West diall ; being drawn through the center , will produce a North East diall of the same declination . And Again , the reall houre lines of a South East diall being drawn on the other side of the paper , and the hours named by their Complements to 12 , that is , 10 for 2 , 9 for 3 , 8 for 4 , &c. will make a South West diall of the same declination . CHAP. VII . How to place any upright diall truly . ALL upright dialls , in what oblique latitude soever have the Meridian perpendicular to the horizon , wherefore to set your diall exact , hang a line with a plummet at the end thereof , and with a nail fixed in the line of 12 towards the top thereof , to hang the plummet upon , apply the diall to the place where it is to be fixed , so that the line and plummet may hang just down upon the line of 12 , neither inclining on one side or the other , the diall thus fixed if the declination were truly taken , and the dial rightly made , by the former directions , shall at all times ( the Sun shining upon it ) give you the true hour of the day . CHAP. VIII . How to insert the halve and Quarters of hours in all dialls . THe halves and quarters of hours are drawn in all plaines by the same rules , and the like reason , that the hours are inserted . Therefore take notice that if you would insert the halfe hours into any diall , you must divide your Equinoctiall Circle into 24 equall parts instead of 12 , and if you would insert the quarters , then you must divide it into 48 parts , and then proceed in all respect , as you did for the whole hours . CHAP. IX . How to finde the declinatioon of any upright Wall. THe declination of a plain is an arch of the horizon comprehended between the pole of the plains horizontall line , and the meridian of the place . To finde this declination , two observations must be made , the Sun shining , and both at one instant of time ( as neer as may be . ) The first is the horizontall distance of the Sun from the pole of the plain . The second is the Suns Altitude . First , to finde the horizontall distance . Apply the side of a Quadrant to your plain , holding it ( as neer as may be ) horizontall , that is to say , levell , Then holding up a thrid and plummet , which must hang at full liberty , so that the shadow of the thrid may passe directly through the center of the Quadrant , then diligently note ● through what degree of the Quadrant the shadow passed , and count those degrees from the side of your Quadrant which is perpendi●cular to the plain , for those degrees are the Horizontall distance . Secondly , At the same instant , take the Suns a●●itude , these two being heedfully taken , will help you to the plains declination by th rules following . By the 17 or 18 Chapters of the Second Book find the Suns Azimuth . Then observe whether the Sun be between the pole of the plains horizontall line and the North or South points , or not . If the Sun be between them , adde the Azimuth and horizontall distance together , and the sum of them is the declination of the plain . If the Sun be not between them , substract the lesser of them from the greater , and the difference shall be the declination of the plain . These rules sh●w you the quantity of your plains declination . But , CHAP. X. Shewing how to know whether your plain declin from the Meridian towards either the East or West . YOu must take notice in your observation , that if the Meridian point fall between the Azimuth and the pole of the plains horizontall line , then doth the plain decline to the Coast contrary to that wherein the Sun is , that is to say , if the Sun be to the Eastward of the Meridian , the plain declines to the Westward , But if the Meridian point be not between the forementioned distance and the pole of the plain , then doth the plain decline to the same Coast in which the Sun was at the time of observation . CHAP. XI . Concerning Polar Dials . A Polar diall is made in all respects as an East or West Diall is made , onely the line of 6 a clock in the East or West Diall , is 12 a clock in the Polar Diall , the houre of 7 is 1 , of 8 is 2 , of 9 is ● , of 10 is 4 , and of 11 is 5. Also the houre of 5 in the East or West Diall , is 11 in the Polar , of 4 is 10 , of 3 is 9 , of 2 8 , of ● is 7 , &c. The Cock of this Diall is a plate of Iron or Brasse made of the breadth between 12 and 3 a cloock , and set perpendicular upon the line of 12 , as in the East or West Diall it is upon the line of 6. In these Dialls the Equinoctiall line is to lie parallel to the Horizon , and not to be elevated according to the complement of the Latitude of the place , as in the East or West Diall it is . CHAP. XII . Concerning Equinoctiall Dialls . AN Equinoctiall Diall is of all other Dialls , the most easie to make , for if you describe a Circle , and divide it into 24 equall parts , and draw lines from the center through eve●● one of those equall parts , the lines so drawn shall be the true houre lines . For the Stile of these Dialls , it is no other but a streight Wyre of any length set perpendicular in the Center of the Circle , whose shadow shall give the true houre of the Day . CHAP. XIII . Of such Plains as decline very far from the East or West towards the Meridian as 75 , 80 , or 85 , deg ▪ above which plains the Pole hath small Elevation . SUch plains as decline above 60 degrees the houre lines will come very close together , so that if they be ▪ not extended very far from the center , there will be no sensible distance between hour and hour ▪ To remedie this inconvenience , there are severall wayes , I will instance only in one which is familiar and easie , and that is this . When 〈◊〉 have 〈…〉 your diall on a large sheet of paper , fix it on some large Table or smooth Floor of a Room , if the Diall you are to make be very large , as 5 , 6 , or 7 ▪ foot square , then by the side of a long Rular laid to the Center and every hour line , as also to the Stile and Substile , draw lines to the full extent of the Table or Flour , and you shall finde them to be of a competent largnesse . Then according to the bignesse of your plain , cut off the houres . Stile and Substile , leaving the center quite ou● , and yout work is finis●ed . CHAP : XIIII Concerning Declining Reclining and Inclining Dials . VVE should now shew the manner of drawing houre lines upon declining reclining and inclining plains , of which there are severall varieties , and many cautions , which in this place and at this time , would be too many to ennumerate : but if this which hath been already delivered concerning Upright decliners shall be kindly accepted , it shall animate me to do the like for all other plains whatsoever . FINIS . ADVERTISEMENT . NOte , that this Scale and all other Instruments for the Mathematicks , are made by Walter Hayes , at the Crosse dagers in Moore , Fields next doore to the Popes head Tavern , London . 
A35744	----	Mr. De Sargues Universal way of dyaling, or, Plain and easie directions for placing the axeltree and marking the hours in sun-dyals, after the French, Italian, Babylonian, and Jewish manner together with the manner of drawing the lines of the signs, of finding out the height of the sun above the horizon, and the east-rising of the same, the elevation of the pole, and the position of the meridian ... / [edited] by Daniel King, Gent. Maniére universelle pour poser l'essieu. English Desargues, Gérard, 1591-1661. This text is an enriched version of the TCP digital transcription A35744 of text R17188 in the English Short Title Catalog (Wing D1127). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 146 KB of XML-encoded text transcribed from 72 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2017 A35744 Wing D1127 ESTC R17188 13154958 ocm 13154958 98167 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A35744) Transcribed from: (Early English Books Online ; image set 98167) Images scanned from microfilm: (Early English books, 1641-1700 ; 414:7) Mr. De Sargues Universal way of dyaling, or, Plain and easie directions for placing the axeltree and marking the hours in sun-dyals, after the French, Italian, Babylonian, and Jewish manner together with the manner of drawing the lines of the signs, of finding out the height of the sun above the horizon, and the east-rising of the same, the elevation of the pole, and the position of the meridian ... / [edited] by Daniel King, Gent. Maniére universelle pour poser l'essieu. English Desargues, Gérard, 1591-1661. King, Daniel, d. 1664? Bosse, Abraham, 1602-1676. [17], 108 p. : ill. Printed by Tho. Leach, and are to be sold by Isaac Pridmore ..., London : 1659. Translation of: Maniére universelle pour poser l'essieu. Added illustrated t.p., engraved. The diagrams are reproductions of the engravings by Abraham Bosse who published the original French edition. Advertisement on p. [17]. Reproduction of original in Cambridge University Library. eng Dialing. Sundials. A35744 R17188 (Wing D1127). civilwar no Mr. De Sargues Universal way of dyaling. Or plain and easie directions for placing the axeltree, and marking the hours in sun-dyals, after t Desargues, Gérard 1659 29465 52 0 0 0 1 0 21 C The rate of 21 defects per 10,000 words puts this text in the C category of texts with between 10 and 35 defects per 10,000 words. 2006-06 TCP Assigned for keying and markup 2006-06 Apex CoVantage Keyed and coded from ProQuest page images 2007-04 Robyn Anspach Sampled and proofread 2007-04 Robyn Anspach Text and markup reviewed and edited 2008-02 pfs Batch review (QC) and XML conversion Mr. DESARGVES . vniversall way of makeing all manner of Sun dialls ▪ Published by Daniell king ▪ & Sold by Isaake Pridmore at y , golden falcon in y strand ▪ A● 1659 Mr. De SAKGVES UNIVERSAL WAY OF DYALING . OR Plain and easie directions for placing the Axeltree , and marking the hours in Sun-dyals , after the French , Italian , Babylonian and Jewish manner . Together with the manner of drawing the lines of the signs , of finding out the heighr of the Sun above the Horizon , and the East rising of the same , the Elevation of the Pole , and the position of the Meridian . All which may be done in any superficies whatsoever , and in what situation soever it be , without any skill at all in Astronomy . By DANIEL KING Gent. LONDON , Printed by Tho. Leach and are to be sold by Isaac Pridmore at the Golden Faulcon in the Strand , near the New Exchange , 1659. TO THE ILLVSTRIOVS GEORGE VILLIERS , Duke , and Marquess of Buckingham , Earl of Coventry , Viscount Villiers , Baron Whaddon , and Ros , Knight of the most noble Order of the Garter , &c. Sir , HAving had the honour to observe your Graces great affection , and love to Sciences and Arts , and your own excellency being most eminent therein , together with your unparallel'd love and inclination to the splendour of your Native Country , in promoting Learning and Ingenuity . These high merits with my own particular obligations and attendance , encourage my endeavours of the patronage to a new birth never presented to the English Nation ; presuming by Gods assistance to bring forth something of worth that hathnot yet seen light , and if your Grace shall please to pardon my observant presumption , you will hereby more strictly engage him ever to honour your Heroick worth , who is , The very humblest of Your Servants , Daniel King . The Preface . Concerning the particulars of this TREATISE . WHereas the Superficies or outsides whereon Dyals may be made , may be either flat , bowed , or crooked , plain or rugged , & situated diversly , the most part of the books treating of this matter , contain severally the manner of making flat Dyals in all kinds of positions , Horizontal , Vertical , Meridional , Septentrional , Oriental , Occidental ; Declining , Inclining ; Inclining , and Declining ; and accordingly in all other kinds of superficies . They may also shew , for those that are ignorant of it , the way to find the elevation of the Pole , the Meridian line , the Declinings , Inclinings , and other particularities . But Monsieur de Sargues intention being to publish nothing , if it be possible , that is to be found in another Book , and to give you only the general Rule to make , and not to copy out a number of examples all differing one from another ; I will give you but one example only in this volume , by this universal manner , the discourse whereof may be applyed generally to all kinds of superficies , and in what situation soever they be , without having any knowledge of the Pole , nor of the height of the Sun , nor of the declining or inclining , nor of the Meridian line , nor of any other thing in Astronomy , and without a needle touch'd , nor of any kind of thing that may give a beginning to that , as you shall see yet better when we shall treat of the practice . And in practising this general Rule , you shall find at one and the same time , the elevation of the Pole , and the position of the Meridian , you shall know how to place the needle of your Dyal , and so you shall come to find the equal hours , which are called hours after the French way , alias Astronomical . The rest being more curious than necessary , I thought to set down nothing else but those two things ; But I have been perswaded , for the satisfaction os some , to add also the manner of drawing upon the same superficies , the lines of the signs , the hours after the Italian and Babylonian way , and of the Antients , the height of the Sun , and the situation thereof , in respect of the Horizon . Of the Practice of Sun-Dyals . MAny diverse things are represented in Sun-Dyals by the shadow of the Sun , to divers ends , the hour is shewn by them , and serves only for that purpose every day . Other things are represented also by it , as the signs , and other particularities , whereby it may serve sometime for the divertisement of a few In antient time the hours were not counted as they are now a dayes , and in Italy at this day they are counted otherwise than in France The manner that they count their hours in France is according to Astronomy , and here is at length a generall way of framing and making Dyals with houres equal to the Sun , according as the hours are now counted in France , after which way one may come to shew if need be , by the shadow of the Sun , and also of the Moon , whatsoever can be shewn concerning other circumstances to satisfie curiosity . There are two things which together do compose those kinds of Dyals of equal hours , after the French way , the one is as piece that shoots out or sallies out of the superficies of the Dyal , the shadow whereof falling upon this superficies , shews what a clock it is , the other are the lines drawn upon the superficies of the Dyal , each of them representing one of the hours after the French way . They make Dyals after the French way wherein there is only the shadow of one portion alone , as might be a button of the piece that shoots out , that shews what a Clock it is . But in this general way , there is still the whole shadow of all the length , in a direct line of this piece that shoots out , shewing continually what a Clock it is , of which piece or length , you may take if you will a button , and mark the hours with that button only , together with all the other particularities that may be added to such Dyals . Some call by one and the same name these two kinds of pieces the shadow whereof shews the hour , as well that same whose shadow shews continually the hour at length , as that which hath but the shadow of a button to shew it . But to the end we may distinguish both kinds of pieces one from the other , that same whose shadow shews the hour at length , and is the original spring of all the others , I call it the axeltree of the Dyal . This axeltree may be made as well with a straight , round , and smooth rod of yron or brasse , as with a flat piece , and cut of one side in a straight line . There are often other rods in the Dyals , which serve to bear up the axeltree that shoots out , and those kind of rods I call the supporters of the axeltree . The lines that are drawn upon the superficies of the Dyal , and that shew each of them one of the hours after the French way , I call them lines of the hours after the French way . In the innumerable number of such kinds of Dyals as may be made after the French way , it happens that the superficies of the Dyal , is either all flat , or is not so altogether . When the superficies of the Dyal is all flat , every line of the hours is a straight line . And when the superficies of the Dyal is not altogether flat , it may be that every line of hours is not all straight also . To make one of those Dyals of equal hours after the French way , by this universal way , there are two things to be done one after another . The first is to place the axeltree as it ought to be , that is , shooting out of the superficies of the Dyal ; The second is , to draw the lines of the hours as they must be upon the same superficies . And by means of this general way , you shall do those two things without knowing in what day , nor in what time of the year , nor in what Country you are , without knowing what the superficies of the Dyal is , whether it is plain or rough , nor which way it looks , without knowing any thing concerning the making and the placing of the parts of the world , or without any skill in Astronomy , without any needle touch'd with the loadstone , or any instrument or figure that may serve for a beginning towards the making of a Dyal : But by the means only of the Beams of the Sun , by one general rule you shall place the axeltree , and draw the lines of those hours upon one of those Dyals whatsoever the superficies may be , and which way soever it looks , with all the celerity and exactnesse that is possible in art ; and if you are equally exact in every operation , you shall make by this means many Dyals upon different superficies , and turned towards several parts of the world , which shall agree plainly among themselves , and if you do not do it , you may be sure that the fault is on your part , and not in the rules , since that others do succeed well in it . There are some pieces that are requisite for the framing of a Dyal , and whereof it is composed , such are the axel-tree rod with its supporters . There are other pieces that must be used in the making of a Dyal , as Rules , Compasses , a Squirt , a lead with his two frames , one to mark with , and the other to level . There are some other things that you shall use also , as pegs , and rods , either of yron , or of brasse , or of wood , some sharp at both ends , the others sharp at one end only , a table , either of wood , or of slate , or of any other stuff , flat and solid , to draw upon if need be , some straight lines with the rule ; and in case the superficies of the Dyal were plain and even , you must use some fine strings , supple and strong , some mastick , cement , or plaster , or such like stuff fit to seal with , &c. all which things you must have in readinesse , whensoever you will go about the making of a Dyal . And though you would learn to make but one of these Dyals well , it is fit you should have some models of all those pieces , and when you are upon those chapters that concern them , as you shall understand an Article , it will be requisite that with the models of those pieces , you work at the same time an actual model of the thing which that article shall teach you to do , and so you must work from one end to another , till you have at last every way compleated an actual model of this kind of Dyals , and you shall need to make but few of such models of Dyals upon any superficies , turned towards several parts of the world , to bring you acquainted with the practice of making Dyals to the life , or after the natural , in what kind or odd situation of superficies soever they may be . Lastly , you shall find the precepts and the descriptions to be more troublesome , than the actual making or working , study only to be as exact in every one of these operations of making of these Dyals , as in the practice of other Arts . The Epistle to the READER . Courteous Reader , THis Treatise being originally written in French , and generally approved of all those that have any skill in the Art of Dyalling , I have thought it my duty to lay hold upon this occasion , to shew how desirous I have ever been to procure any good unto my Country . Therefore I have caused it to be carefully translated into English , and have set it forth for the good and utility of all such , as are curious , and true lovers of that Art ; Reputing my self most happy to meet with any occasion , whereby I may contribute any thing towards the advancement of learning , and of the publick good , — Non enim nobis solum nati sumus ; We are not meant to be wholely and soly for our selves . As for the work it self , I am so confident it will so gain the attentive Readers approbation , as that I shall forbear to say any more in commendation of it , than that it is an expedite , and sure way of obtaining the site of the Axis , and of other requisites in the framing of all sorts of Dyals , of no lesse curiosity , than use , performed without the ordinary rules and presupposals of the spiritual calculations and practice ; I need premise no more , but advise to follow the directions that are set down through all the book , for effecting that which is promised , and thou shalt see the same plainly and readily performed . Accept then , Courteous Reader , this small labour , the undoubted Testimony of my Love , as kindly ▪ as I offer it cordially unto thee , hoping that God will enable me to give thee hereafter some thing of more consequence , So Farewell . Vtere & fruere . Thine D. K. To all Lovers of Ingenious Practices . THe French have excelled all other Nations in the Art of Perspective for this last Age , their many Books and curious Writings so excellently composed do witnesse for them . Dyalling I accompt one kind of Perspective , for that glorious Body the Sun , the Eye of the world , traceth out the lines and hour-points by his Diurnal Course , and upon the resubjected Plane by the laws of Picture , Scenographically delineates the Dyal . Many have writ upon this subject of several Countryes , in several Ages , many are the Rules and Practices set down ; But among all those of forein Parts , none hath performed the same with more ease , and lesse trouble , than Monsieur du Sargues the Author ; as wholy laying aside those tedious observations of Azimuth's , Declination , Reclination , Inclination , Meridian ; Substile , &c. and performing the operation only by three observations of the Suns shadow from a Point . It will not be amisse to give the Reader a small consideration hereof ; the point B of the pin AB , in all the figures is alwayes one part of the Axis , or Gnomon of the Dyal , and may be used to shew the hour : this point B , you must imagine to be the Center of the Earth ( for the vast distance to the Sun , maketh the space betwixt the Center and superficies of the Earth to be insensible ) and from it at all times of the year ( excepting the Aequinoctial day ) the Sun in its course forms two Cones , whose Apex is the point B , that next the Sun termed Conus luminosus or the light Cone , the other whereof our Author makes use , termed Conus umbrosus the dark Cone , now in this dark Cone , if by any three points equally distant from the Apex B , the Cone be cut , the Section will be a Circle parallel to the Equinoctial : And thereby , as the Author shews many wayes , the position of the Axis or Gnomon may be found out , and the Dyal easily made . Now it rests , courteous Countryman , that we be very gratefull , and every way forward to encourage Mr. D. King , one very industrious in the studies of Antiquities and Heraldry ; who out of his desire to serve his Country , hath caused this piece speak English , hath been very carefull to see the Cutts well done , and will ( no doubt ) proceed to cause some of those rare pieces of perspective in French to be translated . Then prosper King , untill thy worthy hand , The Gallick learning make us understand . JONAS MOORE Mathesios Professor . Books Printed for Isaac Pridmore , and are to be sold at the Golden Faulcon near the New-Exchange . THE Rogue , or the life of Gusman de Alpherache the witty Spaniard , written in Spanish by Matthew Aleman , Servant to his Catholick Majesty ; the fifth and last Edition Corrected . A Physical discourse , exhibiting the cure of Diseases by signatures , whereunto is annexed a Philosophical discourse , vindicating the Souls prerogative in discerning the truths of Christian Religion with the eye of reason , by R. Bunworth . Seif-Examination or Self-Preparation for the worthy receiving of the Lord ▪ Supper ; delivered in a Sermon concerning the Sacrament , by Daniel Cawdrey , sometimes Preacher there , with a short Chatechism : the third Edition . The Obstinate Lady , a Comedy written by Sir Aston Cockaine . Sportive Elegies written by Samuel Holland Gent. A New discovery of the French Disease , and running of the Reins , with plain and easie directions for the perfect curing the same , by R. Runworths . The Vnspotted high Court of Iustice , erected and discovered in three Sermons Preached in London and other places , by Thomas Baker , Rector of St. Mary the More in Oxon. A Chain of Golden poens , imbellished with wit , Mirth , and Eloquence , together with two most exelent Comedies , viz. the Obstinate Lady , and Trapolin suppos'd a Prince , by Sir Aston Cockaine . The Ascent to blisse by three steps , viz. Philosophy , History , and Theology , in a brief discourse of Mans felicity , with many rem●●keable examples of divers Kings and Princes . The Heroical Loves , or Anthcon & Fidelta a poem , by Thomas Bancroft . Advice to Balams Asse , or Momus Catechised , in Answer to a certain scurrulous and abusive scribler by , Iohn Heydon A●●hor of advice to a daughter , by T. P. Gen● . The Analysis of all the Epistles of the new Testament , wherein the chief things of every particular chapter are reduced to heads , for the help of the Memory , and many hard places explained , for the help of the understanding , by Iohn Dale Master of Arts , and fellow of Magdal 〈…〉 in Oxford . 1 I Figure , To all sorts of People . I come now to the first of those two things that you are to doe for to make one of those Dyals , which is the manner how to find the position , or the placing of the Axeltree . WHen you have a mind to find out the right placing of the Axeltree of one of those Dyals by this general way , mark first which way the light of the Sun comes to the place where you will make your Dyal , and which way it goes out again . Then make fast upon the place , as the figure above doth shew , with cement , plaister , mastick , or the like , a peg or pin , AB , by the great end A , putting the other small or sharp end B as far out of the superficies of that place as you can , In such sort that while the Sun doth shine upon that place , the shadow of the end of the pin B may fall always upon this superficies , and for the rest it is no matter how this pin or peg be framed , or placed , or turned , you are only to look to the small end thereof , that must be in such a manner , that you may set or apply upon it one of the feet of the Compass . Then in a fair Sunshiny day , when the light is very clear , and the shadow very clean , whilst it falls upon this superficies in the figure below , mark in it , as the figure shews , in one and the same day , at three several times as far asunder as you can , three several points CDF , each of them at the end of the shadow of this pin AB that answers to the small end of it B. You must nore that there is a certain time and place in which you cannot mark the points of the shadow ; That is when this superficies is flat , and situated after such a manner , that the ground plot thereof being stretch'd at length , answereth and reacheth into the center of the Sun . For in that case how short soever the peg or pin B may be , the shadow hereof cannot goe and fall in this superficies , but at the end of an extreme length . Therefore when the days are equal with the nights , or very near , you cannot mark in this manner three points of shadow in a flat superficies , which is situated in that manner called parallel to the Equator . When you have thus mark'd three points of shadow , you have no more need of the light of the Sun , and you may make an end of the rest in any other time and season , as well by night as by day , as I shall say three times together for one and the same manner , in three several ways to be expressed , after I have briefly satisfied the Theoriciens that take pleasure to see the reasons of the precepts , or rules of the practice of the Arts , before they see the precepts themselves . 1 To the Theoriciens . This Resolution will serve you . AFter you have conceived that the Sun in his full revolution of a natural day makes a Circle parallel to the Equator , and the rest of this Hypothesis , for Dyals . The three beams of the Sun or straight lines , BC , BD , BF , make in their point or common end B. some angles or corners equal one to another ; with an other straight one that makes the fourth , which is the Axeltree of the Dyall . Now the position or placing of these three straight lines BC , BD , BF , is given out ; Therefore the placing of this fourth which is the Axeltree of the Dyal is given also . You shall have hereafter in the fourth figure an other resolution of this kind , before you have the way to compose some problemes , or propositions about it . I said to the Theoriciens , because if you were not at all versed in any kind of practice , either of Geometry or Art , you might hardly understand me at first concerning the 2d . & third figures following , because of the short & compendious way whereby I expresse my self unto those that are skilled in Geometry : but I can assure you , that when you have understood what is written in order for all sorts of people , if you come again to these second and third figures , you shall know at the very first sight what they mean . For the Theoriciens , And for those that are skilled in Geometry . THe I figure is a plate of some thin , flat , smooth , and solid stuff ▪ as Iron tinned , or the like , being round , and having a hole just in the Center , greater or lesser , according to the occasion . The II figure is a straight rod , round , smooth , and solid , as of Iron , or the like , of the bigness of the hole in the plate . The III figure is as it were a whirl made of the plate , and of the rod put thorow the plate , in such sort that it is perpendicular to the said plate , as the squire that turns round about doth represent unto you , and is so fast that it cannot stir or move . In the V figure AB is the peg or pin that hath mark'd unto you the points of the shadow CDF , the rods or sticks BC , BD , BF , are solid and strong , as of wood , or the like , having each of them a slope edge in a direct line all along , going from the point of the peg B to each point of the shadow CDF , and are so turned or ordered , that in applying the whirl unto them , the edge of the plate may goe andtouch the three slope edges of the rods all at once ; and the rods or sticks are made fast in this situation , in such sort that they cannot move nor stir . The rule that crosseth over the three slope edges , BC , BD , BF , toucheth them all three , or else two at the time only , whereby it shews whether those slope edges are all three in one and the same situation , or upon one and the same ground or no , and on which side is their hollowness when there is any . The hand applies the whirl unto it , and keeps it there till the Axeltree BO● come to touch the end of the peg or pin B , and that at the same time the edge of the plate EDH touch the three slope edges of the rods . And when the whirl is placed or setled after this manner , the rod is the Axeltree of the Dyal , and placed as it ought to be , and there remains nothing else but to make it fast in this situation or position . The IV figure doth shew , that if you goe to make use of thin and supple strings in this practice or working , in pulling those two mark'd with Ie , and Ih , to make them fast in direct lines , they would make the two strings mark'd with bc , bf to bend , so that you can doe nothing exactly with them , which is the reason that Monsieur Desargues hath not thought fit to make use of them for the Beams of the Sun , but rather of the slope edges of the rods that are both stiff and strong . 2 3 To the Theoriciens , And others that are skill'd in Geometry . THis foregoing figure shews to the eye that all the pieces of the Instrument are made so strong and firm , that they cannot bend . AB is the pin , by whose point B , you have had the points of shadow C , D , F. The three sticks or rods BC , BD , BF , have each of them a slope edge in a direct line at length , going from the point of the pin B to the three points of shadow , C , D , F. The slope edges of the two longest sticks or rods , BC , BF , have some portions made in them , equal every one to the third and shortest stick BD. The three sticks IH ▪ ID , IE , are every one longer than BD , and all three made even , then they are joyned all by the end to one of the points EDH , of the slope edges of the other sticks , BC , BD , BF , and their other ends I , are brought together in one and the same point , I. The rod BI ▪ is straight , round , smooth and strong as of yron or the like , it hath a straight line BI , drawn from one end to an other , and one of the points B , of this line of the said rod toucheth the point of the pin ; And with an other point I of the same it toucheth the point I of the three rods or sticks . This being so , the rod BI comes to be the Axeltree of the Dyal rightly placed , there remains nothing else but to make it fast in this position or situation . The figure shews in the rods that goe from the point of the pin B , to the points of shadow CDE , how one may make fast those rods at one end to the pin , and also all together to one point , by binding them to it ; And how they may be made f●●t at the otherend to one point of the superficies of the Dyal , by fastning them to it with mastick , plaster , cement , or thelike . This way is more sure than that with the strings ; But yet it is not the easiest ▪ nor the least troublesome , in my judgement . To the Theoriciens , Another resolution of the same kind with the former . THe position or placing is given of the four points BC DF , and the placing of the two straight lines BE , BH , that divide in two the angles CBD , and DBF , and of the two ground plots that passe unto those two straight lines BE , and BF , and that are perpendicular to the ground plots of those Angles CBD and DBF , are given out ; Therefore the intersection or intercutting of these two ground plots so perpendicular is given . But this intercutting is the axeltree of the Dyal , therefore the position of the axeltree of the Dyal is given . Any one may frame at his pleasure upon that which is granted concerning this composition , many other resolutions , and divers compositions of problemes , and divers general ways of practice . In the mean time you shall have here three several ways one after another , to see which is the most advantagious for the actual practizing of the Art , and to induce you to seek or try if there is any other shorter . 4 For the Theoriciens . The Composition of the Probleme , or Proposition , in Consequence of the Resolution made upon the lowermost figure of the first draught THe first figure is the place of the Dyal , with the pin and the points of shadow , CDF . Make a ground plot of it , II upon one straight line BD , and with one point B , three Angles DBN , DBR , DBH , equal to the three Angles of the first figure , that are between the beams of the Sun , DBC , DBF , CBF , every one to his own respectively . From the Center B , II figure , and from any space BD , draw a half circle that may meet in the points , DNRH , the straight lines BD , BN , BR , BH . Make in the third figure a triangle DGV , with three spaces , equal to the three spaces DH , DR , DN , every one to his respectively , as having the condition necessary for that purpose . Find the Center O of the circle EVGD , drawn about this triangle VGD . Draw two Diametters DOE , POB , of this circle , perpendicular one unto another . Lengthen one POB sufficiently of one side and on the other . From one of the ends D , from the other EOD draw as far as that which is lengthened POB one straight line DB , even with the straight line DB of the II figure , for it must reach unto it , viz. In the Equinoctial at the point O , and in an other place at an other time , lengthen sufficiently III figure this straight line BD. Make in it the segments or cuttings even with the beams of the Sun of the I figure BD , BF , BC. Take in the III figure , in the straight line POB , conveniently a point I other than B. Make in the IIII figure three rods or sticks CI , DI , FI , each of them sharp at both ends , and equal with the three spaces CI , DI , FI , of the third figure . Draw a straight line along the Axeltree rod , mark in this line of the Axeltree conveniently figure IIII , one Cut BI equal with the space BI of the III figure . Set figure IIII one of the ends of the stick CI to the point of shadow , C , one of the ends of the stick DI to the point of shadow , D , and one end● of the stick FI to the point of shadow F. Let the ends of those sticks or rods be so well fastened to the points of shadow CDF , that they cannot stir . Bring together the other ends I , of those sticks in one point I. Put one of the point ▪ B of the Axeltree rod to the point of the pin B , and the other point I with the three ends of the sticks CI , DI , FI , set or joyned together . And if you have been very exact in the work , the point I of the pin will go and place it self with the three ends of the sticks set together in the point I , if not , you have not wrought exactly . 6 To the Theoriciens . It is no matter whether the Figures come right to the Compasses , you are only to take notice what this insuing Discours ordains you to do . MAke figure I with three straight lines CQRD , DIPE , and CF , a Triangle even and like unto the Triangle figure III , of the three points of shadow CDF , upon the straight line CQRD figure I. make a Triangle CBD , both like and equal with the Triangle figure III. of the Sun-beams CBD , and upon the straight line FPID figure I. make a Triangle FDB , like to the Triangle figure III. of the Sun-beams FDB , make longer if need be figure I. on the side of D the straight lines CQRD and FPID . By the points B and B draw a straight line BRAYH perpendicular to CQRD , and a straight line BIAKL perpendicular to the straight line FPID , find out the end or point A , common to these two straight lines BRAYH , BIAKL , and by this end A draw a straight line AE perpendicular to the straight line BRAYH , and a straight line AG , perpendicular to the straight line BIAKL , from the point R draw as far as the straight line AE a straight line RE even with RB , from the point I draw as far as the straight line AG a straight line IG even with IB. From the point E carry to the straight line BRAIH a straight line EH , perpendicular to the straight line RE , from the point G , carrry to the straight line BIAKL a straight line GL , perpendicular to the straight line IG , from the points B and B carry a straight line BQ that may divide in half the Angle CBD and a straight line BP that may divide in half the Angle DBF . By the points Q and H draw a straight line QOH , and by the points P and L draw a straight line POL , find the end or the point O common to the two straight lines QOH and POL , and from the point A for center and space AO draw an half circle that may meet with the straight lines AL in K and AH in Y. Now make in some other place even or flat , as in the second figure in one and the same line BDFC three cuts BC , BD , BF , even with the Sun beams , figure III. BC , BD , BF , each of them to his own from the point B of this second figure for center , and from the interval or space EY or GK , of the first figure , draw an half Circle O from the point C figure II for center , and from the space CO , of the first figure draw an other half Circle O from the point D of the II figure for center and space DO of the first figure , draw an other half Circle O , and from the point F also of the II figure for center and space FO of the I figure , draw an other half Circle O , and if you have done right , all these half Circles will meet in the same point O , if not , you have not been exact in working . By the points B and O draw a straight line BO , take in this line a point at discretion , first make three rods even with the spaces CI , DI , FI of the second figure , and every one sharp at both ends , make in the length of the axeltree rod figure III the space BI , even with the space BI of the II figure . Lastly set these rods to the axeltree figure III as I have said at the end of the fifth table , and the axeltree of the Dyal is placed . There are some situations of superficies of Dyals , where practising this manner of drawing one or the other of the points LH or O comes so far from the straight line CF , that you should have need of too great a space to come to it . But in what manner soever the superficies of the Dyal may be situated , and at all times or seasons of the year , I mean , in any strange or odd kind of example that may be found , you may work or practise these kinds of draughts with as much ease as in the most easy pattern . 5 And by means of these three angles even with those in the air between the beams of the Sun , you may chuse at pleasure within the lines that represent those beams , other points CDF and otherwise disposed between them , then those which the shadow of the point of the pin hath given upon the superficies of the Dyal , and upon those three points chosen out at pleasure , you may make an other triangle CDF , and practise afterwards this manner of drawing as far as the triangle CBO figure II than in this triangle ; and in the straight line BC , make BC , BD , BF , even with the beams in the air , BC , BD , BF , of the third figure , contained from the point of the pin B , to the points of shadow CDF in the superficies of the Dyal each of them to his own , and after you have taken , as it is said , the point I in the straight line BO , you must make use of the points CDF , last made in the triangle OCB , for to set the rods CI , DI , FI , to the axeltree BI then to work on as before . To make other points instead of those of the superficies of the Dyal , you need only to make some at the two extremities or furthest ends CF , and make BC , and BC , equal one to the other , and unequal with the middlemost BD but a little bigger , more or lesse according as the angles DBC , DBF , are more or lesse unequal among themselves , and instead of making figure I the triangle CDF of the spaces between the points of shadow CDF of the superficies of the Dyal , you shall make it of the spaces between the points that are set in the place of these points of shadow . 6 To the Theoriciens . MAke in one and the same plain , as in the first figure , vith three right lines CgkD , CrtF , DieF , a triangle , CDF equal and like to the triangle of the three points of shadow , fig. IV. CDF make upon the said three straight lines CgkD , CrtF , DieF , three other triangles CBD , CbF , DBF , equal and like to the triangles in the air of the beams of the Sun , III. fig. CBD , CBF , DBF every one to his own . By the points B and b I. fig. draw a straight line Bqg that may part in two the Angle DbF. Draw out of the point C at your discretion a straight line aqkty perpendicular to the straight line CgkD , and out of the point F , draw a straight line hPirx perpendicular to the straight line FeiD , make in the triangle Fcb the section or cutting Cl , equal with Ca , of the triangle CBD , and the section Fs , with Fh , of the triangle FbD , from the point t center , and space tl , draw a bow lm , from the point k , center and space ka , draw a bow am , that may meet with the bow lm , in m ▪ and draw along the straight line km , from the point r center and space rs , draw a bow sn , from the point i center and space i , h , draw a bow in ( hn ) that may meet with the bow ( sn ) in ( n ) draw along the straight line ( in ) : Make in the straight line ( km ) the section or cutting ( ku ) equal with kq. By the point ( u ) bring to the straight line aqkty , a straight line ( uy ) perpendicular to the straight line ( km ) ; make in the straight line ( in ) a section ( iz ) equal with iP ; by the point ( zx ) carry to the straight line ( hPirx ) a straight line ( zx ) perpendicular to the straight line ( in ) finde out the butt end ( y ) common to the two straight lines ( aqkty ) and uy . And also the butt end x common to the straight lines hPirx and zx , draw the straight lines goy , and eox ▪ find the butt end o common to these straight lines gov , eox . Make in an other place figure II. a Triangle gqy , of the three straight lines , as , gq , gy , and yu of the first 7 figure ; make in the II. fig , and in the straight lines gy and gq , the section ( go ) equal to go of the I. fig. And the section gb also equal to go of the I. figure ; draw if you will the straight line ( bo ) of the second figure . Make again in another place fig , 3. a Triangle ( cbo ) of the three straight lines ( bo ) of the Triangle ( gbo ) of the second figure . And of CB and CO of the first figure ; and upon bc fig. 3. make the cuts bc , bd , bf , equal to the lines BC , BD , bF , of the first figure , every one to his own respectively . And if you have done rightly , the spaces fo , do , co of the Triangle cbo , fig. 3. are equal with the spaces FO , DO , CO of the first figure , every one to his own respectively . Take fig. 3. in the straight line ( bo ) according to your discretion the point ( i ) other then ( b ) make three sticks sharp at both ends , and equal to the three spaces ci , di , si , of the third figure : mark along upon`the rod or Axeltree the space ( BI ) equal to the space ( bi ) of the third figure : work as I have said , and as the fourth figure doth shew you , and you shall find the Axeltree of the Dial placed in his right place . You may after this manner , as in others substitute , or bring in other points CDF in stead of those of shadow of the superficies , or face of the Dyal and work by this mean , every where with the like ease . Figure 8 , For those that have skill in Geometry . THe higher figure is the place of the Dyal with the face unequal to the pin AB , and to the three points of shadow CDF , all markt , as it is said . Get a flat and solid thing , as a slate , a board , paceboard , or the like . Draw upon it in the lower figure a straight line BDFC , make in that line three cuts BC , BF , BD , equal with the three spaces BC , BF , BD , of the place of the Dyal , each of them to his own respectively , then from the point B of the lower figure for the center , and from the spaces BC , BF , BD , draw some circles DH , FE , CG . By this means you see whether the spaces BC , BD , BF , of the higher figure or Dyal , are equal or unequal one unto another , and when these spaces are unequal among themselves , as it happens in this example , you see which is the least , and which is the biggest , as in this example , the space BD , comes to be the shortest of the three . Now from the point C of the lower figure for the center , and from the space between the two points of shadow C and F of the higher figure , draw a circle E , that may meet in one point E , the circle of the space BF , viz. the circle FE , for it must meet with it , then draw the straight line FB , that may go and meet in one point H , the circle of the shortest space BD , viz. the circle DH . Again from the point C of the lower figure for center , and space between the two points of shadow C and D of the higher figure , draw a circle N , that may go and meet in the point N , the circle of the shortest space BD , viz. the circle DH , for it must meet with it . From the point F in the lower figure for center , and from the space between the two points of shadow FD of the higher figure , draw a circle that may meet in the point R , the circle of the shortest space BD , viz. the circle DH , for it must meet with it ▪ By this means the three spaces or straight lines DH , DR and DN , of the circle DH , which is that of the shore●t space BD , have the conditions that are requisite for the making of a triangle . Figure 9 , For those that are skilled in Geometry . MAke in another place , as in the lower figure , a triangle DGV of three straight lines , equal with the three spaces , DH , DR , DN , of the higher figure , every one to its own . Find in the lower figure the center O of a circle , the edge whereof may reach to the points VDG , according as the lower figure doth declare . Draw a straight line DOE , through the Diameter or midd'st of this circle . By the point O in the lower figure , draw a straight line POQ , perpendicular to this Diameter DOE . From the point D in the lower figure , for the center and space BD of the higher figure , draw a circle that may meet as in B , the straight line QOP , for it must meet with it in one or two points , viz. In the times of the Equinoxe in one point only , which is the point O , and all the rest of the year , in two points divided on both sides from the point O. And that you may be exact in working , do as much on the other part , and of E for center . Draw in the lower figure the straight line BD , which in the times of the Equinoxe is joyned with the straight line OD , and all the rest of the year is divided from it , and draw along this straight line BD beyond the point D. Make in the straight line BD of the lower figure two Sections BC and BF , equal to the two Sections BC , & BF of the higher figure , each of them to his own respectively . Take in the straight line QOP of the lower figure of one side or other of the point B , at your discretion , a point I , besides the point B , and that may stand as far from the point B , as the occasion may give you leave . Figure 10 , To those that are skilled in Geometry . THen according as the lower figure shews you , cut three sticks CI , FI , DI , sharp at both ends , and equal to the three spaces CI , FI , DI , of the higher figure , each of them to its own space respectively , and upon the rod below , whereunto you mean to make the Axeltree of the Dyal , make a Section BI , equal with the space BI of the higher figure . 8 9 10 11 Figure 11 , For those that are skilled in Geometry . AFterwards as you see in the lower figure set to the points of shadow CDF upon the place of the Dyal , the three ends CDF of the sticks CI , DI , FI , each of them to his own point ; and the point B of the Axeltree rod to the point of the pin AB , then bring into one point alone in the air I , the three other ends I of these three sticks CI , DI , FI , and set them to the point I of the Axeltree rod . For the ends of these three sticks , and the point I of the Axeltree ought to meet all four together in one point in the air I , and then you shall find the Axeltree rod placed as it must be in the Dyal . So that you need no more but to make it fast afterwards in this position or placing , or else to place an other in an other place that may be a parallel to it . If the matter was only about that which is sufficient to shew Geometrically the truth of the proposition , it were sufficient to have either the three sticks only without the spaceBI of the Axeltree rod , or the spaceBI of the Axeltree rod with two sticks , without a third . But to make the operation sure and effective , you can not be confident that you have done rightly without a fourth stick that may serve for a proof ; This is that which Monsieur de Sargues had a mind to impart unto you . Figure 8 , The same thing over again , but in other Terms . To the workmen of many sorts of Arts . VVHen you have markt the three points of shadow CDF in the place where you mean to make one of these Dyals , draw with the rule in some even or flat place , as you see in the lower figure , a line BD , FC , and make in that line a prick or point B where you shall think fit , or at your discretion . Then go to the place of the Dyal in the higher figure , take with the Compasses the space from the point B , of the ●pin●●… B to the point of shadow C. And with that space come back to the line BD , FC of the lower figure , set one of the feet of the Compasses to the point B , and with the other foot go and mark in that same line BD an an other point or prick C , then with the same space , give a stroak with the Compasse CG about the point B. Go back again to the place of the Dyal above , take with your Compasses the space from the point B , of the pin AB to the point of shadow F , and with this space come back to the lower figure ; Set again one of the feet of the Compasse to the point B , and with the other foot go and mark in that same line BC an other prick or stay F , and draw again about the point B with the same space of the Compasse this half circle FE . 8 Then look in the lower figure which of the three stroaks CG , FE , DH , is nearest the point B , and which are the furthest off , as in this example you see that the stroak DH is nearer to the point B than any of the two others FE and CG , and if they were either two or three together it were no matter . When you know which of these stroaks of the lower figure CG FE and DH , is the nearest to the point B , and which are the farthest , as here , the stroak DH is the nearest , and the two CG , FE are the farthest off . Go to the higher figure to the points of shadow C , and F , which are even with the two stroaks below CG , and FE , which are the furthest from the point B , and open your Compasses upon the points of shadow C and F , and remember well the two letters or cotes upon which you have opened your Compasses , and with this space come back to the lower figure , and set one of the feet of the Compasses to the point C , and with the other foot go and mark a point E , upon the stroak of the Compasse FE , for it must reach to it . Then draw with the rule by the two points E and B , a line EB , that may go and make a point H upon the stroak DH , which is the nearest to the point B. Go back again to the higher figure , and open your Compasses upon the points of shadow C and D , and with this space come back to the lower figure ▪ set one foot of the Compasse upon the point C , and with the other foot go and mark a point N ▪ upon the stroak DH , which is the nearest to the point B , for it must reach to it . Go back again to the higher figure , and open your Compasses upon the point of shadow F and D , and with this space come again to the lower figure , and set one foot of the Compasse upon the point F , and with the other foot go and make a point R upon the stroak DH , which is the nearest to the point B , for it must reach to it . After that , you have no more to do upon the place of the Dyal , till you place the Axeltree as it ought to be , and you have in the lower figure upon the stroak DA , which is the nearest to the point B , four several points or stayes DNRH to make three point perdus with , as you shall see , in the mean time remember when you open the Compasse upon the points of shadow in the place of the Dyal , to take great notice upon what letters you have opened your Compasse , that you may apply the same space below upon the two stroaks which are equal with the two points of shadow upon which you have opened your Compasse , and set one foot upon one of the stroaks , and the other upon the other stroak . And moreover that the points NR , may well come out from betwixt the points D and H ; and that I have caused them to come in so betwixt them , by reason of the smallnesse of the place , and what way soever they come to be disposed , it is but one and the same thing still . Fig. 9 , To the workmen of many sorts of Arts . SEt your Compasse upon the points D , and H , of the higher figure , and with that space go to some flat or even place in the lower figure , and make two points D , and V , so that the space DV below , may be even with the space DH above . Then go to the figure above , and set your Compass upon the points D and R , and with this space come back to the figure below , and set one foot of the Compasse to the point V , and with the other foot draw a line from the point D , to the point G , so that the space VG below , may be even with the space DR above . Go back again to the figure above , and set your Compass to the points D & N , and with this space come back to the lower figure , & set one foot of the Compass to the point D , and with the other foot draw a line from the point V , to the point G , so that the space DG below , may be even with the space DN above , and may meet in G the other circular line that you have drawn about the point V , for it must meet with it . And so you have made in the lower figure three points VGD that will be perdus or lost . Now find a center O , upon which having set one of the feet of the Compass , and the other upon D , let this foot in turning the Compass about , go and passe by those three points perdus VGD , then draw with the rule by the points , as it were O and D , a line DOE , and setting again one foot of the Compass to the point O , and turning the other foot to E , make in the line DOE , the side OE , even with OD. Then by the point O , draw a line QOP , that may cut the line DOE , in two equal parts ; again set your compass to the points B , and D , of the figure above , and with this space go to the figure below , set one foot of the compass to the point D , & with the other foot draw from the point E , a line B , that may meet as it were in the point B , the line QOP , and make with this other foot of the compass a point B , in the line QOP , for it must meet with it , if you have done exactly . 9 9 When the dayes and the nights are equal , it meets with it in one point alone , viz. O , and at some other times it meets in two points , one of one side of the O , and the other on the other side , as in the point E. Then remove your Compass out of his place , and with the same space of the points B and D of the figure above , set one foot of the Compass to the point E , and with the other foot draw from the point D , with your Compass another line B , that may go and meet the line ▪ QOP , with the line that you have traced with the Compass about the point D , and both of them in one and the same point B , for it must do it if you have been exact : And that serves to mark more exactly this point B , in the line QOP , how neer soever it is to the Point O. After that , Whether the Point B of the lower figure meets with the Point O or not , draw with the rule by the points B and D , the line BD , and draw this line BD , as you see beyond the point D. That being done , open your Compass upon the points B and C , of the higher figure , and carry this space to the line BD of the lower figure , and from B into C. Set your Compass again upon the points B and F of the higher figure , and bring this space to the line BD of the lower figure , and from B into F. And finally make in the line QOP , a point I , at your discretion of one side or other of the point B , and let it be as far distant from the point B , as occasion will give you leave . And so you have in this lower figure from the point I to every one of the points BDFC , all the measures that are necessary for the placing of the axeltree or needle in your Dyal , in the manner hereafter following . Figure 10 , To the workmen of many sorts of Arts . CUt three rods or sticks sharp at both ends as you see below , one CI of the length that is betwixt the point C , and the point I of the figure● above ▪ the other FI of the length that is betwixt the point F , and the same point I of the higher figure , the other DI of the length that is betwixt the point D , to the same point I of the higher figure , then open your compasse upon the points B and I of the higher figure , and bring down this space upon the Axeltree rod , and make in the same as you see , two points B and I with this same space BI of the figure above . 10 11 Figure 11 , To the workmen of many sorts of Arts . GOe to the place of the Dyal below which I have expressed again , a purpose to avoid the confusion of lines , and put the end of the rod CI to the point of shadow C , the end F of the rod FI to the point of shadow F , and the end D of the rod DI to the point of shadow D , and set the point B of the Axeltree rod to the point B of the pin AB . Then bring together into one point in the Air I , the three other ends of the three rods CI , DI , FI , for they must come in there together , and bring the point I , of the Axeltree rod , to the same point I in the Air together with the three other ends of the rods I , for these four things must come alltogether into one and the same point in the Air I , if so be you have been exact in working . And when these three ends of the rods and the point I of the Axeltree rod , are all four gathered together into one and the same point in the Air I , the Axeltree rod will come to be placed directly as it must be in the Dyal , and so you need no more but to make it fast in that place , or to fasten an other either near it or farr from it , that may be even with it , or parallel to , or equally distant from it . If the four points I should go and meet in the body of the Dyal , you must but take in it's figure the point I nearer , or in the other side of the point B , and make an end of the rest as I have said . 8 Figure 8 , I will say the same thing over again , but more at large . To all sorts of People that have neither skill in Geometry nor in Arts ; but are apt and sit to learn them both ▪ BEfore you undertook to make this Dyal , you had nothing about you , nor knew nothing wherewith to further you in it , and going about it , you have made use of the pin AB , as it were at a venture . Now you must consider that having placed the pin AB , in this manner , you have given out of your self in the end thereof a point alone unmovable and fixed in the air . Then by means , of this fixed point in the air B , and of the Sun-beams , you have found out three other unmovable and fixed points of shadow CDF , on the outward face of the place where you have a mind to make your Dyal ▪ So you see that by means of this end of pin B , and of the Sun-beams , you have established upon the place where you intend to make your Dyal four points fixed and divided one from another , viz , one in the Air , which is the point or the end B of the pin AB , and three in the superficies of the Dyal which are the three points of shadow CDF . Whereby you have found also six spaces , that is to say , the lengths of six straight lines unmoveable , fixed , distinct , and divided one from the other . For if you consider well , you shall see that you have found out by this means the spaces , or lengths , or distances that are from the point B of the pin AB , to every one of the three points of shadow CDF , viz. the space from the point of the pin B to the point of shadow C , the space from the same point of the pin B to the point of shadow D , and the space again from the same point of the pin B to the point of shadow F. And for your better instruction , if you will make these three lines visible to the eye , set unto every one of them either a ruler or a string stretch'd out in a direct line from the point of the pin B , to every one of the points of shadow CDF , as the points do shew it unto you ; And so you may see the three lines BC , BD , BF , which otherwise are invisible in the Air . And besides these three spaces or lengths , you have also found out the three spaces or lengths that are from every one of the three points of shadow CDF , unto the other , viz. the space from the point of shadow C , to the point of shadow F , the space from the point of shadow C , to the point of shadow D , and the spaces from the point of shadow F to the point of shadow D , as you may see by the points that are there . So you have six spaces or lengths BC , BD , BF , CF , CD , DF , which you have already found unmovable , and fixed to the place wherein you intend to make your Dyal , which are so great a furtherance unto your work , that there remains nothing else to do , but by the help and means of the said six spaces or lengths , to find also three or four more , that you may have all that is requisite for the placing of the Axeltree rod of your Dyal as it ought to be . You must know that there are several wayes whereby these six spaces which you have found already , viz , BC , BD , BF , CF , CD , DF , are made use of to find out those three or four more , which you must have to inable you to place the Axeltree rod of your Dyal as it must be . And that of all those several wayes a man may have a liking to one for one reason , and another man to an other for some other reason , and of those several wayes Monsieur de Sargues hath shewed me three or four at the most , viz. that which he hath set down in the figure of his model or project page , and of the others for which you must know how to make sometimes somekind of alteration , and which I have set down in short , there is one in the sixth figure , and another in the seventh . As for this it is such , that there is no occasion but you may practise it in effectually , and with the like ease every where , without you need either to add or alter any thing , as you shall see presently . Draw with the rule , as you see in the figure below in some flat or even place a straight line BD , FC , then go to the figure above , and open your Compasse , and set one of the feet to the point B of the pin AB , and the other foot to the point of shadow C , and by that means you shall take with your Compasse the space or the lengths that are from the point of the pin B , to the point of shadow C , whereof you will be pleased to remember , to the end that when I shall bid you for brevity sake , take after the same manner with your Compasse such a space , you may be able to do with your Compasse even as I told you just now of the space BC in the figure above . Now with this space BC of the figure above , come back to the figure below , and set at your discretion one of the feet of the Compasse upon the straight line that you have drawn there , as for example set it to the point B , then turning the Compasse about upon this point B , draw with the other foot a circular line CG , which circle by this means shall have a space BC equal with the space BC of the higher figure , and will meet the line BD for example in the point C. Go back again to the figure above , take there after the same manner the space from the end or point of the pin B to the point of shadow D , and with this space come back to the figure below , and set again one of the feet of your Compasse to the point B , and holding it still upon this point B draw with the other foot a second circular line DH , that will be equal with the space above BD , and that may meet the line BC , for example in the point D. Go back again to the figure above , and take with your Compasse the space betwixt the point of the pin B and the point of shadow F , and with this space come back to the figure below set one of the feet of the Compasse to the point B , and draw with the other foot a third circular line FE , with the space BF of the figure above , and that may meet the line BD for example in the point F. By this means you have set away and transported the three spaces BC , BD , BF , from the rise or place which they had in the place of the Dyal above in a flat and even place below ; and all of them united together in one single line BDFC , in which you may see whether those spaces be equal amongst themselves , as they may be in some occasions which is indifferent , or whether they be unequal , by seeing whether the points CDF , are united together two or three in one single point , or whether they are disunited or divided one from another , and when these three points CDF , are disunited or divided one from the other which happens most commonly , and that these three spaces BC , BD , BF , are unequal amongst themselves , as it falls out in this example , you see which of these spaces are the greatest and which is the least , considering which of these three points CDF is the nearest , and which is the furthest from the point B , that is to say also that by this means you see which of these circular lines CG , EF , DH , is nearest to the point B , and which are furthest as in this example you see that of the three spaces BC , BD , BF , those two BC and BF , are the greatest , and BD is the least ; and of the three half circles CG , FE , DH , you see that DH is the nearest to the point B , and that the circle FB is nearer to it than the circle CG . When you have thus found out which of the three spaces BC , BD , BF is the least , and which of the three circles CG , FE DH is the nearest of the point B. Go back again to the higher figure to the points of shadow CDF , and take with the compasse the space betwixt the points of shadow C and F , which are at the ends of the two greatest spaces BC and BF , and with this space CF of the higher figure , go to the lower figure to the same point CF , and set one foot of the compasse upon either of those points C and F , that is the furthest from the point B , viz. C , and holding still the point of the compasse upon this point C , go and mark with the other foot a point for example E , in the circle of the other of these two points C and F , viz. in the circle of the point 〈◊〉 which is the circle FE , for this other foot of the compasse must reach to this circle of the point F , as for example in the point E ; This being done ▪ draw by this point E to the point B a straight line EB , that may go and meet , in one point H the circle of the point D which is the nearest to the point B , and mark in it this point H. Then go to the figure above , and take with the compasse the space betwixt the two points of shadow C and D , and with this space CD of the figure above go to the figure below , to the same points C and D set one foot of the compasse to that point of these two C and D , which is the furthest from the point B , viz. C , and keeping this foot of the compasse upon this point C , go and mark with the other a point , for example N in the circle of the other of the two points C and D , viz. in the circle of the point ▪ D with the circle DH , for this other foot of the compasse must reach to the circle of the point D , for example in the point N. Go back to the figure above , take with your compasses the space betwixt the two points of shadow F and D , and with this space FD of the figure above , come back to the figure below to the like points F and D set one foot of the compasse upon that point of these two F and D , which is furthest from the point B , viz. F and holding still the point of the compasse upon this point F , go and mark with the other foot a point for example R , in the circle of the other of these two points F and D to wit , in the circle of the point D which is the circle DH , for this other foot of the compasse must reach to the circle of the point D , for example in the point R. This being done , you have no more to do in the place of the Dyal , till you go and place the Axeltree rod in it as it must be , and in the figure below in the circle of the point D , which is nearest to the point B , you have found by this means four points DN , RH , different and divided one from the other , and when the two points N and R should be found united together , it were no matter . Now by the means of these four points you have three spaces amongst others from the point D , to every one of the three points HR and N to wit , the space from D to H , the space from D to R , and the space from D to N , of which spaces you see which is the greatest and which is the least , when they are all three unequal , as in this example , for it may happen that there will be two found equal amongst them . And with the help of these spaces DH , DR , DN , you shall find presently those four that you must have for the placing of the Axeltree rod of your Dyal , as it ought to be . Figure 9 , To all sorts of People ▪ TAke with the Compasse in the figure above , of these three spaces DH , DR DN , that same that is the greatest of all , as in this example here , the space DH and with this space DH of the higher figure , go to some place that is flat , as in the figure below , and set at the same time the two feet of the Compasse upon it , as for example , to the two points D and V ▪ and mark these two points as it were D and V , which by this means will be distant one from the other , the length of the space DH of the figure above . Go back again to the figure above , take there with the Compasse the space from D to R , and with this space go to the figure below , set one of the feet of the Compasse to the point V , and from thence draw with the other foot towards the point D a circle G , which by this means shall be made of the space DR of the figure above . Go back again to the figure above , take with the Compasse the space from D to H , and with this space go to the figure below , set one of the feet of the Compasse to the point D , and from thence with the other foot draw towards the point V another circular line that may meet , for example , in G the other circle , which you have drawn about the point V , for this other foot of the Compasse must meet the other circle in one or two points , and for example , in the point G for one . 9 Figure 9 , To the workmen of many sorts of Arts . SEt your compasse upon the points D and H of the higher figure , and with that space go to some flat or even place in the lower figure , and make two points D and V , so that the space DV below , maybe even with the space DH above . Then go to the figure above , and set your compasse upon the points D and R , and with this space come back to the figure below , and set one foot of the compasse to the point V , and with the other foot , draw a line from the point D to the point G , so that the space VG below , may be even with the space DR above . Go back again to the figure above , and set your compasse to the points D and N , and with this space come back to the lower figure , and set one foot of the compasse to the point D , and with the other foot draw a line from the point V to the point G , so that the space DG below , may be even with the space DN above , and may meet in G the other circular line that you have drawn about the point V , for it must meet with it . And so you have made in the lower figure three points VGD , that will be perdus or lost . Now find a Center O , upon which having set one of the feet of the compasse , and the other upon D , let this foot in turning the compasse about go and passe by those three points perdus VGD , then draw with the rule by the points , as it were , O and D , a line DOE , and setting again one foot of the compasse to the point O , and turning the other foot to E , make in the line DOE , the side OE , even with OD. They by the point O , draw a line QOP , that may cut the line DOE in two equal , parts . Again , set your compasse to the points B and D of the figure above , and with this space go to the figure below , set one foot of the compasse to the point D , and with the other foot draw from the point E a line B , that may meet as it were in the point B the line QOP , and make with this other foot of the compasse a point B in the line QOP , for it must meet with it , if you have done exactly . When the dayes and the nightes are equal , it meets with it in one point alone , viz. O , and at some other times it meets in two points , one of one side of the O , and the other of the other side , as in the point E , then remove your compasse out of his place , and with the same space of the points B and D , of the figure above set one foot of the compasse to the point E , and with the other foot draw from the point D with your compasse another line B , that may go and meet the line QOP , with the line that you have traced with the compasse about the point D , and both of them in one and the same point B , for it must do it if you have been exact , And that serves to mark more exactly this point B in the line QOP , how near soever it is to the point O. After that , whether the point B of the lower figure meets with the point O or not , draw with the rule by the points B and D the line BD , and draw this line BD , is you see , beyond the point D. That being done , open your compasse upon the points B and C of the higher figure , and carry this space to the line BD of the lower figure , and from B into C. Set your compass again upon the points B and F of the higher figure , and bring this space to the line BD of the lower figure , and from B into F. And finally make in the line QOP a point I , at your discretion , of one side or other of the point B , and let it be as far distant from the point B , as occasion will give you leave ; And so you have in this lower figure from the point I , to every one of the points BD , FC , all the measures that are necessary for the placing of the axeltree or needle in the Dyal , in the manner hereafter following . Figure 9 , To all sorts of People . THat being done open your compasse at your discretion , and the more that the occasion will permit you to open it , it will be so much the better , and with this opening , set one of the feet of the compasse to the point G of the figure above , then turning about this point of the compass upon this point G , draw with the other foot four circles H , L , M , S , about the point G. viz. two H , and L from the point D , and two M and S from the point V , then remove your compass , and set one of the feet upon the point D , and with the other foot draw from the point G , two circles that may meet in two points , viz. Land G , those two circles that you have drawn about the point G , from the point D , if this same foot of the compass could not meet with these two circles HL , that you have drawn about the point G & from the point D , it is because you had not opened your compass enough , before you did set it upon the point G , and in such a case you shall open it more , and set it again upon this point G , and when this same foot meets these two first circles , for example , in H and in L , matk these two points L , and H. Afterwards remove the compass and keeping still the same open , set one foot to the point V , and turning the compass upon the point V , draw with the other foot from the point G , two circles that may meet likewise in two points as S , and M , the two circles that you have drawn about the point G , from the point V , and mark those two points S and M , in which those two circles meet with the other two : And therefore note that before you make yout compass to turn upon the point G ▪ you must open it in such a manner , that when you shall set it afterwards upon the points D and V , the other foot may meet with the circles that you have drawn about the point G. 9 Then draw by the two points H and L , a long straight line H , L , O , and by the two points S and M , draw an other long line and as straight , M , S , O , and these two lines HL and MS , being sufficiently drawn at length , will meet in one and the same point O. By the two points D and O , draw a straight line DO , and draw it at length , as you see from the point O , then set one of the feet of the compass upon the point O , and the other foot upon the point D , and turn that foot of the compass which is upon the point O. The other foot which is upon the point D must go and touch every one of the points G and V , and when this other foot hath gone over the point D , G , V , go and mark at the very same time with it a point , as E in the line DOE , and so you shall make the portion OE of the line OD , equal with the portion OD. Then open your compass at discretion more than from the space OD , and as much as the occasion will give you leave , the more the better , and the compass being so open at discretion , set one of the feet to the point D of the lower figure , and turning that foot of the compass upon the point D , draw with the other foot from the point O , two circles , as P and Q then remove the compass , keeping still the same distance , set one foot upon the point E , and turning it about , draw with the other foot from the point O , two other circles , that may meet in two points with the two circles that you have drawn about the point D , and as for example in the two points Q and P , and draw with the rule by these two points , as Q and P , a long straight line QP , that must reach to the point O , if you have been very exact in the working ; if it doth not reach to it you have not been very exact , and I advise you to begin it again : If it reaches to it , go back to the figure below , and take with the compass the distance between B and D , then with this space go to the figure below , set one foot of the compass upon the point D , and turning it about , draw with the other foot from the point O , a circle that may meet the line QOP , as for example in the point B , for this other foot of the compass must go and meet that straight line POQ either in one or in two points , because the space from B to D of the higher figure ought never to be smaller or lesser than the space DO of the figure above . It is true that twice in the year , viz. in Autumne and in the Spring , when the dayes and nights are equal , that space BD of the figure above comes to be equal with the space DO , of the figure below , and in those times that other foot of the Compasse that tutns about the point D of the figure below , meets the line QOP just in the point O. But at all other times the space BD of the figure above is somewhat bigger than the space DO of the lower figure ; And then the other foot of the Compasse that turns about the point D , meets the line QOP , in two points , one of each side of the point O , as for example in B , for one . And that you may be the more exact , remove the Compasse from one part of the straight line BO unto the other , and with the same opening of the space BD of the higher figure , set one of the feet upon the point E of the figure below , and turning this foot upon this point E , draw with the other and from the point D an other circle that will meet ( if you have been exact in the working ) the straight line QOP , and the circle also that you have drawn about the point D , and both in one point ; as for example in the point B , which will inable you to discern well the point B in the straight line POQ ; mark this point B in the line POQ , whether you find it united with the point O , and so both of them making but one and the same point , as it falls out , when the days and nights are equal , or whether you find it divided from the point O , as it falls out in other seasons , and as you see in this example ▪ Then draw with the rule by these two points B and D a straight line BD , which you shall stretch out sufficiently beyond the point D. When the days and nights are equal , as in Autumne and in the Spring , and that the point B is found to be united with the point O , the line BD comes likewise to be united with the line OD , and both together make but one line ; But at any other time , as the two points B and O are two several points and divided one from the other , so the two lines BD and OD , are two several lines ▪ and divided one from the other , This being done , go to the figure above , mea●ure with your Compasse the space from B to C , and with this space go to the figure below , set one foot of the Compasse upon the line BD , to the point B , and set the other foot in any place of the same line BD where it may light upon ; as for example in the point C , by this means you shall make the portion BC , of the line BD of the lower figure , equal with the portion BC , of the line BD of the higher figure , make after the same manner with the Compasse , the portion BF , of the line BD of the lower figure , equal to the Portion BF , of the line BD of the figure above . Finally , in the same figure below , and in the line QOP , mark at your discretion another point , I , of one side or other of the point B , according as you shall find it most convenient for the place of the Dyal , and as far from this point B , as occasion will permit , the further the better , and so you have found the four spaces that you wanted for the perfect placing of the Axeltree of your Dyal . For in so doing , you have found in this figure below the distances that are from every one of the four points BDF C , to one and the same point I , that is to say the space from B to I , the space from D to I , the space from F to I , and the space from C to I , which distances BI , DI , FI , and CI , will serve you to place the Axeltree of the Dyal in the manner following . Figure 10 , To all sorts of People . CUt ( as you see in the lower figure , ) three sticks sharp it both ends , one CI of the length of the point C , to the point I , otherwise of the space CI of the figure above : The other FI , of the length of the space FI of the higher figure , and take with your Compasses the space BI of the figure above , and being so open , see both feet at once upon a straight line , along the Axeltree rod of the lower figure , for example , in two points as B and I , and mark these two points B , I , in the Axeltree rod . 10 11 Figure 11 , To all sorts of People . THat being done , go to the place of the Dyal , the which , to avoid the confusion or multiplicity of lines , I have set below in the lower figure , set in this lower figure one of the ends of the stick CI , to the point of shadow C , one of the ends F , of the stick FI , to the point of shadow F , and one of the ends D , of the stick DI to the point of shadow D , and one of the points B of the Axeltree rod , set it to the point B , of the pin AB . And holding thus the three ends CDF , of the three sticks to the points of shadow CDF , every one respectively to his own , and the point B of the Axeltree rod , to the point of the pin B , bring together the three other ends I , of the three sticks or rods CI , DI , FI , into one point in the air I , for they must meet there ▪ then bring the point I of the Axeltree rod , also to the point in the air I , with the three ends I of the sticks , for it must come and meet there exactly , if you have done right , or if the straightnesse of the place hath not hindered you . If the straightnesse of the place of the Dyal hinders the three ends I , of the sticks from meeting together in one point in the air I , take the point I in the figure below in the ninth cut , or that above in the tenth figure in an other place , than in that where you had taken it and according to the occasion , then bring the sticks to it as before ( for you may take it anywhere , or in any place of the line POQ , of one side or other of the point B ; ) but the further you can take it from the point B , will be better , and take it in so many-places , that having set the sticks of the points CDF , to this point I , and mark'ed the space BI upon the Axeltree rod , the four points I , may at last meet together in one point in the air I. And when the point B of the Axeltree rod , is at the point B , of the pin AB , and when the three ends I , of the sticks , and the point I of the Axeltree rod are met , as you see in the lower figure , all four together in one and the same point in the air I. The Axeltree rod will come then to be placed , just as it ought to be in the Dyal . That if you do not care to be sure that your Dyal must be as just , as it is possible for art to do , in such a case , you may spare one of the four lengths CI , DI , FI , BI , and content your self with three only , as being sufficient for the Theorie : But the fourth will serve you for a proof , to see whether or no you have been very exact in working , and will justifie the three others . Figure 12 , To all sorts of People . THe figure above shews you how that which you have done with three sticks , may be done either with many Compasses , with the help of some body , or else with other kinds of branches tyed or fastened one with another . The same figure above , as also the figures below , shew how every one of those branches may be of two several pieces , which go in by couples into one hoop or ring , and slide along one by another , and are made fast with a screw to the measure where you will have them to stand upon , and these pieces may be made of tinn'd yron , or of yron , if you are afraid that their points will grow dull by often using them . Or otherwise they shew you that insteed of one stick , you may have two , both sharp at one end , which you shall fasten and bind together at the other end , of what length or measure you please . The same figures do shew you also , that two divers branches , viz. CI , and FI , may be fastened together in the place where you will have them to stand together , with a presse and a screw to fasten them with . The higher figure shews you besides , that you may ●●●●en or bind with strings or threds , the Axeltree rod with the point B , of the pin AB , and the two branches CI , FI , with the Axeltree rod , to make them stand fast of themselves in their place . When you have found thus the placing of the Axeltree rod , it is in your choice , either to seal it and fasten it in that place , or to place another insteed of it , that may go the same way , and that may be every way equally distant from it ; But that you may be the more exact , it will be as good to seal or fasten that in the place , where the practice of the Draught hath caused it to meet , than to place another , unlesse there was some occasion or necessity for it . 12 Figure 3 , To those that have understood what hath been said before . HAving understood what I have said before , concerning those many wayes of finding the position of the axeltree of the Dyal , you may compose others besides , making use partly of that of one figure , and partly of that of an other . For example , here is one way composed of two of those that are afore . OVt of the third or fifth figure , you shall take in the Sun-beams or sticks BC , BD , BF , three spaces equal each unto the other ; And out of the 5 and 6 figures , you shall make a triangle of three lines equal to the three spaces HE , DE , DH of the third figure , and you shall find the center O of the circle , circumscribed about this triangle . You shall find also within the ground plot of the points HDE , the points like to A & O of the 6 figure or cut , which in this case come to be united together in one and the same point O. That is to say , having found one of these two points A & O , you have found also the other , because they are united or gathered together into one . So you have in the second figure of the third cut , the spaces DO , and DI , for two sides of a triangle with straight angles or corners ODI , whose side DI , holds up the straight angle , and the sides DO , and DI , do contain or comprehend it . Make this triangle ODI , with three sticks , or with any other thing that may be strong and small as you will , so that you may at your need lengthen the side IO , from the right angle O. 3 Set the point D , of this triangle DIO , to the point of shadow D , and holding this point of triangle to this point of shadow D , make the side IO , of this triangle ( drawn at length if need be ) come and touch the point B , of the pin AB for if you have been very exact in working , it must touch it . Take a stick HI , of the length of DI , set one of the ends to the point H , and bring the other end to the point I , of the triangle ODI , without the side IO , leave the end B of the pin AB , for it must be so , if you have wrought exactly as you ought . You may have also an other stick EI , of the length DI , and set one end to the point E , and bring the other likewise to the point I , of the triangle DIO , without the side OI , leave the end B of the pin AB . That being done the rod BI , comes to be the axeltree of the Dyal , and placed as it ought to be , and so of all the other wayes that you find besides . You may , if you will make use of a triangle rectangular EOI. and of the stick HI , content your self only with the three equal lengths EI , DI , HI , to find out the point thereby , that you may draw from thence a line to the point B , without making use of any thing else to know if you have done exactly or no , you can not be sure whether you have done well or ill . But when you have together with that , either a fourth length BI ▪ or the straight angle DOI , that will serve you to try whether or no you have been exact in your operations , for as concerning an effectual execution , unlesse you have from time to time such a kind of proof , to shew whether you have wrought exactly as you ought , you cannot assure your self that your work is as well as it can be done . One thing I must tell you , that in some certain occasions according to the times and the placing , or according as the superficies of your Dyal is , the shadow of the pin comes to be of such a length , and the extremity or end thereof so weakned , and so diminished in strength , and so confuse in the superficies of the Dyal , that it is very hard to find out Figure 13 , To all sorts of People . I come now to the next and second thing that you are to do , which is to trace out the lines of the hours . IN this example I suppose that the Axeltree rod doth not meet the superficies of the Dyal , about the place that you work in ; and therefore I represent it suspended in the air , with two or three supporters as you see , I suppose also that the superficies of the Dyal is not smooth , but rough and uneven as I have said . When you have placed the Rod BI , which is the Axeltree of the Dyal , as you see both in the higher and lower figure , you have made an end then of the first of those two things that you were to do , for the making of your Dyal : Now there remains but the second to be done , which is the finding and the tracing out of the lines of the hours in the Dyal ; and for that purpose . Consider in your higher figure , that the superficies and the axeltree of your Dyal are two divers things , and differing one from an other , and there is no such communication from the one to the other , as that with them alone you may find out directly the place of the lines of the hours , without making use of a third thing that may be a means betwixt those two . The meanest and the least thing that you can have to be a means betwixt the superficies and the axeltree of the Dyal , is a Ruler . 13 To the end that this middle rule may serve you alike in all occasions , it must have all the conditions that you see represented in the figure below . First it must be as long as the place will give you leave , and it must crosse over if need be the whole superficies of the Dyal , and reach over on both sides if it be possible . Secondly it must be in the air , and suspended between the superficies and the axeltree of the Dyal . Thirdly it must be placed as far from the axeltree rod , as possible may be . Fourthly it must be placed like a crosse , in regard of the same axeltree rod . Figure 14 , To all sorts of People . TO place this middle rule well , and as it ought to be betwixt the superficies and the axeltree of the Dyal . Chuse along the axeltree rod BI , of the higher figure , some fit or convenient place , as in the point O , and make a round and fixed stay in that place , by winding , or tying some strong thing about this axeltree rod , as the figure doth shew . Tye a string to the axeltree rod BI , by the means of a ring that may be so big , that you may turn the string with it about the axeltree rod easily , as the lower figure shews you ; Then with the corner of a squire ED , in the lower figure , thrust on the ring where the string is , and put it close to this stay O , and holding the string fast between the stay O , and the squire ED , set the back of one of the sides OE of this squires length , to the axeltree rod BI , and by this means , the other side DO , of this squire , will shoot out into the air like a wing from the axeltree rod BI , then stretch out the string in a straight line from the stay O of the axeltree , along the back of the other side OD of the squire . And holding still in this manner the ring close to the stay of the axeltree , by means of the squire , and the back of one of the sides joyned at length to the axeltree rod , and the other side of the squire like a wing , and the string stretcht out in a straight line along this wing , turn both the squire and the string altogether still in this same manner about the axeltree rod , as the lower figure doth shew . 14 When you have found out those two places that are farthest one from another , in which this string turning in this manner with the squire along the side like a wing , may go and meet the superficies of the Dyal , as here the places G and H. Make with mastick , or plaster , or cement , or such like stuff , a little knob flat at the top in each of these places , as for example , one in G , and another as in H , which two knobs may shoot out of the superficies of the Dyal , in such sort , that you may lay a Ruler on the top of them , going from one of the knobs to the other , as you see here in the lower figure . 15 Figure 15 , To all sorts of People . VVHen you have thus made those two knobbs G , H , in the lower figure , take the squire again and the string , and set them again close to the stay O , of the axeltree rod , as you know they were . And make them go about again as before , both together about the axeltree BI , and while you are turning about , the string will fall right over against the two knobs , shorten or lengthen it so , that it may go and touch a point , at the top of each one of the two knobs , one after another ▪ viz. a point as P , at the top of the knob G , and a point as Q , at the top of the knob H , and mark these two points Q and P , upon these two knobs . When you have mark'd two points in this manner , set a Ruler in the lower figure upon these two knobs , and place it so , that it may passe from one to the other , by those two points Q and P , and make the Ruler fast in this place with cement or plaster , or the like , in such wise that it may not stir any way . And this rule so placed , is the third and middle piece between the superficies , and the axeltree of the Dyal , by means whereof you shall cause , as I shall say hereafter , this superficies and this axeltree to have what communication soever you please one with another . After you have placed this middle rule in this manner , between the superficies and the axeltree of the Dyal . Consider that in France now they reckon 34 hours , for one day and a night , and that these 24 hours , are divided in twice twelve hours , and that every one of these 12 hours is subdivided in twice 6 hours . So that in the 24 hours of one day and one night , as they are now reckoned in France , there are two hours that are each of them of 12 , that is to say , one hour of 12 in the middest of the night , and another hour of twelve in the middest of the day , these two hours of 12 , are called midnight and midday , then there are two other hours , each of them of 6. viz. an hour of 6 in the evening , and another of 6 in the morning . Where you must note , that both the two hours of 12 , and the two hours of 6 , come alwayes to meet together in one and the same line , though it may be lengthened if need be , viz. the two of twelve in one line , and the two of 6 in an other . And you shall know , that it is an infallible thing , that within the Compasse of the superficies of the Dyal where you work in , if you have placed the axeltree pretty near it , there must needs be either one of the hours of 12 , or one of the hours of 6 , and sometimes they meet there both at once , that is to say , one of the hours of 12 , and one of the hours of 6. There be many situations of superficies of Dyals , in which , within the Compasse , where one may trace in the hours , there is only the line of the hours of 12 , and there is found not any one of the hours of 6 , and others in which there is found only the line of the hours of 6 , and not any one of the hours of 12. But there is no Dyal in which , within the Compasse where it is traced in , but the hours of 12 , and any one of the hours of 6 are found in it , I mean that one may find in it , either one or other of the hours of 12 , and of the hours of 6 , by setting the axeltree near enough to the superficies of the Dyal . And now since you are sure , that there is without doubt in your Dyal , either one of the hours of 12 , or one of 6. You shall begin to seek in it , first the place of that sort of hours of twelve or of six , that may be in it . And when you have found the point , either of one of the hours of six , or of one of twelve ; You shall find afterwards the points of the other hours , that meet with it in the Dyal it is in your choice to begin to seek the point , of which of the two sorts of hours of 6 , or of 12 , you will , and I will show you two wayes of seeking them out , both one after another , that when they come to be both in the Dyal , you may find them out both there if you will , for they serve for a proof one unto another , if you have been exact in your operation . That you may finish your Dyal as you ought , seek in the middle rule the point which is there to be found , either of the hours of twelve , or of the hours of six , for it is set there on purpose to serve for that chiefly . For example , seek out first in it the point of one of the hours of six , as I am going to shew , then I will shew you the way to find out the point of one of the hours of 12 , and afterwards I will shew you the way to find out the points of all the other hours of the Day . 16 Figure 16 , To all sorts of People . TO seek out in the middle rule , whether the point of one of the hours of 6 be there . Take the string that is made fast to the axeltree , set it very close again to the point or stay O , as it was when you made it turn about the axeltree , then stretch it out in a straight line from the point O of the stay of the axeltree , to the middle rule PQ , and holding still this string so stretcht out , make it turn about the point O , carrying it from one to the other of the points Q and P ▪ along the rule QP , and making it longer or shorter if need be , and set a Carpenters level or triangle over it , to see , while it turns thus , stretch't out in a straight line about this point O , keeping along the middle rule , whether there is any place , wherein it comes to be found levell , as for example , you see in the higher figure ( over the leaf ) and when you find it to be level , you must make it fast there . And that your level may be more fast , you may set it by the middle upon the axeltree rod close to the stay , or you may set close to the stay a ruler notch't at one end , as you see the ruler N , is notcht , then guide it with the string , And it will serve to fasten the level upon it . Or to say it again otherwise , you must , as you know , cause to go about the axeltree rod the squire and the string stretcht out , as I have said , in a straight line , and made longer if need be , this string will go and passe all along the ruler PQ . And if it happens that the string , so guided or carryed along the middle rule PQ , come to be found level , as the higher figure shews , mark in this middle rule the point 6 , where this string toucheth or reacheth unto when it is so level , and remember that this point 6 , is the point of one of the hours of 6 , either of the evening , or of the morning . This is that which concerns one point of one of the hours of 6 , that if the string in turning thus , comes to passe from one end of the middle ●●…e to the other , without falling to be level it is a sign , that not one of the hours of 6 , comes to be found in this Dyal , to frame it from the point that you have taken for a rest or stay . Now that you may seek out the point of the hours of 12 , in the lower figure . Fasten the center of a hanging plummet , with a string S , to the middle of the body of the axeltree rod , above it or under , as the lower figure shews , it matters not , and as the occasion will permit or require , and set this plummet so , that it may come and fall as near the middle rule as you can . Then tye to the axeltree rod , as far as you can from the superficies of the Dyal , another string I with a loose knot , and when the plummet of the first string S , comes to be at rest , make it so fast that it may not stir , then stretch out this second string I in a straight line , in such sort , that comming from the axeltree rod , it may go and touch the string of the hanging plummet S , without breaking ( or stirring ) the string nor the lead , and so holding this second string I , stretcht out close to the string with the plummet on S , see whether this second string stretcht out in this manner can , being made shorter or longer , if need be , go and meet the middle rule , in a point , or not . And when this second string so stretch'd out comes to meet the middle rule , in one point , as 12 , mark in the middle rule this point 12 , in which this string so stretcht out doth meet with it , and note that that point 12 , is the point of one of the hours of 12. When you have found out and markt in the middle rule the point of one or other of the hours , either of 6 , as you see in the figure above , or of 12 , as you see in the figure below , if you have them both , they shall serve for a proof one to another , if you have but one , you may make use of that alone . Let us suppose first , that it is the point of one of the hours of 6 , as the point , 6 ▪ 〈◊〉 you shall go on in finding out the points of the other hours , which may be found in your Dyal , in this manner following . Figure 17 , To all sorts of People . MArk at your discretion in the rule PQ . two several points MN , and consider the point in the middle of the body of the axeltree , close by the stay O , that is the Point about which you have turned the string with the corner of the squire . You see there three several points unmoveable , and fixed , viz. the point M , and the point N , in the middle rule , and the point O , in the middle of the body of the axeltree rod , close to the stay . And so having those three points fixed , M , N , O , you have by this means the three several distances , viz. the measures of the distances that are from one of these three points , to the two others , viz. the space or distance from the point M , to the point N , the distance from the point M to the point O and the space from the point N , to the point O. Remember two things , one is , that the point O , is in the middle of the body , that is to say of the bignesse , and not in the out side of the axeltree rod . The other is , that these two points MN , that you have mark'd at discretion in the middle rule , are not for all that certainly the points of hour , and that they are to serve you to find out the points of hour , and perhaps they may chance to be some of them ; and may be not , and perhaps you must blot them out after you have found out the points of hour . This being done so , take with your Compasse upon the middle rule , the distance from the point M , to the point N , and with this space go to some place that is flat or smooth , and set both the feet of your Compasse therein at once , as in the figure below in the points M and N , and by these two points , draw a straight line MN , as long at either end as the rule PQ . Then go back to the Dyal above , take with the Compasse the distance which is from the posnt M , to the middle of the bigness of the axeltree close by the stay O , or else otherwise , take the distance which is from the point M , to the axeltree towards the stay O , and adde unto it half of the bigness of the Axeltree ; and with this space MO , come back to the figure below , set one of the feet of the Compasse to the point M , and turning this foot about upon this point M , trace with the other foot a line crooked like a bow O , go back to the figure above , take again with your Compass the distance which is from the point N , to the middle of the bigness of the axeltree close by the stay O , and with this space come back to the lower figure , set one of the feet of the Compass to the point N , and turning this foot upon this point N , trace with the other foot another crooked line that may meet with the other in one point , as O , for it must meet with it . Then open yout Compass at discretion , rather more than lesse , and set one of the feet of the Compass so open at discretion to the point O , and turning this foot of the Compass upon this point O , trace with the other foot a round RGSH . Go back to the Dyal in the figure above , take with your Compass upon the rule QP , the distance which is from one of the points M or N , to the point of 6 hours , and with this space , for example of M6 , come back to the lower figure , set one of the points or feet of the Compass upon this point M , go and mark with the other foot in the line M , a point as 6 , of the same side upon the rule . And so you have in the line MN , one and the same thing as you have in the Dyal in the middle rule , viz. the three points MN and 6 , of the same distance , in each of these two straight lines . This being done , draw in the figure below by the two points O and 6 , a straight line O , 6 , which may divide the round RGSH , in two halfs RGS , and RHS. Open the Compass at your discretion , and as much as the space will give you leave , and keeping your Compass so open at discretion , set one of the feet to the point S , and turning this foot upon this point S , trace with the other foot , two crooked lines L and D , then with the same space , remove your Compass out of his place , and set one of the feet to the point R , and turning this foot about upon this point R , trace with the other foot two other crooked lines , that may meet in two points L and D , the two crooked lines that you have drawn about the point S , and mark those two points L and D , and draw by those two points a straight line LD , which may passe by the point O , if you have been exact in your operations . So you have divided this round , into four quarters of a round , with the two straight lines SOR , LOD , and if the straight line LOD , drawn in length comes to meet the line MN , in a point as 12 , it shews that there is also the point of the hours of 12 , in your Dyal , viz. in the middle rule between the superficies and the axeltree ; now divide with your Compass every one of these quarters of the round , into six parts equal , as you see in the points that are upon the brim of the round RGSH , and by the center or middle point of this round O , and by every one of the points of these divisions of the edge or brim of the round , draw some lines or beams , as you see some drawn already , that may go and meet the straight line MN , as in the points , 5 , 4 , 3 , 2 , 1 , 11. and these points are the points of the other hours , that are to be found in your Dyal . 18 17 Figure 18 , To all sorts of People . NOw take with the Compass in the figure above , the space from 6 to 5 , and with this space go to the Dyal in the figure below , set one of the feet of the Compass to the point 6 , and keeping this foot of the Compass upon this point 6 , go and mark with the other foot in the middle rule another point 5 , and by this means you shall transport with your Compass the space 6 , 5 , ●●rom the line of the figure above , which represents your table , or the flat place in the Dyal of the figure below , upon the middle rule MN , & so accordingly take with your Compass every one of the other spaces , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1● , 12 , 11 , from the higher figure , and bring them in this manner to the Dyal upon the middle rule , in the lower figure , and so you have done in this middle rule in the Dyal of the lower figure , all and the same spaces as those are , that are upon the table of the lower figure : And those points of the middle rule of the lower figure , are as many points of hours that will be in the Dyal , among which you know that the point 6 , is the point of one of the hours of six , either of the evening , or of the morning , whereby you shall come to know which are the other hours , whereof you have the points so mark'd in the rule of the Dyal . As for to let you know , whether this point of 6 hours , is either of those in the morning , or of those of the evening , I will not trouble this paper with it , because that is plain enough of it self : And you see well enough , whether the shadow of the axeltree rod will fall upon this point , either in the morning about the beginning of the day , or else in the evening about the shutting up of the day . By this means you may see well enough , whether the hours of your Dyal are of those of the forenoon , or of the afternoon , for to mark them accordingly , without speaking any further about it . If you have found upon the middle rule MN , the point of one of the two hours of 12 , and not the point of one of the hours of 6 , you are but to do with that point of hour of twelve , the same thing that I said you should do with the point of one of the hours of 6. When you have so transported the points of the hours from the Table of the figure above , to the Dyal in the lower figure upon the middle rule ▪ That which remains to do , is to transport those points of hours from the middle rule , into the superficies of the Dyal in the manner following , and to trace in it afterwards the line of the hours , as I will shew you . You see , there are two strings tyed to the axeltree rod , in the figure below , set the rings of these two strings , as far as you can one from another , and as from R in S , then take one of these two strings , as that which comes from the point R , carry it stretcht out in a direct line from the axeltree rod , to one of the points of hours that are mark'd in the middle rule , for example in the point of hour 12 , and cause this string comming so from the axeltree , to passe to this point of hour 12 , of the middle rule , and to go altogether in a straight line as far as the superficies of the Dyal ; and mark in the superficies of the Dyal the point in which this string so carryed , meets it ; By this means you shall transport this point or hour 12 , from the middle rule into the superficies of the Dyal to the point XII . And after the same manner , you shall transport one after another the points of hour 11 , 12. 1 , 2 , 3 , 4 , 5 , 6 , from the middle rule , into the superficies of the Dyal to the points XI , XII ▪ I , II , III , IV , V , VI . Figure 19 , To all sorts of People . How to trace the lines of the hours upon the superficies of the Dyal . OF the two strings , fig. above , that are made fast to the axeltree rod , stretch out one in a straight line , from the axeltree rod , as from the point R to a Point of hour of the middle rule , as to the point of hour I ▪ and holding this string so stretcht , take the other or second string comming from the point S , and stretching it likewise in a straight line , make it crosse over the first string BI ▪ and let it touch it without breaking his straight line , and let it go in a straight line from thence , to the superficies of the Dyal as to the point D , and mark the point D , in the superficies of the Dyal , wherein this second string so carried , comes to touch it ; then make this second string to go and touch again the first in another place , and with this second string go and touch in the same manner , another point E , in the superficies of the Dyal E , And so remove this second string along the first string , as many times as you shall have need to mark any several points , as D , E , L , in the superficies of the Dyal , to trace the line of that hour there , then draw a line as fine and delicate as you can by all those points DEL , in the superficies of the Dyal , that line shall be the line of that hour I. And after this manner , you shall trace in the superficies of the Dyal , the lines of all the other hours that are in the middle rule , and your Dyal will be finished . The lower figure shews you to the eye , how that after you have transported as above said , all the points of hour from the middle rule into the superficies of the Dyal , you may take away this middle rule , and the two knobs that hold it up , and make an end of tracing the rest of the lines in the superficies of the Dyal , as I have said , with the strings comming from R and I , and by means of the points of hours XI , XII , I , II , III , IV , V , VI . Figure 20 , To all sorts of People . AFter that you have placed the axeltree of the Dyal as it must be , if you desire to find the points of the hours in the superficies , with some extraordinary instruments , that which is the plainest of all , viz. a round flat plate , and stiff , as of tinn'd yron , or the like , and divided into 24 parts equal one unto another , and set up in the manner of a rotunda or whirl , by the squire , or with right angles about the axeltree of the Dyal , as the figure below doth shew , is the most common and the shortest way of all . The figure H , shews you this round alone , and how it is open of one side , that the center thereof may be placed with the center of the axeltree . The 1. figure shews the neck that may be applyed unto this round about the center , to the end that one may with this neck , set the round to the axeltree of the Dyal by the squire , or with angles straight between themselves , as you see in the second figure . When you have thus set this round to the axeltree of the Dyal , the lower figure shews you how you must place the string of the plummet , hanging upon the axeltree by a point of one of the divisions of the edge of this round , that it may give you the points of the hours in the superficies of the Dyal . The strings which comming from the axeltree passe afterwards to the points of the division of this round in it's 24 parts , shew you , how you must afterwards bring the strings from the axeltre , by the points of the division of this round in 24 parts , equal to the superficies of the Dyal , that you may have the points of the hours in this superficies . The string LS , XII , that passeth to the string with the plummet RS , gives the point of the hours of 12. The string LVI , that passeth to one of the points of this division in 24 , and is found to be level , gives the point of the hours of 6. The other strings shew you , that the way of tracing the points of the other hours is the same as above . 20 Figure 21 , To all sorts of People . VVHen you have brought , as I have said , by means of this rotunda and the strings , all the points of the hours , into the superficies of the Dyal ; You may take away the rotunda if you will , and make an end of tracing the lines of the hours as before with the strings , and by means of the points of hours , which you have brought into the superficies of the Dyal , as you see in the figure below , the line DELKPQSYIZG . And for this purpose by means of the said strings , carry a string in a straight line from the axeltree to the point of hour , for example I , and holding it there stretcht in a straight line , carry of one side or other according to the occasion , an other string comming also from the axeltree , as from I , or from B , that may go in a straight line as far as the superficies of the Dyal , and let it go and touch , and crosse over the string IR , several times in several places , and at every time go with this second string to touch and mark a point in the superficies of the Dyal , until you have enough , as you see the points D , E , L , K , P , Q , Y , I , Z , G , and carry by these points a line sweetned , that shall be a line of hour , do the like for the lines of the other hours , and you have done . When you have mark'd in the superficies of the Dyal , a point of every one of the hours that are to be found in it : If you desire to trace the lines of the hours every one at once , without making use of the strings , as in the figure above , you may do it when it is dark , as by night , with the light of a candle , in that manner as it is exprest in the lower figure . Set a light behind the axeltree rod of the Dyal , and turn the same lightabout this axeltree , untill the shadow of this axeltree come to one of the points of hour I , and trace in the superficies of the Dyal a line DELKPQSYIZG , all along this shadow of the axeltree , that line shall be a line of hour , do the like for every one of the other points of hour , and you have finish'd your Dyal . 22 Instruments to work with all , 21 Figure 22 , Several Instruments to work withall in these occasions hereafter specified . I Did not intend to burden my memory with any thing in this matter , but with Monsieur de Sargues universal rules for the placing of the axeltree , and for the tracing in a Dyal the hours after the French way , without medling with the rest , which is more curious than useful . But to follow the advice of many considerable persons whom I do honour , I have set down also the way to mark that which is commonly called the signs : The hours after the Italian or Babylonian way : The hours after the manner of the Ancients : The elevations of the Sun above the Horizon ; and the rising of the same . And for as much as none can do any of these things universally , without using these instruments more or lesse ; This table following shews to the eye all the pieces that are used in those several occasions . These instruments are first a circle , a half circle , or the quarter of a circle , which is all one which is made to turn about its Diameter set fast in it 's due and convenient place , or down right , as in the fourth figure , or level , as in the second or third figure , or else inclining or hanging downward , as in the first figure . The way to make this circle to move in all kinds of positions , is to set two rings in it's Diameter , through which one may put in a stick straight , round , and smooth , about which this circle may turn round like a weather cock about his needle or spear , as in the second figure , and there must be within those hoop rings , a screw to fasten this circle in that place , or which way soever you will have it to stand . The sticks or rods are represented by the 7th . figure , with a fork at the end of every one , bored in the cheek , to put a pin through , as you see , that one may be set plum or down right , and the other level , being made fast at one end to the axeltree rod as in the 5th . figure . And for this purpose also the axeltree rod is bored in O. The 8th . figure shews the axeltree rod by it self bored with O , and the pin Q , put through the hole , to shew more plainly that which the 5th figure represents , viz. all the pieces set or joyned together being mark'd with qo . You see that the hoop rings are near the edge or brim of the circle , a purpose to leave the center O , and a space about it free , having commonly a piece taken off , that this circle may turn freely about the forked end , that is to say about his center , without any let or hinderance at all . 23 for the Signes . Figure 23 , How to mark the Signs . GEt , figure 2 below , a half circle both thin and stiff ctsrd , draw there a beam OZS , perpendicular to the Diameter CPOQD , take on both sides of this beam OS , 23 degrees and half , for example 23 degrees and half from s towards t , and as much from s towards r , draw the straight line r , t , make upon Diameter t , r , a half , circle tzr , divide the edge or brim of that half circle in six equal parts , as in the points that you see there ; draw by those points as far as the half circle CtsrU , some straight lines that may be perpendicular to the straight line r , t , bring from the center O , by the points that those perpendicular have made upon the edge of the half circle CtsrD , some straight lines , as you see that the strings shew you , and with those lines drawn out sufficiently , you shall mark the signs in the Dyal , as I shall say . You see that the half circle is cut thorow , or made hollow from the point P , to the point Q , round about the center O , according to the circumference PZQ , which is notch'd also in the points that you see in it , which are betwixt every degree of the half circle ; And the center O , and these notches , are there a purpose to fasten a string upon them , insteed of bringing it from the center O. The two figures s32tgez , s45rbcz , both on the right and on the left side of the half circle CtsrD , shew as you may judge by their letters or coats , each of them one half of the figure tsrz , of the half circle CtsrD , which I have made thus bigger than each half of this figure , that one may set in the letters ge , cb , and some figures , 2345 , about the edges of the two half circles , and also the signs , as you see , which I could not do in the middle figure without confusion . The lines comming from the points t23s , s45r , towards the lowest past of the figure or plate , comming near one an other , go and seek the center of the half circle t23s45r , Every one of the three saces between those straight lines is to hold two signes , mark them there in the same order that you see them , close by these straight lines , one of one side , and an other on th'other side . And by this meanes the straight line of the half circle t23s45r , which from the center of the half circle passeth to the points , is that of the signes of Aries , and of Libra ; that which passeth to the point 3 , is that of Taurus , and of Virgo ; that which passeth to the point 2 , is that of Gemini , or the Twinnes , and of Leo ; that which passeth to the point t , is that of Cancer ; that which passeth to the point r , is that of Capricornus ; that which passeth to the point 5 , is that of Sagittarius and Aquarius ; that which passeth to the point 4 , is that of the Scorpion , and of the Fishes . The two figures s32cgez , s45rbcz , shew you also that with one quarter of circe mark'd on both sides with 6 signes in one part , and 6 others in the other part , you may do the same thing as well as with the half circle , by turning this quarter of circle , as you see in the said figures , once of one side , and once of th'other . I shall for all that speak to you alwayes as if you had the half circle in your hand . Therefore when you will mark the lines of the signes in the Superficies of the Dyall , The first figure shewes you how you must set up your half circle with the axeltree , for to turn it about the same , without going up or down along the axeltree . The first figure above shewes how you must make your half circle , viz. about the axeltree , set then the half circle to the axeltree of the Dyall , as you see in the figure that is under the first . Tye a string with a loose knot , just in the center of the circle . Turn the half circle about the axeltree , cause at the same time the string comming from the center , to passe by one of the lines of the signes , holding it longer or shorter as need shall require , go and touch with the string many several points in the superficies of the Dyall one after an other . Draw a line sweetned , by all those points , and it is the line of the signes that are markt along the straight line of the half circle which the string doth cover , in turning with it about the axeltree . Do the like for every line of the signes , mark the signes in the Dyall by the lines so drawn according to the situation , in regard of the Country and of the place of the Dyall , and as the figure shews , you have marked the signes in the Dyall . And if the strings could not come from the center , fasten them with a knot to the beames , comming from the center , in the notches or clefts of the circumference PZQ . Set a button or an other mark in the axeltree , in the place where the center of the half circle CtsrD hath been , and the shadow of the button will go and mark the signe that the Sun is in . Figure 24. To mark the houres after the Italian or Babylonian way . THE first figure shews how you must set on your half circle , and how to make it turn about the axeltree . Moreover , the line NO ( shews you what kind of line comming from the center O of this half circle , you must make use of , in making the half circle to turn about . When you have drawn the lines of the houres after the French way at length , in the superficies of the Dyall , as the figure below doth shew . Set on , as the same figure shewes you also , the half circle O t rto the exeltree of the Dyall with a string ON in it's center . Let this half circle hang down right or Plum , and when this half circle is just down and at rest , draw the string ON , in a straight line comming from the center O , and closing with the half circle , in such sort as it may go , and touch it all at length , turn this string as a beam of the half circle about the center O , till it be very levell , as the figure shewes by the setting on of the Carpenters levell A. When the string ON is stretcht out very levell close to the half circle , mark exactly upon the edge of this half circle , the point wherein the string ON toucheth it , as doth the letter E. Then let the half circle turn about the axeltree . Make in the mean time the string ON , to passe by the point t , which you have markt upon the edge of the half circle , and making it shorter or longer according as need shall require , go and touch with this string many several points one after an other in the superficies of the Dyall in divers places , 1 , 2 , 3 , 4 , 5 , 6. Draw an obscure line by those points , as you see the line bowed or crooked , 23 , 24 , 1 , 2 , 3 , 4 , 5 , 6 , and which reacheth beyond the axeltree towards h. This line crosseth over the superficies of the Dyall , out of the Equinoctial line PQ , and meets by the way all the lines of the houres after the French way , as you see it doth in 23 , 24 , 1 , 2 , 3 , 4 , 5 , 6. There Remaines to trace the lines of these houres after the Italian or Babylonian way in the superficies of the Dyall and when you know how to trace one , you shall be able also to trace the other . Therefore to trace a line of those kinds of houres , It is no matter which you begin to trace first ; Count upon the Equinoctiall line PQ fixs paces of houres equall , after the French way , one after an other , as from XII . to VI . Afterwards follow the lines of the houres , after the French way , that passe by the points XII . & VI the two extremities of these six spaces every one to the aforefaid line of which you have found the place , by turning the string with the half circle about the axeltree by the point t , as you see , as farre as the points 24. And 6. Take conveniently in these two lines of hours after the French way , In each of them one of the points wherein it meets with the Equinoctial line PQ , or else the said line found with the string 1 , 2 , 3 , 4 , 5 , 6 , that is to say in one , the point that the line placed with the string makes in it , and in the other ; the point that the Equinoctial line makes in it , for example , In the line , XII , 24 , take in it the point 24. Wherein it meets with the line found with the string : and in the other , VI , take in it the point VI . wherein it meets with the Equinoctiall line . Set either a string or a rule by these two points so taken 24. and VI , as you see the line , 24. 6. Then with the string comming from the center of the half circle Q go razing or laying even the string 24. VI by making it longer or shorter as need requires ; as you see in O , g , mark many several points in the superficies of the Dyall one after an other , as for example 24 , g , VI , more or lesse , according as the superficies of the Dyall is more or lesse uneven . Draw an obscure line by the points 24 , g , VI , and it will be a line of houres after the Italian or Babylonian way , and so of all the rest . The string h , OH shews that you may if need be , do the like both of one and of the other side of the center O , to go and place of one part or other according to the occasion , the line , as 2 , 3 , 4 , 5. And if you have a straight line , as might be O , q , which may turn about the center O , and be perpendicular to the Axeltree BI , and you hold the half circle with this straight line , set one at a convenient or reasonable distance from the other : And let it be alwayes exactly of the distance of six hours after the French way : First of all , this string describes the Equinoctial line in the superficies of the Dyall , Secondly when one of the two , either the half circle or the straight line O q , is found in one of the points of the hours of the Equator , th'other is likewise found in it , in an other point of hour , then drawing with a string coming from the center O , a straight line that may go from the point , as t , to the end of the straight line O q which you shall go drawing with this string made shorter or longer , as need requires , and mark some points of line of hour , after the Italian or Babylonian way in the superficies of the Dyall . And for this purpose there is nothing so easie as to have a circle of Equator , that may be fitted to the half circle , and where you may have alwayes a space ready made for it's hour . 25 To mark the houres after the Manner of the Iewes . 24 for the houres after the Italian way . Figure 25 , To mark the hours after the manner of the Ancients or the Jewes . YOu must know first that it would be very troublesome to draw in the superficies of the Dyall , the lines of this kind of hours in such a manner as that they might be alwayes just and right in theory , all the year long . And therefore it is sufficient to draw them just by demonstration in three points onely , viz , in their points of both ends , and of the middle , which are the points of those circles that appear the greatest above the Horizon being parallel to the Equator , and of the Equator it self . The rest goes as it may , and therefore it may be said , that the lines of such hours traced in this manner are false in the rest of their length , yet Curiositie makes them passe for current . Wherefore to mark this kind of lines of hours . The higher figure 4 shewes which way you must make this half Circle to turn about , viz. about a straight axeltree line placed levell in the center of the Axel-tree of the Dyall . And to be short , set up and make very fast a rod in a straight line passing to the center O , and let it be first within the joynt of the axeltree rod , secondly let it be level , as the figures do shew of a plummet P. and of a level A , this being done , tye some strings with a loose knot to this rod so levelled NL as you see NR , and LT. Take the string from about the center O , stretch it out in a direct or straight line from the center O to one of the points of hour , after the French way , of the Equinoctial line of the Dyall , for example to the point of 1 hour , as you see the string OI . This string being thus strecht out , take the other strings of one or th'other end NL , and Crosse over this string OI with them , and so go and mark many points in the superficies of the Dyall , as TIR . Draw an obscure line by those points as TIR , it is a line of hours after the manner of the Ancients or the Jews , do the like with the other hours and half hours of the Equinoctial line . If you leave a rod in the Dyall , as NOL , the shadow thereof will go and shew these hours continually at length , if you will not leave it in , the button or center O of the axeltree of the Dyall will shew them . 26 for the hight of the sun . Figure 26. How to mark the Elevation of the Sun above the Horizon . THE higher fig. 3. shews which way you must turn the half circle , viz. about a straight axel-line hanging down right . Set up your half circle , so that it may turn like a weather-cock about a rod hanging down right , or plum , above or below the axeltree of the Dyal , it matters not which . VVhilest you turn it thus as it is above said , cause in the mean time the string comming from the center O to passe by one of the degrees of the edge of the circle ; and make the string shorter or longer as need shall require , mark with it many several points in the superficies of the Dyal , according as you see them rankt one by an other , in four places . Draw a small or obscure line through all these points , and this will be one of the lines of the Elevation of the Sun . Count the Degrees in the edge of the circle , beginning at the first of the beam which is level , and ending at the 90. Beam which is down right or plum . Mark in the line of the Dyal , the number of the Degrees of the border of the circle , where the string passes that hath mark't the points of that line , and so of all the others , and the shadow of the button of the axeltree which is in the center of the circle , will shew the Elevation of the Sum above the Horizon . Figure 27 : How to mark the Sun rising , or East rising of the Sun . THE figure 2 above shews how you must place the half circle , viz. parallel unto the Horizon , I would not put a levell to it to avoid confusion . It shews also that one of the Diameters of the circle must be set within the center of the Dyal , that is to say , thar it must go directly from the south to the North , and accordingly the Diameter which is perpendicular to it , will go from East to West . When your circle is set fast in this position , let a plummet op in the lower figure hang from the center O. This being done , from each point of Degree of the edge of the circle , as from x and from z. mark with a string XT or ZR . many points in the superficies of the Dyal . Draw a small or obscure line through these points , as TY or SR. it is a line of the Suns Eastrising - Mark in it the number of Degrees of the point of the circle from whence the string comes , according as you will count them , to begin either from the East , or from the South . And so of all the other Degrees accordingly . And the shadow of the button O will shew which way the sight of the Sun comes upon the Dyal . I will take here occasion to tell you , that if for some reason or other , you could observe , in one and the same day , but two shadows of the Sun in stead of three , as we have said in the placing of the axeltree in the Dyal , the declining of the Sun in that day , will serve you for a third shadow , or else two other shadows observed in an other day . I mean you may find equally the placing of the axeltree by one or other of those ways above mentioned , and with 3 shadows ; and with 2 shadows , and the declining of the Sun in that day , and with 4 shadows , two of one day , and two of an other , which are three wayes that come all to one . 27 for the Eastrising of the sun . 28 Figure 28. I do not specifie in this volum these kinds of flat Dyals , wherein you may work without knobs or middle rule : And where you may draw the Equinoctial line ; trace out and divide the circle Equator : in a word , where you may do all : yea and in the very superficies of the Dyal , you may easily come to know them you self , by putting this universall way into practice . Here is onely a way how to trace out all the twelue lines of the hours equal , after the French way , in the flat Dyals where the axeltree meets the superficies athwart in the space that you work in , so that you shall have no need of a greater place . And what I have already said , and what I am now going to say again , will serve to find out the way to do the like in all kinds of Dyals universally . When you have drawn upon your Dyal the Equinoctial line M 12 M , drawn conveniently and divided the circle Equator Q 12 Q bring to the Equinoctial line , the beam of the 12 hours Q , 12. Draw of both sides of the Equator , and from the Equinoctial line a straight line MQ parallel to the beam of 12. hours O 12. bring the beams of the other hours to the first , which they shall find of the Equinoctial in rt : and of MQ in c , d●g , Q , Bring in the Dyal the line of the twelve hours B. 12. draw by the point M , of the Equinoctial line , and from the center of the Dyal B a straight line ML parallel to the line of twelve hours B 12 ; make upon this line ML and upon the point M a triangle LMN like to the triangle in the aire OB 12. and let the angles of these triangles in the points L and B be equal one unto an other , Carry the spaces Mq , Mg , Md , Mc , from the straight line MQ into the straight line MN , viz. from M into N , into u , into i , into o , bring by the points N , u , i , o , some straight lines NL , ub , if , oh , parallel to the side , NL of the triangle LMN : Carry from the center of the Dyal B by the points r , t , h , f , o , L , Some straight lines , BL , Bh , Bf , Bh , Bt , Br ; These are such lines of hours as you may continue beyond the center B , and mark them according to their orders . THE END . Notes, typically marginal, from the original text Notes for div A35744e-3300 i.e. That are made without any aim , or heed . 
A33999	----	Geometrical dyalling, or, Dyalling performed by a line of chords onely, or by the plain scale wherein is contained two several methods of inscribing the hour-lines in all plains, with the substile, stile and meridian, in their proper coasts and quantities : being a full explication and demonstration of divers difficulties in the works of learned Mr. Samuel Foster deceased ... : whereto is added four new methods of calculation, for finding the requisites in all leaning plains ... : also how by projecting the sphere, to measure off all the arks found by calculation ... : lastly, the making of dyals from three shadows of a gnomon ... / written by John Collins ... Collins, John, 1625-1683. This text is an enriched version of the TCP digital transcription A33999 of text R17003 in the English Short Title Catalog (Wing C5373). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 180 KB of XML-encoded text transcribed from 64 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2017 A33999 Wing C5373 ESTC R17003 12394868 ocm 12394868 61099 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A33999) Transcribed from: (Early English Books Online ; image set 61099) Images scanned from microfilm: (Early English books, 1641-1700 ; 958:23) Geometrical dyalling, or, Dyalling performed by a line of chords onely, or by the plain scale wherein is contained two several methods of inscribing the hour-lines in all plains, with the substile, stile and meridian, in their proper coasts and quantities : being a full explication and demonstration of divers difficulties in the works of learned Mr. Samuel Foster deceased ... : whereto is added four new methods of calculation, for finding the requisites in all leaning plains ... : also how by projecting the sphere, to measure off all the arks found by calculation ... : lastly, the making of dyals from three shadows of a gnomon ... / written by John Collins ... Collins, John, 1625-1683. [6], 82, (i.e. 80) p. : ill., plates. Printed by Thomas Johnson for Francis Cossines and are to be sold at his shop ... : also to be sold by Henry Sutton ..., London : 1659. Reproduction of original in the University of Illinois (Urbana-Champaign Campus). Library. Errata: p. 82. eng Foster, Samuel, d. 1652. Dialing -- Early works to 1800. A33999 R17003 (Wing C5373). civilwar no Geometrical dyalling: or, dyalling performed by a line of chords onely, or by the plain scale. Wherein is contained two several methods of i Collins, John 1659 31860 3 0 0 0 0 0 1 B The rate of 1 defects per 10,000 words puts this text in the B category of texts with fewer than 10 defects per 10,000 words. 2005-07 TCP Assigned for keying and markup 2005-08 Aptara Keyed and coded from ProQuest page images 2005-09 Andrew Kuster Sampled and proofread 2005-09 Andrew Kuster Text and markup reviewed and edited 2005-10 pfs Batch review (QC) and XML conversion Geometricall DYALLING or , Dyalling by a line of CHORDS onely by IOHN COLLINS Accomptant Philomath London Printed for Francis Cossinet in Tower Street and for H : Sutton behind Exchang Generall Scheme Projection of ye Sphere Anno 1659 In a Circle In a Parralellogram Chords fitted to ye Schemes in ye Booke ruler Chords in time fitted to the Booke ruler Geometricall Dyalling : OR , DYALLING Performed by a Line of CHORDS onely , Or by the PLAIN SCALE . Wherein is contained two several Methods of Inscribing the Hour-Lines in all Plains , with the Substile , Stile and Meridian , in their proper Coasts and Quantities ; Being a full Explication and Demonstration of divers difficulties in the Works of Learned Mr. Samuel Foster deceased , late Professor of Astronomy in Gresham Colledge , ; Also a Collection of divers things from the Works of Clavius and others . Whereto is added four new Methods of Calculation , for finding the Requisites in all Leaning Plains , with full directions suited to each Method for placing them in their proper Coasts , without the help of any Dclinations . Also how by Projecting the Sphere , to measure off all the Arks found by Calculation , and to determine what hours are proper to all kinde of Plains , omitting superfluity . Lastly , the making of Dyals from three Shadows of a Gnomon placed in a Wall at random , with a Method of Calculation suited thereto , and divers ways from three Shadows , to finde a Meridian-line . Written by John Collins of London Accomptant , Philomath . London , Printed by Thomas Johnson for Francis Cossinet , and are to be sold at his Shop at the Anchor and Mariner in Tower-street , at the end of Mincing-lane , with other Mathematical Books ; Also to be sold by Henry Sutton Mathematical Instrument-Maker , living in Threed-needle Street behinde the Exchange . 1659. To the Reader . BEing in conference with my loving friend Mr. Thomas Rice , one of the Gunners of the Tower , much exercised in making of Dyals in many eminent places of the City , he was pleased to communicate unto me the knowledge of a General Scheme , for Inscribing the Requisites in all Oblique Leaning Plains , which he added , was the useful Invention of the Learned Master Samuel Foster , Professor of Astronomy in Gresham Colledge London , deceased , from whom he received Instructions concerning the same , in the year 1640. And the said Mr. Rice not having his Papers about him , did dictate to me from his Memory the Construction and Practice of the said Scheme , which I afterwards Methodized just as it is delivered in page 25 and 26. At the same time also I received directions for Inscribing the Substile and Stile in upright Decliners , and East or West Reclining , or Inclining Plains , but they were of another mans Invention , and did not seem to be derived from the former general Scheme , and therefore I have not used them in this Treatise , but derived the performance thereof from the said general Scheme : Moreover , Mr. Rice added , that in regard of the death of the Author , and since of his Executor , who had the care and inspection of his Papers , I should do well to Study out the Demonstration of the former Scheme , and make it publick , the rather , because it hath been neglected . The manner of Inscribing the hour-lines being already published in a Treatise of the Authors , Intituled , Posthumi Posteri ; this desire of his , which was also furthered by Mr. Sutton and others , I am confident is fully effected in the following Treatise , and much more then was in his Request , and here let me adde , that I have had no other light from the Endeavors of the Learned Author , then what was as above communicated unto me , all which might be expressed in half a page of paper , or little more , to which I have made this large Access and Collection , nam facile est inventis adore ; not that I would hereby any thing endeavor to ecclipse the Authors Work , the excellency of whose Inventions in this and other kindes , will speak forth his Renown to all posterity . And though in probability I have not performed so much , nor so well , as was obvious to the knowledge of the Learned Author , yet I am confident when the Reader understands what is written , he would be as loath to be without the knowledge thereof , as my self ; and I am induced to believe , that the Author left nothing written about many particulars in this Treatise ; throughout which , we suppose the Reader , furnished with the common Rudiments of Geometry , that he can raise Perpendiculars , draw Parallels , describe a Parralellogram , bring three points into a Circle , understands Definitions and Tearms of Art , knows what a Line of Chords is , can prick off an Arch thereby ; all which , with Delineations for all the usual Cases of Triangles , both from Projection and Proportions , the Reader will meet with in my Treatise , called , The Mariners Plain Scale new plain'd , now in the Press . Vale & fruere . I remain thy friend , and a Well-willer to the Publique Advancement of Knowledge , JOHN COLLINS . The Contents : DYals distinguished Page 1 , 2 To take the Suns Altitude without Instrument p. 3 To finde the Reclination of a Plain p. 4 Also the Declination thereof p. 5 A general proportion and Scheam for finding the Suns Azimuth or true Coast p. 6 To draw a Horizontal Dyal p. 7 Also a South Dyal p. 8 A new way to divide a Tangent liue into five hours and their quarters p. 10 , 11 A direct South Polar Dyal p. 12 To prick off the Requisites of upright Decliners p. 13 To prick off an Arch or Angle by Sines or Tangents p. 14 The Scheam for placing the Requisites of upright Decliners demonstrated p. 15 , 16 To inscribe the hour-lines in an upright Decliner p. 17 The Demonstration thereof p. 18 to 21 An East Dyal p. 22 Requisites placed in East or West leaning Plains p. 23 The Demonstration thereof p. 25 The Construction of the general Scheam for placing the Requisites in Declining Re-Inclining Plains p. 25 to 27 The first Method of Calculation for Oblique Plains p. 28 to 30 And directions for the true placing the Requisites suited thereto p. 30 , 31 The general Scheam demonstrated p. 32 to 34 The hour-lines inscribed in an Oblique Plain ibid. The general Scheam fitted for Latitudes under forty five degrees p. 35 To draw the hour-lines in a Declining Polar Plain p. 36 Also how to delineate the hour-lines in Plains having small height of Stile p. 37 , 38 Another way to perform the same p. 39 to 41 A second Method of Calculation for Oblique Plains p. 42 Proportions for upright Decliners p. 43 A third Method of Calculation for Oblique Plains p. 44 Directions for placing the Requisites suited thereto p. 46 A fourth Method of Calculation for Oblique Plains p. 47 Through any two points assigned within a Circle to draw an Arch of a Circle that shall divide the primitive Circle into two Semicircles . p. 49 To measure the Arks of upright Decliners by Projection p. 50 Also the Arks of leaning East or West Plains thereby p. 51 To project the Sphere for Oblique Plains — To measure off all the Arks that can be found by Calculation With the Demonstration of all the former Proportions — from p. 52 to 59 To determine what Hours are proper to all Plains p. 60 to 61 Another Method of inscribing the Hour-lines in all Plains by a Parallelogram p. 62 To draw the Tangent Scheme suited thereto p. 63. The Hour-lines so inscribed in a Horizontal and South Dyal p. 63 , 64 As also in an upright Decliner p. 65 With another Tangent Scheme suited thereto for pricking them down without the use of Compasses p. 66 , 67 A general Method without proportional work , for fitting the Parallelogram into Oblique Plains that have the Requisites first placed p. 69 , 70 By help of three shadows to finde a Meridian-line p. 70 , 71 Another Scheme suited to that purpose p. 73 A Method of Calculation for finding the Azimuth , Latitude , Amplitude , &c. by three shadows p. 75 From three shadows to inscribe the Requisites and Hour-lines in any Plain p. 77 Which is to be performed by Calculation also p. 97 Characters used in this Book . + Plus , for more , or Addition . - Minus less , or Substraction . = Equal . q For square . □ Square . ▭ The Rectangle or Product of two terms . ∷ for Proportion . If the Brass Prints in this Book be thought troublesome to binde up , they may be placed at the end thereof , for the Pages to which they relate , are graved upon them . A Quadrant of a Circle being divided into 90d , all that is required in this Treatise , is to prick off any number of the said degrees with their Sub-divisions , which may be easily done anywhere from a Quadrant drawn and divided into nine equal Parts , and one of those parts into ten sub-divisions , called degrees , but for readiness the equal divisions of a whole . Quadrant are transferred into a right line ( as in the Frontispiece ) called a Line of Chords , serving more expeditly to prick off any number of degrees or minutes in the Arch of a Circle . An Advertisement . The Reader may possibly desire to be furnished with such Scales to any Radius , these and all manner of other Mathematical Instruments either for Sea or Land , are exactly made in Brass or Wood , by Heury Sutton in Threed-needle-street , behinde the Exchange , or by William Sutton in Upper Shadwell , a little beyond the Church , Mathematical Instrument-Makers . A direct South Diall Reclining 60d Lat 51d 32′ An erect direct North Diall Lat 51d 32′ A South Diall Declin 40d East Reclining 60d Lat 51d 32′ A North Diall Declin 40d West Reclining 75d Latitude 52d 32′ THE DISTINCTION OF DYALS . THough much of the subject of Dyalling hath been wrote already , by divers in diverse Languages , notwithstanding the Reader will meet with that in this following Treatise , which will abundantly satisfie his expectation , as to the particulars in the Preface , not yet divulged in any Treatise of this nature . In this Treatise , that we might not be too large , divers definitions are passed over , supposing that the Reader understands what a Dyal is , what hour Lines are , that that part of the Stile that shews the hour , ought to be in the Axis of the World , that the hour Lines being projected on any Regular Flat , will become straight Lines . Dyals are by Clavius in the second Chapter of his Gnomonicks distinguished into Seventeen Kindes . 1. The Horizontal , being parallel to the Horizon . 2. South and North direct , by some called a Vertical Dyal , because parallel to the prime Vertical or Circle of East and West . 3. 4. South direct , Reclining or North Inclining less more then the Pole . 5. 6. South direct , Inclining or North Reclining less more then the Equinoctial 7. South direct Reclining , or North Inclining to the Pole , called a direct Polar Plain , because Parallel thereto 8. South direct Inclining , or North Reclining to the Equator , called an Equinoctial Plain , because parallel thereto . 9. South and North , Declining East or West . 10. East and West direct . 11. East and West direct , Reclining or Inclining . 12. A South North Plain declining East or West Reclining Inclining to the Pole , called a Polar Decliner . 13. A South North Plain Declining East or West Inclining Reclining to the Equinoctial , called an Equinoctial Decliner . 14. 15. A South North Plain Declining East or West Reclining Inclining Whereof are two sorts , the one passing above , the other beneath the Pole 16. 17. A. South North Plain Declining East or West Inclining Reclining whereof are 2. sorts , the one passing above the Equinoctial , the other beneath it . Of each of these we shall say something , but before we proceed , it will be necessary in the first place to shew how to finde the Scituation of any Plain . 1. To finde whether a Plain be Level or Horizontal . The performance hereof is already shewed by Mr. Stirrup , in his Compleat Dyallist , Page 58. where he hath a Scheme to this purpose , I shall onely mention it : Get a smooth Board , let it have one right edge , and near to that edge , let a hole be cut in it for a Plummet to play in ; draw a Line cross the Board Perpendicular , to the streight edge thereof passing through the former hole ; if then setting the Board on its smooth edge , and holding it Perpendicularly , so that the Plummet may play in the hole , if which way soever the Board be turned , the threed will fall ( being held upright with the Plummet at the end of it , playing in the former hole ) directly on the Perpendicular Line drawn cross the Board , the said Line no wayes Reclining from it , the Plain is Horizontal , otherwise not . 3. To take the Altitude of the Sun without Instrument . Upon any Smooth Board , draw two Lines at right Angles , as AB and AC , and upon A as a Center with 60d of a Line of Chords describe the Arch BC , and into the Center thereof , at A drive in a Pin or Steel Needle , upon which hang a threed and Plummet , then when you would observe an Altitude , hold the Board so to the Sun , that the shaddow of the Pin or Needle may fall on the Line AC , and where the threed intersects the Arch BC , set a mark , suppose at E then measure the Ark BE on your Line of Chords , and it shews the Altitude sought , for want of a Line of Chords , you may first divide the Arch BC into 9 parts , and the first of those Divisions into 10 smaller parts . 3. To draw a Horizontal and Vertical Line upon a Plain . The readyest and most certainest way to do this , especially if the Plain lean downwards from the Zenith , will be by help of a threed and Plummet held steadily , to make two pricks at a competent distance in the shadow of the threed on the Plain , projected by the eye , and a Line drawn through those , or parallel to those two points , shall be the Plains perpendicular or Vertical Line , and a Line drawn perpendicular to the said Line , shall be the Plains Horizontal Line . To finde the Reclination of a Plain . The Reclination of a Plain , is the Angle comprehended between the Plains perpendicular , and the Axis of the Horizon , and the same Definition may serve for the Inclination , onely the upper face of a Plain , leaning from the Zenith , is said to recline , and the under face to incline . In this sense Mr. Oughtred in his Circles of Proportion , Mr. Wells in his Dyalling , Mr. Newton in his Institution , and Mr. Foster in his late writings , understand it , and so it is to be taken throughout this Treatise ; but here it will not be amiss to intimate that Mr. Gunter , Mr. Wingate , and Mr. Foster in a Treatise of a Quadrant , published in Anno 1638. account the Inclination from the Horizon ; the complement of the Angle here defined , and accordingly have suited their Proportions thereto , in which sense , the Reclining side is said to be the upper face of an Incliner ; but for the future it will be inconvenient to take it any more in that Acception . Now to finde the Reclination of a Plain , apply the streight edge of the Board we used for trying of Horizontal Plains , to the Plains perpendicular , holding the threed and plummet so in the grooved hole , that it may intersect the line , parallel to the streight edge of the Board , and make two pricks upon the Board where the threed passeth , and it will make an Angle with the said Line , equal to the Plains Reclination , to be measured with Chords , according as was directed for taking of Altitudes ; and if need require , a Line may be drawn parallel to that drawn on the Board , which represented the threed , which will make the same Angle with the Plains perpendicular , the former Line did . The Inclination will be easily got , by applying the streight edge of the Board , to the Plains perpendicular , and holding the threed and plummet so at liberty , as that it may cross the Line parallel to the Boards streight edge , and it will make an Angle therewith , equal to the Plains Inclination . Of Declining Plains . A Plain hath its Denominations from the Scituation of its Poles : by the Poles of a Plain , is meant a Line imagined to pass through the Plain at right Angles thereto , the Extremities of which Line on both sides the Plain , are called its Poles . If a Plain look full South , without swerving , either to the East or West ; it is said to be a direct South Plain , if it swerve towards East or West , it is said to decline thereto ; if it stand upright without leaning , it is said to be erect Now a South Plain that declines Eastward , the opposite face thereto , is said to be a North Plain declining as much Westward , by which means the Declination of a Plain will never exceed 90d , and to which sense the following directions toroughout this Treatise are suited . Now we may define the Declination of a Plain to be the Arch of the Horizon , contained between the true Points of East or West , and the Plain , equal whereto is the Arch of the Horizon between the true North or South , and that Vertical Circle or Azimuth that passeth through the Plains Poles . To finde the Declination of a Plain . Upon a Board , having one streight smooth edge , draw a Line parallel thereto , and thereon describe a Semicircle , with the Radius of the Chords , which distinguish into two Quadrants by a perpendicular from the Center ; then holding the Board parallel to the Horizon , and applying the streight edge to the Wall , hold up a threed and Plummet , so that the shadow thereof may pass through the Center , and in the shadow , make a mark near the Limb , from which a Line drawn into the Center shal represent the shadow of the thread . In this Scheam , let ☉ C represent the shadow of the threed passing thorow the Center ; at the same instant , take the Suns Altitude , and by the following directions , finde the Suns true Azimuth . Admit the Sun be 75d to the Eastwards of the South , prick it from ☉ to S , then the distance between P and S shews the quantity of the Plains Declination , and the Coast is also shewn , for if P fall to the Eastwards of S , the Declination is Eastwards , if to the Westwards of it , it is Westward ; in this Example it is 30d Eastwards . We may suppose that the Reader knoweth which way from the Line of shadow to set of the Suns true Coast , that he may easily do by observing the Coast of his rising or setting , or whether his Altitude increase or decrease , and needs no directions . A General Proportion for finding the Azimuth . As the Cosine of the Altitude , Is to the Secant of the Latitude , Or as the Cosine of the Latitude : Is to the Secant of the Altitude ; So is the difference of the Versed Sines of the Suns distance from the elevated Pole , and of the Ark of difference between the Latitude and Altitude . To the Versed Sine of the Azimuth from the North in this Hemisphere . Declination 23d 31′ , North . Latitude 51d 32′ Altitude 41d : 34′ Azimuth 105d from the North Upon C as a Center , describe the Semicircle EBM , and draw ECM the Diameter , and CB from the Center perpendicular thereto , from B set off the Declination to D , the Latitude to L , the Altitude to A ; and draw the Line CA continued . The nearest distance from D to BC when the Declination is North , place from C towards M when South towards E , and thereto set K , assuming the Diameter EM to represent the Secant of the Latitude , the Radius of the said Secant shall be the Cosine of the Latitude doubled ; wherefore the nearest distance from L to CM doubled , shall be the Radius to the said Secant , which prick on the Line of Altitude from C to Z , then the nearest distance from Z to CM shall be the Cosine of the Altitude to that Radius , which extent prick from E to F , and draw the Lines EF and FM ; then for the third term of the proportion , take the distance between A and L , and prick it from M to G , from which point take the nearest distance to BC , and place it from C to X , so is the distance KX , the difference of the Versed Sines sought , which prick from F to Q , and draw QR parallel to FM , so is RM the Versed Sine of the Suns Azimuth from the North sought , and CR the Sine of it to the Southwards of the East or West , and BP the measure thereof in the Limbe , found by drawing a LIne from R parallel to GB ; in this Example the Suns true Goast is 15d to the Southwards of the East or West . Note also , that the Point R may be found without drawing the Lines EF , QR by entring the extent KX , so that one foot resting on the Diameter , the other turned about , may but just touch FM . To make a Horizontal Dyal . A Plain is then said to be Horizontal , when it is parallel to the Horizontal Circle of the Sphere proper to the place of Habitation ; these are such kinde of Dyals as stand commonly upon posts in a Garden . By the former Directions , a true Meridian Line may be found , let it here be represented by the Line FM . A Horizontal Dyal for the Latitude of London , 51 degrees , 32 minutes . Cross the same with a perpendicular , to wit , the Line of Six , and draw an Oblong or square figure as here is done , wherein to write the hours , from M to C set off such a Radius as you intend to draw a Circle withall , which for convenience may be as big as the Plain will admit , and therewith upon C as a Center describe a Circle , and set off the Radius of the said Circle from F to d , from M to k , and draw the right Line Fd , then prick the Poles height or Latitude of the place from f to L , and from L to S , and through the point S draw the Line MS , and it shall represent the Cock or Stiles height from the point L , take the nearest distance to FM , and prick that extent from F to N on the Line Fd , then a Ruler laid from N to k , where it intersects the Meridian is the Regulating point , or point ☉ , then from M divide the Circle into twelve equal parts for the whole hours , setting the Letters of the Alphabet thereto , and lay a ruler from the Regulating point to each of those Divisions , and it will intersect the Circle on the opposite side , from which Intersections , Lines drawn into the Center at M , shall be the hours Lines required , to be produced beyond the Center , as many as are needful , which shall be the hours before or after 6 in the Summer half year , the halfs and quarters are after the same manner to be inscribed , by dividing each equal division of the Circle into halfs and quarters . This Circular Dyalling , was in effect , published and invented by Mr. Foster in his book of a Quadrant , in Anno 1638. for the demonstration of this Work will demonstrate the truth of those Circular performances , which he Operates on the back of that Quadrant , but is more expresly hinted in his Posthuma , the demonstration whereof shall follow . Those that have plain Tables with a Frame , may have Tangent Lines put on the sides of their Frame ; and then if a Center be found upon the paper under the Frame by the Intersection of a Ruler laid over those Tangents , the requisite Divisions of a Circle to any Radius that can be described upon the paper , will be most readily given without dividing any Circle or setting of Marks or Letters thereto , the Frame keeping fast that paper , on which the Draught of the Dyal is made , which may also be supplyed from a Circle divided on Pastboard cut out , which is to be Laced upon a Board over the paper whereon the Draught is to be made . An upright full South Dyal . This is no other then a Horizontal Dyal , in that Latitude which is equal to the complement of the Latitude of the place , you are in , onely the hours must not be continued beyond the Center , and the Delineation requires no other directions then the former . On a direct North Dyal , in this Latitude , there will not be above two hours above , and two hours beneath each end of the Horizontal Line to be expressed , and the Stile will have the same Elevation and point upward . Horizontall Diall An Vpright South Diall In all upright North and South Plains , the Meridian or hour line of 12 is perpendicular to the Plains Horizontal line , if the Plain be direct the height of the Stile above the substile , is equal to the complement of the Latitude . If the Plain be South Reclining , or North Inclining , the height of the Stile above the substile , is equal to the Ark of difference between the complement of the Latitude and the Ark of Re-Inclination , and if this latter Ark be greater then the former , the contrary Pole is elevated . If the Plain be South Inclining , or North Reclining , the height of the Stile above the substile , is equal to the sum of the Colatitude , and of the Re-Inclination , and when this latter Ark is greater then the Latitude , the Stiles height will be greater then a Quadrant . Such Horizontal or direct South Dyals , both upright and leaning , whereon the stile hath but small Elevation , are to be drawn with a double Tangent Line without a Center , wherein the following directions for direct Polar plains , and those for other Oblique plains , whereon the stile hath but small elevation , will fully direct you . All Plains that cut the Axis of the World , have a Center ; but if they be parallel thereto , the stile hath no Elevation . Such direct South Reclining , or North Inclining Plains , whose Arch of Reclination is equal to the complement of the Latitude , are parallel to the Axis , and are called Polar plains ; in these the Hour Lines will be Tangent Lines of any assumed Radius , and the parallel height of the stile above the plain , must be made equal to the Radius of the Tangent , by which the hours were set off . First draw the Tangent line CG , and perpendicular thereto ACB , upon C as a Center , with any Radius describe the Quadrant of a Circle HB , and prick off the Radius from C to A , from H to E , from B to D , from D to F , from F twice to G. A Ruler laid from A to E Findes the point 1 on the Tangent CG for the hour line of one , and laid from A to D findes the point 2 for the hour line of two , the hour line of three is at the point H , of four at the point F , of five at the point G ; thus are the whole hours easily inserted . Now to insert the halfs and quarters . Divide the Arches BE , ED , DH into halfs , and a Ruler laid from A to those divisions wil finde points upon CH , where all the half hours under 45d are to be graduated ; and if a Ruler be laid from B to those respective divisions of the Quadrant BH , it will finde points on the Tangent CG , where the whole hours and halfs are to be graduated above 45d , after the same manner are the quarters to be inserted ; But in regard the halfs and quarters above 45d will by this direction be found with much uncertainty , I have added the help following . First , Divide the halfs and quarters under 45d as now directed , then for those above make use of this Table . In a direct South Polar Dyal . The distances of these hour lines . Are equal to the distances of these hour lines . 3 and 3 ¼ 3 ½ and 4 2 ¼ and 3 ¾ 4 : and 4 ¼ 4 ¼ and 4 ½ 4 ¼ and 4 ¼ ¾ and 1 ¼ 1 ¼ and 2 ½ 1 ½ and doubled ¾ and 1 ¾ 3 ½ and 3 ¾ doubled : otherwise , 1 ¾ and 2 ¾ 2 and 3 ¾ This I do not assert to be absolutely true , but so neer the truth , that there will not arive above one thousandth part of the Radius difference in the greatest Dyal that is made , and will be more certain then any Sector , though of a Vast Radius , or then they can with convenience be prickt down the common way , by a Contingent Line , the meaning of the Table will be illustrated by one or two examples . The distance between the hour lines of two and a quarter , and three and three quarters will be equal to twice the distance between the hour lines of twelve , and one and a half . Also the distance between four and a quarter , and four and a half , is equal to twice the distance between three and a half , and three and three quarters ; or it is equal to the distance between one and three quarters , and two and three quarters , but the former is nearer the truth . It will be inconvenient in such plains , as also in direct East or West Dyals , to express any hour line from the substile beyond 75d or 5 hours . To these I may add some other observations which were communicated by Doctor Richard Sterne , to Mr. Sutton , which may be of use to try the truth of these kinde of Plains . 1. The distance between the hours of 3 and 4 , is equal to the distance between the hours of 1 and 3. 2. The distance between the hours of 2 and 4 , is double to the distance between the hours of 12 and 2. 3. The distance between the hours of 11 and 4 , is doubte to the distance between 12 and 3 , or to that between 9 and 12 , and so equal to the distance between 9 and 3 , also equal to the distance between 4 and 5. 4. The distance between the hours of 11 and 5 , is double to the distance between the hours of 11 and 4 , as also to the distance between the hours of 4 and 5 , and between 9 and 3 ; and quadruple to the distance between 12 and 9 , or between 12 and 3. These are absolutely true , as may be found by comparing the differences of the respective Tangents from the natural Tables . To draw a Polar direct South Dyal . Having drawn the Plains perpendicular in the middle of the Plain , let that be the hour line of 12 , then assuming the stile to be of any convenient parallel height , that will suit the plain , making that Radius , divide a Tangent line into hours and quarters by the former directions , and prick them down on the Plain , upon a line drawn perpendicular to the Meridian or hour line of 12 on each side thereof , and through the points so prickt off , draw lines parallel to the Meridian line , and they shall be the hour lines required , as in the Example . To describe an Equinoctial Dyal . Such direct North Recliners , or South Incliners , whose Re-Inclination is equal to the Latitude , are parallel to the Equinoctial Circle , and are therefore called Equinoctial Dyals , there is no difficulty in describing of these : Divide a circle into 24 equal parts for the whole hours , & afterwards into halfs & quarters , and place the Meridian line in the Plains perpendicular , assuming as many of the former hours as the Sun can shine upon for either face , and then placing a round Wyre in the Center for the Stile , perpendicular to the Plain , and the Dyal is finished . A South plaine Declining 30d East Latitude 51-32′ page 13 page 17 A South Diall Declin 30d East Latitude 51-32′ To prick off the Substile and Stiles height on upright Decliners in their true Coast and quantity . On such Plains draw a Horizontal Line , and cross the same with a perpendicular or Vertical , at the intersection set V at the upper end of the Vertical Line set S , at the lower end N , at the East end of the Horizontal Line set E , at the West end W , prick off the Declination of the Plain in its proper Coast from S or N to D , and draw DV through the Center , the same way count off the Latitude to L , and from it draw a Line into the Center ; in the same quarter make a Geometrical square of any proportion at pleasure , so that two sides thereof may be parallel to the Horizontal and Vertical Line , at the intersection of one of the sides thereof , with the Vertical set A , and of the other side , with the Horizontal Line , set B , and where the Latitude Line intersects the side of the square , set F. To prick off the Substilar . For Latitudes above 45d take BF , and prick it on the Line of Declination beyond the Center from V to O , and from the point O , draw the Line OC parallel to the Vertical Line , and produced beyond O , if need require , and thereon from C to I prick the side of the Square , and a line drawn into the Center , shall be the Substilar line ; but for Latitudes under 45d prick the side of the square from V to O , and draw OC as before , and make CI equal to AF , and a line from I drawn into the Center , shall be the Substilar line . Stiles Height . From the Point I erect the extent OC perpendicularly to the substilar OI , at the extremity thereof set K , from whence draw a line into the Center , and the Angle IVK will shew how much the stile is to be elevated above the substilar line . In order to the Demonstration hereof , let it be observed that an Angle may be prickt off by Sines or Tangents in stead of Chords . To prick off an Angle by Sines or Tangents . As in the Scheme annexed , let BC be Radius , and let there be an arch prickt off with Chords , as BE ; I say , if the Tangent of the said Ark BA be taken out of a line of Tangents to the same Radius , and be erected perpendicular to the end of the Radius , as BA , a line drawn from A into the center , shall include the same Angle as was prickt off by Chords , as is evident from the definition of a Tangent . In like manner an Angle may be prickt off by Sines , the nearest distance from E to BC is the Sine of the Arch BE , so in like manner the nearest distance from B to EC , is the Sine of the same Arch ; wherefore if with the Sine of an Arch from the end of the Radius be described another Ark , as F , and from the Center or other extremitie of the Radius , a line drawn just touching the same , the Angle included between the said line , and the Radius shall be an Ark equal to the Ark belonging to the said Sine ; and what is here done by a line of natural Sines or Tangents , by help of a Decimal line of equal parts , equal to the Radius ' may be done by help of the natural Tables without them . Any Proportion relating to the sixteen cases of Sphoerical Triangles , amongst which the Radius is always ingredient , may be so varyed , that the Radius may be in the third place , and a Tangent or a Sine in the fourth place , and then if the Ark belonging to the fourth Proportional be known , an Angle equal thereto may be prickt off by Tangents or Sines according to the nature of the fourth term , as beforth ; if it be unknown , notwithstanding an Angle equal thereto may be prickt off with Sines or Tangents according to the nature of the fourth term , from the two first terms of the Proportion , because they are in such Proportion , the first to the second , as the Radius or third term is to the fourth ; upon this Basis follows the demonstration of this Scheme . The Demonstration of the former Scheme for upright Decliners . 1. That the Substilar Line is true prickt off . Assuming VB equal to VA to be Radius , then BF becomes the Co-tangent of the Latitude , whereto VO is made equal , which becoming Radius , OC is the Cosine of the Declination , and VC the Sine . Now one of the Proportions for finding the Substile distance from the Meridian is this following : As the Sine of the Plains Declination , Is to the Tangent of the Latitude : So is the Radius to the Tangent of the Substiles distance from the Horizontal Line . From whence it follows , that if the Sine of the Declination be pricked on the Horizontal Line from the Center , as VC , and the Tangent of the Latitude , erected perpendicularly thereto , as CI equal to VA , it shall give a point from whence a Line drawn into the center , shall be the Substilar Line . Now I am to prove that VA is the Tangent of the Latitude , thus it is made good : In a Tangent Line of 45d , if the Tangent of any part or ark of it be assumed to be Radius ( as here FB the Cotangent of the Latitude equal to VO , which before was made Radius ) then doth the said whole Tangent line ( which in this case is any side of the Square ) become the Tangent of that arks complement : thus , VA becomes the Tangent of the Latitude , because the Radius is a mean Proportional between the Tangent of an Ark , and the Tangent of that Arks complement ; For , As the Cotangent of the Latitude BF , Is to the Radius BG , So is the Radius BF , to the Tangent of the Latitude BG , there being the like Proportion between the two latter , as the two former terms . 2. That the Stiles Height is true prickt off . To perform this , the Cosine of the Declination OC , to that lesser Radius , was erected perpendicularly from the point I , in the Substilar 〈◊〉 , and IK made equal thereto . Now one of the Proportions for calculating the Stiles height is : As the Secant of the Latitude , Is to the Cosine of the Declination : So is the Radius , To the Sine of the Stiles Height . Consequently the Stiles height may be prickt off from the two first terms of the Proportion , if it can be proved that VK is the Secant of the Latitude ; here let it be remembred that VP is the Tangent of the Latitude , IP the Sine of the Declination , and IK the Cosine . By construction the Angles VPI and VIK are right Angles , therefore the square of VI is equal to the two squares of VP , and IP by 47 Prop. 1 Euclid ; again the square of VK is equal to the two squares of IK and IV , wherefore the square of VK is equal to the sum of the three squares of IK , IP and VP ; but the squares of PI a Sine , and IK its Cosine are equal to the square of the Radius . But the square of the Radius , and the square of VP , the Tangent of the Latitude , is equal to the square of VK , therefore VK is the Secant of the Latitude , because the square of the Radius , more the square of the Tangent of any Ark , is equal to the square of the Secant of the said Ark . 3. That VK is equal to VF . Let VF be an assumed Radius , so will FB be the Cosine of the Latitude . Again observe that in any line of Sines if the Sine of any Ark be made Radius , the whole line becomes the Secant of that Arks complement , so here VK being the Secant of the Latitude is equal to VF the Radius , for let VT be made equal to VF , I say it holds as VO the Cosine of the Latitude , is to the Radius VT , so is VO the Radius to VT the Secant of the Latitude , which is therefore equal to VK . Whence the trouble of raising a perpendicular in this Scheme , because the side of the square passeth through the complement of the Latitude , is shunned the point K falling in the outward circumference . Lastly , from all this it unavoidably follows that VI is the Cosine , and IK the Sine of the Stiles height to the Radius VF , which was to be proved . An upright South Dyal , Declining 30 degrees Eastwards , Latitude 51 degrees , 32 minutes . To draw the hour Lines of the former upright Decliner . First draw the Plains perpendicular CO , which for upright Plains is the Meridian line , and with the Radius VF of the former Scheme , draw an occult Ark upon C as a center , and therein set off NV equal to TN of the former Scheme , and draw the line IVC for the substile , and with the Radius VC describe a circle thereon . 1. The first work will be to finde the Regulating Point in the Substile , called the Point Sol. Prick the Radius of the said Circle from I to P , and from C to Q , and draw the Line IP , and therein make IK equal to IK in the former Scheme , then a ruler said from K to Q , where it intersects the Substilar is the Regulating Point ☉ . 2. To finde a Point from whence the Circle is to be divided into 12 equal parts . Lay a Ruler from O to the point ☉ , and it intersects the Circle on the Opposite side , at it set M. 3. To divide the Circle . From the point M before found , lay a ruler through the Center V , and it will finde a Point on the other side the circle , at it set f , the points Mf divide the Circle into halfs , and each half is to be divided into six parts , the Radius VM will easily divide each half into three parts , and then it will be easie to divide each of those parts into halfs , and so the whole Circle will be divided into 12 equal parts , and if it be desired to inscribe the half hours and quarters , then must each of those parts be divided into halfs and quarters . Having thus divided the Circle into 12 parts , distinguish them with the Letters of the Alphabet . 4. To draw the hour lines , and to number them . Lay a Ruler to each of those Letter divisions , and upon the Regulating Point ☉ in the Substile , and it will finde points on the Opposite side of the Circle from which if lines be drawn into the Center at C ' they shall be the hour lines required . 5. An inconvenience shunned . When the points through which the hour lines are to be drawn , fall near the Center , the hour lines cannot be drawn from the said points with certainty . In this case let any hour line near the said point be produced , if need require , and upon C as a Center with the Radius CV describe the Ark of a Circle from the produced hour-line , take half the distance between the point , where the produced hour line , cuts the Circle , and the point through which the hour line proposed is to be drawn , and prick that extent off in the Arch swept , setting one foot where the Arch swept cuts the hour line already drawn , from whence half the distance was measured , and the other foot will finde a Point in the said Arch , through which the hour line desired is to pass : Thus we inscribed the hour line of three in the former Dyal . When the North Pole is elevated , the Center of the Dyal must be below , but when the South Pole is elevated , as in this example , it must be above , and the Point ☉ must be alwayes found in the other part of the Diameter most remote from the Center . 6. The Stiles height may be transferred from the former Scheme into this , by pricking the Arch TK twice in this Circle , and it findes a point , from whence a line drawn into the Center , shall represent the Stile . That the hour lines are true Delineated . The Point Sol found in the Substile , divides the Diameter of the Circle drawn on the said Substile in such Proportion , as the Radius is to the Sineof the Stiles height , by reason of the equiangled Triangles , whose Bases bear such proportion as their perpendiculars . The next work performed by dividing the Circle into 12 equal parts , and finding points on the opposite side , by laying a ruler over those equal parts through the point Sol , carrieth on this Proportion . As the Radius is to the Tangent of the hour from the substile , So is the Sine of the Stiles height To the Tangent of the hour line , from the Substile , being the proportion used in all Dyals , whereby to set off the hours . In the third figure following , let BA represent the Substile , and let the Regulating Point be at M , so that BM bears such Proportion to MA , as the Radius doth to the Sine of the Stiles height , let the perpendicular BHE represent a line of Tangents , whose Radius is equal to BC , and the perpendicular AD , another line of Tangents to the same Radius both infinitely produced , and let BE represent the Tangent of some particular Arch or hour to be drawn , from E through the point M draw a right line to D , and the Proportion lies evident in right lines . As the Radius BM : Is to the Tangent of the hour line from the Subtile BE ∷ So is the Sine of the Stiles height MA To the tangent of the hour line from the substile AD . If from every degree of a Tangent line , lines be drawn into the Center of the Circle proper to that Tangent , they shall divide that Quadrant to which they belong into 90 equal parts , this follows from the definition of Tangents , but if the lines be drawn from every degree of the said tangent to the extreamity of the diamer , they shall divide a Semicircle into 90 equal parts by 20 Prop. 3. Euclid . because an Angle in the Circumference , is but half so much as it is in the Center . Thus the Semicircle BGA is supposed to be divided into 90 equal parts from the Tangent BE , by lines drawn to A , as is the line EGA , so is the other Semicircle from the other Tangent AD , by lines drawn to B , as is BFD ; Now if it can be demonstrated that the points GMF are in a right line , it follows that the same Proportions may be carried on from the equal divisions of these Semicircles , as were done in the right Tangent lines BE and AD , this Proposition being very well Geometrically demonstrated by my loving friend Mr. Thomas Harvy ; his Demonstration thereof shall hereafter follow . Having found the distance of the hour line , from the subsistle AB , if B be made the Center of the Dyal , the right line BF drawn into the Center , shall represent the hour line proposed , making the same Angle with the Substile , as was found by the proportion ; note that no hour can be further distant on one side or other of the Substile then 90d , and the said Ark of distance will be found in one of the Semicircles , and if the Center were not placed in the Circumference , the Angle found would be extended beyond its due quantity , in all upright Decliners or leaning Plains : The first Proportion carried on , is this ; As the Sine of the Stiles height : Is to the tangent of the distance between the Meridian and substile : So is the Radius : to the tangent of the inclination of Meridians : Whereby is found the point from whence the Circle is to begin to be divided into 12 parts for the whole hours with their halfs and quarters , and those equal divisions give the Angles between the Meridian of the plain , and the respective hours , called by some the Angles at the Pole , and then the work is the same as before : Now to the Proposition . Construction . The right lines AD , BE are parallel , and touch the Circle in the extreams of the Diameter AB , DE is drawn at pleasure , cutting the Tangents in D and E , and the Diameter in M , BD and AE cutteth the Circumference in F and G , it is required to be proved that the Points F , M , and G , are in a right line . Draw the lines MF , MG , first if AD be = BE then AM is = MB , for the Triangles ADM and BEM are equiangeld , therefore M is the Center . Again AD being = BE and the Angle DAB = EBA & AB common to both Triangles DAB , EBA therefore the Angles DBA , EAB are equal , and the double of them FMA , GMB the Angles at the Center , are also equal , wherefore FM and MG is one and the same right line which was to be proved . Secondly , if AD be not equal to BE , as in the second , third , & fourth Scheams let BE be greater , and make BH = AD and BI = AM , and draw the right line AH cutting the Circumference in K ; draw likewise IK , IH , GK , and extend GK infinitely both wayes , which shall be either parallel to AB ( as in the second figure ) or cut it in the point P one side or the other , as in the third and fourth figures , in each from the points A and B , and the Center C at right Angles to GK , draw the lines AO , BN , CL lastly draw BG , from which Construction it willfollow , That 1. HI is parallel to ED for HB is = AD and BI = AM and the Angles HBI and DAM are equal , therefore the angle AMD = BME is = BIH wherefore HI is parallel to ED. 2. IK is parallel to MF for HB is = AD and the Angle HBA = DAB , and AB common to both the Triangles HBA and DAB , therefore the Angles HAB and DBA are equal , wherefore AK = BF , but AK being = BF and IA = MB , and the Angle KAI = MBF , therefore KI is parallel to MF . 3. The Triangles AHE and AGK , are equiangled , so also are the Triangles AHB and AGO , because the Angle AKG = ABG is = AEB and EAH common to both the Triangles AHE and AGK , therefore they are equiangled ; again the Angles AHB and and AGO being equal , the right Angled Triangles ABH and AOG are also equiangled . 4. NL is equal to LO , because BC is equal to CA and NK is = GO , because KL is = LG ; now in the equiangled Triangles ABH , AOG as BH : HA ∷ OG : GA and in the Triangles AHE , AGK as HA : HE ∷ GA to GK , therefore ( ex equo ) it will be as BH : HE ∷ OG : GK , but as BH : HE ∷ BI : IM for IH is parallel to EM , therefore as BI : IM ∷ OG : GK but NK is = OG , therefore BI : IM ∷ NK : KG , but IC is half of IM and KL half of KG , therefore BI : IC ∷ NK : KL , therefore by Composition of Proportion , as BC : IC ∷ NL : KL , alternately , as BC : NL ∷ IC : KL . In the second figure BC is = NL , therefore IC is = KL , wherefore ( because IC is also parallel to KL ) IK is parallel to CL , by like reason GM is also parallel to CL , but in the third and fourth figures the sides PC , PL of the Triangle PCL are cut Proportionally by the parallel BN , and as before , as BC : NL ∷ IC : KL , and as PC : PL ∷ BC : NL , therefore PC : PL ∷ IC : KL , wherefore IK is parallel to CL . Again as before , BC : NL ∷ IC : KL , But CM is = IC and IG is = KL , therefore as BC : NL ∷ CM : LG , but as hath been said , as BC : NL ∷ PC : PL , therefore as PC : PL ∷ CM : LG , wherefore because the sides PM : PG , of the Triangle PMG are cut proportionally in C and L , CL and MG are parallel one to another . Now in all the three figures it hath been proved that IK and MG are either of them parallel to CL , therefore they are parallel one to another , wherefore the Angle GMI is equal to the Angle KIB which before was proved to be equal to AMF , therefore the Angles GMI and AMF are equal one to another , wherefore FM and MG is one and the same right line which was to be proved . To draw an East or West Dyal . Let the hour line of six , which is also the Substilar line , make an Angle equal to the Latitude of the place , with the Plains Horizontal line , above that end of it that points to the Coast of the elevated Pole , then draw a line perpendicular to the Substilar line , which some call a Contingent line , and with such a Radius as you determine the Stile shall have parallel height above the Substile : divide a Tangent line of hours and quarters according to the direction for direct Polar Recliners , then through those divisions draw lines parallel to the Substile and they shall be the hour lines required ; thus the hours of 5 and 7 , are each of them Tangents of 15d from six , and so for the rest , let them be numbred on each side of six ( being the Substile ) for an East dyal with the morning hours , for a West dyal with the afternoon hours , the one being the complement of the other to 12 hours , and therefore we have but one example , namely an East dyal for the Latitude of London . How to fill this or any other Plain , with any determined number of hours , shall afterwards be handled . A West Diall Reclining 50d Latitude 51-32′ An East Diall Inclining 40d Latitude 50d To prick off the requisites of an East or West Reclining or Inclining Dyal in their true Scituation and quantity . In these Plains , first draw the Plains perpendicular , and cross it with a Horizontal line , which is also the Meridian line ; in any one of the quarters , make a Geometrical square , as before directed for upright decliners , and from the Plains Vertical in the said quarter , count off the Latitude to L , and from the Horizontal line in the said quarter count off the Reclination or Inclination to R , and from L and R , draw lines into the Center , and where the Latitude line intersects the side of the square , set F. 1. To prick off the Substile . For Latitudes above 45d place BF from V the Center in the Horizontal line for Recliners Northwards , for Incliners Southwards , and thereto set O , and through the said point , draw a line parallel to the Vertical , and place the nearest distance from A to RV on the former parallel , from O for Recliners upwards to I , but for Incliners downwards , and a line thence drawn into the Center , shall be the Substile . 2. The Stiles height . Place the nearest distance from B to RV on a perdendicular raised from the point I in the Substile , and it findes the Point K , whence a line drawn into the Center , shall represent the Stile . In the other Hemisphere the words Northwards and Southwards , must be mutually changed . For Latitudes under 45′ The side of the square must be placed from V to O and AF must be placed on the line of Reclination or Inclination from V to C , the nearest distance from C to VA placed on the perpendicular , passing through O for Recliners upwards , but for Incliners downwards , findes the Point I , through which the Substile is to pass , and the nearest distance from C to VB raised perpendicularly on the point I in the Substile , findes the point K for the Stile , as before . Demonstration . 1. For the Substile . In these Plains VB or VA being Radius FB is the Contangent of the Latitude and the nearest distance from A to RV , is the Cosine of Reclination or Inclination to the same Radius , and the nearest distance from B to RV , the Sine thereof ; this for Latitudes above 45 , but for lesser Latitudes . AF the Tangent of the Latitude was made Radius , and thereupon the Radius or side of the square be-became the Cotangent of the Latitude , and the nearest distance from C to VA was the Cosine , and from C to VB the Sine of the Reclination or Inclination , so that the prescribed Construction in both cases erects the Cosine of the Re-Inclination on the Cotangent of the Latitude , which is made good from this proportion . As the Cotangent of the Latitude Is to the Cosine of the Re-Inclination from the Zenith So is the Radius : To the tangent of the Substilar from the Meridian . 2. For the Stile . A proportion that will serve to prick it off , is , As the Cosecant of the Latitude , is to the Sine of the Re-Inclination , So is the Radius : To the Sine of the Stiles height . page 25 A West Diall Reclining 50d Latitude 51 32′ A South Plaine Declining 40d East Inclining 15 deg A West Plain , Reclining 50 degrees , Latitude 51 degrees , 32 minutes . The Point Sol , Substile , Stile , and hour-lines , are all found after the same manner as in an upright Decliner , the Substile being set off from the Meridian line here , after the same manner as it was from the Meridian there . A South Plain Declining East 40 degrees , Inclining 20 degrees , Latitude , 51 degrees , 32 minutes . To prick off the Requisites in all Declining , Reclining , and Incliing Plains in their true Coast and quantity . Now followeth those Directions inlarged , which as I said in the Epistle , I received from Mr. Thomas Rice . UPon any Plain first draw a true Horizontal line , at the East end thereof set E , and at the West end W , cross the said line with a perpendicular , which may be called the Plains perpendicular , by some termed the Plains Vertical line , or line of Reclination , at the intersection of these two lines set V , and upon it as a Center describe a Circle ( the Radius whereof may be equal to 60d of a line of Chords ) at the upper end of the Vertical line set S , and at the lower end N. As the Declination is , set it off with Chords from S or N towards the true Coast , and at it set D , from whence draw a line through the Center ; also set off the Latitude of the place the same way and in the same quarter , and at it set L , from which draw a line to the Center . From that end of the Horizontal line , towards which the declination was counted , set the Inclination ( which as well as the Reclinais reckoned from the Zenith , the former being the Denomination of the under , the latter of the upper face ) upwards towards S , and the Reclination downwards towards N , and at it set R , from whence draw a line through the Center to the other side , of the Circle . In the same quarter of Declination , draw HA parallel to the Horizontal line , and FG parallel to the Vertical line , in a Geometrical square , of like and of any convenient distance from the Center at pleasure , and where the Latitude line intersects the side of the square , let the letter F be placed . On the line of Declination beyond the Center , make VO equal to FG , and draw OB parallel to the Horizontal line , continued till it meet with the side of the square FG produced , at the point of concurrence , set B , and where it intersects the Plains perpendicular , set P , and draw OC parallel to the Vertical line cutting WE at C , and make HA equal to OC or BG , now by help of the three points A , B , C thus found , the requisites will be easily prickt off . 1. The Substilar line . The nearest distance from A to RV , set on the line CO ( produced if need be ) from C to I the same way the distance was taken from A , that is , if downward or upward , the other must be so too , will shew where VI the Substilar Line is to be drawn . 2. The Stiles height . The least distance from B to RV , set on a perpendicular raised upon the Substile from the point I will finde the point K , from whence draw a Line into the center at V , and the Angle IVK will shew how much the Stile is elevated above the Substile , and if the work be true , VK and VF will be equal , whence it follows , that the trouble of raising the mentioned perpendicular may be shunned . 3. The Meridian line . The least distance from C to RV , set upon the Line OPB , from P on that side , which is farthest from the Line RV , will finde the Point M , from whence a Line drawn into the Center , shall be the meridian Line . And I adde that on all North Recliners in the Northern Hemisphere , the meridian Line must be drawn through the center on the other side ; and then the construction of the Scheam will place it below the Plains Horizontal Line , which is its proper Scituation for the said upper face , and for the under face the Scheam placeth it true without caution . 4. A Polar Plain how known . If the line RV fall just into B , the Plain is a Polar Plain , in such a Plain the Stile hath no height , but is parallel to the Axis , in this case the Inclination of Meridians must be known , directions for such Plains must afterwards follow . But if the line RV fall between the Points B and P , then must the Substile Stile and Meridian be all drawn through the Center , and stand beyond on the other side . Annotations on the former Scheam . 1. That for Latitudes under 45d this construction of the Scheam , supposeth the sides of the square produced , which will therefore be lyable to large excursions or other inconveniences , wherefore for such Latitudes , I shall somewhat vary from the construction prescribed . 2. In finding the Substilar line , in stead of erecting CI upon VC , you may prick the same on the Vertical line VN , and thereto erect VC , and get the point I possibly with more certainty by finding the intersection of two Arks where the said Point is to pass . 3. In pricking off the Meridian line , the distance of C from the Center may be doubled or tripled , but so must likewise VP , and the nearest distance from C to RV erected on a line drawn parallel to WE , passing through the Point P so found , and in stead of drawing such a parallel , the Point M may be found by the intersection of two Arks . 4. That this Scheam placeth the requisites of all Dyals in their true coast and quantity , yet notwithstanding if this Scheam be held before a Looking-Glass , the Effigies thereof in the Glass shews how the Scheam would happen and place the Requisites , namely , the Stile , Substile and Meridian , for a Plain of the same Denomination , but declining to the contrary Coast . And if the face of the said Scheam be laid upon a Window , and the Substile , Stile and Meridian be continued through the Center on the backside thereof , it shews you how these requisites are to be placed on the opposite side of the Plain , which being done , may be held before a Looking-Glass as before , and will be represented for the contrary Declination of that opposite face : the truth of all which will be confirmed from the Scheam it self . This Scheam for Declining , Reclining , or Inclining Plains , useth a new method of Calculation , derived from an Oblique Triangle in the Sphere , wherein there is two sides with the Angle comprehended , given to finde both the other Angles , which is reduced by a perpendicular to two right Angled Triangles , from which the following proportions are derived . I shall therefore first deliver the said method , then demonstrate that the said proportions are carried on in the Scheam ; and lastly , from the Sphere , shew how those proportions do arise . 1. To finde a Polar Plains Reclination or Inclination . As the Radius , Is to the Cosine of the Plains Declination , So is the Cotangent of the Latitude , To the Tangent of the Reclination or Inclination sought . 2. To finde the distance of the Substile or Meridian line , from the plains perpendicular for a Polar plain . As the Radius , Is to the Sine of a Polar Plains Reclination , So is the Tangent of the Declination , To the Tangent of the Substilar line from the Plains perpendicular . 3. The Inclination of Meridians . As the Radius , Is to the Sine of the Latitude , So is the Tangent of the Declination , To the Tangent of the Inclination of Meridians . Affections of a Polar Plain . The Substilar on the upper face , lies above that end of the Horizontal line , towards the Coast of Declination , and the Meridian lyes parallel to the Substile beyond it , towards that end of the Horizontal line that is towards the Coast of Declination . For Declining , Reclining , or Inclining Plains . First finde a Polar Plains Reclination for the same Declination . Then for South Recliners and North Incliners , get the difference , but for North Recliners and South Incliners , the sum of a Polar plains Reclination , and of the Re Inclination of the plain proposed , and then it holds . 1. For the Substile . As the Cosine of the said Ark of difference or sum according as the Plain leans Northward or Southwards , Is to the Sine of the Polar Plains Reclination , So is the Tangent of the Declination , To the Tangent of the Substilar from the Plains perpendicular . 2. For the Stiles height . As the Radius , Is to the Cosine of the Substiles distance from the Plains perpendicular , So is the Tangent of the Sum or difference of Reclinations , as before limited , To the Tangent of the Stiles height . 3. Meridians distance from the plains perpendicular . As the Radius , Is to the Sine of the Re Inclination , So is the Tangent of the Declination , To the Tangent of the Meridian from the Plains perpendicular . 4. Inclination of Meridians . As the Sine of the Stiles height , Is to the tangent of the distance between the Meridian and Substile , So is the Radius , to the tangent of the Inclination of Meridians . For South Recliners or North Incliners , the difference between the substiles distance from the plains perpendicular , and the Meridians distance therefrom , is equal to the distance between the Meridian and Substile ; the like for such North Recliners or South Incliners , as Recline or Incline more then an Equinoctial Plain , having the same declination , but if they lean above it , or have a lesser Reclination , the sum is the distance between the Meridian and the Substile . The three first Proportions , besides the finding of a Polar Reclination , are used in the Scheam for the placing of the Requisites , and the latter proportion in the Circular Scheam for drawing the hours . Another proportion for finding the Inclination of Meridians by Calculation , is : As the Cosine of the Latitude , Is to the sine of the substiles distance from the Plains perpendicular , So is the Cosine of the Re Inclination of the Plain , To the sine of the Inclination of Meridians . The Reclination of an Equinoctial Plain to any assigned Declination , is necessary for the determining of divers affections : The Proportion to finde it , is : As the Radius : Is to the Cosine of the Plains Declination : So is the Tangent of the Latitude , To the Tangent of the Reclination sought . The upper face of an Equinoctial Plain , is called a North Recliner , the Meridian descends from the end of the Horizontal line opposite to the Coast of Declination , the Substilar line is the hour-line of six , and maketh right Angles with the Meridian line . Directions for the true Scituating of the Meridian and Substile suited to the former method of Calculation . 1. For Plains leaning Northwards . If a South Plain recline more then a Polar Plain , having the same Declination , the Plain passeth beneath the Pole of the World , the North Pole is elevated upon the upper face , the Substile and Meridian line lye above that end of the Plains Horizontal line , towards the Coast of Declination , the Substilar line being next the Plains perpendicular . For the under face being a North Incliner , the South Pole is elevated , the lines lye in the same position below the plains Horizontal line , and on the contrary side of the plains perpendicular . If a South Plains Reclination be less then the Polar Plains Reclination , the Plain passeth above the Pole , and the North Pole is elevated on the under face , being the inclining side . The Substile and meridian lye above that end of the Plains Horizontal line that is opposite to the Coast of Declination , the meridian being nearest the Plains perpendicular , for the upper face being a South Recliner , the South Pole is elevated , and the lines lye in the same Position below the Plains horizontal line , but on the contrary side of the plains perpendicular descending below that end of the Horizontal line , opposite to the Coast of Declination . 2. For Plains leaning Southwards generally on the upper face the North Pole is elevated on the under face the South Pole . To place the Substile . Such North Recliners whose Reclination is less then the complement of a Polar plains Reclination , the Substile is elevated above the end of the Horizontal line contrary to the Coast of Declination , and on the under face , being a South Incliner , the Substilar is depressed below the end of the Horizontal line , opposite to the Coast of Declination . But when the Reclination is more then the complement of the Reclination of a Polar plain , the Substile is to lye below the plains Horizontal Line , from that end opposite to the Coast of Declination . But for South Incliners , being the under face , the Substile is elevated above the end of the Horizontal line , opposite to the Coast of Declination . To place the Meridian . On all North Recliners the Meridian lies below the Horizontal Line from that end thereof , opposite to the Coast of Declination , because at noon the Sun being South , casts the shadow of the Stile to the Northwards . On the under face , being a South Incliner , it must always be placed below the Horizontal Line , below that end of it toward the Coast of Declination . These Directions suppose the Declination to be denominated from the Scituation of that face of the plain on which the Dyal is to be made , and the Horizontal line for all Dyals that have Centers , is supposed to pass through the same . Now to the Demonstration of the former Scheam . 1. 'T is asserted that if RV fall into the point B , the plain is a Polar plain , in which case the Stile is parallel to the Axis of the world . Demonstration . Every Declining plain may have such a Reclination found thereto , as shall make the said plain become a Polar plain , and the proportion to finde it , may be thus : As the Tangent of the Latitude , Is to the Cosine of the Declination , So is the Radius , To the Tangent of the Reclination sought . In the former Scheam . if we make FG the Cotangent of the Latitude Radius , the side of the square will be the Tangent of the Latitude , now VO equal to FG , being Radius , OC equal to GB , is the Cosine of the Declination ; wherefore a Line drawn into the Center from B , shall include the Angle of a Polar plains Reclination agreeable to the two first terms of the proportion , and to the directions for pricking off an Angle by Tangents . 2. That the substile is true prickt off . Upon V as a Center with the Radius VB , imagine or describe a Circle then is BG equal to VP , the Sine of a polar plains Reclination , which is equal to HA , and the Ark comprehended between A and B , will be a Quadrant . But in a Quadrant any line being drawn from the Limbe passing through the Center , the nearest distance from the end of one of the Radij will be the Sine of the Ark thence counted , and the nearest distance from the other Radius thence counted , will be the Sine of the former Arks complement ; so in this Scheam the nearest distance from B to RV , when a plain Reclines , is the Sine of the Ark of difference , but when it inclines of the sum of the Reclination of the Plain proposed , and of the Reclination of a Polar plain , and the nearest distance from A to RV , is the Cosine of the said Ark . Make VQ on the plains perpendicular equal to IC , which is equal to the nearest distance from A to RV , and from the point Q erect the perpendicular QT , whereto the line PO will be parallel , and consequently there will be a proportion wrought : Thus it lyes , As VQ the Cosine of the sum of the Polar plains Reclination , and of the Reclination of the plain proposed , Is to PV equal to BG , the Sine of a Polar Plains Reclination . These two terms are of one Radius , namely VB , So is the tangent of the Declination QT , To the tangent of the substiles distance from the plains perpendicular PO , These two terms are to another Radius , namely QV , then if the Tangent of an Ark be erected on its own Radius , as here is QI equal to CV , and a line be drawn from the extreamity into the Center , the Angle belonging to that Tangent shall be prickt off agreeable to the general direction . 3. That the stiles height is true prickt off . The Proportion altered to bring the Radius in the third place will be , As the Secant of the substiles distance from the Plains perpendicular , Is to the tangent of the sum or difference of Reclinations , as before limited , So is the Radius , To the tangent of the stiles height . In the Scheam making VQ Radius , VI becomes the Secant of the substiles distance from the Plains perpendicular , and the nearstdiestance from B to RV , is the tangent of the difference or sum of the Reclinations , which when VB was Radius , was but the Sine thereof , the reason why it now becomes a tangent , is because the Consine of the said Ark VQ , is made Radius : But , As the Cosine of any Ark , Is to the Radius , So is the Sine of the said Ark , To the tangent of the said Ark . Therefore the nearest distance from B to RV equal to IK , being erected thereon , and a line from the extreamity drawn into the Center , shall prick off the stiles height suitable to the two first terms of the former proportion , and to the general direction for pricking off an Angle by Tangents . 4. That the Meridian is true prickt off , the Proportion to effect it : Is , As the Cotangent of the Plains Declination , Is to the Sine of the Re Inclination , So is the Radius , To the tangent of the Meridian from the Plains perpendicular . If VC be made Radius , then is CO equal to VP the Tangent of the complement of the Plains Declination , and the nearest distance from C to RV , is the Sine of the Re Inclination to the same Radius , which is erected perpendicularly on VP suitable to the two first terms of the proportion and the General Direction . Lastly , that VK is equal to VF , these Symbols are used , q signifieth square , + for more or Addition , = equal . VBq = VQq + IKq , the reason is because VB being Radius , VQ is the Cosine of an Ark to that Radius , and IK the Sine by construction . VBq = VGq + GBq , therefore these two squares are equal to the two before . If to the latter part of each of these Equations , we adde QIq or rather its equal POq , the sum shall be equal to VKq . I say then VQq + IKq + POq = VKq , this will be granted from the former Demonstration for upright Decliners . Again , VGq + GBq + POq = VFq , therefore VF is equal to VK , this cannot be denyed , because , VGq + FGq = VFq . And the two squares GBq , or rather VPq + POq are equal to GFq , which is equal to VOq by Construction . To draw the hour-lines for the former South Plain , Declining Eastwards 40 degrees , Inclining 15 degrees Latitude , 51 degrees , 32 minutes . First having assigned the Center of the Dyal , through the same draw the plains perpendicular , represented by the prickt line CN , and with the Radius of the former Scheam upon C as a Center , describe an Occult Ark , and therein set off NV equal to the substiles distance from the plains perpendicular , and through the point V draw the Substilar , and upon V as a Center , describe the Circle , and prick off the Meridians distance from the the Substile in the former Scheam , namely YX twice in this Circle from I to O , and draw CO for the Meridian , after the same manner set off the Stile , then finde the Regulating point Sol , divide the Circle , and draw the hour-lines according to former Directions , and when hour-lines are to lye both above and below the Center , they are to be drawn through . To fit the Dyalling Scheam , for Latitudes under 45 degrees . A South Plain Declining 50 degrees East , Inclining 20 degrees , Latitude 30 degrees . The former Construction would serve , if the sides of the square were produced far enough , but to shun any such excursion , make VO equal to the side of the square , and through the point F , draw a parallel to the Plains perpendicular , and where the parallel OP produced interesects it , is the point B , upon V as a Center , with the extent page 34 A South Diall Declin 40d East Inclin 15′ Latit 52-32 A South plain Declin 50d East Inclining 20d Lat 30d A South plain Declin 30d East Reclining 34-31 Lat 51-32 page 35 A South plaine Declin 30d East Reclining 25d Lat 51-32 page 37 VB , draw the Arch ZB , and prick off a Quadrant thereof from the point B , and it will finde the point A , the point C is found no otherwise then before ; and now having these three points , the whole work is to be finished according to those Directions , to which when the Stile hath a competent height , nothing need more be added , unless it be some examples . How to draw such Dyals whereon the Stile hath no elevation as Polar Plains , or but very small elevation , as in upright far Decliners , and many leaning Plains . These plains are known easily , for if the Re-Inclination pass through the point B , the plain is a Polar plain , and the Substile is to be found by the former Construction , which the Scheam makes the same with the Meridian : moreover , another Ark is to be found , called the Inclination of Meridians : The proportion to finde it , is : As the Radius , Is to the Sine of the Latitude , So is the tangent of the Declination , To the tangent of the Inclination of Meridians . If CO be made Radius , CV will be the Tangent of the Declination , which enter on VL from V to K , and the nearest distance from K to SV shall be the Tangent of the Inclination of Meridians , which is to be prickt on its own Radius from P to M , and draw a line from the Center , passing through M to the Limbe , whereto set F , so is the Arch NF the Inclination of Meridians sought , to wit , the Arch of time between the substile and Meridian . To draw the hour-lines . First draw a perpendicular on the Plain VN , and upon V as a Center , describe the Arch of a Circle , and from the Dyalling Scheam prick off the substiles distance , and draw IV which shall represent the same , as also the Inclination of Meridians from I to f , then upon V as a Center , describe as great a Circle as the plain will admit , and finde the point f , therein also , by laying a Ruler over V the Center , and f in the former Circle , and from the said point divide the Circle into twenty four equal parts for the whole hours ( but we shall not need above half of them ) then determine what shall be the parallel height of the Stile above the Substile , and prick the same on the Substilar line , from V the Center to I , through which point draw a Contingent line at right Angles to the Substile , and laying a Ruler from the Center over each of the divisions of the Circle , through the points where it intersects the Contingent line , if lines be drawn parallel to the Substile , they shall be the hour-lines required , the hour-line that belongs to the point f being the Meridian-line or hour-line of 12 , after the same manner are the halfs and quarters to be inscribed , if the Contingent line be too high , the Center V may be placed lower , if it be required , to fit so many hours precise to the Plain ; first draw it very large upon some Floor , and then it may be proportioned out for a lesser Plain at pleasure , as was mentioned for East or West Plains . A South Plain , Declining Eastwards , 30 degrees , Reclining 34 degrees , 31 minutes . The Scheam placeth all things right for Equinoctial Reclining Plains , without any further caution . To draw the Hour-lines on such Plains , where the Stile hath but small elevation . A South Plain Declining 30 degrees Eastwards , Reclining 25 degrees , Latitude 51 degrees 32 minutes . In these Plains , because the hour-lines will run close together , the Dyal must be drawn without a Center , by help of two Contingent lines , and first of all the Inclination of Meridians must be known , thereby is meant the Arch of time between the Substile and the Meridian line or hour line of 12 , and that may be found several ways ; here I shall follow the proportion in Sines before delivered . Making VI Radius prick the same on the Latitude line VL , and from the point found , take the nearest distance to the Horizontal line , place this extent from V to F , and draw CF ; then take the nearest distance from R to the Plains Vertical SN , which place from V to Q , then draw QT parallel to FC , so is VT the Sine of the Inclination of Meridians , which may be easily measured in the Limbe by the Arch SY , by drawing a line parallel to the Plains perpendicular from the point T , or by pricking the same on HA produced , and laying a Ruler thereto , or by drawing the touch of an Arch with VT upon S as a Center , then a Ruler laid from V touching the outward Extremity of that Arch , findes the point Y. Moreover , we need not make VI Radius , but prick the nearest distance from L to VE , from V upwards , then the nearest distance to the Plains perpendicular from the Intersection of the Substile , with the Limbe must be placed from V towards W , and from the two points thus found a line drawn , and the rest of the work , as before . To draw the Hour-lines . First draw the Plains perpendicular CN , and draw an Occult Arch , wherein prick down NY and NK , and draw the Substile and Stile as before , making the same Angles . Through any two points in the Substile , as at A and B , draw two right lines continued , making right Angles therewith . Draw a line parallel to the Stile at any convenient distance , which is to represent the new Stile , as here DE . Take the nearest distance from B to DE , and set it on the Substile from B to V , also the nearest distance from A to DE , and set it from V to C , through which point draw another line perpendicular to the Substile . Upon V as a Center , describe the Arch of a Circle of as large a Radius as the plain will admit , and from the substile on the same side thereof the Meridian happened in the former Scheam , set off the Inclination of Meridians , and it findes the point M , from whence divide the Circle into 24 equal parts , and draw lines from the Center V through those parts , cutting both the Contingent lines B and C , the respective divisions of the Contingent line C , must be transferred into the Contingent line A , and there be made of the like distance from the substile as in the said line C , then lines drawn through the Divisions of the two Contingent lines A and B , shall be the respective hour-lines required . A South Plain Declining Eastwards 30 degrees , Reclining 25 degrees , Latitude 51 degrees , 32 minutes . After the same manner must such East and West Recliners or Incliners , that have small elevation of the Stile , and upright far decliners be pricked down , and in these plains the Meridian many times must be left out , the proportion to finde the Inclination of Meridians for upright Decliners : Is , As the Radius , Is to the Sine of the Latitude , So is the Cotangent of the Plains Declination , To the Cotangent of the Inclination of Meridians . In the Dyalling Scheam , making CV Radius , CO is the Cotangent of the Declination , which enter on the Latitude line VL , and take the nearest distance to SV , which extent prickt upon CO , and it findes a Point , through which a line drawn from the Center to the Limbe , shall shew the inclination of Meridians to be measured from N. Otherwise : That we may not transfer large Divisions on the Contingent line from a small Circle , and that the Plain may be filled with any determined number of hours , such as by after Directions shall be found meet , draw any right line on a Board or Floor that shall represent the Plains perpendicular , as CN , and from the same set off the Substile and Stiles height from the general Scheam as before , drawing a line parallel to the Stile as DE , also a line perpendicular to the Substile , which I call the Floor Contingent , and that it may be large , let it be of a good distance from the Center ; from the point of Intersection at Y , take the nearest distance to the parallel Stile , which prick from Y to V , and upon V as a Center describe as large a Circle as may be with convenience , and from the Substile set off the Inclination of Meridians therein to M ( which Ark refers to V as its Center ) and from the said Point divide it into 24 equal hours ( or fewer , no more then are required ) and laying a Ruler over V , and those respective divisions graduate them on the large Contingent line , I say from this large Contingent line thus drawn and divided , we may proportion out the Divisions of two ( or many ) Contingent lines that are lesser , and thereby fill the Plain with any proper number of hours required . Then in Order to drawing the hour-lines on the Plain . The first work will be to draw the larger contingent line on the plain , which may be drawn anywhere at pleasure : for performing whereof , note , that what Angle the Substile makes with the Plains perpendicular , the Contingent line is to make the same with the Horizontal line , and the complement thereof , with the Vertical line ; also draw another Contingent line above this , parallel thereto at any convenient distance . In the bigger Contingent line assume any two points to limit the outward most hours that are intended to be drawn on the Plain , as admit on the former Plain , I would bring on hours from six in the morning , to two in the afternoon , between the space A and B of the greater Contingent , take the said extent AB , and upon the point B at 2 of the Floor Contingent , describe an Ark therewith , to wit , L , then from A Draw the line AL , just touching the outward extreamity of the said Ark . I say the nearest distance from D to AL , being pricked on the Plains greater Contingent from A to D , findes a point therein through which the Stile is to pass . Also the nearest distance from Y to AL , findes the space AY on the Plains greater Contingent , and through the point Y a line drawn perpendicular to AB , shall be the Substilar line . The Stile is to be drawn through the Point D , making an Angle with the Plains Contingent line equal to the complement of its height above the Substile in this example 80d 57′ , to wit , the Arch NY. The respective nearest distances to AL , from each hour-point , in the Floor Contingent , being pricked on the Plains greater Contingent from A towards B , findes Points therein , through which the hour-lines are to pass . The next work will be to limit one of the extream hours on the Plains lesser Contingent , and that must be done by proportion . As the Distance between the parallel stile and substile on the Floor Contingent , Is to the distance between the stile and substile on the Plains lesser Contingent , So is the distance between the substile and either of the extream or outward hours on the Floor Contingent , To the distance between the substile , and the said outward hour on the Plains lesser Contingent . A South Diall Declining 30d East Reclining 25d Lat 51-32 This proportion is to be carryed on in the Draught on the Floor , place DY from A to G on the Floor Contingent , and with the extent EG taken from the Plains lesser Contingent , upon G on the Floor Contingent , draw the Arch O , and from A draw a line touching the outward extreamity of the said Arch , and let it be produced . The nearest distance from Y to AO , being prickt on the Plains lesser Contingent , reaches from G to C , the point limiting the outward hour of six . Then if the nearest distances to the line AO , be taken from all the respective hours on the Floor Contingent , and placed on the Plains lesser Contingent from C towards F , you will finde all the hours points required , through which and the like points on the Plains greater Contingent , the hour-lines are to be drawn . Here note , that the extent DY on the Floor Contingent , may be doubled or tripled ; if it be tripled it reacheth to H , also the extent EG on the plains lesser Contingent , is to be encreased after the same manner , and an Ark therewith described on H before found , as Q , and by this means the line AO will be drawn and produced with more certainty , then by the Ark O near the Center . And what is here done by help of the Stiles distance from the Substile , may be done by help of the outward hour , if the distance of the said hour-line from the Substile be found Geometrically or by Calculation , for the said hour-line will make an Angle with the Plains Contingent lines , equal to the complement of the Ark of its distance from the Substile . This Plain is capable of more hours which cannot conveniently be brought on . After the same manner are upright far Decliners to be dealt withall , and all other Plains having small height of Stile . But to limit the outward hours on Polar plains , and East or West Plains , the trouble will not be half so much . A South Plain Declining Eastwards 30 degrees , Reclining 25 degrees , Latitude 51 degrees , 32 minutes . A second method of Calculation for Oblique Plains . By the former method the Meridians distance from the Plains perpendicular is to be found , and the Polar Reclination Calculated . Then for South Recliners , or North Incliners , get the difference , but for North Recliners or South Incliners the sum , of the Polar Plains Reclination , and of the Reclination of the Plain proposed , and it holds . As the Cosine of the Polar Plains Reclination , Is to the Sine of the former sum or difference , So is the sine of the Latitude , To the Sine of the stiles height . Which Pole is elevated is elevated is easily determined , by comparing the Reclination of the proposed Plain with the Polar Reclination , and all other affections are to be determined , as in the first Method . Then for the Substile , and Inclination of Meridians . As the Cosine of the stiles height , Is to the Sine of the Plains Declination , So is the Cosine of the Latitude , To the Sine of the substiles distance from the Plains perpendicular : And so is the Cosine of the Reclination , To the Sine of the Inclination of Meridians . This method ariseth from the aforementioned Oblique Triangle in the Sphere , in which by help of two sides , and the Angle comprehended , the third side is first found , and the other Requisites by the proportions for Opposite sides and Angles . Proportions for upright Decliners . 1. To finde the Substiles distance from the Meridian . As the Radius , Is to the Sine of the Declination , So is the Cotangent of the Latitude , To the Tangent of the Substile from the Meridian . 2. Angle of 12 , and 6. As the Radius , Is to the Sine of the Plains Declination , So is the Tangent of the Latitude , To the Tangent of the Angle between the Horizontal line and six . 3. Inclination of Meridians . As the Sine of the Latitude , Is to the Radius , So is the Tangent of the Declination , To the Tangent of the Inclination of Meridians : 4. Stiles height . As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Plains Declination , To the Sine of the stiles height . These Arks are largely defined in my Treatise , The Sector on a Quadrant . For East and West Re-Incliners , The complement of the Latitude of the place , is such a new Latitude : in which they shall stand as upright Plains , and the complement of their Re-Inclination is their new Declination in that new Latitude , having thus made them upright Decliners , the former proportions will serve to Calculate all the Requisites . In all upright Plains , the Meridian lyeth in the plains perpendicular , and if they Decline from the South ( in this Hemisphere ) it is to descend or run downward ; if from the North it ascends , and the Substile lyeth on that side thereof opposite to the Coast of Declination . In East or West Re-Incliners , it lyeth in the plains horizontal line , on the Inclining side the South Pole is elevated , but on the upper side the North Pole , and the Substile lyeth above or below that end of the Meridian line , which points to the Pole elevated above the Plain . On all plains whatsoever to Calculate the hour distances . As the Radius , Is to the Sine of the stiles height above the substile , So is the tangent of the Angle at the Pole , To the tangent of the hour-lines distance from the substilar line . By the Angle at the Pole , is meant the Ark of difference between the Ark called the Inclination of Meridians , and the distance of any hour from the Meridian for all hours on the same side of the Meridian the Substile falls , and the sum of these two Arks for all hours on the other side the Meridian . All hours on any Plain go to the contrary Coast of their Scituation in the Sphere , thus all the morning or Eastern hours , go to the Western Coast of the plain , and all the evening or Western hours , go to the Eastern Coast of the Plain . A third Method of Calculation for leaning Plains , that is , for all sorts of Plains that do both Decline , and also Incline or Recline . They may be referred to a new Latitude , in which they shall stand as upright Plains , and then they will have a new Declination in that new Latitude ; which two things being found , the former Proportions for upright Decliners will serve to Calculate all the Arks required . How this may be done on a Globe , is not difficult to apprehend , having set the Globe to your Latitude , let one of the Meridians of the Ecliptick or Longitude in the heavens , represent a Declining Reclining Plain , this Circle intersects the Meridian of the place in two Points , the one above , the other beneath the Horizon : Imagine the Globe to be so fixed , that it cannot move upon its Poles , then elevate or depress the Globe so in the Meridian that the point of Intersection above the Horizon may come under the Zenith , then will the Pole of the world be elevated above the Horizon to the new Latitude sought , and where the Meridian of Longitude that represents the Plain intersects the Horizon it shews the new Declination . Or it may be thus apprehended : The distance between the Pole of the world , and that point of Intersection that represents the Zenith of the new Latitude , is the complement of the said new Latitude , and the distance between that point , and the Equinoctial is the new Latitude it self ; the new Declination is the complement of the Angle between the plain and the meridian of the place , an Ark usually found in Calculation under this denomination . To finde these Arks by Calculation . As the Radius , Is to the Cosine of the Plains Declination , So is the Cotangent of the Re-Inclination from the Zenith , To the tangent of the Meridional Ark , namely the Ark of the Meridian between the Plain and the Horizon . And this is the first thing Master Gunter and others finde ; for South Recliners North Incliners the one being the upper , the other the under face , get the difference between this Ark and the Latitude of the place , the complement of the said residue , remainder , or difference , is the new Latitude sought ; but for North Recliners or South Incliners , the difference between this fourth Arch and the complement of the old Latitude is the new Latitude . To finde the new Declination : As the Radius , Is to the Cosine of the Re-Inclination , So is the sine of the old Declination , To the sine of the new . This method is hinted to us in Mr. Fosters Posthuma , also in his Book of Dyalling in Anno 1638 , where he refers leaning plains to such a Latitude wherein they may become East or West Recliners , but that method is to be deserted , as multiplying more proportions then this , and doth not afford that instrumental ease for pricking down the hours that this doth . Affections determined . Such South Recliners , whose meridional Arch is less then the Latitude , pass beneath the Pole , and have the North Pole elevated above them , but if the meridional Ark be greater then the Latitude , they pass above the Pole , the North Pole is elevated on the under face , all other affections are before determined . If the meridional Arch be equal to the Latitude , the plain is a Polar plain ; for plains leaning Southwards , if the meridional Arch be equal to the complement of the Latitude , the plain is an Equinoctial plain , if it be more , the plain hath less Reclination then an Equinoctial plain , if it be less it hath more , and all affections necessary for placing ( and Calculating ) the meridian line were before determined . This method of Calculation findes the Substiles distance from the meridian , not from the plains perpetdicular , wherefore it must be shewed how to place it in Plains leaning Southwards , for plains leaning Northwards use the former directions . To place the Substile in North Recliners . In these plains the Meridian and Substilar are to meet at the Center , and not being drawn through , will make sometimes an Acute , sometimes an obtuse Angle . When the Plains Meridional Ark is greater then the Colatitude , they make an Obtuse Angle , in this Case , having first placed the Meridian line , above it prick off the complement of the distance of the Substile from Meridian to a Semicircle . But when the Meridional Ark is less then the Colatitude , prick off the said distance it self above the Meridian line . In South Incliners . When the Plains Meridional Ark is greater then the Colatitude , the Substile and Meridian make an Acute Angle , when it is equal to the Colatitude , they make a right Angle , when it is less then the Colatitude they make an Obtuse Angle , and must be prickt off by the complement of their distance to a Semicircle , the Substile always lying on that side of the Meridian , opposite to the Coast of Declination . A fourth Method of Calculation for leaning Plains . An Advertisement . In this method of Calculation for all Plains leaning Northward , both upper and under side their Declination is the Arch of the Horizon between the North and the Azimuth of the plains South Pole , so that their Declination is always greater then a Quadrant ; But for all Plains leaning Southwards , both upper and under face , their Declination is the Arch of the Horizon between the North and the Plains North Pole , wherefore it is always less then a Quadrant ; in this sense Declination is used in the following Proportions . As the Sine of half the sum of the complements , both of the Latitude and of the Reclination , Is to the Sine of half their difference , So is the Contagent of half the Declination , To the Tangent of a fourth Arch. Again , As the Cosine of half the sum of the former Complements , Is to the Cosine of half their difference , So is the Contangent of half the Declination , To the tangent of a seventh Arch. Get the sum and difference of the fourth and seventh Arch , then if the Colatitude be greater then the complement of the Reclination , the sum is the Substiles distance from the plains perpendicular , and the difference the Inclination of Meridians . But if it be less , the difference is the Substiles distance from the Plains perpendicular , and the sum the Inclinations of Meridians . To place the Substile . For Plains leaning Southwards , when the Angle of the Substile from the Plains perpendicul is less then a Quadrant , it will on the upperface lye above that end of the Horizontal line that is opposite to the Coast of Declination , and on the under face lye beneath it , but when it is greater , it will lye below the said end , on the upper face , and above it on the under face , but this will not be till the Reclination be more then the complement of the Reclination of a Polar plain that hath the same Declination ; for plains leaning Northwards the Directions of the first Method suffice . To place the Meridian . Either Calculate it , and place it according to the directions of the first and second Method , or else Calculate it by this Proportion . As the Radius , Is to the Sine of the Stiles height , So is the tangent of the Inclination of Meridians ( when it is Obtuse , take its complement to a Semicircle ) To the tangent of the Meridian line from the substilar . For Plains leaning Northward , the first directions must serve , but for Southern Plains the second , because the distance of the Meridian is Calculated from the Substile supposed to be placed , and here the work is converse to that , for in that we supposed the Meridian placed , and not the Substile . For the Stiles height . As the Sine of the fourth Arch , Is to the Sine of the seventh Arch , So is the tangent of half the difference of the complements both of the Latitude and Reclination , To the tangent of an Arch sought . How much the said Ark being doubled wants or exceeds 90d , is the Stiles height . In South Recliners , if the said Ark being doubled , is less then 90d , its complement is the elevation of the North Pole , and the Plain falls below the Pole . But if the said Arch exceed 90d the Plain passeth above the Pole , and the excess is the elevation of the North Pole on the under face of a South Recliner , called a North Incliner , and the affections were determined in the first method where the Declination hath its Denomination from that Coast of the Meridian to which the Plain looketh . These methods of Calculation may not precisely agree one with another , though all true , unless the parts Proportional be exactly Calculated from large Tables in every Operation , which to do as to the Examples in this Book , my leisure would not permit ; This last method is derived also from the former Oblique Triangle , the Proportions here applyed , being demonstrated in Trigonometria Brittanica by Mr. Newton . The Demonstration of the former Proportions In projecting the Sphere , it is frequently required to draw an Arch through any two different Points within a Circle , that shall divide the said Circle into two equal Semicircles Construction . 1. Draw a line from one of the given points through the Center , for conveniency through that point which is most remote . 2. From the Center raise a Radius perpendicular to that line . 3. And from the said point draw a line to the end of the Radius . 4. From the end of the Radius raise a line perpendicular to the line last drawn , and where it intersects the former line drawn through the first point and Center , is a third point given , describe a Circle through these three Points , and the Proposition will be effected . Example . Let it be required to draw the Arch of a Circle through the two points E and F that shall divide the Circle BD into two equal parts . Operation . From E draw EG , through the Center A , make AD perpendicular thereto , joyn ED , and make DG , perpendicular to ED cutting EG in G , through E , F , and G , draw the Arch of a Circle which will divide the Circumference BDC into two equal parts in B and C , that is , if CA be drawn , it will pass through B , if not , let it pass above or below , as let it pass below and cut BFE in H. Demonstration . By construction EDG and DAG are right Angles ; therefore □ AD = ▭ EAG by 13 Prop. 6 Euclid . because EDG being a right Angle , AD is a mean Proportional between EA and AG , but ▭ EAG should be = ▭ CAH by 35. Prop. of 3 Euclid . therefore ▭ CAH = □ AD . But □ AD = □ AI that is = ▭ CAI , therefore ▭ CAH = ▭ CAI which is absurd , therefore CI cannot pass below B , the same absurdity will follow if it be thought to pass above it , therefore CA produced , will fall in the point B , wherefore BDC is a Semicircle , which was to be proved : And hereof I acknowledge I have seen a Demonstration by the Learned teacher of the Mathematicks , Mr. John Leak , to this effect . To project the Sphere and measure off the Arks of an upright Decliner . Upon Z as a Center , describe the Arch of a Circle , and cross it with two Diameters at right Angles in the Center , whereto set NESW to represent the North , East , South and West . Prick off the Latitude from N to L , and lay a Ruler to it from E , and where it cuts NZ , set P to represent the pole . Prick off the Declination of the plain from E to A , and from S to D , and draw the Diameter AZB , which represents the plain , and DZC , which represents the Poles thereof . Through the three points CPD , draw the Arch of a Circle , and there will be framed aright Angled Triangle ZHP right Angled at H , in which there will be given the side ZP the complement of the Latitude , with the Angle PZH the complement of the Declination . Whereby may be found the Stiles height represented by the side PH , the Substiles distance from the plains perpendicular represented by ZH , and the Angle between that Meridian which makes right Angles with the plain , and the Meridian of the place represented by the Angle ZPH , shewing the Arch of Time between the Substile and meridian , called the Inclination of Meridians , from which Triangle are educed those proportions delivered for upright Decliners . To measure off these Arks . 1. The Substile . A Ruler laid from D to H , findes the point F in the Limbe , and the Arch CF is the measure of the Substiles distance from the meridian , to wit , 21d 41′ . 2. The Stiles height . Set off a Quadrant from F to G , lay a Ruler from G to D and where it intersects BZ , set ☉ which is the Pole of the Circle CPD , lay a Ruler from ☉ to P , and it intersects the Limbe at I , so is the Arch AI the measure of the Stiles height , to wit , 32d 32′ 3. Inclination of Meridian . Lay a Ruler from P to ☉ , and it intersects the Limbe at T , and A South plaine Declin 30d East Latitud 51d 32′ page 51 A West plaine Reclining 50d Latitude 51d 32′ the Arch WT is the measure of the Inclination of meridians , to wit , 36d 25′ . 4. Angle of 12 and 6. In like manner we may draw a Circle passing through the points WPE , as the prickt Arch PE doth , then in the Triangle ZPQ right Angled at P , we have the side ZP given , and the Angle PZQ to finde the side ZQ , lay a Ruler from D to Q , and you will finde a Point in the Limbe , the distance whereof from C is the measure of the Arch sought , to wit , 57d 49′ , to be measured by projection as the Inclination of Meridians . Lastly the hour-lines , these are represented by meridians drawn through the Poles of the World , as in the Triangle HPQ there will be given the Stiles height PH , and the Angle HPQ , to wit , the Ark of difference between the Inclination of Meridians and the hour from noon , for all hours on that side of the meridian the Substile falls , but on the other side the sum of these two Arks , and this Angle is called the Angle at the Pole ; the side required is HQ , the distance between the Substile and the hour line proposed , which must be any hour , though in this Scheam it represents the horary distance of six from the Substile . To project the Sphere for an East or West Reclining or Inclining Plain , Latitude 51 degrees , 32 minutes , a West Plain , Reclining 50 degrees . Having drawn the Fundamental Scheam , and therein set off the Latitude as before , count the Reclination from N to R , and by laying a Ruler , finde the point A , through it and the North and South points draw the Arch of a Circle which shall represent the plain , finde the pole thereof by setting off a Quadrant from R to G , then through the pole of that Circle ☉ , and the pole of the World P , draw the Arch ☉ PR , then in the Triangle NHP right angled at H , we have given the side NP the Latitude , and the Angle PNH the Reclination , to finde PH the Stiles height , and NH the Substiles distance from the Meridian , and the Angle NPH the Inclination of Meridians , which is also represented by the Angle BPS . 1. To measure the stiles height . Set off a Quadrant from B to C , and draw a Line through the Center , and where it intersects the plain at F , is the Pole of the plains Meridian , lay a Ruler from F to H , and it cuts the Limbe at I. Also lay it thence to P , and it cuts the Limbe at K , the Arch IK is the measure of the Stiles height , to wit , 36d 50′ 2. The Substile . A Ruler laid from ☉ to H , cuts the Limbe at M , and the Arch NM is the measure of the Substiles distance from the Meridian , to wit , 38d 59′ . 3. The Inclination of Meridians . Set off a Quadrant from K to O , and lay a Ruler from it to F , and it intersects the Meridian of the plain at Q , then a Ruler laid from P to Q , findes the point T in the Limbe , and the Arch ST is the measure of the Inclination of Meridians , to wit , 53d 26′ . Otherwise with less trouble lay a Ruler from P to F , and it intersects the Limbe at V , and the Arch EV is the measure of the Inclination of Meridians , as before . To project the Sphere to represent a Declining Reclining Plain . A South Plain Declining 40 degrees East , Reclining 60 degrees , Latitude 51 degrees , 32 minutes . Having drawn the fundamental Circle , prickt off the Declination , and found the pole point as before , prick the Reclination from B to I , and laying a Ruler to it from A , finde the point R , and through the three points BRA describe a Circle , representing the plain , also from K finde the pole thereof ☉ , and through the two points P and ☉ draw the Arch of a Circle FG , representing the plains Meridian , at the intersection of the plain with the Meridian set Z , and draw the Arch of a polar plain through P to S , and there will be several Triangles Constituted , from which were derived the several Methods of Calculation . In the right Angled Triangle ANZ there is given NA the complement of the Declination , and the Angle NAZ the complement of the Reclination , whereby may be found ZN the plains meridional Ark , which taken from NP rests ZP the complement of the new Latitude ; also the Angle NZA which is the complement of the new Declination , and hence were derived the Proportions for the third Method , likewise in the same Triangle may be found ZA the meridians distance from the Horizon . In the Oblique Angled Triangle CP ☉ , there is given the side CP , the complement of the Latitude , the side C ☉ , the complement of the Reclination with the Angle PC ☉ , the complement of the Declination from the South to a Semicircle , whereby may be found the Angle CP ☉ , the Inclination of Meridians , and the Angle C ☉ P whereof the measure is RH the distance , of the Substile from the Plains perpendicular , and the third side P ☉ , the complement whereof is PH the Stiles height , and from hence was derived the third method of Calculation suited to Proportions for finding both the unknown Angles of an Oblique Spherical Triangle at two Operations , when there is given two sides with the Angle comprehended between them . The first method of Calculation is built upon the perpendicular Trigonometrie , for the perpendicular PS reduceth the former Oblique Triangle , to two right Angled Triangles , to wit , the right Angled Triangle , PSC , and the right Angled Triangle PS ☉ , both right Angled at S. In the right Angled Triangle PSC , we have CP given the Colatitude , and the Angle PCS the Declination to finde SC a Polar plains Reclination thereto . Again , In Oblique Spherical Triangles , reduced to two right Angled Triangles by the demission of a perpendicular , it is a common inference in every book of Trigonometry , when two sides with the Angle comprehended are given , to finde one of the other Angles : That , As the Sine of the Side between the Angle sought and perpendicular , Is to the Tangent of the given Angle , So is the Sine of the Side between the Angle given , and perpendicular , To the Tangent of the Angle sought . And so in that Oblique Triangle , the difference between the Reclination of the plain proposed , and the Polar plain is RS , then because R ☉ is a Quadrant , S ☉ is the complement of the former Ark , therefore it holds : As s SO : t SCP ∷ s SC : t S ☉ P which is the very proportion delivered delivered in the said Method for finding the Substiles distance . Then in the right Angled Triangled PS ☉ , we have S ☉ , and the Angle S ☉ P to finde the side P ☉ , whereby is got the Stiles height ; the Inclination of Meridians is found in the Oblique Spherical Triangle by the proportion of Opposite sides and Angles . Lastly , in the right Angled Triangle ZRC , there is given RC , and RCZ to finde RZ the distance of the Meridian from the plains perpendicular . To measure the respective Arks abovesaid . 1. The New Latitude . A Ruler laid from W to Z , and P will give you the Arch MO in the Limbe , the complement of the new Latitude , to wit , 27d 40′ . 2. The New Declination . The Ruler laid from the Plains Zenith at Z , to its Pole at ☉ , findes the point Q in the Limbe , and the Arch SQ is the new Declination , to wit , 18d 56′ . 3. The Substiles distance from the Plains perpendicular . A Ruler laid from ☉ to H , findes the point T in the Limbe , and the Arch CT being 26d 26′ is the Substiles distance from the Plains perpendicular . 4. The Meridians distance from the Plains perpendicular . A Ruler laid from ☉ to Z , findes the point V in the Limbe , and the Arch DV being 35d 56′ is the Meridians distance from the Plains perpendicular . 5. The Stiles height . Set off a Quadrant from G to X , and draw XC , where it intersects the plain as at Y , is the Pole of the Arch FG , then laying a Ruler from Y to H and P , you shall finde the Stiles height in the Limbe to be the Arch 2 , 3 , namely , 26d 6′ . A South plaine Declin 40d East Reclining 60d Lat 51-32′ page 55 A South plaine Declin 30d East Reclining 25d Lat 51 32′ 6. The Inclination of Meridians . A Ruler laid from P to Y , intersects the Limbe at ¶ , the Arch W ¶ is 20d 58′ , and so much is the Inclination of Meridians . The Polar Reclination CS is 31d 19′ . The Scheam determineth all the affections of the plain . 1. It shews that H the point of the Substilar lies on that side the plains perpendicular , that is towards the Coast of Declination . 2. That Z the point for the place of the Meridian , lyes towards the same Coast as before , but below the Substilar Line . 3. The Arch PH shews you that the North Pole is elevated above the upper or Reclining face . After the same manner may all the Requisite Arks be measured , and affections determined for all plains whatsoever . A South Plain Declining 30 degrees Eastwards , Reclining 25 degrees , or a North Plain Declining 30 degrees Westwards , Inclining ●● degrees . In this Scheam we have the same Oblique Triangle PC ☉ reduced to two right Angled Triangles PSC and PS ☉ , SC is the Inclination of a Polar plain , and RC the Inclination of the plain proposed , the difference is SR , and the complement of it , is the complement of S ☉ to a Semicircle , because S ☉ is greater then a Quadrant , and the proportions are wholly the same , though the Triangle have sides greater then a Quadrant . The North Pole is elevated on the Inclining face , the Meridian Z lyes from the plains perpendicular towards that end of the Horizontal Line , opposite to the Coast of Declination , the same way and beneath it lyeth the Substilar . The complement of the new Latitude ZP is 10d 10′ The new Declination , viz. the complement of NZA is 26d 57′ The meridians distance from the Plains perpendicular RZ 13d 43′ The Substiles distance therefrom RH 18d 22′ The Stiles height PH is — 9d — 3′ The Inclination of Meridians , to wit , the Angle CP ☉ 27d 18′ The Polar Reclination — CS — 34d 31′ A South Plain Declining 40 degrees East , Inclining 15 degrees , or rather a North Plain Declining 40 degrees West , Reclining 15 degrees . Here again the Oblique Triangle CP ☉ is reduced to two right Angled Triangles PSC and PS ☉ , and SR is the sum of the Polar Reclination SC , and the Re Inclination of the Plain proposed CR , and S ☉ is the complement hereof , because ☉ R is a Quadrant , finde the Equinoctial point AE . The Polar Reclination CS 31d 19′ . The new Latitude ZAE is 32d 15′ . The new Declination being the complement of NZA is 38d 23′ The Meridians distance from the plains perpendicular ZR 12d 33′ The Substiles distance from the plains perpendicular RH 32d 16′ The Stiles height PH is — 41d 30′ The Inclination of Meridians ☉ PC rather the Acute Angle ☉ PN is 55d 58′ The Meridian Z lies from the plains perpendicular towards the Coast of Declination on the Reclining side , but must be drawn through the Center , because the Sun at noon casts his shadow Northwards , unless in the Torrid or Frozen Zone , and the Substile Hlyes on the other side the plains perpendicular . A North Plain Declining 40 degrees Eastwards , Reclining 75 degrees . In this plain likewise the Oblique Triangle C ☉ P is reduced to two right Angled Triangles PS ☉ and PSC by the perpendicular PS which is part of the Arch of a polar plain , here CR more CS is equal to the sum of the Plains Reclination proposed , and of the Polar plains Reclination , which is greater then a Quadrant for the Arch R ☉ is a Quadrant ; now the Cosine of an Ark greater then a Quadrant is the Sine of that Arks excess above a Quadrant , wherefore the Sine of S ☉ is the Cosine of the sum of both the Reclinations , and the Case the same as before . A South plaine Declin 40d East Inclining 15d Lat 51d 72′ page 56 A North plain Declin 40d East Reclin 75d lat 51d 32′ Prick off a Quadrant from G to X , and draw XC , it cuts the plain at Y , a Ruler laid from Y to H , and P findes the points 2 , 3 in the Limbe , and the Arch 2 , 3 being 61d 31′ , is the Stiles height ; orther complement thereof to a Semicircle might be found by measuring the Arch PF . A Ruler laid from ☉ to Z , findes the point V in the Limbe for the Meridian Line , from which draw a Line through the Center on the other side , and it will be placed in its true Coast and quantity from the Plains perpendicular at A , to wit , 39d 2′ . The Inclination of Meridians , to wit , the Angle CP ☉ is 20d 30′ , The new Latitude ZAE 26d 38′ . The new Declination is 9d 35′ to wit , the complement of KRB. The Polar Reclination CS is 31d 19′ . The truth of this Stereographick Projection is fully handled by Aguilonius in his Opticks , and how to determine the affection of any Angle of an Oblique Spherical Triangle , I have fully shewed in a Treatise , called the Sector on a Quadrant . For the Resolution of Spherical propositions , Delineations from proportions or the Analemma , will be more speedy and certain ( though they may also be thus resolved ) which I have handled at large in the Mariners Plain Scale new plain'd . To determine what hours are proper to all kinde of Plains . To do this it will be necessary to project upon the Plain of the Horizon , the Summer and Winter Tropicks . Get the Sum and difference of the Colatitude , and of the Suns greatest Declination , so we shall obtain his greatest and least Meridian Altitudes . The depression of the Tropick of Cancer under the Horizon , is equal to the least Meridian Altitude , and the depression of the Tropick of Capricorn to the greatest . Example : 38d 28′ Colatitude 23d 31′ 61d 59′ greatest 14d 57′ least Meridian Altitude , having drawn the Primitive Circle , &c. as before . Prick 14d 15′ from S to C , and 61d 59′ from S towards W , a Ruler laid from the points found , will intersect the meridian ZS at the point L for the Winter Tropick , and K for the Summer Tropick , through which the Circles that represent them are to pass , to finde the Semidiameters whereof , set off their depression from N towards E , thus 14d 57′ the depression of the Summer Tropick terminates at O , a Ruler laid from E to ☉ , findes the point X in the meridian SZN produced , so is XK the Diameter of the Summer Tropick , which being divided into halfs , will finde the Center thereof whereon to describe it . In like manner is the Diameter of the Winter Tropick to be found ; or if the Amplitude be given ( or found as elsewhere is shewed ) which at London is 39d 54′ , and set off both ways from G and E we shall have three points given through which to draw each Tropick , and the Centers falling in the Meridian Line will be found with half the trouble , as to finde a Center to three Points . Also Project the Pole Point P as before , being thus prepared FKG will represent the Summer and HLI the Winter Tropick . Let it be required to know what hours are proper for a South plain Declining 30d Eastwards , through the three points BPA describe the the Arch of a Circle BQP , then laying a Ruler from B to Q , finde the point R in the Limbe , and from it set off a Quadrant to M , then a Ruler laid from B to M findes the point ☉ the Pole of the hour Circle BQP , then laying a Ruler from P to ☉ , it findes the point T in the Limbe , and the Arch ET being 65d 40′ is the measure of the Angle BPS , which turned into Time is 4 ho . 23′ prope , and sheweth that at no time of the year the Sun will shine longer on the South side of this plain , then 23 minutes past 4 in the afternoon . In like manner if the Arch of a Circle be drawn through the two points PV , we may finde the time when the Sun will soonest in the morning begin to shine on the South side of this plain . A South plaine Declin 30d East lat 51d 32′ page 61 A South plaine Declin 60d East Reclining 40d Lat 51-32′ So if there were a South plain Declining 60d Eastwards , Reclining 40 degrees here represented by BRA , if it were required to know what hours are proper for the upper , and what for the under face , then where the plain intersects the Tropicks as at I and K , draw two Meridians into the Pole at P , to wit , IP and KP , and first finde the Angle IPZ , as was before shewed , to wit , 53d 14′ which in time is 3 hours 33 minuutes , shewing that the Sun never shines longer on the upper face of the Plain , then 33 minutes past 3 in the afternoon , which is capable of receiving all hours from Sun rising to that period of time , and the Angle KPZ , to wit , 39d 50′ in time 2 hours 39 minutes , shews that the Sun never begins to shine sooner on the under face then 39 minutes past 2 in the afternoon after , which all the hours to Sun-set may be expressed . To finde these Arks of time by Calculation , there must be given the Stiles height above the Plain 15d 22′ , PH , and the complement of the Inclination of Meridians to a Semicircle , 136d 32′ , to wit , HPS , then in the right Angled Triangle PHI there is given PH the Stiles height , PI the complement of the Declination , besides the right Angle at H , to finde the Angle IPH 83d 18′ which taken from the complement of the Inclination of Meridians HPS , there rests the Angle IPS the Arch of time sought , to wit , 53d 14′ . The ascensional difference may be found by drawing the Arch of a Circle through the three points TPF , and thereby the length of the longest day determined that no hours be expressed , on which the Sun can never shine . Another manner of Inscribing the hour-lines in all Plains having Centers . The method here intended , is to do it in a parallelogram from the Meridian line , whence the hour-lines may be prickt down by a Tangent of three hours , with their halfs and quarters from a Sector , without collecting Angles at the Pole , or by help of a Scheam which I call the Tangent Scheam , the foundation of this Dyalling supposeth the Axis of the world to be inscribed in a Parallelipiped on continued about the Axis , the sides whereof are by the plains of the respective hour Circles in the Sphere divided into Tangent-lines , that is to say , each side is divided into a double tangent of 45d set together in the middle , and the said parallelipiped on being cut by any Plain , the end thereof supposed to be intersected , shal be either a right or Oblique Angled parallelogram , and then if from the opposite tangent hour points on the sides of the intersected parallelipipedon , lines be drawn on the Plain , they shall cross one another in a Center , and be the hour-lines proper to the said plain , but of the Demonstration hereof , I shall say no more at present , the inquisitive Reader will finde it in the Works of Clavius . To draw the Tangent Scheam . I have before in Page 10 shewed how to divide a Tangent Line into hours and quarters , which in part must be here repeated ; draw any right Line , as MABH , from any point therein as at B , raise a perpendicular , and upon B as a Center , describe the Quadrant CH , and prick the Radius from B to A , from C to G , from H to F , and laying a Ruler from A to F , and G , you will finde the points D and E upon the perpendicular CB , I say the said perpendicular is divided into a Tangent line of three hours , and the halfs and quarters may be also divided thereon , by dividing the Arches CF , FG , and GH into halfs and quarters , then from those subdivisions , laying a Ruler to A , the halfs and quarters may be divided on , as were the whole hours . Being thus prepared , draw the Lines MB , LE , KD , and IC , all parallel one to another , passing through the points B , 1 , 2 , 3. In this Scheam they are perpendicular to BC , but that is not material , provided they pass through the same points , and are parallel one to another , yet notwithstanding the points A and H must be in a right Line perpendicular to CB. This Scheam thus prepared , I call the Tangent Scheam , because a Line ruled any way over it , shall be divided also into a Tangent of the like hours and quarters , whence it follows that one of these Scheams may serve to inscribe the hour-lines into many Dyals , which I shall next handle . To inscribe the Hour lines in a Horizontal Dyal . Having drawn the Meridian line M , XII , and perpendicular thereto the hour-line of six , Let it be observed that the sides of the Horizontal Dyal in page 8. to wit VI , IX , and VI , III and IX , III , are a right Angled Parallelogram , the one side whereof being the Diameter of the Circle being Radius , the other side thereof must be made equal to the Sine of the Latitude , in that Scheam the nearest distance from L to MF was the Sine of the Latitude or Stiles height , the Semidiameter of the inward Circle , being Radius , and that extent being doubled and pricked from M to VI on each side , as also from F twice to IX and III , by those extents the Parallelogram was bounded , the side III , F IX , being parallel to the Horizontal Line . Then take the extent MF from the Horizontal Dyal , and place it in the Tangent Scheam from M to O , and draw the Line MO , and the respective Divisions of the said Line being cut by the parallels of the Tangent Scheam are the same with the Divisions of the hour-lines on the inward sides of the Horizontal Dyal VI , IX , and VI , III , and from the tangent Scheam they are to be transferred thither with Compasses . Also place F , IX or F , III from the Horizontal Dyal into the Tangent Scheam from M to N , and draw the line MN , which being cut by the parallels of the tangent Scheam , the Distances of those Divisions from M are to be pricked down in the Horizontal Dyal , the first from F to XI , and I the second from F to X and II , &c. To inscribe the hour-lines in a direct erect South Dyal . The Diameter of the Circle in the South Dyal in page 8 , is the same as in the Horizontal , and the nearest distance from L to FM was the Consine of the Latitude , and was pricked twice on the Horizontal Line from M to VI on each side , whereby that inward Parallelogram was limited . Wherefore the divisions of the Line MO in the Tangent Scheam , are the same with the hour distances in this Dyal on the sides VI , III , and VI , IX ; then for the divisions of the hours on each side of XII , take the extent XII , IX or XII , III , and because it is less then the outward parallel distance of the sides of the Tangent Scheam , having therein made MI perpendicular to BM , place this extent from I to P , and draw the line MP , then prick the extent RL from the Tangent Scheam , from F to XI and I , and the extent QK , from F in the Dyal , to X , and II , and from the hour points so found , draw lines into the Center at M , and they shall be the hour-lines required . To delineate an upright Decliner in an Oblique Parallelogram . An upright South Dyal Declining 30 degrees West , Latitude 51 degrees , 32 minutes . First draw the Meridian or Plains perpendicular CN , and upon C as a Center , with the Radius of the Dyalling Scheam , describe A South plained Declin 40d East Reclining 60d Lat 51d 32′ page 64 A North plaine Declin 60 West Inclining 60d Lat 51-32′ place this anywhere page 65 A South plaine Declin 30d West Latitude 51d 32′ A South Diall Declin 40d East Inclin 15d Lat 51d 32′ In Latitudes under 45d the side of the square AV must be assumed to be the Cotangent of the Latitude , the Radius whereto will be AF the Tangent of the Latitude , to which Radius being prickt on the side of the square from the Center , the Sine of the declination must be taken out as before , and erected on the Cotangent of the Latitude , and this work must be performed on that side of the Center on which the Substile lyes . To fit in the Parallelogram . Produce the Lines WE , VD , and VL , in the Dyalling Scheam far enough , then assuming any extent to be Radius , enter it on the lines VD and VL from the Center to Y and Z , the nearest distance from Y to VE is the Cofine of the Latitude to that Radius which enter on the Line of 6 or GC , so that one foot resting thereon , the other turned about may just touch CN , at the point found set H , and make CI on the other side equal to CH. The nearest distance in the Dyalling Scheam from Z to VE , is the Cosine of the declination to the former Radius , which prick on the Meridian line from C to L. And draw a Line through L parallel to HI , and therein make LP , LQ each equal to CH , and draw HP and IQ , and there will be an oblique Parallelogram constituted , the sides whereof will be Tangent Lines . Nota , we might assume the point G in the hour-line of six , for the Parallelogram to pass through , and the nearest distance from D to VE in the dyalling Scheam , would be the Cosine of the declination to the same Radius to be prickt on the Meridian Line as before . To inscribe the hour-lines . In the following Tangent Scheam made as the former , produce CB , and make BQ equal to BC , then take the extent PQ on the dyal , and upon Q as a Center describe the Ark Y therewith , and draw the Line YC just touching the extreamity , then you may proportion out the hours in this manner . If a line be drawn in the Dyal from H to L , and from L to I , the hour-lines being drawn shall divide each of these lines into a double Tangent , and consequently the hour-lines may also be prickt off on the said lines , after the method now prescribed . For upright far Decliners and such plains as have small height of Stile , recourse must be had to former directions for drawing them with a double Contingent line , each at right Angles to the Substile . The foundation whereof is this , Any point being assumed in the Substilar line of a Dyal , the nearest distance from that Point to the Stile , is the Sine of the Stiles height , the Radius to which Sine is the distance of the assumed point in the Substilar line from the Center of the Dyal ; Then in all Dyals the hour distances from the Substilar line are Tangents of the Angle of the Pole , the Sine of the Stiles height being made the Radius thereto , having finished the delineation of the Dyal , the Stile is to be placed directly over the Substilar line , without inclining to either side of the Plain , making an Angle therewith equal to its height above the same , for the Substilar line is elsewhere defined to be such a line over which the Stile is to be placed in its nearest distance from the Plain , therefore if the Stile incline on either side , it will be nearer to some other part of the Plain then the Substilar line , whence it comes to pass in places near the Equinoctial , if an upright Plain decline but very little , the Substile is immediately cast very remote from the Meridian . Another way to prick down the hour-lines in Declining leaning Plains . Every such Plain in some Latitude or other will become an upright Decliner : First therefore by the former directions prick off the Substile , Stile and Meridian , in their true Coast and quantity , and perpendicular to the Meridian , draw a line passing through the Center , and A South Diall Declin 40d East Inclin 15d Lat 51d 32′ A North Diall Declin 40d East Reclining 75d Lat 51d 32′ it shall represent the Horizontal line of the Plain in that new Latitude as here VS ; from any point in the Stile as K , let fall a perpendicular to the Substile at I , and from the point I , in the Substile let fall a perpendicular to the Meridian at P. To finde the new Declination . Prick IP on the substilar line from I to R , and draw RK , so shall the Angle IRK be the complement of the new declination , and the Angle IKR the new Declination it self . To finde the new Latitude . Upon the Center V with the Radius VK , describe a Circle , I say then that VP is the Sine of the new Latitude to that Radius which may be measured in the Limbe of the said Circle , by a line drawn parallel to VS , which will intersect the Circle at F , so is the Arch SF the measure of the new Latitude . To prick off the Hour-line of Six . This must be prickt off below the Horizontal line , the same way that the substilar lyes : The proportion , is , As the Cotangent of the Latitude , Is to the Sine of the Declination , So is the Radius , To the Tangent of the Angle between the Horizon and six . If RK be Radius , then is VP the Tangent of the new Latitude , but if we make VP Radius , then is RK the Cotangent of the new Latitude . Wherefore prick the extent RK on the Horizontal line from V to N , and thereon erect the Sine of the Declination to the same Radius perpendicularly , as is NA , and a line drawn into the Center shall be the hour-line of six ; the proportioning out of the Sine of the Declination to the same Radius , will be easily done , enter the Radius VP from K to D , and the nearest distance from D to IK , shall be the Sine of the new declination to that Radius . To fit in the Parallelogram . This is to be done as in upright Decliners , for having drawn a line from the new Latitude at F into the Center , if any Radius be entred on the said line from V , the Center towards the Limbe , the nearest distance from that point to VP the Meridian line , shall be the Cosine of the new Latitude to that Radius . Again if the same Radius be entred on RK produced if need be , the nearest distance to VR ( produced when need requires ) shall be the Cosine of the new Declination , and then the hour-lines are to be drawn as for upright decliners , nothing will be doubted concerning the truth of what is here delivered , if the demonstration for inscribing the Requisites in upright decliners be well understood , it being granted that Oblique Plains in some Latitude or other will become upright decliners . There are two Examples for the Latitude of London suited to these directions , in both which the Letters are alike , the one for a South Plain declining 40d Eastwards , Inclining 15d , the other for a North Plain declining 40d East , Reclining 75d . To finde a true Meridian Line . For the true placing of an Horizontal Dyal , as also for other good uses it will be requisite to draw a true Meridian Line , which proposition may be performed several ways , amongst others the Learned Mathematician Francis van Schooten in his late Miscellanies demonstrates one , performed by help of three shadows of an upright Stile on a Horizontal Plain , published first without Demonstration in an Italian book of dyalling by Mutio Oddi . But if all three be unequal , as let AC be the least , erect three lines from the point A , perpendicular to AP , AC , AD as is AF , AG , and AH equal to the Stiles height AE , and draw lines from the extreamities of the three shadows to these three points as are FB , GC , and HD ; then because AC is less then AB , therefore GC will be less then FB , by the like reason GC will be less then HD , wherefore from FB and HD cut off or Substract FI and HK equal to GC , and from the points I and K let fall the perpendiculars IL , KM , upon the Bases AB , AD , afterwards draw a line joyning the two points M , L , and from the said points let fall the perpendiculars LN equal to LI , and MO equal to MK . Then because the two shadows AB and AD are unequal , in like manner FB and HD will be unequal ; but forasmuch as FI and HK , are equal by construction , it follows that LI , KM , or LN and MO will be unequal , and forasmuch as these latter lines are parallel a right line that connects the points O and N , being produced will meet with the right line that joyns M , L produced , as let them meet in the point P , from whence draw a Line to C , and it shall be a true line of East and West , and any Line perpendicular thereto shall be a Meridian line , thus the perpendicular AQ let fall thereon , is a true Meridian Line passing through the point A , the place of the stile or wyre . Whereto I adde that if MO & LN retaining their due quantities be made parallel it matters not whether they are Perpendicular to ML or no , also for the more exact finding the Point P , the lines MO and LN , or any other line drawn parallel to them , may be multipyled or increased both of them the like number of times from the points M and L upwards , as also from the points O and N downwards , and Lines drawn through the points thus discovered , shal meet at P without producing either ML or ON . See 15 Prop. of 5 Euclid . and the fourth of the sixth Book . The greater part of van Schootens Demonstration is spent in proving that ML and ON produced will meet somewhere , this for the reasons delivered in the construction I shall assume as granted , then understand that the three Triangles ABF , ACG and ADH stand perpendicularly erect on the plain of the Horizon beneath them , upon the right Lines AB , AC and AD , whence it will come to pass that the three points F , G , and H meet in one point , as in E the top of the Stile AE , and that the right lines FB , GC and HD are in the Conique Surface of the shadow which the Sun describes the same day by his motion , the top of which Cone being the point E. Wherefore if from those right Lines we substract or cut off the right Lines FI , GC and HK being each of them equal to one another , then will the points I , C and K , fall in the circumference of a Circle , the plain whereof is parallel to the plain of the Equator ; and therefore if through the points K and I such a Position a right Line be imagined to pass , and be produced to the plain of the Horizon , it will meet with ML produced in the point P , where ON being produced , will also meet with it ; so that the point P being in the Plain of the Circle , as also in the plain beneath it , as also the point C being in each Plain , a right line drawn through the Points P and C will be the Common Intersection of each plain , and the line PC will be parallel to the Plain of the Equator , and is therefore a true Line of East and West , which was to be proved . On all Plains though they Decline , and Recline , or Incline , after the same manner may be found the Line PC , which will represent the Contingent Line of any Dyal , and a perpendicular raised upon the Line CP shall be the Substilar Line , which in Oblique Plains is the Meridian of the Plain , but not of the place , unless they are both Coincident from this manner of finding a Meridian Line on a Horizontal plain , nothing else can be deduced without more Scheams : From the three shadows , may be had the three Altitudes , and the Meridian line being given , the Azimuths to those three shadows are likewise given , which is more then need be required in order to the finding of the Latitude of the place , and the Declination and Amplitude of the Sun , which because this Scheam doth not perform of it self , I shall adde another to that purpose . By three Altitudes of the Sun , and three shadows of an Index on an Horizontal Plain , to finde a true Meridian Line , and consequently the Azimuths of those shadows , the Latitude of the place , the Suns Amplitude and Declination . Let the three shadows be CA , the Altitude whereto is AF 22d , 28′ The second shadow CB , the Altitude whereto is BG 59d , 21′ both these in the morning , The third shadow in the afternoon CD the — Altitude whereto is — DH 18d , 20′ Let the Angle ACB be 70d , and the Angle BCD 135d , by following Operations we shall finde that the shadow CA , is 10d to Southwards of the West , that the shadow CB is 60d to Northwards of the West , that the shadow CD is 15d Southwards of the East . Having from the three shadows prickt off the three Altitudes to F , G , and H , from those points to the shadows belonging to them , let fall the perpendiculars FI , GK , HL , which shall be the Sines of those Altitudes , and the Bases IC , KC , and LC shall be the Cosines , from the point K in the greater Altitude , draw Lines to the points I and L in the lesser Altitudes and produce those lines . From the points K and I , the Sines of the two Altitudes , are to be erected perpendicularly , thus KM is made equal to KG , and IN is made equal to IF , then producing MN and KI , where they meet as at W , is one point , where the plain of the Suns parallel of Declination intersects the plain of the Horizon ; in like manner on the Base KL , the Sines of two Altitudes KG , and LH , ought to stand perpendicularly from the points K and L in their common Base , but if they retain the same height , and are made parallel to one another , a line joyning the points of the tops of those Sines produced , shall meet with the line joyning the points of their Bases produced , in the same point as if they were perpendicular . Thus KP and LQ are drawn at pleasure through the points K , and L parallel one to another , and KG is the Sine of the greater Altitude , and LO is equal to the Sine of the lesser Altitude , and these two points being joyned with the Line OG produced , meets with the Line KL produced , in the point E , another point where the plain of the Suns parallel intersects the plain of the Horizon . Or you may double KG , and finde , the point P , as also double LO and finde the point Q , a line drawn through P and Q , findes the point E , as before , but with more certainty . If you joyn EW it shall be a true Line of East and West , and a perpendicular let fall thereon from C the Center , shall be a true Meridian line to the perpendicular Stile or wyer , as is CS . The Arch SR is the Suns Amplitude from the North 50d 6′ the shadow being contrary to the Sun , casts his parallel towards the South . Draw KT parallel to SC , and it shall be the perpendicular distance between the Sine of the Suns greatest Altitude , and the Intersection of his parallel with the Horizon . Upon K erect KV equal to the Sine of the Suns greatest Altitude of the three KG , and the Angle KTV shall be equal to the complement of the Latitude , for the Angle between the plain of any parallel of declination and the Plain of the Horizon , is always equal to the complement of the Latitude , and if upon T as a Center with the Radius CS , you describe the Ark KX , the said Ark shall measure the complement of the Latitude in this example 38d 28′ which being given together with the Sine of the Amplitude CY , it will be easie to draw a Scheme of the Analemma , whereby to finde the Suns declination , the time of rising , and setting , his height at six the Azimuth thereto , the Vertical Altitude and hour thereto , &c. and many other propositions depending on the Suns motion , as I have elsewhere shewed . The whole ground hereof is , that a right line extended through the tops of the Sines of any two Altitudes of the Sun taken the same day before his declination very , shall meet with the Plain of the Horizon in such a point where the Plain of the Suns parallel intersects the Plain of the Horizon , and finding of two such points , a line drawn through them , must needs represent the intersection of those two Plain ; in the former Scheme the Sines of the two Altitudes are KM and IN , a line drawn through the bottomes of those Sines , as KI extended shall be in the Horizontal Plain , and the Line MN extended through the tops of those Sines is in the plain of the Suns parallel , as also in the Horizontal plain ; now whether these Lines stand erect or no is not material , provided they retain their Parallelisme and due length , and pass through the points of their Bases I , K , for the proportion of the perpendiculars to their Bases , will be the same notwithstanding they incline to the Horizon . To Calculate the Latitude , &c. from three shadows . A usual and one of the most troublesome propositions in Spherical Trigonometry , is from three shadows to finde the Latitude of the place : thus Maetius propounds it , and with many operations both in plain and Spherical Triangles resolves it . The first operations are to finde the Suns 3 Altitudes to those shadows , and that will be performed by this proportion , As the Length of the shadow , Is to the perpendicular height of the Gnomon , So is the Radius , To the Tangent of the Suns Altitude above a Horizontal Plain , which proportion on other Plains will finde the Angle between the Sun and the Plain or Wall . Next Maetius gives the distances between the points of the three shadows , and then by having three Sides of a Plain Triangle , he findes an Angle , to wit , the differences of Azimuth between the respective shadows , which Angle may be measured off the Plain with Chords . But propounding it thus , Three Altitudes of the Sun above a Horizontal Plain , with the differences of Azimuth between the three shadows belonging to those Altitudes , being given , let it be required to finde the Suns true Azimuth , the Latitude of the place , and the Suns Amplitude . And how this may be Calculated from the former Scheme , I shall now shew . In the Triangle ICK of the former Scheme , the two sides IC and CK , represent two shadows , and the Angle ICK is the Angle or difference of Azimuth between them , and the said sides IC and CK , are the Cosines of the Altitude proper to those shadows ; now by seven Operations in right lined Triangles , we may finde the proper Azimuth or true Coast of any of those shadows . 1. In the right lined Triangle ICK , having the two sides IC and CK , with the Angle between them ICK , at one operation may be found both the other Angles CIK and IKC . 2. In the same Triangle by another operation , may be found the Side IK . Then to proceed , draw Nf parallel to IK , and fM will be the difference of the Sines of both the Altitudes belonging to those shadows , whereby may be found KW . 3. The proportion lyes , As fM the difference of the Sines of both those Altitudes , Is to fN equal to IK before found , So is KM the Sine of the greater Altitude , to KW sought . 4. 5. 6. By three like Operations may be found in the other shadow Triangle , the Angles CKL , and KLC , with the Side KE . Having proceeded thus far to the Angle IKC , adde the Angle CKL , the sum is equal to the Angle WKE . 7. Then in the Triangle WKE , we have the two Sides thereof WK , and KE given , and the Angle comprehended by them , and at one Operation we may finde both the other Angles EWK , and KEW , the complement of the Angle TWK , is the Angle WKT . The difference between the Angles WKT , and IKC , is the Angle CKZ , which shews the Suns Azimuth from the Meridian proper to that shadow , which may be otherways found , for the difference between EKT and TKC , also shews it . By two other Operations the Latitude may be found . 1. As the Radius : Is to WK before found ∷ So is the Sine of TWK to KT : 2. As VK the Sine of the greater Altititude , Is to KT before found , So is the Radius , To the Tangent of the Latitude . By another Operation may be found the Amplitude CY , having found the Angle ZKC , the Angle CKp is the complement thereof , then it holds : As the Radius , Is to CK the Cosine of the greater Altitude , So is the Sine of the Angle CKp to CP , the difference between which , and Yp equal to TK , is YC the Sine of the Amplitude sought . To make any Dyal from three Shadows . The Geometrical performance of the former Proposition , is insisted upon by Clavius in his Book of the Astrolable , but ▪ he mentioneth no method of calculation as derivable from it : from this proposition Monsieur Vaulezard a French Mathematician educeth a general method for making of Dyals , from three shadows of a Gnomon stuck into a wall at randome , whereof he doth not so much as mention any demonstration ; I shall endeavour to deliver the method thereof with as much perspicuity as I can . Every Plain in some place or other , is an horizontal Plain ; admit an Oblique Plain in our Latitude , the Substilar line represents the meridian of that place , and any contingent line drawn at right Angles , thereto will represent a true line of East and West in reference to that horizon , and if the hours did commence at 12 , from each side the substile , the Dyal here would shew the true time of the day there . In every oblique Plain assuming any Point in the Stile , and crossing the Stile with a perpendicular to that Point , which shall meet with the Substile , the Point so found in the Substile , is the Equinoctial Point , in respect of the assumed Point in the Stile ; hence we may inferre that if these two Points and the Substilar line were given the Center of the Dial might be easily found , now the former construction applyed to an oblique Plain , assumed to be an Horizontal Plain , in respect of some unknown place , will find the Equinoctial Point and the substilar Line , the Stile Point being assumed in the extremity of a Gnomon any wayes placed or stuck in a wall at randome . From the former Scheme it may be observed that the Sine of any of the three Altitudes being erected on the Perpendicular between the foot of that Sine and the Suns parallel ▪ gave an Angle equal to the elevation of the Equinoctial above the Horizon , which is the thing sought in the following work , but the Sine of the greatest Altitude performs the proposition best . If a stick or pin be stuck into a wall for this purpose , and doth not make right Angles therewith , a Perpendicular must be let fall from the extremity thereof into the wall , which is called the perpendicular Stile , and the distances of the shadowes ; from the foot thereof must be measured thence , in respect of the tip of this perpendicular Stile the Equinoctial Point must be found , wherefore it will be convenient to assume the said perpendicular Stile to be the Sine of the greatest of the three Altitudes of the Sun above the Plain , which properly in respect of Us are Angles between the wall and the Sun . Up on this assumption it will follow , that the shortest shadow will be the Cosine of that Angle , and the distance between the tip of the Stile , and the extremity of that shadow will be the Radius : now from the lengths of the three shadows , and the height of the perpendicular Stile , it will be easie to find the Sines and the Cosines of the Angles between the wall and the Sun . fig. 1a fig. 2a fig. 3a fig. 4a Being thus prepared , draw three Lines elsewhere as in the third figure meeting in a Center , and making the like Angles as the shadows did , and let them be produced beyond that Center , and have the same Letters set to them : make CH CG in this Scheme , equal to the nearest distances from H I to CB , and make CL equal to EC , in the former Scheme . Draw the lines IG , IH produced ; and make HP GO equal to nearest distances from I H to BG , and parallel to HP and GO , draw the lines IN , IM , each of them made equal to CB , then draw the Hipotenusals MO , NP , meeting with the Bases at K and L , draw the Line KL , and it shall represent the Plains Equinoctial or contingent Line . From I , let fall the perpendicular IQ produced , and it shall be the Substilar line on the Plain and the Point Q is the Equinoctial Point sought . To place the Substile . Then repair to the Plain whereon you would make the Dyal represented by the first Scheme , and place CQ therein , so that it may make the same Angle with the shadow CE , as it doth with the Line IC in the third Scheme , and it shall be the Substilar Line , which is to be produced , also prick IQ from the third Scheme from C to Q on the Plain in the Substilar Line , and upon the point C , perpendicular to the Substile , raise the Line AC , and make it equal to CB , drawing the Line AQ , then will the Angle AQC be equal to the complement of the Stiles height ; for it is the Angle between the plain and the Equinoctial ; just as before in the Horizontal plain , the Sine of the greatest Altitude was erected on a Line falling perpendicular from the foot of the said Sine , to the intersection of the plain of the Suns parallel with the plain of the Horizon , and thereby gave the Angle between the Suns parrallel and the Horizon , equal to the Complement of the Latitude . To find the Center of the Dyal and Stiles height . If from the point A you raise the Line AV perpendicularly to AQ , where it cuts the Substilar Line , as at V , is the Center of the Dial , and he Line AV represents the Stile . 3. To draw the Meridian Line . If the Plain recline , a thread and plummet hanging at liberty from the Point B , and touching the Plain , will find a Point therein , suppose K a line drawn from K into the Center of the Dial , shall be the Meridian line of the place . A broad ruler with a sharp pin at the bottome of it , being in a right line , with a line traced through the length of that ruler , whereto the plummet is to hang , having a hole cut therein for the bullet to play in , will find this Point in a reclining plain . On an inclining Plain , if the thread and plummet hang upon the Stile at liberty , to some part of the Stile more remote from the Center , fasten another thread , which being extended thence to the Plain just touching the former thread and plummet hanging at liberty , will find many Points upon the Plain , from any of which , if a line be drawn into the Center it shall be the Meridian line required ; See the fourth figure : or in either of these cases hold a thread and plummet so at liberty that it may just touch the Stile , and bringing your sight so , as to cast the said thread upon the Center at the same by the interposition of the thread ; the eye will project a true Meridian-line on the Plain , for the thread represents the Axis of the Horizon , and the Plain of the Meridian , is in the said Axis . If the Center fall inconvenient upon the Plain , it will be necessary to draw the Plains perpendicular , passing through the foot of the perpendicular Stile C , and measure the Angle of some of the Shadowes from it , and accordingly so place it in the third figure or draught on the floor , on which find the Center , and afterwards assign the Center on the Plain where it may happen convenient , and from the said draught , by help of the Plains perpendicular , set off the substile and Stiles height ; On plains on which probably the Stile hath but small height , the perpendicular Stile in this work must be assumed the shorter . 4. To draw the Hour Lines . These may be set off in a Paralellogram after the Substile , Stile ' and Meridian are placed which I have handled before . Or they may be inscribed by the Circular work , if we assume the perpendicular Stile AC , to be the Sine of the Stiles height , the Radius thereto will be AV , prick the said Radius on the Substile from V the Center to , Z , and upon Z as a Center , describe the Circle as in the first figure , and find the regulating point ☉ by former directions , over which from O the point where the Meridian of the place cuts the said Circle , lay a ruler , and it finds M , on the opposite side , being the Point from whence the Circle is to be divided into 12. parts , for the houres with subdivisions for the halfes & quarters . Here note that the arch MV , is the inclination of Meridians , the whole Semicircle being divided but into 90d , and if that be first given ( as hereafter ) we may thereby find the Point O , whence the Meridian Line is to be drawn into the Center . 5. A Method of Calculation suited hereto . This is altogether the same as for finding the Azimuth of the shadowes on the Horizontal Plain and as easie , whereby in the third figure , the Angle of the Substile CIQ , must be set off from the shadow ICE , just as the Meridian Line CP , might be set off from the shaddow CK on the Horizontal . And the complement of the Stiles height AQV , in the first figure here , as the complement of the Latitude KTV , was found there . Lastly , for placing the Meridian Line of the place by calculation , the Substiles distance from the plains perpendicular , and the reclination of the Plain must be found , which may be got easily at any time without dependence on the Sun , and then in the often-mentioned oblique Triangle in the Sphere , we have two sides with the Angle comprehended given , to wit , the complement of the Stiles height , the complement of the reclination , and the Substiles distance from the Plaines perpendicular whereby may be found the inclination of Meridians : and consequently the Meridian line of the place , also the Latitude thereof , with the Plains declination , if they be required . To perform all this other wise Geometrically and instrumentally , ( but not by calculation ) there was an entire considerable quarto treatise , with many excellent Prints from brass-Plates thereto belonging , printed at Paris in France , in An. 1643. by Monsieur Desargues of Lions , which Treatise I have seen but not perused , a year after was published the small Treatise of Monsieur Vaulezard before mentioned . The inscribing of the Signes , Azimuths , parallels of the longest day , &c. are lately handled by Mr. Leybourn in his Appendix to Mr. Stirrups Dyalling , as also by Mr. Gibson in his Algebra , who thinks in many cases that they deform a plain , and are seldome understood by the vulgar , wherefore it will not be necessary to treat thereof . The directions throughout this Book are suited to the Northern Hemisphere and are the same in the Southern Hemisphere , if the words South for North , and North for South be mutually changed . Since the Printing of this Treatise , I have not had time to revise it with the Copy , and so cannot give thee a full account of what faults may have escaped , which I think are not many ; these few following be pleased to correct . Errata . Page 17 line 3 upon M read upon C. p. 15 l. 34 for Substilar Sine r. Substilar Line . p. 25. l. 12 for 20d r. 15d . p. 36 l. 17 for 51′ . r. 31′ . FINIS . 
A29756	----	The description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of Gunters quadrant applyed thereunto ... / contriv'd & written by J. Brown, philomath. Brown, John, philomath. 1661 Approx. 191 KB of XML-encoded text transcribed from 105 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A29756 Wing B5038 ESTC R33265 13117505 ocm 13117505 97766 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29756) Transcribed from: (Early English Books Online ; image set 97766) Images scanned from microfilm: (Early English books, 1641-1700 ; 1545:10) The description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of Gunters quadrant applyed thereunto ... / contriv'd & written by J. Brown, philomath. Brown, John, philomath. [24], 168 p., [8] leaves of plates : ill., charts. Printed by T.J. for J. Brown and H. Sutton, and sold at their houses, London : 1661. Woodcut illustration of man sighting with sextant: T.p. verso. Errata: p. 168. Imperfect: pages stained and tightly bound with slight loss of print. Reproduction of original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Quadrant. Dialing. Mathematical instruments. 2004-01 TCP Assigned for keying and markup 2004-02 Aptara Keyed and coded from ProQuest page images 2004-04 Mona Logarbo Sampled and proofread 2004-04 Mona Logarbo Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion The Description and Use OF A JOYNT-RULE : Fitted with Lines for the finding the Hour of the Day , and Azimuth of the Sun , to any particular Latitude ; Or to apply the same generally to any latitude Together with all the uses of Gunters quadrant applyed thereunto , as Sun-rising , Declination , Amplitude , true place , right Ascension , and the hour of the Night by the Moon , or fixed Stars ; A speedy and easie way of finding of Altitudes at one or two stations ; Also the way of making any kinde of erect Sun-Dial to any Latitude or Declination , by the same Rule : With the Description and Use of several Lines for the mensuration of Superficies , and Solids , and of other Lines usually put on Carpenters Rules : Also the use of Mr. Whites Rule for measuring of Board and Timber , round and square ; With the manner of using the Serpentine-line of Numbers , Sines , Tangents , and Versed Sines . Contriv'd & written by J. Brown , Philom . London , Printed by T. I. for I. Brown , and H. Sutton , and sold at their houses in the Minories , & Thredneedle-street . 1661. TO THE READER . Courteous Reader , AMong the multitude of Books which are printed and published , in this scribling Age , some serious , some seditious ; some discovering or savouring of Art , others of ignorance , possibly every one endeavoring to bring their Male : Among the rest of the crowd , I , like the widow , throw in my mite . If it be ( or seem to be ) little , it . is like the Giver , and therefore I presume will of some be accepted , as little as it is ; and as little worth as it is , it is like enough to be challenged : but I shall endeavor to prevent prejudice , by the following Discourse . Having for some time been enquiring to find out a way , whereby Work-men might on their Rules ( their constant Companions ) have a way easily and exactly to finde the hour of the Day , and Suns Altitude and Azimuth , and the like ; and have at several times for several men , at their request , used one and the other contrivance , to finde the Hour : as that of the Cillender , Quadrant , or the like , as the Altitude by a Tangent on the inside of a Square , or Joynt-Rule , and the line of Sines on the flat side ; but still one inconvenience or other of trouble in adding of complements , or difficulty of taking of Aititudes , or trouble to the memory , did accrue to the work ; or else the Radius was small , and so much the more short of exactness : at last there came to my sight a Quadrant made by Mr. Thomson , and as I was informed , was first drawn or contrived to that form , by Mr Samuel Foster , that ingenious ●rtist , and laborious Student , and Reader of the Mathematicks in Gresham Colledge : And considering of the ease and speed in the using thereof , I set my self to the contriving thereof to a more portable form , at last took some pains in delineating one , and another in several forms , and enquired after the uses thereof , and in effect have done , as Mr. Gunter with Stofflers Astrolabe , and Nepeirs Logarithms , and as Mr. Oughtred with Gunters Rule , to a sliding and circular form ; and as my father Thomas Brown into a Serpentine form ; or as Mr. Windgate in his Rule of Froportion , and as of late Mr. Collins with Mr. Gunters Sector on a Quadrant , so may this not unfitly be called , The Quadrant on a Sector . And in fine , the Invention will be valued for the learned Authors sake , and never a whit the worse for the new Contrivers sake : For first , hereby it is made large in little room , and as well on wood as on brass , which is an incommunicable property to broad Quadrants , though of never so good matter , as experienced Workmen know right well ; and by a Tangent of 30 degrees laid together , is gotten all kinde of Angles or Altitudes under 90 degrees , and to be afforded for a low price , in comparison of other Instruments , which will not perform the same operations any better . Having made some Joynt-Rules in the manner following , and exposing them to sale , I have been many times solicited to write somewhat of the use , and now at last after near a years suspence , have committed the following Discourse to publique view , partly to save the labor of tedious transcribings , and also to make so useful , cheap , and exact an Instrument , ( if it be truly made ) to be more known or occupied . In which business , I desire to disclaim all vain-glorious os●entation , and therefore have nakedly and plainly asserted the manner how , and why it comes to be published to the world by me . It is a mechanical thing , and mechanically applied , and of mechanical men will be humanically accepted , I doubt not . Having begun to write , I could not break off so short and abruptly , as at first I did intend to do ; therefore have added this short Discourse of ordinary Dyalling , the exact Method of which I finde in no other Author that ever I met with , ( and indeed I have not time to read many ) yet I dare presume , that for speed , ease , convenience , and exactness , inferior to none , especially the way of making far declining Dyals ; As for other declining reclining Dyals , I referre you to other Authors , or to a Discourse thereof by it self : if I finde encouragement , and ability to perform the same , a Copy whereof I have had a long time by me , written by a very ingenious Artist ) the demonstration of which Dyals is most excellently and easily shewed by the Figure inserted , page 77. As for the other part for taking of Altitudes and Angles , it may also be very conveniently done , if the Rule be fitted to a three-leg staff , with a small Ball-socket to set it level , or upright , as other Surveighing-instruments be , as will be amply found , if a tryal be made thereof . That of Ma●er White 's Rule is a thing that hath given very good content to several Gentlemen in the Counties of Essex , Suffolk , and Norfolk , and indeed is a very neat and accurate way of operation , well becoming a Gentleman ; for while a Workman shall take measure , his Rule keeps the count of length , or breadth ; and having the length first given , the girt or squareness is no sooner agreed on , but you have the content without Pen or Compasses . As for the other lines , as Decimal-board , and Timber-measure , Inches , and Foot , in the way of Reduction , Girt-measure , Circles , Diameter , Circumference , Squares inscribed and equal : The Use of them will be very grateful to many a learner . Lastly , this brief touch of the Serpentine-line I made bold to assert , to see if I could draw out a performance of that promise , that hath been so long unperformed by the promisers thereof . These Collections , courteous Reader , I have printed at my friends and my own proper charges , and if they prove to be ( as I do hope they will ) of publique benefit , I shall enjoy my expectation , and be ready at all times to serve you further , as I may , in these or other Mathematical instruments , at my house at the Sun-dyal in the Minories , and remain to you much obliged , February 8 ▪ 1660. Iohn Brown. A TABLE of the things contained in this BOOK . CHAP. I. Page THe description of the rule for the hour onely 1. 2. 3 CHAP. II. To rectifie or set the Rule to his true angle for observation 4 To finde the Suns altitude 5 To finde the hour of the day 6 To finde the Suns rising and setting 7 To finde a level or perpendicular 7 CHAP. III. A further description of the rule for hour and Azimuth generally 10 , 11 , 12 CHAP. IIII. To finde the Suns declination 13 To finde the Suns true place , and right ascention 14 To finde the Suns amplitude 14 CHAP. V. To finde the Suns Azimuth at any altitude and declination , in this particular latitude in Summer 16 To finde the Azimuth in Winter , and Equinoctial 17 , 18 CHAP. VI. To finde the hour of the night by the Moon . 19 To finde the Moons Age 20 To finde the Moons place 21 To finde the Moons hour by the 11 chap. 2 and 3 Proposition 23 To finde the true hour of the night thereby 24 CHAP. VII . To finde the hour of the night by the sixed Stars 25 Three examples thereof 26 , 27 , 28 CHAP. VIII To finde the Amplitude , Azimuth , rising and so●●hing of the fixed Stars , and Examples thereof in page 29 , 30 , 31 , 32 CHAP. IX . To finde the hour and Azimuth , &c. in any latitude 33 To finde the Suns rising , setting , and ascentional difference ibid. To finde his amplitude in any latitude 34 To finde the Suns altitude at six in any latitude . 35 To finde the hour when the Sun is in the Equinoctial 36 To find the hour in any l●titude , altitude , and declination 37 To finde the Suns Azimuth in any latitude , at any declination , and altitude in summer 38 To finde the same in winter 39 CHAP. X. To finde the inclination of Meridians , substile , stile , and angle , between 12 and 6 for erect decliners three ways , one particular , and two general 40 To finde the substile , stiles elevation , inclination of Meridians 41 To finde the angle between 12 and 6 for a particular latitude 42 To perform the same in general for any latitude by the general scale of altitudes 43 , 44 Five canons to finde the same by the artificial sines and ta●gents , and how to work them on the rule 45 , 46 CHAP. XI . To draw a Horizontal Dyal to any latitude 47 To draw a vertical direct North or South Dial to any latitude 48 , 49 To draw a direct erect East or West Dial 50 , 51 CHAP. XIII . To finde the declination of a Plain 52 To do it by the needle 53 , 54 To finde the quantity of an angle the sector or rule stands at 55 An example of the work 56 To finde a declination by the Sun at any time 57 , 58 Some precepts and examples for the same 59 , 60 , 61 To finde a declination at two choice particular times , viz. when the Sun is in the Meridian of place , or Plain 62 , 63 How to supply a deficiency in one line of the rule , by another line on the rule 64 CHAP. XIIII . To draw a vertical declining Dial to any declination and latitude . 65 To perform it another way 66 , 67 To supply a defect on the parallel contingent 68 CHAP. XV. To draw the hour-lines on an upright declining Dial , declining above 60 degrees 96 To make the table for the hours 70 To finde the substile , stiles augmentation , and Radius , to fit and fill the Plain with any certain number of hours 71 , 72 , 73 , 74 , 75 An advertisement relating to declining reclining Plains 76 , 77 , 78 , 79 CHAP. XVI . To finde a perpendicular altitude at one or two stationi 80 To do it at one station 81 To perform it at two stations 82 , 83 , 84 To work it by the sector lines 85 , 86 CHAP. XVII . The use of several lines inserted on rules for the use of several workmen , for the mensuration of superficial , and solid measure , and reduction , &c. 87 The use of inches , and foot measure laid together , in giving the price of one , to know the price of a hundred , or the contrary 88 The use in buying of timber , knowing the price of a load , or 50 foot , to know the price of one foot , or the contrary 89 , 90. The like work for the great hundred or 112 l. to the C. 90 , 91 The use of the line of decimal board measure 92 The use of the line of decimal Timber measure 93 The use of the line of decimal yard measure 94 Or that which agreeth with feet and inches 95 The use of the line of decimal round or Girt measure 96 The use of the line of decimal sollid measure by the diameter 97 The table of decimal superficial under measure 98 , 99 The table for decimal sollid under measure 100 , 101 The table of under-yard measure for foot measure or inches 102 , 103 The table of under girt measure to inches 104 The table of under diameter measure to inches and quarters 105 A table of brick measure 106 The use of the lines of Circumference , diameter , and squares equal and inscribed 108 CHAP. XVIII . The use of Mr. Whites rule or the sliding rule in Arithmetique , and measuring superficial and sollid measure from 110 to 118 CHAP XIX . To finde hour and Azimnth by that sector To finde the Azimuth by having latitude , Suns declination and altitudes , cemplements , and the hour from noon 119 To finde the hour , by having the same complements , and the Azimuth from south 120 Having the complements of the latitude and altitude , and the Suns distance from the pole , to finde the Suns Azimuth 121 Having the same complements , to finde the hour by the sector 122 Having latitude , declination , meridian , and present altitude , to finde the hour of the day . 123 Having latitude , declination , and altitude , to finde the Suns Azimuth at one operation by the sector 124 Having the length of the shadow , to find the altitude , and the contrary 126 Having latitude and declination , to find the Suns rising and setting 127 To finde the Suns altitade at any hour generally 129 To finde when the Sun shall be due East or West 130 To finde the quesita in erect Dials 131 , 132. CHAP. XX. The use of the serpentine line . The description thereof . 134 , 135 , 136 , 137 Some observations in the use thereof 138. 139 , 140 To finde the hour of the day thereby , according to Mr. Gunter 142 To finde the hour thereby . according to Mr. Collins 145 To finde the Azimuth thereby Mr. Collins his way 146 To finde the Azimuth Mr. Gunters way 148 To finde the hour and Azimnth at one operation , by help of the natural sines and versed sines 149 to 154 To finde the Suns altitude at any hour or Azimuth 155 , 156 To finde the hour by having the Azimuth , and the contrary 157 Five other useful propositions 159 To square and cube a number , and to finde the square , and cube root of a number 162 , 163 To work questions of interest and annuities 164 The use of the everlasting Almanack 167 The right Ascension and Declination of 12 principal fixed Stars in the heavens ; most of which are inserted on the Rule : or if room will allow , all of them .   R. Asc. Declina . Stars Names H. M. Deg. M. Pleiades , or 7 Stars 03 24 23 20 Bulls-Eye 04 16 15 48 Orions Girdle 05 18 01 195 Little Dog 07 20 06 08 Lyons Heart 09 50 13 40 Lyons Tayl 11 30 16 30 Arcturus 14 00 21 04 Vultures Heart 19 33 08 00 Dolphins Head 20 30 14 52 Pegasus Mouth 21 27 08 19 Fomahant 22 39 31 17s Pegasus lower wing 23 55 13 19   1   3   5 7 4     6   8       Moneths 9   11   2 10 12   1 2 3 4 5 6 7   8 9 10 11 12 13 14 Days 15 16 17 18 19 20 21   22 23 24 25 26 27 28   29 30 31         Week-days S M T W T F Sat Dom. Letter d c b a g f e Leap years 68 80 64 76 60 72 84 E●acts 26 9 12 25 28 11 23 THE Description and Use OF A JOYNT-RULE . CHAP. I. The Description of the Lines on the Rule , as it is made onely for one Latitude , and for the finding the hour of the day onely . FIrst open the ( Joynt of the ) Rule , then upon the head-leg , being next to your right hand , you have a line beginning at the hole , which is the Center of the quadrantal lines , and divided from thence downward toward the head , into as many degrees as the Suns greatest altitude in that latitude will be , which with us at London is to 62 degrees ; which line I call the Scale of Altitudes , divided to whole , halfs , and sometime quarters of degrees . 2. Secondly , On the other leg , and next to the inside is the line of hours , usually divided into hours , quarters , and every fifth minute , beginning at the head with 4 , and so proceeding to 5 , 6 , 7 , 8 , 9 , 10 , 11 , and 12 at the end , and then back again with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , for the morning and afternoon hours . 3. Next to this is a Kalendar of Moneths and Days in two lines ; the uppermost contains that half year the days lengthen in , and the lowermost the shortning days , as by the names of the moneths may appear ; the name of every moneth standing in the moneth , and at the beginning of the moneth : and all but the two moneths that have the longest , and the shortest days , viz. Iune and December , are divided into single days , the tenth day having a figure 10 , or a point or prick on the head of the stroke , and the fifth onely a longer stroke without a prick , and the beginning of every moneth a long stroke , and every single day all alike of one shortness , according to the usual manner of distinguishing on lines . 4. And lastly you have a line of degrees , for so they be most properly called , and they are the same with the equal limb on quadrants , and serve for the same use , viz. for taking of Altitudes , or Horizontal Angles , and are divided usually to whole , and half degrees of the quadrant , and figured with 30 , 40 , 50 , 600 , 7010 , 8020 , and 90 , just on the head , cutting the center or point , where the Scale of Altitudes and the Line of Hours meet ; which point , for distinction sake , I call The rectifying point . And the reckoning on this line , as to taking of Altitudes , is thus : At the number 600 is the beginning , then towards the head count 10 , 20 , 30 , where the 90 is ; then begin at the end again , & count as the figures shew you to 90 at the head , as before . CHAP II. The Uses of the Rule follow . 1. To Rectifie or set the Rule to his true Angle . OPen the Rule to 60 degrees , which is done thus , ( indifferently : ) make the lines on the head , and the lines on the other leg , meet in a streight line ; then is the Scale of Altitudes and the line of Hours set to an Angle of 60 degrees , the rectifying point , being the center of that Angle ; Or to do it more exactly , do thus : put one point of a pair of Compasses into the rectifying point , then open the other to 10 , 20 , 30 , or 40 , on the Scale of Altitudes , the Compasses so opened , and the point yet remaining in the rectifying point , turn the other to that margenal line in the line of hours , that cuts the rectifying point , and there stay it ; then remove the point that was fixed in the rectifying point , and open or shut the Rule , till the point of the Compasses will touch 10 , 20 , 30 , or 40 , being the point you set the Compasses too in the Scale of Altitudes , in the innermost line that cuts the center , and the rectifying point , then is it set exactly to 60 degrees , and fitted for observation . 2. To finde the Suns Altitude at any time . Put a pin in the center hole , at the upper end of the Scale of Altitudes , and on the pin hang a thread and plummet ; then if the Sun be low , that is to say , under 25 degrees high , as in the winter it will always be , then lift up the moveable leg , where the moneths and the degrees be , till the shadow of the end fall just on the meeting of that leg with the head , then the thread shall shew the Suns altitude , counting from 600 towards the head , either 10 , 20 , 25 , or any degree between . But if the Sun be above 25 or 30 degrees high , lift up the head leg till the shadow of that play as before , or make the shadow of the pin in the center hole play on the innermost line of the Scale of Altitudes where the pin standeth , then the thread will fall on the degree , and part of a degree that his true altitude shall be . But if the Sun be in a cloud , and can not be seen so as to give a shadow , then look up along by the head-leg , or moveable leg , just against the middle of the round body of the Sun , and the thread playing evenly by the degrees , shall show the true altitude required . The like must you do for a Star , or any other object , whose altitude you would find . 3. Having found the Suns altitude , and the day of the moneth , to finde the hour of the day . Whatsoever you finde the altitude to be , take the same off from the Line of Altitudes , from the center downwards with a pair of Compasses , then lay the thread ( being put over the pin ) on the day of the moneth , then put one foot of the Compasses in the line of hours , in that line that cuts the rectifying point , and carry it further off , or nigher , till the other foot of the Compass being turned about , will just touch the thred , at the nearest distance , then the point of the Compasses on the line of hours , shall shew the true hour and minute of the day required . Example on the 2. of July . 1. I observe the altitude in the morning , and I finde it to be 30 degrees high , then laying the thread on the day of the moneth , and taking 30 degrees from the Scale of Altitudes , and putting one point in the line of hours , till the other point turned about , will but just touch the th●ead , and I finde it to 23 minutes past 7 , but if it had been in the afternoon , it would have been 37 minutes past 4. 2. Again , on the tenth of August in the Afternoon , at 20 degrees high , I take 20 degrees from the Scale of Altudes , and laying the thread on the day of the moneth , viz. the tenth aforesaid , counting from the name at the beginning of August , toward September , and carrying the Compasses in the line of hours , till the other point doth but just touch the thread , and you shall finde it to be 54 minutes past 4 a clock . 3. Again , on the 11. of December at 15 degrees high , work as before , and you shall finde it to be just 12 a clock ; but to work this , you must lay the Rule down on something , and extend the thread beyond the Rule , for the nighest distance will happen on the out-side of the Rule . 4. Again , on the 11 of Iune at noon I finde the altitude to be 62 degrees high , then laying the thread on the 10 th or 11 th of Iune , for then a day is unsensible , and working as before , you shal finde the point of the Compasses to stay at just 12 a clock , the time required for that altitude . 4. To finde the Suns rising any day in the year . Lay the thread on the day of the month , and in the line of hours it sheweth the true hour and minute of the Suns rising or setting ; for the rising , count the morning hours ; and for the setting , count the evening hours . 5. To finde if any place lye level , or nor . Open the rule to his true angle of 60 degrees , then set the moveable leg upon the place you would make level , and if the thread play just on 60 degrees , it is a true level place , or else not . 6. To try if any thing be upright or not . Hang a thread and plummet on the center , then aply the head leg of the rule to the wall or post , and if it be upright , the thread will play just on the innermost line of the scale of altitudes , or else not . CHAP. III. A further description of the Rule , to make it to shew the Suns Azimuth , Declination ; True place , right Ascention , and the hour of day or night , in this , or any other Lattitude . 1. FIrst in stead of the scale of Altitudes to 62 degrees , there is one put to 90 degrees in that place , and that of 62 is put by in some other place where it may serve as well 2. The line of hours hath a double margent , viz , one for hours , and the other for Azimuths , & then every 5 th minute is more properly made 4 , or else every 2 minutes , and in a large rule to every quarter of a degree of Azimuth , or to every single minute of time . 3. The degrees ought to be reckoned after 3 maner of wayes : first as before is exprest ; secondly from 60 toward the end , with 10 , 20 , 30 , 40 , 50 , 60 , &c. to be so accounted in finding the Azimuch for a particular latitude ; and and thirdly from the head or 90 , toward the end , with 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , &c. for the general finding of Hour and Azimuth in any latitude , and many other problems of the Sphere besides ; to which may be added , where room will alow , a line of hours , beginning at 6 at the head , and 12 at the end , but reckoning 15 degrees for an hour , and 4 minutes for every degree , it may do as well without it . 4. To the Kalender of moneths and days , is added a line of the Suns true place in the Zodiack , or where room fails , the Characters of the twelve Signs put on that day of the moneth the Sun enters into it , and counting every day for a degree , may indifferently serve for the use it is chiefly intended for . 5. Under that is a line of the Suns right Ascension , to hours and quarters at least , or rather every fifth minute , numbred thus : 12 and 24 right under ♈ and ♎ , or the tenth of March , and so forward to the tenth of Iune , or ♋ , where stands 6 , then backwards to 12 where you began , then backwards still to the eleventh of December , with 13 , 14 , 15 , 16 , 17 , 18 , to ♑ , then from thence forward to 24 where you first began : but when you are streightned for room , as on most ordinary Rules you will be , then it may very well suffice to have a point or stroke , shewing when the Sun shall gradually get an hour of right Ascension , and from that for every day count four minutes of time , till it hath increased to an hour more , and this computation will serve very well ; and in stead of saying 13 , 14 , 15 hours of right Ascension , say 1 , 2 , 3 , &c. which will perform the work as well , and reduce the time to more proper terms . 6. There is fitted two lines , one containing 24 houres , and the other 29 days , and about 13 hours , and they serve to finde the time of the Moons coming to the South , before or after the Sun , and by that , the time of high-water at London-bridge , or any other place , as is ordinary . CHAP. IV. The Uses follow in order . 1. To finde the Suns Declination . LAy the Thread on the day of the moneth , then in the line of degrees you have the declination . From March the tenth toward the head is the Declination Northward , the other way is Southward , as by the time of the year is discovered . Example : On the tenth of April it is 11d 48 ' toward the North ; but on the tenth of October it is 10d 30 ' toward the South . 2. As the thread is so laid on the day of the moneth , in the line of the Suns place , it sheweth that ; and in the line of the Suns right Ascension , his right Ascension also , onely you must give it its due order of reckoning , as thus : it begins at ♈ Aries , and so proceeds to ♋ , then back again to ♑ at the eleventh of December , then forwards again to ♈ Aries , where you began . 3. To finde the Suns right Ascension in hours and minutes . Lay the thread as before , on the day of the moneth , and in the line of right Ascension you have the hour and minute required , computing right according to the time of the year , that is , begin at the tenth of March , or ♈ Aries , and so reckon forwards and backwards as the moneths go . Example . On the tenth of April the Suns place is 1 degree in ♉ Taurus , and the Suns right Ascension 1 hour 55 minutes : on the tenth of October 27d 1 / 4 in ♎ Libra , and his right Ascension is 13 hours and 42 minutes . 4. To finde the Suns Amplitude at rising or setting . Take the Suns Declination out of the particular Scale of Altitudes , and lay it the same way as the Declination is , from 90 in the Azimuth Scale , and it shall shew the Amplitude from the east or west , counting from 90. Example : May the tenth it is 33. 37. CHAP. V. Having the Suns Declination , or day of the moneth , to finde the Azimuth at any Altitude required for that day . FIrst finde the Suns Declination , by the first Proposition of the fourth Chapter , then take that out of the particular Scale of Altitudes , or scale to 62 degrees ; then whatsoever the Altitude shall happen to be , count the same on the degrees from 60 toward the end of the Rule , according to the second maner of counting , in the third Proposition of the third Chapter , and thereunto lay the thred , then the Compasses set to the Declination , carry one point along the line of hours on the same side of the thread the Declination is ; that is to say , if the day of the moneth , or Declination be on the right side the Aequinoctial , then carry the Compasses on the right side ; but if the Declination be on the South side , that is , toward the end , ( counting from the tenth of March , or Aries or Libra , then carry the Compasses along the line of hours and Azimuths on the left side of the th●ead , as all win●er time it will be , and having set the Compasses to the least distance to the thread , it sh●ll stay at the Suns true Azimuth from the South required , counting as the figures are numbred ; or from East or West , counting from 90. Example 1. On the tenth of Iuly I desire to finde the Suns Azimuth at any Altitude , first on that day I finde the Suns Declination to be 20. 45 , which number count from the beginning of the particular Scale of Altitudes toward 62 , and that distance take between your Compasses , then are they set for all that day ; then supposing the Suns height to be ten degrees , lay the thread on 10 , counted from 60 toward the left end , then carrying the Compasses on the right side of the thread , ( because it is summer or north declination ) on the line of Azimuths , it shall shew 110. 40 , the Azimuth from the south required ; but if you count from 90 , it is but 20. 40. from the east , or west point northward , according to the time of the day , either morning or evening . Example 2. Again , on the 14. of November , or the 6. of Ianuary , when the Sun hath the same declination south-ward , and the same Altitude , to work this you must lay the Rule down on something , then lay the thread on the Altitude , counted from 60 toward the end ( as before ) and carrying the Compasses on the south-side of the Aequinoctial , along the Azimuth-line , till the other point do but just touch the th●ead , and it shall stay at 36. 45 , the Azimuth from south required ; if it be morning , it wants of coming to south ; if it be after-noon , it is past the south . Example 3. But if the Sun be in the Aequinoctial , and have no declination , then it is but laying the thread to the Altitude , and in the line of Azimuths the thread shall shew the true Azimuth required . As for instance : at 00 degrees of altitude , the Azimuth is 90 , at 10 degrees it is 77. 15 , at 20 degrees 62. 45 , at 30 degrees high 43. 15 , at 35 degrees high 28. 10 , at 38 degrees 28 ' high , it is just south , as by practice may plainly appear . But if the Suns altitude be above 45 , then the degrees will go beyond the end of the Rule : To supply this defect , do thus : Substract 45 out of the number you would have , and double the remainder , then lay the Rule down with some piece of the same thickness , in a streight line with the moveable leg ; then take the distance from the tangent of the remainder doubled ( counted from 60 to the end of the Rule , in the line next the edge ) to the Center , lay that distance in the same streight line from the tangent doubled , and that shall be the tangent of the Angle above 45 , whereunto you must lay the thread for the finding the Azimuth , when the Sun is above 45 degrees high . CHAP. VI. To finde the hour of the Night by the Moon . FIrst by the help of an Almanack , get the true time of the New Moon , then compute her true place at that time , which is always the place of the Sun ( very nigh ) at the hour and minute of conjunction ; then compute how many days old the Moon is , then by the line of Numbers say : If 29 dayes 13 hours , ( or on the line 29. 540 ) require 860 degrees , or 12 signs , what shall ●ny less number of days and part of a day require ? The answer will be : The Moons true place at that age . Having ●ound her true place , then take her al●itude , and lay the thred on the Moons place found , and work as you did for the Sun , and note what hour you finde ; then consider if it be New Moon , the hour you finde is thētrue hour , likewise in the Full ; but if it be before or after , you must substract by the Line of Numbers thus : If 29 days 540 parts require 24 hours , what shall any number of days and parts require ? The answer is : What you must take away from the Moons hour found , to make the true hour of the Night which was required . But for more plainness sake , I will reduce these Operations to so many Propositions , before I come to an Example . PROP. 1. To finde the Moons Age. First , it is most readily and exactly done by an Ephemerides , such a one as you finde in Mr. Lilly's Alman●ck , or ( as to her Age onely ) in any book or Sheet-Almanack ; but you may do it indifferently by the Epact thus , ( by the Rules of the 152 page in the Appendix to the Carpenters Rule . ) Adde the Epact , the moneth , and the day of the mone●h together , and the sum , if under 30 , is the Moons age ; but if above , consider if the moneth have 30 or 31 days , then substract 29 or 30 out , and the remainder is the Moons age in days . Example . August 2. 1660. Epact 28. Month 6. day 2. added makes 36. Now August or sixt moneth , hath 31 days , therefore 30 being taken away , 6 days remains for the moons age required . PROP. 2. To finde the Moons place . By the Ephemerides aforesaid in Mr. Lilly's Almanack , you have it ser down every day in the year ; but to finde it by the Rule , do thus : Count six days back from August 2. viz. to Iuly 27. there lay the thread , and in the line of the Suns place , you have the Moons place required , being then near alike ; then in regard the Moon goes faster than the Sun , that is to say , in 29 days 13 hours , 12 signs , or 360 degrees ; in 3 days 1 sign , 6 degr . 34 min. 20 sec. in one day o signs , 12 degr . 11 min. 27 sec. in one hour , 30 min. 29 sec. ( or half a minute : ) adde the signs , and degrees , and minutes the Moon hath gone in so many days and hours , if you know them together , and the Sun shall be the Moons true place , being added to what she had on the day of her change ; but far more readily , and as exactly by the line of Numbers ( or Rule of Three ) say , if 29. 540. require 360. what 6 facit 73 1 / 4 , that is , 2 signs , 13 degrees , 15 minutes , to be added to 14 degrees in ♌ , and it makes 27 1 / 4 in ♎ , the Moons place for that day . Or thus , multiply the Moons age by 4 , divide the product by 10 , the quotient sheweth the signs , and the remainder multiplied by 3 , sheweth the degrees which you must adde to the Suns place on the day required , and it shall be the Moons true place required for that day of her Age. Example . Iuly 27. the Sun and Moon is in Leo 14 degr . 0. August 2. being 6 days , adde the Moons motion 2. 13d 9 ' . makes being counted , Virgo ♍ , Libra ♎ . 27 deg . 9 the place required ; which on the Rule you may count without all this work or trouble : but for plainness sake I am constrained so to do . Or thus . PROP. 3. To finde the Moons hour . To do this , you must do the work of the second Chapter , and second and third Proposition ; where note that the Moons place found , is to be used as the day of the moneth , or Suns declination . Example . The Moon being 27 degrees in Libra , and 20 degrees high , I finde the our to be 31 ' past 9 , if on the east-side ; or 29 ' past 2 , if on the west-side ●f the Meridian . PROP. 4. To finde the true hour of the Night . Having found the Moons hour , as before , consider the Moons age , then say by the Line of Numbers , or Rule of Three , if 29. 540 part require 24 , what shall 6 days require facit 4 hours and 52 minutes ? which taken from 9. 31. rest ● . 29. the true time required . Example . Moons hour — 9 31 Time to be subst . ● 52 True time remain 3 39. the time required . This work is done more readily by the two lines fitted for that purpose ; for look for 6 the Moons age in one , and you shall finde ● hours 52 minutes , the time to be substracted in the other . CHAP. VII . To finde the hour of the Night by the fixed Stars . FOr the doing of this , I have made choice of twelve principal fixed Stars , all within the two Tropicks ; many more might be added , but these will very well serve the turn : The names of them , and their right Ascension in hours and minutes , is set on the Rule , and the star is placed in his true Declination on , or among the moneths ; and for to know the stars next to a Tutor , a Celestial Globe , or a Nocturnal , of all the chief stars from the Pole to the Equinoctial , and to be had at the Sun-Dial in the Mi●nories , is the best ; the uses whereof do follow . First know the star you observe , then observe his Altitude , and laying the thread on the star by the second Chapter , second and third Proposition , get the stars hour , then out of the righ● Ascension of the star , take the righ● Ascension of the Sun , ( found by the fifth Proposition of the third Chapter ) for that day , and note the difference for this difference added to the star hours found , shall shew the hour o● the night . Example . On the first of November I observe the Altitude of the Bulls-eye , and find it to be 30. then by second and third of the second Chapter , I finde the hour to be 7. 54 past the Meridian the Suns right Ascension that day finde to be 15 hours 8 minutes , the stars right ascension is 4 hours 16 minutes ; which taken , the greatest from the least , by adding of 24 hours , re● 13. 08. then 7. 54. the stars hou● added , makes 21. 02. from which taking 12 hours , rest 9. 02. the hour o● the night required . For more plainness , note the work of two or thre● Examples . Stars right Ascension being set on the Rule — 4 — 16 Suns right Asc. Nov. 1. — 15 — 08 — Substraction being made , by adding 24 , remain — 13 — 08 To which you must adde the stars hour found — 7 — 54 Then the remainder , taking away 12 hours , is — 9 — 02 the true hour . Again May 15 by Arcturus , at 50 0 ' high westwards . The right Asc. of Arcturus , is 14h● — 0 ' Suns right Asc. May 15. is — 4 — 10 — The right Asc. of the Sun taken from the R. Asc. of the star , rest ● — 50 The stars hour at 50 degr . high , found to be — 02 — 13 — Which being added to the difference before , makes — 1● — 03 or — 11 — 03. Again January 5. by the Great-dog at 15 degrees high east . The right Asc. of Great-dog , is 06-29 The right Asc. of the Sun , is 19 — 50 — Substr . made , by adding 24 h. is 10-39 Stars hour at 15 degr . high , is 9 — 32 Ante M. or P. Septen . Which added to the difference found , and 12 substracted , remains 8. 11 ' . for the true Hour of the Night required to be found , and so of any other star se● down in the Rule , as by the trial and pr●ctice , will prove easie and ready to the ingenuous practitioner . But by the line of 24 , or twice 12 hours , and the help of a pair of Compasses , you may perform it withou● writing it down , thus : Take the righ● Ascension of the Sun , out of that lin● of hours between your Compasses ( being always counted under 12 ) an● set the same from the right Ascensio● of the star , toward a lesser number , o● the beginning of the hours , and th● point shall stay at the remainder that is to be added to the stars hour found , then open the Compasses from thence to the beginning of the hours , and adde that to the stars hour found , and it shall reach to the hour of the night required . Example . Feb. 6. 1660. by Arcturus 20 degrees high : Take 10 hours 1 minute between your Compasses , and set it from 14 hours , or 2 hours beyond 12 , and it shall stay at 3. 59 : then take 3. 59 , and adde it to 6. 24 , the stars hour at 20 degrees high , and it shall be 10. 23 , the hour of the night required . CHAP. VIII . To finde the Amplitude or Azimuth of the fixed Stars ; also their rising , setting , and southing . 1. FIrst for the Amplitude , take the stars declination from the particular Scale of Altitudes , and lay it from 90 in the Azimuth-line , and it shall shew his amplitude from East or West toward South or North , according to the declination , and time of day , morning or evening . The same work is for the sun . Example . The Bulls-eye hath 25. 54 degrees of amplitude , so hath the Sun at 15 degrees 48 minutes of declination . 2. To finde the Azimuth , work as you did for the Sun at the same declination the star hath , by Chapter 5. and you shall have your desire . Example . December 24. at 6 degrees high , by the Bulls-eye I finde the Azimuth to be 107. 53. from the south . 3 To finde the stars rising and setting , lay the thread on the star among the moneths , and in the line of hours it shews the stars rising and setting , as you counted for the Sun ; but yet note this is not the time in common hours , but is thus found : Adde the complement of the Suns Ascension , and the stars right Ascension , and the stars hour last found together , and the Sun , if less than 12 ; or the remain 12 being substracted , shall be the time of his rising in common hours ; but for his setting , adde the stars setting last found to the other numbers , and the sum or difference shall be the setting . Example . For the Bulls-eye on the 23 of December , it riseth at 2 in the afternoon , and sets at 4. 46 in the morning . 4. To finde the time of the southing of any star on the Rule , or any other whose right ascension and declination is known , Substract the Suns right ascension from the stars , increased by 24 , when you cannot do without , and the remainder , if less than 12 , is the time required , in the afternoon or night before 12 ; but if there remain more than 12 , substract 12 , and the residue is the time from mid-night to mid-day following . Example . Lyons-heart on the tenth of March , the Suns Ascension is 0 2 ' . Lyons-heart whole right asc . is 9 50 ' Time of southing is 9 48 ' at night . 5. To finde how long any Star will be above the Horizon . Lay the thread to the star , and in the hour-line it sheweth the ascensional difference , counting from 90 ; then note if the star have North declination , adde that to 6 hours , and the sum is half the time ; if south , substract it from 6 , and the residue is half the time ; and the complement of each to 24 being doubled , is the whole Nocturnal Arch under the Horizon . Example . For the Bulls-eye , his Ascensional difference will be found to be one hour , 23 minutes , which added to 6 hours and doubled , makes 14. 46 , the Diurnal Ark of the Star , and the residue from 24 is 9. 14. for the Nocturnal Ark , or the time of its being under the Horizon . CHAP. IX . To perform the fore-going work in any latitude , as rising , amplitude , ascensional difference , latitude , hour , and azimuth , wherein I shall give onely the rule , and leave out the examples for brevity sake . 1. FOr the rising , and setting , and ascensional difference , being all one , do thus : Take the Suns declination out of the general Scale of Altitudes , then set one foot of the Compasses in the colatitude on the same scale , and with the other lay the thred to the nighest distance ; then the thred so laid , take the nighest distance from the latitude to the thread , with that distance set one foot in the Suns declination , counted from 90 toward the center , and the thread laid to the nearest distance , shall in the degrees shew the ascensional difference required , counting from 90 at the head toward the end of the Rule ; and if you reduce those degrees and minutes to time , you have the rising and setting before and after 6 , according to the declination and time of the year . 2. To finde the Suns amplitude . Take the Suns declination , and setting one foot in the colatitude , with the other lay the thread to the nearest distance , and on the degrees it sheweth the Suns amplitude at rising or setting , counting as be●ore from 90 to the left end of the Rule . 3. Having amplitude and declination , to finde the latitude . Take the declination from the general scale , and set one foot in the amplitude , the thread laid to the nearest distance in the line of degrees , it sheweth the complement of the latitude required , or the converse . 4. Having latitude , Suns declination and altitude , to find the height at 6 , and then at any other time of the day and year . Count the declination in the degrees from 90 toward the end , thereto lay the thread , the least distance from which to the latitude in the general Scale , shall be the Suns height at 6 in the summer , or his depression in the winter . The Compasses standing at this distance , take measure on the general Scale of altitudes , from the beginning at the pin towards 90 , keeping one point there , open the other to the Suns altitude , thus have you substracted the height at 6 , out of the Suns altitude ; but in winter you must adde the depression at 6 , which is all one at the same declination with his height at 6 in summer , and that is done thus : Put one point of the Compasses so set in the general Scale to the Suns Altitude , then turn the other outwards toward 90 , there keep it , then open the Compasses to the beginning of the Scale , then have you added it to the Suns altitude ; having this distance , set one foot in the colatitude on the general Scale , lay the thread to the nearest distance ; the thread so laid , take the nearest distance from 90 to the thred , then set one foot in the declination , counted from 90 , and on the degrees it sheweth the hour from 6 , reckoning from the head , or from 12 , counting from the end of the Rule . I shall make all more plain , by making three Propositions of it , thus : Prop. 1. To finde the hour in the Aequinoctial . Take the Altitude from the beginning of the general Scale of altitudes , and set one foot in the colatitude , the thread laid to the nearest distance ( with the other foot ) in the degrees , shall shew the hour from 6 , counting from 90 , and allowing for every 15d 1 hour , and 4 min : for every degree . Prop. 2. To finde it at just 6. Is before exprest by the converse of the first part of the fourth , which I shall again repeat . Prop. 3. To finde it at any time do thus . Count the Suns declination in the degrees , thereunto lay the thred , the least distance , to which from latitude in the general Scale , shall be the Suns altitude at 6 ; which distance in summer you must substract from , but in winter you must add to the Suns present altitude ; having that distance , set one foot in the coaltitude , with the other lay the thread to the neerest distance , take again the neerest distance from 90 to the thread , then set one foot in the Suns diclination counted from 90 , and lay the thread to the neerest distance , and in the degrees it shall shew the hour required . Example . At 10 declination north , and 30 high , latitude 51. 32 , the hour is found to be 8. 25 , counting 90 for 6 , and so forward . Again at 20 degrees of declination South , and 10 degrees of altitude , I finde the hour in the same latitude to be 17 minutes past 9. Having latitude , delination , and altitude , to finde the Suns Azimuth . Take the sine of the declination , put one foot in the latitude , the thread laid to the neerest distance : in the degrees , it sheweth the Suns height at due East or West , which you must in summer substract from the Suns altitude , as before on the general Scale of Altitudes , with which distance put one foot in the colatitude , and lay the thread to the neerest distance , then take the neerest distance from the sine of the latitude , fit that again in the colatitude , and the thread laid to the heerest distance , in the degrees shall shew the Suns Azimuth required . 6. But in winter you must do thus : By the second Proposition of the ninth Chapter , finde the Suns Amplitude for that day , then take the altitude from the general Scale of altitudes , and putting one point in colatitude , lay the thread to the neerest distance , then the neerest distance from the latitude must be added to the Suns Amplitude ; this distance so added must be set from the coaltitude , and the thread laid to the highest distance , and in the line of degrees , it gives the Azimuth from south , counting from the end of the rule , or from the East or West , counting from the head or 90 degrees . Example . At 15 degrees of declination and 10 altitude , latitude 51. 32. the Azimuth is 49. 20. from the South , or 40 degrees and 40 ' from East or West . CHAP. X. To finde all the necessary quesita for any erect declining Sun-dial both , particularly and general , by the lines on the Dial side , also by numbers , sines , and tangents artificial , being Logarithms on a Rule . 1. First a particular for the Substile . COunt the plains declination on ●he Azimuth scale , from 90 toward the end , and thereunto lay the thread , in the line of degrees it shews the distance of the substile from 12. Example . At 10 degrees declination , I find 7. 51. for the substile . 2. For the height of the stile above the substile . Take the Plaines Declination from 90 in the Azimuth line , but counted from the South end , between your compasses : and measure it in the particular scale of altitudes , and it shall give the height of the stile required . Example . At 30 declination is 32. 35. 3. For the inclination of Meridians : Count the substile on the particuler scale of Altitudes , and take that distance between your compasses , measure this distance on the Azimuth line from 90 toward the end , and counting that way it sheweth the inclination of Meridians required . Example . At 15 the substile , the inclination of Meridians will be found to be 24. 36. 4. To finde the Angle of 6 from 12. Take the plaines declination from the particular scale of altitudes , and lay it from 90 on the Azimuth scale , and to the Compasses point lay the thread : then on the line of degrees you have the complement of 6 from 12 , counting from 60 toward the end . Note this Rule ( as this line is drawn ) doth not give this Angle exactly , neither will it be worth the while to delineate another line for this purpose . But if it be required it may be done , but I rather prefer this help , the greatest error is about the space of 45 minutes of the first degree in the particular scale of altitudes ; so that if you conceive those 45 minutes to be divided as the particular scale of altitudes is , like a natural sine , and if your declination be 30 , then take half the space of the 45 minutes less , and that shall be the true distance to lay on the Azimuth line from 90 whereunto to lay the thread . Example . A plaine declining 30 degrees , the angle will be found to be 32. 21. whose complement 57. 49. is the angle required . 5. To perform the same generally by the general scale of altitudes ; and first for the stile . Lay the thred to the complement of the latitude , counted in the degrees from the head toward the end , then the nightest distance from the complement of the plaines declination to the thread , taken and measured on the general scale , from the center , shall be the stiles height required . 6. To finde the inclination of Meridians Take the plaines declination , from the general scale , and fit it in the complement of the stiles elevation , and lay the thread to the neerest distance , and on the degrees it sheweth the inclination of Meridians required . 7. For the substile , Count in the inclination of Meridians on the degrees from 90 , and thereto lay the thread , then take the least distance from the latitudes complement to the thread , set one foot of that distance in 90 , and lay the thread to the neerest distance , and in the degrees it shall shew the substile from 12 required . 8. For the angle of 6 from 12. Take the side of the square , or the measure of the parallel from 12 , and fit it in the cosine of the latitude , and lay the thread to the nighest distance , then take out the nearest distance from the sine of the latitude to the thread , then fit that over in the sine of 90 , and to the nearest distance lay the thread , then take the nearest distance from the sine of the plains declination to the thread , and it shall reach on the parallel line , or side of the square , from the Horizon to 6 a clock line required . Four Canons to work the same by the artificial sines & tangents . Inclination of Meridians . As the Sine of the latitude , To the Sine of 90 : So the Tangent of the Declination , To the Tangent of inclination of Meridians . Stiles Elevation . As the Sine of 90 , To the Cosine of the Declination : So the Cosine of the latitude , To the Sine of the Stiles elevation , Substile from 12. As the Sine of 90 , To the Sine of the Declination : So the Cotangent of Latitude , To Tangent of the Substile from 12 ▪ For 6 and 12. As the Co●tangent of the Latitude , To the Sine of 90 : So is the sine of Declination , To the Cotangent of 6 from 12. For the hours . As the Sine of 90 , To the Sine of the Stiles height : So the Tangent of the hour from the proper Meridian , To the Tangent of the hour from the Substile . The way to work these Canons on the Sines and Tangents , is generally thus : As first , for the inclination of Meridians , set one point in the Sine of the latitude , open the other to the Sine of 90 , that extent applied the same way , from the Tangent of the Plains declination , will reach to the Tangent of the inclination of Meridians required . CHAP. XI . To draw a Horizontal Dyal to any latitude . FIrst draw a streight line for 12 , as the line A B , then make a point in that line for a Center , as at C , then through the Center C , raise a perpendicular to A B , for the two six a clock hour-lines , as the line D E ; then draw two occult lines parallel to A B , as large as the Plain will give leave , as D E , and E G then fit C D in the Sine of the Latitude , in the general Scale , and lay the thread to the nighest distance , then take the nearest distance from 90 to the thread , and set it from D and E in the two occult lines , to F and G , and draw the line F and G parallel to the two sixes , ( or make use of the Sines on the other side , thus : Fit A D , or C D in the Sine of the latitude , and take out the Sine of 90 , and lay it as before from D and E , ) then fit D F , or E G in the Tangent of 45 degrees , on the other side of the Rule , and lay off 15 , 30 , and 45 , for every whole hour , or every 3 degrees and 45 minutes for every quarter , from D and E , toward F and G , for 7 , 8 9 , and for 3 , 4 , and 5 a clock hour points . Lastly , set C D , or B E in the Tangent of 45 , and lay the same points of 15 , 30 , 45 , both wayes , from B or 12 , for 10 , 11 , and 1 , 2 , and to all those points draw lines for the true hour-lines required , for laying down the Stiles height ; if you take the latitudes complement , out of the Tangent-line as the Sector stood , to prick the noon hours , and set it on the line D F , or E G , from D or E downwards from D to H , it will shew you where to draw C H for the Stile , then to those lines set figures , and plant the Dial Horizontal , and the Stile perpendicular , and right north and south , and it shall shew when the sun shineth , the true hour of the day . Note well the figure following . CHAP. XII . To draw a Vertical , Direct , South , or North Dyal . FIrst draw a perpendicular line for 12 a clock , then in that line at the upper end , in the south plain : and at the lower end in a north plain , appoint a place for the center , through which point cross it at right angles , A Horizontall Diall A South Diall for 6 and 6 , as you did in the Horizontal Plain , as the lines A B , and C D , on each side 12 make two parallels , as in the Horizontal , then take A D the parallel , and fit it in the sine of the latitudes complement , and take out the sine of 90 and 90 , and lay it in the parallels from D and C , to E and F , and draw the line E F , then make D E , and B E tangents of 45 , and lay down the hours as you did in the horizontal , and you shall have points whereby to draw the hour lines . For the north you must turn the hours both ways for 4 , 5 , 8 and 7 in the morning and 4 , 5 , 7 and 8 , at night the height of the stile must be the tangent of the complement of the of the latitude when the sector is set to lay off the hours from D , as here it is laid down from C to G , and draw the line A G for the stile . For illustration sake note the figure . CHAP. XII . To draw an erect East or West Dial. FIrst by the fifth Proposition of the second Chapter draw a horizontal line , as the line A B at the upper part of the plaine . Then at one third part of the line A B , from A the right end if it be an East plaine , or from B the left end , if it be a West Plain , appoint the center C , from which point C draw the Semicircle A E D , and fit that radius in the sine of 30 degrees , ( which in the Chords is 60 degrees ) then take out the sine of half the , latitude , and lay it from A to E , and draw the line C E for six in the morning on the East , ( or the contrary way for the West . ) Then lay the sine of half the complement of the latitude , from D to F , and draw the line C F , for the contingent or equinoctial line , to which line you must draw another line parallel , as far An East Diall . A West diall . assunder as the plaine will give leave , then take the neerest distance from A to the six a clock line , or more or less as you best fancy , and fit it in the tangent of 45 degrees , and prick down all the houres and quarters , on both the equinoctial lines , both ways from six , and they shall be points , whereby to draw the hoor lines by , but for the two houres of 10 and 11 there is a lesser tangent beginning at 45 , and proceeding to 75 , which use thus : fit the space from six to three , in the little tangent of 45 , and then and then lay of 60 in the little tangents from 6. to 10 and the tangent of 75 from 6 to 11 , and the respective quarters also if you please , so have you all the hour●s in the East , or west Diall , the distance from six to nine or from six to three , in the West , is the height of the stile , in the East and West Diall , and must stand in the six a clock line , and parallel to the plaine . CHAP. XIII . To finde the declination of any Plain . FOr the finding of the declination of a Plain , the most usual and easie way , is by a magnetical needle fitted according to Mr. Failes way , in the index of a Declinatory ; or in a square box with the 90 degrees of a quadrant on the two sides , or by a needle fitted on the index of a quadrant , after all which ways , you may have them at the Sign of the Sphere and Sun-Dial in the Minories , made by Iohn Brown. But the work may be very readily and exactly performed by the rule , either by the Sun or needle in this manner following , of which two ways that by the Sun is always the best , and most exact and artificial , and the other not to be used ( if I may advise ) but when the other failes , by the Suns not shining , or as a proof or confirmation of the other . And first by the needle because the easiest . For this purpose you must have a needle well touched with the Loadstone of about three or four inches long , and fitted into a box somewhat broader then one of the legs of the sector , with a lid to open and shut ; and on the inside of the lid may be drawn a South erect Dial , and a wire to set the lid upright , and a thread to be the Gnomon or stile to that Dial : it will not be a miss also to extend the lines on the Horizontal part for the same thread is a stile for that also . Also on the bottom let there be a rabbit , or grove , made to fit the leg of the rule or sector ; so as being pressed into it , it may not fall off from the rule , if your hand should shake , or you cease to hold it there . This being so fitted , the uses follow in their order . Put your box and needle on that leg of the rule , that will be most fit for your purpose , and also the north end of the needle toward the wall , if it be a south wall ; and the contrary , if a north , as the playing of the needle will direct you , better then the way how in a thousand words , then open or close the Rule , till the needle play right over the north and south-line , in the bottom of the Box : then the complement of the Angle that the Sector standeth at ( which may always be under 90 degrees ) is the declination of the Plain . But if it happen to stand at any Angle above 90 , then the quantity thereof above 90 , is the declination of the wall . To finde the quantity of the Angle the Sector stands at may be done two ways : first by protraction , by laying down the Rule so set on a board , and draw two lines by the legs of the Sector , and finde the Angle by a line of Chords . Secondly , more speedily and artificially , thus : By the lines of Sines being drawn to 2. 4 , or 6 degrees asunder . The Sector so set , take the parallel sine of 30 and 30 , and measure it on the lateral sines from the center , and it shall reach to the sine of half the Angle the line of sines stands at , being more by 2 , 4 , or 6 degrees then the sector stands at , because it is drawn one , two , or three degrees from the inside . Or else take the latteral sine of 30 from the center , and keeping one foot fixed in 30 , turn the other till it cross the line of sines on that line next the inside , and counting from 90 , it shall touch at the Angle the line of sines standsat , being two degrees more then the Sector stands at ; the lines being drawn so , will be ( as I conceive ) most convenien . Take an Example . I come to a south-east-wall , and putting my box and needle on my Rule , with the cross or north-end of the needle toward the wall , and the Rule being applyed ●lat to the wall , on the edge thereof , on the evenest place thereof , and held level , so as the needle may play well with the head of the Rule toward your right hand , as you shall finde it to be in an east-wall most convenient ; then I open or close the Rule , till the needle play right over the north and south line in the bottom of the box ; then having got the Angle , ( take off the box , or if you put it on the other side that labor may be saved . ) I take the parallel sine of 30 , and measuring it from the center it reaches , suppose to the sine of 20 , then is the line of sines at an Angle of 40 degrees , but the Sector at two degrees less , viz. 38 degrees , whose complement 52 is the declination ; then to consider which way , minde thus : First it is south , because the sun being in the south , shines on the wall . Secondly consider , the sun being in the east , it shines also on the wal , therefore it is east plain : thus have you got the denomination which way , and also the quantity how much that ways . Or if you take the latteral sine of 30 from the center , and turn the point of the Compasses from 30 towards 90 , on the other leg you shall finde it to reach to the sine of 50 degrees , whose complement , counting from 90 , is 40 , or rather 38 , for the reason before-said , or else adde 2 to 50 , and you have the angle required , without complementing of it , being the true declination sought for . Thus by the needle you may get the declination of any wall , which in cloudy weather may stand you in good stead , or to examine an observation by the Sun , as to the mis-counting or mistaking therein ; but for exactness the Sun is alwayes the best , because the needle , though never so good ▪ may be drawn aside by iron in the wall , and also by some kinde of bricks , therefore not to be too much trusted unto . To finde a Declination by the Sun. First open the Rule to an Angle of 60 degrees , as you do to finde the hour of the day , and put a pin in the hole , and hang the thread and plummet on the pin ; also you must have another thread somewhat longer and grosser then that for the hour in a readiness for your use . Then apply the head leg to the wall , if the sun be coming on the Plain , and hold the Rule horizontal or level , then hold up the long thread till the shadow falls right over the pin , or the center hole , at the same instant the shadow shall shew on the degrees , how much the sun wants of coming to be just against the Plain , which I call the Meridian or Pole of the Plain , which number you must write down thus , as suppose it fell on 40 , write down 40-00 want : then as soon as may be take the Suns true altitude , and write that down also , with which you must finde the Suns Azimuth ▪ then substract the greater out of the less , and the remainder is the declination required . But for a general rule , take this : if the Sun do want of coming to the Meridian of the place , and also want of coming to the Meridian of the Plain , then you must alway substract the greater number out of the less ▪ whether it be forenoon or after-noon , so likewise when the Sun is past the Meridian , and past the Plain also . But if the signes be unlike , that is to say , one past the Plain , or Meridian , and the other want either of the Plain , or Meridian , then you must add them together , and the sum is the declination from the South . Which rule for better tenaciousness sake take in this homely rime . Signs both alike substraction doth require , But unlike signs addition doth desire . The further illustration by two or three Examples . Suppose on the first of May , in the forenoon , I come and apply the Rule , being opened to his Angle of 60 degrees to the wall , ( viz. the head leg , or the leg where the center is ) and holding up my thickest thread and plummet , so as the shadow of it crosseth the center , and at the same instant also on 60 degrees , then I say the sun wants 60 deg . of coming to the Meridian of the Plain ; at the same instant , or as soon as possible I can , I take the suns altitude , as before is shewed , and set that down , which suppose it to be 20 degrees , then by the rules before , get the Suns azimuth for that day , and altitude ; which in our example will be found to be 94 degrees from the ( south or ) Meridian , then in regard the signs are both alike , i. e. want , if you substract one out of the other , there remains 34 the declination required ; but for the right denomination which way , either north or south , toward either east or west , observe this plain rule : First , if the Sun come to the Meridian or Pole of the Plain , before it come to the Meridian or Pole of the place , then it is always an East-plain ; but if the contrary , it is a West-plain , that is to say , if the Sun come to the Meridian or Pole of the place , before it comes to the Meridian or Pole of the Plain , then it is a West-plain . Also if the sum or remainder , after addition or substraction , be under 90 , it is a South-plain ; but if it be above 90 , it is a North-plain . Also note , that when the sum or remainder is above 90 , then the complement to 180 , is always the declination from the north toward either east or west ; So that according to these rules in our example it is 34 degrees South-east . Again in the morning , Iune 13. I apply my rule to the wall , and I finde the Sun is past the Pole or Meridian of the Plain 10 degrees , and the altitude at the same time 15 degrees , the Azimuth at that altitude , and day in this latitude , will be found to be 109 degrees want of south or pole of the place ; therefore unlike signs , and to be added , and they make up 119 degrees , whose complement to 180 is 61 ; for 61 and 119 added , make up 180 , therefore this Plain declineth 61 degrees from the north toward the east . Again the same day in the afternoon , I finde the Azimuth past the south or meridian of the place 30 degrees , and at the same time the Sun wants in coming to the meridian or pole of the Plain 10 degrees , here by addition I finde the declination to be 40 degrees south-west . Note what I have said in these three examples , is general at all times ; but if it be a fair day , and time and opportunity serve , to come either just at 12 a clock , when the Sun is the meridian or pole of the place ; or just when the Sun is in the meridian or pole of the Plain , then your work is onely thus : First if you come to observe at 12 , then applying your rule to the wall , and holding up the thread and plummet , how much so ever the Sun wants or is past the pole of the Plain , that is the declination , if it be past it is east-wards ; if it wants , it is south-west-wards ; if neither , a just South Plain , and then the poles , or Meridians of place and Plain , are the same . But secondly , if you come when the Sun is just in the pole of the Plain , then whatsoever you finde the Suns Azimuth to be , that is the declination ; if it wants of south , it declineth East-wards ; if it be past , it declineth West-wards . Thus I have copiously , ( and yet very briefly ) shewed you the most artificial way of getting the declination of any wall , howsoever situated . Note if the Sun be above 15 degrees wanting of the Meridian of the Plain , your rule will prove defective in taking the Plains Meridian when the center leg is next the wall , then you must turn the other leg to the wall , and then you finde a supply for all angles to 45 degrees past the Plain . But for the supply of the rest which is 45 degrees , do thus : open the rule till the great line of tangents & the outside of the leg make a right angle , for which on the head you may make a mark for the ready setting , then making the inside of the leg at the end of 45 , as a center , the tangents on the other leg supply very largely the defect of the othersides . Or if you set on the box and needle on the rule , and open or shut the rule till the shadow of the thread shew just 12 , then the Angle the Sector stands at , is the complement of what the sum wants , or is past the meridian or pole of the Plain . CHAP. XIV . To draw a vertical declining Plain to any declination . FIrst draw a perpendicular-line for 12 , as A B then design a point in that line for the center , as C , at the upper end , if it be a South Plain ; or contrary , if it be a North Plain , then on that center describe an Arch of a Circle on that side of 12 , which is contrary to the Plains declination , as D E , and in that Arch lay off from 12 the substile , and on that the stiles height , and the hour of 6 , being found by the tenth chapter , and draw those lines from C the center , then draw two parallels to 12 , as in the direct south : then fit the distance of the parallels in the secant of the declination , and take the secant of the latitude , and set it from the center C on the line of 12 , to F , and on the parallel from 6 to G , and draw a line by those two points F and G , to cut the other parallel in H : then have you found 6 , 3 , and 9 , then fit 6 G in the tangent of 45 , and prick off 15 , 30 , and the respective quarters both ways from 6 , for the morning , and afternoon hours , then fit F G in 45 , and lay off the same points from F both ways for 10 , 11 , 1 , and 2 , and the quarters also , if you please , and those shall be points to draw the hour-lines by . The stile must be set perpendicular over the substile , to the Angle found by the rules in the tenth chapter , and then the Dial shall shew the true hour of the day , being drawn fit to his proper declination . Another way to perform this Geometrically for all erect Dyals with centers . When you have drawn a line of 12 , and appointed a center , make a Geometrical square on that side 12 , as the stile must stand on , as A B C D , the perpendicular side of which square may also be the parallel as before . Again , fit the side of the square in the cosine of the latitude , and take out the sine of the latitude , and fit that over in the sine of 90 , then take out the 〈◊〉 sine of the declination , and lay it from D to G for the hour of 6 , and draw the line A G for the 6 a clock hour line . Then again , fit the side of the square ( or the distance of the parallel in the other way , when you want a secant , or your secant too little ) in the sine of 90 , and take out the cosine of the declination , fit that in the cosine of the latitude , then take out the sine of 90 , and lay it from the center on the line of 12 , and 〈◊〉 6 in the side of the square , and by those two points draw the contingent line ▪ and then fit those points or distances in the tangent of 45 , and lay down the hours as in the former part of this chapter ; but if you want the hours before 9 in west-decliners , or the hours after 3 in east-decliners , and the 6 fall too high above the horizontal line on that side , you may supply this defect thus : Take the measure with your Compasses from 3 to 4 and 5 , upwards in the west-decliners , or from 9 to 7 and 8 in east-decliners , and lay it upwards from 9 or 3 on the deficient side , in the parallel as it should have been from 6 downward in south or upward in north Plains , and you shall see all your defective points to be compleatly supplied , whereunto draw the hour-lines accordingly . CHAP. XV To draw the Hour-lines on an upright declining Dial , declining above 60 degrees ▪ IN all erect Decliners , the way of finding the stile , substile 6 and 12 , and inclination of Meridians , is the same ; but when you come to protract , or lay down the hour-lines , you shall finde them come so close together , as they will be useless , unless the stile be augmented . The usual way for doing of which , is to draw the Dial on a large floor , and then cut off so much and at such a distance , as best serveth your turn , but this being not always to be affected , for want of conveniency , and large instruments , it may very artificially be done by a natural tangent of 75 or 80 degrees , fitted on the legs of a sector in this manner . The example I shall make use of , let be a Plain declination 75 degrees , South , West . First on the north edge of the Plain draw a perpendicular-line A B , representing 12 a clock line ; then on the center draw an occult Arch of a circle as large as you can , as B D , therein lay off an angle of 37. 30 for the substile , ( though indeed this line will not prove the very substile , yet it is a parallel to it : ) then cross it with two perpendiculers for the two contingent lines , at the most convenient places , one in the upper part , and the other in the lower part of the Plain , as the two lines C E , and F G do shew , then by help of the inclination of Meridians , make the table for the hours in this manner ; for this declination , it is 78 : 09 , Now every hour containing 15 degrees , and every quarter three degrees 45 ' I find that the substile will fall on neer a quarter of an hour past five at night , therefore if you take 75 , the measure of five hours out of 78. 9 , there remaines 03. 09. for the first quarter from the substile next 12. Again , if you take 78. 09 , the inclination of Meridians out of 78. 45 , the measure of 5 hours one quarter , and there remaines 00. 36. for the first quarter on the other side of the substile , then by continual adding of 03. 45 , to 03. 09 , and to 00. 36 , I make up the table as heer you see or else you may against 12 , set down 78. 09 , and then take out 3. 45 , as often as you can , till you come to the substile , and then what the remainder wants of . 3. 45 , must be set on the other side of the substile , and 3. 45 added to that , till you come to as late as the sun will shew on the Plain , as here you see to 1 / 4 past 8. 45 36 . sub . 48 09 2 41 51 8 03 095 51 54 . 38 06 . 06 54. 55 39 . 34 21 . 10 39. 59 24 . 30 36 . 14 24. 63 09 1 26 51 7 18 094 66 54 . 23 06 . 21 54. 70 39 . 19 21 . 25 39. 74 24 . 15 36 . 29 24. 78 09 12 11 51 6 33 093     08 06 . 36 54.     04 21 . 40 39.     00 36 . 44 24.     The stile or gnomon is to be erected right over the substile , to the angle of 9. 16 , and augmented as much as from H to ● , and from I to ● , at the nighest distance from the point H and I , drawnin the substile line on the Plain , this way is as easy , speedy , artificial and true , as any extant , ( if your scales be true , ) and improveth the Plain at the first , to the most possible best advantage that can be . Note also , if in striving to put as many hours as you can on the Plain , the sum of the two natural numbers added together , comes to above 40000 : then you must reject something of one of them , for they will not be comely nor convenient , nor far enough asunder . I might have enlarged to declining , reclining plaines , out my intention is not to make a business of it , but onely to give a taste of the usefulness , and convenience of a Joynt-rule , as now it is improved . An Advertisement relating to Dialling . These directions are sufficient for any Horizontal , direct , and declining vertical Dials , but for all other , as East or West inclining and reclining , direct , or reclining and inclining , and declining polars ; and all declining and inclining : and reclinimg Plains , the perfect knowledge of their affections and scituations is very hard to conceive , but much more hard to remember by roat , or knowingly . A representation of which instrument take in this place , let the circle E S W N , represent the Horizon S AE P N the Meridian , AE W AE E the Equinoctial , P C P the Pole , being the points of an Axes made of th●ead , and passing through the center C W C E being shaddowed , represents the Plain , set to a declination , and to a reclination 45 degrees , F the foot that doth support it ; moreover the Plain hat● an Arch fitted to it , to get the stiles height , and to set it to any angle ; also the Horizon and Plain is made to turn round , which could not well be expressed in this figure , or representation of the instrument : a more liveley and easie help cannot likely be invented , and for cheapness the like may be in a half circle , or made of Past-board . The use and application of which figure , I shall not now speak of , partly because of the facillity in the using thereof , and partly because of the difficulty in description thereof ; and lastly , wheresoever you shall buy the same , at either of the three places , the use will be taught you gratis . Also note , that half of the Instrument may be made either in b●ass , or pastboard , and be made to fold down in a book , and perform the uses thereof indifferently well , for many purposes , as to the Affections of Dials . Then having discovered the affections , you may by the Canno●s in the seventh Chapter of Mr. Windgates R●le of proportion , finde all the requisites , and then to speed the laying down the hours , you may do it by help of the tangent of 45 , as you did in erect decliners ; for after you have drawn the hours of 6 , and three , or nine , and made a parallelogram by two lines parallel to 12 , and one parallel to six , then making the distance from six to nine , or three in the line parallel to 12 , a tangent of 45 , and the distance from 12 to 9 , or 3 in the parallel to 6 , likewise a tangent of 45 , in the sector , and laying off 15 , 30. 45 , or the respective quarters from 6 , and 12 , in those two lines , and they shall be the true points to draw the hour lines , by laying a ruler to them , and the center of the Dial ; but for those that have no centers , the rule of augmenting the stile in far decliners , serveth for these also . CHAP. XVI To finde a perpendicular altitude at one or two stations , and observations by by the degrees on the rule . OPen the rule to his Angle of 60 degrees , then looking up to the top or point of altitude you would observe the height of , as you do when you take the Suns altitude , and note the degrees and parts the thread cuts , and write it down in chaulk or ink , that you forget it not . Then measure from the place you stand to the foot or base of the object , ( being right under the top of the object , whose height you would measure ) in feet , yards , or any other parts . Thirdly , consider this being a right angled Plain triangle , if you have the angle at the top , the angle at the base is always the complement thereof . These things being premised , the proportion holds , as the fine of the angle opposite to the measured side , ( or base ) being the complement of the angle found , is to the base or measured side : so is the angle found , to the height required . Always remember to adde the height of your eye from the ground , ( at the time of taking the angle ) to the altitude found . For the operation of this , extend the compasses from the sine of the complement of the Angle found to the number of the measured side , on the line of numbers that distance applied the same way , from the sine of the Angle found , shall reach on the line of numbers to altitude required . Example at one station . I open my rule , and hang on the thread and plummet on the center , and observing the Angle at C , I finde it to be 41. 45 , and the Angle at B the complement of it 48. 15 , and the measure from C to A 271 feet : then the work being so prepared , is thus : As the Sine of 48. 15 , Is to 271 the measure of the side opposite to it : So is the Sine of the Angle 41. 45 , To 242 the measure of the side A B , opposite to the Angle at C , the height required . Again , at the station D , 160 foot from A , I observe and finde the Angle D to be 56. 30 , the Angle at A is the complement thereof , viz. 33. 30. This being prepared , I extend my Compasses from the sine of 33. 30 to 160 , on the line of numbers the same extent will reach from the sine of 56. 30 , to 242 on the line of numbers , lacking a small fraction , with which I shall not trouble you . An example at two stations . As the Sine of the difference , which is the Angle C B D 14. 45 , Is to the side measured , viz. D C 111 feet on the numbers : So is the sine of the Angle at C 41. 45 , To the measure of the side B C , the hypothenusa , or measure from your eye to the top of the object , viz. 290 feet . Again for the second Operation . As the sine of 90 the right angle at A , Is to 290 the hypothenusa B D : So is the sine of 56. 30 , the Angle at the first station D , To 242 , the Altitude B A , the thing required : : So also is the sine of the Angle at B 33. 70 , the complement of 56 , 30. To 160 the distance from D to A. To perform the same by the line of sines , drawn from the center on the flat-side , and the line of lines , or equal parts or inches in ten parts . To work these or any other questions by the line of natural Sines and Tangents , on the flat-side , drawn from the center , it is but changing the terms , thus : As the measured distance taken out of the line of lines , or any scale of equal parts , is to the sine of the angle , opposite to that measured side , fitted across from one leg to the other , the Sector so standing , take out the parallel sine of the angle opposite to the enquired side , and that measure shall reach on the line of equal parts , to the measure of the Altitude requi●●● ▪ Example as before . Take out of the lines or inches 2. 71 , and fit it in the sine of 48. 15 , across from one legge to the other , which I call A parral sine , ( but when you measure from the center onwards the end , I call it A latteral sine , ) then take out the parallel sine of 41. 45 , and measure it on the line of inches , or equal parts , and it shall reach to 2 inches 42 parts , or 242 , the Altitude required . After the same manner may questions be wrought on the line of lines , sines , or tangents alone , or any one with the other , by changing the Logarithmetical Canon from the first to the second , or third , and the second or third to the first to second ; as the case shall require , from a greater to a less , and the contrary ; for the fourth is always the same ; of which in the use of the Sector , by Edmund Gunter , you may finde many examples , to which I refer you . Also without the lines of sines either natural or artificial , you may find altitudes , by putting the line of quadrat , or shadows , on the Rule as in a quadrant , then the directions in the use of the quadrat , page ▪ 146 of the Carpenters Rule , will serve your turn , which runs thus : As 100 ( or 50 according as it is divided ) to the parts cut by the thread , so is the distance measured , to the height required ; which work is performed by the line of numbers onely . Or again , As the parts cut , to 100 or 50 : so is the height to the distance required . But when the thread falls on the contrary shadow , that is , maketh an Angle above 45 , then the work is just the contrary to the former . What is spoken here of taking of Altitudes , may be applied to the taking of distances ; for if the Sector be fitted with a staff , and a ball-socket , you may turn it either horizontal , or perpendicular , and so take any Angle with it , very conveniently and readily by the same rules and directions as were given for the finding of Altitudes . CHAP. XVII . The use of certain lines for the mensuration of superficial and solid bodies , usually inserted on Ioynt-Rules for the use of Work-men , of several sorts and kindes . FIrst the most general and received line for mensuration of Magnitudes , is a foot divided into 12 inches , and those inches into 8 , 10 , 12 , or more parts ; but this being not so apt for application to the numbers , I shall not insist of it here , but rather refer you to the Carpenters Rule ; yet nevertheless those inches , laid by a line of foot measure , doth by occular inspection onely , serve to reduce foot measure to inches , and inches likewise to foot measure , and some other conclusions also . 1. As first , The price of any commodity at five score to the hundred , either tale or weight , being given , to finde the price of one in number , or one pound in weight . As suppose at two pence half-peny a pound , ( or one ) I demand to what cometh the hundred weight , ( or five-score , ) counting so many pound to the hundred weight ? If you look for two inches and a half , representing two pence two farthings , right against it on the foot measure , you have 21 very near ; for if you conceive the space between 20 and 21 , to be parted into 12 parts , this will be found to contain ten of them , for the odde ten pence . But for the more certain computation of the odde pence , look how many farthings there is in the price of one pound , twice so many shillings , and once so many pence is the remainder , which if it be above 12 , the 12 or 12s , being substracted , the remainder is the precise number of pence , above the shillings there expressed ; and on the contrary , at any price the C hundred , or 5 score , to finde the price of one , or 1l . As suppose at 40s . the C. or 5 score , look for 40 in the foot measure , and right against it in the inches , you have 4 inches , 3 quarters , and 1 / 4 of a quarter , which in this way of account is 4 pence 3 farthings , and about a quarter of one farthing . Thus by the lines , as they are divided , it proceeds to 12 pence a pound ; but if you conceive the inches to be doubled , and the foot measure also , you shall have it to 24 d. or 48 d. the pound , or one in tale , of any commodity . As at 18 d. a piece , or pound , the price comes to 7l . 10s . the C. for then every ten strokes is 20s in the foot measure , and every inch is 2 pence , and every eighth one farthing . 2. Secondly , for the buying of Timber at 50 foot to a Load , at any price the load , how much a foot . Here in resolving this , the inches are to be doubled , and the foot measure taken as it is : As at 40 shillings the Load , 40 in the foot measure stands right against 4 inches 3 quarters and better , which being doubled , is 9d . 2 far . 1 / 2 far . near , for the price of one foot ; and on the contrary , at 5d . a foot , is 41s . 8 d. a load , &c. 3. For the great Hundred of 112l . to the Hundred , let the space of 12 inches be divided into 112 parts , then the like rule holds for that also . For the inches being divided into quarters , every quarter is a farthing , and every eighth half a farthing , and every division of the 112 is a shilling , and every alteration of a farthing in the price of a pound , makes a groat in the Hundred , as thus : At 3 pence a pound is 28s . the C. At 3d. 1 q. a pound , 30s . and 4d . the C. At 3d. 1 / 2 the pound , 32s . 8 d. the C. At 3 d. 3 farthings , 35s . the C. Thus you see that every fraction at a farthing advance , is 4 pence in the Hundred ; but for any other account , as 3 pence farthing half farthing , then count the fraction , as 1 , 12th part of a shilling , and nearer you cannot come by a bare occular inspection ; but the price of the Hundred being given , the price of the pound you have as near by this occular inspection , as any usual Coin is reducable , viz. to the 32 part of a peny , or nigher if you please . Again note here also , you may double , or quadruple the price : as to 24d . or 48d . the pound , or any price between . As for example : At 13d . a pound , is 6l . 1s . 4d . the C. At 32d . or 2s . 8d . the pound , is 14l . 19s . the C. and the like by dupling and quadrupling the inches , and the 112 parts , that layeth by it . 4. These lines of equal parts serve as Scales , for the protracting of any Draught of house or field , or the like ; also for addition or substraction of any small number . 5. Note that the line of foot measure , may be applied for the reducing of any odde fraction to a decimal fraction , as you may fee it in page 64. of Mr. Windgats Arithmetick made easie . 2. The use of the lines of decimal Timber and Board measure . The lines of decimal Timber and Board measure , are fitted to agree with the tenths , or foot measure , as those lines in the first chapter of the Carpenters Rule , are fitted to the inches , and the use of them is thus : And first for the decimal Board measure . Suppose a Board is 1 foot 50 broad , I look for 150 on that line , and from that place to the end of the Rule forwards , toward 100 , so much in length must you have to make a foot of superficial or board measure . 2. Or else thus : If you apply the end of the Rule next 100 , to one edge of the breadth of a board , or glass , then right against the other edge of the board , on that line of decimal board measure , you shall finde the 10ths and 100s , ( or feet 10ths & 100 ) parts of a foot , that you must have in length to make a foot superficial at that breadth . Example . I come to a board and applying the upper end next 100 , even to one edge of the board , the other reacheth to 0. 8 tenths , then I say that 8 tenths of a foot length at that breadth , makes a foot . 3. The use of decimal Timber measure . The use of this is much like the Board measure , onely here you must have a respect to the squareness of the piece , and not to the breadth onely ; for after you know how much the piece of timber or stone is square , in feet and 100 parts , then look that number on the line of decimal Timber measure , and from thence to the end of the Rule , is the length that goes to make a foot of timber . Example . At 14 , or 1. 40. parts of a foot square , look the same on the rule , and from thence to the end where 40 is , is the length of a foot of Timber , at that squareness , being about 51 parts of a foot divided into a 100 parts . 5. The use of the line of decimal yard measure , also running yard measure , according to the inches or decimal parts of a foot . The decimal yard measure , is nothing else but a yard or 3 foot divided into a 100 parts , and used in the same manner as the foot measure is , for if you take the length , and the breadth in that measure , and multiply it together , you shall have the content , in yards and 100 parts of a yard . Example . Suppose a peece of plastering is 4 yards 78 parts one way ; and 7. 35 parts another way : being multiplied together makes 35 yards , and 9954. of 10000 , which is very neer 36 yards . 5. But the decimal running yard measure is fitted to the foot measure , and the use is thus : Suppose a room is to be measured that is 7 foot 8 tenths high , and I would know how much makes a yard , at that breadth or height ; look foor 7f . 8 10ths on the line of decimal running yard measure , and the space on the rule , from thence to the end next 100 is the true length , that goeth to make up a yard of superficial measure , at that breadth or height . But if the peece be between 4 foot 5 10 broad , and 2 foot , then the table at the end of the line , will supply the defect : or you may change the terms , and call the length the breadth , and the contrary . But if it be under 2 foot broad , then if you do as you did with the board measure , you shall have your desire . Example . At 1 foot 3 10th broad , 6 foot 9 10ths make a yard . 6. But if the running yard measure be made to agree with the inches , then measure the height of the room in feet and inches , and if you take a pair of compasses , and measure from that place , to the end of the rule , then turn the compasses set at that distance as many times as you can about the room , so many yards is there in the room . 7. The use of the line of decimal round measure , commonly called Girt-measure , which is when the circumference of a round Cillender , or piller given in inches or ten parts of a foot . First ( for Girt-measure according to inches , being the most usual measure ) now much the pillar is about , then look for the same number on the line of Girt-measure , and from thence to the end of the rule , is the length that goeth to make a foot of Timber . But if it be under 30 inches about , then you must have above two foot in length , and then a table at the end of the line , or a repetition in another line , will supply the defect . But if the line of Girt-measure be divided according to foot measure , then use it as before , seeking the decimal part on the line , and from thence to the end is a foot . 8. The use of a line of solid measure , by having the Diameter of a round piece given in inches , or foot measure . Take the diameter with a rule , or a pair of Callipers , and learn the measure either in inches , or foot measure according as your line of Diameter is divided . Then look for the same number on the line of Diameter , and from thence to the end of the rule forward , is the length that makes a foot of timber at that diameter ( or measure cross the end of the round piece of Timber or stone . ) The Tables of all the under measure for all these lines follow . Decimal Superficiall under M.   10th . F. 1000   10 F. 1000 p.   1 100. 00     3. 848   2 50. 000     3. 706   3 33. 300     3. 570   4 25. 000     3. 450   5 20. 000 3 3 3. 332   6 16. 600     3. 217   7 14. 300     3. 115   8 12. 500     3. 025   9 11. 120     2. 940 1 1 10. 000   5 2. 850   1 9. 100   6 2. 780   2 8. 340   7 2. 700   3 7. 720   8 2. 628   4 7. 150   9 2. 560   5 6. 670   4 2. 500   6 6. ●60     ● . 440   7 5. 888     2. 382   8 5. 5●5     2. 336   9 5. 260     2. 273 2 2 5. 000     2. 213     4. 760     2. 173     4. 5●6     2. 127     4. 350     ● . 083     4. 170     2. 042   5 4 ▪ 000   5 2. 000 Decimall Superficiall . M.   F. 1000. p. 01 F. 1000. p.   1. 962   1. 320   1. 923   1. 304   1. 816   1. 286   1. 850   1. 268   1. 820 8 1. 250   1. 785   1. 237   1. 756   1. 220   1. 726   1. 207   1. 697   1. 192 6 1. 669   1. 178   1. 640   1. 164   1. 615   1. 151   1. 589   1. 138   1. 563   1. 125   1. 538 9 1. 112   1. 516   1. 100   1. 493   1. 087   1. 472   1. 076   1. 450   1. 063 7 1. 430   1. 052   1. 409   1. 041   1. 391   1. 030   1. 373   1. 020   1. 353   1. 011   1. 337 10 1. 000 Decimal Solid under Measure .   F. 1000. p. 10 F. 1000. p. 1 10000. 000   14. 805 2 2500. 000   13. 735 3 1100. 000   12. 780 4 630. 000   11. 916 5 400. 000 3 11. 125 6 277. 900   10. 415 7 200. 430   9. 760 8 150. 660   9. 125 9 120. 350   8. 625 1 100. 0000   8. 150   82. 800   7. 700   96. 500   7. 310   59. 390   6. 900   51. 100   6. 565   44. 500 4 6. 250   39. 150   5. 945   34. 650   5. 664   30. 850   5. 404   27. 750   5. 465 2 25. 000   4. 938   22. 700   4. 720   20. 675   4. 530   18. 920   4. 342   17. 400   4. 162   16. 000 5 4. 000 Decimall Solid under measure .   F. 1000. p. 01. F. 100. p.   3. 825   1. 738   3. 7●0   1. 694   3. 524   1. 651   3. 430   1. 608   3. 310 8 1. 568   3. 188   1. 528   3. 078   1. 493   2. 968   1. 458   2. 873   1. 420 6 2. 780   1. 390   2. 688   1. 356   2. 602   1. 323   2. 521   1. 297   2. 442   1. 266   2. 366 9 1. 236   2. 294   1. 208   2. 227   1. 185   2. 160   1. 160   2. 100   1. 131   2. 043   1. 109 7         1. 985   1. 084   1. 93●   1 , 061   1. 878   1. 041   1. 830   1. 021   1. 781 10. 1. 000 Vnder Yard-measure for feet and inches from one inch'to four feet six inches F. F. 1000. F. F. 1000. In.   In.   1 108.000   3. 850 2 54. 000   3. 720 3 36. 000   3. 600 4 27. 000 6 3. 482 5 21. 600   3. 373 6 18. 000   3. 271 7 15. 420   3. 175 8 13. 520   3. 085 9 12. 000 3 3. 000 10 10. 300   2. 922 11 9. 820   2. 842 1. 9. 000   2. 769   8. 320   2. 710   7. 740   2. 633   7. 201 6 2. 572   6. 760   2. 512   6. 350   2. 455 6 6. 000   2. 400   5. 680   2. 345   5. 400   2. 298   5. 140 4 2. 250   4. 906 1 2. 203   4. 695 2 2. 160 2 5. 500 3 2. 119   4. 320 4 2. 073   4. 160 5 2. 037   4. 000 9 2. 000 Vnder yard measure according to Decimal or Foot measure F. 10. F. 1000. p.   F. 1000. p. 1 90. 000 4 3. 7●0 2 45. 000 5 3. 600 3 30. 000 6 3. 461 4 22. 500 7 3. 332 5 18. 000 1 3. 211 6 15. 000 9 3. 104 7 12. 880 3 3. 000 8 11. 200 1 2. 903 9 10. 000 2 2. 812 1 9. 000 3 2. 728 1 8. 190 4 2. 648 2 7. 510 5 2. 572 3 6. 930 6 2. 502 4 6. 430 7 2. 435 5 6. 000 8 2. 370 6 5. 625 9 2. 310 7 5. 290 4 2. 250 8 5. 000 1 2. 195 9 4. 735 2 2. 142 1 4. 500 3 2. 093 1 4. 285 4 2. 046 2 4. 092 5 2. 000 3 3912     Vnder Girt-measure . Inc. about F. in . 100.   F. in . 100 1 1809.6.81 24 3.1.87 2 452. 4. 74 25 2. 10. 74 3 201. 0. 77 26 2. 8. 12 4 113. 1. 18 27 2. 5. 87 5 72. 4. 60 28 2. 3. 70 6 50. 3. 19 29 2. 1. 83 7 39. 3. 22 30 2. 0. 13 8 28. 4. 00 31 1. 10. 60 9 22. 4. 09 32 1. 9. 21 10 18. 1. 15 33 1. 7 : 94 11 14. 11. 46 34 1. 6. 78 12 12. 6. 80 35 1. 5. 72 13 10. 8. 09 36 1,4,75 14 9. 2. 79 37 1. 3 , 86 15 8. 0. 51 38 1 , 3 , 04 16 7. 0. 82 39 1 , 2 , 28 17 6. 3. 14 40 1 , 1 , 57 18 5. 7. 30 41 1 , 0 , 92 19 5. 0. 15 42 1 , 0 , 31 20 4. 6. 28 43 0 , 11 , 74 21 4. 1. 24 44 0 , 11 , 22 22 3. 8. 87 45 0 , 10 , 72 23 3. 5. 04     Vnder measure for the Diameter in Inches and quarters . In. over F. in 10   F. in 100 , 29 3 0 , 0 , 00 , 4 , 8 , 31 , 73● , 1 , 00 , 4 , 4 , 06 , 326 , 0 , 10 , 4 , 0 , 27 1 183 , ● , 80 7 3 , 8 , 89 , 118 , 2 , 16 , 3 , 5 , 85 , 81 , ● , 00 , 3 , 3 , 10 , 59 , 10 , 24 , 3 , 0 , 61 2 45 , 10 , 90 8 2 , 10 , 36 , 36 , 2 , 41 , 2 , 8 , 31 , 29 , 3 , 93 , 2 , 6 , 44 , 24 , 3 , 86 , 2 , 4 , 73 3 20 , 4 , 40 9 2 , 3 , 15 , 17 , 4 , 25 , 1 , 1 , 70 , 15 , 0 , 00 , 1 , 0 , 37 , 13 , 1 , 86 , 1 , 11 , 13 4 11 , 5 , 14 10 1 , 9 , 99 , 10 , 1 , 78 , 1 , 8 , 93 , 9 , 0 , 62 , 1 , 7 , 95 , 8 , 1 , 80 , 1 , 7 , 04 5 7 , 3 , 98 11 1 , 6 , 18 , 6 , 7 , 80 , 1 , 5 , 38 , 6 , 0 , 72 , 1 , 4 , 63 , 5 , 6 , 53 , 1 , 3 , 93 6 5 , 1 , 10 12 1 , 3 , 27     13 1 , 1 , 02 A Table of the number of bricks in a rodd of Walling at any Feet high , from 1 to 20 for 1 and 1 1 / 2 Feet high . at 1 brick thick . at 1 brick & 1 / 2 thick 1 176 264 2 352 528 3 528 792 4 704 1056 5 880 1320 6 1136 1704 7 1232 1848 8 1408 2112 9 1584 2376 10 1760 2640 11 1936 2904 12 2112 3168 13 2288 3432 14 2464 3696 15 2640 3960 16 2816 4224 16 1 / 2 2904 4356 17 -2992 4488 18 -3168 4752 19 -3344 5010 20 -3520 5280 If you would have this Table for 1 / 2 a brick , take the half of the table for one brick . If for two bricks then double it . If for two and a 1 / 2 then ad both these together ; if for three , double that for one brick and 1 / 2. If you have any number of feet of brick work , at half a brick , one brick , or two bricks , or more , and you would reduce it to one brick and half , then say by the line of numbers as 1. 2. 4. 5 or 6 is to three , so is the number of feet at 1 / 2 1. 2. 2 1 / 2 or three bricks to the number of feet at one and 1 / 2. The use of four scales , called Circumfence , Diameter , Square equal , Square inscribed . Suppose you have a circle whose diameter is 10 inches , or 10 feet : and to this circle you would finde the Circumference , or the side of a square equal , or inscribed , or having any one of the three , to finde the other three , do thus : Take the measure of the Circumference , Diameter , or either of the squares , which is first given , and open the compasses to the number of the given measure , in its respective scale : the compasses so set , if you apply it in the scale whose number you would know , you shall have your desire . Example . Suppose a circle whose diameter is 10 inches , and to it I would know the Circumference , take 10 out of the diameter scale , and in the Circumference scale it shall reach to 31 42 , and on the line of square equal 8 86 , and on square inscribed 7. 07. For illustration sake , note the figure . The use of the line to divide a circle into any number of parts . Take the Semidiameter , or Radius of the circle between your compasses , and fit it over in six and six of the line of circles , then what number of parts you would have , take off from that point by the figure in the line of circles , and it shall divide the circle into so many parts . As suppose I would have the former circle divided into nine parts , take the measure from the center to the circle as exactly as you can , fit that over in 6 and 6 , then take out 9 and 9 , and that shall divide it into so many parts ; but if you would divide a wheel into any odde parts , as 55. 63. or 49 parts , you shall finde it an almost impossible thing , to take a part so exact that in turning about so many times , shall not miss at last : to help which the parts the rule giveth shall fit you exact enough for all the odde parts , then the even will easie be had by dividing , therefore usually the rule is divided but to 30 or 40 parts . So that for this use as the finding the side of an 8 or 10 square piece , as the mast of a ship , or a newel , or a post , this will very readily , and exactly help you . CHAP XVIII . The use of Mr. Whites rule , for the measuring of Timber and Board , either by inches or foot measure . 1. ANd first for superficial or board measure , by the inches , the breadth and length being given in inches , and feet and parts , slide or set 12 on any one side , to the breadth in inches or parts on the other side , then just against the length found on the first side , where 12 was on the second side you shall have the content in feet , and 10ths , or 100 parts required . Which by the rules of reduction by the foot measure , you may reduce to inches and 8 parts . Example . At three inches broad , and 20 foot long , you shall finde it to be 5 foot just : but at 7 inches broad , and the same length it will come to 11 foot 7 10th fere , or 8 inches . 2. The breadth being given in inches , to finde how many inches in length goes to make a foot of board or flat measure . Set 12 on the first side , to the breadth in inches on the second side , then look for 12 on the second side , and right against it on the first side , is the number of inches , that goes to make a foot Superficial , at that breadth . Example . At three inches broad you shall finde 48 inches to make a foot . 3. To work multiplication on the s●iding , or Whites rule . Set one on the first side , to the multiplicator on the second side , then seek the multiplicand on the first , and right against it on the second , you shall finde the product . Example . If 9 be multiplied by 16 , you shall finde it to be 144. 4. To work division on the same rule . Set the divisor on the first side to one on the second , then the dividend on the first , shall on the second shew the quotient required . Example . If 144 be to be divided by 16 , you shall finde the quotient to be 9. 3. To work the rule of 3 direct . Set the first term of the question , sought out on the first line , to the second term on the other , ( or second line : ) then the third term sought on the first line , right against it on the second , you shall finde the fourth proportional term required . Example . If 15 yard 1 / 2 , cost 37s . 6d . what cost 17 3 / 4 ? facit 42s . 10d . 3 q. for if you set 15 1 / 2 right against 37 1 / 2 , then look for 17 3 / 4 on the first line , ( where 15 1 / 2 was found , and right against it on the second line , is neer 42 the fractions are all decimal , and you must reduce them to proper fractions accordingly To work the rule of 3 reverse . 4. Set the first term sought out on the first line , to the second being of the same denomination or kind to the second line , or side . Then seek the third term on the second side , and on the first you shall have the answer required . Example . 5. If 300 masons build an edifice in 28 days , how many men must I have to perform the same in six days , the answer will be found to be 1400. 6. To work the double rule of 3 direct . This is done by two workings : As thus for Example . If 112 l. or 1 C. weight , cost 12 pence the carraiage for 20 miles , what shall 6 C , cost , 100 miles ? Say first by the third rule last mentioned , as 1 C. weight to 12 , so is 6 C. weight to 72. pence , secondly say if 6 C. cost 72 pence or rather 6s , for 20 miles ? what shall 100 miles require ? the answer is 30 s. for if you set 20 against 6 then right against 100 is 30 , the answer required . The use of Mr. Whites rule in measuring Timber round , or square , the square or girt being given in inches , and the length in feet and inches . 1. The inches that a piece of Timber is square , being given : to finde how much in length makes a foot of Timber , look the number of inches square on that side of the Timber line , which is numbred with single figures from 1 to 12 , and set it just against 100 on the other or second side , then right against 12 at the lower , ( or some times the upper ) end , on the first line , in the second you have the number of feet and inches required . Example . At 4 1 / 2 inches square , you must have 7 foot 1 inch 1 / 3 to make a foot of Timber . But if it be above 12 inches square , then use the sixth Problem of the 5th chapter of the Carpenters Rule , with the double figured side and Compasses . 2. But if it be a round smooth stick , of above 12 inches about , and to it you would know how much in length makes a true foot , then do thus . Set the one at the beginning of the double figured side , next your left hand , to the feet and inches about , counted in the other side , numbred with single figures from 1 to 12 , then right against three foot six 1 / 2 inches , in the single figures side next the right hand , you have in the first side the number of feet , and inches required . Example . A piece of 12 inches about , requires 11 f 7 in : fere to make a foot . Again a piece of 15 inches about , must have 8 foot 1 / 2 an inch in length , to make a foot of timber . 3. But if you would have it to be equal to the square , made by the 4th part of a line girt about the piece , then instead of three foot 6 1 / 2 inches make use of four foot , and you shall have your desire . 4. The side of a square being given in inches , and the length in feet , to find the content of a piece of timber . If it be under 12 inches square then work thus : set 12 at the beginning or end of the right hand side , to the length counted on the other side , then right against the inches square on the right side is the content on the left side Example . At 30 foot long , & 9 inches square , you shall find 16 foot 11 inches for the working this question , 12 at the end must be used . But if it be above 12 inches square , then ser one at the beginning , or 10 at the end of the right hand side , to the length counted on the other side , then the number of inches or rather feet and inches , counted on the first side , shall shew on the second the feet and parts required . Example . At 1 f. 6 inch . square , and 30 foot long , you shall finde 67 feet and about a 1 / 2. 5. To measure a round piece by having the length , and the number of inches about , being a smooth piece , and to measure true , and just measure , then proceed thus : Set 3 f. 6 1 / 2 inches on the right side , to the length on the other side , then the feet and inches about , on the first side , shall shew on the second or left , the content required . As at 20 inches about , and 20 foot long , the content will be found to be about 4 foot 5 inches . But if you give the usual allowance , that is made by dupling the string 4 times , that girts the piece : then you must set 4 foot on the right side , to the length on the other , then at 1 foot 8 inches about , the last example you shall finde but three foot 6 inches . 6. ● astly , if the rule be made fit for foot measure , onely then the point of 12 is altogether neglected , and one onely made use of as a standing number : and the point at three foot 6 1 / 2 will be at three foot 54 parts ; and the four will be the same , and the same directions in every respect , serve the turn . And because I call it Mr. Whites rule , being the contriver thereof , according to feet and inches , I have therefore fitted these directions accordingly , and there are sufficient to the ingenious practitioner . CHAP. XIX . Certain Propositions to finde the hour , and the Azimuth , by the lines on the Sector . PROP. 1. HAving the latitude , and complements of the declination , and Suns altitude , and the hour from noon , to finde the Suns Azimuth , 〈◊〉 that time . Take the right sine of the complement of the Suns altitude , and mak 〈◊〉 it a parallel sine in the sine of th 〈◊〉 hour from noon , ( counting 15 degree 〈◊〉 for an hour , and 1 degree for for minutes ) counted from the center . The Sector so set , take the right sine of the complement of the declination , and carry it parallel till the compasses stay in like sines , and the sine wherein they stay shall be the sine of the Azimuth required . Or else thus : Take the right sine of the declination , make it a parallel in the cosine of the Suns altitude , then take the parallel sine of the hour from noon , and it shall be the latteral or right sine of the Azimuth from the south required . If it be between six in the morning , and 6 at night ; or from the north , if it be before or after six : and so likewise is the Azimuth . PROP. 2. Having the Azimuth from south or north , the complement of the Suns altitude , and declination , to finde the hour . Take the latteral , or right sine of the complement of the Suns altitude , make it a ga●●llel in the cosine of the declination : the sector so sett , ake out the parallel sine of the Azimuth , and measure it from the center , and it shall reach to the right sine of the hour from noon required . Or else as before . As the right sine of the complement of the Suns declination : is to the parallel sine of the Azimuth , so is the right sine of coaltitude , to the parallel sine of the hour from noon , counting as before . PROP. 3. Having the complements of the latitude , Suns altitude , and declination . to finde the Suns Azimuth from the north part of the horizon . 1. First of the complement of the latitude , and Suns present altitude finde the difference . 2. And secondly count it on the line of sines from 90 toward the center . 3. Take the distance from thence , to the sine of the Suns declination ; but note when the latitude and declination differ , as in winter you must counte the declination beyond the center , and you must call it the Suns distance from the pole . 4. Fourthly , make that distance a parallel sine in the complement of the latitude . 5. Fifthly , then take out the parallel sine of 90. 6. And sixthly , make it a parallel sine in the coaltitude . 7. Seventhly , then the sector so set , take out parallel , Radius , or sine of 90. And eighthly , measure it on the line of sines from 90 towards , ( and if need be beyond ) the center : and it shall reach to the versed sine of the Suns Azimuth from the north , or if you count the other way from the south , note that in working of these , if the line of sines be too big , then you have two or three smaller sines on the rule , where on to begin and end the work . Example . Latitude 51. 32 , Declination 18 ▪ 30 , Altitude 48. 12 , you shall finde the Azimuth to be 130 from the North , or 50 from the South . PROP. 4. Having the complements of the latitude and declination , or Suns distance from the Pole , and the Sun altitude given , to finde the hour from East or West , or else from noon . 1. First of the complement of the latitude , and Suns distance from the Pole , finde the difference . 2. Count this from the sine of 9● toward the center . 3. Take the distance from thence to the sine of the Suns altitude . 4. Make that distance a parallel sine of the complement of the latitude . 5. Take out the parallel sine of 90 degrees , and 6. Make that a parallel sine in the codeclination , then 7. Take out the parallel sine o● 90 again , and 8. Measure it from the sine of 90 toward the center , and it shall shew the versed sine of the hour from the North , or the sine of the hour from East or West ; or if you reckon from 90 , the hour from noon required . Example . Latitude 51. 32 , Declination North 20. 14 , Altitude 50. 55 , you shall finde the hour from the North to be 10 houres , or 10 a clock in the forenoon , or 4 hours past 6 , or two short of noon , according to each proper reckoning . PROP. 5. Having the latitude , the complement of the Suns declination , the Suns present altitude , and Meridian altitude for that day , to finde the hour . Make the lateral secant of the latitude , a parallel sine in the codeclination , then take the distance from the Suns Meridian altitude , to his present altitude , and lay it from the cer on both sides of the line of sines , and take the parallel distance between those two points , and measure it from 90 on a line of sines , of the same Radius the secants be , ( as the small adjoyning sine is , ) and it shall shew the versed sine of the hour from noon , or the right sine before or after 6 , towards noon or night . PROP. 6. Having the Latitude , Declination , and Suns Altitude , to finde the Suns Azimuth . Take the latteral secant of the latitude , and make it a parallel in the complement of the Altitude : Then take the distance , between the sun● of the complement● of of Museum the latitude and altitude , ( if under 90 , ) and the sons declination , and lay it from the center on the line of sines , that parallel distance taken , and measured on the sines ( of the same Radius the secant was ) from 90 , shall shew the versed sine of the Azimuth from noon . But if the sum of the colatitude , and coaltitude exceed 90 , then take the excess above 90 , out of the natural sine from the center toward 90 , and add that to the sine of the Suns declination towards 90 , and then the parallel distance between those two points , shall be the Azimuth required , from noon , But when the latitude and declination are unlike , as with us ( in the northern parts ) in winter , then you must take the declination out of the excess , or the lesser out of the greater , and lay the rest from the center , and the parallel distance , shall be versed sine of the Azimuth from noon . Example . At 18 15 altitude , latitude 51 32 , declination 13 15 South . The sum of the colatitude , and the coaltitude is 109. 37 , then count the center for 90 , the right sine of 10 for a 100 and 19 for 109 , and 37 minutes forwarder , there set the point of the Compasses , then take from thence to the right sine of the declination , and lay this distance from the center on the line of sines , and the parallel space between , is the versed sine of the Azimuth required . PROP. 7. Having the length of the shaddow of any object standing perpendicular , and the length of that object , to finde the altitude . Take the Tangent of 45 , and make it a parallel in the length of the shaddow in the line of lines , then the parallel distance between the length of the object that casts the shaddow , taken from the line of lines , and measured on the line of tangents from the center , shall reach to the Suns altitude required . Example . If the object be 40 parts long , and the shaddow 80 parts , the altitude will be found to be 26. 35. But if you have the altitude , and shaddow , and would know the height of the object , then work thus : Take the length of the shaddow out of the line of lines , or any other equal parts , and make it a parallel tangent of 45 , then take out the parallel tangent of the Suns altitude , and measure it on the line of lines , ( or the same equal parts ) and it shall shew the length of the object that caused the shaddow : the same rule doth serve in taking of altitudes by the rule , as in the 18 chapter , accounting the measure from the station to the object , the length of the shaddow , and the Suns altitude , the angle at the base . PROP. 8. To finde the Suns rising and setting in any latitude . Take the latteral cotangent of the latitude , make it a parallel in the sines of 90 and 90 , then take the latteral tangent of the Suns declination , and carry it parallel in the sines till it stay in like sines , that sine shall be the asentional difference between six , and the time of rising before or after 6 , counting 15 degrees to an hour . PROP. 9. To finde the Amplitude in any latitude . Take the latteral sine of 90 , and make it a parallel in the cosine of the latitude , then the parallel sine of the declination , taken and measured in the line of sines from the center , shall give the amplitude required . PROP. 10. To finde the Suns height at six in any latitude . Take the lateral or right sine of the declination , and make it a parallel in the sine of 90 , then take out the parallel sine of the latitude , and measure it in the line of sines from the center , and it shall reach to the altitude required . Note in working of any of these Propositions , if the sines drawn from the center , prove too large for your Compasses , or to make a parallel sine or Tangent to a small number of degrees , then you may use the smaller sine or tangent adjoyning , that is set on the Rule , and it will answer your desire . And note also in these Propositions , the word right , or latteral sine or tangent , is to be taken right on from the center or beginning of the lines of sines , or tangents ; and the word parallel always across from one leg to the other . PROP. 11. To finde the Suns height at any time , in any latitude . As the right Sine of 90 , Is to the parallel cotangent of the latitude : So is the latteral or right Sine of the hour from 6 , To the parallel tangent of a fourth ark ; which you must substract from the suns distance from the Pole , and note the difference . Then , As the right of the latitude , To the parallel cosine of the fourth ark : So is the parallel cosine of the remainder , To the latteral sine of the Altitude required . PROP. 12. To finde when the Sun shall come to due East , or West . Take the tangent of the latitude from the smaller tangents , make it a parallel in the Sine of 90 , then take the latteral tangent of the declination from the smaller tangents , and carry it parallel in the Sines , till it stay in like Sines , and that Sine shall be the Sine of the hour required from 6. PROP. 13. To finde the Suns Altitude at East or West ( or Vertical Circle . ) As the latteral sine of declination , Is to the parallel sine of the latitude : So is the parallel sine of 90 , To the latteral sine of the Altitude required . PROP. 14. To finde the Stiles height in upright declining Dials . As the right Sine of the complement of the latitude , To the parallel sine of 90 : So the parallel sine complement of the Plains declination , To the right sine of the Stiles elevation PROP. 15. To finde the Substiles distance from the Meridian . As the lateral tangent of the colatitude , To the parallel sine of 90 : So the parallel sine of the declination , To the latteral tangent of the Substile from the Meridian . PROP. 16. To finde the Inclination of Meridians . As the latteral tangent of the declination , To the parallel sine of 90 : So is the parallel sine of the latitude , To the latteral cotangent in the inclination of Meridians . PROP. 17. To finde the hours distance from the Substile in all Plains . As the latteral tangent of the hour from the proper Meridian , To the parallel sine of 90 : So is the parallel sine of the Stiles elevation , To the latteral tangent of the hour from the substile . PROP. 18. To finde the Angle of 6 from 12 , in erect Decliners . As the latteral tangent of the complement of the latitude , To the parallel sine of the declination of the Plain : So is the parallel sine of 90 : To the latteral tangent of the Angle between 12 and 6. Thus you see the natural Sines and Tangents on the Sector , may be used to operate any of the Canons that is performed by Logarithms , or the artificial Sines and Tangents , by changing the terms from the first to the third , and the second to the first , and the third to the second , and the fourth must always be the fourth , in both workings being the term required . CHAP. XX. A brief description , and a short-touch of the use of the Serpentine-line , or Numbers , Sines , Tangents , and versed sine contrived in five ( or rather 15 ) turn . 1. FIrst next the center is two circles divided one into 60 , the other into 100 parts , for the reducing of minutes to 100 parts , and the contrary . 2. You have in seven turnes two in pricks , and five in divisions , the first Radius of the sines ( or Tangents being neer the matter , alike to the first three degrees , ) ending at five degrees and 44 minutes . 3. Thirdly , you have in 5 turns the lines of numbers , sines , Tangents , in three margents in divisions , and the line of versed sines in pricks , under the line of Tangents , according to Mr. Gunters cross staff : the sines and Tangents beginning at 5 degrees , and 44 minutes where the other ended , and proceeding to 90 in the sines , and 45 in the Tangents . And the line of numbers beginning at 10 , and proceeding to 100 , being one entire Radius , and graduated into as many divisions as the largeness of the instrument will admit , being from 10 to 50 into 50 parts , and from 50 to 100 into 20 parts in one unit of increase , but the Tangents are divided into single minutes from the beginning to the end , both in the first , second , and third Radiusses , and the sines into minutes ; also from 30 minutes to 40 degrees , and from 40 to 60 , into every two minutes , and from 60 to 80 in every 5th minute , and from 80 to 85 every 1oth , and the rest as many as can be well discovered . The versed sines are set after the manner of Mr. Gunters Cross-staff , and divided into every 10th minutes beginning at 0 , and proceeding to 156 going backwards under the line of Tangents . 4. Fourthly , beyond the Tangent of 45 in one single line , for one turn is the secants to 51 degrees , being nothing else but the sines reitterated beyond 90. 5. Fifthly , you have the line of Tangents beyond 45 , in 5 turnes to 85 degrees , whereby all trouble of backward working is avoided . 6. Sixthly , you have in one circle the 180 degrees of a Semicircle , and also a line of natural sines , for finding of differences in sines , for finding hour and Azimuth . 7. Seventhly , next the verge or outermost edge is a line of equal parts to get the Logarithm of any number , or the Logarithm sine and Tangent of any ark or angle to four figures besides the carracteristick . 8. Eightly and lastly , in the space place between the ending of the middle five turnes , and one half of the circle are three prickt lines fitted for reduction . The uppermost being for shillings , pence , and farthings . The next for pounds , and ounces , and quarters of small Averdupoies weight . The last for pounds , shillings , and pence , and to be used thus : If you would reduce 16 s. 3 d. 2 q. to a decimal fraction , lay the hair or edge of one of the legs of the index on 16. 3 1 / 2 in the line of l. s. d. and the hair shall cut on the equal parts 81 16 ; and the contrary , if you have a decimal fraction , and would reduce it to a proper fraction , the like may you do for shillings , and pence , and pounds , and ounces . The uses of the lines follow . As to the use of these lines , I shall in this place say but little , and that for two reasons . First , because this instrument is so contrived , that the use is sooner learned then any other , I speak as to the manner , and way of using it , because by means of first , second , and third radiusses , in sines and Tangents , the work is always right on , one way or other , according to the Canon whatsoever it be , in any book that treats of the Logarithms , as Gunter , Wells , Oughtred , Norwood , or others , as in Oughtred from page 62 to 107. Secondly , and more especially , because the more accurate , and large handling thèreof is more then promised , if not already performed by more abler pens , and a large manuscript thereof by my Sires meanes , provided many years ago , though to this day not extant in print ; so for his sake I claiming my intrest therein , make bold to present you with these few lines , in order to the use of them : And first no●e . 1. Which soever of the two legs is set to the first term in the question , that I call the first leg always , and the other being set to the second term , I call the second leg . 2. Secondly , if one be the first or second term , then for the better setting the index exactly , you may set it to 100 , for the error is like to be the least neerest the circumference . 3. Thirdly , be sure you keep a true account of the number of turnes between the first and second term . 4. Fourthly , observe which way you move , from the first to the second term . To keep the like from the third to the fourth , exept in the back rule of three , and in such cases as the Canon requires the contrary . 5. Fifthly in multiplication , one is always the first term , and the multiplycator or multiplycand the second , and the product always the fourth . Also note that in multiplycation the product of two numbers multiplyed , shall be in as many places as both the multiplycator , and multiplycand , except the least of them , be less then the two first figures of the product ; moreover , for your more certain assinging of the two last figures of four or six , which is as many as you can see on this instrument , multiply the two last in your minde , and the product shall be the figure , as in page 28 of the Carpenters Rule . 6. In division , the multiplicator is always the first term , and one the second , the dividend the third , and the quotient the fourth ; also the quotient shall have as many figures as the dividend hath more then the divisor , except the first figures of the divisor be greater then the dividends , then it shall have one less . Also note , the fraction after division , is a decimal fraction , and to be reduced as before . 7. Note carefully whether the fourth proportional ought to be a greater or a less , and resolve accordingly , and note if one cometh between the third and fourth term , then must the fourth be raised a Radius or a figure more , and be careful to set the hairs exactly over the part representing the number or minutes of any degree . 8. Always in direct proportion , and Astronomical calculation , set the first leg to the first term , and the second leg to the second term , and note how many circles is between , then set the first leg to the third term , and right under the second leg , the same way , and so many turnes between the third and fourth , is always the fourth term required . Example . As 1 , to 47 , so is 240 , to 11280. As the sine of 90 , to 51 degrees 30 minutes , so is the sine of 80 to 50. 26. And so of all other questions according to their respective Canons by the Logarithms in other books as in Mr. Oughtreds Circles of Porportions , from page 62 to 107 , and others . Here followeth the working of certain Propositions by the Serpentine-line . Those that I shall insert , are onely to shew the manner of working , and knowing of that once , all the Canons for all manner of questions , either in Arithmetick , Geometry , Navigation , or Astronomy , by any other Author , as Mr. Gunter , Mr. Oughtred , Mr. Windgate , Mr. Norwood , or others , may be speedily resolved , and as exactly as by the Tables , if the instrument be well and truly made . And first ●or the hour , according to Gunter . PROP. 1. Having the Latitude , Declination , and the Suns Altitude , to finde the hour . Add the complement of the altitude , the complement of the Suns present altitude , and the distance of the Sun , from the elevated Pole together , and nore their sum , and half sum , and find the difference between their half sum and the complement of the Suns present altitude , then work thus : for 36 42 degrees high , at 23. 32 declination , lat . 51. 32. Lay the first leg , viz. that next your right hand being here most convenient , on the Sine of 90 , keeping that fixed there , lay the other leg to the cosine of the latitude , viz. 38. 28. and note the turns between , which here is none between , but it is found in the next over it , then set the first leg to the Sine of 66 , 28 , the suns distance from the Pole , and in the circle just over it you shall have the Sine of 34. 47. for the fourth Ark or Sine . Then in the second Operation . Set the first or left leg to the sine of 34. 47. of the fourth Sine last found , then keeping that fixed there , set the other leg to the Sine of the half sum , viz. 79. 37. then remove the first leg to the Sine of the difference between the half sum , and the coaltitude , viz. 25. 49. and then in the next circle , the other leg shall shew the sine of 48. 34. whose half distance toward 90 , being found by the Scale of Logarithms on the outermost circle , will discover the Sine of an Ark , whose complement being doubled , and turned into time ( by counting 15 degrees to an hour ) will give the hour required ; but by help of the versed Sines all this trouble is saved ; for when the index or second leg cuts the Sine of 48. 34. at the same instant it cuts the versed Sine of 60 , the hour from noon required , being 8 in the morning , or 4 in the afternoon , at 23. 32. of declination , in the latitude of 51. 32. PROP. 2. To work the same another way , according to Mr. Collins . The Latitude and Declination given , to finde the Suns height at 6 a clock , Dec. 23. 31. Lat. 51. 32. Lay the left leg on the Sine of 90 , and the other to the Sine of 23. 32 , and you shall finde one turn between upwards , then the first leg laid on the Sine of the latitude 51 , 32 , the other leg shall shew the Sine of 18. 3. ( in the second circle above 51 , 32. ) for the Suns height at 6 required , and this is fixed for one day . Then in summer time , or north declinations , by help of the line of natural sines , in the second line , finde the difference between the Suns present altitude , and the latitude at 6 , but in Winter or Southern ( signs or ) declinations , add the two altitudes together , in this manner . Lay one leg of the index to the natural sine of the altitude at 6 , and the other to the altitude proposed , the two legs so set , bring one of them ( viz the right ) to the beginning of the line of natural sines , and the other shall stay at the difference required , but in Winter set one leg to the beginning of the sines , and open the other to the height at 6 , or rather depression under the Horizon at 6 , ( which is all one at like declinations , North and South ) then set the first leg to the present altitude of the Sun , and the other shall shew the Sine of the sum of both added together ; which sum or difference is thus to be used : Lay the left leg to the Cosine of the declination , and the other to the secant of the latitude , counted beyond 90 , as far as the secant of 9 , 40 ; or rather lay the left leg on the Co-sine of the latitude , and the other to the secant of the suns declination , then the first leg laid on the sine of the sum in winter , or difference in summer , shall cause the other leg to fall on the sine of the hour from 6 , toward noon in winter and summer , except the altitude in summer be less then the altitude at 6 , then it is the hour from 6 , toward mid-night . PROP. 3. Having the latitude , Suns altitude and declination , to finde the Suns Azimuth from east or west . Lat. 51. 32. Declin . 23. 30. Alt. 49 56. First you must get the Suns altitude , or depression in the vertical Circle by this Cannon . Lay the first leg to the sine of the latitude , and the second to the sine of 90 , and you shall finde them both to be on the same line , then the first leg laid on the sine of the declination , shall cause the second ( being carried with the first , without moving the Angle first set ) to fall on the sine of 30. 39. the Suns altitude in the Vertical Circle ; with which you must do , as you did before with the altitude at 6 , and the present altitude , to finde the sum and difference by help of the line of natural Tangents , then this proportion holds . Lay the first leg to the cosine of the Altitude , ( by counting the Altitude from 20 ) and the second leg to the tangent of the latitude , and observe which way , and the turns between ; then the first leg removed and laid to the sine of the sum ( before found ) in winter , or the difference in summer , shall cause the second leg to fall on the sine of the Azimuth of the Sun , from east or west toward noon , if winter ; and also in summer , when the Suns altitude is more than his altitude at the Vertical Circle ; but if less from the east or west , toward north or mid-night meridian : Thus in our Example , it will be found to be the sine of 30 degrees , or 60 from the south , the sine of the difference being found to be 14. 49. PROP. 4. To finde the Azimuth , according to Mr. Gunter , by having the Latitude , Suns Altitude , and Declination given . First by the Suns declination get his distance from the Pole , which in summer or North declinations , is always the complement of the declination , ( likewise in south latitudes , and south declinations ) but when the latitude and declination is unlike , then you must adde 90 to the declination , and the sum is the distance from the elevated Pole. Having found the distance from the Pole , adde that and the complement of the latitude , and Suns altitude together , finde the difference between their half sum , and the Suns distance from the Pole , then the proportion will be thus , as in this Example : 13 Declination , 41. 53. Alt. Lat. 51. 32. Lay one leg on the sine of 90 , and the other to the sine of 38 , 28 , being so set , remove the first leg to the sine complement of the altitude 48. 07. and the second legg shall fall on a fourth sine , which will be found to be 27. 36. then set the first or left leg to 27. 36. the fourth sine , and the second to 81. 47. the sine of the half sum , then removing the first leg to the sine of the difference , shall cause the second to shew two circles lower , the versed sine of 130 , the Azimuth required , being counted from the North part of the horizon , whose complement to 180 from the South is 50 degrees . Two other Canons to finde the hour of the day , and Azimuth of the Sun , by one operation , by help of the natural sines : and first for the hour . Having the latitude , the Suns Meridian , and present altitude , and declination , to finde the hour from noon . First lay one leg to the Meridian altitude , in the line of natural sines , and the other to the sine of the altitude in the same line , then bring the right leg to the beginning of the line of sines , and the other 〈…〉 difference , which difference 〈…〉 keep . Then lay one leg on the 〈◊〉 the declination , and the other 〈◊〉 secant of the latitude , and note the turnes between , or rather lay the first leg to the cosine of the latitude , and the other to the secant of the declination , then the legs being so set , bring the first or left leg to the sine of the difference first found , and the other leg shall shew the versed sine of the hour from noon , if the versed sines had been set thus , i. e. the versed sine of 90 , against the sine of 90 , as in some instruments it is : But to remedy this defect , do thus : keep the right leg there , and open the other to the versed sine of 0. or sine of 90 , and note the turnes between , then lay the leg that was on the sine of 90 , ( or versed sine of 0 , ) to the versed sine of 90 , and the other leg shall shew the versed sine of the hour from noon , counting from 90. Example . At 45 13 degrees altitude , declination North 23 32. latitude 51 32 , the Meridian altitude is 62 , ( being found by adding colatitude , and declination together , and in southern declinations by substraction . ) Then the natural sine of 45. 42 , taken from 62 , shall be the sine of 9. 38. then as the cosine of the latitude 38. 28 , is to the secant of declination 23. 32 , so is the sine of 9. 38. to the sine of 17. 05 ) or versed sine of 45 ▪ if they were placed and numbred , as in some instruments they be : but to help i● in this , say : as the versed sine of 0 , is to the verses of 114. 26 , so is the versed sine of 90 , to the versed sine of 135 , whose complement to 180 is the angle or hour from noon required . Secondly , for the Suns Azimuth . Having the latitude , declination , and Suns altitude , to finde the Azimuth from South or North. First add the complements of the coaltitude and cola●itude together , then if the sum be under 90 , take the distance between the cosine of it , and the sine of the declination , in the line of natural sines , and measure it in the line of sines from the beginning , and it shall give the sine of the difference ; but if the sum exceed 90 , then when the latitude and declination is alike , add the excess to the declination ; but if contrary substract one out of other , and measuring the sum or remainder from the beginning of the sines , you have the difference which you must keep . Then lay the first or left leg to the cosine of the altitude , and the second to the secant of the latitude , or else lay the first to the cosine of the latitude , and the other to the secant of the altitude , and more the turnes between , then lay the first leg to the sine of the difference before found , and the other shall shew the versed sine of the Azimuth from noon required , if the versed sines be set as before is expressed , that is to say 90 of right sines , and versed sines together , and numbred forwards as the sines be : but in the use of this instrument , the remedy aforesaid supplyeth the defect . Example . At 10. 19. altitude , 23. 32. declination , 51. 32. latitude , to finde the Azimuth . The sum of the coaltitude and co-latitude is 118.09 , the excess above 90 , with right sine of declination added is 60. 30 , found by natural sines , then say , As the cosine of latitude , to the secant of altitude , so is the sine of the difference 60. 30 , to the versed sine of the Azimuth , but here to the versed sine of an ark beyond Radius unknown , then as the versed sine of 0 , to that ark , so is the versed sine of 90 , to the ver sine of 60 , the azimuth from noon , whose complement to 180 , is 125 the Azimuth from Sonth required . Having been so large in these , I shall in the rest contract my self as to the repetition , and onely give the Canon for the propositions following , the way of working being the same in all other , as in these before rehearsed , and note also what is to be done by the Serpentine-line , is to be done by the three same lines of numbers , sines and Tangents on the edge of the Sector , by altering the term leg , to to the point of the Compasses . The Canons follow . PROP 4. Having latitude , declination , and hour given , to finde the Suns altitude at that hour or quarter . And first for the hour of 6. As the sine of 90 , To the sine of the latitude 51. 30 : So is the sine of the declination 23. 30 , To the sine of the altitude at 6. 18. 13. Secondly , for all hours in the Equinoctial . As the Radius or sine of 90 , to the cosine of the latitude 51. 32 : so is the sine of the Suns distance from 6 ( in hours and minutes , being turned into degrees and minutes 30 , for 8. or 4 , ) To the sine of the altitude of the Sun at the time required 18. 07 , But for all other times say , As the sine of 90 , To the cotangent of the latitude 38. 28. So is the sine of the Suns distance from 6 , 30 , 0 , To the tangent of the 4 arke 21 40. Which fourth arke must be taken out of the Suns distance from the Pole 66 , 21 , ( in Cancer ) leaveth a residue 44 48 , which is called a fift ark . But for the hours before and after 6 , you must add the fourth arke to the Suns distance from the Pole , and the Sum is the fifth ark . Then say , As the cosine of the fourth ark 78 , 20 , Is to the sine of the latitude 51 , 32 : So is the cosine of the residue 45 , 12 , To the sine of the Suns altitude at 8 , 36 , 42 , at that declination . PROP. 6. Having the latitude , declination , and Azimuth , to finde the Suns altitude at that Azimuth . And first to finde the Suns altitude at any Azimuth in the aequator . Then , As the sine of 90 , to the cosine of the Suns Azimuth from the South 50 , 0 , So is the cotangent of the latitude 38 , 28 , to the tangent of 27 , 03 ▪ the Suns altitude , at that Azimuth required . Secondly , to find it at all other times , do thus : As the sine of the latitude 51 , 32 , To the sine of the Suns declination , 23 , 32 : So is the cosine of the Suns altitude in the aequator , at the same Azimuth from the vertical , viz. 30 , to the sine of a 4th . ark 28 , 16. Which fourth ark must be added to the Suns altitude at the aequator in all Azimuths under 90 , from the meridian , where the latitude and declination are alike . But in Azimuths more then 90 from the meridian , take the altitude in the aequator out of the fourth ark , and the sum or remainder shall be the altitude required , viz. 42 , 56. But when the latitude and declination are unlike , as with us in winter time , then take the fourth ark out of the altitude at the aequator , and you shall have the altitude belonging to that Azimuth required . PROP. 7. Having the hour from noon , and the altitude to find the Suns Azimuth at that time . As the cosine of the altitude , To the sine of the hour , So is the cosine of the Suns declination , To the sine of the Azimuth required . PROP. 8. Having the Suns Azimuth , Altitude , and declination , to find tho hour of the day . As the cosine of the declination , To the sine of the Suns Azimuth : So is the cosine of the altitude , To the sine of the hour . PROP. 9 , Having the latitude and declination , to find when the Sun shall be due East or West . As the tangent of the latitude , To the tangent of the Suns declination , So is the sine of 90 , To the cosine of the hour from noon . PROP. 10. Having the latitude and Suns declination , to find the Amplitude . As the cosine of the latitude , To the sine of the declination : So is the sine of 90 , To the sine of the amplitude from the East or West , toward North or South , according to the time of the day and year . PROP. 11. The latitude and declination given , to find the time of the Suns rising before or after 6. As the cotangent of the latitude , To the sine of 90 : So is the tangent of the Suns declination , to the sine of the Suns ascentional difference between the hour of 6 and the Suns rising . PROP. 12. Having the Suns place , to find his declination , and the contrary . As the sine of 90 , To the suns distance from the next equinoctial point , So is the sine of the suns greatest declination , To the sine of his present declination required . PROP. 13. The greatest and present declination given , to find the Suns right ascension . As the tangent of the greatest declination , To the sine of 90 : So the tangent of the present declination , To the right ascension required . Onely you must regard to give it a right account by considering the time of the year , and how many 90s . past . PROR . 14. To find an altitude by the length , and shadow of any perpendicular object . Lay the hair on one legg to the length of the shadow found on the line of numbers , and the hair of the other leg to the length of the object that caused the shadow found on the same line of the numbers ; then observe the lines between , and which way when the legs are so set , bring the first of them to the tangent of 45 , and the other leg shall ●hew on the line of tangents , so many turns between , and the same way the tangent of the altitude required . Thus may you apply all manner of quest . to the Serpentine-line & work them by the same Canons , that you use for the Logarithms in all or most Authors . PROP. 15. To square , and cube a number , and to findethe square root , or cube roat of a number . The squaring of a number , is nothing else but the multiplying of the number by it self , as to square 12 is to multiply 12 by 12 , and then the cubing of 12 , is to multiply the square 144 by 12 , & that makes 1728 , and the way to work it , is thus : Set the first leg to 1 , and the other to 12 , then set the first to 12 , and then the second shall reach to 144 , then set the first to 144 , and the second shall reach to 1728 , the cube of 12 required : but note , the number of figures in a cube , that hath but one figure is certainly found by the line , by the rule aforegoing : but if there be more figures then one , so many times 3 must be added to the cube , and so many times two to the square . To find the square root of a number , do thus : Put a prick under the first , the third , the 5th , the 7th , & the number of pricks doth shew the number of figures in the root ; and note if the figures be even , count the 100 to be the unit , if odde as 3 , 5 , 7 , 9 , &c. the 10 at the beginning must be th● unit , as for 144 , the root consists of two figures , because there is two pricks under the number , and if you lay the index to 144 in the numbers , it cu●s on the line of Logarithms 15870 , the half of which is 7915 whereunto if you lay the index , it shall shew the 12 the root required ; but if you would have the root of 14+44 , then divide the space between that number , and 100 you shall finde it come to 8 , 4140 that is four turnes , and 4140 for which four turnes , you must count 80000 , the half of which 8,4140 , is 4,2070 , whereunto if you lay the index , and count from 1444 ●r 100 , at the end you shall have it cut at 38 lack four of a 100. To extract the cubique root of a number , set the number down , and put a point under the 1 , the 4th , the 7th , and 10th , and look how many pricks , so many figures must be in the root , but to finde the unity you must consider , if the prick falls on the last figure , then the 10 is the unit at the beginning of the line , as it doth in 1728 , for the index laid on 1728 , in the Log●rithms , sheweth 2,3760 , whose third part 0,7920 counted from 10 , falls on 12 the root , but in 17280 , then you must conceive five whole turnes , or 1000 to be added , to give the number that is to be divided by three , which number on the outermost circle in this place , is 12 , +3750. by conceiving 10000 to be added , whose third part counted from 10 , viz. two turnes or 4.125 , shall fall in the numbers to be near 26. But if the prick falls of the last but 2 , as in 172800 then 100 at the end of the line , must be the unit , and you must count thus : count all the turnes from 172830 to the end of the line , and you shall finde them to amount to 7,6250 , whose third part 2 , 5413 counted backward from 100 , will fall on 55,70 the cubique root required . PROP. 16. To work questions of interest or progression , you must use the help of equal parts , as in the extraction of roots , as in this question , if 100 l. yield 106 in one year , what shall 253 yield in 7 year ? Set the first leg to 10 at the beginning , in this case representing a 100 , and the other to 106 , and you shall finde the legs to open to 253 of the small divisions , on the Logarithms , multiply 253 by 7 , it comes to 1771 , now if you lay the hair upon 253 , and from the place where the index cuts the Logarithms count onwards 1771 , it shall stay on 380 l. 8 s. or rather thus : set one leg to the beginning of the Logarithms , and the other to 1771 either forward or backward , and then set the same first leg to the sum 253 , and the second shall fall on 380. 8 s. according to estimation ; the contrary work is to finde what a sum of money due at a time to ●ome , is worth in ready money : this being premised here , is enough for the ingenious to apply it to any question of this nature , by the rules in other Authors . However you may shortly expect a more ample treatise , in the mean time take this for a taste and farewell . The Use of the Almanack . Having the year , to finde the day of the week the first of March is on in that year , and Dominical letter also . First if it be a Leap-year , then look for it in the row of Leap-year , and in the column of week-days , right over it is the day required , and in the row of dominical letters is the Sunday letters also : but note the Dominical letter changeth the first of Ianuary , but the week day the first of March , so also doth the Epact . Example . In the year 1660 , right over 60 which stands for 1660 , there is G for the Dominical or Sunday letter , beginning at Ianuary , and T for thursday the day of the week the first of March is on , and 28 underneath for the epact that year , but in the year 1661. being the next after 1660 the Leap-year , count onwards toward your right hand , and when you come to the last column , begin again at the right hand , and so count forwards till you come to the next Leap-year , according to this account for 61 , T is the dominical letter , and Friday is the first of March. But to finde the Epact , count how many years it is since the last Leap-year , which can be but three , for every 4th is a Leap-year , and adde so many times 11 to the epact in the Leap-year last past , and the sum , if under 30 , is the Epact ; if above 30 , then the remainder 30 or 60 , being substracted is the Epact for that year . Example for 1661.28 the epact for 1660 , and 11 being added makes 39 from which take 30 , and there remaineth 9 , for the Epact for the year 1661 the thing required . Note that in orderly counting the years , when you come to the Leap-year , you must neglect or slip one , the reason is , because every Leap-year hath two dominical letters , and there also doth the week day change in the first of March , so that for the day of the month , in finding that the trouble of remembring the Leap-year is avoided . To find the day of the Month. Having found the day of the week , the first of March is on the respective year ; then look for the month in the column , and row of months : then all the daies right under the month are the same day of the week the first of March was on , then in regard the days go round , that is change orderly every seven days , you may find any other successive day sought for . Example . About the middle of March 1661 on a Friday , what day of the month is it ? First the week day for 1661 is Friday , as the letter F on the next collumn beyond 60 she●et● ; then I look for 1 among the months , and all the days right under , viz. 1 , 8 , 15 , 22 , 29. in March , and November 61 , are Friday , therefore my day being Friday , and about the middle of the month , I conclude it is the 15th day required . Again in May 1661. on a Saturday about the end of May , what day of the month ? May is the third month , by the last rule I find that the 24 and 31 are Fridays , therefore this must needs be the 25 day , for the first of Iune is the next Saturday . FINIS . ERRATA . PAge 23. l. 4. adde 1660 p. 24. l. 6. for 5 hours r. 4. l. 9. for 3. 29. r. 4. 39. 1. 12 for 5. 52. r. 4 , 52. l. 13. for 3. 39. r. 4. 39. l. 17 for 5 hours 52. r. 4. 52. p. 27. l. ult . dele or 11. 03. p 31. l 4. for sun r. sum . p. 50. l. 8. for B r. A. p. 50 d CHAP. XII . p. 51. r. 16. for 6. 10. 1. 6 to 10. p. 71. l. 6. for 7 / 4 r. 1 / 4. l. penult . for 2 afternoon , r. 1. p. 74 l. ult . for 1. r. 1 , 2. p. 83. l. 18. for BC r. BD. p. 69 l. 17. add measure , p. 129 l. 24. for right of , r. right sine of . p. 114 l. 9 for 18 3. r 18 13. p. 147 1. 2 for 20 , r. 90 p. 163. l. 16. for of , r. on . 
A64223	----	The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T. Taylor, John, 1666 or 7-1687. 1667 Approx. 163 KB of XML-encoded text transcribed from 77 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-10 (EEBO-TCP Phase 1). A64223 Wing T533B ESTC R221720 99832990 99832990 37465 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A64223) Transcribed from: (Early English Books Online ; image set 37465) Images scanned from microfilm: (Early English books, 1641-1700 ; 2155:11) The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T. Taylor, John, 1666 or 7-1687. [8], 144 p. printed for William Tompson, bookseller at Harborough in Leicestershire, London : 1667. J.T. = John Taylor. In two books; Book II has caption title (the first word of which is in Greek characters); register and pagination are continuous. Pages stained with some loss of print. Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematics -- Early works to 1800. Navigation -- Early works to 1800. Dialing -- Early works to 1800. 2000-00 TCP Assigned for keying and markup 2001-00 Aptara Keyed and coded from ProQuest page images 2001-08 Sean Norton Sampled and proofread 2005-03 Olivia Bottum Text and markup reviewed and edited 2005-04 pfs Batch review (QC) and XML conversion THE SEMICIRCLE ON A SECTOR : In two Books . CONTAINING The description of a general and portable Instrument ; whereby most Problems ( reducible to Instrumental Practice ) in Astronomy , Trigonometry , Arithmetick , Geometry , Geography , Topography , Navigation , Dyalling , &c. are speedily and exactly resolved . By J. T. LONDON , Printed for William Tompson , Bookseller at Harborough in Leicestershire . 1667. To the Reader . ALL that is intended in this Treatise , is to acquaint thee with an Instrument , that is both portable and general , of no great price , easie carriage , yet of a speedy and Accurate dispatch in the most difficult Problems in Astronomy , &c. The lines for the most part have been formerly published by Mr. Gunter , the famous Mr. Foster , Mr. White , &c. The reduction of the 28. Cases of Spherical Triangles unto II. Problems , I first learned from the reverend Mr. Palmers Catholick Planisphere . Many of the proportions in the Treatise of Dyalling are taken from ( though first compared with the Globe ) my worthy Friend ( to whom I am indebted in all the Obligations of Civility , and without whose encouragement this had never adventured the publick Test ) Mr. John Collins . The applying Mr. Fosters Line of Versed Sines unto the Sector was first published by Mr. John Brown , Mathematical Instrument-Maker , at the Sphere and Sun-Dial in the Minories , London , Anno 1660. who bath very much assisted me , by making , adding unto , and giving me freely the perusal of many Instruments , according to any directions for Improvement , that was proposed to him . After this account , what hath been my part in this Work , I hazard to thy censure ; and when I see others publish a more convenient , speedy , accurate , and general Instrument , I assure them to have as low thoughts of this , as themselves . But here is so large a Catalogue of Errata's as would stagger my confidence at thy pardoning , had they not been irrevocably committed before I received the least notice of them . The Printer writing me word ( after I had corrected so much as came to my sight ) that he could alter no Mistakes until the whole Book was printed : By which means he enforced me to do pennance in his Sheets , for his own Crimes : Did not one gross mistake of his become my purgation , viz. in lib. 2. ( throughout Chap. 3. ) where instead of the note of equality ( marked thus = ) he hath inserted the Algebraick note of Subtraction , or Minoration ( marked thus - ) Nor hath the Engraver come behinde the Composer , who so miserably mangleth Fig. 13. that ( at first sight ) it would endanger branding of a mans Brains to spell the meaning thereof , either in it self , or in reference to the Book . All that I can help thee herein , is this ; Whereas the Book mentions that Figure for an East Dyal , if you account it ( as now cut ) a West Dyal , and alter the names of the hours , by putting Figures for the afternoon , in the place of those there for the morning ; you will then have a true West Dyal of that Figure . The correction of Punctations would be an endless task ; for I finde some to be resolved , ever since Valentine , to recreate themselves at Spurn-point . What other material mistakes are in the Book ( which ought to be corrected before reading thereof ) you will finde mentioned in the Errata . Farewel . March 29. 1667. J. T. Errata . PAge 4. line 12. signs , r. sines . p. 8. l. 5. seconds , r. secants . p. 12. l. 9. all , r. allone . p. 13. l. 9 , and 10 . sec . r. min. p. 14. l. 17. sec. r. min. p. 22. l. 14. any , r. what . p. 23. l. 3. exact , r. erect . p. 24. l. ult . adde lib. 2. p. 25. l. 2. signs , r. sines . l. 5 . sign . r. sine . p. 35. l. 3 . a mark , r. an ark . p. 37. l. 22. 20. r. 22. and 42. r. 20. p. 44. l. 5. At , r. At. p. 49. l. 1 . divided by , r. dividing . p. 62. l. 15. dele , a. In lib. 1. chap. 9. the pages are false numbred . But in chap. 9. p. 62. l. 11 . next , r. exact . l. 15 . gauger , r. gauge . l.24.the , r. what . p. 73.l . 5 . whereas , r. where I. p. 80. l. 13 . wherein , r. whereof . l.22.pont , r. point . p. 91 . l.1 . the co-tangent , r. half the co-tangent . l. 19 . L. r. P. p. 92. l. 21 . PZ . r. PS . p. 93. l. 12 . NSP , r. NSZ . 6.angle , r. ark . throughout page 96. Fig. 3. r. Fig. 7. and Fig. 4. r. Fig. 8. p. 98. l. 13. serve , r. scrue . p. 99 . l.12 . places , r. plates . l. 13. proportion , r. perforation . l.9.serve , r. scrue . l.4.serve , r. scrue . 21.serve , r : scrue . p. 111. l. 20. lay in , r. laying . l.14.of , ●● and. p. 128. l. 3. Fig. 12. r. Fig. 13. l. 22 . ED. r. EC . In Fig. 12. C. r. A. at the end of the line G. To the Right Honorable , The Lord SHERARD , Baron of Letrim . My Lord , SInce the trifling Treatise of an Almanack hath usurped a custom to pinnion some Honourable Name to the Patronage of the Authors Follies ; had we not certain evidence from the uncertainty of their Predictions , that their Brains ( like their great Oracles the Planets ) are often wandring ; it might be deemed a Crime , beyond the benefit of the Clergy , to prefix before any Book , a dedication to a Noble Person : Or when I read the unreasonableness of others in those Addresses , imploring their Patrons to be their Dii Tutelares , and prostrate their reputes to the unmannerly mangling of every Censurist , under the notion of protecting ( that is adopting ) the Authors Ignorance , or negligence ; it s enough to tempt the whole world to turn Democritians , and hazard their spleens in laughing at such mens madness . My present design is only to give your Lordship my observance of your Commands about the Description and Improvement of the Sector ; and wherein I have erred ( through mistake , or defect ) I despair not , but from your Honor I shall meet with a pardon of Course to be granted unto Your Lordships most humble Servant J. T. THE SEMICIRCLE ON A SECTOR . LIB . I. CHAP. I. A Description of the Instrument , with the several Lines inscribed thereon . THe Instrument consists of three Rulers , or Pieces ; two whereof are joyned together by a River that may open and shut to any Angle , in fashion of the Sector ; or to use a courser comparison , after the manner of Compasses . The third piece is loose , or separable from them , to be put into the Tenons at the end of the inward ledge of the joyned pieces , and thereby constituting an aequilateral triangle . On these Rulers ( after this manner put together ) we take notice ( for distinction sake ) of the sides , ledges , ends , and pieces . The sides are thus differenced , one we call the quadrantal , the other the proportional , or sector side . The ledges are distinguished by naming one the inward , the other the outward ledge . The ends are known in terming one the head ( viz. where the two pieces are riveted together ) the other the end . The pieces are discovered by styling one the fixed piece , viz. that which hath the rivet upon it ; the second the movable piece , which turns upon that rivet , and the last the loose piece , to be put into the tenons , as before expressed . The quadrantal side of the joyned pieces is easily discerned , having the names of the Moneths stamped on the movable piece , and Par. Scale on the fixed piece . The quadrantal side of the loose piece is known by the degrees on the inward and outward limb . These directions are sufficient to instruct you how to put the Instrument together . Imagining the Instrument thus put together , the lines upon the quadrantal side are these . First on the fixed piece , next the outward limb , is a line of 12 equal parts ; and each of those parts divided into 30 degrees , marked from the end towards the head with ♈ ♉ ♊ , &c. representing the 12 Signs of the Zodiack ; the use of this line , with the help of those under it , was intended to find the hour of the night by the Moon . The next line to this is a line of twice 12 , or 24 equal parts , each division whereof cuts every 15th degree of the former line : and therefore if the figures were set to every 15th degree on this side the former line , this second line would be useless , and the former perform its office more distinctly . This line was intended an assistant for finding the hour by the Moon ; but is very ready to find the hour by any of the fixed Stars . The third line is a line of 29● equal parts , serving for the dayes of the Moons age ; in order to find the hour of the night by the Moon . But the operation is so tedious , and far from exactness , that I have no kindness for it ; and should place some other lines in the room of this and the former , did I not resolve to impose upon no mans phantasie . The fourth line is a line of altitudes for a particular latitude , noted at the end , Par. Scale , &c. This helpeth to find the hour and azimuth of the Sun , or any fixed Star very exactly . The fifth line is a line of natural sines ; at the beginning whereof there is a pin , or else an hole to put a pin into , whereon to hang the thread and plummet for taking of altitudes . To this line of sines may be joyned a line of tangents to 45 degr . The use of the sines alone , is to work proportions in signs . The use of the sines with the tangent line may be for any proportion in trigonometry ; but that I leave to liberty . The sixth line , and last on this side the fixed piece is a line of versed sines , numbred from the centre at the head to 180 at the end . On the quadrantal side of the movable piece , the first line next the inward edge , is a line of versed sines answering to that on the fixed piece . The use of these versed sines is various at pleasure . The second line from the inward edge , is a line of hours and Azimuths serving to find the hour by the Sun or Stars , or the Azimuth of the Sun , or any fixed Star from the South . The third and fourth lines are lines of Moneths , marked with the respective names , and each Moneth divided into so many parts as it contains dayes . The fifth line is a line of signs marked ♈ Taurus ; Gemini ; , &c. each sign being divided into 30 degr . and proceeding from ♈ , or Aries ( which answers to the tenth of March ) in the same order as the Moneths . The use of this line , with help of the Moneths , is to find the Suns place in the Zodiack . The sixth line is a line of the Suns right ascension , commonly noted by hours from 00 to 24. but better if divided by degrees or sines from 00 to 360 , and both wayes proceeding backward and forward as the signs of the Zodiack , or dayes of the Moneth . Lastly the outward edge or limb of the movable and loose piece both , is graduated unto 180 degrees , or two quadrants ; whose centre is the pin , or pin-hole before mentioned , at the beginning of the sines on this side the fixed piece . The perpendicular is at 0 / 60 upon the loose piece ; from whence reckoning along the outward edge of the loose piece , till it intersects the produced line of sines at the end of the fixed piece , you have 90 degrees . Or counting from 0 / 60 on the loose piece , and continuing it along the degrees of the outward limb of the movable piece , until they intersect the produced line of sines on the fixed piece at the head , you have again 90 degrees , which compleat the Semicircle . The use of this line is for taking of altitudes , counting upon the former 90. degr . when you hold the head of the fixed piece toward the Sun : and numbring upon this latter , when you hold ( which is best because the degrees are largest ) the end of the fixed piece toward the Sun. There are other wayes of numbring these degrees for finding of the Azimuth , &c. which shall be mentioned in their proper places . On the quadrantal side of the loose piece , the inward edge or limb is graduated unto 60 degrees ( or twice 30 , which you please ) whose centre is a pin at the head . The use of this is to find the altitude of the Sun , or any Star , without thread and plummet ; or to perform some uses of the Cross-staff . This is for large rules or instruments , and therefore not illustrated here . In the empty spaces upon the quadrantal side may be engraven the names of some fixed Stars , with their right ascensions and declinations . On the proportional side the lines issuing from the centre are the same upon the fixed and movable piece , but happily transplaced ( thanks to the first contriver ) after this manner : The line that lies next the inward edge on the fixed piece hath his fellow or correspondent line toward the outward edge on the movable piece ; by which means these lines all meeting at the centre , stand all at the same angle , and give you the freedom from a great deal of trouble , in working proportions by sines and tangents , or laying down any sine or tangent to any Radius given , &c. The lines issuing from the centre toward the outward edge of the movable piece , whose fellow is next the inward edge of the fixed piece , is a line of natural sines on the outward side , marked at the end S , and on the inward side a line of lines or equal parts , noted at the end L ; the middle line serving for both of them . The lines issuing from the centre next the inward edge of the movable piece , whose fellows are toward the outward edge of the fixed piece , are lines of natural tangents , which on the outward , side of the line is divided to 45 the Radius ; and on the inward side of the lines ( the middle line serving for both ) at a quarter of the former Radius from the centre is another Radius noted 45 at the beginning , and continued to the tangent 75. These lines are noted at the end T. The use of these you will find Chap. 3,4,5 . Betwixt the lines of sines and tangents , both upon the fixed and movable pieces , is placed a line of seconds , continued unto 60 , and marked at the end . Se. Next to the outward edge on the fixed and movable piece ( which is best discerned when those pieces are opened to the full length ) is a line of Meridians divided to 85 ; whose use is for Navigation , in describing Maps or Charts , &c. In the vacant spaces you may have a line of chords , sines , and tangents , to any Radius the space will bear ; and what other any one thinks best of , as a line of latitudes and hours , &c. On the proportional side of the loose piece are lines for measuring all manner of solids , as Timber , Stone , &c. likewise for gaging of Vessels either in Wine or Ale measure . On the outward ledge of the movable and fixed piece , both ( which in use must be stretched out to the full length ) is a line of artificial numbers , sines , tangents , and versed sines . The first marked N , the second S , the third T , the fourth VS . On the inward ledge of the movable piece is a line of 12 inches divided into halfs , quarters , half-quarters . Next to that is a prick'd line , whose use is for computing of weight and carriages . Lastly a line of foot measure , or a foot divided into ten parts ; and each of those subdivided into ten or twenty more . On the inward ledge of the loose piece you may have a line of circumference , diameter , square equal , and square inscribed . There will still be requisite sights , a thread and plummet . And if any go to the price of a sliding Index to find the shadow from the plains perpendicular , in order to taking a plains declination , and have a staff and a ballsocket , the Instrument is compleated with its furniture . Proceed we now to the uses . Onely note by the way that Mr. Brown hath ( for conveniency of carrying a pair of Compasses , Pen , Ink , and Pencil ) contrived the fixed piece and movable both to be hollow , and then the pieces that cover those hollows do , one supply the place of the loose piece for taking altitudes ; the other ( being a sliding rule ) for measuring solids , and gaging Vessels without Compasses . CHAP. II. Some uses of the quadrantal side of the Instrument . PROBL. 1. To find the altitude of the Sun or any Star. HAng the thread and plummet upon the pin at the beginning of the line of sines on the fixed piece , and ( having two sights in two holes parallel to that line ) raise the end of the fixed piece , toward the Sun until the rayes pass through the sights ( but when the Sun is in a cloud , or you take the altitude of any Star , look along the outward ledge of the fixed piece , until it be even with the middle of the Sun or Star ) then on the limb the thread cuts the degree of altitude , if you reckon from 0 / 60 on the loose piece toward the head of the movable piece . PROBL. 2. The day of the Moneth given , to find the Suns place , declination , ascensional difference , or time of rising and setting , with his right ascension . The thread laid to the day of the Moneth gives the Suns place in the line of signs , reckoning according to the order of the Moneths ( viz. forward from March the 10th . to June , then backward to December , and forward again to March 10. ) In the limb you have the Suns declination , reckoning from 60 / 0 on the movable piece towards the head for North , toward the end for South declination . Again , on the line of right ascensions , the thread shews the Suns right ascension , in degrees , or hours , ( according to the making of your line ) counting from Aries toward the head , and so back again according to the course of the signs unto 24 hours , or 360 degrees . Lastly on the line of hours you have the time of Sun rising and setting , which turned into degrees ( for the time from six ) gives the ascensional difference . Ex. gr . in lat . 52. deg . 30 min. for which latitude I shall make all the examples . The 22 day of March I lay the thread to the day in the Moneths , and find it cut in the Signs 12 deg . 20 min. for the Suns place , on the limb 4 deg . 43 min. for the Suns declination North. In the line of right ascensions it gives 46 min. of time , or 11 deg . 30 min. of the circle . Lastly , on the line of hours it shews 28 min. before six for the Suns rising ; or which is all , 7 deg . for his ascensional difference . PROBL. 3. The declination of the Sun or any Star given to find their amplitude . Take the declination from the scale of altitudes , with this distance setting one point of your Compasses at 90 on the line of Azimuth , apply the other point to the same line it gives the amplitude , counting from 90 Ex. gr . at 10 deg . declination , the amplitude is 16 deg . 30 min. at 20 deg . declination , the amplitude is 34 deg . PROBL. 4. The right ascension of the Sun , with his ascensional difference , given to find the oblique ascension . In Northern declination , the difference betwixt the right ascension and ascensional difference , is the oblique ascension . In Southern declination take the summ of them for the oblique ascension , Ex. gr . at 11 deg . 30 sec. right ascension , and 6 deg . 30 sec. ascensional difference . In Northern declination the oblique ascension will be 5 deg . in Southern 18 deg . PROBL. 5. The Suns altitude and declination , or the day of the Moneth given to find the hour . Take the Suns altitude from the Scale of altitudes , and laying the thread to the declination in the limb ( or which is all one , to the day in the Moneths ) move one point of the compasses along the line of hours ( on that side the thread next the end ) until the other point just touch the thread ; then the former point shews the hour ; but whether it be before or after noon , is left to your judgment to determine . Ex. gr . The 22 day of March , or 4 deg . 43 min. North declination , and 20 deg . altitude , the hour is either 47 minutes past 7 in the morning , or 13 minutes past 4 afternoon . PROBL. 6. The declination of the Sun , or day of the Moneth , and hour given to find the altitude . Lay the thread to the day or declination , and take the least distance from the hour to the thread , this applyed to the line of altitudes , gives the altitude required . Ex. gr . The 5 day of April or 10 deg . declination North , at 7 in the morning , or 5 afternoon , the altitude will be 17 deg . 10 sec. and better . PROBL. 7. The declination and hour of the night , given to find the Suns depression under the horizon . Lay the thread to the declination on the limb ; but counted the contrary way , viz. from 60 / 0 on the movable piece toward the head for Southern ; and toward the end for Northern declination . This done take the nearest distance from the hour to the thread , and applying it to the line of altitudes , you have the degrees of the Suns depression . Ex. gr . at 5 deg . Northern declination , & 8 hours afternoon , the depression is 13 deg . 30 min. PROBL. 8. The declination given to find the beginning and end of twilight , or day-break . Lay the thread to the declination counted the contrary way , as in the last Problem , and take from your Scale of altitudes 18 deg . for twilight , and 13 deg . for day-break , or clear light ; with this run one point of the Compasses along the line of houres ( on that side next the end ) until the other will just touch the thread , and then the former point gives the respective times required . Ex. gr . At 7 deg . North declination , day breaks 8 minutes before 4 : but twilight is 3 houres 12 minutes in the morning , or 8 hours 52 minutes afternoon . PROBL. 9. The declination and altitude of the Sun or any Star , given to find their Azimuth in Northern declination . Lay the thread to the altitude numbred on the limb of the moveable piece from 60 / 0 toward the end ( and when occasion requires , continue your numbring forward upon the loose piece ) and take the declination from your line of altitude ; with this distance run one point of your Compasses along the line of Azimuths ( on that side the thread next the head ) until the other just touch the thread , then the former point gives the Azimuth from South . Ex. gr . at 10 deg . declination North , and 30 deg . altitude , the Azimuth from South is 64 , deg . 40 min. PROBL. 10. The Suns altitude given to find his Azimuth in the aequator . Lay the thread to the altitude in the limb , counted from 60 / 0 on the loose piece toward the end , and on the line of Azimuths it cuts the Azimuth from South . Ex. gr . at 25 deg . altitude the Azimuth is 53 deg . At 30 deg . altitude the Azimuth is 41 deg . 30 min. fere . PROBL. 11. The declination and altitude of the Sun , or any Star given to find the Azimuth in Southern declination . Lay the thread to the altitude numbred on the limb from 60 / 0 on the moveable piece toward the end , and take the declination from the Scale of altitudes ; then carry one point of your Compasses on the line of Azimuths ( on that side the thread next the end ) until the other just touch the thread , which done , the former point gives the Azimuth from South . Ex. gr . at 15 deg . altitude and 6 deg . South declination the Azimuth is 58 deg . 30 min. PROBL. 12. The declination given to find the Suns altitude at East or West in North declination , and by consequent his depression in South declination . Take the declination given from the Scale of altitudes , and setting one point of your Compasses in 90 on the line of Azimuths , lay the thread to the other point ( on that side 90 next the head ) on the limb it cuts the altitude , counting from 60 / 0 on the moveable piece . Ex. gr . at 10 deg . declination the altitude is 12 deg . 40 min. PROBL. 13. The declination and Azimuth given to find the altitude of the Sun or any Star. Take the declination from the Scale of altitudes ; set one point of your Compasses in the Azimuth given , then in North declinanation turn the other point toward the head , in South toward the end ; and thereto laying the thread , on the limb you have the altitude , numbring from 60 / 0 on the moveable piece toward the end . Ex. gr . At 7 deg . North declination , and 48 deg . Azimuth from South , the altitude is 35 deg . but at 7 deg . declination South , and 50 deg . Azimuth the altitude is onely 18 deg . 30 min. PROBL. 14. The altitude , declination , and right ascension of any Star with the right ascension of the Sun given , to find the hour of the night . Take the Stars altitude from the Scale of altitudes , and laying the thread to his declination in the limb , find his hour from the last Meridian he was upon , as you did for the Sun by Probl. 5. If the Star be past the South , this is an afternoon hour ; if not come to the South , a morning hour ; which keep . Then setting one point of your Compasses in the Suns right ascension ( numbred upon the line twice 12 or 24 next the outward ledge on the fixed piece ) extend the other point to the right ascension of the Star numbred upon the same line , observing which way you turned the point of your Compasses , viz. toward the head or end . With this distance set one point of your Compasses in the Stars hour before found counted on the same line , and turning the other point the same way , as you did for the right ascensions , it gives the true hour of the night . Ex. gr . The 22 of March I find the altitude of the Lions heart 45 deg . his declination 13 d. 40 min. then by Probl. 5. I find his hour from the last Meridian 10 houres , 5 min. The right ascension of the Sun is 46 m. of time , or 11 d. 30 m. of the Circle , the right ascension of the Lions heart , is 9 hour 51 m. fere , of time , or 147 deg . 43 m. of a circle ; then by a line of twice 12 , you may find the true hour of the night , 7 hour 13 min. PROBL. 15. The right ascension and declination of any Star , with the right ascension of the Sun and time of night given , to find the altitude of that Star with his Azimuth from South , and by consequent to find the Star , although before you knew it not . This is no more than unravelling the last Problem . 1 Therefore upon the line of twice 12 or 24 , set one point of your Compasses in the right ascension of the Star , extending the other to the right ascension of the Sun upon the same line , that distance laid the same way upon the same line , from the hour of the night , gives the Stars hour from the last Meridian he was upon . This found by Probl. 5. find his altitude as you did for the Sun. Lastly , having now his declination and altitude by Probl. 8. or 10. according to his declination , you will soon get his Azimuth from South . This needs not an example . By help of this Problem the Instrument might be so contrived , as to be one of the best Tutors for knowing of the Stars . PROBL. 16. The altitude and Azimuth of any Star given to find his declination . Lay the thread to the altitude counted on the limb from 60 / 0 on the moveable piece toward the end , setting one point of your Compasses in the Azimuth , take the nearest distance to the thread ; this applyed to the Scale of altitudes gives the declination . If the Azimuth given be on that side the thread toward the end , the declination is South ; when on that side toward the head , its North. PROBL. 17. The altitude and declination of any Star , with the right ascension of the Sun , and hour of night given to find the Stars right ascension . By Probl. 5. or 14. find the Stars hour from the Meridian . Then on the line twice 12 , or 24 , set one point of your Compasses in the Stars hour ( thus found ) and extend the other to the hour of the night . Upon the same line with this distance set one point of your Compasses in the right ascension of the Sun , and turning the other point the same way , as you did for the hour , it gives the Stars right ascension . PROBL. 18. The Meridian altitude given to find the time of Sunrise and Sunset . Take the Meridian altitude from your particular Scale , and setting one point of your Compasses upon the point 12 on the line of hours ( that is the pin at the end ) lay the thread to the other point , and on the line of hours the thread gives the time required . PROBL. 19. To find any latitude your particular Scale is made for . Take the distance from 90 , on the line of Azimuth unto the pin at the end of that line , or the point 12 : this applyed to the particular Scale , gives the complement of that latitude the Instrument was made for . PROBL. 20. To find the angles of the substile , stile , inclination of Meridians , and six and twelve , for exact declining plains , in that latitude your Scale of altitudes is made for . Sect. 1. To find the distance of the substile from 12 , or the plains perpendicular . Lay the thread to the complement of declination counted on the line of Azimuths , and on the limb it gives the substile counting from 60 / 0 on the moveable piece . Sect. 2. To find the angle of the Stile 's height . On the line of Azimuths take the distance from the Plains declination to 90. This applyed to the Scale of altitudes gives the angle of the stile . Sect. 3. The angle of the Substile given to find the inclination of Meridians . Take the angle of the substile from the Scale of altitudes , and applying it from 90 on the Azimuth line toward the end ; the figures shew the complement of inclination of Meridians . Sect. 4. To find the angle betwixt 6 and 12. Take the declination from the Scale of altitudes , and setting one point of your Compasses in 90 on the line of Azimuths , lay the thread to the other point and on the limb it gives the complement of the angle sought , numbring from 60 / 0 on the moveable piece toward the end . This last rule is not exact , nor is it here worth the labour to rectifie it by another sine added ; sith you have an exact proportion for the Problem in the Treatise of Dialling Chap. 2 . Sect. 5. Paragr . 4. CHAP. III. Some uses of the Line of natural signs on the Quadrantal side of the fixed piece . PROBL. 1. How to adde one sign to another on the Line of Natural Sines . TO adde one sine to another , is to augment the line of one sine by the line of the other sine to be added to it . Ex. gr . To adde the sine 15 to the sine 20 , I take the distance from the beginning of the line of sines unto 15 , and setting one point of the Compasses in 20 , upon the same line , turn the other toward 90 , which I finde touch in 37. So that in this case ( for we regard not the Arithmetical , but proportional aggregate ) 15 added to 20 , upon the line of natural sines , is the sine 37 upon that line , and from the beginning of the line to 37 is the distance I am to take for the summe of 20 and 15 sines . PROBL. 2. How to substract one sine from another upon the line of natural sines . The substracting of one sine from another , is no more than taking the distance from the lesser to the greater on the line of sines , and that distance applyed to the line from the beginning , gives the residue or remainer . Ex. gr . To substract 20 from 37 I take the distance from 20 to 37 that applyed to the line from the beginning gives 15 for the sine remaining . PROBL. 3. To work proportions in sines alone . Here are four Cases that include all proportions in sines alone . CASE 1. When the first term is Radius , or the Sine 90. Lay the thread to the second term counted on the degrees upon the movaeble piece from the head toward the end , then numbring the third on the line of sines , take the nearest distance from thence to the thread , and that applyed to the Scale from the beginning gives the fourth term . Ex. gr . As the Radius 90 is to the sine 20 , so is the sine 30 to the sine 10. CASE 2. When the Radius is the third term . Take the sine of the second term in your Compasses , and enter it in the first term upon the line of sines , and laying the thread to the nearest distance , on the limb the thread gives the fourth term . Ex. gr . As the sine 30 is to the sine 20 , so is the Radius to the sine 43. 30. min. CASE 3. When the Radius is the second term . Provided the third term be not greater than the first , transpose the terms . The method of transposition in this case is , as the first term is to the third , so is the second to the fourth , and then the work will be the same as in the second case . Ex. gr . As the sine 30 is to the radius or sine 90 , so is the sine 20 to what sine ; which transposed is As the sine 30 is to the sine 20 , so is the radius to a fourth sine , which will be found 43 , 30 min. as before . CASE 4. When the Radius is none of the three terms given . In this case when both the middle terms are less than the first , enter the sine of the second term in the first , and laying the thread to the nearest distance , take the nearest extent from the third to the thread : this distance applyed to the scale from the beginning gives the fourth . Ex. gr . As the sine 20 to the sine 10 , so is the sine 30 to the sine 15. When only the second term is greater than the first , transpose the terms and work as before . But when both the middle tearms be greater than the first , this proportion will not be performed by this line without a paralel entrance or double radius ; which inconveniency shall be remedied in its proper place , when we shew how to work proportions by the lines of natural sines on the proportional or sector side . These four cases comprizing the method of working all proportions by natural sines alone , I shall propose some examples for the exercise of young practitioners , and therewith conclude this Chapter . PROBL. 4. To finde the Suns amplitude in any Latitude . As the cosine of the Latitude is to the sine of the Suns declination , so is the radius to the sine of amplitude . PROBL. 5. To finde the hour in any Latitude in Northern Declination . Proport . 1. As the radius to the sine of the Suns declination , so is the sine of the latitude to the sine of the Suns altitude at six . By Probl. 2. substract this altitude at six from the present altitude , and take the difference . Then Proport . 2. As the cosine of the latitude is to that difference , so is the radius to a fourth sine . Again Proport . 3. As the cosine of the declination to that fourth sine , so is the radius to the sine of the hour from six . PROBL. 6. To finde the hour in any Latitude when the Sun is in the Equinoctial . As the cosine of the latitude is to the sine of altitude , so is the radius to the sine of the hour from six . PROBL. 7. To finde the hour in any latitude in Southern Declination . Proport . 1. As the radius to the sine of the Suns declination , so is the sine of the latitude to the sine of the Suns depression at six ; adde the sine of depression to the present altitude by Probl. 1. Then Proport . 2. As the cosine of the latitude is to that summe , so is the radius to a fourth sine . Again , Proport . 3. As the cosine of declination is to the fourth sine , so is the radius to the sine of the hour from six . PROBL. 8. To finde the Suns Azimuth in any latitude in Northern Declination . Proport . 1. As the sine of the latitude to the sine of declination , so is the radius to the sine of altitude at East , or West . By Probl. 2. substract this from the present altitude , then , Proport . 2. As the cosine of the latitude is to that residue , so is the radius to a fourth sine . Again , Proport . 3. As the cosine of the altitude is to that fourth sine , so is the radius to the sine of the Azimuth from East or West . PROBL. 9. To finde the Azimuth for any latitude when the Sun is in the Equator . Proport . 1. As the cosine of the latitude to the sine of altitude , so is the sine of the latitude to a fourth sine . Proport . 2. As the cosine of altitude to that fourth sine , so is the radius to the sine of the Azimuth from East , or West . PROBL. 10. To finde the Azimuth for any latitude in Southern Declination . Proport . 1. As the cosine of the latitude to the sine of altitude , so is the sine of the latitude to a fourth . Having by Probl. 4. found the Suns amplitude , adde it to this fourth sine by Probl. 1. and say As the cosine of the altitude is to the sum , so is the radius to the sine of the Azimuth from East or West . The terms mentioned in the 5th . 7th . 8th . 10th . Problems are appropriated unto us that live on the North side the Equator . In case they be applyed to such latitudes as lie on the South side the Equator . Then what is now called Northern declination , name Southern , and what is here styled Southern declination , term Northern , and all the proportion with the operation is the same . These proportions to finde the hour and Azimuth , may be all readily wrought by the lines of artificial sines , only the addition and substraction must alwayes be wrought upon the line of natural sines . CHAP. IV. Some uses of the Lines on the proportional side of the Instrument , viz. the Lines of natural Sines , Tangents , and Secants . PROBL. 1. To lay down any Sine , Tangent , or Secant to a Radius given . See Fig. 1. IF you be to lay down a Sine , enter the Radius given in 90 , and 90 upon the lines of Sines , keeping the Sector at that gage , set one point of your Compasses in the Sine required upon one line , and extend the other point to the same Sine upon the other Line : This distance is the length of the Sine required to the given Radius . Ex. gr . Suppose A. B. the Radius given , and I require the Sine 40. proportional to that Radius . Enter A. B. in 90 , and 90 keeping the Sector at that gage , I take the distance , twixt 40 on one side , to 40 on the other , that is , C. D. the Sine required . The work is the same , to lay down a Tangent to any Radius given , provided you enter the given Radius in 45 , and 45 , on the line of Tangents . Only observe if the Tangent required be less than 45. you must enter the Radius in 45. and 45 next the end of the Rule . But when the Tangent required exceeds 45. enter the Radius given in 45 , and 45 'twixt the center and end , and keeping the Sector at that Gage , take out the Tangent required . This is so plain , there needs no example . To lay down a Secant to any Radius given , is no more than to enter the Radius in the two pins at the beginning of the line of Secants ; and keeping the Sector at that Gage , take the distance from the number of the Secant required on one side , to the same number on the other side , and that is the Secant sought at the Radius given . The use of this Problem will be sufficiently seen in delineating Dyals , and projecting the Sphere . PROBL. 2. To lay down any Angle required by the Lines of Sines , Tangents , and Secants . See Fig. 2. There are two wayes of protracting an Angle by the Line of Sines , First if you use the Sines in manner of Chords . Then having drawn the line A B at any distance of your Compass , set one point in B , and draw a mark to intersect the Line B A , as E F. Enter this distance B F in 30 , and 30 upon the Lines of Sines , and keeping the Sector at that Gage , take out the Sine of half the Angle required , and setting one point where F intersects B A , turn the other toward E , and make the mark E , with a ruler draw B E and the Angle E B F is the Angle required , which here is 40. d. A second method by the lines of Sines is thus , Enter B A Radius in the Lines of Sines , and keeping the Sector at that Gage , take out the Sine of your Angle required with that distance , setting one point of your Compasses in A , sweep the ark D , a line drawn from B by the connexity of the Ark D , makes the Angle A B C 40 d. as before . To protract an Angle by the Lines of Tangents is easily done , draw B A the Radius upon A , erect a perpendicular , A C , enter B A in 45 , and 45 on the Lines of Tangents , and taking out the Tangent required ( as here 40 ) set it from A to C. Lastly , draw B C , and the Angle C B A is 40 d. as before . In case you would protract an Angle by the Lines of Secants . Draw B A , and upon A erect the perpendicular A C , enter A B in the beginning of the Lines of Secants , and take out the Secant of the Angle , with that distance , setting one point of your Compasses in B , with the other cross the perpendicular A C , as in C. This done , lay a Ruler to B , and the point of intersection , and draw the Line B C. So have you again the Angle C B A. 40. d. by another projection . These varieties are here inserted only to satisfie a friend , and recreate the young practitioner in trying the truth of his projection . PROBL. 3. To work proportions in Sines alone , by the Lines of natural Sines on the proportional side of the Instrument . The general rule is this . Account the first term upon the Lines of Sines from the Center , and enter the second term in the first so accounted , keeping the Sector at that Gage , account the third term on both lines from the Center , and taking the distance from the third term on one line to the third term on the other line , measure it upon the line of Sines from the beginning , and you have the fourth term . Ex. gr . As the Radius is to the Sine 30 , so is the Sine 40 to the Sine 18. 45. There is but one exception in this Rule , and that is when the second term is greater than the first ; yet the third lesser than the first , and in this case transpose the terms , by Chap 3. Probl. 3. Case 3. But when the second term is not twice the length of the first , it may be wrought by the general Rule without any transposition of terms . Ex. gr . As the Sine 30 is to the Sine 50 , so is the Sine 20 to the Sine 31. 30. min. And by consequent , when the third term is greater than the first , provided it be not upon the line , double the length thereof , it may be wrought by transposing the terms , although the second was twice the length of the first . Ex. gr . As the Sine 20 is to the Sine 60 , so is the Sine 42 , to what Sine ? which transposed is , As the Sine 20 is to the Sine 42 , so is the Sine 60 to the Sine 35. 30. This case will remove the inconveniency mentioned , Chap. 3. Probl. 3. Case 4. of a double Radius . I intended there to have adjoyned the method of working proportions by natural Tangents alone , and by natural Sines , and Tangents , conjunctly : But considering the multiplicity of proportions when the Tangents exceed 45. I suppose it too troublesome for beginners , and a needless variety for those that are already Mathematicians . Sith , both may be eased by the artificial Sines and Tangents on the outward ledge , where I intend to treat of those Cases at large , and shall in this place only annex some proportions in Sines alone , for the exercise of young beginners . PROBL. 4. By the Lines of Natural Sines to lay down any Tangent , or Secant required to a Radius given . In some Cases , especially for Dyalling , your Instrument may be defective of a Tangent , or Secant for your purpose , Ex. gr . when the Tangent exceeds 76 , or the Secant is more than 60. In these extremities use the following Remedies . First , for a Tangent . As the cosine of the Ark is to the Radius given , so is the sine of the Ark to the length of the Tangent required . Secondly , for a Secant . As the cosine of the Ark is to the Radius given , so is the Sine 90 to the length of the Secant required . PROBL. 5. The distance from the next Equinoctial Point given to finde the Suns declination . As the Radius to the sine of the Suns greatest declination , so is the sine of his distance from the next Equinoctial Point to the sine of his present declination . PROBL. 6. The declination given to finde the Suns Equinoctial Distance . As the sine of the greatest declination is to the sine of the present declination , so is the Radius to the sine of his Equinoctial Distance . PROBL. 7. The Altitude , Declination , and Distance of the Sun from the Meridian given to finde his Azimuth . As the cosine of the altitude , to the cosine of the hour from the Meridian , so is the cosine of declination to the sine of the Azimuth . CHAP. V. Some uses of the Lines of the Lines , on the proportional side of the Instrument . PROBL. 1. To divide a Line given into any Number of equal parts . See Fig. 3. SUppose A B a Line given to be divided into nine equal parts . Enter A B in 9 , and 9 on the lines of lines , keeping the Sector at that gage , take the distance from 8 , on one side , to 8 on the other , and apply it from A upon the line A B , which reacheth to C ; then is C B a ninth part of the line A B. By this means you may divide any line ( that is not more than the Instrument in length ) into as many parts as you please , viz. 10 , 20 , 30 , 40 , 50 , 100 , 500 , &c. parts according to your reckoning the divisions upon the lines , Ex. gr . The line is actually divided into 200 parts , viz. first into 10 , marked with Figures , and each of those into twenty parts more . Again , if the line represents a 1000 , then every figured division is 100 , the second or shorter division is 10 , and the third or shortest division is 5. In case the whole line was 2000 , then every figured division is 200 , every smaller or second division is 20 , every third or smallest division is 10 , &c. Suppose I have any line given , which is the base of a Triangle , whose content is 2000 poles , and I demand so much of the Base as may answer 1750 poles . Enter the whole line in 10 , and 10 at the end of the lines of lines , and keep the Sector at that gage . Now the whole line representing 2000 poles , every figured division is 200 ; therefore 1700 is eight and an half of the figured divisions , and 50 is five of the smallest divisions more ( for in this case every smallest division is 10 , as was before expressed ) wherefore setting one point of the Compasses in 15 of the smallest divisions beyond 8 on the Rule . I extend the other point to the same division upon the line on the other side , and that distance is 1750 poles in the base of the Triangle proposed . How ready this is to set out a just quantity in any plat of ground , I shall shew in a Scheam , Chap. 12. PROBL. 2. To work proportions in Lines , or Numbers , or the Rule of three direct by the Lines of Lines . Enter the second term in the first , and keeping the Sector at that gage , take the distance 'twixt the third on one line , to the third on the other line , that distance is the fourth in lines , or measured upon the line from the centre , gives the fourth in numbers , Ex. gr . As 7 is to 3 , so is 21 to 9. PROBL. 3. To work the Rule of Three inverse , or the back Rule of Three by the Lines of Lines . In these proportions there are alwayes three terms given to finde a fourth , and of the three given terms two are of one denomination ( which for distinction sake I call the double denomination ; ) and the third term is of a different denomination from those two , which I therefore call the single denomination , of which the fourth term sought must also be . Now to bring these into a direct proportion , the rule is this . When the fourth term sought is to be greater than the single denomination ( which you may know by sight of the terms given ) say , As the lesser double denomination is to the greater double denomination , so is the single denomination to the fourth term sought . The work is by Probl. 2. If 60 men do a work in 5 dayes , how long will 30 men be about it ? As 30 is to 60 , so is 5 to 10. The number of dayes for 30 men in the work . Again , when the fourth term is to be less than the single denomination , say , As the greater double denomination is to the lesser double denomination , so is the single denomination to the fourth term sought . If 30 men do a work in 5 dayes , how long shall 60 be doing of it ? As 60 is to 30 , so is 5 to 2½ . The time for 60 men in the work . PROBL. 4. The length of any perpendicular , with the length of the shadow thereof given , to finde the Suns Altitude . At the length of the shadow upon the lines of lines , is to the Tangent 45 , so is the length of the perpendicular numbred upon the lines of lines , to the tangent of the Suns altitude . PROBL. 5. To finde the Altitude of any Tree , Steeple , &c. at one station . At any distance from the object ( provided the ground be level ) with your Instrument , look to the top of the object along the outward ledge of the fixed piece , and take the angle of its altitude . This done , measure by feet or yards , the distance from your standing to the bottom of the object . Then say , As the cosine of the altitude is to the measured distance numbred upon the lines of lines , so is the sine of the altitude to a fourth number of feet or yards ( according to the measure you meeted the distance ) to this fourth , adde the height of your eye from the ground , and that sum gives the number of feet or yards in the altitude . CHAP. VI. How to work proportions in Numbers , Sines , or Tangents , by the Artificial Lines thereof on the outward ledge . THe general rule for all of these , is to extend the Compasses from the first term to the second ( and observing whether that extent was upward or downward ) with the same distance , set one point in the third term , and turning the other point the same way , as at first , it gives the fourth . But in Tangents when any of the terms exceeds 45 , there may be excursions , which in their due place I shall remove . PROBL. 1. Numeration by the Line of Numbers . The whole line is actually divided into 100 proportional parts , and accordingly distinguished by figures , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , and then , 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100. So that for any number under 100 , the Figures readily direct you , Ex. gr . To finde 79 on the line of numbers , count 9 of the small divisions beyond 70 , and there is the point for that number . Now as the whole line is actually divided into 100 parts , so is every one of those parts subdivided ( so far as conveniency will permit ) actually into ten parts more , by which means you have the whole line actually divided into 1000 parts . For reckoning the Figures impressed , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , to be 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , and the other figures which are stamped 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , to be 100 , 200 , 300 , 400 , 500 , 600 , 700 , 800 , 900 , 1000. You may enter any number under 1000 upon the line , according to the former directions . And any numbers whose product surmount not 1000 , may be wrought upon this line ; but where the product exceeds 1000 , this line will do nothing accurately : Wherefore I shall willingly omit many Problems mentioned by some Writers to be wrought by this line , as squaring , and cubing of numbers , &c. Sith they have only nicety , and nothing of exactness in them . PROBL. 2. To multiply two numbers , given by the Line of Numbers . The proportion is this . As 1 on the line is to the multiplicator , so is the multiplicand to the product . Ex. gr . As 1 is to 4 , so is 7 , to what ? Extend the Compasses from the first term , viz. I unto the second term , viz. 4. with that distance , setting one point in 7 the third term , turn the other point of the Compasses toward the same end of the rule , as at first , and you have the fourth , viz. 28. There is only one difficulty remaining in this Problem , and that is to determin the number of places , or figures in the product , which may be resolved by this general rule . The product alwayes contains as many figures as are in the multiplicand , and multiplicator both , unless the two first figures of the product be greater than the two first figures in the multiplicator , and then the product must have one figure less than are in the multiplicator , and multiplicand both . Ex. gr . 47 multiplied by 25 , is 2175 , consisting of four figures ; but 16 multiplied by 16 , is 240 , consisting of no more than three places , for the reason before mentioned . I here ( for distinction sake ) call the multiplicator the lesser of the two numbers , although it may be either of them at pleasure . PROBL. 3. To work Division by the Line of Numbers . As the divisor is to 1 , so is the dividend to the quotient . Suppose 800 to be divided by 20 , the quotient is 40. For , As 20 is to 1 , so is 800 to 40. To know how many figures you shall have in the quotient , take this rule . Note the difference of the numbers of places or figures in the dividend and divisor . Then in case the quantity of the two first figures to the left hand in your divisor be less than the quantity of the two first figures to the left hand in your dividend , the quotient shall have one figure more than the number of difference : But where the quantity of the two first figures of the divisor is greater than the quantity of the two first figures of the dividend , the quotient will have only that number of figures noted by the difference . Ex. gr . 245 divided by 15. will have two figures in the quotient ; but 16 divided 〈…〉 ●●ve only one figure in the quotient . PROBL. 4. To finde a mean proportional 'twixt two Numbers given by the line of Numbers . Divide the space betwixt them upon the line of numbers into two equal parts , and the middle point is the mean proportional . Ex. gr . betwixt 4 and 16 , the mean proportional is 8. If you were to finde two mean proportionals , divide the space 'twixt the given numbers into three parts . If four mean proportionals divide it into five parts , and the several points 'twixt the two given numbers , will show the respective mean proportionals . PROBL. 5. To work proportions in Sines alone , by the Artificial Line of Sines . Extend the Compasses from the first term to the second , with that distance set one point in the third term , and the other point gives the fourth . Only observe that if the second term be less than the first , the fourth must be less 〈…〉 or if the second term exceed 〈…〉 fourth will be greater than the third . This may direct you in all proportions of sines and tangents singly or conjunctly , to which end of the rule to turn the point of your Compasses , for finding the fourth term . Ex. gr . As the sine 60 is to the sine 40 , so is the sine 20 to the sine 14. 40. Again , As the sine 10 is to the sine 20 , so is the sine 30 to the sine 80. PROBL. 6. To work proportions in Tangents alone by the Artificial Line of Tangents . For this purpose the artificial line of tangents must be imagined twice the length of the rules , and therefore for the greater conveniency , it is doubly numbred , viz. First from 1 to 45 , which is the radius , or equal to the sine 90 : In which account every division hath ( as to its length on the rule ) a proportional decrease . Secondly , it s numbred back again from 45 to 89 , in which account every division hath ( as to its length on the line ) a proportional encrease . So that the tangent 60 you must imagine the whole length of the Rule ; and so much more as the distance from 45 unto 30 or 60 is . This well observed , all proportions in tangents are wrought after the same manner of extending the Compasses from the first term to the second , and that distance set in the third , gives the fourth , as was for sines and numbers . But for the remedying of excursions , sith the line is no more than half the length , we must imagine it . I shall lay down these Cases . CASE 1. When the fourth term is a tangent exceeding 45 , or the Radius . Ex. gr . As the tangent 10 is to the tangent 30 , so is the tangent 20 , to what ? Extending the Compasses from 10 on the line of tangents to 30 , with that distance I set one point in 20 , and finde the other point reach beyond 45 , which tells me the fourth term exceeds 45 , or the radius ; wherefore with the former extent , I set one point in 45 ; and turning the other toward the beginning of the line , I mark where it toucheth , and from thence taking the distance to the third term , I have the excess of the fourth term above 45 in my compass : wherefore with this last distance setting one point in 45 , I turn the other upon the line , and it reacheth to 50 , the tangent sought . CASE 2. When the first term is a tangent exceeding 45 , or the Radius . Ex. gr . As the tangent 50 is to the tangent 20 , so is the tangent 30 , to what ? Because the second term is less than the first , I know the fourth must be less than the third . All the difficulty is to get the true extent from the tangent 50 to 20. To do this , take the distance from 45 to 50 , and setting one point in 20 , the second term , turn the other toward the beginning of the line , marking where it toucheth , extend the Compasses from the point where it toucheth to 45 , and you will have the same distance in your Compasses as from 50 to 20 , if the line had been continued at length unto 89 tangents , with this distance , set one point in 30 the third term , and turn the other toward the beginning ( because you know the fourth must be less ) and it gives 10 the tangent sought . CASE . 3. When the third term is a Tangent exceeding 45 , or the Radius . As the tangent 40 is to the tangent 12 , 40 min. so is the tangent 65 , to what ? Extend the Compasses from 40 to 12 d. 40 min. with distance , setting one point in 65. turn the other toward 45 , and you will finde it reach beyond it , which assures you the fourth term will be less than 45. Therefore lay the extent from 45 toward the beginning , and mark where it toucheth , take the distance from that point to 65 , and laying that distance from 45 toward the beginning it gives 30 , the tangent sought . These Cases are sufficient to remove all difficulties . For when the second term exceeds the Radius , you may transpose them , saying , as the first term is to the third ; so is the second to the fourth , and then it s wrought by the third Case . I suppose it needless to adde any thing about working proportions by sines and tangents conjunctly , sith , enough hath been already said of both of them apart , in these two last Problems ; and the work is the same when they are intermixed . Only some proportions I shall adjoyn , and leave to the practice of the young beginner , with the directions in the former Cases . PROBL. 7. To finde the Suns ascensional difference in any Latitude . As the co-tangent of the latitude is to the tangent of the Suns declination , so is the radius to the sine of the ascensional difference . PROBL. 8. To finde at what hour the Sun will be East , or West in any Latitude . As the tangent of the latitude is to the tangent of the Suns declination , so is the radius to the cosine of the hour from noon . PROBL. 9. The Latitude , Declination of the Sun , and his Azimuth from South , given to finde the Suns Altitude at that Azimuth . As the radius to the cosine of the Azimuth from south , so is the co-tangent of the latitude , to the tangent of the Suns altitude in the equator at the Azimuth given . Again , As the sine of the latitude is to the sine of the Suns declination , so is the cosine of the Suns altitude in the equator ( at the same Azimuth from East or West ) to a fourth ark . When the Azimuth is under 90 , and the latitude and declination is under the same pole , adde this fourth ark to the altitude in the equator . In Azimuths exceeding 90 , when the latitude and declination is under the same pole , take the equator altitude out of the fourth ark . Lastly , when the latitude and declination respect different poles , take the fourth ark out of the equator altitude , and you have the altitude sought . PROBL. 10. The Azimuth , Altitude , and Declination of the Sun , given to finde the hour . As the cosine of declination is to the sine of the Suns Azimuth , so is the cosine of the altitude to the fine of the hour from the Meridian . Proportions may be varied eight several wayes in this manner following . 1. As the first term is to the second , so is the third to the fourth . 2. As the second term is to the first , so is the fourth to the third . 3. As the third term is to the first , so is the fourth to the second . 4. As the fourth term is to the second , so is the third to the first . 5. As the second term is to the fourth , so is the first to the third . 6. As the first term is to the third , so is the second to the fourth . 7. As the third term is to the fourth , so is the first to the second . 8. As the fourth term is to the third , so is the second to the first . By thesse any one may vary the former proportions , and make the Problems three times the number here inserted . Ex. gr . To finde the ascensional difference in Problem 10 , of this Chapter , which runs thus . As the co-tangent of the latitude is to the tangent of the Suns declination , so is the radius to the sine of ascensional difference . Then by the third variety you may make another Problem , viz. As the radius is to the co-tangent of the latitude , so is the sine of the Suns ascensional difference to the sine of his declination . Again , by the fourth variety you may make a third Problem , thus , As the sine of the Suns ascensional difference is to the tangent of the Suns declination , so is the radius to the co-tangent of the latitude . By this Artifice many have stuffed their Books with bundles of Problems . CHAP. VII . Some uses of the Lines of Circumference , Diameter , Square Equal , and Square Inscribed . ALL these are lines of equal parts , bearing such proportion to each other , as the things signified by their names . Their use is this , Any one of them given in inches or feet , &c. to finde how much any of the other three are in the same measure . Suppose I have the circumference of a Circle , Tree , or Cylinder given in inches , I take the same number of parts ( as the Circle is inches ) from the line of circumference , and applying that distance to the respective lines , I have immediately the square equal , square inscribed , and diameter , in inches , and the like , if any of those were given to finde the circumference . This needs no example . The conveniency of this line any one may experiment in standing timber ; for taking the girth , or circumference with a line , finde the diameter ; from that diameter abate twice the thickness of the bark , and you have the true diameter , when it s barked , and by Chap. 9. Probl. 5. you will guess very near at the quantity of timber in any standing Tree . CHAP. VIII . To measure any kinde of Superficies , as Board , Glass , Pavement , Walnscot , Hangings , Walling , Slating , or Tyling , by the line of Numbers on the outward ledge . THe way of accounting any number upon , or working proportions by , the line of numbers , is sufficiently shewn already , Chap. 6. which I shall not here repeat , only propose the proportions for these Problems , and refer you to those directions . PROBL. 1. The breadth of a Board given in inches , to finde how many inches in length make a foot at that breadth , say , As the breadth in inches is to 12 , so is 12 to the length in inches for a foot at that breadth . Ex. gr . At 8 inches breadth you must have 18 inches in length for a foot . PROBL. 2. The breadth and length of a Board given to finde the content . As 12 is to the length in feet and inches , so is the breadth in inches to the content in feet . Ex. gr . at 15 inches breadth , and 20 foot length , you have 25 foot of Board . PROBL. 3. A speedy way to measure any quantity of Board . The two former Problems are sufficient to measure small parcels of Board . When you have occasion to measure greater quantities , as 100 foot , or more , lay all the boards of one length together , and when the length of the boards exceeds 12 foot , use this proportion . As the length in feet and inches is to 12 , so is 100 to the breadth in inches for an 100 foot . Ex. gr . At 30 foot in length 40 inches in breadth , make an 100 foot of board ( reckoning five score to the hundred . ) This found with a rule or line , measure 40 inches at both ends in breadth , and you have 100 foot . When one end is broader than another , you may take the breadth of the over-plus of 100 foot at both ends , and taking half that sum for the true breadth of the over-plus by Probl. 2. finde the content thereof . When your boards are under 12 foot in length , say , As the length in feet and inches is to 12 , so is 50 unto the breadth in inches for 50 foot of board , and then you need only double that breadth to measure 100 foot as before . In like manner you may measure two , three , four , five , a hundred , &c. foot of board speedily , as your occasion requires . PROBL. 4. To measure Wainscot , Hangings , Plaister , &c. These are usually computed by the yard , and then the proportion is . As nine to the length in feet , and inches , so is the breadth or depth in feet and inches to the content in yards , Ex. gr . At 18 foot in length , and two foot in breadth , you have four yards . PROBL. 5. To measure Masons , or Slaters Work , as Walling , Tyling , &c. The common account of these is by the rood , which is eighteen foot square , that is 324 square foot in one rood , and then the proportion is . As 324 to the length in feet , so is the breadth in feet to the content in roods . Ex. gr . At 30 foot in length , and 15 foot in breadth , you have 1 rood 3 / 10 and better , or one rood 126 / 124 parts of a rood . CHAP. IX . The mensuration of Solids , as Timber , Stone , &c. by the lines on the proportional side of the loose piece . THese two lines meeting upon one line in the midst betwixt them ( for distinction sake ) I call one the right , the other the left line , which are known by the hand they stand toward when you hold up the piece in the right way to read the Figures . The right line hath two figured partitions . The first partition is from 3 , at the beginning to the letters Sq. every figured division representing an inch , and each subdivision quarters of an inch . The next partition is from the letters Sq. unto 12 , at the end , every figured division signifying a foot , and each sub-division the inches in a foot . The letters T R , and T D , are for the circumference and Diameter in the next measuring of Cylinders . The letters R and D. for the measuring of Timber , according to the vulgar allowance , when the fourth part of the girt is taken , &c. The letters A and W are the gauger points for Ale and Wine Measures . Lastly , the figures 12 'twixt D and T D. are for an use , expressed Probl. 2. The left line also hath two figured partitions , proceeding first from 1 at the beginning to one foot , or 12 inches , each whereof is sub-divided into quarters . From thence again to 100 , each whereof to 10 foot , is sub-divided into inches , &c. and every foot is figured . But from 10 foot to 100 , only every tenth foot is figured ; the sub-divisions representing feet . The method of working proportions by these lines ( only observing the sides ) is the same as by the line of Numbers , viz. extending from the first to the second , &c. PROBL. 1. To reduce Timber of unequal breadth and depth to a true Square . As the breadth on the left is to the breadth on the right , so is the depth on the left , to the square on the right line . At 7 inches breadth , and 18 inches depth , you have 11 inches ¼ and better for the true square . PROBL. 2. The square of a Piece of Timber given in Inches , or Feet , and Inches to finde how much in length makes a Foot. As the square in feet and inches on the right is to one foot on the left , so is the point Sq. on the right to the number of feet and inches on the left for a foot square of Timber . At 18 inches square , 5 inches ¼ and almost half a quarter in length makes a foot . When your Timber ( if it be proper to call such pieces by that name ) is under 3 inches square , account the figured divisions on the right line from the letters Sq. to the end , for inches , and each sub-division twelve parts of an inch . So that every three of them makes a quarter of an inch . Then the proportion is , as the inches and quarters square on the right is to 100 on the left , so is the point 12 'twixt D , and T D , on the right , to the number of feet in length on the left to make a foot of Timber . As 2 inches ½ square you must have 23 foot 6 inches , and somewhat better for the length of a foot of Timber . PROBL. 3. The square and length of a plece of Timber given to finde the content . As the point Sq. on the right is to the length in feet and inches on the left , so is the square in feet and inches on the right to the content in feet on the left . At 30 foot in length , and 15 inches square , you have 46 foot ½ of Timber . At 20 foot in length , and 11 inches square you have 16 foot , and almost ¼ of Timber . When you have a great piece of Timber exceeding 100 foot ( which you may easily see by the excursion upon the rule ) then take the true square , and half the length , sinde the content thereof by the former proportion , and doubling that content , you have the whole content . PROBL. 4. The Circumference , or girth , of a round piece of Timber , being given , together with the length , to finde the content . As the point R. on the right , is to the length in feet and inches on the left , so is the circumference in feet and inches on the right , to the content in feet on the left . At 20 foot in length , and 7 foot in girth , you have 60 foot of Timber for the content . This is after the common allowance for the waste in squaring ; and although some are pleased to quarrel with the allowancer , as wronging the seller , and giving the quantity less , than in truth it is ; yet I presume when they buy it themselves , they scarcely judge those Chips worth the hewing , and have as low thoughts of the over-plus , as others have of that their admonition . If it be a Cylinder that you would take exact content of , then say , As the point T R , on the right , is to the length on the left , so is the girt on the right to the exact content on the left . At 15 foot in length , and 7 foot in girt , you have 59 foot of solid measure . The Diameter of any Cylinder given , you may by the same proportion finde the content , placing the point D , instead of R , in the proportion for the usual allowance , and the point T D , for the exact compute . PROBL. 5. To measure tapered Timber . Take the square or girt at both ends , and note the sum and difference of them . Then for round Timber , as the point R. on the right , is to the length on the left , so is half the sum of the girt at both ends on the right , to a number of feet on the left . Keep this number , and say again , As the point R. on the right , is to the third part of the former length on the left , so is half the difference of the girts on the right , to a number of feet on the left ; which number added to the former , gives the true content . The same way you may use for square Timber , only setting the feet and inches square instead of the girt , and the point Sq. instead of the point R. At 30 foot in length , 7 foot at one end , and 5 at the other in girt , half the sum of the girts is 6 foot , or 72 inches ; the first number of feet found 67 , half the difference of the girts is 1 foot , or 12 inches , the third part of the length 10 foot ; then the second number found will be 7 foot , one quarter and half a quarter . The sum of both ( or true content ) 74 foot , one quarter , and half a quarter . For standing Timber , take the girt about a yard from the bottom , and at 5 foot from the bottom , by Chap. 7 , set down these two diameters without the bark ; and likewise the difference 'twixt them . Again , by Chap. 6. Probl. 4. finde the altitude of the tree , so far as it bears Timber ( or as we commonly phrase it , to the collar ) this done , you may very near proportion the girt at the collar and content of the tree , before it falls . In case any make choice of the hollow contrivance mentioned , Chap. 1. they need no compasses in the mensuration of any solid ; provided the lines for solid measure , and gauging vessels , be doubly impressed ( only in a reverted order , one pair of lines proceeding from the head toward the end , and the other pair from the end toward the head ) upon the sliding cover , and its adjacent ledges . This done , the method of performing any of the Problems mentioned in this Chapter , is easie . For whereas you are before directed to extend the Compasses from the first term to the second ; and with that distance setting one point in the third term , the other point gave the fourth , or term sought . So here , observing the lines as before , slide the cover until the first term stand directly against the second ; then looking for the third on its proper line , it stands exactly against the fourth term , or term sought on the other line . Only note , that when the second term is greater than the first , it s performed by that pair of lines proceeding from the head toward the end : But when the first term is greater than the second , it is resolved by that pair of lines which is numbred from the end toward the head . CHAP. X. The Gauge Vessels , either for Wine , or Ale , Measure . PROBL. 1 . The Diameter at Head , and Diameter at Boung , given in Inches , and tenth parts of an Inch , to finde the mean Diameter in like measure . TAke the difference in inches , and tenth parts of an inch , between the two diameters . Then say by the line of numbers , As 1 is to 7 , so is the difference to a fourth number of inches , and tenth parts of an inch . This added to the Diameter at head , gives the mean Diameter . Ex. gr . At 27 inches the boung , and 19 inches two tenths at the head , the difference is 7 inches , 8 tenths . The fourth number found by the proportion will be 5 inches , 4 tenths , and one half , which added to the diameter at the head , gives 24 inches , 6 tenths , and one half tenth of an inch for the mean diameter . PROBL. 2. The length of the Vessel , and the mean Diameter given in Inches , and tenth parts of an Inch , to finde the content in Gallons , either in Wine or Ale measure . Note first , that the point A. on the right is the gauge point for Ale measure , and the point W. on the right , is the gauge point for Wine measure . Then say , As the gauge point on the right to the length in inches , and tenth parts of an inch on the left , so is the mean diameter in inches and tenth parts of an inch on the right , to the content in gallons on the left . CHAP. XI . Some uses of the Lines on the inward ledge of the moveable piece . THe Line of inches and foot measure do by inspection ( only ) reduce either of the measures into the other . Ex. gr . Three on the line of inches stands directly against 25 cents of foot measure . Or 75 cents of foot measure is directly against 9 on the inches . Another use of these lines ( welcome perhaps to them that delight in instrumental computations ) is to know the price of carriage for any quantity , &c. by inspection . For this purpose , the line of inches represents the price of a pound , every inch being a penny , and every quarter a farthing . The prickt line is the price of an hundred pound at five score and twelve to the hundred , every division signifying a shilling . The line of foot measure is the price of an hundred pound at five score to the hundred , every division standing for a shilling . Ex. gr . At 3 pence the pound on the inches , is 28 shillings on the prickt line , and 25 shillings on the foot measure , for the price of an hundred , the like for the converse . Otherwise , the price of a pound being given , the rate of an hundred is readily computed without the rule . For considering the number of farthings is in the price of a pound , twice that number of shillings , and once that number of pence , is the price of an hundred , reckoning five score to the hundred ; of twice that number of shillings , and once that number of groats , is the price of an hundred , at five score and twelve to the hundred . Ex. gr . At three half pence the pound , the number of farthings is six . Therefore , twice six shillings , and once six pence ( that is 125. 6d . ) is the price of an hundred , at five score to the hundred . Again , twice six shillings , and once six groats ( that is 14s . ) is the price of an hundred , at five score and twelve to the hundred . But of this enough , if not too much already . CHAP. XII . To divide a plot of Ground into any proposed quantities . See Fig. 4 SUppose A B C D E F G H I K L , a plot of ground , containing 54 acr . 2 roods , 28 poles , from the point A , I am required to shut off 18 acres , next the side B C. Draw A D. and measure the figure A B C D , which is 14 acr : 2 r. 3 p. that is 3 acr . 1. r. 37 p. or 557 poles too little . Draw again A F , and measure A D F , which is 1309. poles . Then by Chap. 5. Probl. 1. entring D F , the base of the triangle in 1309 on the lines of lines , and taking out 557 , set it from D to E , and draw A E. So have you the figure A B C D E 18 acres . Again , from the point G , I would set off 20 acres next the side A E , draw A G , and measure A E F G 15 acr . or . 2 pol. whereof want 4 acr . or . 38 pol. that 678 poles , then draw G K , and measure G A K 1113 poles . Lastly , by Chap. 5. Probl. I. enter A K , the base of the triangle in 1113 upon the lines of lines , and taking out 678 , set it from A , and draw G L. So have you the figure A E F G L , twenty acres . How ready the instrument would be for surveying with the help of a staff , Ballsocket , and Needle , is obvious to any one that considers its graduated into 180 degrees . CHAP. XIII . So much of Geography as concerns finding the distance of any two places upon the Terrestrial Globe . HEre are three Cases , and each of those contains the same number of propositions . CASE 1. When the two places differ in latitude only . PROP. 1. When one place lies under the Equator , having no Latitude . The latitude of the other place turned into miles , ( reckoning 60 miles , the usual compute , for a degree ) is the distance sought . PROP. 2. When both places have the same pole elevated , viz. North , or South . Take the difference of their Latitudes , and reckoning 60 miles for a degree ( as before ) you have their distance . PROP. 3. When the two places have different poles elevated , viz. one North , the other South . Adde two latitudes together , and that sum turned into miles is the distance . CASE 2. When the two places differ in Longitude only . PROP. 1. When neither of them have any Latitude , but lie both under the Equator . Their difference of Longitude turned into miles ( as before ) is their distance . PROP. 2. When the two places have the same Pole elevated . The proportion is thus . As the Radius is to the number 60 , so is the cosine of the common latitude to the number of miles for one degree of longitude . Multiply this number found by the difference of longitude , and that product is the distance in miles . PROP. 3. When the two places have different poles elevated . As the Radius is to the cosine of the common latitude , so is the sine of half the difference of longitude to the sine of half the distance . Wherefore this sine of half the distance doubled and turned into miles , is the true distance . CASE 3. When the two places differ in Longitude and Latitude both . PROP. 1. When one of the places lies under the equator , having no Latitude . As the radius is to the cosine of the difference of longitude , so is the cosine of the latitude to the cosine of the distance . PROP. 2. When both the places have the same pole elevated . As the radius is to the cosine of the difference in longitude , so is the co-tangent of the lesser altitude , to the tangent of a fourth ark . Subtract this fourth ark out of the complement of the lesser latitude , and keep the remain . Then , As the cosine of the fourth ark is to the cosine of the remain , so is the sine of the lesser latitude to the cosine of the distance . PROP. 3. When the two places have different poles elevated . As the radius is to the cosine of the difference in longitude , so is the co-tangent of either latitude to the tangent of a fourth ark . Subtract the fourth out of the latitude not taken into the former proportion , and note the difference . Then , As the cosine of the fourth ark is to the cosine of this difference , so is the sine of the latitude first taken , to the cosine of the distance . CHAP. XIV . Some uses of the Instrument in Navigation , or plain Salling . HEre it will be necessary to premise the explication of some terms , and adjoyn two previous proportions . 1. The Compass being a circle , divided into 32 equal parts , called rumbs ; one point or rumb is 11 d. 15 min. of a circle from the meridian : two points or rumbs is 22 d. 30 min. &c. of the rest . 2. The angle which the needle , or point of the compass under the needle , makes with the meridian , or North and South line is called the course or rumb ; but the angle which it makes with the East , and West line , or any parallel , is named the complement of the course or rumb . 3. The departure is the longitude of that Port from which you set sail . 4. The distance run , is the number of miles , or leagues ( turned into degrees ) that you have sailed . 5. When you are in North latitude , and sail North-ward , adde the difference of latitude to the latitude you sailed from ; and when you are in North latitude , and sail Southward , subtract the difference of latitude from the latitude you sailed , and you have the latitude you are in . The same rule is to be observed in South latitude . 6. To finde how many miles answer to one degree of longitude in any latitude . As the radius is to the number 60 , so is the cosine of the latitude to the number of miles for one degree . 7. To finde how many miles answer to one degree of latitude on any rumb . As the cosine of the rumb from the Meridian , is to the number 60 , so is the radius to the number of miles . The most material questions in Navigation are these four . First , To finde the course . Secondly , The distance run . Thirdly , The difference of latitude . Fourthly , The difference in longitude ; and any two of these being given , the other two are readily found by the Square and Index . These two additional rulers were omitted in the first Chapter of this Treatise ; sith they are only for Navigation , and large Instruments of two , or three foot in length , which made me judge their description most proper for this place : because , such as intend the Instrument for a pocket companion , will have no use of them . The square is a flat rule , having a piece , or plate fastened to the head , that it may slide square , or perpendicular to the outward ledge of the fixed piece . It hath the same line next either edge on the upper side , which is a line of equal parts , an hundred , wherein is equal to the radius of the degrees on the outward limb of the moveable piece . The Index is a thin brass rule on one side , having the same scale as the square . On the other side is a double line of tangents , that next the left edge , being to a smaller ; that next the right edge , to a larger radius . For the use of these rulers , you must have a line of equal parts adjoyning to the line of sines on the fixed piece , divided into 10 parts , stamped with figures , each of those divided into 10 parts more ; so that the whole line is divided into 100 parts , representing degrees . Lastly , let each of those degrees be sub-divided into as many parts , as the largeness of your scale will permit , for computing the minutes of a degree . The Index is to move upon the pin on the fixed piece ( where you hang the thread for taking altitudes ) and that side of the Index ( in any of the four former questions ) must be upward , which hath the scale of equal parts . The square is to be slided along the outward ledge of the fixed piece . Then the general rules are these . The difference of latitude is accounted on the line of equal parts , adjoyning to the sines on the fixed piece . The difference of longitude is numbred on the square . The distance run is reckoned upon the Index . The course is computed upon the degrees on the limb from the head toward the end of the moveable piece . But when any would work these Problems in proportions , let them note , The distance run , difference of longitude , and difference of latitude , are all accounted on the line of numbers ; the rumb or course is either a sine or tangent . This premised . I shall first shew how to resolve any Problem by the square and Index ; and next adjoyn the proportions for the use of such as have only pocket Instruments . PROBL. 1. The course and distance run given , to finde the difference of latitude , and difference of longitude . Apply the Index to the course reckoned on the limb from the head , and slide the square along the outward ledge of the fixed piece , until the fiducial edge intersect the distance run on the fiducial edge of the Index . Then at the point of Intersection you have the difference of longitude upon the square , and on the line of equal parts on the fixed piece , the square shows the difference of latitude . The proportion is thus , As the radius is to the distance run , so is the cosine of the course to the difference of latitude . Again , As the radius is to the distance run , so is the sine of the course to the difference in longitude . PROBL. 2. The course and difference of latitude given to find the distance run , and difference in longitude . Slide the square to the difference of latitude on the line of equal parts upon the fixed piece , and set the Index to the course on the limb . Then at the point of intersection of the square and index , on the square is the difference of longitude , on the index the distance run . The proportion is thus . As the cosine of the course is to the difference of latitude , so is the radius to the distance run . Again , As the radius is to the sine of the course , so is the distance run to the difference of longitude . PROBL. 3. The course and difference in longitude given , to finde the distance run , and difference of latitude . Apply the Index to the course on the limb , and the difference of longitude on the square to the fiducial edge of the Index . Then at the point of intersection you have distance run on the index , and upon the line of equal parts on the fixed piece , the square shows the difference of latitude . The proportion is thus . As the sine of the course is to the difference of longitude , so is the radius to the distance run . Again , As the radius is to the distance run , so is the cosine of the course to the difference of latitude . PROBL. 4. The distance run , and difference of latitude given to finde the course , and difference in longitude . Slide the square to the difference of latitude on the line of equal parts on the fixed piece , and move the index until the distance run numbred thereon , intersect the fiducial edge of the square ; then at the point of intersection you have the difference of longitude on the square , and the fiducial edge of the Index on the limb shows the course . As the distance run is to the difference of latitude , so is the radius to the cosine of the course . Again , As the radius is to the distance run , so is the sine of the course to the difference of longitude . PROBL. 5. The distance run , and difference of longitude given to finde the course , and difference of latitude . Apply the distance run numbred on the fiducial edge of the index , to the difference of longitude , reckoned on the fiducial edge of the square . Then on the line of equal parts upon the fixed piece , the square shows the difference of latitude , and the index shows the course on the limb . The proportion is thus . As the distance run is to the difference of longitude , so is the radius to the sine of the course . Again , As the radius is to the distance run , so is the cosine of the course , to the difference of latitude . PROBL. 6. The difference of latitude , and difference of longitude given , to finde the course , and distance run . Apply the square to the difference of latitude on the scale of equal parts upon the fixed piece , and move the index until its fiducial edge intersect the difference of longitude , reckoned on the square . Then at the point of intersection you have the distance run upon the index , and the fiducial edge of the index upon the limb shows the course . The proportion is thus . As the difference of latitude is to the difference of longitude , so is the radius to the tangent of the course . Again , As the sine of the course is to the radius , so is the difference of longitude to the distance run . PROBL. 7. Sailing by the Ark of a great Circle . For this purpose the tangent lines on the index will be a ready help , using the lesser for small , and the greater tangent line , for great latitudes . The way is thus , Account the pont 60 / 0 , on the outward limb of the moveable piece to be the point , or port of your departure ; thereto lay the fiducial edge of the index , and reckoning the latitude of the Port you departed from upon the index , strike a pin directly touching it , into the table your instrument lies upon . This pin shall represent the Port of your departure . Therefore hanging a thread , or hair , on the center , whereon the index moves ; and winding it about this pin . Count the difference of longitude 'twixt the port of your departure , and the Port you sail toward , from 60 / 0 on the moveable piece toward 0 / 60 , on the loose piece ; and thereto laying the same fiducial edge of the index , reckon the latitude of this last Port upon the index , directly touching of it , strike down another pin upon the table , and draw the thread strait about this pin fastening it thereto . This done , the thread betwixt the two pins represents the ark of your great Circle ; and laying the fiducial edge of the index to any degree of difference of longitude accounted from 60 / 0 on the moveable piece , the thread shows upon the index what latitude you are in , and how much you have raised , or depressed the pole since your departure . On the contrary , laying the latitude you are in ( numbred upon the index ) to the thread , the index shows the difference of longitude upon the limb ; counting from 60 / 0 on the moveable piece . So , that were it possible to sail exactly by the ark of a great Circle , it would be no difficulty to determine the longitude in any latitude you make . But I intend not a treatise of Navigation ; wherefore let it suffice , that I have already shown how the most material Problems therein , may easily , speedily , and ( if the instrument be large ) exactly , be performed by the instrument without the trouble of Calculation , or Projection . CHAP. XV. The Projection and Solution of the sixteen Cases in right angled Spherical Triangles by five Cases . See Fig. 5. THe fundamental Circle N B Z C , is alwayes supposed ready drawn , and crossed into quadrants , and the diameters produced beyond the Circle . CASE 1. Given both the sides Z D , and D R , to project the Triangle . By a line of chords , prick off Z D , upon the limb , and draw the diameter D A E. Again , by a line of tangents , set half co-tangent D R , upon A D , from A to R , then have you three points , viz. N R Z , to draw that ark , and make up the triangle . The center of which ark always lies on AC , ( produced beyond C , if need requires ) and is found by the intersection of the two arks made from R , and Z. CASE 2. Given one side Z D , and the hypothenuse Z R , to project the Triangle . Prick off Z D , and draw D A E , by Case 1. Again , set half the co-tangent Z R , on the line A Z , from A to F , and the tangent Z R , set from F to P , with the extent F P , upon the center P , draw the ark V F I , and where it intersects the diameter D A E , set R ; then have you three points N R Z , to draw that ark , as in the former Case . CASE 3. Given the Hypothenuse Z R , and the Angle D Z R , to project the Triangle . Prick half the co-tangent D Z R , from A to S , and the secant D Z R , from S to T , upon the center T , with the extent T S , draw the ark N S Z. Again , by Case 2 , draw the ark H F I , where these two arks intersect each other , set R. Lastly , lay a ruler to A R , and draw the diameter DRAE , and your triangle is made . CASE 4. Given one side Z D , and its adjacent Angle D Z R , to project the triangle . Prick off Z D , and draw D A E , by Case 1. Again , by Case 3 , draw the ark N S Z , where this ark intersects the diameter DAE , set R , and your triangle is made . Or , Given the side D R , and its opposite angle D Z R , you may project the triangle . Draw the ark N S Z , as before . Again , take half the co-tangent D R , and with that extent upon A , the center , cross the ark N S Z , setting R , at the point of intersection . Lastly , lay a ruler to A R , and draw the diameter D R A E , which makes up your triangle . The triangle projected in any of the four former Cases , to measure any of the sides , or angles , do thus . First , the 〈◊〉 Z D , is found by applying it to a line of chords . Secondly , R A , applyed to a line of tangents , is half the co-tangent D R. Thirdly , A S , applyed to a line of tangents is the co-tangent D Z R. Fourthly , set half the tangent D Z R , from A to L , then is L the pole point , and laying a ruler to L R , it cuts the limb at V , and the ark Z V , upon the line of chords , gives the hypothenuse Z R. Fifthly , prick Z D from C to K , a ruler laid from R to L , cuts the limb at G , then G K , upon a line of chords , is the quantity of Z R D. CASE . 5. The two oblique angles , D Z R , and Z R D given , to present the Triangle . See Fig. 6. This is no more than turning the former triangle . Thus , Draw the ark N S Z , by Case 1 , and set half the tangent of that ark from A to L. Again , set half the co-tangent D R Z , from A to F , and the secant of D R Z , from F to L , upon the center A , with the extent A P. Draw the ark P G , and with the extent F P , from L , cross the ark P G in G. Lastly , upon the Center G , with the extent G L. Draw the ark R D F L , and your triangle is made . The triangle projected you may measure off the sides and hypothenuse . Thus , First , the hypothenuse Z R , is measured by a line of chords . Secondly , a ruler laid to L D , cuts the limb at H , and Z H , upon a line of chords , is the measure of the ark Z D. Thirdly , draw A G , and set half the tangent D R Z , from A to V , apply a ruler to V D , it cuts the limb at E , then R E , upon a line of chords , measure the ark R D. Note . The radius to all the chords , tangents , and secants , used in the projection , and measuring , any ark or angle , is the semidiameter of the fundamental circle . CHAP. XVI . The projection and solution of the 12 Cases in oblique angled spherical triangles in six Cases . See Fig. 7. THe fundamental circle N H Z M , is alwayes supposed ready drawn , and crossed into Quadrants , and the Diameters produced beyond the Circle . CASE 1. The three sides , Z P , P Z , and Z S , given , to project the Triangle . By a line of chords prick off Z P , and draw the diameter P C T , crossing it at right angles in the center with AE C E , set half the co-tangent P S , from C to G , and he secant P S from C to R , upon the center R , with the extent R G , draw the the ark FGL . Again , set half the co-tangent Z S , from C to D , and the tangent Z S , from D to O , with the extent O D , upon the center O , draw the ark B D P , mark where these two arks intersect each other as at S. Then have you three points T S P , to draw that ark , and the three points N S P , to draw that ark , which make up your triangle . CASE 2. Given two sides Z S , and Z P , with the comprehended Angle P Z S , to project the Triangle . Prick off Z P , and draw PCT , and AECE , and the ark B D P , by Case 1. Again , set the tangent of half the excess of the angle P Z S above 90 , from C to W , and co-secant of that excess from W to K , upon the center K , with the extent K W ; draw the ark N W Z , which cuts the ark B D P in S. Then have you the three points T S P , to draw that ark which makes up the triangle . CASE 3. Two Angles S Z P , and Z P S , with the comprehended side Z P , given , to project the Triangle . Prick off Z P , and draw the lines P C T , and AE C E , by Case 1 , and the angle NWZ . by Case 2. Lastly , set half the co-tangent ZPS from C to X , and the secant Z P S , from X to V , upon the center V , with the extent V X , draw the ark T X S P , and the triangle is made . CASE 4. Two sides , ZP , and PS , with the Angle opposite to one of them SZP , given , to project the Triangle . Prick off ZP , and draw PCT , and AECE , by Case 1. and the angle SZP by Case 2. Lastly , by Case 1. draw the ark FGL , and mark where it intersects NWZ , as at S , then have you the three points TSP , to draw that ark , and make up the triangle . CASE 5. Two Angles SZP , and ZPS , with the side opposite to one of them ZS , given , to project the Triangle . Draw the ark BDP , by Case 1. and the ark NWZ , by Case 2. at the intersection of these two arks , set S , with the tangent of the angle ZPS , upon the center C. sweep the ark VΔI . Again , with the secant of the ark ZPS upon the center S , cross the ark VΔI , as at the points V and I. Then in case the hypothenuse is less than a quadrant ( as here ) the point V , is the center , and with the extent VS , draw the ark TSP , which makes up the triangle . But in case the hypothenuse is equal to a quadrant , Δ , is the center ; if more than a quadrant , I , is the center ; in which cases the extent from Δ , or I , to S , is the semidiameter of the ark TSP . CASE 6. Three Angles ZPS , and PZS , and ZSP , given , to project the Triangle . See Fig. 7. and 8. The angles of any spherical triangle may be converted into their opposite sides , by taking the complement of the greatest angle to a Semicircle for the hypothenuse , or greatest side . Wherefore by Case 1. make the side ZP , in Fig. 4. equal to the angle ZSP , in Fig. 3 , and the side ZS , in Fig. 4. equal to the angle ZPS , in Fig. 3. and the side PS , in Fig. 4. equal to the complement of the angle PZS , to a Semicircle in Fig. 3. Then is your triangle projected where the angle ZPS in Fig. 4. is the side ZS , Fig. 3. Again , the angle ZSP , Fig. 4. is the side ZP , in Fig. 3. Lastly , the complement of the angle PZS to a Semicircle in Fig. 4. is the measure of the hypothenuse , or side P S , in Fig. 3. The Triangle being in any of the former Cases projected , the quantity of any side or angle may be measured by the following rules . First , The side Z P , is found by applying it to a line of chords . Secondly , CX , applyed to a line of tangents , is half the co-tangent of the angle ZPS . Thirdly , CW applyed to a line of tangents , is half the co-tangent of the excess of the angle SZP , above 90. Fourthly , set half the tangent of the angle ZPS , from C , to Π , a ruler laid to ΠS , cuts the limb at F ; then PF , applyed to a line of chords , gives the side PS . Fifthly , take the complement of the angle PZS , to a Semicircle , and set half the tangent of that complement from C , to λ , a ruler laid to λS , cuts the limb at B , and ZB , applyed to a line of chords , gives the side ZS . Sixthly , a ruler laid to Sλ , cuts the limb at L. Again , a ruler laid to SΠ , cuts the limb at φ , and L φ , applyed to a line of chords , gives the angle ZSP . The end of the first Book . An Appendix to the first Book . THe sights which are necessary for taking any Altitude . Angle , or distance ( without the help of Thread or Plummet ) are only three , viz. one turning sight , and two other sights , contrived with chops , so that they may slide by the inward or outward graduated limbs . The turning sight hath only two places , either the center at the head , or the center at the beginning of the line of sines on the fixed piece ; to either of which ( as occasion requires ) it s fastened with a sorne . The center at the head serving for the graduations next the inward limb of the loose piece . And the center at the beginning of the line of sines serving for all the graduations next the outward limb of the moveable and loose piece both . Yet because it is requisite to have pins to keep the loose piece close in its place . You may have two sights more to supply their place ( which sometimes you may make use of ) and so the number of sights may be five , viz. two sliding sights , one turning sight , and two pin sights , to put into the holes at the end of the fixed and moveable piece , to hold the tenons of the loose piece close joynted . Every one of these sights hath a fiducial ( or perpendicular ) line , drawn down the middle of them , from the top to the bottom , where this line toucheth the graduations on the limb , is the point of observation . The places of these sights have an oval proportion , about the middle of them , only leaving a small bar of brass , to conduct the fiducial line down the oval cavity , and support a little brass knot ( with a sight hole in it ) in the middle of that bar , which is ever the point to be looked at . There are two wayes of observing an altitude with help of these sights . The one when we turn our face toward the object . This is called a forward observation in which you must alwayes set the turning sight next your eye . This way of observation will not exactly give an altitude above 45 degrees . The other way of observing an altitude is peculiar to the Sun in a bright day , when we turn our back toward the Sun. This is termed a backward observation ; wherein You must have one of the sliding sights next Your eye , and the turning sight toward the Horizon . This serves to take the Suns altitude without thread , or plummet , when it is near the Zenith . PROBL. 1. To finde the Suns altitude by a forward observation . Serve the turning sight to the center of those graduations you please to make use of ( whether on the inward or outward limb ) and place the two sliding sights upon the respective limb to that center ; this done , look by the knot of the turning sight ( moving the instrument upward or downward ) until you see the knot of one of the sliding sights directly against the Sun , then move the other sliding sight , until the knot of the turning sight , and the knot of this other sliding sight be against the horizon ; then the degrees intercepted 'twixt the fiducial lines of the sliding sights on the limb , shew the altitude required . PROBL. 2. To finde the distance of any two Stars , &c. by a forward observation . Serve the turning sight to either center , and apply the two sliding sights to the respective limb ( holding the instrument with the proportional side downward ) and applying the turning sight to your eye , so move the two sliding sights either nearer together , or further asunder , that you may by the knot of the turning sight see both objects even with the knots of their respective sliding sights , then will the degrees intercepted 'twixt the fiducial lines of the object sights on the limb show the true distance . By this means you may take any angle for surveying , &c. PROBL. 3. To finde the Suns Altitude by a backward Observation . Serve the turning sight to the center at the beginning of the line of sines , and apply one of the sliding sights to the outward limb of the loose piece , and the other to the outward limb of the moveable piece ; and turning your back toward the Sun , set the sliding sight upon the moveable piece next your eye ; and slide it upward or downward toward the end , or head , until you see the shadow of the little bar , or edge , of the sight on the loose piece fall directly on the little bar on the turning sight ; and at the same time the bar of the sight next your eye , and the bar of the turning sight to be in a direct line with the Horizon . Then will the degrees on the limb intercepted 'twixt the fiducial lines of the sliding sights ( if you took the shadow of the bar ) or 'twixt the fiducial line of the sliding sight next your eye , and the edge of the other sliding sight ( when you took the shadow of the edge ) be the true altitude required . ΣΚΙΟΓΡΑΦΙΑ , OR , The Art of Dyalling for any plain Superficies . LIB . II. CHAP. I. The distinction of Plains , with Rules for knowing of them . ALL plain Superficies are either horizontal , or such as make Angles with the Horizon . Horizontal plains are those , that lie upon an exact level , or flat . Plains , that make Angles with the Horizon are of three sorts . 1. Such as make right angles with the Horizon , generally known by the name of erect , or upright plains . 2. Such as make acute angles with the horizon , or have their upper edge leaning toward you , usually termed inclining plains . 3. Such as make obtuse angles with the horizon , or have their upper edge falling from you , commonly called reclining plains . All these three sorts are either direct , viz. East , West , North , South . Or else Declining From South , toward East , or West . From North , toward East , or West . All plain Superficies whatsoever are comprized under one of these terms . But before we treat of the affections , or delineation of Dials for them ; it will be requisire to acquaint you with the nature of any plain , which may be found by the following Problems . PROBL. 1. To finde the reclination of any Plain . Apply the outward ledge of the moveable piece to the Plain with the head upward , and reckoning what number of degrees the thread cuts on the limb ( beginning your account at 30. on the loose piece , and continuing it toward 60 / 0 on the moveable piece ) you have the angle of reclination . If the thread falls directly on 60 / 0 upon the moveable piece , it s an horizontal ; if on 30. on the loose piece , it s an erect plain . PROBL. 2. To finde the inclination of any Plain . Apply the outward ledge of the fixed piece to the plain , with the head upward , and what number of degrees the thread cuts upon the limb of the loose piece , is the complement of the plains inclination . PROBL. 3. To draw an Horizontal Line upon any Plain . Apply the proportional side of the Instrument to the plain , and move the ends of the fixed piece upward , or downward , until the thread falls directly on 60 / 0. upon the loose piece ; then drawing a line by the outward ledge of the fixed piece , its horizontal , or paralel to the horizon . PROBL. 4. To draw a perpendicular Line upon any Plain . When the Sun shineth hold up a thread with a plummet against the plain , and make two points at any distance in the shadow of the thread upon the plain , lay a ruler to these points , and the line you draw is a perpendicular . PROBL. 5. To finde the declination of any Plain . Apply the outward ledge of the fixed piece to the horizontal line of your plain , holding your instrument paralel to the horizon . This done , lift up the thread and plummer , until the shadow of the thread fall directly upon the pin hole on the fixed piece ( where you hang the thred to take altitudes ) Then observe how many degrees the shadow of the thread cuts in the limb , either from the right hand , or from the left hand 0 / 60. upon the loose piece ; and immediately taking the altitude of the Sun. By lib. 1. cap. 2. Probl. 9 , 10 , 11. finde the Suns Azimuth from South . And , When you make this observation in the morning , these Cases determine the declination of the plain . CASE 1. When the shadow of the thread upon the limb falls on the right hand 0 / 60 on the loose piece , take the difference of the shadow , and Azimuth ( by subtracting the lesse out of the greater ) and the residue or remain is the plains declination . From South toward East , when the Azimuth is greater than the shadow . From South toward West , when the shadow is greater than the azimuth , when the shadow and azimuth are equal , it s a direct South plain . When the difference is just 90. its a direct East , when above 90. subtract the difference from 180. and the remain is the declination from North toward East . CASE 2. When the shadow falls on the left hand 0 / 60. Adde the azimuth and shadow together , that sum is the plains declination ; from South toward East , when under 90 ; if it be just 90 , it s a direct East . If above 90. subtract it from 180. the remain is the declination from North toward East . When the sum is above 180. subtract 180 from it , and the remain is the declination from North toward West . CASE 3. When the shadow falls upon 0 / 60. the azimuth is the plains declination . When under 90 , its South-East , when equal to 90. direct East , when above 90. subtract it from 180. the remain is the declination from North toward East . If you make the observation afternoon , the following Cases will resolve you . CASE 4. When the shadow falls on the left hand 0 / 60. the difference 'twixt the shadow and azimuth is the declination ; when the shadow is more than the azimuth it declines South-East , when less , South-West . When the shadow and azimuth are equal , it s a direct South plain , when their difference is equal to 90. it s a direct West ; when the difference exceeds 90. subtract it from 180. the remain is the declination North-West . CASE 5. When the shadow falls on the right hand 0 / 60. take the sum of the shadow and azimuth , and that is the declination from South toward West , when under 90. when just 90. its a direct West plain ; when more than 90. subtract it from 180. the remain is the declination North-West ; when the sum is above 180. subtract 180 from it , and the remain is the declination from North toward East . CASE 6. When the shadow falls upon 0 / 60. the azimuth is the quantity of declination . From South toward West , when under 90. when equal to 90. its direct West ; when more than 90. subtract it from 180. The remain is the declination from North toward West . PROBL. 6. To draw a Meridional Line upon a Horizontal Plain . Draw first a circle upon the plain , and holding up a thread and plummet ( when the Sun shines ) so that the shadow of the thread may pass through the center of the circle , make a point in the circumference where the shadow intersects it . At the same time finding the Suns Azimuth from South , by a line of chords , set it upon the limb of the circle from the intersection of the shadow toward the South , and it gives the true South point . Wherefore laying a ruler to this last point and the center , the line you draw is a true Meridian . CHAP. II. The affections of all sorts of Plains . Sect. 1. The affections of an horizontal Plain . 1. THe style , or cock of every horizontal Dial , is an angle equal to that latitude for which the Dial is made . 2. The place of the style is directly upon the meridional , or twelve a clock line , and the angular point must stand in the center of the hour lines . 3. The rule for drawing the hour lines before six in the morning , is to draw the respective hour lines afternoon beyond the center ; or for the hour lines after six at evening , draw the respective hour lines in the morning beyond the center . 4. To place an horizontal Dial upon the plain ; first draw a Meridian line upon the plain by Cap. 1. Probl. 6. and lay in the line of 12. exactly thereon , with the angular point of the style toward the South , fasten the Dial upon the plain . Sect. 2. The affections of erect , direct South and North Plains . 1. The style in both these is an angle equal to the complement of the latitude for which the Dial is made to stand upon the Meridian line , or perpendicular of your plain , with the angular point in the center of the hour lines , and that point in South alwayes upward , in North alwayes downward . 2. To prick off the Dial from your paper draught upon the plain , lay the hour line of 6. and 6. upon the plains horizontal ; and applying a ruler to the center , and each hour line , transmit the hour lines from your paper draught to the plain . Sect. 3. The affections of erect , direct East and West Plains . 1. In both these the style may be a pin or plate , equal in length or heighth to the radius of the tangents , by which you draw the Dial. 2. The style in both of them is to stand directly upon the hour of six , and perpendicular to the plain . Sect. 4. The affections of erect declining Plains . 1. These are of two sorts , either such as admit of centers to the hour lines and style , or such as cannot with conveniency ( because of the lowness of the style , and nearness of the hour lines to each other ) be drawn with a center to those lines . Of this latter sort are all such plains , whose style is an angle less than 15 degrees . For where the angle of the style is more than 15 degrees ; those Dials may be drawn with a center to the hour lines . 2. In all erect declining plains with centers the Meridian is the plains perpendicular . In those that admit not of centers in their delineation , the meridian is parallel to the plains perpendicular . 3. In all declining plains the substile , or line whereon the style is to stand , must be placed on that side the meridian , which is contrary to the coast of declination ; and also in such decliners as admit of centers , the angle 'twixt 12. and 6. is to be set to the contrary coast to that of declination . 4. In all decliners , without centers , the inclination of meridians is to be set from the substyle toward the coast of declination . 5. The proportions in all erect decliners , for finding the height of the style , the distance of the substyle from the meridian , the angle of twelve and six , with the inclination of meridians ( all which may be wrought either by the canon , or exactly enough for this purpose by the instrument ) are as followeth . To finde the Styles height above the Substile . As the radius is to the cosine of the latitude , so is the cosine of declination , to the sine of the styles height . To finde the Substyles distance from the Meridian . As the radius is to the sine of declination , so is the co-tangent of the latitude to the tangent of the substyle from the Meridian . To finde the angle of twelve and six . As the co-tangent of the latitude is to the radius , so is the sine of declination , to the co-tangent of six from twelve . To finde the inclination of Meridians . As the sine of the latitude is to the radius , so is the tangent of declination to the tangent of inclination of Meridians . 6. All North decliners with centers have the angular point of the style downward , and all South have it upward . 7. All North decliners without centers have the narrowest end of the style downward , all South have it upward . 8. In all decliners without centers take so much of the style as you think convenient , but make points at its beginning and end upon the substyle of your paper draught , and transmit those points to the substyle of your plain , for direction in placing your style thereon . 9. In all North decliners the Meridian , or inclination of Meridians is the hour line of twelve at mid-night : in South decliners , at noon , or mid-day . This may tell you the true names of the hour lines . 10. In transmitting these Dials from your paper draught to your plain , lay the horizontal of your paper draught , upon the horizontal line of the plain , and prick off the hours and substyle . Sect. 5. The affections of direct reclining Plains inclining Plains . For South Recliners , and North Incliners . 1. The difference 'twixt your co-latitude , and the reclination inclination is the elevation , or height of the style . 2. When the reclination inclination exceeds your co-latitude , the contrary pole is elevated so much as the excess . Ex. gr . a North recliner , or South incliner 50. d. in lat . 52. 30. min. the excess of the reclination inclination , to your co-latitude is 12. d. 30. min. and so much the North is elevated on the recliner , and the South pole on the incliner . 3. When the reclination inclination is equal to the co-latitude , it s a polar plain . For South incliners , and North recliners . 1. The Sum of your co-latitude , and the reclination inclination is the styles elevation . 2. When the reclination inclination is equal to your latitude , it s an equinoctial plain , and the Dial is no more than a circle divided into 24. equal parts , having a wyer of any convenient length placed in the center perpendicular to the plain for the style . 3. When the reclination inclination is greater than your latitude , take the summe of the reclination inclination of your co-latitude from 180. and the residue , or remain is the styles height . But in this case the style must be set upon the plain , as if the contrary pole was elevated , viz. These North recliners must have the center of the style upward , and the South incliners have it downward . Note . In all South re-in-cliners North re-in-cliners , for their delineation the styles height is to be called the co-latitude , and then you may draw them as erect direct plains , for South , or North ( as the former rules shall give them ) in that latitude , which is the complement of the styles height . For direct East and West recliners incliners . 1. In all East and West recliners incliners , you may refer them to a new latitude , and new declination , and then describe them as erect declining plains . 2. Their new latitude is the complement of that latitude where the plain stands , and their new declination is the complement of their reclination inclination : But to know which way you are to account this new declination , remember all East and West recliners are North-East , and North-West decliners . All East and West incliners are South-East , and South-West decliners . 3. Their new latitude and declination known , you may by Sect 4. par . 5. finde the substyle from the Meridian , height of the style , angle of twelve and six , and inclination of Meridians , using in those proportions the new latitude and new declination instead of the old . 4. In all East and West recliners incliners with centers , the Meridian lies in the horizontal line of the plain ; in such as have not centers , its paralel to the horizontal line . Sect. 6. The affections of declining reclining Plains inclining Plains . The readiest way for these , is to refer them to a new latitude , and new declination , by the subsequent proportions . 1. To finde the new Latitude . As the radius is to the cosine of the plains declination , so is the co-tangent of the reclination inclination to the tangent of a fourth ark . In South recliners North incliners get the difference 'twixt this fourth ark , and the latitude of your place , and the complement of that difference is the new latitude sought . If the fourth ark be less than your old latitude , the contrary pole is elevated ; if equal to your old latitude , it s a polar plain . In South incliners North recliners the difference 'twixt the fourth ark , and the complement of your old latitude is the new latitude . If the fourth ark be equal to your old co-latitude , they are equinoctial plains . 2. To finde the new declination . As the radius is to the cosine of the reclination inclination , so is the sine of the old declination to the sine of the new . 3. To finde the angle of the Meridian with the Horizontal Line of the Plain . As the radius is to the tangent of the old declination , so is the sine of reclination inclination , to the co-tangent of the angle of the meridian with the horizontal line of the plain . This gives the angle for its scituation . Observe , in North incliners less than a polar , the Meridian lyes . above That end of the Horizontall Line contrary to the Coast of Declination .     below   South recliners more than a polar , the Meridian lyes . below That end of the Horizontall next the Coast of Declination .     above   North recliners less than an equinoctial , the Meridian lyes above That end of the Horizontalnext the Coast of Declination .     below In North recliners this Meridian is 12. at midnight .   equal to an equinoctial the Meridian descends below the Horizontal at that end contrary to the coast of Declination , and the substyle lies in the hour line of six .     South incliners more than an equinoctial , the Meridian lyes . below That end of the horizontal contrary to the declination .     above In South incliners this Meridian is only useful for drawing the Dial , and placing the substyle , for the hour lines must be drawn through the center to the lower side . After you have by the former proportions and rules found the new latitude , new declination , the angle and scituation of the meridian , your first business in delineating of the Dial will be ( both for such as have centers , and such as admit not of centers ) to set off the meridian in its proper coast and quantity . This done , by Sect. 4. Paragr . 5. of this Chapter , finde the substyles distance from the Meridian , the height of the style , angle of twelve and six ( and for Dials without centers , the inclination of Meridians ) in all those proportions , using your new latitude and new declination , instead of the old , and setting them off from the Meridian , according to the directions in Paragr . 3. and 4. you may draw the Dials by the following rules , for erect declining plains . In placing of them , lay the horizontal line of your paper draught upon the horizontal line of your plain , and prick off the substyle and hour lines . Only observe . That such South-East , or South-West recliners , as have the contrary pole elevated must be described as North-East , and North-West decliners , and such North-East , and North-West incliners as have the contrary pole elevated , must be described as South-East , and South-West decliners , which will direct you which way to set off the substyle , and hour line of six from the Meridian in those oblique plains , which admit of centers , or the substyle from the Meridian , and the inclination of Meridians from the substyle in such as admit not of Centers . 4. Declining polar plains must have a peculiar calculation for the substyle and inclination of Meridians , which is thus . To finde the Angle of the Substile with the Horizontal Line of the Plain . As the radius is to the sine of the polar plains reclination , so is the tangent of declination to the co-tangent of the substyles distance from the horizontal line of the plain . To finde the inclination of Meridians . As the radius is to the sine of the latitude , so is the tangent of declination to the tangent of inclination of Meridians . 5. The reclining declining polar hath the substyle lying below that end of the horizontal line that is contrary to the coast of declination . The inclining declining polar hath the substyle lying above that end of the horizontal line , contrary to the coast of declination . CHAP. III. The delineation of Dials for any plain Superficies . HEre it will be necessary to premise the explication of some few terms and symbols , which for brevity sake we shall hereafter make use of . Ex. gr . Rad. denotes the radius , or sine 90. or tangent 45. Tang. is the tangent of any arch or number affixed to it . Cos. is the cosine of any arch or number of degrees , or what it wants of 90. Ex. gr . cos . 19. is what 19. wants of 90. that is 71. Co-tang - is the co-tangent of any ark or number affixed to it , or what it wants of 90. Ex. gr . co-tangent 30. is what 30. wants of 90. that is 60. = This is a note of equality in lines , numbers , or degrees . Ex. gr . AB = CD . That is , the line AB . is equal unto , or of the same length as the line CD . Again , ABC = EFG . that is the angle ABC . is of the same quantity , or number of degrees , as the angle EFG . Once more AB = CD = FG = tang . 15. That is AB . and CD . and FG. are all of the same length , and that length is the tangent of 15. d. ♒ This is a note of two lines being paralel unto , or equidistant from each other . Ex. gr . FI. ♒ RS. that is the line FL. is paralel unto , or equi-distant from the line RS. Sect. 1. To delineate an horizontal Dial. See Fig. 9. First draw the square BCDE . of what quantity the plain will permit . Then make AF = AG = HD = HE = sine of the latitude , and AH = Radius . Enter HD . in tang . 45. and keeping the Sector at that gage , set off HI = HK = tang . 15. and HO = HL = tang . 30. Again , enter FD. in tang . 45. and set off FQ = GN = tang . 15. and FP = GM = tang . 30. This done , Draw AQ . AP. AD. AO . AI. for the hour lines of 5. 4. 3. 2. 1. afternoon . Again , Draw AK . AL. AE . AM. AN. for the hour lines of 11. 10. 9. 8. 7. before noon . The line FAG . is for six and six . In the same manner you may prick the quarters of an hour , reckoning three tangents , and 45. minutes , for a quarter . How to draw the hour lines before , and after six , was mentioned , Chap. 2. Sect. 1. Sect. 2. To describe an erect , direct South Dial. See Fig. 10. Draw ABCD. a rect-angle parallellogram . Then make AE = EB = CF = FD = cos . of your latitude . And EF = AC = BD = sine of your latitude . Enter CF. in tang . 45. and lay down FK = FL = tang . 15. and FI = FM = tang . 30. Again , enter AC . in tang . 45. and lay down AG = BO = 15. and AH = BN = tang . 30. with a ruler draw the lines EG . EH . EC . EF. EK . for the hour lines of 7. 8. 9. 10. 11. in the morning . and EO . EN . ED. EM . EL. for 5. 4. 3. 2. 1. afternoon , the line AEB . is for six and six . The line EF. for twelve . The description of a direct North-Dial differs nothing from this , only the hour lines from Sun rise to six in the morning , and from six in the evening , until Sun set , must be placed thereon , by drawing the respective morning and evening hours beyond the center as in the horizontal . See Fig. 11. Sect. 3. To describe an erect , direct East Dial. See Fig. 12. Having drawn ABCD. a rectangle paralellogram , fix upon any point in the lines AB . and CD . for the line of six , provided the distance from that point to A. being entred radius in the line of tangents , the distance from thence to B. may not exceed , nor much come short of the tangent 75. This point being found , enter the distance from thence to A. ( which we shall call 6. A. ) radius in the line of tangents , and keeping the sector at that gage , lay down upon the lines AB . and CD . 6. 11. = tang . 75. 6. 10 = tang . 60. and 6. 9 = 6. A. = tang . 45. and 6. 8. = 6. = 4. = tang . 30. Lastly , 6. 7 = 6. 5 = tang . 15. draw lines from these points on AB . to the respective points on CD . and you have the hours . To place it on the plain , draw the angle DCE . = co-latitude , and laying ED. on the horizontal line of the plain , prick off the hours . The same rules serve for delineation of a West Dial , only as this hath morning , that must be marked with afternoon hours . Sect. 4. To describe an erect declining Dial , having a Center . See Fig. 14. Draw the square BCDE , and make AC = AK of quantity what you please . Again , draw AG. 12. ♒ CE. and KHF. ♒ AG. 12. By a line of chords , set off the angles of the substyle , style , and hour of six from twelve ; having first found these angles by Chap. 2. Sect. 4. Paragr . 5. This done , make a mark in A 6. where it intersects KF - as at H. Then enter AK . in the secant of the plains declination , and keeping the Sector at that gage , take out the secant of the latitude , which place from A to G. upon the the line A. 12. and again , from H. ( which is the intersection of the paralel FK . with the line of six ) unto F. This done , lay a ruler to the points F. and G. and draw a line until it intersects CE. as FG. 3. Lastly , Enter GF . in tang . 45. and set off GL = GN = tang . 15. and GM = GO = tang . 30. Again , enter HF. in tang . 45. and set off HR = tang . 15. and HP = tang . 30. A ruler laid to these points , and the center , you may draw the hours lines from six in the morning unto three afternoon . For the other hour lines , do thus , Produce the line EC . and likewise HA. beyond the center , until they intersect each other as at S. Then setting off ST = HR . and S. 4. = HP . you have the hour points after three in the afternoon , until six , although none are proper beyond the hour line of four ; only by drawing them on the other side the center , they help you to the hour lines before six in the morning . Sect. 5. To describe an erect declining Plain without a Center . See Fig. 12. The delineation of these Dials is the most difficult of any , and therefore I shall be the larger in their description . 1. By Chap. 2. S. 4. Paragr . 5. finde the angles of the style , substyle , and inclination of Meridians . 2. Having found the inclination of Meridians , make the following Table for the distance of every hour line and quarter from the substyle . Where I take for example a North plain declining East 72. d. 45. min. in lat . 52. 30. min. Imcl . mer . hou . quart . 76 = 30. 12 00 20 = 15 3 ... 16 = 30 4 00 12 = 45 4 . 09 = 00 4 .. 05 = 15 4 ... 01 = 30 5 00 02 = 15 5 . 06 = 00 5 .. 09 = 45 5 ... 13 = 30 6 00 17 = 15 6 . 21 = 00 6 .. 24 = 45 6 ... 28 = 30 7 00 32 = 15 7 . 36 = 00 7 .. 39 = 45 7 ... 43 = 30 8 00 47 = 15 8 . 51 = 00 8 .. 54 = 45 8 ... 58 = 30 9 00 62 = 15 9 . 66 = 00 9 .. 69 = 45 9 ... 73 = 30 10 00 The manner of drawing this Table is thus . The inclination of Meridians ( which because its a North decliner , is twelve at midnight . ) I finde 76. d. 30. min. Now considering the Sun never riseth till more than half an hour after three in this latitude , I know that one quarter before four is the first line proper for this plain : Therefore reckoning 15. d. for an hour , or 3. d. 45. m. for a quarter of an hour , I finde three hours , three quarters ( the distance of a quarter before four from midnight ) to answer 56. d. 15. min. which being subtracted from 76. d. 30. min. the inclination of Meridians there remains 20. d. 15. m. for the distance of one quarter before four from the substyle . Again , from 20. d. 15. min. subtract 3. d. 45. min. ( the quantity of degrees for one quarter of an hour ) and there remains 16. d. 30. min. for the distance of the next line from the substyle , which is the hour of four in the morning . Thus for every quarter of an hour continue subtracting 3. d. 45. min. until your residue or remain be less than 3. d. 45. min. and then first subtracting that residue out of 3. d. 45. min. This new residue gives the quantity of degrees for that line on the other side the substyle . Now when you are passed to the other side of the substyle , continue adding 3. d. 45. min. to this last remain for every quarter of an hour , and so make up the table for what hours are proper to the plain . 3. Draw the square ABCD. of what quantity the plain will admit , and make the angle CAG equal to the angle of the substyle with twelve . Again , cross the line AG. in any two convenient points , as E. and F. at right angles by the lines KL . and CM . 4. Take the distance from the center unto 45. the radius to the lesser lines of tangents , which is continued to 76. on the Sector side , enter this distance in 45. on the larger lines of tangents , and keeping the Sector at that gage , take out the tang . 20. d. 15. min. ( which is the distance of the first line from the substyle ) set this from 73. d. 30. min. ( the distance of your last hour line from the substyle , as you see by the Table ) toward the end upon the lesser line of tangents , and where it toucheth as here at 75. 05. call that the gage tangent . 5. Enter the whole line KL . in the gage tangent , which in this example is 75. d. 05. min. and keeping the Sector at that gage , take out the tangent 73. 30. min. which is your last hour , and set from L. on the line KL . unto V. Again , take out the tangent 20. d. 15. min. which is your first line , and set it from K. towards V. and if it meet in V. it proves the truth of your work , and a line drawn through V. paralel unto AG. is the true substyle line . Then keeping the Sector at its former gage , set off the tangents of the hours , and quarters ( as you finde them in the Table ) from V. towards K. and from V. towards L. making points for them in the line KL . Lastly , enter the radius of your tangents to these hour points in the radius of secants , and set off the secant of the styles height from V. to T. Thus have you the hour points and style on one line of contingence . To mark them out upon the other line do thus . Set the radius to the hour points upon the former line of contingence , from h. to p. on the line ChM. and entring hV. as Radius in the line of tangents , take out the tangent of the styles height , and set from p. to r. Again , enter hr. radius in your line of tangents , and keeping the Sector at that gage , take out the tangents for each hour , and quarter , according to the table , and lay them down from h. to the proper side of the substyle toward C. or M. and applying a ruler to the respective points on KL . and CM . draw the lines for the hours and quarters . Lastly , enter hr. radius on the lines of secants , and taking out the secant of the styles height , set it from h. to S. and draw the line ST . for the style . Sect. 6. To describe a direct polar Dial. See Fig. 15. Draw BCDE . a rectangle paralellogram , from the middle of BC. to the middle of DE. draw the line 12. or substyle , appoint what place you please in BC. or CD . for the hour point of 7. in the morning , and 5. afternoon . Then , Enter 12. 7 = 12. 5. In the tangent 75. and set off 12. 1. = 12. 11. = tang . 15. and 12. 2. = 12. 10 = tang . 30. and 12. 3. = 12. 9. = tang . 45. Lastly , 12. 4. = 12. 8. = tang . 60. From these points draw the hour lines of 7. 8. 9. 10. 11. 12. 1. 2. 3. 4. 5. which are all the hours proper for these plains . Sect. 7. To draw a declining Polar . See Fig. 16. 1. By Chap. 2. Sect. 6. Par. 4. finde the inclination of Meridians , and distance of the substyle from the horizontal . 2. By Chap. 3. Sect. 5. Par. 2. make a Table for the distance of the hour points from the substyle . 3. Draw the square BCDE . Set off the angle CAG . for the substyle , and cross that substyle line at right angles in any two convenient places , as at H. and K. with the lines PHS. and RKT. for contingent lines . 4. Take any convenient length for your styles height , and enter it radius in your line of tangents , keeping the Sector at that gage , prick off the hours from the substyle ( by your table ) upon both the contingent lines . Draw lines by the points in both contingents , and you have the hours : For all other declining reclining inclining plains , it would be needless ( I presume ) to insist upon the description of them : Sith so much hath been already mentioned , Chap. 2. S. 6. that there can scarcely be any mistake , unless through meer wilfulness , or grandnegligence . CHAP. IV. To determine what hour lines are proper for any plain Superficles . By projection of the Sphere . See Fig. 17. DRaw the Circle NESW . representing the Horizon , and crossing it into quadrants N. is North. S. South , E. East , W. West , NS . the Meridian ( which let be infinitely produced ) Z. the center represents the Zenith . To finde the pole set half the co-tangent of the latitude from Z. toward N. it gives the point P. for the pole or the point through which all the hour lines must pass . The Suns declination in Cancer subtracted from the latitude , and the tangent of half the remain set from Z. to Π. gives the intersection of Cancer with the Meridian . Again , adde the complement of the Suns delineation in Cancer , unto the complement of your latitude , and the tangent of half that sum set from Z. to ψ , gives the diameter of that tropick , half ψ , is the radius to describe it . Half the tangent of your latitude set from Z. to Q. gives that point for the intersection of the equator with the Meridian ; and the co-secant of the latitude set from oe . toward N. gives the point ζ. the center of the equator . Adde the Suns greatest declination ( or his declination in Capricorn ) to the latitude , and the tangent of half that sum set from Z. toward S. gives the point φ , where Capricorn intersects the Meridian . Subtract the Suns declination in Capricorn from your latitude , and that remain subtract from 180. the tangent of half this last remain , set from Z. toward N. gives the point T. the diameter of Capricorn , and half the distance T φ. is the radius to describe it . Set the secant of the latitude from P. towards S. it gives the point H. the center of the hour line of six cross the line ZSH. at right angles in the point H. Then entring PH. Radius on the lines of tangents , set off the hour centers both wayes from H. reckoning 15. d. for an hour . Lastly , setting one point of your Compasses in these center points , extend the other to P. and with that radius describe the hour lines . Thus have you the sphere projected , the following Sections will determine the hours for all plains . Sect. 1. To determine the hour lines for erect direct plains . Fig. 17. The line NS . represents an erect East , and West plain . That side next W. is West , the other side next E. is East , where you may see that the Sun shines upon the East until twelve , or noon , and at that time comes upon the West . The fine WE . represents a direct North and South plain , the side next N. is North , the other next S. is South , where the North cuts the tropick of Cancer ( which in the hour lines you finde 'twixt 7. and 8. in the morning ; and again 'twixt 4. and 5. afternoon ) is the time of the Suns going off , and coming on that plain . Where the South cuts the equator , which is in the points of six , and six is the time of the Suns going off , and coming on that plain . Sect. 2. To determin the hour lines for direct reclining inclining Plains . Fig. 17. NBS . on the convex side is a West incliner , where it cuts Capricorn , is the time of the Suns coming on that plain , afternoon . On the concave side its an East recliner , where it cuts Cancer , is the time of the Suns going off that plain , afternoon . NCS. On the convex side is an East incliner , where it cus Capricorn , is the time of the Suns going off in the morning . On the concave side it is a West recliner , where it cuts Cancer , is the time of the Suns coming on in the morning . WDE. On the convex side is a South incliner , where until D. reach below oe . it hath all hours from six to six , and until D. reacheth below φ. it may have the twelve a clock line . But when D. reacheth below oe . draw a paralel of declination to pass through the point D. and the intersection of that paralel with the limb of the circle NE SW . doth among the hour lines , shew the time of the Suns coming upon that plain in the morning , and going off again afternoon , when D. reacheth below φ. the intersection of the ark WDE. with the tropick of Capricorn , shews the time of the Suns going off that plain before noon , and coming on again , afternoon . And the intersection of the tropick of Capricorn with the limb shews the first hour in the morning the Sun comes on , and the last hour afternoon , that it staves upon that plain . The convex side of WDE. is a north recliner , where it cuts Cancer , is the time of the Suns going off in the morning , and coming on again afternoon . WFE. On the convex side is a North incliner , where it cuts Cancer , is the time of the Suns going off in the morning , and coming on afternoon . On the concave side is a South recliner , where until F. reach beyond P. it enjoyes the Sun only from six to six . When F. reacheth beyond P. where the ark cuts Cancer , you finde how much before six in the morning the Sun comes on , or after six at evening it goes off . To draw any of these arks , Ex. gr . the ark NBS . do thus . Set the tangent of half the reclination inclination from Z. on the line ZW . and it gives the point B. produce ZE. and set the co-secant of the reclination inclination from B. towards E. which reacheth to G. then G. is the center . & GB . the radius to draw that ark . Note . The Semidiameter of the circle SENW . is radius to all the tangents , and secants , which you make use of for placing any oblique plain upon the Scheme . Sect. 3. To determin the hour lines for erect declining plains . Fig. 18. For South-East , or North-West plains . By a line of chords set the angle of declination on the limb from W. toward S. as H. lay a ruler to HZ . and draw HZK. which on that side next NW . represents the North-West , and where the line cuts Cancer , you have the time of the Suns coming on afternoon , and staying until Sun set . But if it cut Cancer twice , then in the morning hours it shews what time the Sun goes off this plain , having all hours from Sun rise to that time , and in the evening hours you have the time of the Suns coming on again , and staying till sun set . That side HZK. next SE. represents a South-East . Where the line cuts the equator in the evening hours , is the time of the Suns going off , where it cuts Cancer , in the morning hours , is the time of the Suns coming on that plain . For North-East , or South-West , set the declination by a line of chords from E. towards S. as L. lay a ruler to ZL . and draw the line LZR. which on that side next NE. is North-East , where it cuts Cancer , is the time of the Suns going off in the morning . If it cuts Cancer twice , you have in the evening hours the time of the Suns coming on again , and staying until Sun-set . On that side next SW . is the South-west . Where it cuts the equator , or Capricorn in the morning hours , is the time of the Suns coming on , where it cuts Cancer in the evening hours , is the time of the Suns going off . Sect. 4. To determin the hours of declining reclining Plains inclining Plains . Fig. 18. First , set in the plain according to its declination . By Sect. 3. Ex. gr . LZR. a North-East , or South-West declining 50. d. 00. min. This done Cross the line LZR. representing the declination of the plain , at right angles in the point Z. as CZBG . Then for North-East incliners , or South-West recliners , set half the tangent of the reclination inclination from Z. toward C. 〈◊〉 T. and set the co-secant of the reclination inclination from T. toward B. as TG . Then G. is the center , and GT . the radius to describe the ark RTL. Whose convex side represents a North-East incliner , where it cuts Libra or Capricorn , is the time of the Suns going off in the morning ; if it cuts Cancer twice , the intersection of Cancer with the evening hours shews what time the Sun comes again upon such a plain afternoon , and continues till Sun setting . The concave side is a South-West recliner , where it cuts Cancer in the morning hours , is the time of the Suns coming on , in case it intersects Cancer twice in the evening hours , you have the time that the Sun goes off . For a North-East recliner , or South-West incliner , set the point T. from Z. toward B. and the point G. from T. ( so placed ) toward C. and draw the ark on that side RZL. toward B. whose convex side will represent a South-West incliner , and where the ark cuts the equator or Capricorn , you have the time of the Suns coming on that plain . The concave side is a North-East recliner , where the ark cuts Cancer , is the time for the Suns going off that plain . When the ark cuts Cancer twice , the Sun comes on again before it sets . For a North-West recliner , or South-East incliner . Enter the declination by Sect. 3. as HZK. Cross it in the point Z. at right angles , as OZD. set half the tangent of the reclination inclination from Z. toward O. as V. and the co-secant of the reclination inclination from V. toward D. as F. then is F. the center , and FV. the radius to draw the ark HVK. Where it cuts Cancer the hour lines , tell you the time of the Suns going off in the morning , and entring again afternoon , upon the North-West recliner . Where it cuts the equator you have the time of the Suns going off the South-East , where it cuts Cancer in the morning hours is the time of the Suns coming on that plain . For a North-West incliner , or South-East recliner , set the point V. from Z. toward D. and the point F. set from V. ( so placed ) toward O. and draw the ark on that side Z. next D. Then where the convex side cuts Cancer , you have the time of the Suns going off in the morning ; and coming on again afternoon upon the North-West incliner . Where the concave side cuts the equator , you have the time of the Suns going off the South-East recliner ; where it intersects Cancer , is the time of his coming on that plain in the morning . Note . All the precedent rules about plains are appropriated to us that live in Northern Hemisphere , In case any one would apply them to the South Hemisphere : What is here called North , there name South , and what we here term South , there call North , and the rules are the same . — Si quid novisti plenius istis , promptius istis , rectius istis , Candidus imperti : Sinon , His u●ere mecum . FINIS . 
A89305	----	Horlogiographia optica. Dialling universall and particular: speculative and practicall. In a threefold præcognita, viz. geometricall, philosophicall, and astronomicall: and a threefold practise, viz. arithmeticall, geometricall, and instrumentall. With diverse propositions of the use and benefit of shadows, serving to prick down the signes, declination, and azimuths, on sun-dials, and diverse other benefits. Illustrated by diverse opticall conceits, taken out of Augilonius, Kercherius, Clavius, and others. Lastly, topothesia, or, a feigned description of the court of art. Full of benefit for the making of dials, use of the globes, difference of meridians, and most propositions of astronomie. Together with many usefull instruments and dials in brasse, made by Walter Hayes, at the Crosse Daggers in More Fields. / Written by Silvanus Morgan. Morgan, Sylvanus, 1620-1693. This text is an enriched version of the TCP digital transcription A89305 of text R202919 in the English Short Title Catalog (Thomason E652_16). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 273 KB of XML-encoded text transcribed from 81 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2017 A89305 Wing M2741 Thomason E652_16 ESTC R202919 99863048 99863048 115230 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A89305) Transcribed from: (Early English Books Online ; image set 115230) Images scanned from microfilm: (Thomason Tracts ; 100:E652[16]) Horlogiographia optica. Dialling universall and particular: speculative and practicall. In a threefold præcognita, viz. geometricall, philosophicall, and astronomicall: and a threefold practise, viz. arithmeticall, geometricall, and instrumentall. With diverse propositions of the use and benefit of shadows, serving to prick down the signes, declination, and azimuths, on sun-dials, and diverse other benefits. Illustrated by diverse opticall conceits, taken out of Augilonius, Kercherius, Clavius, and others. Lastly, topothesia, or, a feigned description of the court of art. Full of benefit for the making of dials, use of the globes, difference of meridians, and most propositions of astronomie. Together with many usefull instruments and dials in brasse, made by Walter Hayes, at the Crosse Daggers in More Fields. / Written by Silvanus Morgan. Morgan, Sylvanus, 1620-1693. Goddard, John, fl. 1645-1671, engraver. [16], 144 p. : ill. (woodcuts, metal cuts) Printed by R. & W. Leybourn, for Andrew Kemb, and Robert Boydell, and are to be sold at St. Margarets Hill in Southwark, and at the Bulwark neer the Tower, London : 1652. With an additional title page, engraved and initialed by John Goddard. The first leaf contains verses "On the frontispiece". Annotation on Thomason copy: The "2" in the imprint date is crossed out and date altered to 1652; "Febr. 4th". Reproduction of the original in the British Library. eng Dialing -- Early works to 1800. Globes -- Early works to 1800. Sundials -- England -- Early works to 1800. A89305 R202919 (Thomason E652_16). civilwar no Horlogiographia optica.: Dialling universall and particular: speculative and practicall. In a threefold præcognita, viz. geometricall, phil Morgan, Sylvanus 1652 34401 50 0 14 0 0 0 19 C The rate of 19 defects per 10,000 words puts this text in the C category of texts with between 10 and 35 defects per 10,000 words. 2007-08 TCP Assigned for keying and markup 2007-08 Apex CoVantage Keyed and coded from ProQuest page images 2008-05 John Pas Sampled and proofread 2008-05 John Pas Text and markup reviewed and edited 2008-09 pfs Batch review (QC) and XML conversion HOROLOGIOGRAPHIA OPTICA Dialing Universall and perticuler . Speculatiue and Practicall together with the discription of the Courte of Arts by a new Method By Sylvanus Morgan JJ. sculp Horologiographia Optica . DIALLING Vniversall and Particular : Speculative and Practicall . In a threefold PRAECOGNITA , viz. Geometricall , Philosophicall , and Astronomicall : and a threefold practise , viz. Arithmeticall , Geometricall , and Instrumentall . With diverse Propositions of the use and benefit of shadows , serving to prick down the Signes , Declination , and Azimuths , on Sun-Dials , and diverse other benefits . Illustrated by diverse Opticall Conceits , taken out of Augilonius , Kercherius , Clavius , and others . LASTLY , TOPOTHESIA , OR , A feigned description of the COVRT OF ART . Full of benefit for the making of Dials , Use of the Globes , Difference of Meridians , and most Propositions of ASTRONOMIE . Together with many usefull Instruments and Dials in Brasse , made by Walter Hayes , at the Crosse Daggers inMore Fields Written by Silvanus Morgan . LONDON , Printed by R & W. Leybourn , for Andrew Kemb , and Robert Boydell ▪ ● and are to be sold at St Margarets Hill in Southwark , and at the Bulwark neer the Tower . 1657. TO WILLIAM BATEMAN , Esqrs. TO ANTHONY BATEMAN , Esqrs. TO THOMAS BATEMAN , Esqrs. Sons to the late Honourable THOMAS BATEMAN , Esq Chamberlain of LONDON , Deceased . GENTLEMEN , YOur late Father being a Patron of this Honourable City , doth not a little invite me to you , though young , yet to patronise no less then the aspiring of Coelum , which , as the Poets feign , was the ancientest of the gods , and where you may see Sol only of the Titans , favouring Jupiters ▪ signe , and by their power and operation hath established Arts or Learning , the fable rather according to that establishment which God hath given them , they are , I say , sought out of those that take their pleasure therein : Pardon my boldness , I beseech you , if like Prometheus I have made a man of clay ; and now come to light my bundle of twigs at the Chariot of the Sun , desiring that you would infuse vigor in that which cannot at all move of it self , & if your benevolence shall but shine upon it , the angles of incidence & reflection shall be all one : your love invites me to be so bold as to think you worthy of my labour , wherein , if faults shall arise in the Cuspis of the Ascendent , they shall also have their fall upon my selfe . And if any shall be offended at this Worke , my device shall be a Dyall with this Mottoe , Aspicio ut aspiciar , only to all favourers of Art I am direct erect plaine , as I am , Gentlemen , to you , and desire to be Yours in the best of my services , S. M. TO THE READER . REader , I here present thee with some Coelestiall operations drawn from the Macrocosmall World , if I should tel you of plurality , it may seem absurd , but I 'le distinguish the word . Mundus the World is somtimes taken Archtypically , and so is God , only in whose divine minde is an example of all things . Mundus the World is somtimes taken Angelicall , and this is the Hierarchicall government of Angels in Ceruphins , Cherubins , and Thrones . Mundus the World is somtimes taken elementary , and this is the Philosophers common place : the Salamander in Fire , the Birds in Air , the Fish in Water , and Men and Beasts on Earth . Somtimes Macrocosmally , considering the whol Universe , as well Aetheriall as Subterene , yea , and every Orb , and this is imaginarily set down in the Praecognita Astronomicall . Somtimes Microcosmally , as in the little World man , and this is described in the last Chapter of the Praecognita Philosophicall . Somtimes Typically , and that either Geographicall or Gnomonicall , or mentally in the minde of the workman . Geographically in Maps or Globes , or Sphears in plano . Gnomonicall in this present Art of Dialling , of which it may be said that Umbra horas Phoebi designat climate nostro Nodus , quod signum Sol tenet arte docet . And by which they must necessarily trace out our times by the orbiculation of the Rady of the circle of the body of the Sunne . Again , the World is mentally considered in the minde of an Artist , as in Painting , Graving , Carving , &c. But having thus defined the word , you may think from hence that I am with Democrates Platonissans , acquainting thee with infinity of Worlds , and in his words , Stanza 20. — — and To speake out though I detest the Sect Of Epicurus for their manner vile . Yet what is true I may not well neglect Of truths incorruptible , ne can the stile Of vicious pen her sacred worth defile . If we no more of truth should deign to speak Then what unworthy mouthes did never soyle , No truths at all mongst men would finde a place But make them spéedy wings , & back to heaven apace . Howsoever thou hast here a field large enough to walke in , which if thou affect the light , thou mayst trace out the truth , and I presume I have done that for thee who art a learner , the most plain wayes that were ever published , and have studyed not to make it the Art of shadows , so much as the shadow of that art whose Gnomons may be said to touch the Poles , and whose planes may be severall Planispheres , a Scale to the Geometrician , a Pole to the Navigator , a Chart to the Geographer , a Zodiaque to the Astronomer , a Table of Houses to the Astrologian , the Meridian and Needle to the Surveyor , a Dyall to us all , to put us in minde of that pretious time which saith to us Fugio , Fuge , and which time shall be swallowed up of Eternity , when there shall be but one day without Tropicall distinctions , where thou shalt not need helps from any other , nor from me who am thine , S. M. In Solarium . HIc tibi cum numero spectantur Nodus & umbra , Quae tria quid doceant , commemorare libet Umbra notat dextrè quota cursitet hora dici , Hincque monet vitam sic properare tuam Ast in quo signo magni lux publica mundi Versetur mira nodulus arte docet Si vis scire , dies quot quilibet occupet horas , Id numerus media sede locatus habet . On my Friend Mr. Silvanus Morgan , his Book of Dialling . THe use of Dials all men understand ; To make them few : & I am one of those . I am not of the Mathematick Band : Nor know I more of Vers , then Vers from Prose . But though nor Diallist I am , nor Poet : I honour those in either doe excell ; Our Author 's skill'd in both alike , I know it , Shadows , and Substance , here run parallel . Consider then the pains the Author took , And thank him , as thou benefit'st by 's Book . Edward Barwick . On the Author and his Book . DAres Zoil or Momus for to carp at thee , And let such Ideots as some Authors be Boldly to prosecute or take in hand Such noble subjects they not understand , Only for ostentation , pride , or fame , Or else because they 'd get themselves a name , Like that lewd fellow , who with hatefull ire , Flinch'd not , but set Diana's Court on fire : His name will last and be in memory From age to age ▪ although for infamie . What more abiding Tombe can man invent Then Books , which ( if they 'r good ) are permanent And monuments of fame , the which shall last Till the late evening of the World be past : But if erroneous , sooth'd with vertues face , Their Authors cridit's nothing but disgrace . If I should praise thy Book it might be thought , Friends will commend , although the work be nought , But I 'le forbeare , lest that my Verses doe Belie that praise that 's only due to you . Good Wiue requires no Bush , and Books will speak Their Authors credit , whether strong or weak . W. Leybourn . ERRATA . REader , I having writ this some years since , while I was a childe in Art , and by this appear to be little more , for want of a review hath these faults , which I desire thee to mend with thy pen , and if there be any errour in Art , as in Chap. 17 , which is only true at the time of the Equinoctiall , take that for an oversight , and where thou findest equilibra read equilibrio , and in the dedication ( in some Copies ) read Robert Bateman for Thomas , and side for signe , and know that Optima prima cadunt , pessimas aeve manent . pag. line Correct . ● 10 equall lines 18 16 Galaxia 21 1 Galaxia 21 8 Mars ▪ 24 12 Scheame 35 1 Hath 38 8 of the Tropicks & polar Circles 40 22 AB is 44 31 Artificiall 46 ult heri 49 4 forenoon 63 29 AB 65 11 6 80 16 BD 92 17 Arch CD 9 ult in some copies omit center 126 4 happen 126 6 tovvard B ▪ 127 26 before 126 prop. 10 ▪ for sine read tang . elev .   Figure of the Dodicahedron false cut pag. 4 LF omitted at end of Axis 25 For A read D 26 In the East and West Diall A omitted on the top of the middle line , C on the left hand , B on the right 55 Small arch at B omitted in the first polar plane 58 For E read P on the side of the shadowed line toward the left hand I omitted next to M , and L in the center omitted 81 K omitted in figure 85 On the line FC for 01 read 6 , for 2 read 12 , line MO for 15 read 11 96 A small arch omitted at E & F , G & H omitted at the ende of the line where 9 is 116 I & L omitted on the little Epicicle . 122 THE ARGVMENT OF THE Praecognita Geometricall , and of the Work in generall . WHat shall I doe ? I stand in doubt To shew thee to the light ; For Momus still will have a flout , And like a Satyre bite : His Serpentarian tongue will sting , His tongue can be no slander , He 's one to wards all that hath a fling His fingers ends hath scan'd her . But seeing then his tongue can't hurt , Fear not my little Book , His slanders all last but a spurt , And give him leave to look And scan thee thorough , and if then This Momus needs must bite At shadows which dependant is Only upon the light . Withdraw thy light and be obscure . And if he yet can see Faults in the best that ever writ , He must finde fault with me . How ere proceed in private and deline The time of th' day as oft as sun shall shine : And first define a Praecognitiall part Of magnitude , as usefull to this art . THE PRAECOGNITA GEOMETRICAL . THe Arts , saith Arnobius , are not together with our mindes , sent out of the heavenly places , but all are found out on earth , and are in processe of time , soft and fair , forged by a continuall meditation ; our poor and needy life perceiving some casual things to happen prosperously , while it doth imitate ▪ attempt and try , while it doth slip , reform and change , hath out of these same assiduous apprehensions made up small Sciences of Art , the which afterwards , by study , are brought to some perfection . By which we see , that Arts are found out by daily practice , yet the practice of Art is not manifest but by speculative illustration , because by speculation : Scimus ut sciamus , we know that we may the better know : And for this cause I first chose a speculative part , that you might the better know the practice ; and therefore have first chose this speculative part of practicall Geometry , which is a Science declaring the nature , quantity , and quality of Magnitude , which proceeds from the least imaginable thing . To begin then , A Point is an indivisible , yet is the first of all dimension ; it is the Philosophers Atome , such a Nothing , as that it is the very Energie of all things , In God it carryeth its extreams from eternity to eternity : in the World it is the same which Moses calls the beginning , and is his Genesis : 't is the Clotho that gives Clio the matter to work upon , and spins it forth from terminus à quo , to terminus ad quem : in the Alphabet 't is the Alpha , and is in the Cuspe of the Ascendant in every Science , and the house of Life in every operation . Again , a Point is either centricall or excentricall , both which are considered Geometrically or Optically , that is , a point , or a seeming point : a point Geometrically considered is indivisible , and being centrall is of magnitude without consideration of form , or of rotundity , with reference to Figure as a Circle , or a Globe , &c. or of ponderosity , with reference to weight , and such a point is in those Balances which hang in equilibra , yet have one beam longer than the other . If it be a seeming point , it is increased or diminished Optically , that is , according to the distance of the object and subject . 'T is the birth of any thing , and indeed is to be considered as our principall significator , which being increased doth produce quantity which is the required to Magnitude ; for Magnitude is no other then a continuation of Quantity , which is either from a Line to a plain Superficies , or from a plain Superficies to a Solid Body : every of which are considered according to the quantity or form . The quantity of a Line is length , without breadth or thicknesse , the forme either right or curved . The quantity of a Superficies consisteth in length and breadth , without thicknesse , the form is divers , either regular or irregular ; Regular are Triangles , Squares , Circles , Pentagons , Hexagons , &c. An equilaterall Triangle consisteth of three right lines & as many angles , his inscribed side in a Circle contains 120 degrees . A Square of four equall right lines , and as many right angles , and his inscribed side is 90 degrees . A Pentagon consisteth of five equall lines and angles , and his inscribed side is 72 degrees of a Circle . A Hexagon is of six equall lines and angles , and his side within a Circle is 60 degrees , which is equall to the Radius or Semidiameter . An Angle is the meeting of two lines not in a streight concurring , but which being extended will crosse each other ; but if they will never crosse , then they are parallel . The quantity of an angle is the measure of the part of a Circle divided into 360 degrees between the open ends , and the angle it self is the Center of the Circle . The quantity of a Solid consists of length , breadth , and thickness , the form is various , regular or irregular : The five regular or Platonick Bodies are , the Tetrahedron , Hexahedron , Octohedron , Dodecahedron , Icosahedron . Tetrahedron is a Solid Body consisting of four equall equilaterall Triangles . A Hexahedron is a Solid Body consisting of six equal Squares , and is right angled every way . An Octahedron is a Solid Body consisting of eight equal Equilaterall Triangles . A Dodecahedron is a Solid Body consisting of 12 equall Pentagons . An Icosahedron is a Solid Body consisting of 20 equal Equilaterall Triangles : All which are here described in plano , by which they are made in pasteboard : Or if you would cut them in Solid it is performed by Mr. Wells in his Art of Shadows , where also he hath fitted planes for the same Bodies . A Parallel line is a line equidistant in all places from another line , which two lines can never meet . A Perpendicular is a line rightly elevated to another at right angles , and is thus erected . Suppose AB be a line , and in the point A you would erect a perpendicular : set one foot of your Compasses in A , extend the other upwards , anywhere , as at C , then keeping the foot fixed in C , remove that foot as was in A towards B , till it fall again in the line AB , then if you lay a Ruler by the feet of your Compasses , keep the foot fixed in C , and turn the other foot toward D by the side of the Ruler , and where that falls make a marke , from whence draw the line DA , which is perpendicular to AB . And so much shall suffice for the Praecognita Geometricall , the Philosophicall followeth . The end of the Praecognita Geometricall . THE ARGVMENT OF THE Praecognita Philosophicall . NOt to maintain with nice Philosophie , What unto reason seems to be obscure , Or shew you things hid in obscurity , Whose grounds are nothing sure . 'T is not the drift of this my BOOK , The world in two to part , Nor shew you things whereon to looke But what hath ground by Art . If Art confirm what here you read , Sure you 'l confirmed be , If reason wonte demonstrate it , Learn somwhere else for me . There 's shew'd to you what shadow is , And the Earths proper place , How it the middle doth possesse , And how heavens run their race . Resolving many a Proposition , Which are of use , and needfull to be known . THE PRAECOGNITA PHILOSOPHICAL . CHAP I. Of Light and Shadows . HE that seeketh Shadow in its predicaments , seeketh a reality in an imitation , he is rightly answered , umbram per se in nullo praedicamento esse , the reason is thus rendred as hath been , it is not a reality , but a confused imitation of a Body , arising from the objecting of light , So then there can be no other definition then this , Shadow is but the imitation of substance , not incident to parts caused by the interposition of a substance , for , Umbra non potest agere sine lumine . And And it is twofold , caused by a twofold motion of light , that is , either from a direct beam of light , which is primary , or from a secondary , which is reflective : hence it is , that Sun Dials are made where the direct beams can never fall , as on the seeling of a Chamber or the like . But in vain man seeketh after a shadow , what then , shall we proceed no farther ? surely not so , for qui semper est in suo officio , is semper orat , for there are no good and lawful actions but doe condescend to the glory of God , and especially good and lawfull Arts . And that shadow may appear to be but dependant on light , it is thus proved , Quod est & existit in se , id non existit in alio : that which is , and subsisteth in it selfe , that subsisteth not in another : but shadow subsisteth not in it selfe , for take away the cause , that is light , and you take away the effect , that is shadow . Hence we also observe the Sun to be the fountain of light , whose daily and occurrent motions doth cause an admirable lustre to the glory of God ; seeing that by him we measure out our Times , Seasons , and Years . Is it not his annuall revolution , or his proper motion that limits our Year ? Is it not his Tropicall distinctions that limits our Seasons ? Is it not his Diurnall motion that limits out our Dayes and Houres ? And man truly , that arch type of perfection , hath limited these motions even in the small type of a Dyall plane , as shall be made manifest in things of the second notion , that is , Demonstration , by which all things shall be made plain . CHAP II. Of the World , proving that the Earth possesseth its own proper place . WE have now with the Philosopher , found out that common place , or place of being , that is , the World , will you know his reason ? 't is rendred , Quia omnia reliqua mundi corpora in se includit . I 'le tell you of no plurality , not of planetary inhabitants , such as the Lunaries ▪ lest you grabble in darkness , in expecting a shadow from the light without interposition , for can the light really without a substance be its own Gnomon ? surely no , neither can we imagine our earth to be a changing Cynthia , or a Moon to give light to the Lunary inhabitants : For if our Earth be a light ( as some would have it ) how comes it to passe that it is a Gnomon also to cast a shadow on the body of the Moon far lesse then it selfe , and so by consequence a greater light cannot seem to be darkned on a lesser or duller light , and if not darkned , no shadow can appear ? But from this common place the World with all its parts , shall we descend to a second grade of distinction , and come now to another , which is a proprius locus , and divide it into proper places , considering it as it is divided into Coelum , Solum , Salum , Heaven , Earth , Sea , we need not so far a distinction , but to prove that the earth is in its own proper place , I thus reason : Proprius locus est qui proxime nullo alio interveniente continet locatum : but it is certain that nothing can come so between the earth as to dispossesse it of its place , therefore it possesseth its proper place , furthermore , ad quod aliquid movetur , id est ejus locus , to what any thing moves that is its place : but the earth moves not to any other place , as being stable in its own proper place . And this proper place is the terminus ad quem , to which ( as the place of their rest ) all heavie things tend , in quo motus terminantur , in which their motion is ended . CHAP III. Shewing how the Earth is to be understood to be the Center . A Center is either to be understood Geometrically or Optically , either as it is a point , or seeming a point . If it be a point , it is conceived to be either a center of magnitude , or a center of ponderosity , or a center of rotundity : if it be a seeming point , that is increased or diminished according to the ocular aspect , as being somtime neerer , and somtime farther from the thing in the visuall line , the thing is made more or lesse apparent . A center of magnitude is an equal distribution from that point , an equality of distribution of the parts , giving to each end alike , and to each a like vicinity to that point or center . A center of ponderosity is such a point in which an unequall thing hangs in equi libra , in an equall distribution of the weight , though one end be longer or bigger than the other of the quantity of the ponderosity . A center of rotundity is such a center as is the center of a Globe or Circle , being equally distant from all places . Now the earth is to be understood to be such a center as the center of a Globe or Sphear , being equally distant from the concave superficies of the Firmament , neither is it to be understood to be a center as a point indivisible , but either comparatively or optically : comparatively in respect of the superior Orbs ; Optically by reason of the far distance of the one from the earth ; as that the fixed Stars being far distant seeme , by the weaknesse of the sense , to be conceived as a center indivisible , when by the force and vigour of reason and demonstration , they are found to exceed this Globe of earth much in magnitude ; so that what our sense cannot apprehend , must be comprehended by reason : As in the Circles of the Coelestiall Orbs , because they cannot be perceived by sense , yet must necessarily be imagined to be so . Whence it is observable , that all Sun Dials , though they stand on the surface of the earth , doe as truly shew the houre as if they stood in the center . CHAP IV. Declaring what reason might move the Philosophers and others to think the Earth to be the center , and that the World moves on an axis , circa quem convertitur . OCular observations are affirmative demonstrations , so that what is made plain by sense is apparent to reason : hence it so happeneth , that we imagine the Earth to move as it were on an axis , because , both by ocular and Instrumentall observation , in respect that by the eye it is observed that one place of the Skie is semper apparens , neither making Cosmicall , Haeliacall or Achronicall rising or setting , but still remaining as a point , as it were , immoveable , about which the whole heavens are turned . These yet are necessary to be imagined for the better demonstration of the ground of art ; for all men know the heavens to be supported only by the providence of God . Thus much for the reason shewing why the World may be imagined to be turned on an Axis , the demonstration proving that the earth is the center , is thus , not in maintaining unlikely arguments , but verity of observation ; for all Gnomons casting shadow on the face of the earth , cast the like length or equality of shadow , they making one & the same angle with the earth , the Sun being at one and the same angle of height to al the Gnomons . As in example , let the earth be represented by the small circle within the great circle , marked ABCD , and let a Gnomon stand at E of the lesser Circle , whose horizon is the line AC , and let an other gnomon of the same length be set at I , whose horizon is represented by the line BD , now if the Sun be at equall angles of height above these two Horizons , namely , at 60 degrees from C to G , and 60 from B to F , the Gnomons shall give a like equality of shadows , as in example is manifest . Now from the former appears that the earth is of no other form then round , else could it not give equality of shadows , neither could it be the center to all the other inferior Orbs : For if you grant not the earth to be the middle , this must necessarily follow , that there is not equality of shadow . For example , let the great Circle represent the heavens , and the lesse the earth out of the center of the greater , now the Sunne being above the Horizon AC 60 d. and a gnomon at E casts his shadow from E to F , and if the same gnomon of the same length doth stand till the Sun come to the opposite side of the Horizon AC , and the Sun being 60 degrees above that Horizon , casts the shadow from E to H , which are unequall in length ; the reason of which inequality proves that then it did not stand in the center , and the equality of the other proves that it is in the center . Hence is also most forceably proved that the earth is compleatly round in the respect of the heavens , as is shewed by the equality of shadows , for if it were not round , one and the same gnomon could not give one and the same shadow , the earth being not compleatly round , as in the ensuing discourse and demonstration is more plainly handled and made manifest . And that the earth is round may appeare , first , by the Eclipses , when the shadow of the earth appeareth on the body of the Moon , darkning it in whole or in part , and such is the body such is the shadow . Again , it appears to be round by the orderly appearing of the Stars , for as men travell farther North or South they discover new Stars which they saw not before , and lose the sight of them they did see . As also by the rising or setting of the Sun or Stars , which appear not at the same time to all Countries , but by difference of Meridians , and by the different observations of Eclipses , appearing sooner to the Easterly Nations then those that are farther West : Neither doe the tops of the highest hils , nor the sinking of the lowest valleys , though they may seeme to make the earth un-even , yet compared with the whole greatnesse , doe not at all hinder the roundnesse of it , and is no bigger then a point or pins head in comparison of the highest heavens . Thus having run over the Systeme of the greater WORLD , now let us say somthing of the Compendium thereof , that is MAN . CHAP V. Of Man , or the little World . MAn is the perfection of the Creation , the glory of the Creator , the compendium of the World , the Lord of the Creatures . He is truly a Cosmus of beauty , whose eye is the Sunne of his body , by which he beholds the never resting motions of the heavens , contemplatively to behold the place of motion ; the place of his eternall rest . Lord , what is man that thou shouldest be so mindefull of him , or the son of man that thou so regardest him ? thou hast made a World of wonder in his face . Thou hast made him to be a rationall creature , endowed ▪ him with reason , so that his intellect becomes his Primum mobile , to set his action at work , nevertheles , man neither moves nor reigns in himselfe , and therefore not for himselfe , but is born not to himselfe , but for his Countrey ; therefore he ought to employ himselfe in such Arts as may be , and prove to be profitable for his Countrey . Man is the Atlas that supports the Earth , A perfect World , though in a second birth : I know not which the compleat World to call , The senslesse World , or man the rationall : One claims compleat in bignesse and in birth , Saith she 's compleat , for man was last brought forth . Man speaks again , and stands in his defence Because he 's rationall , hath compleat sense . Nature now seeing them to disagree , Sought for a means that they united be : Concluded man , that he should guide the Sphears , Limit their motion in Dayes , and Moneths , and Years : He thinking now his Office not in vain , Limits the Sun unto a Diall plain : Girdles the World in Circles , Zones , and Climes , To shew his Art unto the after times . Nature that made him thus compleat in all , To please him more , him Microcosmus call , A little world , only in this respect Of quantity , and not for his defect : Pray , Gentle Reader , view but well their feature , Which being done , pray tell me who 's the greater ? For he hath given me certain knowledge of the things that are , namely to know how the World was made , and the operations of the Elements , the beginning , ending , and midst of times , the alteration of the turning of the Sun , and the change of Seasons , the circuite of years and position of Stars , Wisd. 7. 17. The ende of the Praecognita Philosophicall . THE ARGVMENT OF THE Praecognita Astronomicall . YOu 'r come to see a sight , the World 's the stage , Perhaps you 'l sayt's but a Star-gazing age , What come you out to see ? one use an Instrument ? Can speculation yeild you such content ? That you can rest in learning but the name Of Pegasus , or of swift Charleses Wane ? And would you learn to know how he doth move About his axis , set at work by Jove ? If you would learn the practice , read and then I need not thus intreat you by my pen To tread in Arts fair steps , or to attain the way , Go on , make haste , Relinquent do not stay : Or will you scale Olympick hils so high ? Be sure you take fast hold , ASTRONOMIE : Then in that fair spread Canopie no way From thee is hid , no not Galezia . They that descend the waters deepe doe see Gods wonders in the deepe , and what they be : They that contemplate on the starry skie Do see the works that he hath fram'd so high . Learn first division of the World , and how 'T is seated , I doe come to shew you now . THE PRAECOGNITA ASTRONOMICAL . CHAP I. Of the division of the World , by accidentall scituation of the Circles . COSMUS , the World , is divided by Microcosmus the little World , into substantiall and imaginary parts : Now the substantiall are those materiall parts or substance of which the World is compacted and made a Body , by the inter-folding of one Sphear within another , as is the Sphear of Saturn , Jupiter , Mars , Sol , &c. And these of themselves have a gentle and proper motion , but by violence of the first mover , have a racked motion contrary to their own proper motion : whence it appears , that the motion of the heavens are two , one proper to the Sphears as they are different in themselves , the other common to all . By Phebus motion plainly doth appear , How many dayes doe constitute one yeare . Will you know how many days doe constitute a year , he telleth you who saith , Ter centum ter viginti cum quinque diebus Sex horas , neque plus integer Annus habet . Three hundred sixty five dayes , as appear , With six houres added , make a compleat Year . The just period of the Suns proper revolution . Perpetuus Solis distinguit tempora motus . The Imaginary part traced out by mans imagination , are Circles , such is the Horizon , the Equator , the Meridian , these Circles have of themselves no proper motion , but by alteration of place have an accidentall division , dividing the World into a right Sphear , cutting the parallels of the Sun equally or oblique , making unequall dayes and nights : whence two observations arise : First , Where the parallels of the Sun are cut equally , there is also the dayes and nights equall . Secondly , Where they are cut oblique , there also the dayes and nights are unequall . The variety of the heavens are diversly divided into Sphears , or severall Orbs , and as the Poets have found out a Galazia , the milkie way of Juno her brests , or the way by which the gods goe to their Palaces , so they will assigne to each Sphear his severall god . Goddesse of Heralts . Caliope in the highest Sphears doth dwell , Astrologie . Amongst the Stars Urania doth excell , Philosophie . Polimnia , the Sphear of Saturn guides , Gladnesse , Sterpsicore with Jupiter abides . Historie , And Clio raigneth in mans fixed Sphear . Tragedic . Melpomine guides him that gvids the year Solace . Yea , and Erata doth fair Venus sway . Loud Instruments . Mercury his Orbe Euturpe doth obey . Ditty . And horned Cynthia is become the Court Of Thalia to sing and laugh at sport . Where they take their places as they come in order . The Sphear is said to be right where the Poles have no elevation , but lie in the Horizon , so that to them the Equinoctiall is in the Zenith , that is , the point just over their heads . The Sphear is oblique in regard of its accidentall division , accidentally divided in regard of its orbicular form ; orbicular in regard of its accidentall , equall variation orbicular , it appears before in the Praecognita Philosophicall , his equall variation is seen by the equall proportion of the earth answering to a Coelestiall degree , for Circles are in proportion one to another , and parallel one to another are cut equally , so is the earth to the heavens ; having considered them as before , we will now consider another sort of Sphear , which is called parallel . This parallel Sphear is so that the parallels of the Sun are parallel to the Horizon , having the Poles in their Zenith , being the extream intemperate , colde , and frozen Zone : Ovid in his banishment complaines thus thereof . Hard is the fright in Scythia I sustain , Over my head heavens Axis doth remain . CHAP II. Of the Circles of the Horizon , the Equator , and the Meridian . THe greatest Circle of a Sphear is that which divides it in two equall parts , and that because it crosseth diametrically , and the diameter is the longest line as can be struck in a Circle , and therefore the greatest , which great Circles are represented in the following figure , representing the Circles of a Sphear in an oblique Latitude , according to the Latitude or elevation of the Pole here at London , which is 51 deg. 32 min. being North Latitude , because the North Pole is elevated . The Horizon is a great Circle dividing the part of heauen seen , from where we imagine an Antipodes , the inhabitants being to us an Antipheristasin , our direct opposites , so that while the Sun continues visible to us , it is above our Horizon , and so continues day with us , while it is night with our opposites ; and when the Sun goes down with us it appears to them , making day with them while it remaineth night with us , and according to the demonstration , is expressed by the greot Circle marked NSEW , signifying the East , West , North , and South parts of the Horizon . So now if you imagine a Circle to be drawn from the Suns leaving our sight , through those Azimuth points of heaven , then that Circle there imagined is the Horizon , and is accidentally divided as a man changes his place , and divides the World in a right or oblique Sphear . The Meridian is a great Circle scituated at right angles to the Horizon , equally passing between the East and West points , and consequently running due North and South , and passeth through the Poles of the World , being stedfastly fixed , it is represented by the great Circle marked NDSC , and is accidentally divided , if we travell East or West , but in travailing North or South altereth not , & when the Sun touches this Circle , it is then mid-day or Noon : Now if you imagine a Circle to passe from the North to the South parts of the Horizon , through your Zenith , that Circle so imagined is your Meridian , from which Meridian we account the distance of houres . The Aequinoctiall likewise divides the World in two equall parts , crossing at right angles between the two Poles , and is therefore distant from each Pole 90 degrees , and is elevated from the Horizon on the contrary side of the Poles elevation , so much as the Pole wants of 90 deg. elevation , demonstrated in the Scene by the Circle passing from A to B , and is accidentally elevated with the Poles as we change our Horizon , and when the Sun touches this Circle , the dayes and nights are then equall , and to those that live under this Citcle the dayes and nights hang in equilibra continually , and the Sun doth move every houre 15 degrees of this Circle , making the houre lines equall , passing 15 degrees in one houre , 30 degrees in two houres , 45 degrees in three houres , 60 degrees for four , and so increasing 15 degrees as you increase in houres . This I note to the intent you may know my meaning at such time as I shall have occasion ro mention the Aequinoctiall distances . The Axis of the World is that which the Stile in every Diall represents , being a line imaginary , supposed to passe through the center of the World , from the South to the North part of the Meridian , whose outmost ends are the Poles of the World , this becomes the Diameter , about which the World is imagined to be turned in a right Sphear having no elevation , in an oblique to be elevated above the Horizon and the angle at the center , numbred on the arch of the Meridian between the apparent Pole and the Horizon , is the elevation thereof , represented by the streight line passing from E to F , the arch EN being accounted the elevation thereof , which according to our demonstration is the Latitude of London . The Stars that doe attend the Artick or North Pole , are the greater and lesser Beare , the last star in the lesser Bears tale is called the Pole Star , by reason of its neerness to it : this is the guide of Mariners , as appeareth by Ovid in his exile , thus You great and lesser Bear whose Stars doe guide Sydonian and Graecian ships that glide Even you whose Poles doe view this lesser Ball , Under the Western Sea neere set at all . The stars that attend the Southern Pole is the Cross , as is seen in the Globes . Lord be my Pole , make me thy Style , Lord then Thy name shall be my terminus ad quem . Video Coelos opera manuum tuarum , lunam & stellas que tu fundasti . CHAP III. Of the severall sorts of Planes , and how they are known . DYals are the dayes limiters , and the bounders of time , whereof there are three sorts : Horizontall , Erect , Inclining : Horizontall are alwayes parallel to the Horizon : Erect , some are erect direct , others erect declining : Inclining also are direct or declining : for more explanation the figure following shall give you better satisfaction , where the Horizon marked with diverse points of the Compasse shall explain the demonstration : Now if you imagine Circles to passe through the Zenith A , crossing the Horizon in his opposite points , as from SW through the verticall point A , passing to the opposite point of South-west to North-East , those , or the like circles , are called Azimuthes , parallel to which Azimuthes all erect Sciothericals doe stand . Those Planes that lie parallel to the Horizontall Circle are called Horizontall planes , and his Style makes an angle with the Pole equall to the elevation thereof ; then the elevation of the Pole is the elevation of the Style . Erect Verticals are such which make right angles with the Horizon , and lie parallel to the Verticall point , and these , as I told you before , were either direct or declining . Direct are those that stand in a direct Azimuth , beholding one of the four Cardinall Quarters of the World , as either direct East , West , North , or South , marked with these letters NEWS , or declining from them to some other indirect Azimuth or side-lying points . Erect North and South are such as behold those Quarters , and cuts the Meridian at right angles , so that the planes crosse the Meridian due East and West , and the Poles are their Styles , equally elevated according to the aequinoctiall altitude , being the complement of the Poles elevation . For in all North Faces , Planes , or Dials , the Style beholds the North Pole , and in all South faces , the Style beholds the South Pole : therefore , where the North Pole is elevated , there the North Pole must be pointed out by the Style , and where the South Pole is elevated vice versa . The second sort of Verticals are declining , which ate such that make an acute angle with the Quarter from which they decline ; for an acute angle is lesse then a right angle , and a right angle is 90 degrees : these declining Planes lying in some accidentall Azimuthe . For supposing a Diall to turn from the South or North towards the East or West , the Meridian line of the South declines Eastward , happening in these Azimuthes or between them . South declining East South declining West S by E 11 15 Or to these points of the West decliners , or between them . S by W 11 15 S S E 22 30 S S W 22 30 S E by S 33 45 S W by S 33 45 South-East 45 00 South West 45 00 S. E by E 56 15 S W by W 56 15 E S E 67 30 W S W 67 30 E by S 78 45 W by S 78 45 East 90 00 West . 90 00 Again , North decliners , declining toward the East and West , doe happen in these Azimuthes or between them . North declining East North declining West N by E 11 15 Or to these points of the West decliners , or between them . N by W 11 15 N N E 22 30 N N W 22 30 N E by N 33 45 N W by N 33 45 North-East 45 00 North West 45 00 N E by E 56 15 N W by W 56 15 E N E 67 30 W N W 67 30 E by N 78 45 W by N 78 45 East . 90 00 West . 90 00 By which it appeareth that every point of the Compasse is distant from the Meridian 11 degrees 15 minutes . The third sort of planes are inclining , or rather reclining , whose upper face beholds the Zenith , and in that respect is called Reclining , but if a Diall be made on the nether side , and thereby respect the Horizon , it is then called an incliner , so that the one is the opposite to the other . These planes are likewise accidentally divided , for they are either direct recliners , reclining from the direct points of East , West , North ; and South , and in this sort happens the direct Polar and Aequinoctiall planes , as infinite more according to the inclination or reclination of the plane , or they are as erect planes doe become declining recliners , which looke oblique to the Cardinall parts of the World , and obtusely to the parts they respect . Suppose a plane to fall backward from the Zenith , and by consequence it falls towards the Horizon ; then that represents a Reclining plane , such you shall you suppose the Aequinoctiall Circle in the figure to represent , reclining from the North Southwards 51 degrees from the Zenith , or suppose the Axis to represent a plane lying parallel to it , which falls from the Zenith Northward reclining 38 degrees , one being Aequinoctiall , the other a Polar plane . But for the inclining decliners you shall know them thus , forasmuch as the Horizon is the limiter of our sight , and being cut at right angles representeth the East , West , North , and South points , it may happen so that a plane may lie between two of these quarters in an accidentall Azimuth , and so not beholding one of the Cardinall Quarters is said to decline : Again , the said plain may happen not to stand Verticall , which is either Inclining or Reclining , and so are said to be Inclining Decliners : First , because they make no right angle with the Cardinal Quarters : Secondly , because they are not Verticall or upright . There are other Polar planes , which lie parallel to the Poles under the Meridian , which may justly be called Meridian plains , and these are erect direct East and West Dials , where the poles of the plane remain , which planes if they recline , are called Position planes , cutting the Horizon in the North and South points , for Circles of position are nothing but Circles crossing the Horizon in those points . CHAP IV. Shewing the finding out of a Meridian Line after many wayes , and the Declination of a Plane . A Meridian Line is nothing else but a line whose outmost ends point due North and South , and consequently lying under the Meridian Circle , and the Sun comming to the Meridian doth then cast the shadow of all things Northward in our Latitude ; so that a line drawn through the shadow of any thing perpendicularly eraised , the Sun being in the Meridian , that line so drawn is a Meridian line , the use whereof is to place planes in a due scituation to their points respective , as in the definition of this Circle I shewed there was accidentall Meridians as many as can be imagined between place and place , which difference of Meridians is the Longitude , or rather difference of Longitude , which is the space of two Meridians , which shews why noon is sooner to some then others . The Meridian may be found divers wayes , as most commonly by the Mariners compasse , but by reason the needle hath a point attractive subject to errour , and so overthroweth the labour , I cease to speake any further . It may be found in the night , for when the starre called Aliot , seems to be over the Pole-starre , they are then true North , the manner of finding it , Mr. Foster ▪ hath plainly laid down in his book of Dyalling , performed by a Quadrant , which is the fourth part of a circle , being parted into 90 degrees . It may also be fouhd as Master Blundevile in his Booke for the Sea teacheth , being indeed a thing very necessary for the Sea , which way is thus : Strike a Circle on a plain Superficies , and raise a wire , or such like , in the center to cast a shadow , then observe in the forenoon when the shadow is so that it just touches the circumference or edge of the Circle , and there make a mark ; doe so again in the afternoon , and at the edge where the shadow goes out make another mark , between which two marks draw a line ; which part in halfe , then from that middle point to the center draw a line which is a true Meridian . Or thus , Draw a great many Circles concentricall one within another , then observe by the Circles about noone when the Sun casts the shortest shadow , and that then shall represent a true Meridian , the reason why you must observe the length of the shadow by circles & not by lines is , because if the Sun have not attained to the true Meridian it wil cast its shadow from a line , and so my eye may deceive me , when as by Circles the Sun casting shadow round about , still meetes with one circumference or other , and so we may observe diligently . Secondly , it is proved that the shadow in the Meridian is the shortest , because the Sun is neerest the Verticall point . Thirdly , it is proved that it is a true Meridian for this cause , the Sun , as all other Luminous bodies , casts his shadow diametrically , and so being in the South part casts his shadow northward , and is therefore a true Meridian . But now to finde the declination of a wall , if it be an erect wall draw a perpendicular line , but if it be a declining reclining plane , draw first an horizontall line , and then draw a perpendicular to that , and in the perpendicular line strike a Style or small Wyre to make right angles with the plane , then note when the shadow of the Style falleth in one line with the perpendicular , and at that instant take the altitude of the Sun , and so get the Azimuthe reckoned from the South , for that is the true declination of the wall from the South . The distance of the Azimuthes from the South , or other points , are mentioned in degrees and minutes in the third Chapter , in the definition of the severall sorts of planes : or by holding the streight side of any thing against the wall , as is the long Square ABCD , whose edge AB suppose to be held to a wall , and suppose again that you hold a thrid and plummet in your hand at E , the Sun shining , and it cast shadow the line EF , and at the same instant take the altitude of the Sun , thereby getting the Azimuthe as is taught following , then from the point F , as the center of the Horizon . , and from the line FE , reckon the distance of the South , which suppose I finde the Azimuthe to be 60 degrees from the East or West , by the propositions that are delivered in the end of this Booke , and because there is a Quadrant of a Circle between the South , and the East or West points , I substract the distance of the Azimuthe from 90 degrees , and it shall leave 30 , which is the declination of the wall , equall to the angle EFG : but to finde the inclination or reclination , I shall shew when I come to the use of the Universall Quadrant , or having first found the Meridian line , you may prick down the Azimuthe . CHAP V. Shewing what houre-lines may be drawn upon any Plane . LIght being the cause primary of shadows , shadows being but the imitation of the secondary cause , that is substance , doth delineate unto us the passing away of time , by receiving light on the substance casting shadow . The Sun , though he never moves from the line Ecliptique wherein he hath his annuall or yearly motion , yet have a declination from the Aequinoctiall North or South , making his diurnall or daily motion , altering the dayes and nights according to all the diversities thereof : for the Sun being in the Aequinoctiall hath no declination , but in his diurnall motion still declyning from the Aequinoctiall makes his progresse towards the North or South , describeth many parallel Circles , being parallel to the Aequinoctiall , whose farthest distance from either side is 23 deg. 30 minutes , so that so many degrees that the Sun is distant from the Aequinoctiall , so much is its declination . Now if you imagine the Circle before described to represent the Meridian Circle which crossed diametrically , which diameter shall represent the Aequinoctiall , then laying down the greatest declination , on either side of it , drawing two lines at that distance , on either side of the Aequinoctiall , parallel to it , represent the Tropicks , the upper representing the Tropick of Cancer , marked with GE , the other the Tropick of Capricorn , marked with HI : and if from each severall degree you draw parallels too , they doe represent the parallels of the Sun , which shall shew the diurnall motion of the Sun : now if you crosse these parallels with a line from E to H , that then represents the Ecliptique ; now if you crosse the Aequino-Ctiall at right angles with another line , that line represents the Axis of the World : then if you lay down from the Poles the elevation thereof , to wit , the North and South Poles , according to the elevation of the North Pole downward , where the number of degrees end make a mark ; then account the same elevation from the South Pole upward , and there also make a mark , from which two marks draw a right line , which shall represent your Horizon , and cuts the parallels of the Sun according to the time of his abiding above the Horizon . First , An East and West Diall lies parallel to the Meridian , therefore the Sun in the Meridian cannot shine on them ; neverthelesse , though an East and West Diall cannot have the houre of 12 on it , yet an East or West position may , because it crosseth the Horizon in the North and South . Secondly , A direct North Diall can have but morning and evening houres on it , and then of no use but when the Sun hath North declination , for then his Amplitude or distance from the East and West is Northward , and so at morning or night shines on the face thereof . Thirdly , A North reclining may shew all the houres all the year , if it recline from the North Southward , the quantity of the complement of the least Meridian altitude , but if but the complement of the elevation of the Aequinoctiall , and so become a Polar Plane , it can then but shew while the Sun is in the North Signes , for the Dyall lying parallel to the Aequinoctiall while the Sun is in South declination cannot shine on the plane because it lies under . All upright planes declining from the South may have the houre line of 12 , so also may all North decliners , but not in the Temperate Zone , which is contained between the degrees . South incliners also may have the line of 12 , whose upper face is not below the least Meridian altitude , as also if greater then the greatest Meridian altitude , then doth the upper face want it . Fifthly , all North recliners reclining more then the greatest meridian altitudes complement , may have all the houres but will shew but one part of the yeare . Sixthly , All South declinets or recliners may have the line of 12 on them . And now having proceeded thus far in some theoricall demonstration or grounds of Dials for the Geometricall projection , we will in the next Chapter lay down the theoricall demonstration for the Arithmeticall Calculation , and so proceed to our practicall way of operation as ensueth . CHAP VI . Being the definition of the severall lines of Sines , Tangents , and Secants , to be understood before we can come to Arithmeticall Calculation . A Tangent is a right line without the peripherie to the extremity of the Secant to the Radius being perpendicular eraised , such is represented by the line BC. A Secant is a right line drawn from the center through the circumference to the Tangent , such is represented by the line AB , the Semidiameter of the same Circle is called the Radius . You may furthermore for very convenient uses have those lines placed on a Ruler , for if from one degree of one Quadrant of a Semicircle you draw lines to the same degree of the other Quadrant , cutting the line GA , that line so cut shall be a line of Sines , and if from the centre you draw lines to the Tangent line through every degree of the Quadrant , that line so cut is a Tangent line , whose use is most exquisite and infinite for the solution of many excellent propositions . CHAP VII . Being the fundamentall Diagram for the Geometricall projection of Dials . THe Style being the representation of the Axis of the World , doth become the Gnomon or substance casting shadow on all Planes lying parallel to some Circle or other , as to circles of Azimuthes in all Verticall Dials . So that the figure following is a representation of divers semidiameters , doth plainly shew the theoricall ground of the practick part hereof . Where the line in the demonstration , noted the semidiameter of the Horizon , signifies the Horizon , for so supposing it to represent an Horizontall Diall , the style or Axis must be elevated above it , according to the Poles elevation above the Horizon , and then the semidiameter or Axis of the World represents the style or Axis casting shadow being the line AC . The Geometricall projection of Dials . Where note by the way , that if you set one foot of the Compasses in B , and with the Semidiameter of the Equator , fix the other foot in the line BC , keeping that last foot fast , and at that center draw a Quadrant divided into six parts , & a ruler from the center of the Equator through each division , shall divide the line AB as a contingent line , and if from C to these marks on the line AB you draw lines , it shall be the houre lines of a Verticall Diall . But supposing a Diall to stand verticall , or upright to the Horizon AB , as the line BC , then that is represented by the semidiameter of the Verticall , and his style again represented by the semidiameter or Axis AC , being distant from the Verticall equall to the complement of the Poles elevation , and again , the Aequinoctiall crossing the Axis at right angles , the semidiameter thereof is represented by the line BD , the reason why the angle at A hath to his opposite angle at C , the complement of the angle at A , is grounded on this , the three angles of any right lined triangle are equall to two right angles , and a right angle consists of 90 degrees : now the angle at B is 90 degrees , being one right angle , and the angle at A being an angle of 51 degrees , which wants of 90 39 degrees , which is the angle at C , all which being added together doe make 180 degrees , being two right angles : here you see that having two angles , the third is the complement of 180 degrees . CHAP VIII . Of the proportion of shadows to their Bodies . SEeing the Zenith makes right angles with the Horizon , and a right angle consisteth of 90 degrees , the middle point betwixt both is 45 degrees , the Sun being at that height , the shadow of all things perpendicularly raised , are equal to their bodies , so also is the Radius of a Circle equall to the Tangent of 45 degrees : and if the Sunne be lower then 45 degrees it must necessary follow the shadow must exceed the substance , because the Sun is nigh the Horizon , and this is called the adverse or contrary shadow . Contrarily , if the Sun exceed this middle point , the substance then exceeds the shadow , because the Sun is neerer the verticall point . Mr. Diggs in his Pantometria laying down the manifold uses of his Quadrant Geometricall , doth there shew , that having received the Sun beams through the Pinacides or Sights , that when the Suns altitude cuts the parts of right shadow , then the shadow exceeds the substance erected casting shadow as 12 exceeds the parts cut : But in contrary shadow contrary effects . CHAP IX . To finde the Declination of the Sun . TO give you Orontius his words , it is convenient to take the beginning from the greatest obliquation of the Sun , because on that almost the whole harmony of all Astronomicall matters seeme to depend , as shall be manifest from the discourse of the succeeding Canons . Wherefore prepare of commodious and elect substance , a Quadrant of a Circle parted into 90 equall parts , on whose right angled Radius let be placed two pinacides or sights to receive the beams of the Sun . Then erect it toward the South in the time of the Solsticials , either in Cancer the highest annuall Almicanther , or in Capricorn the lowest annuall ▪ meridian altitude , also observe the equilibra , or equality of day and night in the time of the Aequinoctials , from the Meridian altitude thereof substract the least Meridian altitude , which is , when the Sun enters in the first minute of Capricorn , the remainer is the Declination , or substract the Aequinoctiall altitude from the greatest Meridian altitude , the remainer is the Declination of the greatest obliquity of the Sun in the Zodiaque . The height of the Sun is observed by the Quadrant when the beames are received through the sights by a plummet proceeding from the center , noting the degree of altitude by the thrid falling thereon . You may also take notice that for the continuall variation of the Suns greatest declination it ought to be observed by faithful Instruments : for as Orontius notes that Claudius , Ptolomie found it to be 23 degrees 51 minutes and 20 seconds , but in the time of Albatigine the same number of degrees yet but 35 minutes , Alcmeon found it of little lesse , to wit 33 minutes , Purbachi and some of his Disciples doe affirme the same to be 23 degrees only 28 minutes , yet Johanes Regiomontan . in the tables of Directions , hath alotted the minutes to be 30 , but since Dominick Maria an Italian , and Johannes Varner of Norimburg testifie to have found it to be 29 minutes , to which observation our works doe exactly agree . Albeit all did observe the same well neere by like Instruments , neverthelesse , not justly by exact construction , or by insufficient dexterity of observation some small difference might happen , but not so much as from Ptolomie to our time . Having this greatest Declination , to finde the present Declination is thus , by calculation : As the Radius , is to the Sine of the greatest Declination ; so is the Sine of the Suns distance from the next Aequinoctiall point , that is Aries or Libra , to the declination required : wherefore in the Naturall Sines , as in the Rule of Proportion , multiply the second by the third , divide by the first , the Quotient is the Sine of the Declination . Or by the naturall Sines , adde the second and third , and substract the first , the remainer is the Sine of the present Declination . Degre . ♈ ♎ ♉ ♏ ♊ ♐ Degre . D m D m D m 0 0 0 11 29 20 10 30 1 0 24 11 50 20 23 29 2 0 47 12 11 20 35 28 3 1 11 12 31 20 47 27 4 1 35 12 52 20 58 26 5 1 59 13 12 21 9 25 6 2 23 13 32 21 20 24 7 2 47 13 52 21 30 23 8 3 10 14 11 21 40 22 9 3 34 14 30 21 49 21 10 3 58 14 50 21 58 20 11 4 21 15 8 22 7 19 12 4 45 15 27 22 15 18 13 5 8 15 45 22 23 17 14 5 31 16 3 22 30 16 15 5 55 16 21 22 37 15 16 6 18 16 38 22 43 14 17 6 41 16 56 22 50 13 18 7 4 17 12 22 55 12 19 7 27 17 29 23 0 11 20 7 49 17 45 23 5 10 21 8 12 18 1 23 9 9 22 8 34 18 17 23 13 8 23 8 57 18 32 23 17 7 24 9 19 18 47 23 20 6 25 9 41 19 2 23 22 5 26 10 3 19 16 23 24 4 27 10 25 19 30 23 26 3 28 10 46 19 44 23 27 2 29 11 8 19 57 23 27 1 30 11 29 20 10 23 28 0 De ♓ ♍ ♒ ♌ ♑ ♋ De But I have here added a Table of Declination of the part of the Ecliptique from the Aequinoctiall , the use whereof you may discern is very plain , for if you finde the Signe on the top , and the degrees downward , the common angle shall be the Declination of the Sun that day . As if the Sun being in the 10 degree of Taurus or Scorpio , the declination shall bee 14 degrees 50 minutes , and if you finde the Signe in the bottome , you shall seeke the degrees on the right hand upward , so the 20 degreee of Leo or Aquarius hath the same declination with the former . The ende of the Praecognita Astronomicall . THE ARGVMENT OF Practicall Sciothericy Optical . REader , read this , for I dare this defend , Thy posting life on Dials doth depend , Consider thou how quick the houre 's gone , Alive to day , to morrow life is done : Then use thy time , and alwayes beare in minde , Times hary forehead , yet he 's ball'd behinde , Here 's that that will deline to thee and shew How quick time runs , how fast thy life doth goe : Yet ( festina lente ) learn the praecognit part , And so attain to practice of this art , Whereby you shall be able for to trace Out such a path , where Sol shall run his race , And make the greater Cosmus to appear , Delineating day and time of year . Horologium Vitae . Latus ad occasum , nunquam rediturus ad ortum Vivo hodie , moriar cras , here natus eram . HOROLOGIOGRAPHIA OPTICA . CHAP I. Shewing the making of an Horizontall plane to an Oblique Spheare . FRom the Theoricall Demonstration before , take the semidiameter of the Horizon with your Compasses , then draw the line AB , representing the Meridian or line of 12 , and setting one foot in A , describe the Quadrant CAB , and CA must be at right angles to AB , to which Quadrant draw the tangent line FA , which is the line of contingence , then take from the Theorical demonstration the semidiameter of the Aequator , and placing that on the line AB desctibe a quadrant touching the line of contingence also within the other , represented by the Quadrant H e I which divide into 6 parts , and a Ruler laid to the center e , make marks where the Ruler toucheth the line of contingence , which must be continued beyond F , that so the houre lines may meet with the line BF , where it crosseth that line make marks : then removing the Ruler to the center A of the horizontall Semicircle , draw lines through each mark of the line of contingence which shall be the houres , number the morning houres from the Meridian towards your left hand , and evening or afternoon houres towards the right . The Style must be an angle equall to the elevation of the Pole , the 12 houre must lie under the Meridian Circle . The Arithmeticall Calculation . As the Radius , Is to the Tangent of the Aequinoctiall distance of the houre from the Meridian ; So is the Signe of the elevation of the Pole , To the Tangent of the houres distance from the Meridian . The definition of the Aequinoctiall distance is in the definition of the Aequinoctiall Circle , Chap. 1. Praecognita Astronomicall . The figure of an Horizontall Diall , for the Latitude of London 51d . 30m . South The houres of the afternoon must be the same distance from the Meridian , 1 and 11 , 2 and 10 , 3 and 9 , and so of the rest , this is very plain , neither wants any expositor , only you may on the Horizontal plane , prick down beyond the houre of 6 a clock , the morning houres of 4 and 5 , and the evening houres of 7 and 8 , by reason that the Sun wil shine on the Horizontall plane as soone as it is above the Horizon . The figure of a South Verticall plane , for the Latitude of London , which is parallel to the Prime Verticall . The semidiameter of the Verticall is but the Tangent of the elevation of the Pole to the Radius of the Horizon . And the semidiameter of the Horizon , the Tangent of the elevation of the Equator to the Radius of the Verticall . CHAP II. Shewing the making of a direct Verticall Diall for an Oblique Sphear , that is , a direct North or South Diall plane . EVery plane hath a Verticall point , and for the making of a Verticall Diall for the Latitude of London , out of the theoricall demonstration Chap. 7. praecog. . Astron ▪ take the semidiameter of the Verticall , and with that , as with the semidiameter of the Horizon , describe a Quadrant , & draw the tangent line FG , and with the semidiameter of the Aequator finish all as in the Horizontall : the Style must proceed from the center A , and be elevated from the Meridian line AF , so much as is the complement of the Elevation of the Pole , and must point toward the invisible Pole , viz. the South Pole , and hath but 12 houres on it . The Arithmeticall Calculation . As the Radius , Is to the tangent of the Aequinoctiall distance of the houre from the Meridian ; So is the Co-sine , that is , the complement sine of the elevation , to the tangent of the houre distance from the Meridian required . CHAP III. Shewing the making of a direct North Verticall Diall for an Oblique Sphear , as also a more easie way of drawing the South or Horizontall Planes . THe North Diall is but the back side of the South Diall ; and differeth little from it , but in naming of the houres , for accounting the sixth houre from the Meridian in the direct South verticall , to be the same in the direct North Verticall , and accounting the first houres on the East side of the South , on the West side of the North plane , and so vice versa , the first houres on the West side of the South , on the East side of the North plane , as by the figure appeareth . And because the North Pole is elevated , the Style must point up toward it the visible Pole . It must have but the first and last houres of the South plane , because the Sun never shines but at evening or morning on a North wall in an oblique Sphear , and but in Somer , because then the Sun hath North Declination , but in a right Sphear , it may shew all the houres as a South Diall , but for a season of the yeare . But if you will make the Verticall plane or Horizontall in a long angled Parallelogram , you shall take the Secant of the elevation of the Pole , which is the same with AC in the fundamentall Diagram , and make that your Meridian line , and shall take the Sine of the elevation of the Pole above the Meridian , which in a direct South or North is equall to the elevation of the Aequinoctiall , and in the Fundamentall Diagram is the line DE , and prick it down from A and C at right angles with the line AC , and so inclose the long square BADBCD , it shall be the boūds of a direct North or South Diall ; lastly , if from the fundamental diagram you prick down the several tangents of 15 , 30 45 , from Band D on the lines BB and DD , & the same distances from C toward B and D , & lastly if from the center A , you draw lines to every one of those marks , they shall be the houre-lines of an erect direct South Diall . To make an Horizontall Diall by the same projection you shall take the Secant of 38 deg. 30 min. the elevation of the Equator , which in the fundamentall Scheme is the line AF , for the Meridian , and the Sine of the elevation of the Pole , which in the fundamentall diagram is the same with DA , and prick that down from the Meridian at right angles both wayes , as in the former planes , and so proceed as before from the six of clock houre and the Meridian , with the severall Tangents of 15 , 30 , 45 , you shall have constituted a Horizontall plane . I have caused the pricked line that goes crosse , and the other pricked lines which are above the houre line of six , to be drawn only to save the making of a figure for the North direct Diall , which is presented to you if you turn the Book upside down , by this figure , contained between the figures of 4 , 5 , 6 , the morning houres , and 6 , 7 , 8 , the evening . And because the North pole is elevated above this plane 38 deg. 30 min. the Axis must be from the center according to that elevation , pointing upward as the South doth downward , so as A is the Zenith of the South , C must be in the North . The Arithmeticall calculation is the same with the former , also a North plane may shew all the houres of the South by consideration of reflection : For by Opticall demonstration it is proved , that the angles of incidence is all one to that of reflection : if any be ignorant thereof , I purposely remit to teach it , to whet the ingenious Reader in labouring to finde it . The Figure of a direct East and West Diall for the Latitude of London , ▪ 51 deg. 30 min. East Diall . West Diall . CHAP IV. Shewing the making of the Prime Verticall planes , that is , a direct East or West Diall . FOr the effecting of this Diall , first draw the line AD , on one end thereof draw the circle in the figure representing the Equator ; then draw two touch lines to the Equator , parallel to the line AD , these are they on which the houres are marked : divide the Equator in the lower semicircle in 12 equall parts , then apply a ruler to the center , through each part , and where it touches the lines of contingence make marks ; from each touch point draw lines to the opposite touch point , which are the parallels of the houres , and at the end of those lines mark the Easterly houres from 6 to 11 , and of the West from 1 to 6. These planes , as I told you , want the Meridian houre , because it is parallel to the Meridian . Now for the placing of the East Diall , number the elevation of the Axis , to wit , the arch DC , from the line of the Equator , to wit , the line AD : and in the West Diall number the elevation to B ; fasten a plummet and thrid in the center A , and hold it so that the plummet may fall on the line AC for the East Diall , and AB for the West Diall , and then the line AD is parallel to the Equator , and the Dial in its right position . And thus the West as well as East , for according to the saying , Contrariorum eadem est doctrina , contraries have one manner of doctrine . Here you may perceive the use of Tangent line , for it is evident that every houres distance is ●●t the Tangent of the Aequinoctiall distance . The Arithmeticall Calculation . 1 Having drawn a line for the houre of 6 , whether East or West , As the tangent of the houre distance , is to the Radius , so is the distance of the houre from 6 , to the height of the Style . 2 As the Radius is to the height of the Style , so is the tangent of the houre distance from 6 , to the distance of the same houre from the substyle . The style must be equall in height to the semidiameter of the Equator , and fixed on the line of 6. CHAP V. Shewing the making a direct parallel Polar plane , or opposite Aequinoctiall . I Call this a direct parallel Polar plane for this cause , because all planes may be called by their scituation of their Poles , and so an Aequinoctiall parallel plane , may be called a Polar plane , because the Poles thereof lie in the poles of the World . The Gnomon must be a quadrangled Parallelogram , whose height is equall to the semidiameter of the Equator , as in the East and West Dials , so likewise these houres are Tangents to the Equator . Arithmeticall calculation . Draw first a line representing the Meridian , or 12 a clock line , and another parallel to the said line for some houre which may have place on the line , say , As the tangent of that houre is to the Radius , so is the distance of that houre from the Meridian to the height of the Style . 2 As the Radius is to the height of the style , so the tangent of any houre , to the distance of that houre from the Meridian . CHAP VI . Shewing the making of a direct opposite polar plane , or parallel Aequinoctiall Diall . AN Aequinoctiall plane lyeth parallel to the Aequinoctiall Circle , making an angle at the Horizon equal to the elevation of the said Circle : the poles of which plane lie in the poles of the world . The making of this plane requires little instruction , for by drawing a Circle , and divide it into 24 parts the plane is prepared , all fixing a style in the center at right angles to the plane . As the Radins , is to the sine of declination , so is the co-tangent of the Poles height , to the tangent of the distance of the sub-stile from the Meridian . If you draw lines from 7 to 5 on each side , those lines so cut shall be the places of the houre lines of a parallel polar plane , now if you draw to each opposite from the pricked lines , those lines shall be the houre lines of the former plane . CHAP VII . Shewing the making of an erect Verticall declining Diall . IF you will work by the fundamentall Diagram , you shall first draw a line , such is the line AB , representing the Meridian , then shall you take out of the fundamentall diagram the Secant of the Latitude , viz. AC , and prick it down from A to B , and at B you shall draw a horizontall line at right angles , such is the line CD , then you shall continue the line AB toward i , and from that line , and where the line AB crosseth in CD , describe an arch equall to the angle of Declination toward F if it decline Eastward , and toward G if the plane decline Westward . Then shall you prick down on the line BF , if it bean Easterly declining plane , or from B to G if contrary ; the Secant complement of the Latitude , viz. AG in the fundamentall Diagram , and the Sine of 51 degrees , viz. DA , which is all one with the semidiameter of the Equator , and therewithall prick it down at right angles to the line of declination , viz. BF , from B to H and G , and from F towards K and L , then draw the long square KIKL , and from B toward H and G , prick down the severall tangents of 15 , 30 , 45 , and prick the same distance from K and L towards H and G : lastly , draw lines through each of those points from F to the horizontall line CD , and where they end on that line to each point draw the houre lines from the point A , which plane in our example is a Verticall declining eastward ▪ 45 degrees , and it is finished . But because the contingent line will run out so far before it be intersected , I shall give you one following Geometricall example to prick down a declining Diall in a right angled parallelogram . Now for the Arithmeticall calculation , the first operation shall be thus : As the Radius , to the co-tangent of the elevation , so is the sine of the declination , to the tangent of the substiles distance from the meridian of the place . then , II Operation . Having the complement of the declination and elevation , finde the styles height above the sub-stile , thus , As the Radius , to the co-sine of the declination , so the co-sine of the elevation , to the sine of the styles height above the substyle . III Operation . As the sine of elevation , is to the Radius , so the tangent of declination , to the tangent of the inclination of the Meridian of the plane to the Meridian of the place . IV Operation . Having the styles height above the substyle , and the angle at the pole comprehended between the houre given and the meridian of the plane say . As the Radius , to the sine of the styles height above the substyle ; so is the tangent of the angle at the pole , comprehended between the houre given and the meridian of the plane , to the tangent of the houre distance from the substyle . Thus the Arithmeticall way being laid down , another Geometricall follows . YOu shall first on the semidiameter of the Horizon , viz. AB , describe the arch BC the declination of the plane , and BD the complement of the elevation of the pole , then shall you draw the lines AC and AD , and at B you shall raise the perpendicular DCB . The Figure of an upright plane declining from the South Eastward 30 degrees . Now good Reader , labour to understand my plaine meaning in this , labouring only not to confound thy memory or capacity , & therefore give you also to understand that such are the houre distances of a Westerly declining plane , as are those of an Easterly , only changing the side of the plane , and naming it by the complementall houres , the complemental houres I call those that added together make 12 , as followeth . Forenoon houres of the declining East plane . 6 Complemental houres are 6 are afternoon houres of a declining West plane . 7 5 8 4 9 3 10 2 11 1 So that if the houres of the Easterly declining plane be 6 , 7 , 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , the houres of the Westerly declining Diall is 6 , 5 , 4 , 3 , 2 , 1 , 12 , 11 , 10 , 9 , stil keeping the same distances of the houre lines in one as the other , so that if an Easterly declining be but turned the back side , it represents a Westerly declining Dial as much , and the style must stand over his substyle , and whereabouts the houre lines are closest or neerest together , thereabout is the substyle . Now having shewed you the making of all Horizontall and Verticall , whether direct or declining , Polar or Aequinoctiall , I shall proceed to shew the projecting of those which are oblique , whether declining reclining , or inclining , reclining , &c. whereto , for the more ease , I have calculated to every degree of a Quadrant the houre arches of the Horizontall planes , from one degree of elevation till the Pole is in the Zenith . The Table and use followeth in severall Chapters . Here followeth the Table of the arches of the houre lines distance from the Meridian in all Horizons , from one degree of elevation , till the Pole is elevated 30 degrees , by which is made all direct Murall , whether upright , or reclining Dials .   1 11 2 10 3 9 4 8 5 7 6 6 1 0 16 0 35 1 00 1 44 3 43   2 0 32 1 9 2 00 3 27 7 25   3 0 48 1 44 3 00 5 11 11 3   4 1 5 2 19 4 00 6 54 14 36   5 1 20 2 53 4 59 8 65 18 1   6 1 36 3 27 5 58 10 16 21 19   7 1 52 4 1 6 57 11 55 24 27   8 2 8 4 35 7 54 15 9 30 3   9 2 24 5 9 8 54 15 9 30 3   10 2 40 5 44 9 51 16 45 32 57   11 2 56 6 18 10 48 18 18 35 27   12 3 11 6 51 11 44 19 48 37 49   13 2 27 7 24 12 41 21 17 40 1   14 3 46 7 57 13 36 22 44 42 5   15 3 59 8 30 14 31 24 9 44 0   16 4 14 9 2 15 25 25 31 45 49   17 4 28 9 35 16 17 26 51 47 30   18 4 44 10 7 17 10 28 9 49 4   19 5 15 11 10 18 53 30 39 51 55   20 5 15 11 10 18 53 30 39 51 55   21 5 29 11 41 19 43 31 50 53 13   22 5 44 12 13 21 20 34 5 55 34   23 5 58 12 43 21 20 34 5 55 34   24 6 13 13 13 22 8 35 10 56 37   25 6 28 13 43 22 54 36 12 57 37   26 6 42 14 12 23 40 37 13 58 34   27 6 57 14 41 24 25 38 11 59 27   28 7 10 15 00 25 9 39 11 60 37   29 7 24 15 39 25 52 40 2 61 4   30 7 38 16 6 26 36 40 54 61 49   The continuation of the arches of the Horizontall planes , from 30 to 60 deg. of elevation of the Pole   1 11 2 10 3 9 4 8 5 7 6 6 31 7 51 16 34 27 15 41 44 62 30   32 8 5 17 1 27 55 42 32 63 11   33 8 19 17 27 28 37 43 20 63 49   34 8 31 17 54 29 13 44 5 64 24   35 8 44 18 20 29 50 44 49 64 58   36 8 57 18 45 30 27 45 31 65 30   37 9 10 19 9 31 2 46 12 66 ●0   38 9 22 19 34 31 37 46 50 66 29   39 9 24 19 58 32 11 47 28 66 56   40 9 47 20 22 32 44 48 4 67 23   41 9 58 20 45 33 16 48 39 67 47   42 10 10 21 7 33 47 49 13 68 10   43 10 21 21 30 34 18 49 45 68 33   44 10 32 21 51 34 47 50 16 68 55   45 10 44 21 45 35 16 50 46 69 15   46 10 54 22 33 35 53 51 15 69 34   47 11 6 22 54 36 11 51 43 69 53   48 11 16 23 14 36 37 52 9 70 11   49 11 26 23 33 37 2 52 35 70 27   50 11 36 23 51 37 27 53 0 70 43   51 11 46 24 10 37 52 53 24 71 13   52 11 55 24 57 38 14 53 46 71 24   53 12 5 24 45 38 37 54 8 71 27   54 12 14 25 2 38 58 54 30 71 40   55 12 23 25 19 39 19 54 49 71 53   56 12 32 25 35 39 39 55 9 72 5   57 12 40 25 51 39 59 55 28 72 17   58 12 48 26 5 40 18 55 45 72 28   59 12 56 26 20 40 36 56 2 72 39   60 13 4 26 33 40 54 56 19 72 49   The continuation of the arches of the Horizontall planes , from 60 deg. of elevation , till the Pole is in the Zenith .   1 11 2 10 3 9 4 8 5 7 6 6 61 13 11 26 48 41 10 56 34 72 58   62 13 18 27 1 41 26 56 49 73 7   63 13 25 27 13 41 42 57 3 73 16   64 13 32 27 26 41 57 57 17 73 24   65 13 39 27 37 42 11 57 30 73 32   66 13 45 27 49 42 25 57 42 73 39   67 13 52 27 59 42 38 57 54 73 46   68 13 56 28 9 42 50 58 5 73 53   69 14 3 28 19 43 2 58 16 73 59   70 14 8 28 29 43 13 58 26 74 5   71 14 13 28 38 43 24 58 36 74 11   72 14 18 28 46 43 34 58 44 74 16   73 14 22 28 55 43 43 58 53 74 21   74 14 26 29 2 43 52 59 1 74 25   75 14 30 29 9 44 0 59 8 74 29   76 14 34 29 5 44 8 59 15 74 34   77 14 38 29 22 44 15 59 21 74 37   78 14 41 29 27 44 22 59 26 74 41   79 14 44 29 32 44 28 59 32 74 44   80 14 47 29 37 44 34 59 37 74 47   81 14 49 29 42 44 39 59 41 74 49   82 14 51 29 44 44 43 59 43 74 50   83 14 53 29 49 44 47 59 49 74 53   84 14 55 29 52 44 51 59 52 74 55   85 14 57 29 54 44 53 59 54 74 57   86 14 58 29 56 44 56 59 57 74 58   87 14 59 29 58 44 58 59 58 74 59   88 14 59 29 59 44 59 59 58 74 59   89 14 59 30   44 59 59 59 75     90 15   30   45   60   75     CHAP VIII . Shewing the use of this Table both in Verticall and Horizontall planes . FOr an Horizontall Diall enter the Table with the elevation of the Pole on the left hand , and the arches noted against the houres and the elevation found , are the distance of the houres from the Meridian . For a Verticall or direct South or North , enter the Table with the complement of the elevation on the right side , and the common meeting of the houres at top , and the complement of elevation , is the distance of the houres from the Meridian in the said plane . For every horizontall plane is a direct Verticall in that place whose Latitude or distance of their Zenith from the Aequator , is equall to the complement of the elevation of the Horizontall planes Axis or style . As to make an Horizontall Diall for the Latitude of 51 degrees , I enter the Table and finde these Arches for 1 and 11 , for 2 and 10 , &c. Now the same distances are the distances of the houre lines of a direct South plane , where the Pole is elevated the complement of 51 degrees , that is 39 degrees , for 51 and 39 together doe make 90. So to make a Verticall diall , I enter the Table with 39 , the complement of the elevation of the pole , and finde the arches answering to 1 and 11 , to 2 and 10 , &c. Thus much in generall of the use of the Table , now followeth the use in speciall . CHAP IX . Shewing the use of the Tables in making any Declining or Inclining direct Dials . LEt the great Circle ABCD represent the Meridian , A the North , and C the South , then the line EF represents a South reclining plane , while it fals back from the South Northward , and represents an inclining plane while it respects the Horizon . This is sufficiently discussed before . So much as the plane reclines northward beyond the complement of the elevation of the Pole , so much is the North pole elevated above the plane , as here the plane is represented by EF , the elevation of the style or Axis the arch EG , therefore in this case substract the complement of the reclination of the plane from the elevation of the elevated pole , and the remainer is the arch of the poles elevation above the plane , with which elevation enter the Table in the left margent , and there are the houre arches from the meridian . If the reclination of the plane be lesse then the complement , as is IK , substract the arch of reclination from the complement of the elevation , there is left the elevation of the South pole above the plane , and with the complement of the elevation of the pole above the plane enter the table on the right margent , and there shall you finde the distance of the houres : and herein Mr. Faile failed , for instead of substracting one from the other , he addeth one to another , causing a great errour . The distance of every houre of the North incliner on the back side of the South incliner as much are equall , saving that the houres on the North side must be named by the complement houres to 12 , and as the North pole is above one plane , so is the South pole above the other , you may also conceive the like in making of all South incliners and recliners , by framing the position of the plane on the South side as the figure is on the North : and in North recliners lesse then the elevation of the pole , adde the reclination of the flat , which is the elevation of the North pole above the plane : herein Mr. Fail failed also , as depending on the former , following the doctrine of contraries , which formost well examined would have saved the opening of a gap to this second errour : With the said elevation found enter the Table for the Horizontal arches , and thereby make a Horizontall ▪ plane as is shewed , so is the Diall also prepared . If it recline that it lie between the Horizon and the Equator , then to the elevation of the Pole adde the complement of the reclination , which is the height of the style above the plane , and finish it as a Horizontall plane for that latitude , and not as a Verticall , as Mr. Faile would have it , because every reclining plane is a Horizontall plane where the pole is elevated according to the style . In a given plane oblique to the Meridian , and to the Horizon , and to the prime Verticall , that is , a given plane Inclining declining , to finde as well the Meridian of the place as of the plane , and the elevation of the pole above the plane : Prob. 3 , Petici , Liber Gnomonicorum . TO give you the parallel of Pitiscus his example , we will prosecute it according to the naturall Tangents in his example , and give you his words . Let the Meridian of the place be ABCD ▪ the Horizon AEC , the prime Verticall BED , the Orientall point E , the Verticall declined BKD , and right angled at K , the poles of the World G and I : the poles of the planes H , the Meridian GHI , the angle of declination EBF , the arch of inclination BK . But before all things the arch K , or the distance of the meridian of the place NL is from the Vertical plane KL should be sought by the second Axiome , then the arke BN by the third or fourth Axiome , after these the angle BKN , that is , in one word , the Triangle BKN is found , by which discharged , the arke BN is found either equal to the poles elevation , or greater or lesser . If the arke be equal to the complement of the poles elevation , by it is a token the plane is oblique under the Meridian , to be inclined unto the Pole , in that case the meridian of the place and of the plane , and also the Axis doe concur in the same line G L ▪ if the plane be supposed to fall in the same great circle KN , but if the plane be not supposed , but in some parallel of the same , and the Axis be somwhat carryed away , as necessarily it is done if the Sciotericall be absolved , the Meridian of the plane and place are two lines parallel between themselves , and are mutually joyned together according to the difference of longitude of the place and of the plane , which difference is according to the angle HGC , which is the complement of the angle BNK late found , because the angle KGH is right by 57. p. 1. yea , forasmuch as the meridian of the plane may goe by the poles of the plane , but concurring at G or N are equall to two right , by 20 p. 1. Example , Let the plane meridionall declined to the right hand 29 de . 59 m. inclining toward the pole artick 23 de . 3 m. the elevation of the pole 49 de . 35 m. and there are to be sought in the same the meridian of the place & the plane , and the elevation of the pole or Axis above the plane . The calculation shall be thus . To 67874 the tangent of the arke KN the distance of the meridian of the place from the Verticall of the plane , 34 de . 10 m. per ax . 2 ▪ The sine of the arke NC 49de . 35 m. whose complement is the arke BN 40de . 25 m per axi. 4. To 60388 the sine of the angle BNK 37d 9m . whose complement is the angle HNC , or HGC 52 de . 51. m. the difference of the longitude of the plane from the longitude of the place , or the distance of the meridians of the place and plane . Therefore let the horizon of the place be LC , the verticall of the plane KL , the circle of the plane of the horizon KNC , in which there is numbred from K towards C 34 de . 10m . and at the terme of the numeration N , draw the right line L N E , which shall be the meridian of the plane and place , if the center of the Sciotericie L or F is taken for the center of the World , and the right line L N F for the Axis , but because in the perfection of the Diall , IG remaineth the Axis , with E the center of the world , not in the right line L N F , but above the same , with props at pleasure , but notwithstanding it is raised equall in height with EI and OG , and moreover the plane is somwhat withdrawn frō the axis of the world , therefore the line L N F is now not altogether the meridian of the place , but only the meridian of the plane , or as vulgarly they speake , the substilar . But you may finde the meridian of the place thus , draw IH at right angles to the meridian of the plane , which they vulgarly call the Contingence to the common section of the Equator , which in the plane let E the center of the world be set from the axis IG in the meridian of the plane L N F. Then to the center E , consisting in the line L N E , le the circle of the Equator FK be described , and in the same toward the East , because the horizon of the plane is more easterly then the horizon of the place , and moreover the beame is cast sooner or later upon the meridian of the plane then the place , let there be numbered the difference of longitude of the place and plane 52 de . 51 m. and by K the end of the numeration let a right line be drawn , as it were the certain beams of the Equator EKH , which where it toucheth the common section of the Equator with the plane , to wit , the right line FH , by that point let C the meridian of the place be drawn perpendicular . The second case of the third Probleme of Pitiscus his Liber Gnomonicorum . Sivero arcus BN , repertus fuerit , &c. But if the arke BN shall be found lesse then the complement of the poles elevation , it is a signe the plane doth consist on this side the pole artick , and moreover above such a plane not the pole Artick , but the pole Antartick shall be extolled to such an angle as ILM is , whose measure is the arke IM , to which , out of the doctrine of opposites , the arke GO is equall , which you may certainly finde together with the arke NO thus . As MOG the right angle , to NG the difference between BN and BG , so ONG the angle before found , to OG , per axi. 3. As the tangent ONG to Radius , so the tangent OG , to the sine O N , by axi. 2. Example ; Let the plane be meridionall declined to the right hand 34 de . 30 m. inclined toward the pole artick 16 de . 10 m. and again , let the elevation of the pole be 49 de . 35 m. and there are sought : The meridian of the place : the longitude of the countrey The meridian of the plane : the longitude of the plane ? The elevation of the pole above the plane . The Calculation . 1. As BF Radius , 100000 , to FC tangent complement of declination 55 de . 30 m. 14550 , so 27843 the sine of the inclination 16 de . 10 m. to 40511 , the tangent of K N 22 de . 31 / 3 m. the distance of the meridian of the place from the Verticall of the plane , per axi. 2. The sine of the arke N C 62 de , 532 / 3 m. whose complement is B N 27 de . 61 / 3 m. by which substracted from BG the complement of the poles elevation 40 de . 25 m. there is remaining the arke N G 13 de . 182 / 3 m. by axi. 4. To 61108 the sine of the angle B N K , or O N G 37d . 40 m. per axi. 3. & comp. 1. To 14069 the sine of the arch OG the distance of the axis GL from the meridian of the plane ▪ OL 8de . 51 / 3m . by ax . 3. To 18410 the sine of the arch N O , the distance of the meridian of the plane OL , from the meridian of the place N L 30 deg. 36½ m , by axi. 2. The calculation being absolved , let there be drawn the horizon of the place AC , secondly , the verticall of the plane BQ , thirdly , the horizon of the plane ABCQ , in whose Quadrant AQ , to wit , according to the pole antartique , which alone appeareth above such a plane . First , let be numbred the distance of the meridian of the place from the verticall of the plane 22 de . 3 m. and by the ende of the numeration at P , let the meridian of the plane LP be drawn , then from the point P , let the distance of the meridian of the plane from the meridian of the place be numbered , by the terme of the numeration M , let the meridian of the plane LM be drawn . Finally , from the point M , into whatsoever part , let the proper elevation of the pole be numbered , or the distance of the axis from the meridian of the plane 8 de ▪ 51 / 3m . and by the term of the numeration I , let the axis ▪ LI be drawn , to be extolled or lifted up on the meridian of the plane LM , to the angle MLN . The third case of the third probleme of Pitiscus his liber Gnomonicorum . Si denique arcus BN repertus fuerit major , &c. Lastly , if the arke BN be found greater then the complement of the poles elevation BG , it is a token the plane to be inclined beyond the pole artique , and moreover the pole artique should be extolled above such a plane to so great an angle as the angle GLO , which the arke GO measureth , which arke , together with the arke ON in the end you may find in such sort as in the precedent case . Example , Let there be a meridian plane declining to the right hand 35 de . 54 m. inclining towards the pole artique 75 de . 43 m. and let the elevation of the pole be 49 de . 35½ m. but there is sought the meridian of the plane and place , together with the elevation of the pole above the plane , the calculation shall be thus . to 133874 tangent of the arke KN , the distance of the meridian of the place from the verticall of the plane , 53 de . 14½ m , by axi. 2. The sine of the arke NC 8 de . 29● m. whose complement is BN 81 de . 30½ m. from whence if you substract BG 40 de . 25 m. there remaineth the arke GN 41 de . 5½ m. to 97982 , the sine of the angle BNK , or ONG , by axi. 3. to 64399 the sine of the arch OG , the distance of the axis from the meridian of the plane 40 de . 51 / 3 m. by axi. 3. to 17483 the sine of the arke O N the distance of the meridian of the plane from the meridian of the place , 10 de . 4 m. by axi. & comp. 2. The calculation being finished , let the horizon of the place be AC , the verticall of the plane KD , the horizon of the plane AKCD , in which let be numbered from the vertical point K toward C the distance of the meridian of the place from the vertical of the plane 53 de . 14½ m. and by the end of the numeration let be drawn the meridian of the place LN , then from the meridian of the place , to wit , from the point N backward , let the distance of the meridian of the plane 10 de . 4m . be numbred , and by O the end of the numeration , let LO the meridian of the plane be drawn , from which afterwards let the proper elevation of the pole be numbred , or the distance of the axis from the meridian of the plane 48d . 5½m . and by the term of the numeration G , let the axis LG be drawn , being extolled above the plane BO , to the angle GLO . CHAP X. In which is shewed the drawing of the houre-lines in these last planes not there mentioned , being also part of Pitiscus his example in the fourth Probleme of his liber Gnom . SO then , saith he , Si axis , &c. If the axis be oblique to the plane , as the foregoing are , as in any plane oblique to the Equator many of the houre-lines doe concur at the axis with equal angles , but they are easily found thus . But because Pitiscus is mute in defining which part he takes for the right hand and which the left , we must search his meaning . Pitiscus was a Divine is evident by his own words in his dedication , Celsitudini tuae tota vita mea prolixe me excusarem quod ego homo Theologus ▪ &c. If we take him as hee was a Divine , we imagine his face to be towards the East , then the South is his right hand , and the North is his left hand . That he was an Astronomer too , appeareth by his Books both of proper and common motion , then we must imagine his face representing the South , the East on his left hand , which cannot be , as shall appear . Neither must we take him according to the Poets , whose face must be imagined toward the West . In short , take him according to Geographie , representing the Pole , and this shews the right hand was the East , and left the West , as is evident by the Diall before going , for it is a plane declining from the South to the right hand 30 degrees , that is , the East , because it hath the morning houres not the evening , because the Sun shines but part of the afternoon on the plane . Thus in briefe I have run throngh all planes , and proceed to shew you farther conclusions : But I desire the Reader to take notice that in these examples of Pitiscus . I have followed his own steps , and made use of the Naturall Sines and Tangents . CHAP XI . Shewing how by the helpe of a Horizontall Diall , or other , to make any Diall in any position how ever . HAving prepared a Horizontall Diall as is taught before : on the 12 houre , as far distant as you please from the foot of the style , draw a line perpendicular to the line of 12 , on that describe a Semicircle , plasing the foot of the Compasses in the crossing of the lines , this Semicircle divide into 180 parts , each Quadrant into 90 , to number the declination thereon , let the arch of the Semicircle be toward the North part of the Diall . Then prepare a plane slate , such as will blot out what hath been formerly made thereon , and make it to move perpendicularly on the horizontal plane on the center of the semicircle , which wil represent any declining plane by moving it on the semicircle . Now knowing the declination of the plane turn this slate towards the easterly part , if it decline towards the East , if contrary to the West , if toward the West , and set it on the semicircle to the degree of declination , then taking a candle and moving the Diall till the shadow fall on all the houres of the horizontall plane , mark also where the shadow falls on the declining plane , that also is the same houre on the plane so scituated , drawn from the joyning of the style with the plane . It is so plain it needs no figure . So may you doe in all manner of declining reclining , or reclining and inclining Dials , by framing your instrument to represent the position of the plane . Note also that the same angle the axis of the Horizontal Dial makes with the plane , the same elevation must the axis of that plane have , and where it shadows on the representing plane when the shadow of the horizontal axis is on 12 , that is the meridian of the place . By the same also may you describe all the conclusions Astronomicall , the Almicanthers , circles of height : the parallels of the Sun , shewing the declination : the Azimuthes , shewing the point of the Compasse the Sun is in : and all the propositions of the Sphere . Seeing this is so plain and evident , nay a delightful conclusion , I will not give you farther directions in a matter of so great perspicuity , as to lay down the severall wayes for projecting the Sphere on every severall plane , but proceed to shew the making of a general Dial for the whole World , which we will use as our Declinatorie to finde the scituation of any wall or plane , as shall be required to make a Diall thereon , as followeth in the next Chapter . CHAP XII . Shewing the making of a Diall on a Crosse form , as also a Universall Quadrant drawn from the same projection , as also to describe the Tropicks on Meridian or Polar planes . THis Universall Diall is described by Clavius in his eighth Book de Gnomonicis : But because the Artists of these times have found out a more commodious contrivance of it in the fabrique , I shall describe it according to this Figure . Now to know the houre of the day , you shall turn the plane by the helpe of the needle , so as the end A shall be toward the North , and E toward the South , and elevate the end E to the complement of the elevation , then bringing the Box to stand in the Meridian , the shoulder of the Crosse shall shew you the houre . Upon this also is grounded the Universall Quadrant hereafter described , which Instrument is made in Brasse by Mr. Walter Hayes as it is here described . Prepare a Quadrant of Brasse , divide it in the limbe into 90 degrees , and at the end of 45 degrees from the center draw the line A B , which shall represent the Equator , divide the limbe into 90 degrees , as other Quadrants are usually divided , then number both wayes from the line AB the greatest declination of the Sun from the North and South , at the termination whereof draw the arch CD which shall be the Tropicks , then out of the Table of declination , pag. 45 , from B both wayes let there be numbered the declinatiō of the Signes according to this Table .   G M   ♈ 00 00 ♎ ♉ ♍ 11 30 ♏ ♓ ♊ ♌ 20 30 ♐ ♒ ♋ 23 30 ♑ Now the plane it selfe is no other then an East or West Diall , numbred on one side with the morning houres , and on the other with the evening houres , the middle line AB representing the Equator . And to set it for the houre , you shall project the Tropicks and other intermediate parallels of the Signes upon them as is hereafter shewed , but that the plane may not run out of the Quadrant you shal work thus , opening the Compasses to 15 degrees of the Quadrant , prick that down both wayes , at which distance draw parallels to the line AB , and with the same distance , as if it were the semidiameter of the Equator , describe the semidiameter of the Equator on the top of the line AB , which divide into 12 parts , and laying a ruler through the center and each of those divisions in the semicircle to those parallel lines on each side of AB , marke where they cut , and from side to side draw the parallel houre lines as is taught in the making of an East and West Diall , make those parallel lines also divided as a tangent line on each side AB , so if this Quadrant were held on an East or West wall , and a plummet let fall from the center of the Equator where the style stands ( which may be a pin fitted to take out and in , fitted to the height of the distance between the line A B and the other parallels , which is all one with the Radius of the small Circle ) it shall I say , be in its right scituation on the East or West wall if you let the plummet and threed fall on the elevation of the Pole in that place . But because we desire to make it generall , we must describe the Tropicks and other parallels of declination upon it , as is usuall to be done on your Polar and East and West Diall , which how to doe is thus . Having drawn the houre lines and Equator as is taught from E the height of the style , take all the distances between it and the houre lines where they doe crosse the line AB , and prick them down on the line representing the Equator in this figure from the center B. Then describe an occult arch of a Circle , whereon describe a Chorde of 23 degrees 30 minutes , with such other declinations as you intend on your plane . Then on the line representing the Equator , noted here with the figures of the houres they were taken from , 6 , 7 , 8 , 9 , 10 , 11 , at the marks formerly made , that was taken from E the height of the style , and every of the houres , from these distances I say raise perpendiculars to cut the other lines of declination , so those perpendiculars shall represent those houre lines from whence they were taken , and the distances between the Equator and the severall lines of declination shall be the same distances from the Equator , and the other parallels of declination upon your plane , through which marks being pricked down upon the severall hourelines from the Equinoctiall . If you draw those Hyperbolicall lines , you shall have described the parallels of declination required . But if you will performe the same work a second and easie way , worke by this Table following , which is universall , and is composed out of the Table of Right & Versed shadow . Put this Table before thee , & for the point of each houre line whereby the severall parallels of the Signes shall pass worke thus . The style being divided into known parts ▪ if ▪ into 12 , take the parts of shadow out of the Table in the same known parts by which the style is divided , & prick them down on each houre line as you finde it marked in the Table answering the houre both before and after noon . As suppose that a Polar plane I finde when the Sun is in Aries or Libra at 12 a clock the shadow hath no latitude , but at 1 and 11 it hath 3 parts 13 min. of the parts of the style , which I prick from the foot of the style on the houres of 1 and 11 both above and beneath the Equator : and for 2 and 10 I finde 6 parts 56 min. which I prick down also from the center to the houre lines of 10 and 2 , and so of the other houre lines and parallels , through which if I draw those lines they shall represent the parallels of the Declination . A Table of the Latitude of shadows .   Cancer . Gemini Leo Virgo Taurus Libra Aries   p m p m p m p m p m a m 12 5 13 4 25 2 26 0 0 12 1 6 17 5 35 4 5 3 13 11 2 8 11 8 35 7 27 6 56 10 3 14 5 13 31 12 39 12 0 9 4 23 15 22 45 21 21 20 27 8 5 49 6 47 57 45 45 44 47 7 6 Vmbra infinita . 6 Having promised in the description of the use of this Instrument , to shew how to finde the inclination and reclination of a plane , I shal proceed to give you some cautions ; First then , the quadrant is divided in the limbe , as other quadrants are into 90 degrees , by which is measured the angles of inclination or reclination , for if it be a declining plane onely , the declination is accounted from the North or South toward the East or West , if it decline from the North , the North Pole is elivated above it , and the meridian-line ascendeth , if it decline from the South , the south pole is elivated above that plane , if it decline from the South Eastward , then is the style and sub-style refered toward the west side of the plane , if to the contrary the contrary , and may have the line of 12 except north decliners in the temperate Zone , you may make use of the side of the quadrant to finde the declination , as is taught before page 33 , observing the angle as is cut by the shadow of the thred held by the limbe , & through the center , and that side that lieth perpendicular to the Horizontal line which shal be the angle , as is before taught : And if the south point is between the poles of the plane and the Azimuth , then doth the plane decline Eastward , if it be the afternoon you take the Azimuth in , if it be the forenoon you take the Azimuth in , and the south point be between it and the Poles of the planes horizontal line , it doth decline Westward , if contrary it is in the same quarter where the sun is : For an inclining plane , which is the angle that it maketh with the Horizon ▪ draw a Horizontall line and crosse it again with a square , or verticall line , then apply the side of the quadrant to the vertical line at the beginning of the numeration of the deg. on the quadrant , and the angle contained between the thred & plummet , and the applyed side is the inclination ; in all north incliners the north part of the meridian ascendeth , in south incliners the south part , and in east and west incliners , the meridian lyeth parallel with the Horizon . And for the reclination it being all one with the inclination , considered as an upper and under face of the same plane , if you cannot apply the side of the quadrant , you may set a square or ruler at right angles with the verticall line drawn on the upper face and apply the side of the quadrant to the edge of the ruler , and measure the quantity of the angle by the thred and plummet : but this is of direct , howsoever these are subject to another passion of declining and inclining together , which must be sought severally , and such are those whose Horizontal line declineth toward the north or south and inclination from north or south , towarde the east or west , which must be sought severally . Here followeth the Tables of Right and Contrary shadows . A Table of Right and Contrary shadow , to every Degree and tenth minute of the Quadrant . ☉ Alt 0 1 2 3 4 5 6 7 8 9 ☉ Alti● S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 41378 , 54 687 34 143 44 229 0 171 37 137 10 114 11 97 44 85 23 75 46 60 Verticall shadow 10 4137 , 53 589 16 317 14 216 54 164 44 132 43 111 4 95 26 83 37 74 22 50 20 2065 , 23 515 46 294 31 206 3 158 23 128 33 108 7 93 15 81 55 73 1 40 30 1376 , 6 458 22 274 54 196 13 152 29 124 38 105 19 91 9 80 18 71 43 30 40 1031 , 45 412 29 257 40 187 16 147 1 120 56 102 40 89 9 78 44 70 27 20 50 825 , 13 374 55 242 28 179 6 141 56 117 28 100 8 87 14 77 13 69 14 10 60 687 , 34 343 54 229 0 171 37 137 10 114 11 97 44 85 23 75 46 68 3 0   10 11 12 13 14 15 16 17 18 19   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtall shadow 0 68 3 61 44 56 27 51 59 48 8 44 47 41 51 39 15 36 57 34 51 60 Verticall shadow . 10 66 55 60 47 55 40 51 18 47 32 44 16 41 24 38 51 36 34 34 31 50 20 65 49 95 52 54 53 50 38 46 58 43 46 40 57 38 27 36 13 34 12 40 30 64 45 85 59 54 8 49 59 46 24 43 16 40 31 38 4 35 52 33 53 30 40 63 43 85 7 53 24 49 21 45 51 42 47 40 5 37 41 35 31 33 35 20 50 62 43 57 16 52 41 48 44 45 19 42 19 39 40 37 18 35 11 33 16 10 60 61 44 56 27 51 59 48 8 44 47 41 51 39 15 36 56 34 51 32 58 0   20 21 22 23 24 25 26 27 28 29   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtall shadow 0 32 58 31 16 29 42 28 16 26 57 25 44 24 36 23 33 22 34 21 39 60 Verticall shadow . 10 32 40 31 0 29 27 28 3 26 45 25 52 24 25 23 23 22 25 21 ●0 50 20 32 23 30 44 29 13 27 49 26 32 25 21 24 15 23 13 22 15 21 21 40 30 32 6 30 28 28 58 27 36 26 20 25 10 24 4 23 3 22 6 21 13 30 40 31 49 30 12 28 44 27 23 26 8 24 58 23 54 22 53 21 57 21 4 20 50 31 32 29 57 28 30 27 10 25 56 24 47 23 43 22 44 21 48 20 56 10 60 31 16 29 42 28 16 26 57 25 44 24 36 23 33 22 34 21 39 20 47 0   30 31 32 33 34 35 36 37 38 39   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtall shadow 0 20 47 19 58 19 12 18 29 17 47 17 8 16 31 15 55 15 22 14 49 60 Verticall shadow . 10 20 ●9 19 50 19 5 18 21 17 41 17 2 16 25 15 50 15 16 14 44 50 20 20 31 19 43 18 57 18 15 17 34 16 56 16 19 15 44 15 11 14 39 40 30 20 22 19 35 18 50 18 8 17 28 16 49 16 13 15 38 15 5 14 33 30 40 20 14 19 27 18 43 18 1 17 21 16 43 16 7 15 33 15 0 14 28 20 50 20 6 19 20 18 36 17 54 17 15 16 37 16 1 15 27 14 54 14 23 10 60 19 58 19 12 18 29 17 47 17 8 16 31 15 55 15 22 14 49 14 18 0   40 41 42 43 44 45 46 47 48 49   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtall shadow 0 14 18 13 48 13 20 12 52 12 26 12 0 11 35 11 11 10 48 10 26 60 Verticall shadow . 10 14 13 13 43 13 15 12 48 12 21 11 56 11 31 11 8 10 45 10 22 50 20 14 8 13 39 13 10 12 42 12 17 11 52 11 27 11 4 10 41 10 19 40 30 14 3 13 34 13 6 12 39 12 13 11 48 11 23 11 0 10 37 10 15 30 40 13 58 13 29 13 1 12 34 12 8 11 43 11 19 10 56 10 33 10 11 20 50 13 53 13 24 12 57 12 30 12 4 11 39 11 15 10 52 10 30 10 8 10 60 13 48 13 20 12 52 12 26 12 0 11 35 11 11 10 48 10 26 10 4 0   50 51 52 53 54 55 56 57 58 59   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtal shadow 0 10 4 9 43 9 23 9 3 8 43 8 24 8 6 7 48 7 30 7 13 60 Verticall shadovv . 10 10 1 9 40 9 19 8 59 8 40 8 21 8 3 7 45 7 27 7 10 50 20 9 57 9 36 9 16 8 56 8 37 8 18 8 0 7 42 7 24 7 7 40 30 9 54 9 33 9 12 8 53 8 34 8 15 7 57 7 39 7 21 7 4 30 40 9 50 9 29 9 9 8 50 8 30 8 12 7 54 7 36 7 18 7 1 20 50 9 47 9 26 9 6 8 46 8 27 8 9 7 51 7 33 7 15 6 59 10 60 9 43 9 23 9 3 8 43 8 24 8 6 7 48 7 30 7 13 6 56 0   60 61 62 63 64 65 66 67 68 69   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtall shadow 0 6 56 6 39 6 23 6 7 5 51 5 36 5 21 5 6 4 51 4 36 60 Verticall shadow . 10 6 53 6 36 6 20 6 4 5 49 5 33 5 18 5 3 4 48 4 34 50 20 6 50 6 34 6 17 6 2 5 46 5 31 5 16 5 1 4 46 4 32 40 30 6 47 6 31 6 15 5 59 5 43 5 28 5 13 4 58 4 44 4 29 30 40 6 45 6 28 6 12 5 56 5 41 5 26 5 11 4 56 4 41 4 27 20 50 6 42 6 26 6 10 5 54 5 38 5 23 5 8 4 53 4 39 4 24 10 60 6 39 6 23 6 7 5 51 5 36 5 21 5 6 4 51 4 36 4 22 0   70 71 72 73 74 75 76 77 78 79   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtal shadow 0 4 22 4 8 3 54 3 40 3 26 3 13 3 0 2 46 2 33 2 20 60 Verticall shadovv . 10 4 20 4 6 3 52 3 38 3 24 3 11 2 56 2 44 2 31 2 18 50 20 4 17 4 3 3 49 3 36 3 22 3 8 2 55 2 42 2 29 2 16 40 30 4 15 4 1 3 47 3 33 3 20 3 6 2 53 2 40 2 26 2 13 30 40 4 13 3 59 3 45 3 31 3 17 3 4 2 51 2 37 2 24 2 11 20 50 4 10 3 56 3 42 3 29 3 15 3 2 2 48 2 35 2 22 2 9 10 60 4 8 3 54 3 40 3 22 3 13 3 0 2 46 2 33 2 20 2 7 0   80 81 82 83 84 85 86 87 88 89   S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizōtall shadow 0 2 7 1 54 1 41 1 28 1 16 1 3 0 50 0 38 0 25 0 13 60 Verticall shadow . 10 2 5 1 52 1 39 1 26 1 14 1 1 0 48 0 36 0 23 0 10 50 20 2 3 1 50 1 37 1 24 1 11 0 50 0 46 2 34 0 21 0 8 40 30 2 0 1 48 1 35 1 22 1 9 0 57 0 44 0 31 0 19 0 6 30 40 1 58 1 45 1 33 1 20 1 7 0 55 0 42 0 29 0 17 0 4 20 50 1 56 1 43 1 31 1 18 1 5 0 32 0 40 0 27 0 15 0 2 10 60 1 54 1 41 1 28 1 16 1 3 0 50 0 38 0 25 0 13 0 0 0 CHAP XIII . Of the generall description and use of the preceding Tablein , the pricking down and drawing the circles of declination and Aximuths in any planes . THe Table you see consisteth of 11 columns , the first being the minutes of the Suns altitude , and the greater figures on the top are the degrees of altitude , all the other columns consist of the parts of shadow , and minutes of shadow , noted above with S for shadow , and p m for parts and minutes of shadow , answerable to a gnomon divided into 12 equall parts , and it is , As the sine of a known altitude of the sun , is to the sine complement of the same altitude ; so the length of the Gnomon in 10 or 12 parts , to the parts of right shadow : or for the versed shadow , as the sine complement of the given altitude of the sun , to the right sine of the same altitude ; so the style in parts , to the length of the versed shadow So if we enter the Table with the given altitude of the Sun in the great figures , and if we seeke the minutes in the sides , either noted with horizontall or verticall shadow , according as your plane is , it shall give you the length of the shadow in parts and minutes in the common angle of meeting together . As if we look for 50 de . 40 m. the meeting of both in the Table shall be 9 parts 50 min. for the length of the right shadow on a horizontall plane : But for the versed shadow , take the complement of the altitude of the Sun , and the minutes in the right side of the Table , titled verticall shadow , and the common area of both shall give your desire . By this Table it appeareth first , that the circles of altitude either on the horizontall or verticall planes are easily drawn , consicering they are nothing else but circles of altitude , which by knowing the altitude you will know the length of the shadow , which in the horizontall Diall are perfect circles , and have the same respect unto the Horizon , as the parallels of declination have to the Equator , but in all upright planes they wil be conicall Sections , and by having the length of the style , the altitude of the Sun may be computed by the foregoing Table with much facility , but for the more expediating of the work in pricking down the parallels of declination with the Tropicks , I have here added a Table of the altitude of the Sun for every houre of the day when the Sun enters into any of the 12 Signes . A Table for the altitude of the Sun in the beginning of each Signe , for all the houres of the day for the Latitude of London . Hours . Cancer . Gemini Leo Taurus Virgo Aries Libra Pisces Scorpio Aquar Sagitta . Capric . 12 62 0 58 43 50 0 38 30 27 0 18 18 15 0 11 1 59 43 56 34 48 12 36 58 25 40 17 6 13 52 10 2 53 45 50 55 43 12 32 37 21 51 13 38 10 30 9 3 45 42 43 6 36 0 26 7 15 58 8 12 5 15 8 4 36 41 34 13 27 31 18 8 8 33 1 15     7 5 27 17 24 56 18 18 9 17 0 6         6 6 18 11 15 40 9 0                 5 7 9 32 6 50                 11 37 4 8 1 32                     21 40 This Table is in Mr. Gunters Book , page 240 which if you desire to have the point of the Equinoctiall for a Horizontall plane on the houre of 12 , enter the Table of shadows with 38 de . 30 m. and you shall finde the length of the shadow to be 15 parts 5 m. of the length of the style divided into 12 , which prick down on the line of 12 for the Equinoctiall point , from the foot of the style . So if I desire the points of the Tropick of Cancer , I finde by this Table that at 12 of the clock the Sun is 62 de . high , with which I enter the Table of shadows , finding the length of the shadow , which I prick down on the 12 a clock line for the point of the Tropick of Cancer at the houre of 12. If for the houre of 1 , I desire the point through which the parallel must pass , looke for the houre of 1 and 11 , in this last table under Cancer , and I finde the Sun to have the height of 59 de . 43 m. with which I enter the table of shadows , and prick down the length thereof from the bottome of the style reaching till the other foot of the Compasses fall on the houre for which it was intended . Doe so in all the other houres , till you have pricked down the points of the parallels of declination , through which points they must be drawn Hyperbolically . Proceed thus in the making of a Horizontall Diall , but if it be a direct verticall Diall , you shall then take the length of the verticall shadow out of the said Table , or work it as an Horizontal plane , only accounting the complement of the elevation in stead of the whole elevation . For a declining plane you may consider it as a verticall direct in some other place , and having found out the Equator of the plane and the substyle , you may proceed in the same manner from the foot of the style , accounting where the style stands to be no other wayes then the meridian line or line of 12 in a Horizon whose pole is elevated according to the complement height of the style above the substyle , and so prick down the length of the shadows , from the foot of the style , on every one of the Houre lines , as if it were a horizontal or Verticall plane . But in this you must be wary , remembring that you have the height of the sun calculated for every houre of that Latitude in the entrance of the 12 signes , in that Place where your Plane is a Horizontall plane , or otherwayes , by considering of it as a horizontall or Verricallplane in another latitude For the Azimuths , or verticall circles , shewing one what point of the compasse the sun is in every houre of the day it is performed with a great deale of facility , if first , when the sun is in the Equator , we doe know by the last Table of the height of the sun for every houre of the day and by his meridian altitude with the help of the table of shadows , find out the Equinoctiall line , whether it be a Horizontall or upright direct plane , for having drawn that line at right angles with the meridian , and having the place of the Style , and length thereof in parts , and the parts of shadow to all altitudes of the sun , being pricked down from the foot of the Style , on the Equinoctiall line , through each of those points draw parallel lines to the meridian , or 12 a clock line on each side , which shall be the Azimuths , which you must have a care how you denominate according to the quarter of heaven in which the sun is in , for if the Sun be in the easterly points , the Azimuths must be on the Western side of the plane , so also the morning houres must be on the opposite side . There are many other Astronomical conclusions that are used to be put upon planes , as the diurnall arches , shewing the length of the day and night , as also the Jewish or old unequal houres together with the circles of position , which with the meridian and horizon distinguisheth the upper hemispheare into 6 parts commonly called the houses of Heaven : which if this I have writ beget any desire of the reader , I shall endeavour to inlarge my self much more , in shewing a demonstrative way , in these particulars I have last insisted upon . I might heare also shew you the exceeding use of the table of Right and versed shadow in the taking of heights of buildings as it may very wel appear in the severall uses of the quadrant in Diggs his Pantometria , & in Mr. Gunters quadrant , having the parts of right and versed shadow graduated on them , to which Books I refer you . CHAP XIV . Shewing the drawing of the Seeling Diall . IT is an Axiom pronounced long since , by those who have writ of Opticall conceipts of Light and Shadow , that Omnis reflectio Luminis est secundum lineas sensibiles , latitudinem habentes . And it hath with as great reason bin pronounced by Geometricians , that the Angles of Incidence and Reflection is all one ; as appeareth to us by Euclides Catoptriques ; and on this foundation is this conceipt of which we are now speaking . Wherefore because the direct beams cannot fall on the face of this plane , we must by help of a piece of glasse apt to receive and reflect the light , placed somwhere horizontally in a window , proceed to the work , which indeed is no other then a Horizontall Diall reversed , to which required a Meridian line , which you must endeavour to draw and finde according as you are before taught , or by the helpe of the Meridian altitude of the Sun , your glasse being fixed marke the spot that reflects upon the seeling just at 12 a clock , make that one point , and for the other point through which you must draw your meridian line , you may finde by holding up a threed and plummet till the plummet fall perpendicular on the glasse , and at the other end of the line held on the seeling make another mark , through both which draw the Meridian line . Now for so much as the center of the Diall is a point without , and the distance between the glasse and the seeling is to be considered as the height of the style , the glasse it selfe representing the center of the world , or the very apex of the style , wee must finde out those two Tangents at right angles with the Meridian , the one neere the window , the other farther in , through severall points whereof we must draw the houre-lines . Let AB be the Meridian line found on the seeling , now suppose the Sun being in the highest degree of Cancer should shine into the Glasse that is fixed in C , it shall again reflect unto D , where I make a mark , then letting a plummet fall from the top of the seeling till it fall just on C the glasse , from the point E , from which draw the line A B through D and E , which shall be the Meridian required , if you do this just at noon : Now if you would finde out the places where the hour-lines shall crosse the Meridian , the Center lying without the window EC , you may work thus CHAP X. Shewing the making and use of the Cylinder Dial , whose hour-lines are straight , as also a Diall drawn from the same form , having no Style . THis may be used on a Staff or other round , made like a Cylinder being drawn as is here described , where the right side represent the Tropicks , and the left side the Equinoctial : or it may be used flat as it is in the Book ; the Instrument as you see , is divided into months , and the bottom into signs , and the line on the right side is a tangent to the radius of the breadth of the Parallelogram , serving to take the height of the Sun , the several Parallels downward running through the pricked line , in the midle , are the lines of Altitude , and the Parallels to the Equator are the Parallels of Declination , numbred on the bottom on a Sine of 23 de . and a half . For the Altitude of the Sun . The use of it is first , if it be described on the head of a staff , to have a gnomon on the top , equal to the radius , and just over the tangent of Altitudes , to turn it till you bring the shadow of it at right angles to it self , which shal denote the height required . For the Houre of the Day . Seek the Altitude of the Sun in the midle prick't line , and the Declination in the Parallels from the Equator , and mark where the traverse lines crosse ; through the crossing of the two former lines , and at the end , you shal finde the figures of 2 or 10 , 3 or 9 , &c. only the summer Houres are sought in the right side ▪ where the Sun is highest , and the traverse lines longest ; and in the winter , the Hour is sought on the left side , where the traverse lines are shorter . For the Declination and degree of the Signe . Seek the day of the moneth on the top marked with J. for January , F for February , &c. and by the help of a horse hair or threed extended from that all along of Parallel of Declination , till it cut on the bottom where the signes are numbred : the down right lines that are parallel to the Equator counted toward the right hand , is the degree of the Declination of that part of the Ecliptick which is in the bottom , right against the day of the moneth sought on the top . The pricked line passing through the 18 degree of the Parallel of Altitude , is the line of Twy-light ; this projection I had of my very good friend John Hulet , Master of Arts ▪ and Teacher of the Mathematicks . You may also make a Dyal , by preparing of a hollow Cylinder , and if you doe number on both ends of the Circle , on top and bottom , 15 de . from line to line ; or divide it into 24 parts , and if from top to bottom you draw streight lines , first , by dividing the Cylinder through the middle , and only making use of one half , it shal have 12 houres upon it . Lastly , if you cut off a piece from the bottom at an angle according to the Elevation , and turn the half Cylinder horizontal on that bottom , til the shadow of one of the sides fal parallel with any one of those lines from top to bottom : which numbred as they ought , shal shew the hour without the use of a Style ; So also may you project a Dyal on a Globe , having a round brim on the top , whose projection will seem strange to those that look upon it , who are ignorant of these Arts . CHAP XVI . Shewing the making of a universall Dyall on a Globe , and how to cover it , if it be required . If you desire to cover the Globes , and make other inventions thereon , first learn here to cover it exactly , with a pair of compasses bowed toward the points , measure the Diameter of the Globe you intend to cover , which had , finde the Circumference thus ; Multiply the Diameter by 22 , and divide that product by 7 , and you have your desire . That Circumference , let be the line A B , which divide into 12 equal parts , and at the distance of three of those parts , draw the Parallel C D , and E F , A Parallel is thus drawn , take the distance you would have it asunder , as here it is ; three of those 12 divisions : set one foot in A , and make the Arch at E , & another at B , and make the Arch with the other foot at F , the compasses at the wideness taken , then by the outward bulks of those Arches , draw the line E F , so also draw the line C D. And to divide the Circumference into parts as our example is into 12 , work thus , set your compasses in A , make the Ark B F , the compasses so opened , set again in B , and make the Ark A C , then draw the line from A to F , then measure the distance from F to B , on the Ark , and place it on the other Arch from A to C , thence draw the line C B , then your compasses open at any distance , prick down one part less on both those slanting lines ; then you intend to divide thereon , which is here 11 : because we would divide the line A B into 12 , then draw lines from each division to the opposite , that cuts the line A B in the parts of division . But to proceed , continue the Circumference at length , to G and H , numbring from A toward G9 of those equal parts , and from B toward H as many , which shal be the Centers for each Arch. So those quarters so cut out , shall exactly cover the Globe , whose Circumference is equal to the line A B. Thus have you a glance of the Mathematicks , striking at one thing through the side of an other : for I here made one figure serve for three several operations , because I would not charge the Press with multiplicity of figures . CHAP XVII . Shewing the finding of the Elevation of the Pole , and therewithall a Meridian without the Declination of Sun or Starre . THis is done by erecting a gnomon horizontal , and at 3 times of the day to give a mark at the end of the Shadows : now it is certain , that represents the Parallel of the Sunne for that day ; then take three thin sticks or the like , and lay them from the top of the gnomon , to the places where the shadows fell , and on these three so standing , lay a board to ly on all three flat , and a gnomon in the midle of that board points to the Pole : because every Parallel the Sun moves in , is parallel to the Equinoctial , and that is at right Angles , with the pole of the World . Now the Meridian passeth through the most elevated place of that board or circle so laid , neither can the Sun's Declination make any sensible difference in the so small proportion of 3 or 4 houres time . CHAP XVIII . Shewing how to finde the Altitude of the Sun , only by Scale and Compasses . WIth your Compasses describe the Circle A B C D place it horizontal , with a gnomon in the Center , crosse it with two Diameters ; then turn the board till the shadow be on one of the Diameters , at the end of the shadow , mark , as here at E , lay down also , the length of the gonmon from the Center on the other Diameter to F , from E to F drawe a right line : then take your Compasses , and on the chord of 90 , take out the Radius the Ark of 60 , set the compasses so in E , describe an Arch , then take the distance between the line E F , and the Diameter D B ; which measure on the chord of 90 , and so many degrees as the compasses extend over ; such a quantity is the height of the Sun , in like manner any Angles being given , you must measure it by the parts of a circle . Here followeth the problematical Propositions of the Office of shadow , and the benefit we receive thereof . Prop. 1 By shadow , we have a plain demonstration that the Sphere of Sol is higher than the Sphere of Luna , to confirm such as think they move in one Orbe . Let the Sun be at A , in the great Circle , and the Moon at B , in the lesser , let the Horizon be C D , now , they make one Angle of height , in respect of the Center of the Earth , notwithstanding though they so equally respect the Earth , as one may hinder the sight of the other : yet the shadow of the Sun shall passe by the head of the gnomon E , and cast it to F , and the beames of the Moon shall passe by E to G much longer , which shewes shee is much lower , for the higher the light is , the shorter is the shadow . I call the Moon a feminine , if you ask my reason , shee is cold and moist , participating of the nature of Women ; and we call her the Mother of moisture , but that 's not all , for I have a rule for it , Nomen non crescens . Prop. 2. By shadow , we are taught the Earth is bigger then the Moon ; seeing in time of a total Obscurity , the Moone is quite overshadowed ; for the shadow is cast in this manner . By the same we learn also , that seeing the shadow comes to a point , the Earth is less then the Sun : for if the opacous body be equal to the luminous body , then like two parallels they will never meet , but concurre in infinitum , as these following figures shew . Or if the luminous body were less then the opacous body : then the shadow would be so great in so long a way , as from the Earth to the Starry Firmament , that most of the Starres as were in opposition to the Sun , would not appear : seeing they borrow their light of the Sun . It is also sufficiently proved by shadow , in the Praecognita Philosophical , that the Earth is round , and that it possesseth the middle as proprius locus from which it cannot passe , and to which all heavie things tend in a right line , as their terminus ad quem . From which the semidiameter of the Sun 15 min. substracted doth remain the Altitude of the center of the Sun 50 de . 3 m. the Altitude required , or From this or the former Proposition we may take notice that there is no Dial can shew the exact time without the allowance of the Suns semidiameter : which in a strict acception is true , but hereto Mr. Wells hath answered in the 85 page of his Art of shadows , where saith he , because the shadow of the center is hindered by the style , the shadow of the hour-line proceeds from the limbe which alwayes precedeth the center one min. of time answerable to 15 min. the semidiameter of the Sun ( which to allow in the height of the Style were erroneous ) wherefore let the al●owance be made in the hour-lines , detracting from the true Equinoctial distances of every 15 deg. 15 primes , and so the Arches of the horizontall plane from the Meridian shall stand thus . Prop. 4. By shadow we may finde the natural Tangent of every degree of a quadrant , as appeares by the former example . Houres . Equinoctial distances . True hour distances . 12 0 de . m. de . m. se . 11 1 14 45 11 38 51 10 2 29 45 24 6 31 9 3 44 45 37 4 2 8 4 59 45 53 19 12 7 5 74 45 70 48 6 6 6 89 45 89 40 51 For the Sun being 46 deg , 13 min. of Altitude makes a shadow of 95. parts of such as the gnomon is 100 , so then multiply the length of the gnomon 100 by the Radius , and divide by 95 , and it yeelds 105263 the natural Tangent of that Ark . Prop. 5. By shadow we may take the height of any Building , by the Rule of Proportion ; if a gnomon of 6 foot high give a shadow of 10 foot : how high is that house whose shadow is 25 foot ? resolved by the Rule of Three . Prop. 6. By shadow also we learn the magnitude of the Earth , according to Eratosthenes his proposition . Prop. 7. By shadow we learne the true Equinoctial line , running from East to West , which crossed at right Angles is a true Meridian , where note , that in the times of the Equinoctiall that the shadows of one gnomon is all in one right line . Prop. 8. By shadow we know the Earth to be but as a point , as may appear by the shadow of the Earth on the body of the Moon . Prop. 9. By shadow we may learn the distance of places , by the quantity of the obscurity of an Eclipse . Prop. 10. By gnomonicals we make distinctions of Climates and People , some Hetorezii , some Perezii , some Amphitii . Prop. 11. By shadow the Climates are known , in the cold intemperate Zones the shadow goes round . In the hot intemperate Zones the shadow is toward the West at the rising Sun , and toward the East at the setting Sun , and no shadow at Noones to them as dwel under the Parallels . And to them in the temperate Zones always one way , toward the North , or toward the South . Prop. 12. By shadow we are taught the Rule of delineating painting , according to the perspective way , how much is to be light or dark , accordingly drawn as the center is disposed to the eye : so the Office of shadow is manifold , as in the Optical conclusions are more amply declared ; therefore I referre you to other more learned works , and desist to speak . But for matter of Information , I will here insert certain definitions taken out of Optica Agulion ii lib. 5. First , saith he , we call that a light body from whence light doth proceed ; truly saith he , the definition is plain , and wants not an Expositor , so say I , it matters not whether you understand the luminous body : only that which doth glister by proper brightness as doth the Sun , or that which doth not shine but by an external overflowing light , as doth the Moon . 2. That we call a diaphon body , through which light may pass , and is the same that Aristotle cals perspicuous . 3. It is called Adiopton , or Opacous ; through which the light cannot pass , so saith he , you may easily collect from a diaphon body the definition of shadow : for as that is transparent through which the light may pass : so also is that opacous , or of a dense nature wherein the light cannot pass . 4. That is generated from a shining body , is called the first light , that hath his immediate beginning from the luminous body , it is called the second light , which hath his beginning from the first , the third which hath his beginning from the second , and so the rest in the same order . Whence we make this distinction of day and light , day is but the second light , receiving from the Sun the first , so that day is light , but the Sun is the light . 5. Splendor is light repercussed from a pure polished body ; and as light is called so from the luminous body : so this is called splendent from the splendor . Theor. Light doth not onely proceed from the Center , but from every part of the superficies . Theor. Light also is dispersed in right lines . Theor ▪ Light dispersed about every where , doth collect into a Spherical body . 6. The beames of light , some are equi distant parallels , some intersect each other , and some diversly shaped . Let A be the Light , a beam from A to B , and another from C to D are parallel , A D and C B intersect ; and the other two doe diversly happen , one ascending , the other descending : its plaine . 7. That is called a full and perfect shadow , to which no beam of light doth come . 8. That is called a full and perfect light ; which doth proceed from all parts of that which gives light ; but that which giveth light but in part , is imperfect : this he exemplified by an Eclipse , the Moon interposing her self between the Sun and Earth , doth eclipse the perfect light of the Sun : whereby there appeares but a certaine obscure , dim , glimmering light , and is so made imperfect . Hence we may learn to distinguish day from night ; for day is but the presence of the Sun by a perfect light received , which we count from Sun rising to Sun setting . Twy-light is but an imperfect light from the partial shining or neighbourhood with Sun : whereas Night is a total deprivation or perfect shadow , to which no beam of light doth appettain . Yet from the over-flowing light of the Sun , the Starres are illuminated ; yet because shadow is always in the opposite , those Stars that are in direct opposition to the Sun , are obscure for that season , and hence proceeds the Eclipse of the Moon . Hence it is with the Sciothericalls as it is with the Dutch Emblamist , comparing Love to a Diall , and the Sun with the Motto , Nil sine te , and his comparison to Coelestis cum me Sol aspicit ore sereno , Protinùs ad numeros mens reddit apta suos . Implying that as soone as the Sun shines it returnes to the number , so a Lover seeing his Love on a high Tower , and a Sea between , yet ( protinùs ad numeros ) he will swim the Sea and scale the Castle to return to her : So here lyes the gradation , first , from the Suns light , from the light by the opoacus body , interposition , shadow , and from the shadow of the Axis is demonstrated the houre . Adde also , the beam and shadow of a gnomon , have one and the same termination or ending , toward which I now draw my pen ; desiring you to take notice that the whole Method of Dialling , as may appear by the former discourse , doth seem to be foure-fold , viz. Geometrical , Arithmetical , or by Tables mechanically , or by Observation . So that the Art of shadowes is no other then a certain and demonstrative motion of the Heavens in any Plaine or Superficies , and a Gnomonical Houre is no other then a direct projecting of the hour-lines of any Plain ; so as that it shal limit a Style so to cast its shadow from one line to another , as that it shall be just the twenty-fourth part of the natural day , which consisteth of 24 houres ; and this I have laid down after a most plain manner following : A gnomonical day is the same that the Artificial day is ; which the shadow of a gnomon maketh from the rising of the Sun , till the setting of the same in a concave superficies : which length of the day is also projected from the motion of the shadow of the Style , a gnomonical Moneth is also described on Planes , which is the space that the shadow of a Gnomon maketh from one Parallel of the signe , to an other succeeding Parallel of a signe , again , a gnomonical year is limited by the shadow of a Gnomon , from a point in the Meridian of the Tropick of Cancer , till it shall revolve to the same Meridian Altitude and point of the Tropick , and is the same as is a tropical year : wherefore , above all things we ought first tobe acquainted with the knowledge of the Circles of the Sphaere ▪ Secondly , to have a judicious and exact discerning of those Planes in which we ought to project Dials . Thirdly , to consider the Style , Quality , and Position of the axis or Style , with consideration of the cause , nature and effects in such or such Planes as also an artificial projecting of the same , either on a Superficies by a Geometricall Knowledge , and reducing them to Tables by Arithmetick , which we have afore demonstrated , and come now to the conclusion : So that as I began with the Diall of Life , So we shall Dye-all , For , Mors ultima linea . TO ABRAHAM CHAMBRELAN Esq . S M. consecrateth his Court of Arts . SIR . IF the Originall Light be manifestatiu , by it I have made a double discovery , your genius did so discover it self according to the quality of the Sun , that I am umbrated and passive like the eclipsed Moon , yet cannot but reflect a beame which I have received from the fountain of Light ; 't is you which I make the Patron to my fancy ( which perhaps you may wonder at the Idleness of my head , to tell you a dream , or a praeludium of the several Arts : howsoever knowing you are a Lover of them , I did easily believe you could not but delight in the Scaene ; though in most I have written , I have in some sort imitated Nature it self , which dispenseth not her Light without Shadows , which will truly follow them from whom they proceed , and shall Sir , in time to come render me like Pentheus whose curiosity in prying into Secrets makes me uncertain . Et Solem Geminum duplices se ostendere Thebas , & while I know neither Copernicus , nor Ptolomies Systeme of the World , dare affirmatively reject neither , but run after both ; and submitting my wisdome to the wisest of men , must conclude , that Cuncta fecit tempestatibus suis pulchra , and hath also set the World in their meditation : Yet can not Man find out the Work that God hath wrought . Sir , pardon my boldness , in fastning this on your Patronage , who indeed are called to this Court of Arts , as being Nobly descended , whom only it concernes ; and only whose Vertue hath arrived them to the Temple of Honour , who are all invited as appeareth in the conclusion of this imaginary description , wherein , whilst I seem to be in a dream ; yet Sir , I am certain , I know my selfe to be Yours in all that I am able to serve you , S. M. TOPOTHESIA . OR An IMAGINARY DESCRIPTION of the COVRT of ART . COmming into a Librarie of Learning , where there was more Languages then I had Tongues , that if I had been asked to bring brick I should have brought morter , and going gradually along , as then but passus Geometricus , there I met Minerva , which said unto me ( Vade mecum ) & had not the expression of her gesture be-spoke my company , I should have shunned her ; she then taking me by the hand , led me to the end , where sat one which was called as I did inquire , Clemency , the name indeed I understood , but the Office I did not , whose Inscription was Custos Artis , I being touched now with a desire to understand this Inscription ; began with Desire , & craving leave , used diligence to peruse the Library , and found then a Booke intituled the Gate of Languages , by that I had perused it , I understood the fore-named Inscription , and craving leave of Clemency in what respect she might be called the Keeper of Arts , who answered with Claudanus thus ; Principio magni Custos Clementia Mundi , Quae Jovis incoluit Zonam quae temper at Aethrum , Frigoris & flammae mediam quae maxima natu , Coelicolum : nam prima Chaos Clementia solvit , Congeriem miserata rudem , vultuque sereno , Discussus tenebris in lucem saecula fundit . And arising from a Globe which was then her seat , she began to discourse of the Nature and Magnitude of the Terrestiall body , and propounded to me questions : as first , If one degree answerable to a Coelestiall degree yield 60 miles , what shall 360 degrees yield , the proportion was so plainly propounded , that I resolved it by the ordinary rule of Proportion , she seeing the resolution , propounded again , and said , if this solid Body were cut from the center how many solid obtuse angles might be cut from thence , at this I stumbled , and desired , considering my small practise , that she would reduce this Chaos also , and turne darknesse into light : seeing then my desire and diligence bid me make observation for those three were the wayes to bring me to peace , and resolved , that as from the center of a Circle but three obtuse angles could be struck , so from the center of a Globe , but three such angles could be struck and from thence fell to another question , & asked what I thought of the motion of that body : I answered , Motion I thought it had none , seeing I had such Secretaries of Nature on my side , and was loth to joyn my forces with the Copernicans . She answered , it was part of folly to condemn without knowing the reasons , I said it should stil remain a Hypothesis to me , but not a firme Axiome : for the resolution of which I wil onely sing as sometimes other Poets sang concerning the beginning of the world , and invert the sense onely , as that in another case , so this for our purppse . If Tellus winged bee The Earth a motion round , Then much deceiv'd are they That it before nere found . Solomon was the wisest , His wit ner'e this attain'd ; Cease then Copernicus , Thy Hypothesis vain . And began to discourse of the longitude of the earth , and then I demanded what benefit might incurre from thence to a young Diallist , she answered above all one most necessary Probleme , which we may finde in Petiscus his example , and propounded it thus ; The difference of meridians given , to finde the difference of hours . If the place be easterly , adde the difference of longitude converted into time to the hours given : if it be westerly , substract the easterly places , whose longitude is greater & contra , as in Petiscus his example , the meridian of Cracovia is 45 deg. 30 min. the longitude of the meridian of Heidelberge is 30 degrees , 45 minutes , therefore Heidleberg is the more westerly . One substracted from 45 30 30 45 the other sheweth the difference of longitude , to which degrees and minutes doth answer o ho . 59 m. for as Therefore when it is 2 hours post merid. at Cracovia ; at Neidelberg , it is but 1 hour , 1 minute past noon . For , There is left 1 houre 1 minute . Thus out of the difference of meridians , the divers situation of the heavens is known , and from the line of appearances of the heavens , the divers hours of divers places is known , and this is the foundation of observing the longitude : if it be observed what houre an Eclipse appears in one place , and what in another , the difference of time would shew the longitude , and hereby you may make a dyall that together with the proper place of elevation , shall shew for any other country ; for this Proposition I did hartily gratifie Geographia , and turning , said Astronomy , why stand you so sad ? she answered , Art is grown contemptible , and every one was ready to say ( Astrologus est Gastrologus ) then I said , what though vertue was despised , yet let them take this answer : Thou that contemnest Art And makes it not regarded , In Court of Art shal have no part None there but Arts rewarded . Gnashing the teeth as if ye strive to blame it , Yet know I 'le spare no cost for to obtein it . Perceiving your willingnesse said Astronomy , I will yet extend my charity and lay down the numbers , so that if you add the second and third and substract the first , it shall give the fourth ; the question demanded , and then I being careful of the tuition of what she should say , took a Table-book and writ them as follows . 1 The sine comp. elevation pole 38½ , sine 90 ; sine of the decl. of the sun yields the sine of the amplitude ortive : which is the distance of the suns rising from due East . 2 The sine 90 , the sine ele . pole 51d½ ; the sine of decli. yields the sine of the suns height at six a clock . 3 Sine comp. of altitude of the sun , sine comp. declina . sine 90 ; the sine of the angle of the vertical circle , and the meridian for the Azimuth of the sun at the hour of 6 : The Azimuth is that point of the compasse the sun is on . 4 Sine comp. decli. of the sun : sine compl. eleva . pole 38d½ , sine Altitude of the sun ; the houre distance from six . 5 Sine compl. of decli. sine 90 ; compl , of sine suns amplitude to sine compl. of the Assentional difference . 6 The sine of the difference of assention , Tang. decli. sun ; sine 90 : Tangent complement of the elivation . 7 Sine altitude of the sun , sine declina . of the sun ; sine 90 : Elevation of the pole . 8 Sine 90 , sine com . of distance from 6 ; sine com . declination of the sun : sine comp. of the altitude sun . 9 Sine 90 , sine eleva . pole ; sine alti . of a star : sine decli. of that star . 10 The sine of a stars altitude in an east Azimuth , sine amplitude ortive ; sine 90 : sine of the elevation . 11 The greatest meridian altitude , the lesse substracted sines ; the distance of the Tropicks , whose halfe distance is the greatest declination of the sun ; which added to the least meridian altitude , or substracted from the greater , leavs the altitude of the Equator : the complement whereof , is the elevation of the pole . 12 Tang. eleva . pole , sine 90 ; Tang. decli. of the sun , to the co-sine of the hour from the meridian , when the sun will be due east or west . By these Propositions said Astronomy , you may much benifit your selfe ; but let us now go see the Court of Art : I liked the motion , and we went and behold the sight had like to made me a Delinquent , for I saw nought but a poor Anatomy sitting on the earth naked exposed to the open Ayre , which made me think on the hardnesse of a Child of Art , that it had neither house nor bed , and now being at a pitch high enough resolve never to follow it : this Anatomy also it seems was ruled by many , both Rams , and Buls and Lions , for he was descanted thus on . Anatomy why do'st not make thy moane , So many limbes , and yet can'st govern none ; Thy head although it have a manly signe , Yet art thou placed on watry feminine . 'T is true , yet strong , but prethee let me tell yee , Let not the Virgin always rule your belly : For what , although the Lion rule your heart ; The weakest vessell will get the strongest part . Then be content set not your foot upon A slippery fish , that 's in an instant gone ; A slippery woman , who at Cupids call Will slip away , and so give you a fall : And if Rams horns she do on your head place ; It is a dangerous slip , may spoil your face . Here at I smiled , then said Astronomy , what is your thought ? then said I , do men or Artists so depend on women , as that their strength consists in them ? she said , I misunderstand him , for the Ram that rules the head is a signe masculine , because it is hot and dry , the Fish that rules the feet is cold and moist is therefore called feminine . Pisces the Fish you know's a watery creature , 'T is slippery , and shews a womans nature ; So women in their best performance fail , There 's no more hold then in a Fishes tail . But the more to affect the beholder , I will typigraphe this Court of Art . Under was written these lines , to shew mans misery by the fall , which I will deliver you , as followes : When Chaos became Cosmos , oh Lord ! than How excellent was Microcosmus , Man When he was subject to the Makers will , Stars influence could no way worke him ill : But since his fall his stage did open lye , And Constellations work his destiny . Thus man no sooner in the World did enter , But of the Circumference is the Center . And then came in Vertue , making a speech , and said ; Honour to him , that honour doth belong : You stripling Artist , coming through this throng , Have found out Vertue that doth stand to take You by the hand , and Gentleman you make . For Geometry , I care not who doth hear it , May bear in shield Coat Armor by his merit : We respect merit , our love is not so cold , We love mens worth ( not in love with mens gold ) Not Herald-like to sel , an Armes we give ; Honour to them that honourably live . The noble Professours of the Sciences , may bear as is here blazoned , ( viz. ) the field is Jupiter , Sun and Moon in conjunction proper , in a chief of the second , Saturn , Venus , Mercury in trine or perfect amity ; and Mars in the center of them ; Mantled of the Light , doubled of the night , and on a wreath of its colours a Helitropian or Marigold of the colour of Helion with this Motto , Quod est superius , est sicut inferius ; then did I desire to know , what did each Planet signifie in colour , she then told me as followeth . ☉ Or Gold ☽ Argent Silver ♂ Gules Red ♃ Azure Blew ♄ Sable Black ♀ Vert Green ☿ Purpre Purple And by mantled of Light , she meant Argent and of the night she meant an Azure mantle , powdered with Estoiles , or Stars Silver . I indeed liked the Blazon , and went in , where also I found a fair Genealogie of the Arts proceeding from the Conjunction of Arithmetick and Geometry collected by the famous Beda Dee in his Mathematicall Praeface . Both number and magnitude saith he have a certain originall seed of an incredible property ; of number a unit , of magnitude a point . Number , is the union and unity of Unites , and is called Arithmetick . ☽ Magnitude is a thing Mathematicall ▪ and is divisible for ever , and is called Geometry . Geodesie , or Land measuring Geographia , shewing wayes either in spherick , plane , or other the scituation of Cities , Towns , Villages , &c Chorographia , teaching how to describe a small proportion of ground , not regarding what it hath to the whole &c. Hydrographia , shewing on a Globe or Plane the analogicall description of the Ocean , Sea-Coasts , through the world , &c. N●vigation , demonstrating how by the shortest way , and in the shortest time a sufficient Ship , betweene any two places in passages navigable assigned , may be conducted , &c. Perspective , is an art Mathematicall which demonstrateth the properties of Radiations , direct , broken , & reflected Astronomie demonstrates the distance of magnitudes and naturall motions , apparances and passions proper to the Planets and fixed stars . Cosmographie ; the whole & perfect description of the heavenly , & also elementall part of the world & their homologall & mutuall collation necessary . Stratarithmetrie . is the ki●● appertaining to the War●● , to set in figure any number of men appointed : differing from Tacticie which is the wisdome & foresight . Musick , saith Plato , is sister to Astronomie , & is a Science Mathematicall , which teacheth by sence & reason perfectly , to judge & order the diversity of sounds high & low . Astrologie , severall from , but an off-spring of Astronomie , which demonstrated reasonably the operation and effects of the naturall beams of light , and secret influence of the Stars . Statick ; is an Art Mathematicall , demonstrating the causes of heavinesse and lightnesse of things . ●●thropographie , being the description of the number , weight , figure , s●●uation and colour of every diverse thing conteined in the body of man . Trochilike , descended of number and measure , demonstrating the properties of wheel or circular motions , whether simple or compound , neer Sister to whom is Holicosophie , which is seen in the describing of the severall conicall Sections and Hyperbolicalline in plants of Dyals or other by Spirall lines , Cylinder , Cone , &c. Pneumatithmie , demonstrating by close hollow figures Geometricall , the strange properties of motion , or stay of water , ayr , smoak , fire in their continuity . Menadrie , which demonstrateth how above natures vertue and force , power may be multiplyed ▪ Hypogeodie , being also a child of Mathematicall Arts , shewing how under the sphaericall superficies of the earth at any depth to any perpendicular assigned , to know both the distance and Azimuth from the entrance . Hydragogie , demonstrating the possible leading of water by Natures law , and by artificiall help . H●rometrie , or this present work of Horologiographia , of which it is said , the commodity thereof no man would want that could know how to bestow his time . ●ographie , demonstrating how the intersection of all visuall Pyramids made by any plane assigned , the center , distance and lights , may be by lines and proper colours represented . Then followed Architecture , as chief Master , with whom remained the demonstrative reason and cause of the Mechanick work in line , plane and solid , by the help of all the forementioned Sciences . Thaumaturgike , giving certain order to make strange works , of the sence to be perceived , and greatly to be wondred at . Arthemeastire , teaching to bring to actuall experience , all worthy conclusions by the Arts Mathematicall . While I was busied in this imployment which indeed is my calling , I questioned Caliopie , why she put the note of Illegitimacy upon Astrologie ; she said , it indeed made Astronomy her father , but it was never owned to participate of the inheritance of the Arts , and therefore the Pedegree doth very fitly say , doth reasonably not , quasi Intellectivè ▪ but Imperfectivè ; then did I ask again , why Arithmetick had the distinction of an elder brother the Labell , she told me , because it was the unity of units , and hath three files united in one Lambeaux , and did therefore signifie a mystery , then said I , why do you represent magnitude by the distinction of a second Brother , to which she said , because as the Moon , so magnitude in increasing or decreasing is the same in reason , then did she being the principall of the nine Muses , and Goddesse of Heralds summon to Urania , and so to all the other to be silent , at which silence was heard Harmonicon Coeleste by the various Motions of the Heavens , and Fame her Trumpeter sounded forth the praise of men , famous in their generation ; and concluded with the Dedication and Consecration of the Court of Arts in these words of the learned Vencelaus Clemens . Templum hoc sacrum est , Pietati , Virtuti , Honori , Amori , Fidei , semi Deûm ergò , & Coelo Ductum genus , vos magni minoresque Dei , Vos turba ministra Deorum vos inquam . Sancti Davides , magnanimi Hercules , generosi Megistanes Bellicosi Alexandri , gloriosi Augusti , docti Platones , Facundi Nestores , imici Jonathanes , fidi Achatae . Uno verbo boni Huc adeste , praeiste , prodeste Vos verò orbis propudia Impii Holophernes , dolosi Achitopheles , superbi Amanes , Truculenti Herodes , proditorus Judae , impuri Nerones , Falsi Sinones , seditiosi Catilinae , apostatae Juliani . Adeoque , quicunque , quacunque , quodcunque es Malus , mala , malum , exeste Procul hinc procul ite prophani . Templi hujus Pietas excubat antefores , Virtute & Honore vigilantibus Amore & fide assistentibus Reliqua providente aedituo Memoria , Apud quam nomin● profiteri Fas & Jura sin●nt . Quantum hoc est ? tantum Vos caetera , quos demisse compellamus , praestabitis , Vivite , vincite , valete , Favete . Et vos ô viri omnium ordinum , Dignitatum , Honorum , spectatissimi amplissimi , christianissimi , &c. Which being done , the Muses left me , and I found my self like Memnon , or a youth too forward , who being as the learned Sir Francis Bacon saith , animated with popular applause , did in a rash boldnesse come to incounter in single combate with Achilles the valiantest of the Grecians , which if like him I am overcome by greater Artists , yet I doubt not but this work shall have the same obsequies of pitty shed upon it , as upon the sonne of Aurora's Bright Armour , upon whose statue the sun reflecting with its morning beames , did usually send forth a mourning sound . And if you say , I had better have followed my Heraldry ( being it is my calling ) henceforth you shall find me in my own sphear . FINIS . Notes, typically marginal, from the original text Notes for div A89305e-52990 ☞ ☞ 
A29762	----	Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath. Brown, John, philomath. 1671 Approx. 407 KB of XML-encoded text transcribed from 196 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-05 (EEBO-TCP Phase 1). A29762 Wing B5042 ESTC R17803 12547379 ocm 12547379 63104 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29762) Transcribed from: (Early English Books Online ; image set 63104) Images scanned from microfilm: (Early English books, 1641-1700 ; 955:8) Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath. Brown, John, philomath. 305 [i.e. 304], [6] p. : 40 ill. Printed by John Darby, for John Wingfield ... and by John Brown ... and by John Seller ..., London : 1671. 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Dialing -- Early works to 1800. 2004-01 TCP Assigned for keying and markup 2004-02 SPi Global Keyed and coded from ProQuest page images 2004-03 Judith Siefring Sampled and proofread 2004-03 Judith Siefring Text and markup reviewed and edited 2004-04 pfs Batch review (QC) and XML conversion Horologiographia : OR , The Art of Dyalling , BEING The Second Book of the Use of the Trianguler-Quadrant . Shewing the Natural , Artificial , and Instrumental way , of making of Sun-Dials , on any flat Superficies : With plain and easie Directions , to discover their Nature and Affections , by the Horizontal Projection . With the way of Drawing the usual Ornaments on any Plain : Also , a familiar easie way to draw those Lines on the Ceiling of a Room , by the Trianguler Quadrant . Also , the Use of the same Instrument in NAVIGATION ; Both for Observation , and Operation . Performing the use of several Sea-Instruments still in use . By Iohn Brown , Philomath . London , Printed by Iohn Darby , for Iohn Wingfield , and are to be sold at his house in Crutched-Fryers ; and by Iohn Brown at the Sphear and Sun-Dial in the Minories ; and by Iohn Seller at the Hermitage-stairs in Wapping . 1671. To the Courteous Reader . THou hast here presented to thy view ( Courteous Reader ) in this second Part , a plain discourse of Dialling , both Natural , Artificial , & Instrumental . Natural I call it , from the plain illustration thereof , by the Armilary Sphear of Brass herein described , or by the Poor-man's Dial-Sphear , as I fancy to call it , being only a moving Horizontal-Dial , and a moving Plain , according to the Figure thereof in the Book annexed , whereby all the Arks , Angles , Scituations and Affections , are very plainly represented to an ordinary capacity . Artificial I call it , from the lively delineation of the Horizontal-projection , the fittest in my opinion for the making plain the mystery of Dialling . Instrumental I call it , from the applying of the Trianguler-Quadrant , to the ready resolving all the Arithmetical and Astronomical work , needful thereunto ; and that to competent exactness as in the first Part , and also in this second Part is sufficiently seen , in finding the requisites and delineating the hour-lines to small parts , exactly & speedily by the natural Sines , Tangents and Secants on the Sector and Quadrant . Also , the ready way of finding the Suns Altitude , Hour , Azimuth , Angle of the Plain , and any such business relating to Dyalling , as in the first Part is largely treated on . Further , in this second Part you have Tables of the Suns Declination to every day of the years , 1 , 2 , & 3 , after the Bissextile , as near as any extent . Also , a short , but plain direction , how to use the Trianguler-Quadrant , at any manner of way of Observation used at Sea ; as backward or forward , as the Davis-Quadrant , and the Cross-staff is used ; also , as Gunter's Bow is used both for the Sun or Stars . A figure of the Stereographick Projection Pag ● The Prints of the Lines of Numbers , as you see here inserted , are in part according to Mr. Windgates , as to a single and broken line of Numbers : But the addition of the line of the Fractional parts of a pound , and the several Gage-points , were never before used as I know of ; but do much ease & expedite the Operations by the Line of Numbers , Sines and Tangents . Also , these Scales of Reduction are convenient for the finding the Decimal-fraction , equal to the other Sexagenary-fraction , and are agreeable to those Tables in Mr. Windgates Book of Arithmetick , pag. 82. Also note , that the figure of the Rule at the beginning of the Book , pasted on a Board , is the very same with that spoken of Chap. XV. Use 28. pag. 397 , of the first Part , and will work all Questions wrought by the Trianguler-Quadrant , to exercise them that are out of the way to have them made , and may serve as good directions to the young Instrument-Maker , though these are made too too small a Radius to arrive at exactness . The like may I say of the Gunters-Lines in the Figures annexed , yet as large as the Book will bear . Thus I have given you a brief account of my present Thoughts about this matter , and somewhat more particularly in the First Part , disclaiming all boasting or vain ostentation , knowing that at the next Impression it may be amended in many places ; I shall rest and remain , ready to make amends in the making of these , or any other Mathematical Instruments , at my House at the Sphear and Sun-Dial in the Great Minories , John Browne . February 16. 1670. CHAP. I. The use of the Trianguler Quadrant , IN Making of DIALS . SVn-Dials may be made on any Plain , and all kind of Plains are either Flat , as Horizontal ; or Vpright , or Leaning . The Horizontal hath two faces , the one beholding the Zenith , called the Horizontal-Plain ; the other , beholding the Nadir , as the Ceiling of a Room is . The Upright Plains , are those that make right Angles with the Horizon , and do behold neither the Zenith or Nadir , but are parallel to them . The Leaning Plains are of two sorts generally ; the one called Recliners , beholding the Zenith ; the other sort called Incliners , beholding the Nadir , as the outside , and in-side of a Roof of a House , may represent . The two last sorts , viz. Upright and Leaning , may be Direct , or Declining , viz. beholding the South , or North , or East , or West Point of the Horizon ; or Declining therefrom , viz. Declining from South , or North , toward the East or West . All which Plains , are lively represented by a Sphear , made for that purpose , in Brass or Pasteboard , or by the Projection of the Sphear in Plano , Thus ; Equal to the Radius of the smaller Tangents , describe the Circle ESWN , representing the Horizon , crossing it precisely in the the Center Z , with the Lines SN and EW , denoting the Points of South and North , East and West . Then counting the smaller Tangent on the Sector-side doubly , as thus , calling 5 , 10 ; & 10 , 20 ; & 20 , 40 ; & 30 , 60 ; & 40 , 80 ; & 45 , 90 ; &c. Lay off from Z , towards S , the complement of the Suns Meridian Altitude , in ♋ , in ♈ , and ♑ ; for those Points on the Meridian-line , between Z and S ; and consequently the half Tangent of the complement of the Suns Meridian Altitude in every degree of Declination , ( if you proceed so far ) . Then for the Intersections of all those Lines and Parallels of Declination on the North-side of the Meridian , Observe , That the same number of degrees and minuts , that any Point is above the Horizon on the South part of the Meridian in Summer , just so many degrees and minuts is his opposite Parallel in Winter below the Horizon . As thus for Example . The Sun being in ♋ , or 23 deg . 31 min. of Declination North , hath for his Meridian Altitude 62 degrees , and so many degrees is his opposite Parallel of 23-31 , or ♑ , below the North part of the Horizon , at midnight . As thus ; Let the Center , at the beginning of the Line of Tangents , represent the Center Z ; and let the Tangent of 45 , represent the Horizon in the Scheam , viz. S. and N. Then , As the distance from S. to ♑ , is 15 deg . taken from 45 toward 0 , and laid from S. to ♑ inwards toward the Center Z , as the distance was taken from the Tangent of 45 , toward the beginning of the Line of Tangents , that represents the Center ; So the Point Cancer from N. is 15 deg . counted beyond 45 , toward the end , below or beyond the Horizon . Again , As S. ♋ is 62 degrees from 45 towards 00 ; So is the other Point 62 degrees below N , taken from 45 , viz. at 76 degrees ; which being laid from N , doth over-reach this little page . So that to draw the Tropick of ♑ , the Point ♋ being his opposite , is 28 degrees from Z , or 62 deg . from S ; and the other Point of ♑ , on the North part of the Meridian , is 62 degrees , counting from 45 doubly also ; or 28 degrees from 90 , the supposed end of the Tangent , which is naturally infinite , being the Tangent of 76 degrees , or the Semi-tangent of 152 , reading the Tangents doubly from the Center ; which distance from the Center , to the Tangent of 76 ; or as half-tangents , 152 , laid from Z , gives the Point ♑ on the North-part of the Meridian , below the Horizon ; the midst between which two Points of ♑ on the South and North part of the Meridian , is the Center to draw the Tropick of Capricorn . Again , to illustrate this difficulty , to draw the Tropick of Cancer , the Suns Meridian-Altitude in ♑ , his opposite sign is 15 degrees above the Horizon on the South part of the Meridian , and 15 degrees below the Horizon , on the North-part of the Meridian , viz. the Extent from the Center to the Tangent of 52 deg . 30 min. or the Semi-tangent of 105 , reading it doubly ; being laid from Z , gives the Point ♋ below the Horizon ; the middle between which two Points is the Center to draw the Tropick of Cancer . Again , for the Equinoctial or Parallel of ♈ ; the Meridian Altitude in ♈ , is 38-28 ; and the Meridian Altitude likewise in ♎ , his opposite Parallel is 38-28 also ; so that if you count 38-28 doubly beyond 45 , which will be at the Tangent of 64 degrees and 14 minuts , and take from thence to the Center ; this distance laid from Z , shall give the Point AE below the Horizon , and the the middle between the two Points AE , is the Center to draw the Aequinoctial . Then for the Hour-Lines ; first , set off the Semi-tangent of 38-28 from Z to P ; and the Secant of 38-28 to the same Radius from Z to L , and draw the Line L 45 parallel to EW ; then make PL a Tangent of 45 degrees , and lay off the Tangents of 15-30 , and 45 , from L both-wayes , as you see in the Figure . Also , As the Sector stands , take out the = Tangents of 60 and 75 severally , and turn them four times from L both-wayes , and note those Points with 6 , 7 , 8 , 9 , 10 , 11. Lastly , Set one Point of the Compasses in L , and open the other to P , and draw the Line WPE for the hour of 6. Again , Set one Point in 7-15 degrees from L , and open the other to P , and draw the Hour-line 5 P 5 ; Set the same Extent also in 7 , or 5 , on the other side of L , and draw the Hour-line 7 P 7 , as the Figure sheweth . Then , Set one Point of the Compasses in 8 , 30 degrees from L , and open the other Point to P , and draw the Hour-line 8 P 8 , and remove it to the other side of L , and draw the Hour-line 4 P 4 : And so for all the rest in order . Thus having drawn the Figures ; to draw Lines therein , which shall truly represent any Plain whatsoever , observe the following Rules . 1. The Horizontal-Plain , is represented by the Circle E.S.W.N. 2. A direct South or North-Diall , is represented by the Line E.Z.W. 4. An East or West Plain , is represented by the Meridian-line of 12 , viz. S. & N. 5. A Polar Plain , is represented by the hour of 6 , viz. the Line E.P.W. 6. An Equinoctial Plain , is represented by the Equinoctial-line E.AE.W. 7. Any Direct Reclining , or Inclining-Plain , between the two last , is called , A direct Recliner , whose Poles are alwayes in the Meridian , and are represented by any Reclining Circle , as the two Circles W. ♋ . E. and E. ☉ . W. do shew . 8. An East or West Recliner or Incliner , represented by the Circle N.F.S. 9. A Declining and Reclining , or Inclining Polar-Plain ; that is , it so Declines and Reclines , or Inclines , as to lie parallel to the Pole , as the Circle 8 P 8 doth represent . 10. A Declining Reclining-Plain , that so Declines and Reclines , as not to fall in the Pole or Equinoctial , as generally they will do , as the Circle 60 G 60 doth represent , which Declines from the South-eastwards , and Reclines 62 deg . which kind of Plains are various and infinite , yet confined to six varieties , as afterward . Now , the way of Drawing these Scheams , to represent these varieties , is briefly thus , by the Sector . First , to the Radius of the small Tangents , draw the Circle N. E. S. W. observing this Method , if it be a South Recliner , to set the letter N above , and E on the right hand ; and contrarily , in North Recliners ; for we meddle not with Incliners till afterwards , ( and alwayes observe , that a South Incliner is the same with a North Recliner , and the contrary ) then cross that Circle with two Diameters , precisely in the Center , as the Letters shew ; then according to your Plains Declination from North or South , toward either East or West , set off the Declination with a Line of Chords or Sines , as before is shewed ; and draw that Line for the Perpendiculer Line of the Plain , and laying the same distance as much from E. and W. draw another Line Perpendiculer to the former , representing the Plain ; then , on the first Line , viz. the Plains Perpendiculer , lay off from Z , the half Tangent of the Plains Reclination from Z to E , and the half Tangent of the complement thereof from Z to Q the contrary way ; and the whole Tangent of the complement thereof from Z , contrary to E , on the same Line , extended for a Center , to draw the Reclining Circle that represents the Plain . Lastly , You must draw a Circle through Q and P , ( P being alwayes the Semi-tangent of the complement of the Latitude laid alwayes from Z toward N for the North Pole ) so as to cut the Primitive Circle N.E. S.W. into two equal parts , as is shewed in the 10th Proposition of the third Chapter ; part of which Line , doth represent the Stile-Line of the Dial ; which last work shall be again shewed in the Example . Example . To draw the Scheam for a Plain , Declining from the South to the West 35 degrees ; and Reclining 20 degrees , for the Latitude of 51-30 . First , to the Radius of your small Line of Tangents , being the Latteral distance from the Center to 45 , ( or larger if you please ) draw the Circle N.E.S.W. representing the Horizon , crossing it in the Center with the Lines N.S. & W.E. for the North and South , and East and West Lines . Then , Take out the latteral Tangent of half the latitude , viz. 19-15 , for 38-30 , calling the Tangent of 10 , the half Tangent of 20 ; and lay it from Z at the Center , to P for the Pole-point . Then consider the Declination of your Plain , and which way , as here 35 deg . − 0′ from the South towards the West ; take out the Chord of 35 deg . and lay if from S to C , and from W to A , and from N to D , and from E to B , for the more exact drawing of the Lines AB , CD ; the Lines CD representing the Poles of the Plain , and the Line AB the Declining Plain it self ; then from Z towards D , lay off the Tangent of 10 deg . ( being the half Tangent of 20 degrees , the given Declination ) to E. Also , Take out the Secant of 70 degrees , the complement of 20 , to the same Radius ; and that laid from the Point E , on the Line DC produced , shall be the Center to draw the Circle AFEB , that represents the Declining , Reclining Plain , that declines 35 degrees , and reclines 20 degrees . Also , Lay off half the Tangent of the complement of the Reclination , viz. 35 degrees ( for the Reclination is 20 , the complement whereof is 70 , and the half of 70 is 35 ) from Z to Q. Then to draw the Line QP , do thus ; Observe how many degrees you count from Z to the Point E , counting from the Center , count so many in the manner of half Tangents from 45 ; and the latteral distance from thence to the Center , laid from the Center Z , on the Line CD , gives a third Point , viz. the Point I ; which three Points , QPI , brought into a Circle , will cut the Circle N.E.S.W. into two equal parts . Or thus ; The Semi-tangent of the complement of the Reclination to 180 degrees , laid from Z on the Line CD , will find the Point I. As thus ; The Reclination is 20 , the complement 70 , being taken from 180 rests 110 , whose half is 55 , the Tangent of ZI . Or more briefly thus ; Set one Point of the Compasses in the small Tangent of 45 , and count the Reclination from thence in the way of Semi-tangents , both wayes , both above and under 45 ; and lay one , viz. that under 45 , from C to Q ; and the other , viz. that above 45 from D to I ; then on the middle Point , between Q and I , last found , raise a Perpendiculer to CD , and in that Line will be the Center to draw IPQ. Also , CHAP. II. To Draw the Hour-Lines on all Ordinary Dials ; the easiest in the first place . 1. And first for the first Equinoctial - DIAL . AN Equinoctial Plain , as before is shewed , is that whose plain or flat Superficies lieth parallel to the Equinoctial , and is represented by the Line WAEE in the general Scheam , and therefore needs no other Scheam to represent it ; In which Dial all the Hour-lines are equal one to the other , being just 15 degrees assunder ; so that to draw the Hour-lines here , describe a Circle as the Circle 12.6.12.6 . and fit the Radius in the Sine of 30 degrees ( or the Chord of 60 ) and take out the parallel Sine of 7 degrees 30 minuts , the half of 15 degrees , and number it from 12 round about , and that shall divide the whole Circle into 24 equal parts , for the 24 hours , for the true Hour-lines on the Equinoctial-plain , and is the same in all latitudes ; only in the setting of it , the Poles of it are to be set due North and South ; the Horizontal-line on the Plain , lying Parallel to the East and West-points of the Horizon , and the Stile thereof , only a Wyre or sharp Edge standing perpendiculerly on the Center ; which being so set , must point directly to the North ( and South ) Poles of the World. The reclining Dials-Stile pointing to the North-Pole , and the inclining Dials-Stile pointing to the South Pole ; then is the Dial truly placed . To set a Plain ( or to try whether a Plain be set ) Polar , or Parallel to the Equinoctial , do thus ; But to try the Inclining Plain , apply the Loose-piece to the Plain with the Head-end downwards ; or else apply the Head-leg to the Plain with the Head-end downwards , and the Thred shall cut on 38-30 in London latitude , if the Plain be set parallel to the Equinoctial . 2. To draw a Direct Polar Dial. The next Dial , shall be a Direct Polar-Dial , which is represented in the general Scheam by the Hour-line of 6 , viz. the Line EPW ; And here also the Horizontal-line on the Plain , is parallel to the East & West-points of the Horizon ; and the Pole ( or Point opposite to the Plain ) is in the Equinoctial-point . The Hour-lines , in this Plain , are all parallels , because the Axis , or Stile-line , in all Plains , is parallel to the Poles of the World ; and this Plain it self , being so parallel , the Stile or Axis therein makes no Angle ; therefore the Hour-lines must needs be parallels also . And the way of drawing those Hour-Lines , is thus ; First draw the Perpendiculer-line on the Plain , which is done thus by the Trianguler-Quadrant ; Hang a Plummet and Thred on the Center , and apply the Moveable-leg to the Plain , to and fro , till the Thred falls neatly on 600 , and draw that Line along by the Moveable-leg , which shall be a true Horizontal-line on any reclining Plain ; and a Perpendiculer-line thereunto , is the perpendiculer Line on the Plain . Or else ; When the Sun shineth ( the Sun begins in the Pole of the Plain ) hold up a Thred and Plummet , till the shadow of the Thred fall on the Plain , making two Points in that shadow at the remotest distance asunder ; then a Line drawn through those two points shall be a true Perpendiculer-line , ( this shall need no more Repetition ) . Then , The several = Tangents of 60-45 , 30 & 15 , laid both wayes from 12 on both the Horizontal-lines , shall give you Po●n●s whereby to draw all the Hour-lines in their true places . Also , The = Tangent of 45 , shall be the true breadth of the Plate that must be a S●●●e to this Dial ; or the length of an upright Wyre set any where in the Line 12. Note , That for the hours under 45 , you may take = 45 from the small Tangents , and make it a = Tangent of 45 in the great Tangents ; and then take o● = Tangent of 30 & 30 , for 2 & 10 ; an● the = Tangent of 15 for 11 & 1 ; and if you want them above 45 , then take the = Tangent of 60 & 60 from the small Tangents , and turn that Extent 4 times from 12 both wayes , on both the Horizontal-lines , and those shall be the Points for 8 in the forenoon , and 4 afternoon . And lastly , The = Tangent of 75 , taken and turned 4 times from 12 to 7 in the morning , and to 5 in the afternoon , will fit and fill a Plain of 4 foot in breadth , with a Sector of one foot , shut . 3. To draw a Direct East or West-DIAL . The next Dial , in the third place , is the Direct East or West-Dial , which is represented in the general Scheam by the Line NZS , whose Poles are in the Line EZW , whose Plain also is = to the Pole , & drawn in the same manner as the Polar Dial was ; yet with this difference , the Equinoctial-line , whereon to prick the hours , is not the Horizontal-line , but is thus found . Thus ; You may take the Tangents under 45 , when the Sector is set to the small Tangent , and by turning 4 times , you have the remainder of the great Tangent above 45 ; when the Sector is set to the great Line of 45 , as in the Polar Dial. Or else ; Alter the Sector to the Radius of 45 in the great Tangent that goes but to 45 , and take out the = Tangents of 30 , and lay it from 6 both wayes , for 4 & 8 ; and the = Tangent of 15 , and lay it from 6 , both wayes , for 7 & 5. And lastly , By all those Points , draw Lines = to 6 , for the Hour-lines required ; and number the East-dial with the morning-hours , and the West with the afternoon-hours , the Stile is to be a Plate , or an upright Point ; the top of whose edge , or point , is to be equal to the Tangent of 45 , as the Sector stood to prick down the Hour-lines . 4. To draw the Horizontal-Plain . The fourth Plain next , in a natural order of easiness to apprehend as I judge , is the Horizontal Dial , that lies with its plain = to the Horizon ; and the Zenith is the Pole thereof , represented by the primitive Circle S.N.E.W. in the general Scheam , wherein only the Hour , Arks , and Stile is required . The Stiles Elevation is alwayes equal to the Latitude , and therefore given ; the Substile is alwayes in the Hour 12 , being the Meridian-line . The Hour-lines are found by this general Canon ; As the Sine of 90 , the right Angle PN 1 , to the Sine of PN , a side alwayes equal to the Latitude or Stiles elevation 51-30 ; So is the Tangent of the Angle NP 1 , 15 ; or NP 2 30 , &c. the Angles at the Pole , to the Tangent of N 1 a side , or N 2 a second side , the several Hour-arks on the Plain required ; found by the Artificial Sines and Tangents , as fast as one can write them down . Thus ; The Extent of the Compasses , from the Sine of 90 , the Sine of the latitude 51-30 , being laid the same way from the Tangent of 15 , shall reach to the Tangent of 11-50 ; and if you turn the Compasses the other way from the Tangent of 15 , it shall give the Tangent of 71-6 , for the hour of 5 as well as 11 ; which Numbers being gathered into a Table , and laid off by Chords or Sines in a Semi-circle , shall be the true Hour-points to draw the Lines by . But I shall not insist further thereon , but shew how to draw it more readily , and as truly by the Sector , thus ; First , draw a streight Line ( in the Meridian , if the Plain be fixed ) for 12 , as the Line AB ; then design a Point in that Line to serve for a Center , as at C ; then on the Center C , erect a Perpendiculer-line to AB , and draw it through the Center C , for the two 6 a clock Hour-lines , as the Line DE ; then draw two Lines equally distant from , and = to the first Line AB , on either side , as large as the Plain will give leave , as DF and EG ; ( which may commonly serve for margents to put the figure in ) . Then , Take the distance CD , and make it a = Secant of 00 , and take out the = Secant of the complement of the latitude , and lay it from D to F , and from E to G , on the two Parallel-Lines , and draw the Line FG. Then lastly , For pricking down the Stile , Note , That the = Tangent of 38-30 , the complement of the latitude , as the Sector stands for the Noon hours , laid from D to H , gives a Point to draw it truly by ; or the Sine of 51-30 the latitude , laid from B at nearest distance about H , as the Sector stood for the morning-hours will do as well . The Stile is to be a Plate , or a bended Wyre , cut or bended according to the Angle HCB , and erected Perpendicularly on the Line 12 , so long , as the Sun being 62 degrees high may cause the shadow thereof to reach the hour of 12 ; and then set duly North & South , and Horizontal , the shadow will shew the true hour of the day . Note the Figure . Note also , That a Horizontal Dial drawn for any one latitude , may serve for any other latitude North or South , elevating or depressing the Stile , till it look to the Pole-point ; that is , by making it to recline Northward , or Southward , as much as the difference of the latitudes , viz. that the Dial was made for , and that wherein it is to be used , shall be . 5. To draw a North or South Plain . The next Plain to this , and most like it , is the Direct North and South Dial ; whose Plain lies = to the prime Virtical , or Circle of East and West , and its Poles in the South and North part of the Horizon , and represented by the Line EZW , in the general Scheam , whose Stile is alwayes equal to the complement of the Latitude , as the Horizontals was equal to the Latitude , and consequently given . The Hour Arks on the Plain , are found by the former Canon , viz. As the Sine of 90 , viz. the Angle PZE , is to the Sine of the Side PZ , the Co-latitude or Stiles Elevation ; The Arks on the Plain , found as before by Artificial Sines and Tangents ; and being drawn into a Table , to be laid off by Chords , or Sines , or by the Sector , Thus ; Draw a Perpendiculer-line for the Substile , or 12 a clock Line ; and in that Line design a Point for a Center , as the Point A in the Line AB ; through which Point A , draw another Line , crossing the former at Right Angles , for a Horizontal-line , and the two sixes , as you did in the Horizontal ; then , on each side , and equi-distant from 12 , make two Lines = to AB , as marginal-lines , as CF and DE ; The distance AD , of the Parallel make a = Secant of 00 , and take out the = Secant of 51-30 , the latitude of the place , and lay it from C to F , and from D to E , and draw the Line FE ; then make DF a Tangent of 45 , and lay off the hours and quarters as you did in the Horizontal in all respects . Also , Make BF a = Tangent of 45 , and lay off the = Tangents of every hour and quarters ( if you please ) from B , both wayes , toward E and F ; and by those Points draw Lines for the hours required . The Angle of the Stile may be laid off by Sines , Tangents , or Chords , as before is shewed , to the quantity of the complement of the Latitude , and may be a Plate or Wyre , as you please , as the Angle GAB. The North Dial is the same with the South , for manner of making , only the Noon-hours are neglected , and the Morning and Evening-hours , both before and after 6 on each side only inserted ; and the Center of the Dial for that cause appointed in the middle of the Plain , and not on the upper-part , as in the South , and the Stile-points upwards ( as in the South it points downwards ) . Note the Figures . 6. To draw the Hours on a Direct Recliner . The next Plain to be considered , being also Direct , but not Erect , or Upright , but Leaning from you ; and may be either a North or South Recliner ; That is thus ; As the Poles , of a Direct South Plain , are in the meeting-point of the Meridian and Horizon , viz. the Point S. in the general Scheam ; and the Point N , in the same Scheam , is the Pole of the North Plain ; and the Point Z , is the Pole of the Horizontal Plain . So the Pole of these Plains is a Point in the Meridian , elevated as many degrees above the Horizon , as the Plain shall recline from the Zenith , or upright toward the Horizon . As thus ; Suppose the Hour-circle of 6 in the general Scheam , to represent a Reclining Plain , the Point AE , in the Meridian , is the Pole of it , being as many degrees above the Horizon S , as P is below the Zenith Z. So also is P the Pole to the North Reclining Plain WAEE ; for the Point P , is as much above the Horizon N , as AE is below the Zenith Z. Thus you see what the Pole of a Plain means , viz. a Point 90 degrees every way from it . Now therefore North Direct Recliners have their Poles any where between Z & N , and South Direct Recliners have their Poles any where between Z & S , according to the degrees and minuts of their Reclination . This being premised , for drawing the Hour-lines , observe , That for South Recliners Direct , the difference between the Reclination and the complement of your Latitude , is alwayes the Stiles height for that reclining Plain . But note , That when the Reclination is more than the complement of the Latitude , that then the contrary Pole is elevated , viz. the North Pole on South Recliners . But for North Recliners , the sum of the Co-latitude , and the Plains reclination is alwayes the Stiles elevation ; but note , when the Sum is above 90 , then the complement to 180 is the Stiles elevation ; but it must be turned the other way , viz. contrary to the nature of a North Recliner , for the Stile will look downwards in the North Recliner , and upwards in his opposite South Incliner . Note also by the way , That when the South Recliner reclines equal to the complement of the Latitude , it is called a Direct Polar Dial , or rather an Equinoctial in respect of his Poles , ( but I mind not to be singular ) . And when the Reclination of a North Recliner , is equal to the Latitude , then the Stiles height is just 90 degrees ; and the Plain , called an Equinoctial-plain , or rather Polar , in respect of his Poles , ( being first in order treated on ) . Thus understanding , and right conceiving what the Plains are , the drawing of any of them is the same with the North and South ; for the Stiles height is alwayes to be counted the complement of the Latitude , and by consequence you have the Latitude . As thus for Example . Comparing the reading , and Figure VII . Suppose a Plain recline from the Zenith toward the North part of the Horizon 10 degrees , his Pole is 10 degrees above the Horizon ; and then 10 taken from 38-30 , there remains 28-30 , for the Stiles Elevation ; or the Latitude to draw it as a Horizontal Dial ; and 61-30 for the Latitude for which place you are to draw a Direct Erect South Dial. Again , Suppose a South Recliner , recline 50 degrees , being more than 38-30 , the Co-latitude ; then take the Co-latitude from thence , and there remains 11-30 for the Latitude or Stiles height , to draw a Horizontal Dial by ; and 78-30 for a Latitude to draw a South Dial by ; but the Cock must look up to the North , therefore must be turned the other way . Again , For a North Recliner , reclining 60 degrees ; 60 & 38-30 , added , makes 98-30 , whose complement to 180 , is 81-30 , the Stiles height ; but the contrary way , as you may well perceive by the Horizontal . As once more thus ; Suppose a North Plain recline 85 degrees ; that , and 38-30 , added , makes 123-30 , whose complement to 180 , is 56-30 , the Stiles height : but put the con●●ary way , as a South Incliner , being almost a Horizontal Dial ; so that to draw this Dial , let 56-30 be the Stiles height , or Co-latitude ▪ then , 33-30 , is the Latitude to draw a Direct South Dial by . Or , You may count the Stiles height the Latitude , and then draw it as a Horizontal-Dial , by taking out the Secant of the Co-latitude , and the work will be the same ; As in the Figure North Reclining 60 degrees . 7. To draw a Direct East or West-Recliner . As thus ; With 45 degrees of the small Tangents , draw the Circle N.E.S.W. crossing it in the Center with the Lines WE , SN ; then lay off the half Tangent of 38-30 , from Z to P ; and the half Tangent of the Reclination 45 , from Z to E , and from Z to Q , and draw the Circles NES , and FPQ ; in which Scheam , PF represents the Stile , FG the distance of the Substile from the Meridian , and GPF the Angle between the two Meridians , viz. ZPN of the place , QPF of the Plain . All which requisites are thus found out by the Artificial , or Natural Sines and Tangents . 1. And first for the Stiles Elevation . As the Sine of 90 NZ , to the Sine of ZE , the Reclination 45 ; So is the Sine of NP the Latitude 51-32 , to the Sine of PF , the Stiles height , 33-37 . To work this by the Trianguler Quadrant , or Sector , do thus ; As — sine of NP , the Latitude 51-32 , to = sine 90 ZE ; So is = sine of ZE , the Reclination 45 , to — sine of PF 33-17 , the Stiles height . 2. For the Distance of the Substile from 12 , thus , by Artificial Sines and Tangents . As the Tangent of the Reclination , ZE 45 , to sine of EN 90 ; So is the Tangent of the Stiles elevation , PF 33-37 , to the sine of FG 41-40 , the distance of the Substile from 12. Or , by Natural Sines and Tangents , thus ; As — sine 90 , to = Tangent of the Reclination 45 ; So is = Tangent of the Stiles height 33-37 , to — sine of the Substile from 12 , 41-40 . For , If you only take the Tangent of 33-37 , from the Moveable-leg , and measure it on the Sines from the Center , it shall reach to the sine of 41-40 , the Substiles distance from 12. 3. For the Inclination of Meridians , thus ; As the sine of the Latitude NP 51-32 , to the sine of PFG 90 ; So is the sine of the Substile from 12 , GF 41-40 , to the sine of GPF 58-7 , the Angle between the two Meridians . By Natural Lines thus ; As — sine of the Substile 41-40 GF , to = sine of the Latitude 51-32 NP ; So is = sine of 90 PFG , to — sine of GPF 58-7 , the Incliner . If this Rule fails , use a less Radius , or a = Answer , as is largely shewed before . Thus having found the Requisites , proceed to draw the Dial thus ; First consider the Scheam , where you shall find the Reclining Plain to be represented by the Line SEN , as the upper-edge thereof ; SZN is the Horizontal-line and Meridian-line also ; N or G is the place where the Meridian cuts the Plain , being in the Horizon ; Therefore here the Hour-line of 12 is a Horizontal-line , and the Sun being in the South part of the Meridian , doth cast his shadow Northwards ; and being in the East , casts his shadow Westwards : Therefore laying the Scheam before you , as the Plain reclines from you , you shall see that the Meridian must lie to the right-hand from Z , toward N ; and the Substile upwards from N , towards E , at F. And the the Stile must look upwards , as the Angle GPF doth plainly shew ; and the morning-hours are chiefly fit for the Plain , because the Sun rising Eastward , is opposite to the Plain . Thus the Affections and Scituations of the Cardinal-lines are naturally and demonstratively shewed , the Delineation followes . First , Draw the Horizontal-line SN ; and on Z , ●s a Center , describe a Semi-cirle as SEN , ●nd from N toward E lay off 41-40 , the distance of the Substile from 12 , and draw ●he Line ZF for the Substile ; also beyond ●hat , from F to E , prick off 33-17 , and ●raw that Line for the Stile-line . Then for drawing the Hour-Iines , you ●ust first make the Table of Equinoctial-●istances , or Angles at the Pole , thus ; First , in all Direct Plains , it is orderly ●●us ; 3-45 , for the first quarter of an hour ●●om 12 ; 7-30 , for half an hour ; 11-15 , ●or three quarters ; and 15 degrees for an ●our ; and so successively to 90 : So also ●ill it be in all Plains , whose inclination of Meridians is just 15 , 30 , 45 , 60 , 75 , or ●0 degrees , being even whole hours ; and ●ear as well , when it falls on an even quar●er of an hour also . But when it doth not 〈◊〉 here , then the best Rule or Method I ●now is thus ; First , set down 12 , 11 , 10 , 9 , 8 , 7 , 6 , ● , 4 , 3 , 2 , 1 , & 12 , as in the Table follow●●g . Then right against 12 , set down the In●ination of Meridians ; then substract 15 degrees for every hour ; and 3-45 for every quarter , as often as you can , setting down the remainder ; then draw a Line a-cross , and what the last number remaining wants of 15 , or 3-45 ( for hours , or hours and quarters ) set down on the other side below the Line , as you see in the Table following ; and so proceed , adding of 3-45 to that sum , for every quarter ; or , 15 degrees for every whole hour , till you come to 90 both wayes ; so is the Table of Hour-Arks at the Pole , compleated for all Hours that can come on this Dial , or on any other . The Table .   91 52 8 01 53 2 88 07   05 38   84 22   09 23   80 37   13 08   76 52 7 16 53 1 73 07   20 38   69 22   24 23   65 37   28 08   61 52 6 31 53 12 58 07   35 38   54 22   39 23   50 37   43 08   46 52 5 46 53 11 43 07   50 38   39 22   54 23   35 37   58 08   31 52 4 61 53 10 28 07   65 38   24 22   69 23   20 37   73 08   16 52 3 76 53 9 13 07   80 38   09 22   84 23   05 37   88 08   01 52 2 91 53 Thus you see that the Substile falls on near a quarter past 8 , or 3 hours 3 quarters and better from 12 ; then if you will , by the former Canon , you may find all the Hour-Arks on the Plain . Thus ; As the sine 90 , to the sine of the Stiles height 33-37 ; So is the Tangent of the Hours , in the Table last made , called Arks , at the Pole 31-53 for 6 , to the Tangent of the respective Hour-Arks at the Plain from the Substile 90-0 for 6. More br●●f thus ; As the sine of 90 , to the sine of the Stiles Elevation ; So is the Tangent of the Hour from 12 , to the Tangent of the Hour from the Substile . Which being brought into a Table , may be pricked down in a Semi-circle by Sines , or a Line of Chords , from the Substile on the Plain . But I prefer this Geometrical way before it thus ; Having drawn 12 , the Substile and Stile , ●●aw also a Line at any convenient distance ●●rallel to 12 , as DI ; then at any conve●●●nt distance from the Center Z , draw a 〈◊〉 Perpendiculer to the Substile quite ●●ough the Plain , as the Line KL . Then , Take the nearest distance from that meet●●● Point at F , to the Stile-line , and make 〈◊〉 = Tangent of 45 ; then the = Tangent 〈◊〉 every hour and quarter , as in the Table●●●ken ●●●ken from the Sector , and laid from F the 〈◊〉 way , as the hours go , shall be the true 〈◊〉 whereby to draw the Hour-lines re●●●red . But in regard that this way will some●●●e be troubled with Excursions in some of 〈◊〉 hours , you may help it thus ; Having 〈◊〉 some hours , as suppose 6 & 3 ; or 〈◊〉 9 ; or indeed any 2 hours , 3 hours di●●●●ce assunder , as here 6 & 9 ; take the di●●●nce between 6 and 9 , and lay it from the 〈◊〉 to N on the Meridian , and draw the 〈◊〉 9 N = to 6 at length , beyond 12 ; 〈◊〉 , as before , make 6-9 a = Tangent of 〈◊〉 , and lay off every hour and quarter , as 〈◊〉 the South Erect Dial , both wayes from Also , make N 9 a = Tangent of 45 , ●nd do likewise laying the hours both wayes ●rom N , and you shall have Points enough to draw the Dial by . Otherwise , make these Dials thus . Count the complement of the Latitude where the Dial is to stand , for the Latitude ; And the complement of the Reclination for a new Declination ; and then draw them as Upright Decliners by the following Rules , and you shall do as well and speedily as any way . But note , That all East Recliners , are North-east Decliners ; and West Recliners , are North-west Decliners ; And East and West Incliners ( being the under faces ) are South-east , and South-west Decliners . Also note , That if you draw your Scheam true , and large , you may from thence Geometrically find the Substile , Stile , Inclination of Meridians , and every hours distance on the Plain , by Scale and Compass , thus ; As Captain Lankford hath shewed . First , For the Stiles Elevation set off a Quadrant or quarter , as W.N. from B to I ; then a Rule laid from Z to I , cuts the Plain at A ; then a Rule laid from A , to P & F , cuts the Circle at O and L ; The Ark OL , is the Stiles Elevation , and measured by fit Chords , gives 33-17 . Secondly , For the distance of the Substile from 12 , a Rule laid from Q to F , cuts the Limb or Circle at M , the Ark MN measured on fit Chords , gives 41-40 , the Substile from 12. Thirdly , For the Inclination of Meridians , a Rule laid from P to A , in the Limb gives C ; the Ark WC 58-7 , is the Angle between the two Meridians . Fourthly , To find the Hour-Arks on the Plain ; a Rule laid from Q to the intersecting of every Hours Ark ( in the Scheam ) and the Plain as here , a Rule laid from Q to 6 , cuts the Limb at R , the Ark MR 19-0 is the distance of 6 from the Substile on the Plain ; and so for all others , as 12 & 6 is NR 60-40 , both hours and quarters if you have them truly drawn on a large general Scheam , as Mr. Lankford hath done . Thus much for Direct Plains , both Erect and Reclining , before I come to speak of Decliners ; It will not be amiss to shew how to find the declination of a Plain , both by the Sun-shine , or without , by a Magnetical-Needle , as followeth . As — sine of the Substile 41-40 GF , to = sine of the Latitude 51-32 NP ; So is = sine of 90 PFG , to — sine of GPF 58-7 , the Incliner . CHAP. III. To find the Declination of any PLAIN . FOr finding the Declination of a Plain , the most easie way is by a Magnetical-Needle , fitted according to Mr. Failes way , in the Index of a Declinatory ( as he calls it ) being 180 degrees of a Semi-circle , divided on an Oblong-Board , or Quadrant , or a longer Needle in a square Box , ( or ) fitted with Hinges and a Cover ; after all which wayes , you may have them made at the sign of the Sun-dial in the Minories , by Iohn Brown ; or of any other manner you shall think fit . But , to our Trianguler Quadrant , is a Box and Needle also to be fitted of another form , in some things more convenient . Whose form is thus ; First , in a piece of Box 5 inches long , 2 ½ broad , and 6 tenths of one inch thick , is a hole made near 4 inches long , 1 inch ¾ broad , and 4 tenths deep for a Needle to play in ; about 50 degrees at each end ; with brass-hinges , and a cover , and a brace to keep the lid upright , & an Axis of Th●ed , and a Plummet playing in the lid , and a Horizontal and a South-dial , drawn on the Box and Cover ; also a hasp and glass to keep the Needle close covered , and on the bottom a Grove one tenth of an inch dee● , made just as broad as one leg of the Sector is . The use whereof is thus ; Put your Box and Needle , on that leg of the Sector , as will be most convenient for your purpose ; the North or cross-end of the Needle toward the Wall , when it is a South decliner ; and the contrary when it is applyed to a North decliner , as the playing of the Needle will tell you better than many words ; then open or close the Rule , till the Needle play right over the Line in the bottom of the Box , ( unless there be variation , then you must allow for it Eastwards or Westwards what it is ) . Then , I say , the quantity of the Angle in degrees and minuts the Sector stands at , above or under 90 , is the degrees and minuts of Declination ; being counted from 00 in the little Semi-circle , as complements to the Angle of opening ; as in the 4th Use of the 5th Chapter is largely and plainly shewed . Thus you have the quantity of degrees and minuts of Declination : but to determine which way , consider thus ; If the Needle will stand still in the middle , when the North-end is toward the Wall , then the first denomination is South , if not North. Again , When you know where North and South is , you may resolve which way the East and West is ; For , observe alwayes , if the North be before you , then the East is on the right-hand , and the West on the left ; and contrarily , If the South be before you , the West is on the right-hand , and the East on the left . Then , If the Sun , being in the East-point of the Horizon , can look on the Plain , it is a South-east Plain ; but if it beholds it when in the West-point , it is a South-west Plain . Likewise , If the Cross-end of the Needle will not stand toward the Wall ( the Needle playing well ) and the Sun being due East , beholds the Plain , then it is so many degrees North-east ; but if it cannot look on the Plain , being due East , then it is a North-west Plain , declining so many deg . as the Sector stands at , under or above 90 , being alwayes the complement of the Angle the legs of the Sector stand at , and found by taking the Angle the legs stand at , from 90 , when the Angle is less than 90. Or , Taking 90 out of the Angle , when it stands at an Angle above 90 degrees as a look at the little Semi-circle on the Head sheweth . Example . Suppose I come to a Wall , and putting the Box and Needle on the Leg of the Sector , and applying the other Leg to the Wall ( or on a streight piece of Wood , applied to the Wall , because of the Walls unevenness ) ; and open or close the Legs , till the Needle playes right over the Meridian-line , drawn on the bottom of the Box ; then , I say , the complement of the Angle the Legs of the Sector stands at , being alwayes what it wants of , or is above 90 degrees , is the degrees of Declination ; and the Coast which way , the Needle and Suns being East and West , tells you . For , If the North or Cross-end of the Needle be toward the Wall , it is a South Plain ; and if the Sun , being in the East , can behold it , then it is South-east ; if not , a South-west Plain . A ready way of counting the Angle found , may be thus ; Take the = distance between Center and Center , in the middle of the innermost-lines , and lay it latterally from the Center , and co●nt two degrees more than the Point sheweth , after the manner of Chords from 90 ( at the sine of 45 ) toward the Compass-point , and that shall be the degrees and minuts required . Example . Suppose the Legs are so opened , that the = distance between the two Centers , makes the — sine of 25 ; then , I say , the Lines do stand at an Angle of 50 degrees , and the Legs at 48 , two degrees less , the complement whereof is 42 ; as if you count thus from 45 , you will find , 40 from 45 is 10 , 35 is 20 , 30 is 30 , 25 is 40 , and 2 degrees more makes 42 , the thing desired . But , If you like not the abating of two degrees , then the = distance taken just be●ween the two legs right against the Cen●ers , shall be just the — sine of 24 degrees , ●r 42 , counting after the manner of Chords , viz. every 5 degrees on the Sines , for 10 on ●he Chords backwards from 45 of the Sines , which is 90 in Chords . Or , If you use the first Rule , of the 4th Use●f ●f the 5th Chapter , viz. by taking the — ●ine of 30 , and put one Point of the Com●asses in the middle Center in the Tangent-●●ne , and apply the other to the Line of ●ines , you shall find it reach to the sine com●lement of the Angle the Lines stand at , ●iz . 40 degrees and 2 degrees more , viz. ●2 , is the Angle or thing desired ; as pra●tice with consideration will make easie . Thus , by the Needle , you may find the ●eclination of a Wall , which in cloudy ●eather may stand you in good stead ; or 〈◊〉 prove a declination taken by the Sun , to ●revent mistakes . And if nothing draw the ●eedle from its right position , but that it ●ay well , and you find the Angle truly , ●ou may come to less than half a degree : And this convenience it hath , that it carries the Needle a competent distance from the Wall , to prevent that attraction ; but if it happen to be so near a Meridian , or East and West-plain , that the Angle , by the Sector , cannot well be taken ; then you may only apply the side of the Box and Needle to the Wall , and the Needle it self will shew the Declination , on the degrees on the bottom of the Box. Yet for exactness , the way by the Sun is alwayes the best , where you may come to make a good Observation , and then the Needle only is not to be trusted to ; a better way with opportunity offering it self , To f●nd a Declination of a Wall by the Sun. For this purpose you ( must or ) ought to have another Thred and Plummet , which Thred may be a fine even small Pack-thred , and it is convenient to have it ready hanged up near the Wall , so far off , as the Trianguler-Quadrant may pass along between it and the Wall , that you may not be troubled to hold it up , and lay it down , and be annoyed with the inconveniencies of your hand shaking , and time wasting , to more uncertainty than needs be . Also , You must needs take notice of the two Meridians , viz. one of the place which is the Meridian , or 12 a clock ; to which place , when the Sun or a Star comes , it is said to be in the Meridian . And the other is the Meridian of the Plain , in which Line the Pole-point of every Plain is , being 90 degrees distant from the Plain every way , and in all upright-Dials their Pole is in the Horizon ; and that degree of Azimuth in which the Pole-point lies , counted from South or North toward East or West , is alwayes the declination thereof ; so that by finding the Suns Azimuth at any time , and the distance of the Sun at the same time from the Meridian of the Plain , is gotten the declination . The Azimuth of the Sun from the Meridian of the place , is found by the 26 , 27 , 28 , 30 , 32 , 34 , 39 Uses of the 15th Chapter . But the Azimuth of the Sun from the Meridian of the Plain , is found by applying the Head-leg against the Plain Horizontally , slipping it to and fro , till the shadow of the Thred , hung ( or held ) up , play right over the Center of the Trianguler-Quadrant on the Head-leg ; then what deg . soever the thred cuts , counted from 60●0 on the Loose-piece ( being the Perpendiculer or Pole-point of the Plain ) shall be the Azimuth of the Sun from the Meridian of the Plain . This is the Operation ; the Application or Use is worded several wayes by several men ; I hope I shall do it as fully , and as briefly as some others . The Sun , to our appearance , passeth from East , by the Meridian , to the West every day ; therefore in the morning it wants of coming to the Meridian ; at noon it is for a moment just in the Meridian , and in the afternoon it is past the Meridian of the place . Even so it begins to shine on , and is directly against , and leaveth to shine upon most Plains , when it begins to shine upon , or is not directly against ; I say , it wants of coming to the Pole or Meridian of the Plain . When it is directly against the Plain , then it is in the Meridian or Pole of the Plain ; when it ●s past , it is past , or begins to leave the Plain . Which Th●ee Varieties I intend thus briefly to e●press ; Azimuth Want , or W in the morning only ; Azimuth Direct at noon ; Azimuth past , or P being in the afternoon . The other Three Varieties let be Shadow Want , Shadow Direct , Shadow Past ; all which may be in several Plains at several times ; that is to say , at morning , noon , and night . These Observations , and Cautions premised , the Rule is thus ; 1. If the Azimuth and Shadow are both wanting , or both past ; substract the losser out of the greater , and the residue is the Declination . But if one want , and the other be past , then the sum of them is the Declination . 2. If the Sun come to the Meridian of the Plain , before it come to the Meridian of the place , it is an East Plain . But if it come to the Meridian of the place , before it come to the Meridian of the Plain , it is a West Plain . 3. If the sum or remainder , after Addition or Substraction , be under 90 , it is a South-east , or South-west Plain , declining so many degrees , as the sum or residue is . But if the sum or remainder be above 90 , it is a North-east or a North-west Plain , and the complement of the sum or remainder to 180 , is the quantity of Declination North-east , or North-west . 4. If the sum or remainder be 00 , it is just South ; If 90 , just East or West . But , If it be 180 , it is a direct North Plain . It shall be further Explained by two or three Examples . Suppose that on the first of May , in the forenoon , I come and apply the Head-leg of the Trianguler-Quadrant to the Wall , and holding of it level , the shadow of the Thred , held up steady , cuts the Center and 60 degrees on the Moving-leg ; that is , 60 deg . want ; which I presently set down in a Paper ready prepared , thus ; May 1 , 1669. Forenoon . Shadow 60 00 want . Altitude 20 00   Azimuth 94 00 want . Substract . 34 00 South-east . Then , as soon as possible , or rather by some body else , at the same moment , find the Suns Altitude , which suppose to be 20 degrees ; ( but if you are alone , and have a Thred ready hanged up ; then take the Altitude first , and the shadow will be had presently after , the Thred hanging steadily ) and set that down also , as here you see . Then by the 26th Use of the 15th Chapter , you shall find the Suns Azimuth at that time and Altitude to be 94 degrees , and after Substraction remains 34-0 , for the Walls declination Eastward , becau●e the remainder is under 90 , and the Sun comes to the Meridian of the Plain , before it comes to the Meridian of the place , or South . Again , In a morning , Iune 13 , I observe the Altitude , and find it 15 degrees , and instantly the shadow , and find it to be 10 degrees past the Plain , viz. on the Loose-piece , toward the Head-leg , I set both Altitude and Shadow , with the day and time down thus ; Iune 13 , Forenoon . Altitude 15 0   Shadow 10 0 Past. Azimuth 109 0 Want. Sum is 119 0     180 0     061 0 North-east . And then find the Azimuth at that time and Altitude to be 109 degrees ; here the terms being unlike , I add them together , and the sum being above 90 , I know it must be a North Plain ; and because the Sun comes to the Plain before it comes to the Meridian of the place , it is North-east ; and the complement of 119 to 180 , is 61-0 North-east . Again , Iune 13 , Afternoon ▪ Altitude 15 0   Shadow 20 0 Want. Azimuth 109 0 Past. Sum is 129 0     180 0     051 0 North-west . But if you happen to come when the Sun is in the Meridian of the Plain , then the Suns Azimuth is the Declination , East or West , as the Azimuth is . Also , If you take the shadow , when the Sun is just in the South , or Meridian of the place , the shadow is the Declination ; if it is past the Plain , it is Eastward ; if it wants , it is Westwards . Thus I have ( I hope ) shewed the true manner of finding the Declination of a Wall by the Sun shining on the Plain , as plainly and as briefly as the matter will bear , speaking to young Tiroes therein . It may be done also , by observing when the Sun just begins to shine on a Chimny , or Wall , or high place you cannot for the present come near , conceiving the Sun to be then just 90 degrees from the Meridian of the place wanting , or just when it leaves it being then 90 degrees past the Plain , then take the Altitude and Azimuth , and work accordingly to the former Rules . CHAP. IV. To Draw a South , or North Erect Declining-Dial ▪ FOr better illustrations sake , I will draw a particular Scheam for this Dial also , as I did for the East Recliner ; whose Declination let it be 20 degrees declining from the South toward the West , in the latitude of 51-32 for London . The Scheam is drawn by the former directions ; the Pole of the Plain being at D , declining 20 degrees from S toward W , and the Plain it self is represented by the Line AB ; the Circuler pricked Line DHPC is a certain Meridian drawn through the three given Points DPC , whose Center will be in the intersection of the Plain AB , and the Tangent Line for the hours , which being drawn , whatsoever ZH is in the half Tangents , ZQ is the complement thereof , in the same half Tangents . The Scheam thus drawn , ZH is the Substile , PH is the Stile , QZ the distance of 6 from 12 , HPZ the inclination of Meridians , or Angle between the two Meridians , viz. of the Place PZ , and of the Plain PH , found by the following Canons . By Artificial Sines and Tangents . 1. First for the Substile from 12. As the sine of 90 ZN , to the sine of the Declination NC 20 degrees ; So is the Co-tangent of the Latitude PZ 38-28 , to the Tangent of ZH 15-12 . 2. For the Stiles Elevation . As the sine of 90 ZN , to the Co-sine of the Plains Declination NA 70-0 ; So is the Co-sine of the Latitude ZP 38-28 , to the sine of PH 35-46 , the Stiles Elevation . 3. For the Distance between 6 & 12. As the sine of 90 ZW , to the sine of the Plains Declination WA 20-0 ; So is the Tangent of the Latitude NP 51-32 , to the Tangent of AQ , the Co-tangent of 6 from 12 , 23-18 . Or thus ; As Co-tangent Latitude ZP 38-28 , to the sine 90 ZPQ ; So is S. declination WZA 20-0 , to the Tangent of QA 23-18 . 4. For the Inclination of Meridians . As the sine of the Latitude ZAE 51-32 , to the sine of 90 ZS ; So is the Tangent of the declination SD 20-00 , to the Tangent of AEK 24-56 , the Inclination of Meridians . Or , As Co-sine Latitude 38-28 PZ , to the sine 90 PHZ ; So is the sine of the Substile ZH 15-12 , to the sine of ZPH 24-56 , IM . 5. Then having made a Table of Arks at the Pole , by this Canon you may find the Hour-Arks on the Plain . Thus ; As the sine of 90 PK , to the sine of the Stiles height PH 35 46 ; So is the Tangent of the Hour from 12 , 19-56 for I , AEI , to the Tangent of the Hour from the Substile on the Plain , H 1 , 12-14 . But I prefer the way by Tangents before it , as followeth . All these requisites may be found by the general Scale and Sector , the Canons whereof in brief are thus ; By the Trianguler-Quadrant and Sector . Substile . As — Co-tang . Lat. 38-28 ZP , To = sine 90 90-00 ZN ; So = sine Declination 20-00 NC , To — I Substiler 15-12 ZH . Stile . As — Co-sine Lat. 38-28 ZP , To = sine 90 90-00 ZN ; So = Co-sine declin . 70-00 NA , To — sine Stile 35-46 PH. Distance between 6 & 12. As — sine Declin . 20-0 WA , To = sine of 90 90-0 ZW ; So is = C.T. of Lat. 51-32 NP , To — C.T. 6 & 12 23-18 AQ . Tangent 66-42 .   Inclinations of Meridians . As — I declination 20-0 SD , To = sine Latitude 51-32 ZAE ; So = sine 90 90-00 ZS , To — T. Inclin . Merid. 24-56 AEK . These Requisites are also found by the particular Quadrant , very really and truly , for that Latitude the Rule is made for , in this manner . 1. First , for the Substile . Lay the Thred to the complement of the Plains declination , counted on the Azimuth Line , and on the degrees it giveth the Substile from 12 , counting from 600 on the Moveable-leg . Example . The Thred laid to 70 , the complement of 20 on the degrees , gives 15-12 for the Substile . 2. For the Stiles height . Take the distance between 90 , and the Plains declination on the Azimuth-line , and measure it on the particular Scale from the begin●ing , and it shall give the Angle of the Stiles-Elevation above the Substile , 35-46 . 3. For the Inclination of Meridians . Take the Substile from the particular Scale of Altitudes , and measure on the Azimuth-line from 90 , and it shall give the complement of the Inclination of Meridians , or the Angle counting from 90. viz. here 24-56 . 4. To find the Angle between 12 & 6. Take the Plains Declination from the particular Scale of Altitudes ( less by the sine of the Declination , to a Radius equal to 45 minuts of the first degree on the particular Scale of Altitudes ) , and lay it from 90 on the Azimuth Scale , and to the Compass-point lay the Thred , then on the Line of degrees , the Thred gives the complement of 6 from 12 , counting from 60 toward the end , as here it is in this Dial 23-18 . Also , the requisites may be found Geometrically by the Scheam , thus ; As , 1. A Ruler laid from D to H in the Limb , gives F ; the Ark CF is the Substile . 2. A Ruler laid from Q to P in the Limb gives I , the Ark AI is the Stiles height . 3. A Rule laid from P to Q , cuts the Limb at T ; the Ark TE is the Inclination of Meridians . Or , A Rule laid from P to K , cuts the Limb at L ; then SL is the Inclination of Meridians . 4. A Rule laid from D to Q , cuts the Limb at 6 , the Ark C 6 , is the Angle between 12 and 6. 5. A Rule laid from D , to the intersection of any other Hour-line , with the Plain AB on the Limb , gives Points , whose distances from C , are their Angles from 12 , or their distances from F , or their Angles from the Substile . To Delineate the Dial by the Sector . Thus by any of these wayes , having gotten the Requisites , proceed to draw the Dial thus ; Then , Take out the = Tangent of 30 , & 15 , and the respective quarters , and &c. as before ; then make 6 G a = Tangent of 45 , and do likewise as before , in the Horizontal and South Dials , and to those Points draw the Hour-lines required . 2. To Draw the Hour-lines on a North-Declining Dial. The Requisites , as Substile , and Stile , Inclination of Meridians , 6 & 12 , are found the same way , and by the same Rules , as the South Decliners are done . But when you come to delineate the Dial , there is some alteration ; which I conceive is best seen by an Example , as Northeast declining 35 deg . Lat. 51-32 , at London . First , as be●ore , draw a Perpendiculer-line for 12 a Clock , as AB ; then about the middle , or toward the lower-part of that Line , as at C , make a Point for a Center , as C ; then on the Center C describe the Arch of a Circle , that way , from the Line AB , as is contrary to the Coast of declination , as if the Plain declines Eastward , as here , draw the Arch Westward from AB , as BD ; and the contrary way in North-west plains ; and on that Ark lay down the Substile from 12 , and the Stiles height above the Substile , and the Hour of 6 , by the Angle of 6 & 12 ; and then , by those Points and the Center , draw these Lines . Then , at any distance , draw a Line = to 12 ▪ ( or AB ) as the Line EF , and make that distance a = Secant of 35 , the declination ; the Sector so set , take out the = Secant of the Latitude 51-32 , and lay it on the Parallel-line from 6 to 9 , then make 6-9 , the measure the Compass stands at , a = Tangent of 45 ; and take out the = Tangent of 15-30 , &c. and lay them both wayes from 6 , upwards and downwards ; also , for the hour of 10 , as the Sector stands , take out the = Tangent of 60 , and turn it 4 times from 6 on the Line EF ; and ( when you want it ) the = Tangent of 75 , and turn that also 4 times from 6 , for 11 a Clock-line ; and then by those Points , draw Lines for the Hours required . CHAP. V. To Draw the Hour-Lines on a Dial falling near the Meridian , whose Stile hath but a small Elevation , and therefore no Center . THe former Examples may be sufficient to the considerate , to draw any Erect Declining-Dial having a Center ; but when the Stile happens to be less than 15 deg . of Elevation ; then , if it be not augmented by casting away the Center , the usefulness , and handsomness of the Dial is l●st ; now if you draw the Dial by the former Rules on a Table , and cut off so much , and as many Hours as you care for , the work is performed . Lat. 51-32 d. mi.   S.W. 80-25   Sub. 38-4   Stile 5-56   ● & 12 38-51   I. M. 82-30 12   75●00     67-30 1   60-00     52-30 2   45-00     37-30 3   30-00     22-30 4   15-00     10-30 5   00-00 —   07-30 6   15-00     22-30 7   30-00     37-30 8 First , on or near the North Edge of the Plain , in far South decliners , ( but near to the South-edge of the Plain , in far North-decliners ) draw a perpendicular-Line , representing the Hour-line of 12 , as the Line AB in our Example , being a Southwest declining 80 deg . 25 min. then , in the upper-part of that Line , in South-decliners ; or about the middle , or lower-part in North-decliners , appoint a Center , as here at A ; then upon A , as a Center , as large as you may , draw an Arch as BD ; and in that Arch , or rather by a Tangent-line , lay 〈◊〉 the Substile from B to D , and draw the ●ine AD , as an obscure Line , for the present only to be seen ; and upon that , the Stile-line , as before : Then at any convenient places , as far from the Center as you can , draw two Lines Perpendiculer to the Substile , as the Lines CE , FG , for two contingent Lines , ( antiently and properly so called ) ; then by the Inclination of Meridians , by the directions in the East and West Recliner , being the 7th Dial in the 2d Chapter , make the Table of Hour-Arks at the Pole , by setting down against 12 , 82-30 ; and taking out 7-30 for every half hour , till you come to 00 at the Substile ; and then by adding 7-30 for every half hour , and 15 for every hour , to 8 ½ , as long as the Sun shines ; which in regard it falls on an even half hour , is the most easie , and fits the Points in the Tangent ready made for hours and quarters . The next work , is to resolve what hours shall come on the Plain , as will be best determined by the discreet Orderer , or Surveyor , or experimental Dialist , as here 8 and 1 ; and for those two hours , mark the upper contingent Line in two places where you would have them to be , as at E and C ; then take the — Tangent of 37-30 for 8 , from the small Tangents , and add it to the = Tangent of 67-30 , the Tangent for 1 ; and behold ! it makes the — Tang. of 72-33 . Then , Take the whole space CE , and make it a = Tangent of 72-33 ; then take out the = Tangent of 67-30 , and lay it from C to H ; and take also in the same common-line , right against the small Tangent of 37-30 , which is in the large Tangent 10-50 , the = Tangent of 10-50 , taken for 37-30 , being laid from E , the place for 8 , will meet just at H ; which Point H , is the true place for the Substile , to fit and fill the Plain , with the hours determined . Then , The Sector so set , Take out all the = Tangents above 45 , as in the Table , and lay them the right way from H , toward C , and E ; then take out = Tangent of small 45 ; and setting one Point in H , strike the touch of an Arch , as at I ; then make HI a = Tangent in great 45 , and take out the = Tangents of the rest of the hours under 45 , as in the Table , and lay them both wayes from H , because the Substile falls on an even half hour . Then , Draw the Line HK , = to the first Line AD , for the true Substile ; then make HK Radius , or the Tangent of 45 , and take out the Parallel Tangent of 5-56 , the Stiles height , and lay it from K to L ; then take HI , the first Radius , and setting one Point in L , draw the touch of an Ark as by M ; then draw a Line by the Convexity of the Arches by I and M , for the true Stile-line . Then , Take the nearest distance from the Point K , to the Line IM , and make it a = Tangent of 45 , the greater Radius , and take out the = Tangents , as in the Table , and lay them from K both wayes ; and then lastly , by those Points draw Lines from the hours required . Note , That if in striving to put too many hours , the sum of the two extream hours come to above 76 , it will make the hours too close together , and put you to much more trouble . Also note , If your Rule prove too small , then take the half of the sum of both the Tangents , and turn the Compasses twice . Also , If you be curious , you may use the Natural Logarithm Tangents , instead of the Line of Tangents , but this will serve very well . This is a general way of augmenting all manner of Dials , when the Stiles height is low , as under 15 degrees ; and as ready a way , as you meet with in any Author whatsoever . CHAP. VI. To Draw the Hour-Lines on Declining Reclining Dials . FOR the compleat and true drawing of these Dials , that you may plainly see their Affections and Properties , it will be necessary to have a Scheam for every variety ; in doing whereof , I shall follow the Method that Mr. Wells , of Deptford , used in his Art of Shadows ; which will comprehend any sort of Reclining and Declining Dial , under 6 varieties , viz. 3 South Recliners , and 3 North Recliners ( the Inclining being their opposites , and no other , as afterwards is shewed ) . Wherein I shall be very brief , yet sufficiently plain , to a Mathematical Genius , and render the Canons , by Artificial and Natural Sines and Tangents , and draw the Dial by the Sector , the fittest Instrument for that use ; With other occurrent Observations , as they come in place , and the way by the Scheam Geometrically also . 1. And first for a South-Declining , Reclining Dial , declining from the South toward the West 35 degrees , and reclining from the Zenith 20 degrees , being less than to the Pole , viz. falling from you , between the Zenith and the Pole : As the Circle AEB , representing the Reclining Plain , plainly sheweth P being the Pole , and Z the Zenith . The manner of drawing this Scheam , is plainly shewed before , ( Chap. 1. ) both generally and particularly for the drawing of Dials , and the Example there , is the very Scheam for this Dial ; wherein you may further consider , That the Perpendiculer-line is right before you , and when you look right on this Plain that declines Southwest , the North is before you on the left hand , the South behind you on the right hand 35 degrees , the East on the right hand , the West on the left ; the Line CD the perpendiculer-line right before you , representing the Perpendiculer-line on the Plain , AB the Horizontal-line , ZE the quantity of Reclination , PF the Stiles Elevation above the Plain , having the South Pole elevated above the lower-part of the Plain , because the North-Pole is behind the Plain , EG the distance on the Plain , between the Plains perpendiculer , and the Meridian , ( being to be laid Eastwards , as the Dial-draught sheweth , besides that general Rule before hinted , that whensoever a Plain declines Eastward , the Substile Line must stand Westward , and the contrary ; for the Arch whereon to prick the Substile , and Stile is alwayes to be drawn on that side of the Plain , which is contrary to the coast of declination ) EF the distance from the Substile and Perpendiculer , to be laid the same way ; GF the distance on the Plain , from the Substile to the Meridian , to be laid the same way also ; the Angle FPG , is the Inclination of Meridians ; All which Requisites are found by these Canons Arithmetically , or by the Artificial and Natural Sines and Tangents . 1. To find the Distance of 12 , from the Perpendiculer EG , or Horizon AG , by the second Axiome of Mr. Gellibrand , viz. that the Sines of the Base and Tangent of the Perpendiculer are proportional . By the Sector and Quadrant . As sine 90 Radius , ZD 90 — 00 To Tang. of Declin . Plain ND 35 — 00 So sine of Reclin . Plain ZE 20 — 00 To Tang. of Perp. & 12 EG 13 — 28 Whos 's complement AC 76 — 32 is the distance from the East-end of the Horizon to 12. As — Tangent of ND 35 — 0 To = Sine of ZD 90 — 0 So = Sine of ZE 20 — 0 To — Tangent of EG 13 — 28 , the distance of 12 , from the Perpendiculer . 2. To find the Distance on the Meridian ; from the Pole to the Plain , PG , by the 3 Propositions of Mr. Gellibrand , the Sines of the Sides are proportional to the Sines of their opposite Angles . By the Quadrant and Sector . As the sine of the Perpendiculer from 12 GE 13-28 To the sine of declination GZE 35-00 So is the sine of 90 GEZ 90-0 To the sine of the distance on the Meridian , from the Plain to the Zenith — GZ 23-55 As — sine 90-0 GEP , To = sine 35-0 GZE ; So = sine 13-28 GE , To — sine 23-55 GZ , which taken from 38-28 gives PG 14-33 . Which being taken from 38-28 , the distance on the Meridian from the Pole to the Zenith , leaveth the distance on the Meridian of the place , from the Pole to the Plain , viz. 14-33 , as a help to get the next . 3. To find the Height of the Stile above the Plain PF . In the two Triangles ZGE , and PGF , which are vertical , by the second Consectary of Mr. Gellibrand ; If two Perpendiculer Arks subtend equal Angles , on each side of the meeting , then the Sines of their Hypothenusaes , and Perpendiculers are proportional , ( and the contrary ) ; for the Angles ZGE , and PGF are equal Angled at G , and ZE , and PF , are both two perpendiculer Arks on the Plain AB . Therefore , As the sine of the Hypothenusa GZ , to the sine of the Perpendiculer ZE ; So is the sine of the Hypothenusa PG , to the sine of the Perpendiculer PF , and the contrary . Then thus ; by the Quadrant and Sector . As sine of the Arch of the Merid. from the Zenith to the Plain , ZG 23-57 To sine of the Reclination ZE 20 00 So is the sine of the Arch on the Meridian , from the Pole to the Plain PG 14-33 To sine of the Stiles height PF 12-13 As — sine of PG 14-33 To = sine of ZG 23-58 So = sine of ZE 20-00 To — sine of PF 12-13 4. To find the Distance of the Substile from the Meridian , GF . In the same Vertical Triangle , having the same acute Angle at the Base , the Tangents of the Perpendiculers , are proportional to the Sines of the Base , by the second Axiome of Mr. Gellibrand . Therefore , by the Quadrant and Sector . As the Tang. of the Reclin . ZE 20-0 To the sine of the Distance on the Plain , from the Perpend . to the Merid. GE 13-28 So is the T. of the Stils height PF 12-13 To the S. of the Subst . fr. 12 FG 7-58 As — sine of GE 13-28 To = Tang. of ZE 20-0 So = Tang. of PF 12-13 To — sine of FG 7-58 5. To find the Angle between the two Meridians , of the Place and Plain , viz. the Angle , PFG . By the third Proposition of Mr. Gellibrand , it is proved , That the Sines of the Sides are proportional to the Sines of their opposite Angles , and the contrary . Therefore , by the Quadrant and Sector . As the sine of the Dist. on the Merid. from the Pole to Plain PG 14-33 To the S. of 90 , the opp . Angle PFG 90-00 So is the S. of the Subst . fr. 12 , FG 07-58 To the S. of the Inclin . Merid. FPG 33-28 As = sine of the side 07-58 FG To = sine of the side 14-33 PG So = sine of the Angle 90-00 PFG To — sine of the Angle 33-28 FPG The Angle between the 2 Meridians . By Angle of Inclinations of Meridians , make the Table of the Hour-Angles at the Pole , by the Directions , Chap. 2. which being made as in the Table , draw the Dial in this manner ; 12 33-28   1 18-28   2 3-28       Sub. 3 11-32   4 26-32   5 41-32   6 56-32   7 71-32   8 86-32   9 78-28   10 63-28   11 48-28   12 33-28   Upon AB , the Horizontal-line of your Plain , describe the semi-circle AEB , and from the Perpendiculer-line CE of the Plain , lay off 13-28 Eastward for the 12 a clock Line , on the Plain , or the complement thereof 76-32 , from the East-end at B to + , & draw the Line C + . Again , Set further Eastward from 12 , 7-58 , the distance of the Substile from 12 , to F , and draw the Line CF for the Substile ; and beyond that , set off from F 12-13 , the Stiles height above the Substile to G , and draw CG also . Then , Draw a contingent Line perpendiculer to the Substile CF , as far from the Center as you can , as the Line HI ; then take the nearest distance from the point F , to the Line CG , and make it a = Tangent of 45 ; then the Sector being so set , take out the = Tangents of all the Hour-Arks in the Table , and lay them both wayes from F toward H and I , as they proceed ; then Lines drawn from the Center C , and those Points shall be the Hours required . Or , Having in that manner pricked down 12 , 6 & 3 ( or any other Hours 3 hours distant ) draw two Lines on each side 12 = to 12 , and measure the distance from 6 to 3 in the = , and lay it from C the Center on the Line 12 ; and by those two Points draw a third Line , = to the 6 a clock-line ; then 6-3 , and 12-3 , made a = Tangent of 45 , shall be the two Radiusses to lay off the Hour-lines from 6 & 12 , as before in the former Dials . And the = Tangent of Inclination of Meridians , doth prove the truth of your Work here also , as well as in the Decliners Erect . But note , That this Dial is better to be augmented by the losing the Hours of 8 and 9 in the morning , which makes the Hours more apparent , as you see . Also , the Requisites formerly sound , may Geometrically be found by the Scheam , being large and truly drawn , as before is shewed in the other Dials . Thus , 1. A Rule laid from Q , the Pole-point of the Plain , to G the Point of 12 on the Plain , gives in the Limb the point 12 ; D 12 , 13-28 , is the distance of 12 a clock-line on the Plain from the Plains perpendiculer-line ZD , ( and to be laid from the perpendiculer-line on the Plain Eastwards in the Dial ) ; and the distance on the Limb from A to 12 , is the Meridians distance from the East-end of the Horizontal-line on the Plain , namely 76-32 . 2. A Rule laid from Q to F , on the Limb , gives the Point Sub , for the Substile ; and the Ark Sub. 12 , 7-58 , is the distance from 12 , or the Ark Sub. D 21-26 , the distance from the Perpendiculer . 3. A Rule laid from Q to 6 , the place where the 6 a clock hour-line on the Scheam cuts the Plain , gives on the Limb the Point 6 , the Ark 6 12 , 25-38 , or 6 D , 38-56 , is the distance of the Hour-line of 6 on the Plain , from the Hour-line 12 , or the Perpendiculer . 4. A Rule laid from Y , the Pole-point of the Circle QFP , to P & F , on the limb , gives two points IK , and the Ark IK is the Stiles Elevation 12-13 . 5. A Rule laid from P to Y on the limb , gives the Point M ; EM is the Inclination of Meridians : or , a Rule laid from P , to the intersection of the Circle PFQ , and the Equinoctial-line , gives a Point in the Limb near C , which Ark CS , is more naturally the Angle between the two Meridians , 33-28 . Or , If you like the way of referring this Plain to a new Latitude , and to a new Declination in that new Latitude , Then thus by the Scheam ; 6. A Rule laid from E , to P and G , in the Limb gives L and O ; the Ark LO is the complement of the new Latitude , being the Ark PG , the second requisite , in the former Calculation being 14-33 , the distance on the Meridian from the Pole to the Plain . But note , That this Dial is better to be augmented by the losing the Hours of 8 and 9 in the morning , which makes the Hours more apparent , as you see . Also , the Requisites formerly found , may Geometrically be found by the Scheam , being large and truly drawn , as before is shewed in the other Dials . Thus , 1. A Rule laid from Q , the Pole-point of the Plain , to G the Point of 12 on the Plain , gives in the Limb the point 12 ; D 12 , 13-28 , is the distance of 12 a clock-line on the Plain from the Plains perpendiculer-line ZD , ( and to be laid from the perpendiculer-line on the Plain Eastwards in the Dial ) ; and the distance on the Limb from A to 12 , is the Meridians distance from the East-end of the Horizontal-line on the Plain , namely 76-32 . 2. A Rule laid from Q to F , on the Limb , gives the Point Sub , for the Substile ; and the Ark Sub. 12 , 7-58 , is the distance from 12 , or the Ark Sub. D 21-26 , the distance from the Perpendiculer . 3. A Rule laid from Q to 6 , the place where the 6 a clock hour-line on the Scheam cuts the Plain , gives on the Limb the Point 6 , the Ark 6 12 , 25-38 , or 6 D , 38-56 , is the distance of the Hour-line of 6 on the Plain , from the Hour-line 12 , or the Perpendiculer . 4. A Rule laid from Y , the Pole-point of the Circle QFP , to P & F , on the limb , gives two points IK , and the Ark IK is the Stiles Elevation 12-13 . 5. A Rule laid from P to Y on the limb , gives the Point M ; EM is the Inclination of Meridians : or , a Rule laid from P , to the intersection of the Circle PFQ , and the Equinoctial-line , gives a Point in the Limb near C , which Ark CS , is more naturally the Angle between the two Meridians , 33-28 . Or , If you like the way of referring this Plain to a new Latitude , and to a new Declination in that new Latitude , Then thus by the Scheam ; 6. A Rule laid from E , to P and G , in the Limb gives L and O ; the Ark LO is the complement of the new Latitude , being the Ark PG , the second requisite , in the former Calculation being 14-33 , the distance on the Meridian from the Pole to the Plain . 7. A Rule laid from G to Q on the limb , gives R , the Ark SR is the new declination in that new Latitude , 32-37 . Or else find it by this Rule ; As sine of 90 , to the Co-sine of the Reclination , or Inclination ; So is the sine of the old Declination , to the sine of the new , in this Example , being 32-37 , and generally the same way as the old Declination is . Only observe , That when the North-pole is Elevated on South Recliners , you must draw them as North-decliners ; and North-west and North-east incliners , that have the South-pole Elevated , you must draw them as South-east and West-decliners , which will direct as to the right way of placing the Substile , and Hour of 6 from 12. In this place I shall also insert the general way , by Calculation , to find the new Latitude , as well as new Declination : Which is thus ; As Radius , or Sine of 90 , to the Co-sine of the Plains old Declination ; So is the Co-tangent of the Reclination , or Inclin . to the Tang. of a 4th Ark. Then , In South Recliners , and in North Incliners , get the difference between this 4th Ark , and the Latitude of your place , and the complement of that difference is the new Latitude : if the 4th Ark be less then the old Latitude , then the contrary Pole is Elevated ; but if it be equal to the old Latitude , it is a Polar-plain . But in South Incliners , and in North Recliners , the difference between the 4th Ark , and the complement of the Latitude of the place ( or old Latitude ) shall be the new Latitude , when the 4th Ark and old Latitude is equal , it is an Equinoctial-plain . Thus in this Example ; As sine 90 , to Co-sine of 35 , the old Declination ; So is Co-tangent of 20 , the Reclination to 66-03 , for a 4th Ark ; from which taking 51-32 , the old Latitude , rests 14-31 , the complement of the new Latitude , which will be found to be 75-29 , the new Latitude . By which new Latitude , and new Declination , if you work as for an Erect Dial , you shall find the same Requisites , as by the former Operations you have done ; and the distance of the Perpendiculer and Meridian will set all right . The Second Variety of South Recliners , reclining just to the Pole. 1. The Scheam is drawn , as before , to the same Declination , and the same way , viz. 35 degrees Westward , and reclines 33-3′ , Now , to try whether such a Plain be just a Polar-plain or no , use this Proportion : By the Sector ; As the sine of 90 DA 90-0 To Co-sine of Declin . NA 55-0 So Co-tang . of Reclin . DE 56-57 To Tang. of Latitude NP 51-32 As — Co-sine Declination NA 55-00 To = sine of AD 90-00 So is = Co-tang . of Reclin . DE 56-57 being taken from the small Tangents , To — Tangent of NP 51-32 being measured from the Center on the same small Tangents . Which 4th Ark , if it hit to be right the Latitude , then it is a declining Polar-plain , or else not . 2. If you have a Declination given , to which you would find a Reclination to make it Polar , then reason thus ▪ By the Sector ; As the Co-sine of the Declin . AN 55-0 To the Radius or Sine of AD 90-0 So is the Tang. of the Lat. PN 51-32 To the Co-tang . of the Reclin . DE 56-57 As — Tangent of NP 51-30 To = Sine of AN 55-00 So = Sine of AD 90-00 To — Tangent of DE 56-57 3. If the Reclination were given , and the Declination required to make it a Polar , then the Canon may be thus ; By the Sector ; As the Co-tang . of the Reclin . DE 56-57 To the Radius , or Sine of AD 90-00 So is the Tang. of the Lat. NP 51-32 To the Co-sine of the Declin . NA 55-00 As — Co-tang . Reclination . DE 56-57 To = sine of AD 90-00 So — Tang. of Latitude ND 51-32 To = Co-sine of Declination NA 55-00 But by the Scheam , these three Operations are found by drawing the Scheam . 1. For if the Line or Circle , representing the Plain , cut the Pole P , it is a Polar-Dial . 2. If AB , the Co-declination , be given , then draw the Circle APB , and it gives E ; then ZE is the Reclination , measured by half Tangents ; or a Rule laid from A to E on the Limb , gives an Ark from B ; which measured on fit Chords , is the Reclination . 3. If P , the Pole-point , and ZE the Reclinatin , be given ; then , with the distance ZE , on Z as a Center , draw an Ark of a Circle in that Quadrant which is contrary to the Coast of Declination , observing the letters in the Scheam ; then by the Convexity of that Ark , and the Pole-point P , draw the Circle PE , cutting the Limb into two equal parts , which are the points A & B , the declination required . This being premised , there are two things requisite to be found , before you can draw the Dial. viz. the Substile from the Perpendiculer or Horizon , and the Inclination of Meridians . 1. And first for the Substile , by the Sector . As the sine of PEZ 90-0 To the Co-sine of the Lat. PZ 38-28 So the sine of the Declination PZE 35-00 To the sine of Substile from Perp. PE 20-54 As — sine of Declination PZE 35-0 To = sine of PEZ 90-0 So = sine of Co-latitude PZ 38-28 To — sine of Substile from Perp. FE 20-54 The distance of the Substile from the Perpendiculer , whose complement 69-06 , is the Elevation above the Horizon . Or , A Rule laid from Q to P , gives I ; DI is 20-54 . 2. For the Inclination of Meridians , say , By the Sector . As the Co-sine of the Latitude PZ 38-28 To the sine of PEZ 90-00 So the sine of the Reclin . ZE 33-03 To the Co-sine of Incl. Mer. ZPE 61-15 Whose complement ZPQ 28-45 is the Inclination of Meridians required . As — sine of Reclination ZE 33-3 To = Co-sine of Latitude PZ 38-28 So = sine of 90 PEZ 90-00 To — Co-sine of Incl. Mer. ZPE 61-15 Whose complement QPZ 28-45 is the Inclin . of Meridians required . Or , A Rule laid from P to Y , gives M ; EM is 28-45 , the Inclination of Meridians . Again , 8 88 — 45   81 — 15 9 73 — 45   56 — 15 10 58 — 45   51 — 15 11 43 — 45   36 — 15 12 28 — 45   21 — 15 1 13 — 45   6 — 15 2 1 — 15   8 — 45 3 16 — 15   23 — 45 4 31 — 15   38 — 45 5 46 — 15   53 — 45 6 61 — 15   68 — 45 7 76 — 15 If I take 15 , the quantity in degrees of one Hour , out of 28-15 , the Inclination of Meridians ; there remains 13-45 , for the first Hour on the other-side of the Substile . Then again , by continual addition of 15 degrees to 13-45 , and the increase thereof , I make up the other half . Or else , Against 12 , set 28-45 , and add 15 successively to it , & its increase , till it come to 90 , Then , to 13-45 , the residue of 15 , taken from 28-45 ; add 15 as often as you can to 90 , and thus is the Table made . To draw the Dial. First , Draw a perpendiculer Line on your Plain , as CB , by crossing the Horizontal-line at Right Angles ; then from the perpendiculer-Line lay off from the upper-end , toward the left-hand ( as the Scheam directs , ZD being the Perpendiculer , and ZN the Meridian , and EP on the Plain , the distance between , being toward the left hand ) 20-54 , for the Substile-line , as CD ; then on that Line ( any where ) draw two perpendiculer Lines quite through the Plain , crossing the Substile at right Angles , for two Equinoctial-lines , as EF , & GH . Then consider what hours shall be put o● your Plain , as here is convenient , from 10 in the morning , to 6 afternoon ; ( though the Sun may shine on it from 8 to 7 , bu● then the Lines will be too close together , and the Radius too small ) . And also when you would have those two utmost hours 〈◊〉 be , as at E and F on the upper Equinoctial-line ; or , at G & H on the lower contingent-line . Then , Then , Take the whole distance EF , or GH , and make it a = Tangent of 73-55 ; then the Sector to set , take out the = Tangent of 58-45 , and lay it from the point E to I , on the Equinoctial-line ; Also , take out the = Tangent 61-15 , and lay it from the point F ; and if your work be true , it must needs meet in the point I ; then draw the Line IK for the true Substile , and from thence lay the = Tangent of 45 , to draw a Line near 5 , for the Stiles Elevation , parallel to IK the Substile ; for being a Polar-plain it hath no Elevation , but what you please to augment it to ▪ as here from I to L. Then , As the Sector stands , prick on all the whole hours , halfs , and quarters , according to the Numbers in your Table , at least those that be above 45 ; and for those under 45 , make = Tangent of 45 in small Tangents , a = Tangent of 45 in the great Tangents , and then the Sector shall be set to that Radius , which is most convenient for your use . Note , That this way of Augmenting the Stile , is general in all Dials . 3. The third Variety of South-Recliners . The next and last kind of South Recliners , are such as recline , or fall from you below the Pole , viz have their Plains lying between the Pole and the Horizon , as by the Scheam is more apparent . In which work , the drawing the Scheam , and the things required , are the same as in the first Example , as the Figure , and following words , do make make manifest . The Example here , is of a Plain that declines from the South toward the West 35 degrees , and reclines upon its proper Azimuth ZE , 60 degrees from the Zenith . 1. Having drawn the Scheam , then first for the distance of the Meridian from the Perpendiculer , or Horizon . By the Sector , or Quadrant . As the sine of ZD 90-00 To the Tangent of Declination ND 35-00 So the sine of Reclination ZE 60-00 To the Tang. of Perp. & Merid. EG 31-12 As — Tangent of Declination ND 35-00 To = sine of 90 ZD 90-00 So = sine of Reclination ZE 60-00 To — Tang. of Perp. & Merid. EG 31-12 Whose complement is 58-48 AG , the distance between the West-end of the Horizontal-line , and the Meridian . Or by the Scheam ; A Rule laid from Q to G , cuts the limb at L ; the DL , and AL , are the Arks required ; DL from Perpendiculer , and AL from the Horizon . 2. To find PG , the Ark on the Meridian from the Pole to the Plain . By the Sector . As sine of AD 90-0 To Co-tang . of the Reclin . DE 30-0 So Co-sine of the Declination AN 55-0 To Tang. of dist . Plain & Horiz . NG 25-19 As — Co-tangent Reclin . ED 30-0 To = sine of 90 AD 90-0 So = sine of Reclination AN 55-0 To — Tang. dist . on Mer. P. Hor. NG 25-19 Which being taken from NP 51-32 , leaveth GP 26-13 , the distance on the Meridian from the Pole to the Plain , or the complement of the new Latitude . Or , A Rule laid from E , to P and G , gives on the limb 2 Points , whose distance between , is ab 26-13 , the Ark required . 3. To find the Stiles Elevation above the Plain . By the Sector . As sine dist . Merid. Horizon . GA 58-48 To Co-sine Declination AN 55-00 So sine dist . Pole to Plain GP 26-13 To sine Stiles Elevation PF 25-02 As — sine of GP 26-13 To = sine of GA 58-48 So = sine of AN 55-00 To — sine of PF 25-02 Being found by the Scheam , by laying a Rule from Y , to P and F , on the limb , gives the distance between being 25-02 , the Stiles Elevation , 4. To find the Substile from 12. By the Sector . As Co-tang . of the Declin . AN 55-00 To S. dist . on Mer. fr. Pl. to Hor. NG 25-19 So Tang. of the Stiles height PF 25-02 To S. of the Substile from 12 FG 8-05 As — Co-tang . of Declin . Plain AN 55-00 To = S. dist . on Mer. fr. Pl. to Hor. NG 25-19 So — Tang. of the Stiles height PF 25-02 To = S. of the Substile from 12 FG 08-03 By the Scheam , a Rule laid from Q , to G and F on the limb , gives L and M 8-3 ; Or else , the Ark MD , is the distance of the Substile from the Perpendiculer 23-19 . 5. To find the Inclination of Meridians . By the Sector . As the sine of the distance . on Mer. from Pole to Plain PG 25-19 To the sine of the Angle GFP 90-00 So the sine of dist . of Sub. fr ▪ 12 GF 08-03 To the sine of the Incl. of Mer. GPF 18-27 As — sine GF 08-03 To = sine PG 25-19 So = sine GFP 90-00 To — sine GPF 18-27 By the Scheam , a Rule laid from P to Y , on the limb , gives O , the Ark EO is 18-27 ; the Inclination of Meridians , by help of which , to make the Table of Hour-Arks at the Pole , as before is shewed , and as in the Table following . 12 18 — 27 8-3   10 — 57   1 3 — 27 1-27     Subst .   4 — 03   2 11 — 33 4-58   18 — 03   3 26 — 33 11-55   34 — 03   4 41 — 33 20-35   48 — 03   5 56 — 33 32-45   64 — 03   6 71 — 33 51-45   78 — 03         7 86 — 33 81-52   85-57   8 78 — 27 64-10   70 — 57   9 63 — 27 40-20   55 — 57   10 48 — 27 25-36   40 — 57   11 33 — 27 15-33   25 — 57   12 18 — 27 8-3 To draw the Dial. First , for the Affections , consult the Scheam , wherein , laying the Perpendiculer-line CD right before you , you see that the Substile , and the Meridian , are to be laid from the Perpendiculer toward the left-hand , the Substile lying between the Perpendiculer and the Meridian , and the Stile or Cock of the Dial must look upwards , the North-Pole being Elevated above this Plain , which will guide all the rest . Then , First , draw the Horizontal-line AB , and on C as a Center raise a Perpendiculer , and set off by Chords , Sines , or Tangents , the Meridian or 12 a clock Line , the Substile , and Stile , as exactly as you may ; and draw the Lines 12 C , Substile C , and Stile C. Then , As far from the Center C , as you conveniently may , draw a long Line perpendiculer to the Substile , as the Line EHF ; then setting one Point of a pair of Compasses in H , open the other till it touch the Stile-line at the nearest distance . Then , Make this distance a = Tangent of 45 , and take out the = Tangents of every whole Hour , as in the Table , as far as the Tangent of 76. will give leave ; and then from the Center C , to those Points draw Lines for the even whole Hours ; then to any one whole Hour , as suppose the Hour-line of 3 ; draw two = Lines equally distant on both sides the Line of 3 , as IK , LM . Then , Count any way 3 hours , and 6 hours from 3 , as here 12 , and 9 , so as the = line may cross the 3 remotest hours , as here you see 9 and 12 a clock Hour-lines do cross the = line at I and K ; then take the distance IK , and lay on the Hour-line of 3 from C to N , and draw INL = to 9 C ; Which Work doth constitute the Parallellogram KILM . Then lastly , Make KI , and NI , = Tangents of 45 , and p●ick off every hour , half , and quarter ( and minut if you please ) on the two Lines IK , and IL , from K and N both wayes , as before is already shewed in the Erect Decliners . Note also , That to supply the defect on the other side , when the point M falls out of the Plain , the distance from I , to the Hour-point from 11 , will reach from L to 7 , and from I to 10 , from L to 8. This is general in all Dials . Also note , If you like not to lay off the ●irst Hours by the Tangents , having made the Table , as before , you may soon find the Hour-Arks on the Plain for 3 Hours , as ●ere 3 , 12 , and 9 ; Or , 4 , 1 , and 8 , which ●ould have made the Parallellogram more ●●uare , and consequently more better , and ●●en to draw the rest by the Sector . Thus ●ou may see how your Work accords ; The ●ay by the Table and Contingent-line , and 〈◊〉 way by the Sector on the Parallellogram , 〈◊〉 by Calculation , & at last use the Mystery 〈◊〉 Dialling made plain and ready , to an ●●dinary capacity . Of North Declining Recliners . The other kind , viz. North Declining Recliners , have also three Varieties ; as those , ● . That fall back or recline between the Zenith and Equinoctial : 2d . Those that recline to the Equinoctial : And 3d. Those that recline below the Equinoctial . And first of the first Variety , reclining less then to the Equinoctial . The drawing the Scheam , is the same as in the former , except in the placing of the Points and Letters ; For first , these Plains behold the North-part of the Horizon , and then when you look on the Plain , the South is before you , and the West on your right-hand , and the East on the left ; then the South and North are alwayes opposite , and the point P , representing the Elevated Pole of the place , which with us being North , must be placed towards N downwards , as before in South Recliners it was upwards . Also , It is necessary in the Scheam , to draw the Equinoctial-line , by laying the half Tangent of 51-32 from Z to AE ; then the Secant of 38-28 , the complement of ZE , laid from AE on the Line SN , shall be the Center to draw EAEW for the Equinoctial-Circle . Thus the Scheam being drawn , to find the Requisites , thus ; 1. For the Meridians Elevation , or distance from the Perpendiculer , AG , or GE. By the Secctor . As sine 90 Radius ZD 90-0 To Tangent Declination Plain SD 55-0 So sine Reclination Plain ZE 20-0 To Tangent Merid. & Perpend . GE 26-2 As — Tangent of Declin . SD 55-0 To = sine of Radius ZD 90-0 So = sine of Reclination ZE 20-0 To — Tang. of 12 from Perp. GE 26-02 Whose complement AG , 63-58 , is the Meridians Elevation above the East-end of the Horizon . By the Scheam , A Rule laid from Q to G , on the Limb gives L ; then DL and AL are the Arks required . 2. To find the Distance on the Meridian , from the Pole to the Plain GP . By the Sector . As sine declin . of the Plain GZE 55-0 To sine dist . of Mer. & Perp. GE 26●02 So sine of the Radius GEZ 90●00 To sine of dist . on Merid. from Pole to Plain GZ 32-03 As — sine of GEZ 90-0 To = sine of GZE 55-0 So = sine of GE 26-2 To — sine of GZ 32-03 Which added to 38-28 ZP , makes up GP to be 70-31 . Or , By the Scheam , A Rule laid from E , to P and G , gives on the limb ab ; the Ark ab is 70-31 . 3. To find the Stiles height above the Plain PF . By the Sector . As sine of distance on Mer. from Zenith to the Plain GZ 32-03 To sine of the Plains Reclin . ZE 20-00 So sine of dist . on Mer. from Pole to the Plain GP 70-31 To sine of the Stiles Elevat . above the Plain PF 37-01 As the — sine GP 70-31 To the = sine GZ 32-03 So the = sine ZE 20-00 To the — sine PF 37-01 By the Scheam . A Rule laid from Y , to P and F , on the limb gives c and d , the Stiles height . 4. To find the distance of the Substile from the Meridian GF when it is above 90 deg . take the comp . to 108 deg . ; By the Sector . As Tangent of the Reclin . ZE 20-00 To sine of dist . of 12 from Perp. GE 26-02 So Tang. of the Stiles Elevat . PF 37-01 To sine of the Substile from 12 GF 65-24 As — sine EG 26-02 To = Tangent ZE 20-0 So = Tangent PF 37-01 To — sine GE 65-24 By the Scheam . A Rule laid from Q to G and F , gives on the limb LF , the Ark required . 5. To find the Inclination of Meridians FPG . By the Sector . As sine dist . on Merid. from Pole to Plain GP 70-31 To sine Radius opposite Angle GFP 90-00 So sine dist . on Plain from 12 to Substile GF 65-24 To sine of the Inclin . of Mer. GPF 74-38 As — sine GF 65-24 To = sine GP 70-31 So = sine GFP 90-00 To — sine GPF 74-38 By the Scheam . A Rule laid from P to Y , on the Limb gives g , the Ark Eg is 74-38 , the Inclination of Meridians . Or , A Rule laid from P to K , gives h , Sh is the Inclination of Meridians , by which to make the Table as before is shewed , and as followeth . To draw the Dial. 3 29 — 38 2 44 — 38 1 59 — 38 12 74 — 38 11 89 — 38 10 75 — 22 9 60 — 22 8 45 — 22 7 30 — 22 6 15 — 22 5 0 — 22 4 14 — 38 For drawing the Dial , consult with the Scheam , laying the Plain AEB , and his Perpendiculer CD right before you ; then note , SN is the Meridian-line , ZE the Plains perpendiculer , with the Meridian G on the left-hand , and the Subtile F on the right-hand . Also note , That the Sun being in the South as S , casts ●is beams , and consequently the shadow of ●he Stile into the North ; So that though G be the true Meridian found , yet it is the North-part that is drawn as an Hour-line ; ●ut the Substile , and other Hours , are coun●ed from the South-end thereof , as the Table●nd ●nd the Figure of the Dial , do plainly make ●anifest ; being drawn in this manner . First , draw the Horizontal-line AB , then 〈◊〉 C , as a Center , draw a semi-circle equal 〈◊〉 60 of the Chords , and lay off the Meri●ian , Substile , and Stile , in their right Sci●●ations , as last was declared ; then draw ●●ose lines , and to the Substile erect a Per●endiculer , as DE ; then take the Extent , or nearest distance from the place where the Perpendiculer or Contingent-Line last drawn , cuts 12 and the Stile-line , and make it a = Tangent of 45 ; Then is the Sector set , to lay off all the Hours by the = Tangents of the Arks in the Table , except 11 and 10 , which do excur . For ▪ If you prick the Nocturnal-Hours 12 , 1 , 2 , 3 ; and draw them through the Center , on the other side , they shall be the Hours of 12 , 1 , 2 , 3 , 4 , &c. on the North-part of the Plain , where they are only used . As for the Hours of 10 and 11 , do thus ; Draw a Line = to any one Hour , which = line may conveniently cut those Hour-lines . As , Suppose the Line 6 12 , which is = to the Hour-line of 3 ; then make the distance from 9 to 12 , or from 6 to 9 , in that Line last drawn , a = Tangent of 45 , and lay off hours and quarters , or else the whole Hours , by the distances from 9 to 7 , and 8 for 10 and 11 , turning the Compasses the other way from 9 ; then to all those Points Lines drawn , shall be the Hour-lines required . Or , Having only the hours of 3 , 6 , & 9 , & 12 in a Parallellogram , design the rest by Sector . The Second Variety of North-Recliners , Reclining to the Equinoctial . By the bare drawing of the Scheam , you see , that the Circle AEB , representing the reclining Plain , doth cut the Meridian just in the Equinoctial ; Now to try by Arithmetick , whether it be a just Equinoctial-plain , or no , say : 1. By the Sector . As the sine of 90 AD 90-0 To Tang. of the Reclination DE 54-10 So Co-sine of Declin . Plain AS 35-0 To Co-tang . of the Latitude SG 38-28 As — Tangent Reclination DF 54-10 To = sine 90 AD 90-0 So = Co-sine of Declination AS 35-0 To — Co-tang . of the Lat. SG 38-28 Which happening so to be , it is a declining Equinoctial , or Polar in respect of its Poles , which are in the Poles of the World. 2. If the Declination were given , and to it you would have a Reclination , to make it Equinoctial . By the Sector ; As the Co-sine of the Declin . AS 35-0 To the Co-tang . of the Lat. SG 38-28 So is the sine of 90 AD 90-00 To the Co-tang . of the Reclin . DE 54-10 As the — Co-tang . Lat. SG 38 28 To the = Co-sine Declin . AS 35-00 So the = sine Radius AD 90-00 To the — Co-tang . Reclin . DE 54-10 By the Scheam . The Points AB of Declination , being given , and the Point G on the Meridian , if you draw the Reclining Circle AGB , it will intersect the Perpendiculer at E ; then the measure of ZE is the Reclination , measured by half-Tangents , or by Chords , by laying a Rule from A , to E on the limb , gives a ; the Chord B a , is the Reclination 35-50 . 3. But on the contrary , if the Reclination be given , and a Declination required , to make an Equinoctial Plain ; Then contrarily say thus , By the Sector . As Co-tang . of the Reclin . ED 54-10 To sine of 90 AD 90-00 So Co-tang . of the Latitude SG 38-28 To Co-sine of the Declin . SA 35-00 As — Co-tang . Reclin . ED 54-10 To = sine AD 90-0 So — Co-tang . Latitude SG 38-28 To = Co-sine Declination SA 35-00 But by the Scheam . By the Point G , and the touch of an Arch about E , draw the Circle GE , to cut the limb into two equal parts , and you have the Points AB . 4. The Plain thus made , or proved to be Equinoctial ; to find the Meridians Elevation above the Horizon , AG Or , his Distance from the Perpendiculer EG . ; By the Sector . As sine of 90 ZEG 90-0 To sine of dist . on the Mer. from Z , to the Plain GZ 51-30 So sine of Declin . of the Plain GZE 55-0 To sine of dist . on the Plain from Perpend . to Merid. GE 39-54 As — sine GZE 55-0 To = sine ZEG 90-0 So = sine GZ 51-32 To — sine GE 39-54 Whose complement is AG 50-06 , the Elevation above the Horizon . By the Scheam . A Rule laid from Q to G , gives b on the limb , DB is 39-54 , as before . 5. To find the Stiles Elevation above the Substile on the Plain . By the Sector . As sine of the Latitude GZ 51-32 To sine of the Reclination ZE 35-50 So sine of dist . Mer. Pole to Plain GP 90-00 To sine of the Stiles Elevation PF 48-24 As — sine 90 GP 90-0 To = sine Latitude GZ 51-32 So = sine Reclination ZE 35-50 To — sine Stiles height PF 48-24 By the Scheam . A Rule laid from Y to F on the limb , gives C , NC is 48-24 , the Stiles height . The distance of the Substile from 12 , in these Equinoctial Dials , is alwayes 90 degrees ; for a Rule laid from Q , the Pole of the Plain , to G , on the limb gives b ; a Rule also laid from Q to F , the Substile , on the limb gives d ; the Ark bd , is 90 degrees , both for the distance of the Substile from 12 , and also for the Inclination of Meridians , for the Substile stands on the hour of 6 , being part of the Circle EPW , which is the hour of 6 , 90 degrees distant from the hour of 12. Or , A Rule laid , as before , from Y to P , on the limb , gives N ; the Ark EN , or WN , is 90 , for the Inclination of Meridians . Which being just 90 , the Table is easily made , viz. 15 , 30 ; 45 , 60 ; 75 , 90 ; twice repeated , from 12 to 6 both way s. To draw the Dial. On the Horizontal-line AB , draw an obscure Semi-circle , and set off the Meridian , as the Scheam sheweth , viz. 50 degrees 6 min. above the East-end of the Horizontal-line ; but make visible only the North-end thereof , as the line C 12 ; Then 90 degrees from thence , toward the right-hand , as the Scheam sheweth , when the Perpendiculer-line is right before you , draw a Line that serves both for 6 and the Substile , as C 6. Also , lay off the Chord of 36-47 from 6 to 9 , and draw the Line C 9 also , which is found by Calculation , as before is shewed . Or thus ; Draw a Line = to 12 , or Perpendiculer to 6 , being in this Dial all one , as the Line FEG ; then setting one Point in E the Substile , take the nearest distance to the Stile-line , and it shall reach from E to G , the Point for 9. The same distance EG lay also on the line 12 , from C to H , and draw the line GHI ; then make EG a = Tangent of 45 , and lay off the = Tangents of 15-30-45 , both wayes from E , as hath been often shewed . Also , Make the distance of HG a = Tangent of 45 , and lay the same = Tangents both wayes from H , and to those Points draw the Hour-lines required . The third Variety of North-Recliners . This third and last sort of North-Recliners , are those that recline beyond the Equinoctial , that is , lie between the Equinoctial and the Horizon ; and it differs somewhat from the other five before , in the Scheam and Operation also . For first , the Ark of the Plain is extended below the Horizon , till it meet with the North-part of the Meridian below the Horizon at H ; and the Center of the Ark AQB , is in the Line ZD , as much distance from Q , as the Secant of 65 deg . to the Radius of the Scheam , being the complement of ZQ 25-0 ; Here also the same requisites are to be found as in the other Dials . 1. First , for the Meridians Elevation above the Horizon , AG. By the Sector ; As sine 90 ZD 90-00 To Tang. Declin . Plain SD 55-00 So sine Reclin . Plain ZE 65-00 To Co-tang . Elevation Merid. GE 52-18 As — Tangent Declin . SD 55-0 To = sine 90 ZD 90-0 So = sine Reclination ZE 65-0 To — Co-tang . Merid. Elev . GE 52-18 Whose complement GA 37-42 , is the Meridians Elevation above the Horizon . By the Scheam ; A Rule laid from Q to G , gives on the limb a ; then D a is the distance from the Perpendiculer 52-18 ; and A a the distance from the Horizon 37-42 . 2. To find the Distance on the Meridian from the Pole to the Plain GP . By the Sector . As sine of AD Radius , or sine of AD 90-0 To Co-tang . of Reclin . Plain DE 25-0 So Co-sine of Declin . Plain AS 35-0 To Co-tang . dist . on Merid. from Plain to Zenith SG 14-58 As — Tang. of ED 25-0 To = sine of AD 90-0 So = sine of AS 35-0 To — Tang. of GS 14-58 Whose complement 75-02 GZ , added to ZP , the complement of the Latitude , makes 113-30 , for the distance of the North-pole P , on the Meridian of the place , from ( the North-pole P , to ) the Plain below the Equator at G ; which being more than 90 , find the complement thereof to 180 , viz. 66-30 , being the distance on the Meridian from P the Pole , to the Plain on the North-part of the Meridian , viz. PH , found on the Scheam , by laying a Rule from E or W , to P and H , on the limb gives b and c ; the Ark bc is 66-30 , the distance on the Meridian from the Pole to the Plain . 3. To find th● Stiles height above the Plain PE. By the Sector . As the sine dist . Mer. from Zenith and Plain GZ 75-02 To the sine of the Reclin . Plain ZE 65-00 So the sine dist . Mer. from the Pole to the Plain PH 66-30 To the sine of the Stiles Elev . PF 59-21 As — GZ 75-0 To = ZE 65-0 So — PH 68-0 To = PF 59-21 By the Scheam . A Rule laid from Y , to P and F on the limb , gives d and e ; the Ark de is 59-21 , the Stiles Elevation . 4. To find the Substile from 12 , viz. FG from the South part , or HF from the North part . By the Sector . As Tang. Reclin . of the Plain ZE 65-00 To Co-sine dist . Mer. & Horiz . EG 52-18 So Tang. of the Stiles Elevat . PF 59 21 To sine of the Substile from North part Merid. FH 38-30 As sine dist . Mer. from Perp. EG 52-18 To Tang. of the Reclin . ZE 65-00 So Tang. of the Stiles height PE 59-21 To sine of the Substile from 12 FH 38-30 By the Scheam . A Rule laid from Q , to H and F , on the ●imb , gives f and e ; the Ark f and e , is the Substiles distance on the Plain from 12. 5. To find the Angle between the two Meridians , viz. PF , and PH. By the Sector . As sine dist . Mer. fr. Pole to Plain PH 65-00 To sine of 90 Radius PFH 90-00 So sine of dist . from Subst . & 12 FN 38-30 To sine of Inclin . Merid. FPN 42-45 As — sine of PFN 90-00 To = sine of PN 65-00 So = sine of FN 38-30 To — sine FPH 42-45 By the Scheam . A Rule laid from P to Y , on the limb gives g , then W g is the Angle of the Inclination of Meridians , viz. 42-45 ; by which make the Table , as is several times before shewed , and as followeth . To draw the Dial. 12 42-45   1 57-45   2 72-45   3 87-45   4 77-15   5 62-15   6 47-15 42-55 7 32-15   8 17-15   9 02-15 1-56 10 17-15   11 2-15   12 42-45   Then prick down the Hours of 9 & 6 , by the Table on the Contingent-line , as before , or on the Semi-circle , having Calculated only those two Hour-lines , by the general Canon . Then , Draw a Line = to 12 , at any convenient distance from it , as GH ; Then , take the distance between 6 and 9 in that = Line , and lay it from the Center to I , on the 12 a clock Hour-line , and draw the Line HI ; then make the distances GH , and IH , severally one after another , = Tangents of 45 ; and take out the = Tangents of 45 , 30 , 15 ; and lay them both wayes from 12 and 6 , on those two Lines , as hath been often shewed , in the former Dials ; then lines drawn to those Points , shall be the Hours required . CHAP. VII . Of Declining and Inclining-PLAINS . INclining Plains are but the under faces of Recliners , beholding the Nadir , at the same Angle that the Recliners behold the Zenith . Also , If you turn the Paper , and look against the light , and then the North-east becomes a North-west Decliner 55 , and reclining 35-50 ; and the South-east becomes a South-west , Declining and Inclining as much . Thus you see , that every Draught of a Dial will serve for 4 Plains , that is for the place you draw it , and his opposite ; and for another Plain , declining as many degrees the contrary way , and reclining as much also , and for the opposite thereunto , as by the two Draughts of the two sides , may plainly be seen to appear . And the like holds in all sorts , as Upright Decliners also . As a North-east and a North-west , a South-east and a South-west , declining 30 ; one Dial drawn round about , serves all 4 Dials ; But note , that no South Erect or Inclining Dial , can have the Sun to shine on any Hour-line that falls above the Horizontal-line ; and those hours on the North-recliners , that fall below the Horizontal-line , belong also to the South Dials . But for a plain general Rule , to know what hours belong to any Plain whatsoever in any Latitude , do thus . To know what Hours belong to any Plain . First , draw a general Scheam to your Latitude , as this is done for 51-32 ; and mark the 4 Cardinal-points with E.W.N. & S. as is usual for setting the Scheam right before you . Then , For all Declining Upright Dials , draw only a streight Line for the Plain , Perpendiculer to the Line that doth represent the Pole of the Plain , counting so many degrees as the Declination of the Plain shall happen to be from S. or N. toward E. or W. then all the Hour-lines of the Scheam that that Line of the Plain shall intersect , are the Hour-lines proper to that Plain . 1. Example . Therefore , If you conceive the Sun to be in Cancer , and going of his Diurnal Motion , at his Rising about a quarter before 4 , beholds the North-side of the Line EW , and continueth so to do till 25 minuts after 7 ; and then it shines on the South-plain till 35 minuts after 4 , and then begins again to shine on the North-plain , and so continues till Sun setting . But when the Sun is in the Equinoctial , it beholds the South-plain at the Rising , being at 6 a clock in the morning ; and shines on it all day , till Sun set , being at 6 at night ; and then the North Dial is useless . 2. For a Declining-Plain . Suppose 30 degrees South-east ; first set the Scheam in his right scituation for a South-east Plain ; then if you count 30 degrees from S toward E , for the Pole of the Plain ; and 30 degrees from W toward S , or from E toward N , and draw that Line that shall represent the Plain ; then you shall find that the Sun being in Cancer will begin to shine on this Plain , just a quarter before 5 in the morning , and continue till near half an hour after 2. But about the middle of Ianuary , it will shine on it till a quarter after 4 , viz. till Sun set ; and all the hours after 2 , belong to the North-west Plain that declines 30 degrees , and one hour in the morning also , viz. from a quarter before , till three quarters after 4. The like work serves for any Decliner whatsoever , in any Latitude . 3. But for Decliners and Recliners . Draw a long Line , as AB , and cross it with a Perpendiculer in the Center C , and lay off from C , toward A and B , the Tangent of 45 ; or the Semi-tangent of 90 , equal to the largeness of your Scheam ; then lay off the Semi-tangent of the Reclination from C to D , up and down , both wayes ; then take out the Secant of the complement of the Reclination , which will be a Radius to draw the Arks ADB , which Paper you must cut out , and apply the two Points of the Paper ADBD , to the two Points of Declination of the Plain , noted in the Scheam with A and B ; that is , put A to A , and B to B ; then the round or convex-edge of the Paper , represents the reclining Plain ; and the same edge , on the other part next the Horizon Southwards , represents the South-west Incliner . Example . Suppose I make the Paper ADB , to recline 35-50 , the Reclination of the Equinoctial-plain ; then , first set the Scheam right before you in its right scituation , and putting the Points A , in the Paper , on A on the Scheam ; and B in the Paper ▪ to B on the Scheam ; I shall find it to be even with the reclining Circle AEB ; then following the Tropick of Cancer , I find that it shines on the North Recliner from the Rising till near 2 , at which time it leaves the North-recliner declining Eastward , and begins to shine upon the opposite Plain , viz. the South-west Incliner , declining 55-0 , and reclining 35-50 , and so continues till Sun-set . But note , That if the Line that represents the Plain , cuts the Tropick twice , as the Line EW for a North-plain ; then , though the Sun leave the Plain in the morning , it will shine on it again in the afternoon . Note also , That a North-east Recliner , is represented by the other Convex-edge of the Paper , as here a North-east Decliner 55 , and Inclining 35-50 , the Sun will shine but till 3 quarters after 8 in Cancer ; but in Capricorn it shines till half an hour after 9 , and comes no more on it that day : And note alwayes , That when it leaves any Plain , that then it begins to shine on his opposite , as here the opposite to this North-east Incliner , is the South-west Recliner , being represented by the same Line or Circle ADB , that the North Recliner was : Only , you must count that side of the Line next to the Horizon , the Inclining-plain ; and that side next the Zenith , the Reclining-plain ; For , the Line that represents it , having no bredth , can be no otherwise distinguished , unless you will make a material , Armilary Sphear , of Pastboard or Brass , as the following Discourse doth plainly demonstrate , in these several Operations , for the better conceiving of these Mathematical Excercitations . Thus you have the way of making all manner of Sun Dials , upon any plain Superficies , the Axis of the World being the supposed Stile to all these Plains ; As for those curiosities of Upright Stiles , and Eliptical Dials , and drawing of Dials by the Horizontal , or Equinoctial Dials , you have them in the Works of Mr. Samuel Foster , and others ; and in Kerkers Ars magna , &c. But I intended not a Volumn of Shadows , but only a further improvment of the Trianguler-Quadrant , as you will see in the next Chapter , of drawing the Furniture or Ornament of Dials ; which being but seldom used , I shall here crave an Apology for the brevity therein , fearing , lest that to the young Practitioner it may seem somewhat hard to conceive , though to the exercised in these matters it may be plain enough . Then for a Conclusion , you shall have an easie Mechanick way , to draw a Dial on the Ceiling of a Room , that lieth Flat or Horizontal , which will be very good for Painters or Plaisterers , to Ornament a Room withal , and is not yet treated on that way , as ever I read of . CHAP. VIII . To furnish any Dial , with the usual Mathematical Ornaments by the Trianguler-Quadrant , as Parallels of the Suns Declination , or the Suns place , or length of the Day , to find the Horizontal and Virtical Lines , and Points , to draw the Azimuths , and Almicanters ; the Iewish , Italian & Babylonish Hours , and 12 Houses on any Plain before mentioned . 1. To draw the Tropicks , or Parallels of the Suns Declination , or the length of the Day Artificial , on any Dial. But note , That if it be a Perpendiculer Stile , whose upper Point , or Apex , is to be the Nodus to give the Shadow ; then you must strain a Thred very hard , or apply a Rule for the present whereon to rest the Moving-leg on , instead of the Axis ; or , else you may do it thus , as Mr. Gunter sheweth . First , to make the Trygon , if the Rule or Quadrant prove too large for your small Dial. On a sheet of Pastboard , or Slate , draw a long streight Line , as AB ; to which Line erect two Perpendiculers , one at the upper , and the other at the lower end , as CD , and EF ; then make AB a Tangent of 45 degrees , ( then having first made these little Tables that follow , by the Trianguler-Quadrant , which is only the Suns Declination , at his entrance into the whole Signs , or at an even half-hour of Rising ) ; lay of both wayes from B , the Tangents of the Suns declination at ♈ ♉ ♊ ♋ , as in the Table following ; and draw Lines to these Points from the Center A , as in the Figure annexed ; and then set the marks to them , and this is the Trigon . Figure I. Suns declinations for the Parallels of the length of the Day . Hours Declin 16-26 23-31 16-0 21-41 15-0 16-55 14 11-37 13 5-53 12 0-00 11 5-53 10 11-37 9 16-55 8 21-41 7-34 23-31 For the Signs of the Zodiack . Signs . Declin ♋ 23-31 ♌ ♊ 20-14 ♉ ♍ 11-31 ♈ ♎ 0-00 ♓ ♏ 11-31 ♒ ♐ 20-14 ♑ 23-31 Declinations . 5-0 10-0 15-0 20-0 23-31 both ways Then from the Center A , any way on the Line CD , at such a convenient distance as you think may fit the Plain , set off the Point G ; then making GA Radius of 45 Tang. set off on AB from A , the Tang. of the Stiles Elevation to F , and draw the Line FG , as an obscure Line . Then come to the Dial Plain , and measure from the Center to the place on the Substile-line , where you would have your remotest Line of the sign ♋ or ♑ to pass ; and take this distance between your Compasses , and carry it in , above , or below the Line FG , first drawn and produced to ♋ , or ♑ , till you find one Point to stay in A ♋ , and the other in AG , so as to draw a Line = to FG first drawn ; if that doth not fit , then dele FG , and draw this = to it in its stead , to fit and fill the Plain with the Tropicks to your mind , to make them large and yet convenient . Then note , The point G represents the Center of the Dial ; AG is the length of the Stile from the Center to the Nodus ; a Perpendiculer let fall from A to FG , shews the point H ; GH is the measure on the Substile-line on the Plain from the Center to the Horizontal-line , HA is the Perpendiculer height of the Stile ; A the Apex or top of the Stile or Nodus to give the shadow . Then , Draw a Line from G , = to AB , as LK ; and any where between AB , draw LM = to AG ; and wheresoever FG cuts LM , make a mark as at M ; then make LM a = sine of 90 degrees , and the Sector so set , take out the sine complements of the Arks at the Pole for every hour , and lay them from L towards M , on the Line LM , and to all those Points , draw Lines from G , and mark them with 12 , 1 , 2 , 3 , 4 , &c. as in the Table . Or else , Take the measure from G to F , and lay it on the Dial from the Center on the Substile , and draw that Line precisely Perpendiculer to the Substile , for the true Equinoctial-line on the Plain . Then , The measure from the Center of the Dial , to the crossing of every Hour-line , and the Equinoctial-line , taken and laid from G , to the Line AB , gives Points to draw the Hour-lines on the Trygon ; As in the Figure . Wherein you may note , That if the Substile happens to fall on an even , whole , or half hour , then one Line will serve on both sides of the Substile ; but if not , you must draw as many more , and set Figures to them , to avoid confusion . Then , I say , that the several distances from G , to the crossings of those Hour-lines last drawn on the Trygon ; and the Signs being laid on their correspondent Hour-lines from the Center of the Dial , shall give Points in those Hour-lines , to draw the signs of the Zodiack , with a thin Rule that will bend to those Hyperbolick Sections . The same way serves to draw the Parallels of the length of the Day , if you lay the distunce from G , the crossings of the pricked Lines and Hours on the Trygon , and is as true as any other way by Calculation , which must afterward be performed by protraction in this manner . Thus you have the way to proportion the Height of the Stile , to fit the Plain , and the place of the Horizontal-line in all Erect-Dials , which is alwayes Perpendiculer to 12 , and drawn through that point a-cross the Plain ▪ And this way of drawing the Signs , is general in all Plains whatsoever , that will admit them . II. To find the Horizontal-line in all manner of Plains . First , The Horizontal-plain can have none , nor many other both Reclining and Inclining , whose Reclination or Inclination is above the complement of the Suns Meridian Altitude in ♑ , if the Stile have any considerable Altitude . In all other Plains , the best Mechanick way is thus ; The Dial being set in ( or , as in ) his place , apply the Moveing-leg to the top of the Stile & one corner of it to the Plain ; and at the same time let the Thred play evenly on 600 , and the corner at the Plain will make as many marks as you please to draw it by . Otherwise note , That wheresoever the Hour-line of 6 , and the Equinoctial-line do meet , there is one Point : Then find at what hour and minute the Sun doth Rise or Set at , in the beginning of any other whole Sign , most remote from the first Point , and that shall be another ; and so as many as you please to draw that Line by : This is general for all Plains : To find that Point by the Trianguler-Quadrant . Lay the Thred to the Sign given , and in the Hour-line is the hour and minut required ; Thus the Sun being in ♈ , riseth and setteth at 6 , or 1 quarter of a minut before , or after ; and in ♉ at just 5 , and sets at 7 ; in ♊ at 9 minuts after 4 , or sets 9 minuts before 8 : The like for Winter signs . III. To draw the old unequal Hours . The unequal , Jewish , or Planetary hours , divide the Day , be it long or short , into 12 equal Hours ; for the drawing of which , in the Equinoctial the common hours gives Points . For the Tropicks do thus ; Divide the number of minuts in the longest and shortest dayes by 12 ; viz. divide 986 , the minuts in one day in ♋ at London , by 12 , the Quotient is 82 ½ ; and divide 454 , the number of minuts in one day at London in ♑ , and the Quotient shall be 37′ ¾ ; then if you fasten an Index , or lay a Rule to the Center , and to every 1 hour and 22′ ½ in ♋ from 12 ; and to every 37′ ¾ in ♑ , it shall give Points to draw the Jewish or Planetary hours required , according to this Table , thus made for London , by the Line of Numbers ; against 12 set 6 , and the rest in order as the day proceeds , for our 12 is the 6th hour , according to the Jewes . To make this Table readily by the Line of Numbers . A Table to divide the Planetary hours in ♋ and in ♑ , for London , 51-32 Latitude . H ♑ M hou H ♋ M 8 43 1 5 10 9 28 2 6 31 10 16 3 7 52 10 44 4 9 15 11 22 5 10 37 12 00 6 12 00 12 37 7 1 22 1 15 8 2 44 1 53 9 4 06 2 31 10 5 29 3 8 11 6 51 3 47 12 8 13 Extend the Compasses from 16-26 , the length of the longest day in hours and minuts to 1 , the same Extent shall reach the contrary way from 60 , to 986 , the Number of minuts in one day . Or rather , As 1 hour , to 60 minuts ; So is 16 hours 26′ , to 986 minuts . Then , As 1 , to 82 minuts ¼ ; So is 2 , to 164 minuts ½ ; So is 3 , to 246 minuts ¾ . Or you may say , As 12 , to 1 ; So is 986 to 82-2 , the minuts in 1 hour . Which properly is one hour 22 minuts , the length of one hour in Cancer ; then the second hour , is 2 hours 44 ½ ; the third hour is 4 hours and 6′ ¾ from 12 ; and so for the rest , as in the foregoing Table for London . But if you draw the Parallels , of the length of the day in the Dial , you shall find these hours to cross the even Hour-lines and quarters in the Parallels for 15 and 9 hours , as well as in the Equinoctial . IV. To draw the Italian or Babylonish-Hours . First , draw the common Hours , and the Parallels of the Signs , or rather the length of the day ; Then note , that these Hour-lines meet with the common hours in the Equinoctial ; only the Italians who account from Sun-setting , call our 12 in the Equinoctial 18 ; And the Babylonians , who reckon from the Sun-rising , call our 12 in the Equinoctial 6 hours . Then to mark these in the Tropicks , do thus ; But , if you draw the Parallels of the length of the Day , then you shall find the 18th hour after Sun Setting , to cut the Hour-line of 10 in the Parallel of the Day , being 8 hours long , and 12 in the Parallel of 12 hours long ; and the common Hour-line of 2 in the Parallel of 16 hours long , and so successively for the rest , for so many hours from the last Sun-setting : For , from 6 the last night in the Equinoctial , to 12 this noon , is 18 hours ; but in ♑ , from 47′ after 3 at Sun-set , to the next noon , is 20 hours and 13′ , as in the Figure foregoing . But for the Babylonish-hours , who reckon by equal hours from the Sun Rising , as before , count 2 hours and 13 minuts after 6 in ♑ ; and 2 hours and 13′ before 6 in ♋ ; and just 6 in ♈ , and that shall draw the Line of the Suns rising ; then count 3 hours and 13′ after 6 in ♑ , and 7 in ♈ , and 1 hour 13′ before 6 in ♋ ; and that shall be the first hour after Sun rising , and so successively till night . But if you use the Parallel of the length of the day , the work is easier ; for then 5 , 7 , and 09 , in the Parallels of 16 , 12 , and 8 hours , shall be Points for the first from Sun-rising ; and 6 , 8 , and 10 , shall shew the second hour from Sun-rising , and so forwards , as in the Table following . V. To draw the Azimuth-Lines . For all Erect-Dials , both Direct or Decliners , deal with the Declination of the Plain , as you did with the Inclination of Meridians ; and at the Meridian , or 12 , set the Plains declination ; and then for Rumbs , take 11 deg . 15′ as often as you can ; and what the last number wants of 11-15 , set on the other side of the Substile , and to that add 11-15 till you have enough , as in the Table annexed for a Dial , whose declination was 35 degrees Westwards . Then make the Perpendiculer height of the Stile Radius , or Tangent of 45 , and on the Horizontal-line lay off the = Tangents of the Rumbs last made ▪ in the Table , from the foot of the Stile their right way , and draw Lines through those Points , all Parallel to 12 , for the Rumbs , or Virtical Circles required ; on the Meridian write South , and the rest in their due order . To draw the Azimuth or Virtical-Circles on Reclining , or Inclining-Plains . Points . D. M S. E. 80 00 S. E. by S. 68 45 S. S. E. 57 30 S. by E. 46 15 South . 35 00 S. by W. 23 45 S. S. W. 12 30 S. W. by S. 1 15 Substile   S. W. 10 00 S.W. b. W. 21 15 W. S. W. 32 30 W. by S. 43 45 West . 55 00 W. by N. 66 15 W. N. W. 77 30 N. W. b. W. 88 45 N. W.   In all Reclining , or Inclining Plains , these Azimuths , virtical Circles , or Rumbs , do meet in a Point ( called the Vertical Point ) found in the Meridian , or 12 a clock Line , right over ( in Incliners ) or under ( in Recliners ) the Apex or top of the Stile , that is to give the shadow , when set in its right place , right over the Substile-line ; And as far off the foot of the Stile ( being a Point in the Substile , Square , or Perpendiculer to the Apex or top of the Stile ) in a Vertical Line drawn through the foot of the Stile , = to the Perpendiculer Line of the Plain ) , as the Co-tangent of the Reclination , making the Perpendiculer height of the Stile to be Radius or Tangent of 45 degrees . Also , The Co-tangent of the Reclination of the Plain , to the same Radius , laid from the foot of the Stile , in the same Virtical-Line , shall give the Point in the Vertical-line , to draw the Horizontal-line by ; for a Rule laid to this Point , and the crossing the Equinoctial-line and hour of 6 , shall draw the true Horizontal-line . Then make the distance between this Point , and the meeting of the Equinoctial and 6 , a = Tangent of the West or East Azimuth in the Table , and then the Sector is set , to lay off all the rest , by taking the = Tangents of the Numbers in the Table , and laying them from the Vertical-point in the Horizontal-line , both wayes on the Horizontal-line . For , from hence you may note , That the Sun , being in the Equinoctial , doth rise and set near 6 ; and also doth rise near the East-point , and set near the West ; therefore the same Point in the Dial , must be for the hour 6 in the morning ; and the East Aximuth , or the hour 6 at night , and the West Azimuth , according as the Plain declines Eastwards or Westwards . Then Right Lines drawn from the Vertical-point in the Meridian , and to all these Points in the Horizontal-line , shall be the Azimuth-lines required . A thus for Example in the Figure annexed , being the Third sort of South-Recliners before-going ; Declining 35 degrees Southwest , and Reclining 60 degrees , CH is the Substile , CG the Stile , H the Foot of the Stile , IK the Vertical-line drawn through the foot of the Stile HI the Vertical-point in the crossing of 12 ( and the Vertical-line ) and yet right under G the Apex ( considering the Reclination ) and the raising of G the Apex , Square , or Perpendiculer to H the foot of the Stile ; Then , I say , a Plumb-line let fall from G , will rest in I , the Vertical-point ; The Dial being set in its due place . Then GH , the Perpendiculer height , made a = Tangent of 45 ; HI is the Co-tangent of the Reclination , viz. 30 ; and HK the Tangent o● the Reclination 60 , being the Vertical-point in the Horizontal-line , from whence to lay the = Tangents , of the Rumbs in the Table last made , into the Horizontal-line . Then Lines drawn from the Vertical-point I , to those Points in the Horizontal-line , shall be the Rumbs or Points of the Compass , Vertical Circles , or Azimuths required . Otherwise , When you have made the Tables of the Angles at the Zenith , as before , you may by this Canon make Tables of Angles at the Vertical-point , between the Vertical-line and the Rumb , to be drawn on the Plain . As the sine of 90 , To the Co-sine of Reclination , or Inclination ; So the Tangent of the Angle at Zenith , To the Tangent at the Vertical . This Table being made , you may set one Point in the Vertical-point , and describe a Circle to any Radius , and therein prick off from the Vertical-line , the several Chords of the Rumbs , as in the Table you shall make by the last Canon . A Table , shewing at what Hour and Minute the Sun is in , in an even Azimuth , or Point of the Compass in ♑ , ♈ , ♋ , for 51-32 . Degr. Rumbs . Alt ♋ H. M. Alt. ♈ H. M. Alt ♑ H. M. 00-00 South● 62-00 12-00 38-28 12-00 15-00 12-0 11 15 S. by E. 61 39 11 38 37 58 11 24 14 17 11 16 22 30 S. S. E. 60 33 11 15 36 19 10 48 12 05 10 27 33 45 S.E.b. S. 58 31 10 46 33 29 10 10 8 17 9 33 45 00 S. E. 55 40 10 17 29 22 9 28 3 00 8 39 56 15 S.E.b. E 51 35 9 42 23 51 8 42     67 30 E. S. E. 46 06 9 02 16 56 7 51     78 45 E. by S. 39 03 8 15 8 49 6 57     90 00 East . 30 38 7 21 0 0 6 00     78 45 E. by N. 21 21 6 24         67 30 E. N. E. 11 10 5 19         56 15 N. E.b E 3 48 4 18         45 00 N.E.             33 45 N.E.b. N             22 30 N. N. E             11 15 N. by E             00 00 North.             Lastly , by help of this Table , being general for all Dials in the Latitude 51-32 , it is done thus ; Then you shall see that a Rule laid to the Vertical-point , and any one of those three Points shall cut the other two , if the former Lines be true , and you estimate the minuts well . Note , That this last Table in the Equinoctial , is thus readily made , by Sines and Tangents . As the sine of 90 , to the sine of the Latitude ; So is the Tangents of the Azimuths from the Meridian , being the first column in the Table , To the Tangent of the Angle between the Meridian , and Azimuth Line on the Equator , which are the numbers in the 6th column , reduced into hours and minuts . So that you see the Azimuth of 45 , or Rumb of S.E. will cross the Equinoctial at ●8 minuts past 9 , as in the Table ; which Table is easily made by the Trianguler-Qua●rant , by the Rules in Chap. XV. VI. To describe the Almicanters , or the Parallels of the Suns Altitude above the Horizon . First , on the Equinoctial , these Lines shewing the Suns Altitude , cannot be expressed . On the Horizontal Dial they are Circles , making the Perpendiculer height of the Stile Radius , or Tangent of 45 ; prick off on the Hour-line 12 , from the foot of the Stile , the = Tangent of 10 , 20 , 30 , 40 , 50 , 60 , &c. Then one Point of a pair of Compasses set in the foot of the Stile , and the other opened to 10 , 20 , 30 , &c. draw those Circles for the Parallels of the Altitude required . For all Erect Dials , whether Direct or Declining , they are best done thus ; If the Stile be in , and right set , then the distance from the Nodus , to the crossing of the Horizontal-line , and Azimuth line , on which you would prick down the Altitudes shall be the = Tangent of 45 ; then the Sector so set , the = Tangents of 10 , 20 , 30 , &c. laid from the Horizontal-line on the respective Azimuths , shall be Points to draw the Parallels of Altitude by ( or by applying the Rule to the Nodus and Plain , and the Thred to the Almicanter ) as afterward is plainly shewed . But if the Stile is not in , then the Secants to the same Numbers and Radius , that pricked down the Azimuth Lines , shall be the several Radiusses to use as before ; where you may note , That the Suns Meridian Altitude in the whole even Signs , will help to prove the truth of your work . The East and West Erect Dials , are fitted with Parallels of Altitude in the same manner ; for the Perpendiculer height of the Stile , is a Tangent of 45 , and the = Tangents of 11-15 , 22-30 , 33-45 , &c. laid from the foot of the Stile in the Horizontal-line , draws down-right Lines for the Azimuths ; and the Secant of 11-15 , 22-30 , 33-45 , &c. shall be the several Radiusses to prick off the = Tangents of 10 , 20 , 30 , 40 , 50 , 60 , ( or what you will ) on those Perpendiculer Azimuth Lines , for the Almicanters , or Parallels of Altitude required . But for Declining Reclining Plains , you must first draw the Azimuth Lines , as before is shewed , and then find also the length of the Axis of the Horizon , as Mr. Gunter calls it , which is thus done ; Make the length of the Perpendiculer-Stile a = Tangent of 45 , viz. GH Fig. II. then HI is the Co-tangent of the Reclination , and HK the Tangent of the Reclination ; and then , as the Sector stands , the Secant of the complement of the Reclination , shall be the length of the Axis of the Horizon required , viz. GI , or by the Sines and Tangents Artificial . As the sine of the Reclination , to the sine of 90 ; So is the length of the Stile on the Line of Numbers , being taken in inches and 100 parts , to the length of the Axis in the same parts . Which is an imaginary Diagonal Line , reaching from the Apex to the Vertical-point . This being found , you must find the Angles between this Axis and the Horizontal-line , on every particular Azimuth ; and lastly , the distance between the Vertical-point , and the Parallels of Altitude , on every particuler Azimuth last drawn . For the doing whereof , you must work as you did before , to lay off the Signs , or the Parallels of the length of the Day , for these Almicanters , bear the same respect or proportion to the Horizon , as the Parallels of the length of the day have to the Equator , and are described in the same manner , as followeth . See Figure III. First , draw the Line AB , and make AB a Chord of 60 , and sweep the Arch of a Circle , and lay off 10 , 20 , 30 , &c. and draw the Lines from A the Center , and mark them with 10 , 20 , 30 , 40 , 50 , 60 , the even 10th degrees ; or , 45 for equal , 26-34 for double , 11-19 for 5 times , the length of the shadow and object , or what you please . Then , Draw AC Perpendiculer to AB ▪ and lay off the length of the Axis of the Horizon from A to C ; then make AC the Co-sine of the Reclination , and as the Sector stands , take out the sine of the Reclination , and lay it from A to D , and this will be the distance from the Apex to the Horizon ; Also , the sine of 90 shall reach from C to D , the distance between the Vertical-point and the Horizon ; also the nearest distance from A to CD , is the Perpendiculer height of the Stile AH . Then , Take the distance from I , the Vertical-point point on the Plain , to the Horizon on every particular Azimuth Line , and lay them in the Trygon , or III Figure , from C to the Horizontal-line AD , produced if need be ; and draw those obscure Lines , as in the Figure , and mark them with the Names of the Rumbs , to avoid confusion ; then is your Trygon made ready for use . Then , Take the distance from C in the Trigon , to every crossing of the Azimuth-line and Almicanter , and lay it on the Plain from the Vertical Point I , on its proper Azimuth , finishing one Almicanter before you meddle with another , and the work with patience and diligence will be performed ; the line● are to be drawn from Point to Point , with a steady hand , or a bending thin Ruler , being Conical Sections . Note , That when the Vertical-line of the Plain falls on an even Azimuth , then half the number of Rumbs will serve , being laid each way on both sides at once . Or , Having a Table of the Angles at the Zenith , the same as you made to draw the Azimuth-lines , draw a Line at any convenient distance , Parallel to AC ; the further from AC , the larger and better , as DEF in the Figure ; and note , where CD crosses the last Line EF , as at D ; make DE a Parallel sine of 90 , and lay off the sine complements of the Angles at the Zenith in the Table , from E towards D , and draw and mark the Lines , as in the Figure . Otherwise , The Stile being fixed , and the Dial set in its place where it must be , or at least set to the same Reclination , and Declination that it must be ; then if you apply the side of the Trianguler Quadrant to the Nodus , and the corner at the end of the same edge that toucheth the middle of the Nodus to the Plain ; and at the same time , the Thred and Plummet playing neatly on the Almicanter you would draw , you may find as many Points , and mark them as you please , without all the former trouble , and it may be every whit as true ; if the under-side be in●onvenient , you may use the upper ; only be sure , that the side you apply , and the Thred and Plummet play at the Angle of the Almicanter required . VII . To draw the Circles of Position , or Houses . The Circles of Position , or 12 Houses , meet and cross one another in the crossing of the Meridian and Horizon ; therefore the Horizon is the begining of the 1st and 7th Houses , beginning at the East , and reckoning under the Earth , by Imum Coeli , to the Descendant , or 7th House , at the West-part of the Horizon ; and so to Medium Coeli , the beginning of the 10th House , to the Ascendant , or Horoscope , the beginning of the 1st House . To draw these on the Horizontal-Dial , where they are Parallel Lines to the Hour 12 , do thus ; Take the distance from the Apex to the Equinoctial-line , and make it a = Tangent of 45 ; then the = Tangent of 30 degrees laid both wayes on the Equinoctial , shall give Points to draw Lines by , = to 12 , for the Houses required . For East and West Dials , take the Radius as before , viz. from the Apex to the Equinoctial-line on the Plain , which here is the Meridian ( and but the length of the Stile ) a Tangent of 45 ; then the = Tangents of 30 , 60 , and laid from 6 on the Equinoctial-line , gives Points to draw Lines Parallel to the Horizon , for the Houses required . For East and West Recliners , the Perpendiculer height of the Stile made a Secant of 0 ; then the Secant of the Stiles Elevation , shall be Radius to prick off the = Tangents of 30 , 60 , on the Equinoctial-line from the foot of the Stile , whereby to draw Lines Parallel to the Horizon for the Circles of Position required . All these Lines may most elegantly and ●asily be drawn and expressed , on a large Ceiling , with competent exactness in this manner following . First provide a Quadrant of Brass , or thin Wood , of about a foot Radius , or 14 , 15 , or 16 inches ; also , a Semi-circle of Brass , of about half an inch broad , and about an inch less Radius than the Quadrant : the Semi-circle must have at each end , somewhat more than to make up 180 degrees , to nail to the Transum , or stroke of the Window , where your Glass is to lie . Also , to one Ray of the Quadrant must be fastened two strong Wyres , to fasten the Quadrant to play after the manner of a Casement , one Point in the Ray of the Quadrant next the Center , sticking in the hole where you intend the Glass shall lie ; and the other end fastened to a piece of Wood nailed on the two upright Posts of the Window , so that howsoever you turn the Quadrant , fixed on those two Points , it may be precisely Perpendiculer , the Semi-circle playing all the while through a hole in the other Ray of the Quadrant , that lies Horizontally ; having a Skrew to stay the Quadrant at any Azimuth , as in Figure IV , is plainly expressed to your view . Then having degrees on the Semi-circle , and also on the Quadrant , and having fitted the Quadrant on his Points to play precisely Perpendiculer , which the Plummet in the Quadrant will shew , by turning it round about , and put in the Semi-circle through the hole in the Horizontal Ray of the Quadrant , and nailed it so to the Stoole or Transum of the Window , by putting two little bits of Wood under the ends , that the Quadrant may play evenly and smoothly on the Semi-circle-to almost the half-round , for quite the half-round will not be necessary , or useful . Then is the Instrument set fit for its Operation . Then first , to find the Declination , or rather the true Meridian-line . Turn the Quadrant till the edge be just against the Sun , and at the same instant get the Suns Azimuth ; then if you count so much as the Suns Azimuth is , on the Brass Semi-circle , from the place the Quadrant stands at , the right way , a Line drawn from the Center of the Semi-circle , or Quadrant , to that place , is the true Meridian Line ; which place you must carefully find by two or three tryals , and then mark it with Ink or otherwise , on the Brass Semi-circle to count from thence , in setting the Quadrant to the Suns Azimuth , at every hour and quarter in those Points you intend to draw on the Ceiling ; which a crooked Rule set to 00 , on the Semi-circle ▪ to pass to and fro with the Quadrant , will make easie . Then , having a Table of the Suns Altitude , and Azimuth , at every hour in that Latitude you draw the Dial for ; First , set the Quadrant to the Azimuth at the hour , counted the right way from the marked Meridian-line on the Semi-circle , and there skrew it fast ; Then extend the Thred fastened in the Center of the Quadrant , till it cut the Altitude of the Sun at the same hour and Azimuth , on the degrees of the Quadrant , and extending the Thred to the Ceiling , make a mark for that Hour and Altitude ; that Point at that time , gives the true place where the reflected spot will fall , at that Hour , Azimuth , and Altitude on the Ceiling of the Room . This work repeated as many times as there be hours and quarters in the Summer , and Winter Tropicks , for about 5 hours , ( and in the Equinoctial , and any where between , if you please ) shall give Points enough to draw the Dial , and also the Tropicks , and Azimuths , and Altitudes also , if it were convenient to mark it ; Or , to any other Altitude you mind to have at that Azimuth , all at once , or at most with two slips of the Thred ; the Italian , Babylonish , or Iewish-Hours , as easily drawn by Points found in the other Lines . Also , On the Meridian-line , you may add the day of the month , or any thing that depends on the Suns Meridian Altitude ; which work being well done , and drawn with smoth Lines , and well ornamented , would be a comely & pleasant Ornament to a Ceiling , and far cheaper then some fret Ceilings are done , and more useful . Lastly , When all is done , to put the Glass in right , the Foile being first rubbed off , to to cause it to give but one spot , let the Superficies of the Glass lie just so high as the Center of the Quadrant was , in the drawing the Lines , and put some Putty under it , and the Sun shining , make it to play right on the true Hour , Altitude , and Azimuth ; or , if it be just at noon , then bend it on the Putty with your finger , till it fall just on the Meridian , and day of the Month also in the Meridian-line . Also note , That look what Altitude the Sun hath at any time , the same will the reflected Altitude be , at the same time , if the Glass lie true , which two Observators at the same time may carfully prove . The making of the Tables of the Suns Altitude and Azimuth , is very largely shewed in the 15th Chapter , Vse the 37th and 38th , where you have wayes both general and particular , for any one or more Latitudes . The Figure Explained . A , the place on the Transum for the Glass to lie on , and in the middest thereof one Point , in the Ray of the Quadrant , is to play : IH , a piece of Wood to be nailed fast at H and I , for the other Point to play in at G : L , the hole for the Plummet to play in , being cut through the thin Quadrant : B and C , the ends of the Semi-circle , nailed on the Transum or Stoole : K , the hole in the Quadrant for the Semi-circular-Ring to pass through : FE , the Posts of the Window : D , the beginning of the degrees on the Semi-circle : AM , a Thred extended from the Center of the Quadrant to the Ceiling . Thus you have the usual wayes of Dialing in a competent measure , plainly , and practically handled , which may be useful to many a Learner ; and I hope will be as well accepted , as with free-will ( though with little ability , and less leisure ) readily imparted . A Table of the Suns Azimuth from the South , at every Hour and Quarter , 51-32 .   ♋ ♊ ♌ ♉ ♍ ♈ ♎ ♓ ♏ ♒ ♐ ♑   D. M. D. M. D. M D. M D. M D. M D. M 12 00 00 00 00 00 00 00 00 00 00 00 00 00 00   07 10 05 53 5 37 4 55 03 54 03 53 3 52   14 22 13 24 11 10 9 40 07 57 07 40 7 10   21 27 19 53 16 44 14 13 12 11 11 10 10 37 11 1 27 54 26 00 22 13 18 52 16 22 14 45 14 13   34 14 32 01 27 32 23 22 20 15 18 17 17 38   40 12 37 40 32 41 27 41 24 06 21 53 20 59   45 39 43 04 37 32 32 08 29 44 26 30 24 25 10 2 50 51 47 57 42 05 36 24 31 50 28 52 27 49   55 31 52 41 47 04 40 30 35 35 32 18 31 00   61 43 57 11 50 51 44 26 39 10 35 39 34 17   64 11 61 18 54 33 48 13 42 36 38 50 37 34 9 3 68 10 65 16 58 47 51 56 46 02 42 04 40 36   71 52 69 56 62 33 55 33 49 25 45 16 43 40   75 26 72 37 66 10 58 59 52 43 48 21 46 42   78 48 76 05 69 38 62 22 55 55 51 22 49 40 8 4 82 00 79 20 72 57 65 40 59 00 54 22     85 08 82 42 76 12 68 53 62 08       88 10 85 38 79 23 72 06 65 12       91 09 87 50 82 28 75 03 68 12     7 5 94 05 91 34 85 30 78 06 71 10       96 51 94 25 88 27 81 09         99 38 97 16 91 25 84 06         102 25 100 07 94 22 87 04       6 6 105 08 102 54 97 12 90 00         107 52 104 58 100 02           110 36 108 26 103 00           113 18 111 14 105 54         5 7 116 03 114 03             118 50 116 54             121 41 119 47             124 29 122 37           4 8 127 24               130 28             A Table of the Suns Altitude , at every Hour and Quarter in each Sign , for 51-32 .   ♋ ♊ ♌ ♉ ♍ ♈ ♎ ♓ ♏ ♒ ♐ ♑   D. M D. M D. M D. M D. M D. M D. M 12 62 90 58 42 50 00 38 28 27 00 18 18 14 59   61 50 58 34 49 52 38 24 26 54 18 13 14 55   61 25 58 09 49 31 38 07 26 40 18 00 14 38   60 41 57 29 49 00 37 38 26 15 17 37 14 20 11 59 43 56 34 48 12 36 58 25 40 17 06 13 52   58 32 55 26 47 10 36 07 24 02 16 26 13 10   57 04 54 06 46 02 35 06 23 59 15 38 12 35   55 29 52 36 44 38 33 56 22 58 14 49 11 30 10 53 45 50 55 43 12 32 37 21 51 13 38 10 36   51 52 49 07 41 33 31 10 20 33 12 27 9 20   49 54 47 12 39 46 29 36 19 10 11 09 8 9   47 50 45 12 37 56 27 54 17 35 9 44 6 41 9 45 42 43 06 36 00 26 07 15 58 8 13 5 16   43 30 40 57 33 55 24 14 14 14 9 49 3 41   41 16 38 45 31 50 22 16 12 27 4 54 2 04   39 00 36 30 29 40 20 14 10 32 3 07 0 17 8 36 41 34 13 27 31 18 08 8 35 1 15     34 22 31 55 25 09 15 59 6 30       32 02 29 13 22 56 13 46 4 27       29 42 27 16 20 37 11 33 2 17     7 27 23 24 56 18 18 09 17 0 06       25 03 22 36 15 58 06 58         22 45 20 16 13 38 04 40         20 28 17 58 11 18 02 20       6 18 11 15 41 09 00 00 00         15 56 13 25 06 41           13 44 11 11 04 23           11 35 8 59 02 08         5 9 32 6 50 00 06           7 23 4 44             5 26 2 41             3 36 0 41           4 1 32             The Description and some Uses of the Sphear for Dialling , and for the better understanding of the general and particular Scheams . NExt the Foot and Semi-circle Frame for supporting of it , you may consider ; 1. The fixed Horizon , to which the Foot is fastened with 4 skrews , numbred and divided into 360 degrees , or four 90 deg . whose count begins at the Dividees side of the Meridian-Circle . 2. The Meridian Circle , whose fore-side at the Nadir-point stands in the Center of the Foot ; this is also divided into 4 90s s , and begins to be numbered at the South and North part of the Horizon , upwards toward the Zenith , and downwards toward the Nadir ; which Circle is alwayes fixed as the Horizon is . 3. The Equinoctial Circle , made fast at the East and West Points of the Horizon , moving up and down upon the Meridian-Circle , according to the Elevation of the Equinoctial in any Latitude ; this is divided ●●kewise into four 90s s , & numbred from the Meridian each wayes to the East and West Points of the Horizon . 4. On the Meridian Circle , is set 2 moveable Poles , to be elevated or depressed fit to the Latitude of any place ; on the Fiducial-edge of which , is fastened the Thred , representing the Axis of the World , at any Elevation of the Pole. 5. On the 2 Pole Points , is fastened the Hour Circle , which delineates or represents the motion of the Sun , or any fixed Star , moving in its supposed Diurnal motion about the Poles of the World , and may not improperly be called the moveable Meridian Circle , or Hour Circle , divided as before . 6. The Moveable Horizon , that moveth about to any Azimuth , and slideth or moveth in the fixed Horizon . 7. The Plain , fixed in 2 opposite Points to the moving Horizon ; being set , either Horizontal , when it lies Parallel to the fixed Horizon ; or Erect , when Perpendiculer thereunto ; or set to any Reclination or Inclination , by help of the Semi-circle of Reclination , fastened to the backside of the Plain in the 2 Poles thereof . 8. You have the upper moving Semi-circle , in turning about of which , whateve● degree the fore-side of the Semi-circle cuts the Perpendiculer-point cuts the comple●ment thereof , and to be called the upper Semi-circle , or Circle alwayes Perpendicu●ler to the Plain . 9. There ought to be a Thred fastened in the Center of the Plain , to be extended to any Altitude or Azimuth required . Thus much for Description , repeated again in short thus ; The Horizon ; The Meridian ; The Equinoctial Circles ; The 2 Pole Points , and Axis ; The Hour Circle , or Moveable Meridian ; The Moveable Horizon ; The Plain ; The Semi-circle of Reclination ; The upper Semi-circle , and , The Thred . Note also , Every Circle is divided into 4 times 90 , and numbred the most useful way . Also , on the Plain is set the 12 Months , and every single Day ; on which every respective day , if you extend the Thred , then in the degrees , is the Suns Right Ascention in degrees ( on the innermost Circle , the same in hours and quarters ) from the next Equinoctial-point , on the Line of Declination , his mean Declination ; on the Line of ●he Suns place , his mean true place , sufficiently true for any illustration in Mathematical practice . The Uses whereof in some part follow . 1. To rectifie the Sphear to any Latitude , count the Elevation of the Pole on the Meridian Circle , from the Horizon upwards , and downwards from the North and South parts of the Horizon ; and there make fast , with the help of the small skrew , the Fiducial-edge of the Poles Points , carrying the Hour Circle fixed upon them , then the Pole is rightly elevated . 2. Count the complement of the Poles Elevation on the Meridian , from the South part of the Horizon , and to it set the divided side of the Equinoctial Circle , then is that rectified also ( in the Northern Hemisphere , or in the Southern , if you call the North Pole the South Pole ) . 3. Extend the Thred or Axis passing through the Center to the South Pole , and there make it fast , and then the Sphear is rectified for many Uses in that Latitude . Use I. The Day of the Month being given , to find the Suns true Place . Lay the Thred in the Center of the Plain on the day of the Month , and in the Line of the Suns place , you have his place . Example . On the 5th of November , it is 23 degrees in ♐ ; or if the Suns place be given , look for that , and just against it , in the Months , is the day required . Example . The Suns place being 15 degrees ♌ , I look for it in the Line of his place , and just against it I find Iuly 28 day . Use II. To find his Declination any day . Look for the day given , and right against it in the Line of Declination , is his due Declination required . Example . August the 5th ; The Declination is 14 degrees 5 minuts from the next Equinoctial-point , viz. ♎ . Note , In the Northern Sines , or Summer-time , the Sun hath North declination ; or in Southern Sines , or Winter-months , the Sun hath South declination . Or if you have the Suns declination , find that in the Line Declination , and right against it in the Months is the day required . Example . 21 degrees South declination , beginning from the Equinoctial towards the Winter Solstice , I find Novemb. 15. The like work had been , if the Suns place had been given , to find his declination . Use III. The day given , to find the Suns Right-Ascention . This is usually reckoned from ♈ to ♈ , round , in 24 hours ; but twice 12 is as useful , and then it is thus ; Find the day amongst the Months and Dayes , and just against it , in the time of Hours , is the Suns Right Ascention ; ( but note , it is not right figured for this use ) counting onwards from ♈ , or the 10th of March , to the 13th of Septemb. and from thence to Aries again ; Likewise the degrees are to be reckoned from ♈ onwards , as the Months proceed . Example . On the 12 of May , what is the Suns Right Ascention ? Lay the Thred on the 12th of May , and in the Line of Hours it cuts 9-57′ counting from Aries onwards ; or in degrees 59-15 , counting as before . Thus , if any one of these 4 general things be given , the other may be found . Use IV. The Suns Declination and Latitude being given , to find the Suns Meridian Altitude . The Sphear being rectified , count the declination on the Meridian , from the Equinoctial , that way the declination is , either North or South ; and where the count ends ; there is the Meridian Altitude required for that day , or Declination . Example . Iune 11. Declination 23-30′ ; Count 23-30 , from 38-30 , the place where the Equinoctial stands , for 51-30 Latitude , and the account will end at 62 degrees , the Suns Meridian Altitude at that Declination Northwards : But , if it had been 23-30 South Declination ; then count as much from the Equinoctial downwards , and the count will end at 15 degrees , for the Suns Meridian Altitude , at 23-30 South Declination . Use V. The Suns Declination and Latitude being given , to find the Suns Rising or Setting , and Amplitude , East or West . Count the Suns declination on the Hour-Circle towards his proper Pole , that is South-declination toward the South-Pole , and North-declination towards the North-Pole ; and thereunto lay the Thred that is fastned in the Center ; then bring the Hour-circle and Thred both together , till the Thred touch the Horizon ; then the Thred on the Horizon shews the Amplitude , and the divided-side of the Hour-Circle , shews the Suns Rising and Setting on the Equinoctial , counting the Meridian alwayes 12 , and the 2 East and West-points 6 , and 15 degr . for an Hour , and every deg . 4 min. Example . Iune 11. Declination 23-30 , the Sun Riseth at 13′ before 4 , and the Amplitude is near 40 deg . Again , April 10. Declination 11-30 , the Amplitude is 18-30 from the East to the North , and Riseth at 5 , the Hour-circle cutting 15 degrees on the Equinoctial . Use VI. The Declination & Latitude , & Suns Altitude given , to find both Hour & Azimuth . Rectifie the Sphear , and set the Plain horizontal ; that is , Level or Parallel to the Horizon ; then apply the Thred to the Declination , counted the right way on the Hour-circle ; then turn the Hour-circle and upper Semi-circle about , till the Thred cuts the degrees of Altitude in the upper Semi-circle , and the Hour-circle , shews the hour in the Equinoctial , and the Semi-circle cuts the Suns Azimuth in the deg . on the Horizon or Plain . Example . Declination 10 , Latitude 51-30 , and the Suns Altitude 30 ; the hour will be 8-27 , and the Azimuth 66 , from South Eastwards if in the morning , or the contrary if in the afternoon . Use VII . The Hour , or Azimuth , and the Suns Declination given , to find the Altitude . The Sphear rectified , as before , and the Hour being given , set the Hour-Circle to the hour on the Equinoctial ; then bring the Thred to the Declination , counted on the Hour-circle ; then bring the upper Semi-circle , till the fore-side do just touch the Thred , and the Thred on the Semi-circle , shall shew the Altitude required ; and on the Horizon , the Azimuth at that Hour , and Altitude . But if the Azimuth be first given , then set the upper Semi-circle thereunto , counted on the fixed Horizon ; then the Thred laid to the declination , on the Hour-circle , and turned about till it touch the upper Semi-circle , there it shews the Altitude ; and the Hour-circle on the Equinoctial , shews the hour . Use VIII . To find the Suns Height in the Vertical-Circle . Set the upper Semi-circle to the East or West-Point , or 90 degrees of Azimuth ; then lay the Thred to the declination on the Hour-circle , and then bring it and the Thred together , till it just touch the upper Semi-circle , and it shall there shew the Altitude at East or West required . Example . At 10 degrees Declination North , it will be East at 16 degrees of Altitude . Use IX . To find the Suns Altitude at 6. Set the Hour-circle to 6 on the Equinoctial , and the Thred to the Declination ; then bring the Semi-circle to the Thred , and it shewes the Altitude at 6 required . Example . At 23-30 Declination , the Altitude 18-15 above the Horizon in North-declination ; and as much under in South-declination ; for , you must observe that the surest working is from the upper or divided-sides of the Rings , on every occasion to use it . Use X. To find the Hour of the Day , when the Sun shineth . Rectifie the Sphear , and set the Plain Parallel to the Equinoctial-circle ; then set the Meridian-circle due North and South , and the shadow of the Axis shall on the Plain , shew the true hour . Or , otherwise thus ; At the true place of the Suns Declination , on the Hour-circle make a mark , or stick the point of a Pin , then turn the Sphear about , till the shadow of that mark , fall on the Center ; ( the Sphear standing Horizontal , as near as may be ) then the Hour-circle shall , on the Equinoctial , shew the hour of the day required . Note , A small Bead , or knot on the Thred , will do the business as well as may be . Thus any the like Questions may be wrought for the Stars ; or the manner of raising the Canon for any Spherical Triangle whatever , to work the same exactly by the Logarithms . As thus ; Suppose I would make the Canon , or Proposition , to find the Suns height in the Vertical Circle at any declination . First , The Sphear being rectified , and the Plate set Horizontal , bring the upper Semi-circle to the East-point , and laying the Thred to the declination on the Hour-circle , bring it and the Thred together , till it just touch the upper Semi-circle . The Rings or Circles so standing , and being great Circles of the Sphear , there is constituted a Spherical-Triangle in this form ; Wherein you have , ZAE , 51-30 the Latitude , the Angle at the Equinoctial ; and ZE 90 , the upper Semi-circle ; and AB 23-30 , the Declination , part of the Hour-circle ; to find BE , part of the upper Semi-circle : Now this being a right-angled Spherical-Triangle , and the parts which are given , being one right Angle , viz. The Angle at A , and the Side AB , the Suns Declination ; and the Angle at E 51-30 , to find the Side BE ; Now the Sines of the sides of Spherical-Triangles are proportional to the Sines of their opposite Angles , and the contrary . Therefore , As the sine of the Triangle BEA 51-30 Is to the sine of the Side AB 23-30 So is the sine of the Angle BAE 90-00 To the sine of BE 30-39 And the like for any other , as by comparing the Rules in Mr. Norwood's Trigonometry , and the Circles of the Sphear together , the use and convenience thereof will evidently appear unto you ▪ Only note this plain Observation . That the side of a right-Angled Triangle , which subtends the Right Angle , is most properly called the Hypothenusa ; the other which you make or suppose Radius , the Base . The other , the Perpendiculer . Or more short , The Hypothenusa and Leggs : Therefore if the Hypothenusa and one Leg be given , the proportion is wrought by Sines alone ; but if the two Legs be given , and first and second in the Question , then the Proportion is wrought by Sines and Tangents together . As for Example . As the sine of ♈ ♋ 90-00 To the sine of ♋ AE 23-31 So is the sine of ♈ ♉ 30-00 To the sine of ♉ R 11-31 The Suns Declination in ♉ . Again secondly , As the sine of ♈ AE 90-00 To the Tangent of AE ♋ 23-31 So is the sine of ♈ R 27-54 To the Tangent of R ♉ 11-31 The Declination as before .   But if the one acute Angle , and his opposite Leg or Side be given , then the Proportion is made by Sines only , as in the foregoing Example . Again , In Vertical Triangles that have the same acute Angle at the Base , as the Triangle P ♉ ♋ , and ♈ ♉ R , being equal Angled at ♉ ; the sines of the Bases are proportional to the Tangents of the Perpendiculer , and the contrary . Likewise , The Sines of the Perpendiculers , as proportional to the Sines of the Hypothenusaes , and the contrary . As for Example . Thus for Perpendiculers and Bases . As the sine of the Base ♉ ♋ 60-00 To the Tangent of Perpend . ♋ P 66-29 So the sine of the Base ♉ R 11-31 To the Tangent of Perpend . ♈ R 27-54 Or , As the Tang. P ♋ the Perpend . 66-29 To the sine ♉ ♋ the Base 60-00 So the Tang. ♈ P Perpend . 27-54 To the sine R ♉ the Base 11-31 Also for the Second , viz. Hypothenusaes and Perpendiculers . As the sine of Hypothen . P ♉ 78-29 To the sine of Perpend . P ♋ 66-29 So the sine of Hypothen . ♈ ♉ 30-00 To the sine of Perpend . ♈ R 27-54 Or the contrary thus ; As sine of Perpend . P ♋ 66-29 To sine of Hypothen . P ♉ 78-29 So sine of Perpend . ♈ R 27-54 To sine of Hypothen . ♈ ♉ 30-00 This being premised , when to use Sines alone , and when to use Sines and Tangents together , you may rectifie the Scheam to your present purpose , and see there how the Triangle lies in its Natural parts , very plain and demonstratively to be apprehended . The uses of the Sphear IN DYALLING . TO this purpose , you must take notice , That the Sphear is very excellent to demonstrate that Art ; especially all those Dials whose Stiles have any competent Elevation . Therefore , first to explain the terms . The Sphear being rectified to the Latitude ; Then , first the Plain , or Broad-plate , is to represent any Plain howsoever scituate , either Horizontal , or Erect Direct , or Direct Reclining or Inclining , or East and West Erect , or Reclining , or Inclining , or Erect and Declining , or South Declining , or Reclining or Inclining , less or more than to the Pole or North Declining ; or Re-inclining less , to , or beyond the Equinoctial . Of which in their Order . 1. By Horizontal I mean , when the Plain is set even with the fixed Horizon , and the Notch which the Semi-circle of Reclination passeth in just against the Meridian ; then if you stretch the Axis streight , and bring the upper Semi-circle just to touch the Axis ; then the Axis , on the Semi-circle , sheweth the Stiles height ; and the Edge of the Semi-circle on the Plain , shews the Substile to be in the Meridian . For all the Hour-Arks on the Plain , do thus ; Set the Hour-circle to every hour and quarter on the Equinoctial ; and then if you bring the loose Thred , fastened in the Center of the Plain , along the Plain till it just touch the Hour-circle , then on the Plain it shall shew the Angle from 12 , for that respective hour and quarter the Hour-circle stands at on the Equinoctial , accounting 3-45 for a quarter , and 7-30 for half an hour , and 15 deg . for every whole hour , as was hinted before . 2. For an Erect Direct South or North-Dial . Just as the Plain stood before , that is to say , the Notches of the Moving Horizon against the Meridian ; turn the fixed Semi-circle , till the divided side of the Horizon cuts no deg . on the fixed Semi-circle , then the upper-edge of the Plain respects the Zenith , and the lower the Nadir ; and the two Notches in the Moveing Horizon ( being alwayes the Poles of every Plain ) are just in the Meridian ; therefore it is a Direct Plain , and Erect , because Upright without any Reclination , as the fixed Semi-circle sheweth . Then being so fixed , and made fast there , pull the Axis streight , and bring the upper Moving Semi-circle just to touch the Thred or Axis ; then on the upper and lower Semi-circles , the Axis sheweth the Stiles Elevation ; and on the Plain the Semi-circle cuts the Substiles distance from 12 , viz. 00 , because a Direct Plain . And for all the Hour-Arks on the Plain , set the Hour-circle to every hour , quarter , and half hour on the Equinoctial ; and bring the Thred easily along the Plain , till it just touch the Hour-circle ; then on the Plain it sheweth the Ark from 12 required . Also note , The several Triangles made on the Meridian , Equinoctial , and Hour-circle , at every hour it is set unto . As thus ; Suppose at the Pole , I set P ; at the cutting of the Equinoctial , and Meridian , AE ; at the upper-end , or Zenith , set Z ; on the Meridian , and where the Hour-circle cuts the Equinoctial , at 1 & 11 , set 15 ; at 2 & 10 , 30 ; at 3 & 9 , 45 ; at 4 & 8 , 60 ; at 5 & 7 , 75 ; and at 6 & 6 , 90. Then the Triangle runs thus ; As the whole sine PAE 90-00 To the Tang. of AE 15 15-00 one Hour on the Equinoctial . So the sine of PZ 38-30 To the Tang. of Z 11 11-28 the measure on the Plain for 11-1 . The like work serves for all the rest . But note , Because the Hour-circle cannot pass by 12 , you must turn the other-side , or half , for the afternoon hours . Also note , That if the back-side of the Plain do not well represent the South-side , being the more useful Dial ; then if you hold the Sphear with the foot upward , the Zenith becomes the Nadir , and the North Plain a South Plain , to appear more Plain to the apprehension . 3. For a Direct Reclining Dial. For these Dials , set the Plain Direct , as before , and let the upper part of the Horizon cut the Semi-circle of Reclination , according to the Plains Reclination , and there make it fast ; then the Axis drawn streight , and the upper Semi-circle brought to it , sheweth the Stile and Substile ; and the Thred and Hour-circle , laid as before , giveth the Hour-Arks on the Plain , and sheweth also how the Proportion runs . To find any Requisite also you may observe for all North-Recliners and South-Incliners , that the complement of Latitude and Reclination put together , doth give the Poles Elevation , or Stiles height , for all those Plains , which sometime will be above 90 from the South part of the Meridian ; and then the complement to 180 , is to be set from the North part of the Meridian : But if it be a South-Recliner , then substract the Reclination out of the Comp. Lat. and the remainder is the Stiles Elevation : But if the Reclination be more than the complement Latitude , then substract the complement Latitude out of the Reclination or Inclination , and the remainder is the Stiles Elevation . Note also , That the upper-face of the Plain , that beholds the Zenith , is the Recliner ; and the under-face that beholds the Nadir , is the Inclining-plain . And note , That both Plains , viz. both Incliners and Recliners have the same Requisites in each of them . But , the hours proper to the Recliner , are not to be put on the Incliner ; for when the Sun shines on the one , it can't shine on the other . Therefore to know what hours are fit for these or any Plains whatever , do thus ; The Sphear rectified , and the Plain set to his true scituation , lay the Thred on the Suns declination , on the Hour-circle ( according to what time of year you would know when the Sun begins and ceases to shine on any Plain ) and turn the Hour-circle , with the Thred so laid , till the Thred do but just touch the Plain , and the Hour-circle doth on the Equinoctial , cut the Hour and Minuit required ; when the Sun comes on the East-side , and when it goes off from the West-side of the Plain . Example . Suppose you have a Direct North-plain that Reclines from the Zenith towards the Equinoctial 25 degrees , you shall find the Stiles Elevation to be 63-30 , the Substile from 12. The North-Pole to be elevated on the Recliner , and the South-Pole on the Incliner ; and that the Sun shines on the North-recliner in the longest dayes , viz. 23-31 , declination , from the Rising 13′ before 4 , till 10 ; and then it begins to shine on the South-incliner , and shines till 2 afternoon ; then it comes on the North-recliner again , and continues till it sets . But in the shortest dayes , when the Declination is 23-30 towards South , then on the North-recliner it shines not at all , but only on the South-incliner , from Rising to Setting ; and so doth it all the time the Sun hath South-declination . This Rule serves for all sorts of Dials whatsoever . Note , That the Circles of the Sphear shews the Canon to work this Question exactly , whereof you have a large Discourse in Wells his Art of Shadows , from pag. 391 , to 408 , in 35 Chap. 4. For a Direct East or West Erect-Dial . The Sphear being rectified to the Latitude , bring the Notch in the Moveing Horizon , to the East or West-points on the fixed Horizon , viz. to 90 degrees ; then set the Plain Erect , and make it fast there ; then you shall perceive the Axis lie close to the Plain , it shews the Stile to have no Elevation , but must be set Parallel to the Plain , at any quantity you please , which is to be the Radius of a Tangent-line , whereby to pr●●k down the Hours ; and that the S●bstile or place where the Cock or Stile must stand is in 6 , being the Hour-circle , till it be ●ust against the upper Semi-circle , touch●ng the Thred , and in the Equinoctial it cuts 6 , the true place where the Stile must stand . Also , By the fore-going Rule you shall find the Sun shine all the year from the Rising , till 12 on the East-side ; and on the West-side from 12 , till his Setting . 5. For an East or West-Recliner . Turn the moving Horizon to 90 degrees in the fixed , as before ; then set the Plain to his due Reclination , and make it fast there , and pull the Axis streight , and bring the upper Semi-circle just to touch it , and straitway you have the Stile , and Substile , and 12 , the Inclination , Meridian , and Hour-Arks on the Plain . As for Example . An East-plain reclining from the Zenith towards the Horizon 45 degrees , hath his Meridian , or 12 a clock Line in the Horizon ; for if you extend the Thred from the Center to the fore-side of the Meridian , just there the 12 a clock Line must alwayes be , which in this Plain lies in the Horizon . The Substile doth lie 41-40 from thence upward , as the upper Semi-circle doth shew ; the Inclination Meridian is thus found ; Bring the Hour-circle , till it stand even and parallel to the upper Semi-circle ; then on the Equinoctial it cuts 58-7′ , the Inclination of the Meridian , with which you must make a Table of Hours , or Arks at the Pole , to calculate the Arks on the Plain , if you work Arithmetically . But by the Sphear , Set the Hour-circle to the hours on the Equinoctial , and the Thred being brought along the Plain till it touch the Hour-circle , shall shew on the Plain the Angle from the Horizon or Perpendiculer ; or with some more trouble , from Substile or 12. Also , It shews , that the North-Pole is Elevated on the West-reclining ; and the South , on the East-inclining opposite thereunto ; and that the Recliner in ♋ , shews from 9 in the forenoon , till 8 at night ; and the East Incliner from the Rising , till 9 forenoon in Summer ; and in Winter , till a 11 in the forenoon . Now to make these Plains , as Erect Decliners , let the complement Latitude become a new Latitude ; and the complement Declination a new Declination ; then they may become Erect Decliners , as in the next sort following . 6. Of Erect Decliners East or West . By Declination , I mean the quantity of the Angle that the Meridian or Pole of place makes between the Meridian , or Pole of the Plain ; therefore to set the Sphear to any Declination , do thus ; The Sphear being set to the Latitude , turn the Sphear as well as you can guess , to the scituation of the place ; that is , put the North part of the Meridian towards the North ; and the South part towards the South ; then turn the Notch of the movable Horizon , alwayes to the degrees of the Plains Declination , from North or South , towards either East or West , and Upright also as in Erect Dials : Then is the Plain set to his Declination , viz. the distance of the Horizon between the Meridian , or Pole-place , which is alwayes 12 a clock , and the Meridian , and Pole of the Plain , being alwayes just where the Notch is in the Moving Horizon . Now according to these Rules , A Plain that declines 30 degrees from South to West , The Stiles Elevation is 32-35 . The Substile from 12 , 21-40 . The Inclination of the Meridian 36-24 . The South Pole is elevated on the South-side , and the North Pole on the North-side : And the Sun shines on the North-side from Rising , to 8 ; and on the South-side , from 8 to 7 at night ; and on the North again , till Sun-setting , by working as in the former Directions is expressed . Note , In those Erect Decliners , whose declinations is above 60 degrees , you shall find the Stiles Elevation to be very small ; therefore to make it exact , you must use Arithmetical Calculation ; for the doing of which , the Sphear , with due consideration , gives the best directions , with these Proportions or Canons . As sine 90 ZN 90-00 To sine Declination NC 30-00 So Co-tangent Latitude PZ 38-30 To Tang. Subst . from 12. ZH 21-40 As sine 90 ZN 90-00 To Co-sine Declination NA 60-00 So Co-sine Latitude ZP 38-30 To sine Stiles Elevation PH 32-25 As sine Latitude PN 51-30 To sine 90 PAE 90-00 So Tangent Declination NC 30-00 To Tangent Elevation Merid. AEI 36-24 As Co-tangent Latitude ZP 38-30 To sine of 90 ZPQ 90-00 So sine Declination ZIA 30-00 To Co-tang . 6 from 12 AQ 57-50 Note , If you set P , at the Pole. Z , at the Zenith . N , at the North-end of the Horizon , at the Declination , or Pole-plain . H , on the Plain , just against the moving Semi-circle , or Substile . A , at the Plain on the Horizon . AE , on the Equinoctial . I , at the Hour-circle , cutting the Equinoctial , set just against the upper Semi-circle . Note , Q is to be set on the Plain , right against the Hour-circle , being set to the Hour . Having , I say , by these Rules , and the like , made and found the Requisites , then proceed to draw the Dial thus ; by help of a Sector with Sines and Tangents , to 7-5 ; such as are usually made ▪ But for very far Decliners , use that help as directed in Chap. 4. The like work serves to help all sorts of Dials with low Stiles , Polar , and Meridian . Dials also . The other 6 sorts , yet behind , I shall demonstrate only in two of them , which do properly enough comprehend them all ; and the work of one , is as easie as the work of the other , especially by the help of the Sphear , where the hardest is as plain as the Horizontal . Therefore , 7. Of Declining , Reclining-Dials . 1. For South Recliners , they may recline short of , to , or beyond the Pole , at any Declination , as the putting up and down the Plain , doth plainly demonstrate . Therefore , first , Of one that Declines South-west 35 , and Reclines 20 from the Zenith . Set the Notch , or Pole of the Plain to the Declination , and the Reclining Circle to its Reclination , and there make it fast ; then extend the Axis streight , and bring the upper Semi-circle just to touch it , and the Hour-circle exactly even with the moving Semi-circle . Then , First , The Axis shews the Stiles height on the Semi-circle to be 12-13 . The Thred brought along the Plain while it touches the Meridian , and that shews the Meridians Elevation above the Horizon , on the North Recliner to be 76-32 ; or its Depression below the Horizon in South-Recliners , and that from the East-end , as the Sphear sheweth . Then , 3. The Substile from the Perpendiculer Line of the Plain , is 21-6 , as the upper Semi-circle sheweth ; but from the hour 12 , or Meridian 7-58 , and stands on the East-side of the Meridian . The Inclination of the Meridian is 33-29 , as the degrees on the Equinoctial , between the Meridian and Hour-circle , shew . All the Hour-Arks are easily found from the Plains Perpendiculer Eastwards and Westwards , by applying the Thred to the Hour-circle and Plain , being set to the Hours on the Equinoctial . The South Pole is elevated in the South-Recliner , and the North , on the North Incliner . If you set Letters to the Sides and Angles , according to the former discourse , you will see how all the Canons in the Arithmetical Calculation lie , as I shewed you before in the Declining Dials . And as again thus ; On the Pole set P. On the Zenith Z. At the West-end of the Plain , set A. At the East-end B. At the South Pole of the Plain C. At the North Pole D. At the East-end of the Horizon E. At the West-end W. At the North-end of the Meridian , set N. At the South-end S. Where the Hour-circle cuts the Plain F. Where the Meridian cuts the Plain G. Where the fixed Semi-circle cuts the Plain , set E. As in the Figure before . Then these Canons in short run thus ; As sine Base ZD 90-00 To Tang. Perpend . ND 35-00 So sine of Base ZE 20-00 To Tang. Perpend . GE 13-28 Whose complement AG 70-32 , is the Meridians elevation . As sine of the Side GE 13-28 To sine of the Angle CZE 35-00 So sine of the Angle GFZ 90-00 To sine of the Side GZ 23-57 Which taken from ZP 38-28 , leaves 14-33 , the distance of the Meridians place from the Pole to the Plain , viz. GF . As sine of Hypothen . GZ 23-57 To sine of Perpend . ZE 20-00 So sine of Hypothen . PG 14-33 To sine of Perpend . PF 12-13   the Stile . As Tangent of Perpend . ZF 20-00 To sine of Base GE 13-28 So Tangent of Perpend . PF 12-13 To sine of Base FG 7-58   the Substile to 12. As the sine of the Side ZE 20-00 To the sine of the Side GE 13-28 So is the sine of the Angle PFG 90-00 To the sine of the Angle FPG 33-28   Inclin . Merid. For the Hours in all Dials , say thus ; As sine of 90 , To sine of Stiles height ; So Tangent of the Angle at the Pole , To Tangent of the Angle on the Plain . 8. For North Declining Reclining-Dials . For these Plains also , you must rectifie the Sphear to the Latitude , and set the Plain to his Declination , and Inclination , which is given , and for which you are to make a North Declining Reclining Dial. As you did in the South-Recliner , so work in all respects , as you shall bring forth the Quesita's , either by the Sphear or Arithmetical-Calculation , as is largely shewn . And for a Plain that declines 55 degrees from the North towards the East , and relines 20 from the Zenith , you shall find the Requisites to be as followeth . 1. The Meridians Elevation above the Horizon , is found to be 63 deg . 58 min. But yet observe , You must make use of that part of it which is below the Horizon , because the Sun being Elevated high on the South-part of the Meridian , must needs cast a shadow on the North-part thereof ; therefore in drawing the Dial-part , part is only to be made use of for the Sun to shine on . 2. The Stiles Elevation is 37 degrees 00 minuts . 3. The Substile from 12 , 65-24 ; or from the Plains perpendiculer 39-22 . The North Pole is Elevated ; and in regard the Plain declines to the East , the Stile must be set towards the West , and it shines on the Plain in Summer-time , from the Rising unto 12 : But in the Winter-time , but a few hours . Note also , That these Declining Reclining-plains , may be referred to a new Latitude and Declination , wherein they shall become Upright Decliners , as before is hinted . The Poor-Mans Dial-Sphear ; Or another way to demonstrate the Mystery of Dyalling , both for Declining and Inclining Plains , in a very plain , easie way , for one 6th part of the cost of the other Brass-Sphear . First , as to the Description , and afterward for the Vse . AS to the Description , the Figure annexed , and a few words shall suffice ; wherein consider , First , The plain flat-Board , representing the Horizon , as ABCD. Secondly , The two upright pieces , as East and West-points , as AE , and BF , to support the moving Plain . Thirdly , The Moving-plain , moving to any Inclination , on the two Points E and F , with 180 degrees upon the Plain , and noted by ABEF . Fourthly , Also a Brass-circle as G , fastened to the Plain , to set it to any degree of Inclination ; and a skrew , as at H , that may stay it steady , when set to any Reclination . Fiftly . On the middle of the Horizontal-board , is fastened at the Point M , a true Horizontal-Dial , drawn fit for your Latitude , and to turn round on the Point M , as IMKL . Sixtly , A Thred fastened in L , the Center of the Horizontal-Dial ; and in N , the Center of the Plain ; to be both a Stile for the Horizontal-Dial , and to represent the Axis of the World ; also a small Woodden-Quadrant will be useful , such a one as half the Plain is , to draw Perpendiculers , and measure Angles , as afterwards in the Uses . The Uses follow . Use I. To find the Declination of a Plain by the Sun-shining . Apply the side AB to the Wall , and hold the Instrument level , as by help of a Point Plummer , fastened at N , and the Point playing right on M , it is easie to do ; then by the Trianguler-Quadrant , having first observed the true hour , turn the Horizontal-Dial about on the Point M , till the shadow of the Thred ( or Axis ) shew the same Hour ; then the Point on the North-end of the Horizontal-Dial , shall shew the true Declination of the Plain . For any South Decliner , the use is obvious . But for North-Decliners , you must turn the Plain out of the way of the Thred , still keeping the same side , AB , to the Wall ; and if the Horizontal-Dial hinder , put a Parallel-piece between , as your Rule , or any other thing , and you shall have the Point give the Declination on the Southern Semi-circle on the fixed Horizon . Use II. The Declination of any Erect Decliner given , to find the Substile , and Stile , Inclination of Meridians , and every hour and quarters distance from 12 , being the Perpendiculer Line on the Plain . First , Set the Point at 12 on the Horizontal-Dial , to the Declination of the Plain , toward the East or West , and set the Plain Upright . Then first for the Substile . Apply the side of the Quadrant to the Plain , and cause the shadow of the Thred to play Parallel to the perpendicular Ray of the Quadrant , and at the same time it shall shew on the degrees on the Plain , the true Substiles distance from 12. Example . Suppose the Plain decline 20 degrees South-west , you shall find the Substile to be 15 deg . and 12′ from 12 , and to stand on the East-side of 12 , in a South declining West 20 degrees , Latitude 51-30 . Again , for the Stiles Elevation . Apply the Quadrant to the flat of the Plain , on the Substile Line , so as the Thred may cut the Center of the Quadrant ; and then the Thred shall cut on the Quadrant 35-46 for the Stiles height . Again , for the Inclination of Meridians . The shadow of the Thred when it cuts the Substile 15 deg . 12′ on the Plain , shall on the Horizontal-Dial cut 1 hour 36 min. which reduced to degrees , is 24 deg . 50 min. the Inclination of Meridians . Again , for every Hours distance , in degrees and minuts from 12. Turn the whole Instrument about , ( as it is then first set ) till the shadow of the Thred shall fall on every hour and quarter , and then the shadow shall cut on the degrees on the Plain , the distance of every hour and quarter from 12 , for that declination , in degrees and minuts ; which you may draw into a Table , for your use and purpose ; or hereby examine your more exact Calculation , and prevent all gross mistakes in your former work . Use III. Any Declining North-east , or North-west-Dial being given , to find the former Requisites for those Dials . In the true proper using the Sphear for North-Dials , the Stile should look upwards , which will appear so to do , if you turn the Instrument the bottom upwards , for the further help to your fancy ; but observe that the Hour-Arks , and Angles , are the same for the North , as for the South , only the difference is in the Scituation , as to the contrary-side , and looking upward instead of the South Decliner , looking downward , as by turning the Instrument appears ; so that if you draw the Dial as a South-west , when you would make a North-west ; and set right figures , and the right way , and then your work is effected to your mind , to the right intent and purpose . Example of a North-East , 30 degrees , Latitude 51-30 . Set the Point at 12 , to 30 degrees Westward , and apply the Square to the Plain , till it just touch the Thred ; and on the degrees on the Plain , it cuts 21-40 for the Substile ; and at the same time almost half an hour past 2 for Inclination of the Meridians ; and applying the Quadrant to the Substile-Line on the Plain , and to the Thred ; it cuts 32-35 for the Stiles height , being the same , and the same way found as for the South Decliner East . But observe , That for the Hour-Arks , you must note , That the North-Dial cannot shew 12 at Noon , nor any Hours very near Noon , which will be seen on the South Decliner East ; Therefore 4 in the morning , is here called 8 ; and 5 is called 7 ; and 6 is 6 : 7 in the morning , is called 5 ; and 8 is to be named 4 : And if you turn the Instrument , that the shadow of the Thred may fall on those hours , it will also cut on the degrees on the Plain , the true Hour-Arks required . As thus ; For 8 , it sheweth it not ; at 7 , it sheweth 77-00 ; at 6 , it sheweth 58 deg . 52 minuts ; at 5 , it cuts on the degrees on the Plain 45-38 ; at 4 in the morning , it cuts on the Plain 35-27 ; but the shadow falls then on Hour of 8 , on the Horizontal-Dial . Also note , That these numbers are not laid from the Substile , but from the Plains Perpendiculer , which in all Upright Plains is a Perpendiculer Line ; and in all other Plains , a Perpendiculer to the Horizontal-Line , drawn on the Plain . And thus proceed with any other ; the affections are best seen when you turn the Instrument the upper-part downwards . Use IV. To find the Requisites , and to draw the Hours on a far Declining Erect-Dial , S. W. 80. Set the Point to 80 , as before S. W. then the Thred and the Quadrant shall shew 38-2 for the Substile ; and 82-8 on the Horizontal , for the Inclination of Meridians ; and 6-12 , for the Stiles Elevation ; and the shadow of the Thred on the Horizontal-Dial , will shew you how close and inconvenient the Hours will be , if not helped by the former directions ; and in like manner will the North-East or West be , and likewise helped . Use V. To find the Requisites , and Hour-Arks , from the Perpendiculer of a Declining Inclining Plain , with its Affections . Set the Point at 12 to the Declination , and move the Plain by help of the Arch , or Circle of Brass , to the Inclination , and with the skrew make it fast and steady in that place . Then for the Substile , Apply the Quadrant to the Plain , and also Perpendiculer to the Axis , as the edge of the Quadrant being thick , will neatly shew ; then the Thred will shew on the degrees on the Plain , the distance of the Substile from the Perpendiculer , or the complement thereof from the Horizon ; which Point note with a spot of Ink ; for , when the shadow of the Thred falls on that spot , on the Horizontal Dial , it sheweth the Inclination of Meridians ; that is to say , on what hour and minut , the Cock of the Dial should stand right over . Also , The Quadrant , applied to the Plain and Thred , on the Substile-Line , sheweth the true Stiles Elevation above the Plain . And lastly , making the shadow of the Thred to fall on every Hour on the Horizontal-Dial , it shall at the same time shew how many degrees and minuts on the Plain , that Hour-line ought to be from the Perpendiculer , or from the Horizon ; and also which way , either to the Right or Left , East or West ; or from the Substile , or 12 ; if you will trouble your self to count it , from the place found out for the Substile , or 12. Example of a Plain Declining 30 S.E. and Inclining 20. The Substile , by applying the Square , you shall find to be 30 degrees on the left-hand of the Perpendiculer Westward , and the Inclination of Meridians 48-20 , the Stiles height 51-36 , and the Meridian on the right-hand of the Perpendiculer-line 11-30 Eastward ; and the shadow of the Thred playing on every hour and quarter , on the Horizontal-Dial , will shew on the Plain the quantity in degrees from the Perpendiculer-Line . Use VI. To find the Requisites in a North-east Reclining-Dial , and the Hour-Lines . Set the Instrument as before , and find the Substile , Stile , and Inclination of Meridians as before ; But note , as to the Affections , which way do thus ; Turn the Instrument the bottom upward , and as near as you can guess , turn the Plain to its scituation ; then you shall first see the Stile to look upward in the North-east Recliner , which before was downward in the South-east Incliner . Also , The Substile stands on the right-hand of the Perpendiculer , 30 degrees Westward ( for observe this alwayes , If a Plain declines Eastward , the Substile will stand Westward , and the contrary ) . Also note , That the Meridian-Line is to be drawn quite through the Center on the other-side ; because , when the Sun is in the Meridian above , it must needs cast the shadow of the Axis , or Stile , the contrary way downwards . Use VII . To find what are the most Hours , that the Sun can shine on any Plain , whatsoever . First , on all South Direct , or Declining Inclining-Dials , the mid-day-Meridian is proper to it , unless it incline above 75 degrees , and then it becomes useless in London Latitude ; then what hour soever you can make the Sun to shine on the Plain , and Horizontal-Dial both together , ( the Sun being at that hour above the Horizon ) by bending or turning the Instrument any way , ( when the Point at 12 is first set to the Declination ) that , and all those Hours are proper to that Plain , at one time of the year or other . Also note , That several Hours that serve for the South-plain , do , at some time of the year , belong to the North-plain also ; as by turning the Instrument about , you may plainly see , either by the Sun-shine , or by the Thred , and your Eye cutting the Hour-Lines and the Plain . Also observe , That if you would delineate a South Reclining Plain , you may bring the Plain toward the Thred , till it becomes a Polar-Plain . But if it Reclines below the Pole , then conceive it to become a North Reclining-Dial , and work as is before directed , and you shall obtain your desire ; for the Dials will be the same , the one as the other , as before was hinted at , in the Inclining-Plains . Use VIII . The Declination of any Plain given , to find what Reclination will make it a Polar-Dial , and the contrary . Set the North-point to the Declination , and bring the Plain to touch the Thred ; then on the Brass Circle is cut the Reclination required . Or contrary ; Set the Plain to the Reclination given , and then bring the Thred to the Plain , by turning the Horizontal-Dial , and the Point at 12 shall shew the Declination required , to make it Polar . In like manner you may discover a declining Equinoctial , but not so easily , when the Substile and Meridian are 90 degrees assunder ; the Substile being then alwayes in the hour of 6 , as by moving the Plain , if the Declination be given ; or by moving the Thred , if the Inclination be given , till the Square , touching the Thred , it shall shadow or bourn , just upon 6 on the Horizontal-Dial . Note also , That East and West Recliners , and Incliners , are discovered after the same manner ; So also Direct Recliners , and Incliners , as by moving the Plain to and fro , you shall see the plain and true reason , how the Stile is Elevated or Depressed , and how the Hour-lines are inlarged or contracted , according to the Elevation of the Stile . Also , In East and West-Dials , that the Stile hath no Elevation , but is parallel to the Plain ; and how the Meridian lieth in the Horizon , in East and West Recliners , and Incliners . Many more Uses might be insisted on , which I shall leave to the scruteny of the industrious Practitioner , in the Art of Shadows . CHAP. IX . How to remedy several Inconveniencies in the using of the Artificial Lines of Numbers , Sines and Tangents , as they are usually made . 1. IF the term required happen to be under one degree of Sines and Tangents , then the Line of Numbers will supply it , having due respect to the increase of the Radius , or Caracteristick . As thus ; As the sine of 90 , to the sine of 23-31 , the greatest Declination ; So is the sine of 1 deg . 10′ , the Suns distance from the Equinoctial , to 0-28 , the Declination which falls beyond the end of the Rule . Now to remedy this , the 1 deg . & 10′ , is 70 minuts ; therefore by the Numbers say , So is 70 minuts , the Suns distance from the Equinoctial , to 28 the Suns Declination on the Line of Numbers , observing to extend the same way , as from the first to the second term . 2. When you have occasion to use a sine above 90 degrees , then you must count the sine of 80 , for the sine of 100 ; and 70 , for 110 ; and 60 , for 120. So also , the distance from 90 to 60 in the Sines , is the Secant of 30 degrees ; and the distance from 90 to 50 , is the Secant of 40 ; or the Point beyond 90 , that represents the Secant of 40. 3. If the Extent be too large for your Compasses , as from 45 or 90 , to 3 or 4 degrees ; then instead of 90 or 45 , make use of a Point in the Sines or Tangents right against the middle 1 in the Line of Numbers , where you may have two Brass Center-pins , viz. in the Tangent of 5-43 , and the sine of 5-45 ; and the extent from thence backward or forward , shall reach in the Numbers , to the 4th proportional Number required . Example . As Tang. 45 , to 1-61 in the Numbers ; So is Tang. of 15-0 , to 0-43 in the Numbers . Instead of which , you may say , As the Tang. of 5-43 , to 1-61 on the Numbers ; so is the Tang. of 15 , to 0-43 on the Numbers diminishing a Radius ; for as Tang. 45 to 1-15 , a greater than that ; so is the Tang. of 15 , to a greater than 15 also , viz. 0-43 . Secondly , in Sines & Tangents , or Sines only , where there is another Caution to be observed , As sine 90 , to sine 10 ; so is sine 20 , to sine of 3-24 ⅓ . To work this with small Compasses on a large Line , do thus ; Note , that at 10 on the Line of Numbers , or Sine of 90 , or Tang. of 45 , is one compleat Radius ; but at the middle 1 , on the Line of Numbers , is a place , or Radius , less ; wherein the Logarithm Sines , the Characteristick is 8. Again , at the sine of 0-34 ½ , the Characteristick is 7 , ( and at 3 minuts it is 6 , ) which do note the several decreasings of the Radiusses ; Therefore set the distance from one Number given , to the next nearest place against 1 , or next Radius , as far from a greater or a less Radius , as your occasion serves , and note the place . As thus for Example . In this Operation , the extent from the Point at 5-45 on the Sines , to the sine of 10 degrees , I set the same way from the Point at 0-34 ½ and note the place , which will be at near 1 degree ; then the work is thus ; As the place against the middle 1 , instead of 90 , is to the place last found for 10 ; so is the sine of 20 , to sine of 3 deg . 24′ ⅓ , the 4th term required . But in those Lines of Numbers , Sines , and Tangents , where the Number is double , this is performed by working a-cross only . 4. When the last term in Tangents happens to be above 45 , then the remedy is two wayes , As thus ; As sine of 30 , to sine of 90 ; So is the Tang. of 30 , to Tang. 49-07 . which here happens beyond 45. Apply the end of the Rule , next 90 , close and even with any thing on which the Point of the Compasses may stay , till you take from thence to 45 , for that distance laid from 45 , shall reach to 49-07 , reading the Tangents as numbred beyond 45. Or more neatly thus ; The Compasses being set from the sine of 30 , to the sine of 90 ; set one Point in the Tangent of 45 , and turn the other on the Tangents , and keep it there fixed ; then remove the other from 45 , and close it to the third term , being here the Tangent of 30 ; then this last Extent laid from 45 , shall reach to 49-07 , the Tangent required . 5. When the first term is a Tangent above 45 , and the second under 45. Take the excess of the first Number above 45 , and set it the same way from the second Number ; then the Extent from the second Number to 45 , shall be the true distance between the first and second terms . Example . As the Tangent of 51-30 , to the Tangent of 30 ; So is the Tangent of 40 , to Tangent 21-04′ . For the Extent from 45 , to 51-30 on the Tangents , set the same way from 30 , does reach to about 24-30 ; then the Extent from thence to 45 , shall reach from 40 to 21-04 on the Tangents , the 4th Number required . Or , If it had been from a Tangent above 45 , to a sine , the same way would have remedied the defect . 6. When the third term exceeds 45 of Tangents , then thus ; Example . As sine 90 , to sine 30 ; So is the Tang. of 50 , to Tang. of 30-48 . The Compasses set from the first term sine 90 , to sine of 30 the second , a less ; then set one Point in the Tangent of 45 , and extend the other backwards in the Tangents , and note the place , keeping one Point there close , the other to 50 the third term ( being above 45 , by counting backwards ) Then , I say , that Extent laid from Tangent 45 , shall reach to Tangent 30-48 , the 4th proportional Tangent required . If the Proportion had been increasing , then there had been no trouble at all . Also note , That working a-cross , or changing the terms , is a good remedy also . As thus ; As sine 90 , to Tang. 50 ; which is properly increasing , for the Tang. of 50 being more than the sine of 90 , yet taken on the Rule from 90 to 40 , the complement thereof , as if it were decreasing ; So is sine 30 , to Tang. 30-48 , the contrary way : Therefore , As from the first term , properly counting to the second . 7. Lastly , When one or two Radiusses ( or Alterations of the Characteristick ) falls between the first and second term . As thus for Example . First , By the Line of Numbers only ; As 8000 is to 10 , So is 5000 to 6 ¼ , or 25. To work this properly , and naturally , the unite on the Numbers should be four times repeated , which is seldom more done than twice , as here : But this , and any other , by the Line of Numbers is not interrupted , having a due respect to the Number of Places . For to work this , the best way , is changing of terms thus ; As 8000 , to 5000 in the same Radius ; so is 10 , to 6-25 in the same Radius also . Or , without changing ; As 8000 , to the next 1 ; so is 5000 turning the Compasses the same way , to 6-25 . But to call it so , and not 625 , your reason must guide you more than precepts . But in using Sines and Tangents , the way in the third remedy will fit you . Example . As sine 90 , to 1 degree ( or under ) ; so is sine 30 degrees , to sine 30 minuts . This being too wide an Extent for the Compasses , the third Rule is a remedy for it ; which on a large Radius several times repeated , as in Mr. Oughtred's Circles of Proportion , is as easie as may be ; being sure to remember the number of Radiusses between the first and second term , that you may have so many between the third and fourth term also . Much more might have been said as to this ; but this Observation being alwayes kept , That as the Extent from the first term to the second , is either increasing or decreasing ; So alwayes must the Extent be from the third to the fourth , increasing or decreasing , in like manner , when you use Sines and Tangents ; And Numbers also , except , as before , in a few particuler Rules ; then you will be truly resolved . The end of the Book of Dyalling . AN APPENDIX To the Use of the Trianguler-Quadrant IN NAVIGATION . Where it performs the Uses of the Davis-Quadrant , the Cross-Staff , Bow , Sinical-Quadrant , and Sector , with as much ease and exactness as any , or all of them , will do in Observation or Operation , Naturally or Artificially . Being first thus Contrived , and made by Iohn Brown , dwelling at the sign of the Sphear and Sun-Dial in the Minories , near to Aldgate , London . London , Printed by Iohn Darby , for Iohn Wingfield , and are to be sold at his house in Crutched Fryers ; and by Iohn Brown at the Sphear and Sun-Dial in the Minories ; and by Iohn Selle at the Hermitage-stairs in Wapping . 1671. CHAP. I. The Description thereof FOR SEA-USES . THe Description of the Instrument , is largely and plainly set down in the First Part , and First Chapter . But , in regard that is the general Description of all the Lines that can conveniently be put on , and those necessary for this use being far less , I shall repeat the Description again , as far as concerns the use thereof for Sea-Observations . 1. First for length , it ought to be two foot-long at least , when shut together , and not above 3 foot at any time for Sea-uses ; ( but for Land-uses it may be 6 , 8 , 10 , or 12 foot in length , to find Altitudes or distances to Seconds of a degree certainly ) . 2. The Form of it is the same , as before , viz. an opening Joynt of about an inch and quarter , or half quarter broad each Leg ; and 6 tenth parts of an inch in thickness , with a Loose-piece of the same length , breadth , and thickness , to make it an Equilateral-Triangle . As the Figure sheweth . 3. The Lines necessary for Sea-uses are , first , the 180 degrees upon the moving-Leg and Loose-piece , numbred as before is shewed . Also , 60 degrees on the innermost-edge of the Loose-piece . The Kalendar of Months and Dayes , and degrees of the Suns Place , and Right Ascention , on the moveable-Leg . For the speedy and ready finding the Suns place , and declination , which you may do to a minut at all times , by help of the Rectifying Table , and Astronomical Cautions of Time and Longitude . Also , on the Head-leg , is the general Scale of Sines and Lines , to the great and lesser Radius , as in the Figure . And thus much will serve both for Observation and Operation , as in the following Discourse will fully appear . 4. To this Instrument doth chiefly belong the Sights for the Observations at Sea , where the Horizon is made use of in the taking the Sun or Stars Altitude . And to this Instrument belongs the Index and Square , that makes it a most compleat Sinical-Quadrant , for the plain and easie resolving of all plain Triangles . Also , a weighty Plummet and Thred , and a pair of large Wood or Brass Compasses for Operation . Thus much for Description , being all put on one side only , unless you shall be pleased to add the Artificial Numbers , Sines , and Tangents on the outer-edge , and a Meridian-line , and his Scale on the inner-edge ; and Natural Sines , and Natural Versed-Sines on the Sector-side : But these as you please . CHAP. II. The use of the Trianguler-Quadrant in Observation . THat the Discourse may be plain , and brief , and general ; there are 10 terms to be named and described , before I come to the Vses and Examples , which are as followeth . 1. First , the Head-leg of the Instrument in which the Brass-Rivit is fixed , and about which the other Leg turns , as AB , in the Figure ; on which Leg , the general Scale of Sines and Lines are usually set . 2. The moveable-Leg , on which the Months and Dayes be , as in the Figure , noted by BD ; which Leg turns about the Head-Leg . 3. The Loose-piece that is joyned to the Head , and moving-Leg , by two Tennons at each end thereof , noted by DA in the Figure . 4. The Head-Center , or Center-pin on the round-part of the Head-leg , being Center to the 60 degrees on the in-side of the Loose-piece ; which Point is known by B , in the Figure . 5. The Leg-Center , being near the end of the Head-leg , which is the Center to the degrees on the moving-Leg , and out-side of the Loose-piece , being in all 180 degrees ; and noted in the Figure by the Letter C. 6. The great Radius , or greater Line of Sines , issuing from the Leg-Center toward the Head , having the Tangents on the moveable-Leg to the same Radius ; and the measure from the Leg-Center to the Tangent on the moving-Leg , a Secant to the same Radius ; as CE in the Figure . 7. The little Radius that issues from the Leg-Center toward the end , having the Tangents , on the out-side of the Loose-piece to the same Radius , and the measure from the Center to those Tangents for Secants to the same Radius ; as CF. 8. The Turning Sight alwayes to be skrewed to the Head , or Leg-Center , known by his shape and skrew-hole , as 9. The sliding Horizon-sight to slide on the moving-Leg and Loose-piece , noted with its bigness and hole to look through , as 10. The shadow Sight , and 2 others , to pin the Instrument together , which you may call the Object-Sights , alwayes fixed in the two holes at the ends of the moving-Leg , and the Head-leg ; and the shadow-Sight is to set to and fro to any place required ; noted in the Figure with 〈◊〉 and the other two with 〈◊〉 And Thus you have their Name and Description at large , which in brief take thus for easie remembring . 1. The Head-Leg . 2. The Moveable-Leg . 3. The Loose-Piece . 4. The Head-Center . 5. The Leg-Center . 6. The great Radius . 7. The less Radius . 8. The turning-Sight . 9. The Horizon sliding-Sight . 10. The shadow-Sight , and the two Objest-Sights ; the open-part in one is next to , and the other remoter from the Rule , to answer to the upper or lower-hole in the turning-Sight , according as you please to use them in Observation . Thus much for the Terms , the Vses follow . Use I. To find the Suns , or a Stars Altitude , by a forward Observation , as by a Fore-staff . Skrew the turning-Sight to the Head-Center , and put the object-Sight into the hole at the end of the Head-leg , and put the sliding Horizon-sight on the in-side of the Loose-piece ; Then setting the turning-sight to your eye , and holding the Loose-piece in your right-hand , and the moveable-Leg toward your body , then with your Thumb on the right-hand , thrust upwards , or pull downwards the Horizon-sight , till you see the Sun through the Object-sight , and the Horizon through the Horizon-sight ; then the degrees cut by the Line on the middle of the Horizon-sight , shall shew the true Altitude required . Also observe , That if you like to use the upper or lower-edge of the Horizon-sight , instead of the small bar a-cross the open-hole , after the manner of the ends of a Fore-staff , that then the degrees and minuts cut by the edge of the Brass , is the Altitude required , to be counted as it is figured from the Object-sight , toward the Horizon-sight ; the degrees between them being the Angle required . Note also , That if the Altitude of the Sun , or Star , be above 30 degrees , you will find it a hard matter , to behold the Horizon and Sun with a bare roling the ball of the eye only , and a stirring of the head , will easily cause a stirring of the hand , which will spoil the exactness of Observation , unless the Instrument shall be fixed to a Ball-socket and Three-legged-staff , which is not usual at Sea. Therefore to remedy this , you may observe with the open oval-hole in the turning-sight set to the eye , or taking the turning-sight quite away ; Observe just as you do with a Fore-staffe , setting the round part of the head , to the hollow-part beside your eye , so as the Head-Center-pin may be as near the very sight of your eye as possibly as you can ; which Center is the Center to the degrees now used in a forward way of Observation . Or , rather use this way when the Weather will suffer , by a Thred and Plummet , which I shall add as a second Use. Use II. To observe the Sun or a Stars Altitude , by a forward Observation , using the Thred and Plummet . Skrew the turning-sight to the Head-Center , as before , and put the two Object-sights into the two holes at the two ends of the Rule ; and on the Leg-Center-pin hang the Thred with a weighty Plummet of two pound , or above a pound at least . Then hold up the Trianguler-Quadrant , setting the small-hole on the turning-sight close to your eye ; and if the Sun , or Star , be under 25 degrees high , then look to the Sun or Star through the turning-sight , and that object-sight , which stands in the end of the moveable-Leg , letting the Thred and Plummet play between your Thumb and Fore-finger , as a Brick-layers Plummet in his Plum-Rule doth in a bendid hole , that you may keep it in order whilst you look at the Sun or Star , and the weighty Plummet will pull the Thred streight , and let you know by feeling which way it is playing , till it playeth evenly and truly , whilst you have the Object precisely in the midst thereof , whether it be Sun , Moon , or any Star , or other Object , whose Altitude you would observe ; Then , I say , when the Plummet playes well , and you behold the Object right , bend back the Quadrant , and see what the Thred cuts on the degrees on the moveable-Leg , which shall be the true Altitude required ; And in my opinion , must needs be more exact than any other way of a forward Observation , because you are not troubled to mind the Horizon and Sun both at at once . An Objection may be , The boisterous Winds , and the rouling of the Ship , will hinder such an Observation . Answ. So it will any other way , though happily not so much . Again , I answer , One Object is better and more certainly seen , than two at any time together ; and though the Wind blow hard , if you can stand to observe at all , the heavy Plummet will be sure to draw the Thred Perpendiculer ; and for ought I know , you may come as near this way as any other ; however this , at most times , may confirm and prove the other , and may be useful in Rivers , and Harbours , and misty-Dayes , when you may see the Sun well enough , but not the Horizon at all . Use III. To find the Suns Altitude by a backward Observation , as with a Back-staff , or Davis-Quadrant . Skrew the turning-sight to the Leg-Center ( or Center to the degrees on the moveable-Leg ) ; and set the object-sight to the long stroke by 00-60 on the out-side of the Loose-piece , and put the sliding Horizon-sight on the out-side of the moveable-Leg ; then hold the Object-sight upwards , and the small-hole in the piece turning on the ●dge ( or to the small-hole in the middle ) of the Horizon-sight ( which you please ) close to your eye ; and looking through that hole , and the middle-hole of the turning-sight , to the true Horizon , turning your self about , and lifting up , or pressing down the Horizon-sight , close to the moveable-Leg , till the shadow of the upper-edge of the shadow-sight , being next to the Sun , fall at the same time just on the middle of the turning-sight ; Then , I say , the edge or middle of the Horizon-sight , that you looked through , shall cut the true Altitude of the Sun required . Being the same way as you do observe with a Davis-Quadrant , or Back-staff . Use IV. To find the Suns Distance from the Zenith , by the Trianguler-Quadrant . Skrew the turning-sight to the Leg-Center , and put the Object-sight , whose oval-hole is remotest from the Quadrant , in the hole in the end of the Head-Leg , or rather in a hole on the general Scale , between the turning-sight , and the Sun ; and put the Horizon-sight on the out-side of the moveable-Leg ; then hold the turning-sight toward the Sun , and the small-hole in the edge of the Horizon-sight to your eye ; then look through that hole and the turning-sight , till you see the shadow , the Object-sight , to fall just on the turning-sight , or the shadow of the turning-sight to fall just on the object-sight , which is all one , though the first be more easie , because you shall see the Horizon through the turning-sight , and that , both at once ; Then , I say , the degrees cut by the Horizon-sight , shall be the Suns distance from the Zenith required ; Being the very same work , and done in the same manner , and producing the same Answer , viz. the Suns distance from the Zenith , that the Davis-Quadrant doth . Note , That this way you may observe very conveniently , till the Sun be 20 degrees distance from the Zenith ; and by the adding of a 60 Arch , as in Davis Quadrant , or to 45 will be enough , it will do as well as any Davis Quadrant , being then the same thing . But I conceive , the complement of the Altitude being the same , will do as well ; which Altitude is better found by this Instrument , than the distance from the Zenith by a Davis Quadrant is , as in the next Use will be seen . Use V. To find the Suns Altitude when near the Zenith , or above 90 degrees above some part of the Horizon . In small Latitudes , or in places near the Equinoctial , or under it ; the Sun will be found to be in , or near the Zenith : and if you count from some part of the Horizon , above 90 degrees distant from it ; then instead of setting the sliding Object-sight , to the long stroke at 00 on the Loose-piece , you must set it 30 degrees more towards the Head-leg ; then observe , as you did before , and whatsoever the Horizon-sight cuts , you must add 30 degrees more to it , and the sum shall be the true Altitude required . Example . Suppose that in the Latitude of 10 deg . North , on the 10th of Iune , when the Suns Declination is 23 degrees and 31 min. Northward ; Suppose that at noon , I observe the Suns Meridian Altitude , skrewing the Turning-sight to the Leg-Center , and setting the Object-sight to the 30 degrees on the Loose-piece , near the end of the Head-leg , and the Horizon-sight on the movable-Leg ; then hold up the Quadrant , with the shadow-sight toward the Sun , and the small-hole in the Horizon-sight toward your eye , and look to the Horizon through that , and the turning-sight , the shadow of the right-edge of the shadow-sight , that cuts the degree of 30 , at the same time falling on the middle of the turning-sight , you shall find the Horizon-sight to cut on 46-29 minuts ; to which if you add 30 , the degrees , the shadow-sight is set forwards , it makes up 76-29 , the Suns true Altitude on that day in that Latitude ; 76-29 the Meridian Altitude , and 23-31 the Declination , added together , make 100 deg . 00 ; from which taking 90 , there remains 10 , the Latitude of the place . 1. In this Observation , first you may note this , That if you had stood with your back toward the South , you would have had 103 degrees and 31 minuts , for the sliding Horizon-sight would have stayed at 73 degrees 30 ; to which if you add 30 , it makes 103-31 ; which a Davis Quadrant will not do . 2. In the holding it , you may lean the head of the Rule to your breast , and command it the better , as to steady holding . 3. You may turn the Turning-sight about , to any convenient Angle , to make it fit to look through to the Horizon , and also to receive the shadow of the shadow-sight . If the brightness of the Sun offend the eyes , you may easily apply a red or a blue Glass , to darken the Sun beams , and the Sights may be painted white , to make a shadow be seen better . Use VI. To find the Latitude at Sea , by a forward Meridian Observation of the Altitude , according to Mr Gunter's Bow. Skrew the Turning-sight to the Leg-Center , and set the shadow-sight to the Suns-Declination , and the Horizon-sight to the moving Leg ( or Loose-piece ) , and the Turning-sight to your eye ; then let the shadow-sight cut the Horizon , and the Horizon-sight the Sun , moving it higher or lower till it fits ; then whatsoever the sight sheweth , adding 30 degrees to it , is the Latitude of that place required . Example . Suppose on the 10th of March , when the Declination is only 10′ to the Northward , as in the first after Leap-year it is ; then set the edge , or stroke on the middle of the shadow-sight to 10′ of Declination toward the Head , and the Horizon-sight , on the same Leg toward the end , and slide only the Horizon-sight till it cuts the Sun , and the other the Horizon ; then suppose ●t shall stay at 21-30 : then if you count the degrees between the two Sights , it will amount to the Suns Meridian Altitude ; but if you add 30 degrees to what the Sight cuts , it shall give the Latitude of the place where the Observation was made for 21 and 30 , to which if you add 30 , it makes 51-30 , the Latitude of London , the place where the Observation was made . Note here , That in small Latitudes the Sun will be very high , in Summer time especially , and then the sliding-sight , must be set on the loose peice . As thus for Example . Suppose on the 10th of May 1670 , when the Declination is 20-7 in the Latitude of 30 Degrees , I observe at a Meridian Altitude , I shall find the sliding-sight to stay at 00. on the loose peice ; then it is apparent that 30 added to 0 , makes but 30 degrees for the Latitude ●equired . But if the sliding-sight shall happen to pass beyond 00 on the loose peice , then whatsoever it is you must take it out of 30 , and the remainder is the Latitude required . Example . Suppose on the 11th of Iune 1670 you were in the Latitude of 10 degres to the Northward , and standing with your back to the North , as you must needs do in all forward Observations in more Northern Latitudes , you shall find the sight to pass just 20 deg . beyond 00 on the Loose peice ; therefore 20 taken from 30 the residue is 10 , the Latitude required . Again . Suppose that in the same place you had obsereved on the 11th of December , when the Sun is most Southwards , if you set the one sight to 23-31 Southwards against the 11th of December ; then if you observe forwards with your face toward the South , as before , you shall find the moving-Sight to stay at 20 degrees beyond 00 on the Loose-piece ; then , I say , 20 taken from 30 , rests 10 , the Latitude required , because the sight passed beyond 30 on the Loose-piece . Lastly , if the Moving-sight shall happen to pass above 30 degrees beyond 00 on the Loose-piece , when the other Sight is set to the Suns Declination , and you observe with your face toward the South ( part of the Meridian ) ; then , I say , the Latitude is Southwards as many degrees as the Moving-sight stands beyond 30 , on the Loose-piece toward the Head-leg . So that the general Rule is alwayes , in North Latitudes ( observing the Suns Meridian Altitude , to find the Latitude , by a forward Observation , according to Mr. Gunters Bow ) , your face must be toward the South ; Although that thereby in some Latitudes , the Altitude may seem to be ( as indeed it is above the South-part of the Horizon ) above 90 degrees . Then , If the sliding-sight stay any where on the Moving-leg , or Loose-piece , short of 00 , add it alwayes to 30 , and the sum shall be the true Latitude North ; if it pass beyond 00 , then so much as it doth , take out of 30 , and the remainder is the Latitude North ; but if it shall stay just at 30 on the Loose-piece , then the Latitude is 00 ; but if it pass beyond 30 , then so much as it is , the Latitude is Southward . The same Rule serves , if you were in South Latitude , then you must in forward Observations , to find the Latitude , as with a Gunters Bow , stand with your face to the North , and in setting the fixed-sight to the Declination , you must count South Declination toward the Head ; because those that have Southern-Latitude , have their longest dayes , when those that live in Northern-Latitude have their shortest dayes . The same Rule serves for the Stars also , for being in North-Latitudes ; and observing a Latitude forwards , have your face alwayes toward the South , and set one Sight to the Declination , counting the Stars Northern or Southern Declination ▪ the same way as the Suns , ( and the contrary in South-Latitude ) ; then holding your eye close to the great-hole of the Turning-sight , slide the Moving-sight till its middle-bar or edge ( as is most convenient ) cuts the Star , and the other the Horizon ; then whatsoever the edge of the Moving-sight cuts short of 0 , added to 30 , or beyond 00 , taken out of 30 , shall be the Latitude required . Example . Suppose the middle-Star of Orions-Girdle , whose Declination is 1-28 South , being in the Meridian , I set one Sight to 1-28 of South-declination ; and slipping the other-sight till it cuts the Star , the fixed-sight being set to the Horizon , you shall find it stay in the Latitude of 51-30 , at 21-30 on the Moving-leg , which added to 30 , makes 51-30 . Note , That if the corner of the Instrument be inconvenient for the sight to slide on , as for about 5 degrees it will , then you may remedy it by slipping the Sight set to the Declination 10 degrees more , any way that is convenient , increasing or decreasing ; But then note , That instead of adding even 30 , to what the Moving-sight stayeth at , you must add 10 degrees more , viz. 40 , when you slip it towards the end of the Moving-leg ; or 10 degrees less , viz. 20 degrees , when you slip it 10 degrees more toward the Head , as is easie to conceive of . Use VII . To find the Latitude at Sea , by a backward Meridian Observation , according to the way of Mr. Gunter's Bow. Skrew the Turning-sight to the Leg-Center , and set the sliding-Horizon-sight to the Suns Declination ( the middle or edge of it , as you can best like of ) and the Shadow-sight on the Loose-piece , or Moving-leg , with your face alwayes to the North , in North Latitudes ; or supposing your self to be so , though it may be you are not . Then looking through the hole in the Horizon-sight ( standing at the Declination ) and the Turning-sight to the Horizon , with your hand gently slide the shadow-sight till the shadow fall just on the middle of the Turning-sight , as you do in observing the Altitude with a Davis Quadrant ; then , I say , whatsoever the shadow-sight shall stay at under 0 , add to 30 ; or over 0 , take out of 30 , and the sum or remainder , shall be the Latitude North : but if it happen to stay at just 30 , the Latitude is 00 ; if beyond , it is so much to the Southwards : This is only the converse of the former , and needs no Example , but a few words to demonstrate it ; which may be thus , In the way of an Example . Suppose that on the 11th of December , in sayling toward the East-Indies , about the Isle of St. Matthews , supposing our Ship to be in North-Latitude , I set the Horizon-sight to 23-31 , South Declination ; and the Shadow-sight on the Loose-piece , then standing with my face to the North , as another then would do , as at other times , and looking through the Horizon , and Turning-sight to the North-part of the Horizon , I find the shadow-sight when it playes well over the Turning-sight , to stay at 33 degrees on the Loose-piece . Then , Consider that the distance between the wo Sights , is the Altitude of the Sun above the South-part of the Horizon ; which if you do count on the Trianguler-Quadrant , you will find to be 36-29 , and 33 , which put together , make 69-29 , for the Suns Meridian Altitude ; to which if you add 23-31 , his declination , it makes 93-00 the distance of the North-Pole and Zenith , or 3 degrees of South Latitude ; for had you been just under the Equinoctial , the Altitude would be 66-30 ; or had you been more Northward , it would have been less ; therefore by considering , you may soon see the reason of the Operation . Also , If the Shadow-sight be too near the corner , or too far from the Turning-sight to cast a clear shadow ; then , set the Horizon-sight that stands at the Declination 10 , 20 , or 30 degrees more toward the end of the Moving-leg , and you shall see the inconvenience removed ; but then you must take 10 , 20 , or 30 degrees less than the shadow-sight sheweth , for the reason abovesaid . The reason why even 30 is added , is because that 0 degrees of Declination , stands at 60 on the Moving-leg , instead of 90 , or 00. Note , If you had rather move the lower-sight than the upper , then count like Latitudes and Declinations from 00 on the Loose-piece toward the Head-leg , and unlike the contrary , and then set the shadow-sight fixed there ; then observing , as in a back-Observation , the Horizon-sight shall shew the complement of the Latitude required , without any adding of 30. Thus you see , That the Trianguler-Quadrant , containing 180 degrees in a Triangle , brings the shadow-sight near the Center , and with one manner of figuring , gives the Suns Altitude above the Horizon , backwards or forwards , and his distance form the Zenith , and the Latitude of the place South or North , or North or South , backwards or forwards , by the Sun or Stars , by one side only , as conveniently and with fewer Cautions , and as exactly , if well used , as any other Instrument whatsoever ; So that by this time you see it is a Fore-staff , Quadrant , and Bow. The other Uses follow . Use VIII . To find the Latitude by a Meridian Observation , by the Thred and Plummet , by the Sun or Stars . This way of Observing without a Horizon , must be done by an Astrolabe , which is a Plummet it self , or else with a Plummet fitted to another Instrument , and at some times may do better service than the Horizon , and for an Altitude barely , is shewed already . For the Latitude thus ; Count the Declination , which is the same with the Latitude , from 00 on the Loo●epiece toward the Moveable-leg ; and contrary Declinations , both of Sun or Stars , count the other-way toward the Head-leg , and thereunto set the edge of the Horizon-sight , that hath the small-hole on it . Then let the Sun-beams shine through the small-hole on the Turning-sight , to the small-hole on the Horizon-sight , the Thred and Plummet duly playing , shall shew the Latitude of the place required . But if you look at a Star , having the same Declination , then set your eye to the Horizon-sight , and behold the Star through the Turning-sight , and the Thred shall fall on the Latitude required , when you look toward the South , being in Northern Latitudes . So also , When you turn your face toward the North , in observing those Stars , it is best done when they come to the Meridian below the Pole ; but for their coming to the Meridian above the Pole , then their Declination is increased by the quantity of their distance from the Pole , or the complement of their Declination . As thus ; The Declination of the Pole-Star , when in the Meridian below the Pole , is 87-20 from the Equinoctial ; but when the same Star is on the Meridian above the Pole , then it is 92 deg . 40′ distant from the same Northern part of the Equinoctial . So that if you make 60 on the moveable-leg , to represent the North-pole ; then you may count or observe any Star that is 25 degrees distant from the Pole , both above or below the Pole ; then adding 30 degrees to what the Thred falls on , shall be the true Latitudes complement required ; because you have removed the Pole from 90 to 60 , 30 degrees backward . Example . The declination of the uppermost Star in the great Bears back , is 63-45 ; that is , 26-15 below the Pole ; or , 25 degrees 17 minuts above the Horizon , when on the Meridian below the Pole ; but the same Star , when on the South-part of the Meridian , is 77-47 above the Horizon , or 26-15 above the Pole. Therefore , The Star being below the Pole , you may set the hole in the middle of the Horizon-sight , to the Declination , counting 90 the Pole , and looking up to the Star , as usually , the Plummet will fall on 38-28 , the Latitudes complement required . Again , The same Star being on the South-part of the Meridian , above the Pole , I count 60 on the Moving-leg for the Pole , and 26-15 beyond that Pole further , viz. to 86-15 , which is as far as you can well go , counting 60 the Pole ; then observing , as you did before , you shall find the Thred to play on 08-28 , the Latitudes complement required , for the distance between 08-28 , and 86-15 is 77-47 , adding 30 degrees , because of 60 instead of 90 , for the Pole-point . Note , That the Thred playing near the corner , may prove somewhat troublesome to observe , without help of another person ; but if you will be exact in this or any other Observation , a Staff and a Ball-socket , should be applied to this , as well as to other Instruments , to stand steady and sure in the time of Observation . These wayes are ready and easie , without taking notice of those Regulations and Cautions , which are to be observed in finding the Altitude , barely ▪ as in the Seamans Kalender , and Mr. Wrights Errors in Navigation , is plainly seen . But if you know them all , and had rather use those Rules in those Books ; then , I say , a Thred and Plummet by this Instrument , will do as conveniently as any other , or the three Sights and Horizon , as before is shewed , to find the Altitude . CHAP. III. To Rectifie the Table of the Suns Declination . THus much as for the way of Observation ; now , that your Operation may be true also , it is necessary that you have a Table of the Suns Declination , for the first , second , and third year , after the Leap-year . But in consideration , that the second after the Leap-year , is a mean between the other three ; I have made a Table for that , and the Months on the Trianguler-Quadrant are agreeable thereunto ; and for the first , third , and Leap-year , have added a Rectifying Table to bring it to a minut at least to the real truth , wherein I have followed the Suns place , according to Mr. Streets Table of the Suns place , for 1666. In which Table , you have degrees and minuts ; and a prick after , notes a quarter of a minut ; and two pricks , half a minut ; and three pricks , three quarters of a minut more . Now , by the Rule , you may count to a minut , and the Rectifying Table tells you how many minuts more you must add to , or substract from the degrees and minuts the Table or Rule shall shew it is , in the second year . A Table of the Suns Declination every day at Noon for London in the year 1666 , the second year after the Leap-year , according to Mr. Street's Tables of Longitude . Calculated by Iohn Brown , 1668. Month Dayes . Ianu. Febr. March April . May. Iune . D.M. D.M. D.M. D.M. D.M. D.M. 1 21 45 ... 13 50. . 03 29. 08 31. 18 02 23 11. 2 21 36 13 30. . 03 05. . 08 53. 18 17. 23 14. . 3 21 25. . 13 10. 02 42 09 15. 18 32 23 18. 4 21 14 ... 12 49 ... 02 18. 09 36 ... 19 46 ... 23 21 5 21 03. . 12 29 01 54 ... 09 58 19 01 23 23 ... 6 20 51 12 08. 01 31 10 19. 19 14 ... 23 26 7 20 40 11 47. 01 07 10 40 ... 19 28. 23 27 ... 8 20 27. . 11 25 ... 00 43. 11 01 19 41. 23 29. 9 20 15 11 04. S. 19 ... 11 22 19 54. . 23 30. 10 20 01 ... 10 43 N. 04. . 11 42 ... 20 07 23 30 ... 11 19 48. 10 21 00 27 ... 12 03 20 19 23 31 12 19 34. . 09 59. 00 51. . 12 23 20 31 23 30. . 13 19 20. 09 37 01 15 12 43. 20 42. . 23 30 14 19 06 01 15 01 38. . 13 03 20 54 23 29 15 18 51 08 52 02 02. . 13 22. 21 04. 23 37. . 16 18 35 ... 08 29 ... 02 26 13 42 21 15. 23 15. . 17 18 20. 08 07. . 04 49. 14 01 21 25. 23 23 18 18 04 07 44 ... 03 13 14 20 21 35 23 20. . 19 17 48 07 22 03 36 14 38. 21 44. 23 17. 20 17 31. 06 59 03 59. . 14 57 21 53. 23 14 21 17 14. . 06 36 04 22. . 15 15 22 01 ... 23 10 22 16 57. 06 13 04 45 ... 15 33 22 10. 23 05. . 23 16 40 05 50 05 0● ... 15 50 ... 22 17 ... 23 01 24 16 22. 05 26. . 05 32 16 08 22 25. 22 55 ... 25 16 03 ... 05 03. 05 34. . 16 25. 22 32. . 22 50. 26 15 45 04 39 ... 06 17. 16 42 22 39 22 44. 27 15 27 04 16. 06 40 16 58. . 22 45. . 22 37 ... 28 15 08 03 52 ... 07 02. . 17 14 ... 22 51. . 22 31 29 14 49   07 25 17 30 ... 22 57 22 23 ... 30 14 29. .   07 47 17 47 23 02 22 16. 31 14 10.   08 ●9 .   23 06. .   A Table of the Suns Declination every day at Noon , &c. Month Dayes . Iuly . Augu. Septem Octob. Novem Decem D.M. D.M. D.M. D.M. D.M. D.M. 1 22 09 15 14. 04 26. . 07 12. . 17 37 ... 23 07. 2 22 00. 14 56 04 03 ... 07 35. 17 35 ... 23 12 3 21 51. 14 37. . 03 40 ... 07 58 ... 18 10. 23 16 4 21 42. . 14 19. 03 17. 08 20. 18 25. . 23 19. . 5 21 33 14 00 ... 02 54. 08 42. . 18 41. 23 22. 6 21 23. 13 41. . 02 31 09 04 ... 18 56. 23 25 7 21 13. 13 22. 02 08. . 09 27 19 10 ... 23 27. 8 21 02. . 13 03 01 44. 09 49 19 25. 23 28 ... 9 20 52 12 43. 01 20 ... 10 11 19 39. 23 30 10 20 41 12 23 00 57. 10 32. . 19 53 23 30 ... 11 20 29 12 02 ... N 34 10 53 ... 20 06. 23 31 12 20 17. 11 43 N 10. . 11 15. 20 19 23 30. . 13 20 05 11 21. . S 13. 11 36. 20 31 ... 23 30 14 19 52 ... 11 02 00 36 ... 11 57 20 44 23 29 15 19 39. . 10 41. . 01 00 ... 12 18. 20 55 ... 23 27 16 19 26 ... 10 20. . 01 24 12 38 ... 21 07 23 25. 17 19 13. 09 59. 01 47. . 12 59. 21 18. 23 22. 18 18 59. 09 38 02 10 ... 13 20 21 28 ... 23 19. . 19 18 44 ... 09 16 ... 02 34. . 13 39 ... 21 38 ... 23 16 20 18 30. 08 55. 02 58 13 59. 21 48. ▪ 23 11 ... 21 18 15 ... 08 33. . 03 21. 14 19 21 58 23 07. . 22 18 00. . 08 11. . 03 44. . 14 38 22 07 23 02. . 23 17 45. 07 49 ... 04 08 14 57. . 22 15. . 22 57 24 17 29. 07 27. . 04 31. . 15 16 22 23. . 22 51. . 25 17 31. . 07 05. . 04 54. . 15 35 22 31 22 44 ... 26 16 57 06 43 05 18. 15 53. 22 38. 22 38 27 16 40. . 06 20. . 05 41 16 11. 22 45 22 30. . 28 16 24 05 57. . 06 04 16 29. 22 51. . 22 23 29 16 06. . 05 35. 06 27 16 46 ... 22 57. 22 15 30 15 49. 05 12. . 06 49. . 17 03 ... 23 02 ... 22 06. 31 15 32     17 20.   21 57. A Table of the Suns Declination for every 5th and 10th Day of every Month of the Four years ; Calculated from Mr. Street's Tables of the Suns place , made for the years 1665 , 1666 , 1667 , and 1668 ; the nearest extant . M. D 1. 1665 2. 1666 3. 1667 L.Y. 68 Ianuary . 05 21 01 21 03. . 21 06. 21 09. 10 19 58. . 20 01 ... 20 05 20 08. . 15 18 47. 18 51 18 54. . 18 58. 20 17 27. 17 31. 17 35 17 39. . 25 15 59 ... 16 03 ... 16 08 16 12 ... 30 14 25 14 29 ... 14 34. . 14 39 February . 05 12 24 12 29 12 34 12 39 10 10 39. . 10 43 10 48 10 53. . 15 08 47 08 52. . 08 57 ... 09 03. 20 06 53. . 06 59 07 04 ... 07 10. 25 04 57. . 05 03. 05 08 ... 05 14. . 28 03 47. S 03 52 ... 03 58. . 04 04. March. 05 01 48 ... 01 54 ... 02 00. 01 42. S 10 00 10 N 00 04. N 00 01 ... S 00 16. N 15 02 08 02 02. . 01 56. . N 02 14. . 20 04 05 03 59. . 03 54 .. 04 11. . 25 06 00. 05 54. . 05 49. 06 06. . 30 07 52 ... 07 47 07 42. 07 58. April . 05 10 03. 09 58 09 53 10 09 10 11 47 ... 11 42 ... 11 37 ... 11 35. 15 13 27. 13 22. 13 17. . 13 32. 20 15 01. . 14 57 15 52. . 15 06. . 25 16 29. 16 29. 16 21. 16 34 30 17 50. . 17 47 17 42 ... 17 54. . May. 05 19 04. 19 01 18 57. . 19 08 10 20 10 20 07 20 04 20 13. . 15 21 07 21 04. 21 02 21 10 20 21 55. . 21 53. 21 51 ... 21 57 ... 25 22 34 22 32. . 22 30 ... 22 36 30 23 03 23 02 23 01 23 04. . Iune . 05 23 24 23 23 ... 23 23 23 25. . 10 23 30 ... 23 30 ... 23 30. . 23 31 15 23 27 23 27. . 23 28 23 26 ... 20 23 13 23 14 23 15 ... 23 12 25 22 48 ... 22 50. 22 51 ... 22 47 30 22 19 ... 22 16. 22 18 22 12. A Table of the Suns Declination , &c. M. D 1. 1665 2. 1666 3. 1667 L.Y. 68. Iuly . 05 21 31 21 33 21 35. 21 28 10 20 38 20 41 20 43. 20 35 15 19 36 19 39. . 19 42. . 19 32 ... 20 18 25. . 18 30. 18 34 18 23 25 17 09 ... 17 13. . 17 17. . 17 05 30 15 45. . 15 49. 15 53. . 15 40. . August . 05 13 55. . 14 00 ... 14 05 13 50 10 12 18 ... 12 23 12 28 12 13 15 10 36. . 10 41. . 10 46. . 10 31 20 08 50 08 55. 09 00. . 08 44 25 07 00 07 05. . 07 10. . 06 53 ... 30 05 07 N 05 12. . 05 18 05 00 ... September . 05 02 48. . 02 54. 02 59 ... 02 42. 10 00 51. . N 00 57. N 01 03 N 00 45. N 15 01 07 S 01 00 ... S 00 54 S 01 12. S 20 03 05. . 02 58 02 52. 03 09 ... 25 05 00. 04 54. . 04 49 05 06. . 30 06 55. 06 49 06 44. 07 01. October . 05 08 48 08 42. . 08 38. . 08 54 10 10 37 ... 10 32. . 10 27. 10 43. . 15 12 23 12 18. 12 13. 12 28 ... 20 14 04 13 59. 13 54. . 14 09. 25 15 39. 15 35 15 30. . 15 44 30 16 08. 17 03 ... 16 59 ... 17 12 ... Novemb. 05 18 45 18 41. 18 37. 18 49. 10 19 56 19 53 19 49. . 19 59 ... 15 20 58. 20 55 ... 20 53 21 01. . 20 21 51 ... 21 48. . 21 46. . 21 53. 25 22 32 ... 22 31 22 29. . 22 35 30 23 04 23 02 ... 22 01. . 23 05 ... December . 05 23 20. 23 22. 23 21 23 23 ... 10 23 30 ... 23 30 ... 23 30 ... 23 31 15 23 27 23 27 23 27. . 23 26 20 23 11 23 11 ... 23 13 23 09 25 22 43 22 44 ... 22 46. 22 41 30 22 05 ... 22 06 ... 22 08 ... 22 01 ... A Rectifying Table of the minuts and quarters that are to be added or substracted from the fore-going Table of the Suns Declination , made for the second year after Leap-year , for every day at noon in the Meridian of London . M. D 1 year 3 year L. year . Ianuary . 05 s. 2. . a. 2 ... a. 5 ... 10 s. 3 a. 3. a. 6. . 15 s. 3. . a. 3. . a. 7 20 s. 4 a. 4 a. 8. 25 s. 4. . a. 4. a. 9. . 30 s. 4 ... a. 5. a. 9 ... February . 05 s. 5 a. 5. . a. 10 10 s. 5. a. 5. . a. 10. 15 s. 5. . a. 5 ... a. 10. . 20 s. 5 ... a. 5 ... a. 10 ... 25 s. 5 ... a. 5 ... a. 11 28 s. 6 a. 5 ... a. 11. . March. 05 s. 6. a. 5. . s. 12 10 a. 6. . s 5. . a. 12. . 15 a. 6. s 5. a. 12 20 a. 5 ... s. 5. a. 11 ... 25 a. 5 ... s. 5 a. 11. . 30 a. 5 ... s. 4 ... a. 11. April . 05 a. 5. . s. 4 ... a. 11 10 a. 5. . s. 4. . a. 10. . 15 a. 5 ... s. 4. . a. 10 20 a. 5 s. 4. . a. 09 25 a. 4 ... s. 4 ... a. 08 30 a. 4. . s. 4 a. 07 ... May. 05 a. 3. . s. 3. . a. 7 10 a. 3 s. 3 a. 6 15 a. 2 ... s. 2 a. 5. . 20 a. 2 s. 1 ... a. 4 25 a. 1. . s. 1. . a. 3 30 a. 1. s. 1 a. 2 Iune . 05 a. 0. . s. 0 ... a. 1 10 a. 0 s. 0. . a. 0 15 s. 0. . a. 0 s. 0. . 20 s. 1 a. 1 s. 2 25 s. 1. . a. 1. s. 3 30 s. 1 ... a. 1. . s. 4 Iuly . 05 s. 2. . a. 2. . s. 05 10 s. 3 a. 2 ... s. 06 15 s. 3. a. 3 s. 06 ... 20 s. 3. . a. 3. . s. 07. 25 s. 3 ... a. 4 s. 08. . 30 s. 4 a. 4. s. 09. August . 05 s. 4. a. 4. . s. 09 ... 10 s. 4. . a. 4 ... s. 10 15 s. 5 a. 5 s. 10. . 20 s. 5. a. 5. s. 11 25 s. 5. . a. 5. s. 11. 30 s. 5 ... a. 5. . s. 11. . September . 05 s. 5 ... a. 5 ... s. 11 ... 10 s. 5 ... a. 5 ... s. 12 15 a. 6. . s. 6. . a. 12. 20 a. 6. s. 5 ... a. 12. . 25 a. 5 ... s. 5 a. 11 30 a. 5. . s. 4 ... a. 11. . October . 05 a. 5. . s. 4. . a. 11. . 10 a. 5. s· 4 ... a. 11 15 a. 4 ... s. 5 a. 10. . 20 a. 4 ... s. 5 ... a. 10 25 a. 4 ... s. 5. a. 09. . 30 a. 4. . s. 4. . a. 09 November . 05 a. 4 s. 4 a. 08 10 a. 3 s. 3. . a. 07 15 a. 2. . s. 3. a. 06 20 a. 2 s. 2. . a. 05 25 a. 1 ... s. 2 a. 03. . 30 a. 1. . s. 1. . a. 02 December . 05 a. 1 s. 0 ... a. 01. . 10 a. 0 s. 0 a. 00. 15 a. 0. ● . 0. s. 01 20 s. 0. . ● . 1. s. 02 25 s. 0 ... a. 1. . s. 03 30 s. 1 a. 2 s. 04 ... 1665 1667 1668   1665 1667 1668 1669 1671 1672 For these years . 1669 1671 1672 1673 1675 1676 1673 1675 1676 A Table of the Magnitudes , Right Ascention in Hours and Minuts , and Degrees and Minuts , and the Declination North or South of 33 fixed Stars . N. Names of the Stars . M. R. Asc. R. Asc Decli . N.   D. M. H. M. D. M. S. 01 Pole-Star , or last in little Bear. 2 7 53 0 32 87 33 N. 02 Andromedas Girdles 2 12 31 0 50 33 50 N. 03 Medusaes head 3 41 27 2 46 39 35 N. 04 Perseus right side 2 44 30 2 58 48 33 N. 05 Middle of the Pleides 5 51 22 3 26 23 06 N. 06 Bulls eye 1 64 0 4 16 15 48 N. 07 Hircus or Goat 1 72 44 4 51 45 36 N. 08 Orions left foot 1 74 30 4 58 8 38 S. 09 Mid-star in Orions Girdle 2 79 45 5 19 1 28 S. 10 Orions right shoulder 2 84 5 5 36 7 18 N. 11 Auriga , or Waggoner 2 84 45 5 39 44 56 N. 12 Great Dog 1 97 24 6 30 16 13 N. 13 Castor , or Apollo 2 108 00 7 12 32 30 N. 14 Little Dog 1 110 20 7 21 6 6 N. 15 Pollux , or Hercules 2 110 25 7 22 28 48 N. 16 Hydraes heart 1 137 36 9 10 7 10 S. 17 Lyons heart 1 147 30 9 50 13 39 N· 18 Great Bears fore-guard 2 160 48 10 43 63 32 N. 19 Lyons tayl 1 172 45 11 31 16 32 N· 20 Virgins Spike 1 196 43 13 07 9 11 N· 21 Last in great Bears tayl 2 203 36 13 34 51 5 N· 22 Arcturius 1 209 56 14 00 21 4 N. 23 Little Bears fore-guard 2 222 46 14 52 75 36 N. 24 Brightest in the Crown 3 231 00 15 24 27 43 N. 25 Scorpions heart 1 242 23 16 09 25 37 S. 26 Hercules head 3 254 40 16 59 14 51 N. 27 Lyra , or Harp 1 276 17 18 25 38 30 N. 28 Eagle , or Vulture 1 293 28 19 35 8 1 N. 29 Swans tayl 2 307 30 20 30 44 5 N 30 Dolphins head 3 307 53 20 32 15 0 N. 31 Pegassus mouth 1 321 50 21 27 8 19 N 32 Pomahant 3 339 30 22 38 31 17 S. 33 Pegassus lower wing 2 358 50 23 55 13 22 N. As for Example . To find the Suns Declination for the year 1670 , on the 12th day of May : First , if you divide 70 ( being the tens only of the year of our Lord by 4 , rejecting the 100s s ) you shall find 2 , as a remainder , which notes it to be the second after Leap-year ; and if 0 remain , then it is Leap-year . Then , Look in the Table of Declination for 1666 , the second after Leap-year , as the year 1670 is , and find the Month in the head of the Table , and the day on one side , and in the meeting-point you shall find 20 deg . 31 min. for the Declination on that day at noon required . Or , If you use the Trianguler Quadrant , extend the Thred from the Center over the 12th of May , and you shall find it to cut in the degrees just 20 deg . 31 min. the true Declination for that year and day . Note , That if you have occasion to use the Declination before noon , then observe that the difference between stroke and stroke , is the difference of Declination for one day ; and by consequence , one half of that space for half a day ; and a quarter for a quarter of a day , &c. As thus for Example . Suppose I would have the Suns Declination the 18th of August 1666 , at 6 in the morning ; here you must note , that the 18th stroke from the beginning of August , represents the 18th day at noon just . Now the time required being 6 hours before noon , Lay the Thred one fourth part of the distance for one day , toward the 17th day , and then in the degrees , the Thred shall cut on 9-43′ , whereas at noon just , it will be but 9-38 ; and the next , or 19th day at noon , it is 9 degrees 16 min. and 3 quarters of a min. as the three pricks thus ... in the Table doth plainly shew ; but by the Rule , a minut is as much as can be seen , and so near with care may you come . Note also farther , That if you shall use it in places that be 4 hours , 6 or 8 , 10 or 12 hours more Eastward , or Westward in Longitude , the same Rule will tell you , the minuts to be added in Western-Longitudes , or to be substracted in Eastern-Longitudes , as Reason and Experience will dictate unto you with due consideration . For if being Eastwards , the Sun comes to the Meridian of that place before it comes to the Meridian of London ; then lay the Thred as in morning hours : But if the place be to the Westwards where it comes later , then lay the Thred so many hours beyond the Noon-stroke for London , as the place hath hours of Western-longitude more than London , counting 15 degrees for an hour , and 4 minuts for every degree ; and then shall you have the Declination to one minut of the very truth . But if it happens to be the Leap-year , or the first or third year after the Leap-year , then thus ; Suppose for the 5th of October 1671 , being the third after Leap-year , I would have the Declination . First , if you lay the Thred over the 5th of October , in the degrees , it gives 08 deg . 42 minuts .. , for the Declination in the second year after Leap-year ; then , because this is the third year , look in the Rectifying-Table for the 5th of October , and there you find s. 4 .. , for substract 4 minuts and a half from 8-42 .. rests 8-38 , the true declination required for the 5th of October 1671. The like work serves for any other day or year ; but for every 5th and 10th day , you have the Declination set down in a Table for all 4 years , to prove and try the truth of your Operations ; and by that , and the Line of Numbers , or the Rule of Three , you may continue it to every day by this proportion . As 5 dayes , or 120 hours , to the difference of Declination in the Table , between one 5th day and another ; So is any part of 5 dayes , or 120 hours , to the difference in Declination to be added or substracted to the 5 dayes Declination immediately fore-going the day required . Example . Suppose for the 18th of February 1669 , the first after Leap-year , I would know the Declination by the Table made to every 5th day only ; On the 20th of February , I find 6-53 ½ ; On the 15th day , 8-47 ; the difference between them is 1-53 ½ ; then the Extent of the Compasses from 5 , the Number of dayes , to 1-53 ▪ the minuts difference ( counted properly every 10th for 6 minuts ) shall reach from 3 , the dayes from 15 toward 18 , to 1 degree 7 minuts and a half , which taken from 8-47′ , the Declination for the 15th day , leaves 7 degrees 38 minuts and a half , the true Declination for the 18th day of February , in the first after Leap-year . Or , by the Line of Numbers thus ; The Extent from 5 , the difference in dayes , to 113 ½ , the difference in min. for 5 dayes , shall reach from 3 , the difference in dayes , to 68 , the difference in minuts for 3 dayes , to be added or substracted , according to the increasing or decreasing of the Declination at that time of the year . Proved thus ; If you substract 5′ ½ from 7 deg . 44 ... the declination in the second year , there remains 7 deg . 38′ ½ , the Declination for the 18th of February , 1669. These Tables may serve very well for 30 years , and not differ 6 minuts in Declination about the Equinoctial , where the difference is most ; and in Iune and December not at all to be perceived . Thus you may by the Rule and Rectifying Table , find the Suns Declination to a minut at any time , without the trouble of Calculation . CHAP. IV. The use of the Trianguler-Quadrant in the Operative part of Navigation . Use I. To find how many Leagues , or Miles , answer to one Degree of Longitude , in any Latitude between the Equinoctial and Pole. FIrst , it is convenient to be resolved how many Leagues or Miles are in one Degree in the Meridian or Equinoctial , which Mr. Norwood and Mr. Collins hath stated about 24 leagues , or 72 miles . Or. If you keep the old number , making the miles greater , viz. 60 miles , or 20 leagues ; then the proportion , by the Numbers , Sines and Tangents , runs thus ; As Sine 90 , to 20 on the Numbers for leagues ; So is Co-sine of the Latitude , to the leagues , on the Numbers , contained in one degree of Longitude in that Latitude . But in Miles , to have the Answer , work thus ; As Sine 90 , to 60 on Numbers ; So Co-sine Latitude , to the number of miles . Example , Latitude 51° 32′ . As Sine 90 , to 60 ; So Sine 38-28 , to 37 miles 30 / 100. But by the Trianguler-Quadrant , or Sector , work thus ; Take the latteral 20 for leagues ( or 60 for miles ) from the Line of Lines from the Center downwards ; and make it a parallel in the sine of 90 , laying the Thred to the nearest distance . Then , The nearest distance from the Co-sine of the Latitude , to the Thred , measured latterally from the Center , shall shew the true number of Leagues required . Example , Latitude 51° 32′ . As — 60 , to = sine of 90 ; So = sine of 38-28 , to — 37-30 , on the Lines . As — 20 , to = sine of 90 ; So is = sine of 38-28 , to — 12-40 for Leagues . Or , As — 24 , to = sine of 90 ; So is = sine of Co-lat . to — 15 , the number of Leagues , after the experiment made by Mr. Norwood , of which true measure you may read more in the Second Part of the Plain Scale , by Mr. Collins . Or , If you multiply the Natural Sine of the Co-lat . by 2 , it gives the Leagues ; or by 6 , it gives the Miles in one degree , cutting off the Radius from the Product . Note also , That if you take the Natural-Number of the Secant of the Course or Rumb , and multiply it by 2 , cutting off the Radius from the Product , it shall give the Leagues required , to raise one degree , at the rate of 20 Leagues to one Degree of a great Circle . Use II. To find how many Leagues , or Miles , answer to Raise , or to Depress the Pole one degree on any Rumb from the Meridian . First , by the Artificial Sines , Tangents , and Numbers . As the Co-sine of the Rumb from the Meridian , to 20 Leagues ( or 24 Leagues ) on the Numbers ; So is the sine of 90 , to the number of Leagues required . Which , when you have sayled on that Rumb , you shall raise or depress the Pole one degree . But by the Trianguler-Quadrant , thus ; As — 20 , taken from the Line of Lines , or any equal parts , to the = Co-sine of the Rumb , laying the Thred to the nearest distance . So is the = sine of 90 , or nearest distance from sine 90 , to the Thred , to — Number of Leagues required , to sayl on that Rumb , and to raise the Pole one degree . Use III. To find how many Miles or Leagues answer to any number of degrees in any parallel of Latitude . Suppose you sayling in the Latitude of 48 degrees , have altered your Longitude 30 degrees , and would then thereby know how many leagues you had sailed . First , bring ( or reduce ) the 30 degrees to Leagues , by multiplying them by 20 , or 24 , ( the leagues resolved to be in one degree ) which makes 600 , ( or 720 ) . Then by the Numbers and Sines . The Extent from the sine of 90 , to 42 the Co-sine of the Latitude , shall reach the same way , from 600 on the Numbers , to 400 the leagues required ; or from 720 , to 480 , according to Mr. Norwood . By the Trianguler-Quadrant . Take — 600 from the Line of Lines , or any equal parts , and make it a = in the sine of 90 , laying the Thred to the nearest distance . Then , The nearest distance from the sine of 42 , ( the Co-sine of the Latitude to the Thred ) and it shall give 400 on the Lines , or equal parts , the leagues required . Which is thus more briefly ; As — 600 , to = 90 ; So is = sine 42 , to — 400 , as frequently before . Use IV. To work the six Problems of Plain Sayling by Gunter's Lines on the edge , or the Trianguler-Quadrant . Note , That in this Art of Navigation , or Plain Sayling , that the Angle that any degree of the Quadrant , or Point of the Compass makes with the Meridian , or North and South-line , that is called the Rumb or Course . But the Angle that it maketh with the East and West-line , or parallel , is called the complement of the Rumb or Course . Note , That in plain Triangles , the Sines and Tangents give Angles , and the Numbers give Sides . Note also , That in Plain Sayling , the distance run , or Course , is the same with the Hypothenusa in plain Triangles . Also note , That the difference of Latitude is counted on the Meridian , and the difference of Longitude or Departure from the Meridian , is counted on the Equinoctial , or on a Parallel of Latitudes . One of which Lines , in plain Triangles , is called the Base ; and the other , the Perpendiculer . The Base being a sine , and the Perpendiculer a sine complement . Note also , That in North Latitude , Sailing Southerly , the Latitude doth decrease ; therefore you must substract the difference in Latitude , from the Latitude you parted from ; but if you sayl Northerly , then you must add it to the Latitude you parted from : The like in South Latitudes . But when one Latitude is South , and the other North , then you must add them both together . Note also , That the difference in Latitude and Longitude , ( and Departure ) when given in degrees , are to be reduced to Leagues , by multiplying by 20 , and counted alwayes on the Line of Numbers , or equal-parts , when you use the Trianguler-Quadrant . So then in using the Index and Square in Plain Sailing , the distance sayled , is alwayes counted on the Index from the Center . The Course is counted on the degrees from the Head toward the Loose-piece . The Difference of Latitude on the Head-Leg , from the Leg-center to the Head. The Departure or Longitude , is counted on the Square . The complement of the Course or Rumb is counted on the degrees , beginning at 00 on the Loose-piece . When your number of Leagues exceed 100 , you must double the Numbers on the Index , the Square , and Head-leg , or count 10 for a 100 , &c. Problem I. The Course , and Distance run on that Course , being given , to find the difference in Latitude , and Departure , or difference in Longitude . As sine of 90 , to the distance run ( or Leagues sayled ) on the Line of Numbers ; So is Co-sine of the Course or Rumb , to difference in Latitude on the Numbers . Again , for the Longitude or Departure . So is the sine of the Course , to the Departure , or difference in Longitude . By the Trianguler-Quadrant . As — Leagues sailed , to = sine 90 , laying the Thred to the nearest distance ; So = Co-sine of the Rumb or Course , to — difference in Latitude . Or , So is = sine of the Rumb or Course , to = Departure , or difference in Longitude . By the Index and Square , after the manner of a Synical Quadrant , thus ; Set the Index ( being put over the Leg-Center-pin ) to the Course counted on the degrees from the Head , toward the Loose-piece . Then slide the Square perpendiculer to th● Head-leg , till the divided edge thereof cuts the distance run on the Index ; then shall the Index , on the Square , give the Departure or Difference in Longitude ; and the Square on the Head-leg , shall shew the Difference in Latitude . Problem II. The Course and Difference of Latitude given , to find the Distance run , and Departure . As Co-sine of Course , to the Difference in Latitude ; So is sine 90 , to the Distance run . Then , As sine of 90 , to the Distance run ; So is sine of Course , to the Departure . By the Trianguler-Quadrant , without the Square . As — difference of the Latitude , to = Co-sine of the Course ; So = sine 90 , to — distance run . So is = sine of the Course , to the Departure . With the Index and Square , thus ; Set the Index to the Course , and the Square to the difference in Latitude ; then on the Index , is cut the Distance ; and on the Square , the Departure . Problem III. The Course and Departure given , to find the distance run , and difference of Latitude . As sine Course , to the departure on Numbers ; So is sine 90 , to the distance . Again , As the sine 90 , to distance run ; So is Co-sine Course , to difference in Latitude . By the Trianguler-Quadrant . As — Departure taken from any fit Scale , to = Co-sine of the Course ; So is = sine 90 , to — distance run on the same Scale . So is = sine of the Course , to the difference in Latitude . With the Index and Square . Set the Index to the Course , and slide the Square perpendiculer to the head-leg , till the Index cuts the departure on the Square ; then the Index sheweth the Distance , and the Square the Latitude on the Head-leg , counting from the Center . Problem IV. The Distance run , and difference in Latitude given , to find the Course and Departure . As the Leagues run , to sine 90 ; So the difference in Latitude , to Co-sine Course . Again , As sine 90 , to the distance run ; So is sine of the Course , to the Departure . By the Trianguler-Quadrant . As — Radius , or a small sine of 90 , to = distance run on the Line of Lines ; So is = difference in Latitude , to Co-sine of the Course , measured on the small Sine . So is — sine of the Course , to the = departure , carried = in the Lines . By the Index and Square . Set the Square to the difference in Latitude , and move the Index till the Square cuts the distance run on the Index ; then shall the Index shew on the Square , the Departure ; and on the Degrees , the Course required . Problem V. The Distance run , and Departure given , to find the Course and Difference in Latitude . As the Distance run , to sine 90 ; So is the Departure , to sine of the Course . Then , As sine 90 , to the Distance run ; So is Co-sine Course , to the Difference in Latitude . By the Trianguler-Quadrant . As — distance run , to = sine 90 ; So — departure , to = sine of the Course . So is = Co-sine Course , to — difference in Latitude . By the Square and Index . Slide the Square and Index , till the Index cuts the Departure on the Square , and the Square cuts the Distance run on the Index ; Then , On the Degrees , the Index shall shew the Course ; and on the Head-leg , the Square shall shew the difference in Latitude . Problem VI. The Difference of Latitude , and the Departure given , to find the Course and Distance run . As the Difference in Latitude , to 45 degrees ; So the Departure , to the Tangent of the Course . Again , As sine Course , to the Departure ; So is sine 90 , to the Distance run . By the Trianguler-Quadrant . As — Radius , or Tangent of 45 , to = Difference in Latitude ; So is = Departure , to — Tangent of the Course on the Loose-piece from whence you took 45. Then , As — Departure , to = sine of the Course ; So is = 90 , to — Distance run . By the Index and Square . Set the Square to the Difference in Latitude , on the Head-leg , counted from the Center ; and bring the Index to cut the Departure on the Square ; then the Square on the Index shews the distance ; and the Index , on the degrees , gives the Course required . In all these 6 Problems , which Mr. Gunter makes 12 Problems , of Plain Sayling , I have set no Example , nor drawn no figure , because the way by the Index and Square is so plain ; and of it self makes a figure of the work : For the Index is alwayes the Distance run , the Hypothenusa ▪ 〈◊〉 Secant : The Square sheweth the Departure ; and the Line of Lines on the Head-leg , the difference of Latitude : And you may not only perform the work , but also see the reason thereof , being a help to the fancy of young Learners in these Nautical Operations : And if your Square playes true , you may be more exact than you can by Scale and Compass , and much more quick and ready ; not only in this , but any thing else in right-Angled plain Triangles , as in Heights and Distances , and the like . Use V. The use of the Meridian Line , and his Scale . These six Problems of Plain Sayling for short Distances , may come very near the matter ; as in making a Traverse of the Ships way from place to place Coasting , as in the Streights , and the Channel , and the like : But for great Distances , it is not so certain as the Sayling by Mercators Chart ; therefore to that purpose the Meridian-line was invented , to reduce degrees on the Globe , to degrees in Planò , as Mr. Wright hath largely shewed . On the innermost-edge of the Rule , or Trianguler-Quadrant , you may have a Meridian-line so large , as to have half an inch for one degree of the Equinoctial ; and the inches for measure , to go along by it ; or rather you may have it lie near to the Line of Lines on the Head-leg , as you shall think most convenient , for then it will be the same as Mr. Gunter's is , and perform his very Operations , as near as may be , after his way , by the Thred and Compasses , or Index and Compasses . Problem I. Two places being propounded , one under the Equinoctial , the other in any Latitude , to find their Meridional difference in degrees and minuts , or 100 parts . Look for the Latitude of the place , scituate out of the Equinoctial in the Meridian-line , and right against in the equal-parts is the Meridional difference of those two places . Example . Let the River of Amazones , under the Equinoctial , be one place ; and the Lizard , in the Latitude of 50 degrees North , another place ; look for 50 on the Meridian-line , and right against it , on the equal-parts , is 57-54 , for 57 degrees 54 minuts ; or in Decimal parts of a degree 57-90 . Problem II. Any two places having both Southerly or Northerly Latitude , to find the Meridional difference between them . Extend the Compasses on the Meridian-line , from one of the Latitudes to the other ; the same Extent laid from the beginning of the Scale of equal-parts , by the Meridian-line , shall reach to the Meridional difference required . Or , The measure from the least Latitude , to the beginning on the Meridional-line , shall reach the same way from the greater , to the difference on the equal-parts . Example . If the Latitude of one place be 30 degrees , and the other 50 degrees ; Extend the Compasses from 30 to 50 on the Meridian-line , and that Extent shall reach on the equal-parts , from the beginning of the Line , to 26 degrees 26 minuts . Problem III. When one place hath South Latitude , and the other North Latitude , to find the Meridional difference . Extend the Compasses from the beginning of the Line of Meridians , to the lesser Latitude ; then that Extent applied the same way on the Meridian-line from the greater Latitude , shall shew on the Scale of equal-parts the Meridional difference required . Example . Suppose one Latitude be 10 deg . South , and the other 30 deg . North ; The Extent from 0 to 10 degrees , shall reach from 30 , to 41° 31′ , the Meridional degrees required . Problem IV· The Latitudes of two places , together with their difference in Longitude being given , to find the Rumb directing from one to the other . As the Meridional difference in Latitude , to the difference in Longitude ; So is the Tangent of 45 , to the Tangent of the Rumb or Course . Example . Let one place be in the Latitude of 50 North , the other in 15 deg . and 30 min. North , as the Lizard-point , and St. Christophers ; and let the difference in Longitude be 68 degrees , 30 minuts ; and let the Rumb , leading from the Lizard to St. Christophers , be required· First , by the Meridian-line , and the Scale of Equal-parts , by Problem II. find the Meridional difference in Latitudes , which in our Example will be 42 degrees , and 12 parts of a 100. Then , The Extent on the Line of Numbers , from 42 degrees and 12 minuts , the Meridional difference in Latitude , to 68 degrees and 30 minuts , the difference in Longitude shall reach the same way from the Tangent of 45 , to the Tangent of 58 degrees and 26 minuts , the Rumb from the Meridian of the Lizard Westwards , being two degrees , and better , beyond the 5th Rumb from the Meridian . By the Trianguler-Quadrant thus ; As the — Tangent of 45 , taken from the Loose-piece , is to the = Meridional-difference in Latitudes on the Line of Lines ; So is the = difference in Longitudes , to the — Tangent of the Course 58 degrees 25 minuts . But by the Index and Square , this is wrought very easily and demonstratively thus ; Count the Meridional difference of Latitudes on the Head-leg down-wards from the Center , as 42 and 12 on the Line of Lines , and set the Square to it . Then , Count the difference of Longitudes on the Square , viz. 68-50 , and to that Point lay the Index ; and then the Index on the degrees shall cut the complement of the Course , viz. 31-35 , or 58-25 , if you count from the Head. Having been so large in this , I shall contract the rest . Problem V. By the two Latitudes and the Rumb , to find the Distance on the Rumb . As the Co-sine of the Rumb , to the true difference of the Latitudes , ( on the Numbers ) ; So is the sine of 90 , to the distance on the Rumb required , ( on the Numbers ) . Being given in degrees and Decimal parts , and brought to Leagues by multiplying by 20 , or 24 , according to Mr. Norwood , as before . Note also , That the true difference of Latitudes , is found by Substraction , of the less Latitude out of the greater . By the Quadrant . As — true difference of Latitudes , to = Co-sine of the Course or Rumb ; So is = sine of 90 , to — distance on the Rumb ( in the same Line of Lines ) . The Index and Square is used as in the second Problem of Plain Sayling . Problem VI. By the two Latitudes , and distance between two places given , to find the Rumb . As the distance sayled , in the degrees and 100 parts , counted on the Lins of Numbers , is to the true difference of Latitudes , found as before , by Substraction ; So is the sine of 90 , to the Co-sine of the Rumb required . As — sine of 90 , to = distance sailed ; So is = difference of Latitudes , to — Co-sine of the Course . By the Index and Square , work as in Problem IV. of Plain Sayling . Problem VII . Both Latitudes and the Rumb given , to find the difference of Longitude . As the Tangent of 45 , to the Tangent of the Rumb ; So is the Meridional difference of Latitudes , to the difference of Longitude required . As — Tangent of 45 , to = Tangent of the Rumb , ( first laid on the Lines from the Loose-piece ) ; So is the — Meridional difference of Latitudes , to the difference of Longitudes . By the Index and Square , work as in the 4th Problem last past . Problem VIII . By one Latitude , Distance and Rumb , to find the other Latitude . As sine 90 , to the Co-sine of the Rumb ; So is the distance , to the true difference of Latitude . As — Co-sine of the Course , to = sine 90 ; So is = distance , in degrees and parts , on the Lines , to the — true difference in Latitudes , to be added or substracted from the Latitude you are in , according as you have increased , or depressed the Latitude in the Voyage . By the Index and Square , work as in the 5th Problem last past , or 2d or Plain Sayling . Use VI. To find the distance of places on the Globe of Earth and Sea ; Or , Geography by the Trianguler-Quadrant . Problem I. When two places are scituated under the same Meridian ( or Longitude ) and on the same side of the Equinoctial ; then substract the lesser Latitude out of the greater , and the remainder shall be the distance in degrees required , counting 20 ( or 24 ) Leagues to a degree on every great Circle of the Sphear . Problem II. When one place is on one side of the Equinoctial , and the other on the other side ; and yet both on one Meridian , as was the former ; then the two Latitudes ( viz. the North-latitude , and the South-latitude ) added together , shall give the distance in degrees required . Problem III. When the two places differ only in Longitude , and are both under the Equinoctial , then substract the lesser Longitude from the greater , and the residue is the distance in degrees . Problem IV. When the two places have both one Latitude , or near it , North or South , and differ only in Longitude . Then work thus ; As sine 90 , to Co-sine of the ( middle ) Latitude ; So is the sine of half the difference in Longitude , to the sine of half the distance . By the Trianguler-Quadrant , or Sector . As — Co-sine of the mean Latitude , to the = sine of 90 , laying the Thred to the nearest distance ; So is = sine of half the difference in Longitude , to — sine of half the distance . Problem V. When both places have different Longitudes and Latitudes , as these Three Wayes following ▪ I Way . When one place hath no Latitude , and the other North or South , with difference in Longitude also ; then , As sine 90 , to Co-sine of difference in Longitude ; So the Co-sine of the Latitude , to the Co-sine of the distance required . By the Trianguler-Quadrant , thus ; As — Co-sine of difference in Longitudes , to the = sine of 90 ; So the = Co-sine of the Latitude , to the = Co-sine of the distance . II Way . When both the places have either North or South Latitude , that is , both toward one Pole ; then thus , As the sine of 90 , to the Co-sine of the difference in Longitude ; So the Co-tangent of the lesser Latitude , to Tang. of a 4th Ark. Which 4th Ark , must be taken out of the complement of the greater Latitude , when the difference of Longitudes is less than a Quadrant , or added to it when more , then the sum or difference shall be a 5th Ark. Then , As the Co-sine of the 4th Ark , to Co-sine of the 5th Ark ; So is the sine of the lesser Latitude , to the sine of the distance . By the Trianguler-Quadrant . As — Co-sine of difference in Longitudes , to = sine of 90 ; So is the = Co-tangent of the lesser Latitude , taken from the Loose-piece , and laid from the Center , and from thence taken parallely to the — Tangent of a 4th Ark , which do with , as before is shewed , to find a 5th Ark. And then , As the — Co-sine of the 4th Ark , to the = Co-sine of the 5th Ark ; So is the — sine of the lesser Latitude , to = Co-sine of the distance . III Way . But when one Latitude is on one side the Equinoctial , and the other on the otherside , viz. one having North-latitude , and the other South . Then , As the sine of 90 , to the Co-sine of the difference in Longitude ; So is the Co-tangent of one Latitude , to the Tangent of a 4th Ark. Which taken out of the other Latitude , and 90 deg . added , when the difference of Longitude is less than a Quadrant , but added to it if more than a Quadrant , and that sum or difference shall be the 5th Ark. Then , As the Co-sine of the 4th Ark , to the Co-sine of the 5th Ark ; So is the sine of the Latitude , first taken , to the Co-sine of the distance in degrees . By the Trianguler-Quadrant , or Sector ; As the — Co-sine of the difference of Longitudes , to = sine of 90 ; So the = Co-tangent of one Latitude ( being first taken from the Loose-piece , or Moveable-leg , and laid from the Center downwards , and from thence taken parallely ) to the — Tangent of a 4th Ark. Which 4th Ark you must do with , as before , to obtain a 5th Ark. Then , As — Co-sine of the 4th Ark , to = Co-sine of the 5th Ark ; So — sine of the Latitude , first taken , to = Co-sine of the distance . That is , when the 4th Ark is substracted ; or , to the Co-sine of the comp . distance when added . Example . Suppose I would know how far it is from the Lizard , to the Cape of Good Hope ; the Lizard having 50 degrees of North-latitude , and the Cape of good Hope 35 degrees of South-latitude , and the difference in Longitude 30 degrees . As the sine of 90 , to the Co-sine of the difference in Longitude 30 , being best counted from 90 backwards ; So is the Co-tangent of 50 , ( viz. at 40 ) to 36 degrees 01 minut , a 4th Ark. Then 90 degrees , and 35 degrees , the other Latitude added , makes 125 ; from which sum , taking the 4th Ark , remains 88-59 , for a 5th Ark. Then say , As the Co-sine of the 4th Ark 36-1 , to the Co-sine of the 5th Ark 88-59 ; So is the sine of 50 , the Latitude first taken , to the Co-sine of the distance 89 deg . 3 min. the nearest distance in the Arch of a great Circle . Note , That here you will have occasion to make use of that help mentioned p. 218 , Sect. 3. As thus for instance . The Proportion being as the Co-sine of 36-1 , to the Co-sine of 88-59 ; which is all one , as the sine of 54 and 59 , to the sine of 1 degree and 1 minut , which is too large for ordinary Compasses , on ordinary Gunters Rules ; therefore first lay the distance from the sine of 90 , to the sine of 54 and 59 , the same way from the sine of 5 degrees and 45 minuts , and note the place . Also , Lay the distance from the sine of 90 , to the sine of 50 , the same way from the sine of 5 degrees and 45 minuts ; and note that place also . Then , As the Extent first noted for 54-59 , is to 1 degree and 1 minut , the Co-sine of 88-59 ; So is the second mark noted for 50 , to 89-3 , the distance in degrees required . Which multiplyed by 72 , gives the distance in the Arch of a great Circle , viz. 6412 miles Statute-measure ; Or , 5340 miles , whereof 60 make one degree , on a great Circle on the superficies of the Sea. Use VII . To find the distance of places by the Natural Versed Sines in the way of a Sector on the Trianguler-Quadrant , being much more easie than the two former wayes . First , by the Pen , find the sum and difference of the complements of the two Latitudes , and count that sum and difference on the versed Sines latterally , and take the distance between your Compasses , and make it a parallel versed Sine of 180 degrees . Or , by the Trianguler-Quadrant . If you have not the Line set on from the Leg-center , then the small Line of Sines beyond the Leg-center , being doubled , will do the work , by taking the distance between the sum and difference , and setting one Point in the Center-prick at two times the Radius of the Sines from the Leg-center ; and then laying the Thred to the nearest distance , or the Line of Right Asceneions under the Months , is a fit Line . Then , Take out the = difference of Longitude , and that shall reach latterally from the difference to the distance required . Example . London and Ierusalem , two places in North Latitude ; London 51-32 , Ierusalem 32-0 , whose two complements 38-28 and 58 added , make 96-28 for a sum , and one taken from the other , leave 19-32 for a difference . Now the — distance between the versed Sines of 96-28 , and 19-32 , make a = versed Sine in 180 , keeping the Sector so , or laying the Thred to the nearest distance , ( and noting where it cuts in degrees ) . Then , The = distance between 47 , the difference of Longitude between the two places , shall reach on the versed Sines from 19-32 , the difference to 39-14 , the distance required ; which , at 72 miles to a degree , makes 2805 miles . Note , This one Rule comprehends all the Three last Wayes , and is not troubled with half so many Cautions as the former . Use VIII . Having the Latitudes and Distance of two places , to find their Difference in Longitude . Find the sum and difference of the two Co-latitudes , as before , by Addition and Substraction ; count them on the versed Sines , and take the — distance between , and make it a = versed Sine of 180. Then , The — distance , between the difference and distance on versed Sines , shall stay at the = difference in Longitudes required . Example . Let one place be Burmudas Isle , and the Latitude thereof 32-25 ; let the other place be the Lizard-point , and the Latitude thereof 50 degrees ; the Co-latitudes are 57-35 and 40-0 ; the sum of them is 97-35 ; the difference between them is 17-35 . The distance in the great Circle , according to Mr. Norwood , is 44-30 , or 886 Leagues , counting 20 leagues to one degree . Then , The — distance between the versed Sines of 17-35 , and 97-35 made a = versed Sine of 180 , the Sector is set . Then , The — distance taken between 44-30 , and 17-37 on the versed Sines , and carried parallelly , shall stay at 55 , the difference in Longitude required between those two places . CHAP. V. Of Sayling by the Arch of a great Circle . IN the Book called , The Geometrical Seaman , by Mr. Phillips , is a very ready Figure to shew in a Quadrant , or more , by what Longitudes and Latitudes a Ship is to pass in any long-run , which is contained under 90 degrees , or 120 difference of Longitude , and the two places having both North Latitude . Which Figure , or Quadrant , is neatly and readily performed by the Trianguler-Quadrant , thus ; Upon the back-side of the Index , before spoken of , may be graduated from the Center , two Tangent-Lines , one equal to the Radius on the Loose-piece , the other to the Radius on the Moving-leg ; then in the use , count the fiducial Line in which the Leg-Center-pin stands , alwayes for the Meridian of one place ; and some where in that Line , according to the latitude thereof , counting the Leg-center the Pole of the World ; and the Index being hung thereon , by the Tangents prick down the Latitude ; there , I say , knock in a Pin to stay a Thred for one place ; then , on the degrees , count the difference of Longitude from the Head-leg , and lay the Index to it , and bring the Thred fastened , as before , till on the Index it cuts the degree and part of the other Latitude , and there make the Thred fast with another Pin in the Loose-piece . Then , If you move the Index to any degree of Longitude between those places , the Thred shall cut on the Index the degree of Latitude that answers unto it ; or if you make the Thred cut any degree of Latitude , the Index gives the Longitude required for that Latitude . Note , If the Latitude be small , as between 10 and 30 , the small Tangents are most convenient ; but if it be between 40 and 80 , the greater Tangent Line is best . Note , That two Threds and a pair of Compasses may serve ; but the Index is much better and quicker in Operation . Example ▪ Let the two places be the Summer-Islands and the Lizard-point ; the same Example that you find in Mr. Norwood , pag. 126 ; and in Mr. Phillip's Geometrical-Seaman , pag. 55. that you may the more readily compare the truth thereof by their Operations . The Latitude of the Lizard Point is 50 degrees , the Longitude is 10. The Latitude of the Summer Islands is 32-25 , the Longitude is 300. The Difference of Longitudes is 70 , as is computed by their Observation . Then , Hanging or putting the Center-hole of the Index over the Leg-center-pin , and counting the fiducial-line on the Head-leg for the Meridian of one place , count on the Tangent Line on the Index the Co-tangent of one Latitude , as suppose the Latitude of the Lizard-point ( the Center alwayes counted as 90 ) and there knock in a Pin in a small hole to hang a Thred on . Then count 70 degrees , the difference in Longitude , on the degrees from the Head-leg , and there stay it ; then draw the Thred put over the first Pin , till it cut the complement of the other Latitude , and by help of another Pin stay it there , which you may conveniently do by one of the sliding-sights ; then the Thred being so laid , slide the Index to every single degree , or fifth degree of Longitude , and then the Thred shall shew on the Index , the Co-tangent of the Latitude answerable to that degree of Longitude , as in the Table annexed . Also , If you would have equal degrees of Latitude , and would find the Longitude according to it ; then slide the Index to and fro till the Thred cuts on the Index an even degree of Latitude ; then on the degrees you have the difference of Longitude from either place . Also note , That the drawing of one Line only on the Trianguler Instrument in the beginning , according to the directions of laying of the Thred ; with the Thred and Compasses , will perform this work also . The Table . Long D.L. Latitude 300 09 32 — 25 305 05 35 — 52 310 10 38 — 51 315 15 41 — 24 320 20 43 — 34 325 25 45 — 24 330 30 46 — 54 335 35 48 — 07 340 40 49 — 04 345 45 49 — 47 350 50 50 — 15 355 55 50 — 31 360 60 50 — 33 05 65 50 — 23 10 70 50 — 00 If this work fit not any case that may happen , there is another way mentioned in Page 75 of the Geometrical Seaman , by the Steriographick Projection ; and that Scheam is drawn the same way , as the Horizontal-Projection for Dyalling was , and somewhat easier ; and any two Points given , in a Circle , you may draw a great Circle to cut them , and the first Circle into two equal-parts , by the directions in Page 15 ; And the Application thereof you have very plainly in Mr. Phillips his Book , to which I refer you , having said more than at first I intended , which was chiefly the use thereof in Observation only . So for the present I conclude this Discourse , and shall endeavour a further Advantage in the next Impression , according as Time and Opportunity shall offer . Farewel . The End of the Second Part. The Table of the Things contained in this Second Part. THe difinition and kind of Dials , Page . 7 Directions to draw the Scheam , 9 To draw Lines to represent the several sorts of Plains in the Scheam , 13 To draw a Scheam particularly for one Dial , 14 To draw the Equinoctial Dial , 19 To try when a Plain lies Equinoctial , 20 To draw a Polar-Dial , 21 To draw an Erect East or West-Dial , 24 To draw a Horizontal-Dial , 27 The d●monstration of the Canon for Hours , ib. To draw a Direct Erect South or North-Dial , 30 To draw a Direct Recliner , 33 The use of the Figure , 35 To draw a Direct East or West Recliner , 37 To make the Table of Arks at the Pole , 42 To refer those Dials to a new Latitude , and a new Declination , wherein they may become Erect Decliners , 46 To find the Requisites by the Scheam , ibid. To find the Declination of a Plain by the Needle , or by the Sun , 49 To take off an Angle , or set the Sector to any Angle required , 53 Precepts to find the Declination by the Sun , and Examples also of the same , 58 To draw an Erect Declining-Dial , 62 The Proportions for the Requisites of Erect Decliners , 64 To find the Requisites Three wayes , 66 To draw the Erect South Decliner , 67 To draw the Lines on a North Decliner , 70 To draw the Hour-Lines on a Plain , that declines above 60 degrees ▪ 73 Of Declining Reclining Plains , 77 The first sort of South Recliners , 79 The second sort of South Recliners , being Polars , 90 The third sort of South-Recliners , 98 The first sort of North Recliners , 106 The second sort of North Recliners , being Equinoctial , 114 The third sort of North Recliners , 119 Of Inclining Di●ls , 126 To find the useful Hours in all Plains , 130 To draw the Mathematical Ornaments on all sorts of Dials , 134 To draw the Tropicks , or length of the Day , 136 To make the Trygon , 138 To draw the Planetary or Iewish Hours , 142 To draw the Italian Hours , 144 To draw the Babylonish Hours , 145 To draw the Azimuth Lines , 146 To draw the Almicanters , 154 To draw the Circles of Position , or Houses , 160 To draw the Hours , and all the rest , on the Ceiling of a Room , 165 The Figure of the Instrument , Explained , 166 A Table of the Suns Azimuth , at every Hour and Quarter , in the whole Signs , 168 A Table of the Suns Altitude the same time , 169 The Description and Use of the Armilary-Sphear for Dyalling , several wayes , 172 The Description and Use of the Poor-man's Dial-Sphear for Dyalling , and several Uses thereof , 203 How to remedy several Inconveniences in the use of the Gunter's Rule . 220 The Use , and a further Description of the Trianguler-Quadrant , for Navigation , or Observation at Sea , 227 For a fore-Observation with Sights , 233 For a fore-Observation with Thred and Plummet , 235 For a back-Observation , as a Davis Quadrant , 237 To find the Suns distance from the Zenith , or the Co-altitude , 238 To find the Altitude , when near the Zenith , 239 To find a Latitude by a forward Observation , as with a Gunter's Bow , 241 To find the Latitude by a back Observation , 247 To find the Latitude by a Meridian Observation , with Thred and Plummet , 252 To find the Suns Declination , 254 A Table of the Suns Declination for the second after Leap-year , 256 , 257 A Table of the Suns Declination for every 5th Day the intermediate years , 258 , 259 A Rectifying Table for the intermediate years , 260 A Table of the Magnitudes Declinations and Right Ascentions of 33 fixed Stars , in Degrees , and Hours , and Minuts , 261 The use of the Trianguler-Quadrant , in the Operative part of Navigation , 267 Of Sayling by the Arch of a great Circle , 300 FINIS . Errata for the Second Part. PAge 6. line 10. for too , read to . P. 18. l. 10. f. H , r. the ends of the Arch QP . p. 22. l. 14. f. begins , r. being . p. 15. l. 27. f. Latitude , r. Co-latitude p. 34. l. 8. f. Sun , r. sum . p. 39. l. 24. f. incliner , r. inclination of Meridians . p. 61. l. 23. f. place , r. plain . p. 62. l. 20. f. ☉ r. Q. p. 66. l. 2. f. I , r. T. p. 69. l. 7. f. 12. r. 7. p. 92 l. 22. r. gives a mark near E , whose measure on the Limb from , B. p. 87. l. 8. f. gi●es , r. gives . l. 11. add at R near C. p. 93. l. 25. f. FE . r. PE. p. 100. l. 21. add cd next gives . p. 10● . l. 6. f. 8-5 , r. 8-3 . p. 105. l. 19. f. use , r. have . p. 108. l. 6. f. Pole , r. Zenith . p. 112. l. 3. f. cuts 12 , r. cuts the substile . p ▪ 113. l. 19. f. DF , r. DE. l ▪ 19. f. T. r. CT . Also in l. ●8 . r. CT . p. 122. l. 6. f. E , r. F. p. 122. l. 13 , 14 , 15 , 16 , add Sine . p. 128. l. 26. f. I , r. L. p. 139. l. 21. add , as in this Example . p. 140. l. 6. add to . p 170. l. 10. f. divides , r. divided . p. 181. l. 20. f. popsition , r. proportion . p. 119. l. 27. f. from , r. on . p. 193. l. 6. f. being , r. bring . p. 197 l. 4. f. elevation , r. inclination . p. 200 l. 24. f. C , r. G. p. 204. l. 3. f. F , r. E. p 209. add in the last line , or by the upper part of the Plain . p. 124. l. 18. add , or remove the Thred to turn it further when it reclines beyond the Pole. p. 238. l. 7. add of . l. 20 add but. p. 247. l. 13. r. and much better in small Latitudes . p. 248. l. 5. f. wo , r. two . p. 251. l. 14. f. 20 , r. 33. Also , l. 17. f. 40 , r. 27. Advertisements . The use of these , or any other Instruments concerning the Mathematical Practice , or further Instructions in any part thereof , is taught by Iohn Colson near the Hermitage-stairs . Also by Euclide Speidwel , dwelling near to White-Chappel Church in Capt. Canes Rents , or at the Custom-house . Also , by William Northhall Mariner , dwelling at the Crooked-Billet in Meeting-house Alley on Green-Bank near Wapping . Also the Instruments may be had at the house of Iohn Brown , Iohn Seller , or Iohn Wingfield , as in the Title-page is expressed . There is now extant a large Treatise of Navigation in Folio , describing the Sea-Coasts , Capes , Head-lands , the Bayes , Roads , Rivers , Harbours and Sea-marks in the whole Northern-Navigation ; shewing the Courses and Distances from one Place to another , the ebbing and flowing of the Sea , with many other things belonging to the practick part of Navigation . A Book ( not heretofore printed in England ) Collected from the practice and experience of divers able and experienced Navigators of our English Nation . Published by Iohn Seller , Hydrographer to the Kings most excellent Majesty , and to be sold by him at the sign of the Marine●s Compass at the Hermitage-stairs in Wapping , and by Iohn Wingfield right against St. Olaves Church in Crouched-Fryers . Practical Navigation ; or , an Introduction to that whole Art. Sold by Iohn Seller and Iohn Wingfield aforesaid . Notes, typically marginal, from the original text Notes for div A29762-e2070 See the General-Scheam . 
A29761	----	The description and use of the trianguler quadrant being a particular and general instrument, useful at land or sea, both for observation and operation : more universally useful, portable and convenient, than any other yet discovered, with its uses in arithmetick, geometry, superficial and solid, astronomy, dyalling, three wayes, gaging, navigation, in a method not before used / by John Brown, philomath. Brown, John, philomath. 1671 Approx. 595 KB of XML-encoded text transcribed from 260 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-05 (EEBO-TCP Phase 1). A29761 Wing B5041 ESTC R15524 12392800 ocm 12392800 61037 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29761) Transcribed from: (Early English Books Online ; image set 61037) Images scanned from microfilm: (Early English books, 1641-1700 ; 933:6) The description and use of the trianguler quadrant being a particular and general instrument, useful at land or sea, both for observation and operation : more universally useful, portable and convenient, than any other yet discovered, with its uses in arithmetick, geometry, superficial and solid, astronomy, dyalling, three wayes, gaging, navigation, in a method not before used / by John Brown, philomath. Brown, John, philomath. [16], 483, [13] p., [19] p. of plates : ill. Printed by John Darby, for John Wingfield, and are to be sold at his house ... and by John Brown ... and by John Sellers ..., London : 1671. Reproduction of original in Huntington Library. Does not include 2d pt., "Horologiographia, or, The art of dyalling ..." Wing B5042. Errata: p. [16] Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. 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Dialing -- Early works to 1800. 2004-01 TCP Assigned for keying and markup 2004-02 SPi Global Keyed and coded from ProQuest page images 2004-03 Judith Siefring Sampled and proofread 2004-03 Judith Siefring Text and markup reviewed and edited 2004-04 pfs Batch review (QC) and XML conversion THE Description and Use OF THE TRIANGULER-QUADRANT : BEING A Particular and General Instrument , useful at Land or Sea ; both for Observation and Operation . More Universally useful , Portable and Convenient , than any other yet discovered . With its Uses in Arithmetick ▪ Geometry , Superficial and Solid . Astronomy . Dyalling , Three wayes . Gaging . Navigation . In a Method not before used . By Iohn Brown , Philomath . London , Printed by Iohn Darby , for Iohn Wingfield , and are to be sold at his house in Crutched-Fryers ; and by Iohn Brown at the Sphear and Sun-Dial in the Minories ; and by Iohn Sellers at the Hermitage-stairs in Wapping . 1671. To the Reader . FRiendly Reader , Thou hast once more presented to thy view , a further Improvement and use of the Sector , under the name of the Trianguler Quadrant , so called from the shape thereof . In the year 1660 , it was my lot , first , to apply and improve this former Contrivance of Mr. Samuel Foster on a Quadrant , to a joynt Rule or Sector ; and did , in 1661 , publish my present Thoughts thereof , in a small Discourse , under the name of the Ioynt Rule . Since then , through my perswasions , and assistance , another Piece was published 1667 , by I. T. under the name of the Semi-Circle on a Sector : But neither of these , that is to say , neither my own nor his , spoke what I would have it speak ; neither have I hopes ever to produce a Discourse either for method or matter , worthy or becoming so excellent , universal , and useful an Instrument , for the most Mathematical Occasions , being for acurateness , conveniency , cheapness , and universality , before all others . For , 1. If it is made of Wood , if the Wood keep but streight , it is as true to be made use of as of Metal . 2. It may be made of any Radius or bigness , and yet in little Room in comparison of other Quadrants . 3. More convenient to use whe● large , than other Quadrants . 4. As to the Projection for 〈◊〉 and Azimuth , particularly using on●ly two Lines of Natural Sines , th● Thred and Compasses for those tw● difficult ( and many more easie ) Pro●positions . 5. The neat Conveniency of greater and a less Radius , doubl● treble , or quadruple one to another . 6. The convenient Contrivance that happens to it , of three Instruments in one , viz. A Sector , Quadrant , and Gunter's Rule ; all three conveniently in one . The consideration of these things , and the love and willingness I alwayes had , to the communicating of them to others , hath put me on this hard task of writing this Collection of the use thereof . Wherein I do most heartily beg thy Pardon and Acceptance , to accept in good part the willing endeavours of my poor Ability , which I doubt not but to have from most that know me ; For , first , my insufficiency ●n the Tongues , Arts , and Sciences : Secondly , my Meanness and Poverty in the World , for these Imployments , which take up so much of a mans time , and ability , to perform them to purpose , my plead my excuse ; for first , here is the Product of more than Two years Improvement of more than vacant Hours ; with the great disadvantage of taking three Weeks at times , to do that which three Dayes together might have as well , if not much better , performed ; And at last , to cal● the Assistance of two others , to un●dertake the Charge thereof , to mid●wife it into the World. Thus , as Widows Mites are accepted , which are offered in sinceri●ty ; so I hope will mine , though at●tended with much disorder , as 〈◊〉 Method ; more uncouthness , as 〈◊〉 Stile and Matter ; What it is , it 〈◊〉 as at first Composing , for I could ne●ver get Time nor Liberty , from 〈◊〉 daily Trade and Calling , to tran●scribe it twice . Yet was it not done at any tim● carelesly , but with good will and free intent of plainness and useful●ness for the publick good of others as well as my own recreation 〈◊〉 delight . The Gunters Rule , the Quadrant , and Sector , I need not commend , they are so well known already ; but this I will add , a better Contrivance and more general hath not yet to my knowledge been produced ; nor a Discourse where the use of all the Three together hath so been handled , nor many more Examples , though Mr. Windgate and Mr. Patridge have done sufficiently for the Gunters-Lines , and Mr. Gunter for the Sector , and Mr. Collins with the Quadrant , and all of them distinctly far beyond this ; yet this Discourse of all the Three together , may give content to some others , as well as to me . The Discourse of Dialling , is gathered from Mr. Wells ; and yet those that shall read Mr. Wells , and this , may often-times think otherwise ; for I assure you , I saw not one leaf of his Book all the while it was doing ; but , I hope , it may please in moderate sort , and ordinary capacity , both for plainness , convenience , and variety . The cutting of the Regular Bodies , I learned from Mr. Iohn Leake , and the way is ready , convenient , and exact , and worthy of remembrance . The Theorems , from Mr. Thomas Diggs , as in its due place , is observed . The way of Measuring Superficies and Solids , from Mr. Gunter : and my constant Experience in those Imployments ; and the Learner may here be supplied with what is often complained on , viz. the Interpretation of Hard-words , as much as I could call to mind , or think to be convenient for that purpose . In the 15th Chapter , I have gathered many Cannons from Mr. Collins his Workes , and applied them to the Trianguler Quadrant ; and been more large than needs in some places , yet I hope to the content of some inquiring Persons . The business of Navigation , I fear , may prove most defective ; for my part , I never yet saw Graves-end , much less the Streights of Gibralter ; but for Observation and Operation , the Instrument will do as well as any , if well made and applied . So for the present , I rest and remain , ready to serve you in , and supply defects by well making of these Instruments , at the Sphear and Sun-Dial in the Great Minories . Iohn Browne . The Argument of the Book , and the Authors Apologie . AT length my pains hath brought to pass the things I long intended , And doubt not but in every place , hereafter 't may be mended . To me it hath been of great use , to others more likewise ; Therefore let no man it abuse , before he doth advise . One Part thereof hath had renown , with Artists far and near : The other Part I strive to crown , with use and plainness here . Although my Parts and Time be small , to hold forth Arts aright ; Yet have I plainly set forth all , seemed useful in my sight . And though I have not seen so far , as some perhaps might see ; I doubt not but that some there are , will pleased with it be . For first the Tyroes young may find , some terms to be explained ; Which when well fixed in his mind , time quickly will be gained . In the next place Mechanicks mean , that have small time to spare ; But yet may have a Love extream , to Mathematicks fair . And others that of wordly Means , have little to afford , For various Mathematick Theams , this having they are stor'd ; As first with Gunters Sector , and , his Quadrant eke also ; By Foster altred after , and , with Gunters Rule and Bow. The Traviss Quadrant and Cross-staves , the Davis Quadrant too ; Their uses all to more than halfs , this Instrument will do : With this advantage more beside , of lying in less room , A fault that Saylors must abide , when they on Ship-board come . In the next place , the Rudiments of Geometry exact . The right Sines & ●heir complements , and how they lie compact , Within a Circle , and the rest , the Chords and versed Sines About a Circle are exprest , the Tangents , Secants , Lines . And how their use and place is seen , in Round and Plain Triangles ; Which serve to deck Urania Queen , as Iewels , Beads , and Spangles . In the next place Arithmetick , by Numbers and by Lines ; In wayes that won't be far to seek , by them that use their times ; Because the Precepts are explain'd , by things of frequent use , That for the most part are contain'd , in City , Town , or House ; As Land and Timber , Boards & Stones , Roofs , Chimneys , Walls and Floor , Computed and reduc●d at once , in Thickness , Less or More . The cutting Platoe's Bodies five , which are not yet made six ; And them the best way to contrive , and Dials on them fix : Their Measure and their Magnitude , in Circle circumscribed ; Whose Properties by old Euclide , and Diggs , have been described . Then also in Astronomy , are many Propositions , Which fitly to th' Rule I apply , avoiding repetitions . And after , in the pleasant Art , of Shadows , I do wander , To draw Hour-lines in every part , both upright , over , and under : And all the usual Ornaments , that on Sun-Dials be , Which are describ'd to the intent , Sol's travels for to see ; As first , his Place and Altitude , his Azimuth likewise ; His Right Ascention , Amplitude , and how soon he doth Rise . The same also to Moon and Stars , is moderately appli'd ; Whereby the time of Night appears , the Moons Age , and the Tide . Then Heights and Distances to take , at one , or at two Stations , Performed by those wayes that make , the fewest Operations . And also ready Rules to use , the Logarithmal Table ; Which may prove ready Hints to these , that are in those most able : And many other useful Thing , is scattered here and there , Which formerly by Me hath been , accounted very rare . And lastly , for the Saylors sake , I have spent many an Hour , Th' Trianguler-Quadrant for to make , more useful than all other : Sea - Instruments that they do use , at Sea for Observation ; And sure I am , it won't abuse them in their Operation ; As in the following Discourse , to them that willing be , It will appear with easie force , if they have eyes to see : The Method and the Manner us'd , as neer as I was able , To follow the old Wayes still us'd , and counted warrantable . And in this , having done my best , 〈…〉 up my male ; Ascribing to my self the least , would have the Truth prevail ; And give the honour and the praise , to him that hath us made , Of willing minds his Fame to raise , by his assisting aid . To whom be honour now and eke , henceforth for evermore , Ascribed by all them that seek the Truth for to adore . J. B. ERRATA . PAge 28. line 8. for Rombords , read Romboides . P. 73. l. last . f. 337 , r. 247. p. 75. l. 1. f. 7. r. 8. p 87. l. 14. r. multiplied by . p. 89. l. 14. f. 5 371616. r. 538.1616 ▪ & l. 21. f. 537 ▪ r. 538. p. 90. l. 4. f. 537 , r. 538. & l. 5. add , being better done with a parallel answer . p. 100. l. 2 add , the Thred . p. 128. l , 2. dele 10 min. p. 133. l. 6. f. 60 , r. 16. p. 143. l. 10 , 11. f. from 12 to 7 , r. from 7 to 12. p. 146. l. 22. f 12 Section , r. 13 Section . p. 158 l. last , dele and. p. 160. l. 11. f. 72 , r. 720 , also in line 15 & 23. p. 164. l. 19. f. Diameter , r. Area . p. 165. l. last , add , to 707. p. 184. l. 10 ▪ f. foot , r. brick . & l. 20. f. ½ , r. 1 ½ . p. 187. l. 17. f. Ceiling , r. Tileing p. 201. l. 11. f. 52 Links . r. 55 Links . & l. 12. f. 48 Acres , r. 4 Acres , 3 Roods , & 8000 Links . p 102. l. 5. f. 21 Acres 42 Links , r. 2 Acres , 0 Roods , but 14760 Links ; read so likewise in l. 11. of the same page . p. 204. l. 1. f. 16 ½ r. 18 ½ . p. 205. l. 8. f. 55 , r. 50. & r. 50 f. 55 in l. 21 & 22. p. 206. l. 19. f 4-50 , r. 4-50000 . & l. 21. f. 1 Chain 25 , r. 11 Chains 23. p. 229. l. 16. f. 8-10 th , r. 8-100 . p. 231. l. 15. f of , r. at . p. 234. l. 22. f. 1 of a foot , r. 1.10 th of a foot . p. 236 , the 3 lines over 134-5 , are to come in after 134-5 . Also , the two lines over 3-545 , should come in after 3-545 . p. 257. l. 13. f. ●496 , r. 249-6 . p. 370. l. 3. f. sine r. Co-sine . p. 383. l. 22. add , by the general Scale . p. 384. l. 14. f. = S. ☉ . r. = Co-sine . p. 414. l. 11. f. or r. on . p. 420. l. 22. f. 71 r. 31. p. 429. l. 15. f. Declination , r. Suns Right Ascention . The Description , and some Uses of the Triangular Quadrant , or the Sector made a Quadrant ; being an excellent Instrument for Observations and Operations at Land or Sea , performing all the Uses of the Fore-staff , Davis-Quadrant , Gunter's-Bow , Gunter's-Cross-staff , Gunter's-Quadrant and Sector , with far more conveniency and as much exactness as any , or all of them will do . The Description thereof . 1. FIrst , it is a joynted Rule ( or Sector ) made to what Length or Radius you please , ( as to 6 , 9 , 12 , 18 , 24 , 30 , or 36 inches Length , when it is folded or shut together ; the shorter of which Lengths is big enough for Land uses , or Paper draughts ; the four last for Sea uses , or Observations . ) To which is added , a third Piece of the same length of the Sector , with a Tennon at each end , to fit into two Mortice-holes at the two ends of the inside of the Sector , to make it an Aequilateral Triangle ; from which shape , and its use , it is properly called a Triangular Quadrant . 2. Secondly , as to the Lines graduated thereon , they may be more or less , as your use of them , and as the cost you will bestow , shall please to command : But to make it compleat for the promised Premises , these that follow are necessary to be inscribed thereon , as in the Figure thereof . And first you are in order hereunto to consider , The outer-edges of the ( Sector or ) Instrument , the inner-edges , the Quadrantal-side , the Sector-side , and the third or loose-piece , also the fixed or Head-leg , the moving-leg , the head , and the end of each leg , also the head and leg center ; of which more in its proper place . 1. And first , on the outer-edge is placed the Lines of Artificial Numbers , Tangents , Sines , and versed Sines , to as large a Radius as the Instrument will bear . 2. Secondly , on the in-side or edge , on short Rules is placed inches , foot measure , the line of 112 , or such-like . But on larger Instruments , a Meridian line to one inch , or half an inch ( more or less ) for one degree of the Aequinoctial , for the drawing of Charts , according to Mercator , or any other more useful Line you shall appoint for your particular purpose . 3. Thirdly , in decribing the Lines on the two sides ; first I shall speak to the Sector-side , where the middle Lines all meet at the Center at the head where the Joynt is : the order of which ( went the head or joynted end lyeth toward your left hand , the Sector being shut , and the Sector-side upermost ) is thus : 1. The first pair of Lines , and lying next to you , is the Line of Sines , and Line of Lines , noted at the end with S , and L : for Sines and Lines , the middle Line between them that runs up to the Center , and wherein the Brass center pricks be , is common both to the Sines and Lines in all Parallel uses , or entrances . 2. The Line next these , and counting from you , is , the Line of Secants beginning at the middle of the Rule , and proceeding to 60 at the end , and noted also with Se for Secants , one of which marginal Lines continued , would run to the center as the other did . 3. The next Lines forward , and next the inner-edge on the moving-leg are the Lines of Tangents ; the first of which , and next to you , is the Tangent of 45 , being the largest Radius , ( as to the length of the Rule : ) the other is another Tangent to one fourth part of the length of the other , and proceeds to 76 degrees , a little beyond the other 45 : the middle Line of these also is common to both , in which the Center pricks must be . At the end of these Lines is usually set T. T. for Tangents . 4. On the other Leg of the Sector , are the same Lines again , in the same order counting from you ; wherein you may note , That as the Lines of Sines and Lines on one Leg , are next the outward-edge ; on the other Leg , they are next the inward-edge : so that at every , or any Angle whatsoever the Sector stands at , you have Lines , Sines , and Tangents to the same Radius : and the Secants to just half the Radius , and consequently to the same Radius by turning the Compasses twice ; Also any Tangent to the greater Radius above 45 , and under 76 , by turning the Compasses four times , as afterwards will more appear : Which contrivance is of excellent convenience to avoid trouble , and save time ; and happily made use of , in this contrary manner to the former wayes of ordering them . 5. Fifthly , without or beyond , yet next to the greater Line of Tangents on the head-leg , is placed the first 45 degrees of the lesser Tangents , which begin from the Center at 45 degrees , because of the straitness of the room next the Center , where they meet in a Point : yet this is almost of as good use , as if it had gone quite to the Center , by taking any parallel Tangent from the middle or common Line on the great Tangents , right against the requisite Number counted on the small Tangent under 45. 6. Sixthly , next to this will not be amiss to adde a Line of Sines , to the same Radius of the small Tangent last mentioned , and figured both wayes for Sine and co-Sine , or sometimes versed Sines . 7. Seventhly , next to this a Line of Equal Parts , and Chords , and the Secants in a pricked line beyond the little Tangent of 45 , all to one Radius : To which ( if you please ) may be added , Mr. Fosters Line Soll , and his Line of Latitudes ; but these at pleasure . 8. Eighthly , on the outermost-part of both Legs next the out-side , in Rules of half an inch thick and under , is set the Line of Artificial versed Sines , laid next to the Line of Artificial Sines , on the outer-edge ; but if the Rule be thick enough to bear four Lines , then in this place may be set the Meridian Line , according to Mr. Gunter , counting the Line of Lines as a Scale of Equal Parts . Thus much as for the Sector-side of the Instrument . 4. Fourthly , The last side to be described is the Quadrantal-side of the Instrument , wherein it chiefly is new . Therefore I shall be as plain as I can herein . To that purpose I shall in the description thereof imagine the loose piece , ( or third piece ) to be put into the two Mortise-holes , which position makes it in form of an Aequilateral Triangle , according to the Figure annexed , noted with ABCD ; where in AB is for brevity and plainness sake called the Moveable-leg , DB the Head or Fixed-leg , DA the loose-piece , B the Head , A and D the ends , C the Leg-center , at the beginning of the general Scale ; the center at B the head-center , used only in large Instruments , and when you please on any oother . For the Lines graduated on this side . First , On the outer-edge of the moveable-Leg , and loose-piece , is graduated , the 180 degrees of a semi-circle , C being the center thereof . And these degrees are numbred from 060 on the loose-piece toward both ends , with 10 , 20 , 30 , 40 , &c. and about on the moveable-leg , with 20 , 30 , 40 , 50 , 60 , 70 , 80 , and 90 at the head : Also it is numbred from 600 on the moveable-leg , with 10 , 20 , toward the head ; and the other way , with 10 , 20 , 30 , 40 , 50 , 60 on the loose-piece ; and sometimes also from the Head along the Moveable-leg , with 10 , 20 , 30 , &c. to ●0 on the loose-piece ; and the like also from the end of the Head-leg , and sometimes from 60 on the loose-piece both wayes , as your use and occasion shall require . Secondly , On the Quadrantal-side of the loose-piece , but next the inward-edge is graduated 60 degrees , or the Tangent of twice 30 degrees , whose center , is the center-hole or Pin at B , on the Head or Joynt of the Sector . Which degrees are numbred three wayes , viz. First from D to A for forward Observations ; and from the middle at 30 to A the end of the Moving-leg , with 10 , 20 , 30 ; and again , from D the end of the Head-leg to A , with 40 , 50 , 60 , 70 , 80 , 90 , for Observations with Thred and Plummet . Thirdly , Next to these degrees on the Moving-leg , is the Line of the Suns right Ascention , numbred from 600 on the degrees , with 1 , 2 , 3 , 4 , 5 , 6 , toward the Head , and then back again with 7 , 8 , 9 , 10 , 11 , 12 , &c. 1 , 2 , 3 , 4 , 5 , on the other side of the Line , as the Figure annexed sheweth : The divisions on this Line is ( for the most part ) whole degrees , or every four minutes of time . Fourthly , Next above this is the Line of the Suns place in the Zodiack , noted with ♈ ♉ ♊ ♋ toward the Head ; then back again with ♌ ♍ ♎ over 600 in the degrees , and 12 and 24 in the Line of the Suns right Ascentions ▪ then toward the end , with ♏ ♐ ♑ ; then back again with ♒ and ♓ , being the Characters of the 12 Signes of the Zodiack , wherein you have exprest every whole degree , as the number of them do shew , there being 30 degrees in one Sign . Fiftly , Next above this is a Kalender of Months and Dayes ; every single Day being exprest , and three or more Letters , of the name of every Month being set in the Month , and also at the beginning of each Month , and every 10th day noted with a Prick on the top of the Line representing it , as is usual in such work . Sixtly , Next over the Months , is the Line to find the Hour and Azimuth in a particular Latitude . Put alwayes on smaller Instruments ( and very rarely on large Triangular Quadrants for Sea Observations ) the lowest Margent whereof , and next the Months , is numbred from the end toward the Head , with 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , 110 , 120 , 130 , near the Head Center . For the Semi-diurnal Ark of the Suns Azimuth , and in the Margent next above this , with 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , near the end , for the Morning hours ; then the other way , viz. toward the Head on the other-side the Hour Line , with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , for the Afternoon hours . Seventhly , On the same Quadrantal-side , and Moveable-leg on the spare places , beyond the Months toward the end , is set an Almanack ; and the Names of 12 or more Stars , to find the hour of the Night ; which 12 Stars are noted with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12. among the degrees in small Figures , as in the Figure . Eightly , Next of all to the in-side , is the Line of Natural versed Sines drawn to the Center , with his correspondent Line on the other , or Head-leg . Exprest sometimes in a pricked Line , for want of room . Ninthly , On the Head-leg , and next to the versed Sines last mentioned , is first the Line of Equal Parts , or Line of Lines : and on the same common Line wherein is the Center , is the Line of Natural Sines , whose length is equal to the measure from the center at C to 600 on the moveable-leg ; so that the Line of degrees is a Tangent , and the measure from C to any Tangent , a Secant , to the same Radius of the Natural Lines of Sines , and Lines : Also beyond the Center C on the same common middle Line is another smaller Line of Natural Sines , whose length is equal to the measure from C to 60 on the loose-piece ; then if you count from the Center pin at 60 , on the loose-piece , toward the end of the movable-leg , they shall be Tangents to the same Radius , and the measure from the Center C to those Tangents , shall be Secants to the same Radius , which may be well to be ordered , to a third , or fourth part of the former , from the Center downwards : These two Lines of Sines are best figur'd with their Sines ; and Cosines , the other way with a smaller figure , and the Line of Lines from the Center downward from 1 to 10 where 90 is , which Lines of Sines may be called a general Scale for all Latitudes . Tenthly , Next to this toward the outer-edge is another Line of Natural Sines , fitted to the particular Line of Hour and Azimuths , for one particular Latitude , noted , Pert. Scale of Altitudes ; or Sines . Eleventhly , Next to this is the Line of 29 ½ , for so many dayes of the Moon 's age , in short Rules of the whole length , but in longer not ; being easily known by the single strokes , and Figures annexed to those strokes . Twelfthly , Next the outer-edge is a Line of 24 hours , 360 degrees , or 12 Signs , or in most Rules inches also , used together with the former Line of 29 ½ , and as a Theory of the Sun and Moon , and ready way of finding the Hour by the Moon or fixed Stars . Thirteenthly , To this Instrument also belongs a Thred and Plummet , and Sights , as to other Quadrants ; and a pair of Compasses as to other Sectors ; a Staff and Ball socket also , if you will be curious and accurate . And for large Instruments for Sea , a Square and an Index , which makes it a perfect sinical Quadrant , and two sliding sights also , which makes it a fore and back-staff , and bow , as will appear more at large afterward . Some Uses of the Trianguler Quadrant , for Land and Sea Observations and Operations . CHAP. I. Numeration on the Lines graduated on the Instrument . IN the first place it will not be amiss to hint a few words , as to the reading the Lines , or ( more properly ) Numeration on the Lines ; wherein take notice , That all Lines of Equal Parts , or Lines applicable to Arithmetick , as the Line of Lines , the Line of Numbers , the Line of Foot-measure , and the like ; wherein Fractions of Numbers are requisite : they are most commonly accounted in a Decimal way , and as much as may be , the small divisions are numbred , and counted accordingly . But in the Lines of Sines , Tangents , Secants , and Chords ; being Lines belonging properly to a Circle : in regard that the Sexagenary Fraction is still in use , the intermediate Divisions are , as much as may be , fitted to that way of account , viz. by whole degrees , where they come close together , ( or the Line of no great use . ) And if more room is , to half degrees or 30 minuts , and sometimes to quarters of degrees or 15 minuts ; but toward the beginning of the Line of Natural Sines , or the end of the Natural Tangents and Secants : where the degrees are largest , they are divided to every 10th minute in all large Rules , as by considering and accounting you may plainly perceive . Take two or three Examples of each kind . 1. First , On the Line of Lines , to find the Point that represents 15. In the doing of this , or any the like , you must consider your whole Scale , Radius , or length of the Line , may be accounted as 1 , as 10 , as 100 , as 1000 , or as 10000 ; and no further can be applicable to any ordinary Instrument . Wherein observe , That if the whole Line be one , then the long stroke by every Figure doth represent one tenth of that Integer : and the next shorter without Figures , are hundredth parts of that one Integer ; and a 1000th part is estimated in smaller Instruments , and sometimes exprest in larger : But the hundredth thousand part is alwayes to be estimated by the eye in all Instruments whatsoever . 2. But if the whole Line of Lines shall represent 10 , as it usually doth , and as it is figured , then the long stroke at every Figure is 1 , and the next longer are tenths , and the shortest are hundred parts , and the thousand parts as near as can be estimated . 3. But if the whole Line represents a hundred , as here in our present Example , then the long stroke by every Figure represents 10 , and every shorter stroke is one , and the shortest strokes are tenths , and the hundredth parts as much as can be estimated . 4. But if the whole Line shall represent a 1000 , then the long stroke by the Figure shall represent a hundred , and every shorter 10 , and every of the shortest strokes is one Integer , and a 10th part as near as can be estimated . 5. But lastly , if the whole Line represent 10000 , then every long stroke is 100 , and every shortest cut is ten , and every single Integer is as near as can be estimated by any ordinary Instrument . Now our present Example will properly come under the third Rule , by conceiving the whole Line to represent 100 ; then the first long stroke by 1 is 10 , then the next shorter is for 11 , the next 12 , &c. to 15 ; which is cut up a little above the Line , for the more ready reckoning without telling the parts : which 15 is the Point required to be found . Example the second , to find out 1550 on the Line . This will come under the Notion of the 5th Rule , wherein the whole Line is conceived to represent 10000 ; then the first 1 is for the 1 thousand , then the fifth longer stroke next is for the 500 ; and lastly , the middle between the 500 stroke and the 6000 stroke is for the 50 , being a little beyond the Point for 15 in the first Example . A third Example of 5025. This third Example may su●fice for this work , being so plain after a little due consideration : For first , the whole Line is conceived to represent 10000 , then the long stroke by 5 is for 5000 , then there is no hundreds , therefore the Point required must be short of the next longer stroke , which signifies hundreds , and being it is just 25 , which is ¼ of an hundred , the true Point readily sheweth it self ; If you require a more plainer and larger wording o● this matter , I refer you to the third Chapte● of Mr. Windgates Rule of Proportion ; 〈◊〉 the first Chapter of the Carpenters Rule , b● I. Brown. Lastly , In nameing of any Point found out on the Line , great care and respect mus● be had as to the true value of the Number according to the rate of the question propounded : for the same Point that represents 15 , doth represent 150 ; and also 1500 , or 15000 , ( increasing above the bounds before mentioned ) also it signifies one and a half , or 15 of one hundred , which is usually exprest thus in a Decimal Fraction 15 / 100 , or more readily , 0.15 . Also if it should be a Number with a digit , two ciphers and another digit , as 2.005 , this Number would be found close to the long stroke , by the figure 2 : and may represent either two thousand , and 5 of 1000 more ; or 20 and 5 of a hundred ; or 2 hundred and 5 of another 10 more , or plainly as it is set down , two thousand , no hundred , but five : Thus you see the manner of expressing whole Numbers , or whole Numbers and Decimal Fractions , which on the Lines is one and the same thing ; and thus all Decimal Scales are to be accounted , and in the same manner is the Line of Numbers to be read , as you may see more at large in the two Books before mentioned . SECT . II. But for Numerateon on all Circular Lines , it is much easier : For first , very few Instruments , unless at one part of the Line , can express nearer than minutes of a degree . Secondly , The whole Radius or Line of Lines is but 90 degrees , or but 45 of the Tangents , or 60 of the Chords or Secants : So that in Instruments of 12 or 18 inches Radius , you may express very well every tenth minute , to 60 on the Line of Sines : and every half degree to 75 , and whole degrees to 90. And on the Tangents or Chords , every 10th minute quite through : and the Secants as the Sines . So that any degree or minute being named , t● find the same on the respective Line , count thus ; First , every 10th degree is noted with a long stroke , and figures set thereunto . Secondly , every whole degree is cut between two , or three Lines , and sometimes with a Point or Mark on the end of the stroke ; and every 5th degree cut up higher than the rest , and sometimes with three Points , on the end of the Line , or some other convenient distinction , for readiness sake : and every 10th , 15th , or 30th minute , is cut only between two Lines and no more ; as will appear very plain with a little practice . Example , to find the Sine of the Latitude being at London , 51 degrees , 32 minutes . 1. First , look on any Line of Sines , on the Quadrantal , or Sector side , according a● you have occasion , till you see 50 , which i● 50 degrees ; then one degree forward , toward 60 is 51 degrees , then count thre● 10ths of minutes more for 30 minutes , and then for the odde two minutes , estimate on● fifth part of the next 10 minutes forwarder and that is the precise Point for the Sine 〈◊〉 51 degrees 32 minutes , the latitu●e of London , where sometimes is set a Bra●s Center-Pin . Example the second . 2. To find the Cosine of the Latitude there are two wayes to count the Comple●ment of any Ark or Angle . First , by substracting the Ark or Angle out of 90 by the Pen , and count the residu● from the beginning of the Line of Sines , and that shall be the Sine Complement of the Latitude required . Example . 51 32 taken from 90 , the remainder is 38 28 , now if you count so much from the beginning of the Line of Sines , according to the last Rule that shall be the Point for the Sine of 38 28 , the Complement of 51 32 , or the Sine Complement of the Latitude . Or Secondly , If you count 51 32 from 90 , calling 80 , 10 ; and 70 , 20 ; and 60 , 30 ; 50 , 40 ; 40 , 50 , &c. wheresoever the Number whose Complement you would have shall end , that is the Sine Complement required , which will be at 38 28 , from the Center or beginning , for the Co-sine of 51 32 ; The like work serves for any other Number , or on any other Line , as on the Degrees , Tangents , or Secants , Natural or Artificial , as by practice will more plainly appear , to the willing Practitioner . SECT . III. To find the versed Sine of an Ark or Angle , or the Sine of an Ark or Angle above 90 degrees , or the Chord above 180 degrees , observe these Rules . 1. First , a right Sine , is the measure on the Line of Sines , from the center or beginning of that Line , to the Point that doth represent the Ark or Angle required . 2. The right Sine of an Ark or Angle above 90 degrees , is equal , to the right Sine of the Complement thereof to 180 degrees , being readily accounted , thus ; Count the excess above 90 backwards , from 90 toward the Center ; then the measure or distance from the end of the account to the Center , is the Sine of the Ark above 90 required : Example . Let the Sine of 130 be required , first , if you take 130 from 180 , the remainder is 50 ; then I say that the right Sine of 50 , is also right Sine of 130 ; for if you count backwards from 90 , calling 80 , 100 ; and 70 , 110 ; and 60 , 120 ; and 50 , 130 ; the measure from thence to 00 , or the Center , is the right Sine of 130 degrees . 3. The versed Sine of an Ark or Angle , is the measure on the Line of Sines from 90 toward the Center , counted backwards , as the small figures for Complements shew , counting 90 for 00 , and the Center for 90 , ( as the Azimuth Line is figured ) opening the Line of Sines to a strait Line , and then counting beyond 90 for the versed Sines above 90 , as on the versed Sines is plainly seen in the figure of the Rule . 4. For Chords of any Ark or Angle , do thus : Halve the Ark or Angle required , and take the right Sine thereof , and that shall be the Chord thereof . Example . I would have the Chord of 40 , the half of 40 is 20 ; then I say the right Sine of 20 is the Chord of 40 , to that Radius that is equal to the right Sine of 30 degrees , at the Radius the Rule stands at . 5. To find a Chord to an Ark or Angle above 180 degrees , you must count as you did the right Sines ; for note , the Chord of 180 is equal to the right Sine of 90 doubled , which is the full Diameter of a Circle : and a longer right Line than the Diameter cannot be taken in a Circle ; therefore it must needs follow that Chords of above 180 , are shorter than the Diameter which is the biggest Chord ; therefore the Chord of 260 , is equal to the Chord of 100 degrees , or right Sine of 50 , the Sine of 30 being Radius . 6. In using the Artificial Sines and Tangents , or Secants ; if you are to use a Sine above 90 , then count 80 for 100 , 70 for 110 , 60 for 120 , &c. But for Secants , then count after the manner of versed Sines : Thus the Secant of 60 is as far beyond 90 , as it is from 30 to 90 ; so that when you have occasion to use an Artificial Secant , which is not often , Then set the end of the Rule against a Table , and counting backwards from 90 to the number of the Secant required , turn that distance beyond 90 on the Board or Table , and that shall be your Secant required , as will be afterward hinted , as they come in use . CHAP. II. A brief Description of the Lines of a Circle , and the Explanation of some termes used in the following Discourse . FOR the better understanding of the following discourse , it is needful to understand these Elements or Principles , as the Letters are necessary to be known before reading . 1. A Circle is a figure enclosed in one circular Line , called the Circumference ; in the middle whereof is a Point called the Center : From which Point all right Lines drawn to the Circumference , are equal one to another ; as in the Circle ABCD , E is the Center , ABCD the Circumference , the Lines EA , EB , EC , ED are equal . 2. Any right Line crossing the Circumference , and passing through the Center of a Circle , is called the Diameter ; and it divides the whole Circle into two equal parts , called Semi-circles ( or half-Circles . ) And the half of that Line is called the Semi-diameter or Radius to that Circle . As the Line AC is the Diameter , and EC the half-Diameter or Radius . 3. Any other Right-line crossing the Circumference is called a Chord , or Subtence , as the Line FG , which divides the Circle into two unequal parts : And note , that this Subtence belongs both to the lesser , and also to the greater part of the Circumference ; that is to say , the Chord of 90 deg . is also the Chord of 270 deg . so that FG is Chord to the Ark FBG 90 deg . and also to the Ark FDG being 270 deg . much more than half the Circle . 4. Half the Chord of any Ark , is the right Sine of half that Ark : thus the right Line HG , the half of FG , is the right Sine of the Ark BG the half of FBG . 5. The Sine Complement or Cosine of any Ark is the nearest distance from the Circumference to the Diameter : Perpendiculer to that Diameter from whence you counted the degrees and minutes of the Ark or Angle . As thus , GI is the Cosine of the Ark BG ; and the Right Sine of any Ark is the nearest distance from the Circumference to the Diameter you counted the degrees from , as GH is the Right Sine of BG . 6. The versed Sine of any Ark or Angle , is the Segment of the Diameter between the right Sine of the same Ark and the Circumference . Thus HB is the versed Sine of the Ark BG , and HD the versed Sine of GD . So also is GH the right Sine of the Ark GCD or the Angle GED 45 degrees above 90 , viz. 135 degrees . 7. A Tangent is a right Line drawn perpendiculer to the Diameter , beginning at one extreme of the given-Ark , and terminated by a right Line drawn from the Center to the other extreme , of the given-Ark in the Circumference , till it inter-sect the perpendiculer ; Thus CK is the Tangent of the Ark CG , or the Angle CEG , 45 degrees . 8. A Secant is a right Line drawn from the Center thorow one extreme of the given-Ark , till it meet with the Tangent rais'd perpendicularly from the Diameter , drawn to the other Extreme of the said Ark ; Thus the Line EK is the Secant of the Ark CG , or the Angle GEC . 9. Note , as in a ( Natural ) Sine , the nearest distance from the Ark to one Diameter , from whence you counted the degrees o● the Ark or Angle , was the Right Sine ; and the nearest distance from the same Point to the other perpendiculer Diameter , is the Cosine of that Ark or Angle . So likewise the nearest distance from the Point where the Tangent and Secant meets , to one of the Diameters aforesaid , is the Tangent of the Ark or Angle ; so the nearest distance from the meeting Point of the same Secant-line is the other Tangent-line to the other Diameter abovesaid , is the Co-Tangent of the Ark or Angle abovesaid . Thus the Right-line KC is the Tangent of 45 , and the Right Line KB the Co-Tangent of 45 ; Also the Line LC is the Tangent of 53 , 30 ; and the Line MB is the co-Tangent thereof , viz. the Tangent of 36 , 30. Also the nearest distance from L to EB , is the Tangent of 36 , 30 , to the Radius LC . 10. Every Circle is supposed to be divided into 360 degrees ; the Semi-circle into 180 , the Quadrant or Quarter into 90. 11. Every Degree is supposed to be divided into 60 minutes , and every minute into 60 Seconds , and every Second into 60 Thirds , &c. 12. A Radius , or Semidiameter , is in our Instrumental Practice , supposed to be divided into 10000 parts , and every Chord , Sine , Tangent , or Secant , is to be divided by the Parts of the same Radius , or Radius and Parts more . 13. An Angle is the meeting of two Right Lines , as GE , and EC , meeting at E , do constitute the Angle GEC , called a Right-lined Angle ; or when two Circles cross one another , it is called a Spherical Angle , the Anguler Point being noted alwayes by the middle Letter of three that shew the Triangle . 14. A Plain Triangle is the meeting of three Right Lines crossing one another ; and a Spherical Triangle is constituted by the crossing of three Circles , as in the two Figures noted II and III , you may plainly see . 15. All Angles , Plain and Spherical , are either Acute , Right , or Obtuce . 17. A Right Angle is alwa●●● just 90 degrees , as you may 〈◊〉 in the Figures II , and III , by the Angles at A in both of them . 18. An Obtuce Angle is alwayes more than 90 degrees , as the Angles at D in both Figures shew . 19. A Parallel Line , is any Line drawn by another Line in such a way , that though it were infinitely produced , yet they would never meet or cross one another , as the Lines AB , CD . 20. A Perpendiculer Line , is when one Line so falleth on another Line , that the Angles on each side are equal , as CA falls of the Line BA , Figure VI. 21. All Triangles are either with three equal sides , as Figure IIII , or two equal sides , as Figure V , or all unequal sides , as Figure VI ; the first of which is called Equilateral , the second Isosceles , the third Scalenum . 22. Again , they may be sometimes named from their Angles ; thus : Orthigonium , with one Right Angle , and two Acute Angles . Ambligonium , with one Obtuce Angle , and two Acute Angles . Oxigonium , with three Acute Angles only . 23. The three Angles of every Plain Triangle , are equal to two Right Angles . 24. All Four-sided Figures are either Squares , with four sides , and four right Angles all equal ; or long Squares ( or Oblongs ) with the two opposite sides equal , or the same crushed together , or not Right-Angled , as the Rombus , and Rombords , or else with four unequal Sides , called Trapeziaes . 25. Lastly , many sided-Figures , are some Regular , having every side alike , as 5 , 6 , 7 , 8 , 9 , 10 , &c. Or else unlike , as Fields , and Woods , and Meadows , which being infinite , cannot be comprehended under any Regular Order or Rule . 26. Multiplicator , is a term used in Multiplication , by which any Number is to be multiplied , as is saying 5 times 6 , 5 is the Multiplicator of 6. 27. Multiplicand , is the Number to be multiplied , as 6 by 5 , as above named . 28. The Product , is the Issue or Result of two Numbers multiplied one by the other , as 30 is the Product of 6 multiplied by 5 ; for 6 times 5 is 30. 29. Divisor , is a term used in Division , and is the Number by which another Number is to be divided ; as to say , How many times 5 in 30 ? 5 here is the Divisor . 30. Dividend , is the Number to be divided , as 30 abovesaid . 31. Quotient , is the Answer to the How many times ( as in the abovesaid ) 5 is in 30 ? 6 times : 6 then is the Quotient . 32. Square , is the Product of two Numbers multiplied together , as the Square of 6 multiplied by 6 , is 36. 33. Square-root of any Number , is that number , which being multiplied by it self , shall have a Product or Square equal to the given Number ; thus the Square-root of 36 is 6 ; for 6 multiplied by 6 , is 36 , equal to the first given Number . But if it be a Number that cannot be squared , as 72 , the content of half a Foot of Board ; whose near Square-root is 8 : 4852811 of 10000000 , then is the Square-root to be exprest as near as you may ( or care for ) as here the Square-root of 72 , which is called a Surd Number , that will not be squared . 34. Cube , is a second Product , or power of two Numbers increasing or multiplied together , as thus ; the Square of 6 is 36 , the first power : and the Cube of 6 is 216 , that is to say 6 times 36 , the second power . In Mr. Windgate's Book of Arithmetick , is the way of doing it by Numbers or Figures , being one of the hardest Lessons in Arithmetick . CHAP. III. Certain Geometrical Propositions , fit to be known as Preparatory Rudiments for the following Work. 1. To draw a Right Line between two Points . EXtend a Thred or Hair , from one Point to the other , and that shall be the Line required . But if you use a Rule ( being the fittest Instrument ) to try your Rule , do thus ; apply one end to one Point as to A , and the other end to the other Point at B , and close to the edge draw the Line required : then turn the Rule , and lay the first end to the last Point ( yet keeping the same side of the Rule toward the Paper ) and draw the Line again , and if the two Lines appear as one , the Rule is streight , or else not . Note the Figure I. 2. To draw a Line Perpendiculer to another , on the middle of a Line . On the Point E on the Line AB , I would raise the Perpendiculer Line CE , set one point of the Compasses in E , and open them to any distance , as EB and EA , and note the Points A and B , then open the Compasses wider , and setting one Point in A , make the part of the Arch by C upwards ; and if you have room do the like downwards , near D : Then the Compasses not stirring , set one Point in B , and with the other , cross the former Arks , near C & D : a Rule laid , and a Line drawn , by those two crossings , shall cut the Line AB perpendiculerly just in the Point E , which was required . 3. To let fall a Perpendiculer from a Point to a Line . But if the Point C had been given from whence to let fall the Perpendiculer to the Line AB , do thus : First , set one Point of Compasses in C ; open the other to any distance , as suppose to A and B ; and then ( if you have room upon A and B , strike both the Arks by D , which finds the Point D , if not ) the middle between A and B , give ; the Point E ; by which to draw CED , the Perpendiculer from C desired . Note Figure 2. Note , That if you can come to find the Point D , by the crossing , it doth readily and exactly divide the Line AB in two equal parts , by the Point E. 4. To raise a Perpendiculer on the end of a Line . On the end of the Line AB , at B , I would raise a Perpendiculer : First , set one Point of the Compasses in B ; open them to any distance , as suppose to C ; and set the other Point any where about the middle , between D and E , as suppose at C , then keep that Point fixed there ; turn the other till it cut the Line , as at D , and keep both Points fixed there , and lay a streight Rule close to both Points ▪ and there keep it ; then keep the middle-Point still fixed at C , and turn the other neatly close to the other end and edge of the Rule , to find the Point E ; then a Rule laid to the Points E and B , shall draw the Perpendiculer required . Or else , when you have set the Compasses in the Point C , prick the Point D in the Line , and make the touch of an Ark near to E ; then a Rule laid to DC cuts the Ark last made , at or by E , in the Point E : There are other wayes , but none better than this . Note the Figure 3. 5. From a Point given , to let fall a Perpendiculer to the end of a Line , being the converse of the former . First , from the Point E , draw the Line ED , of which Line find the middle between E and D , viz. the Point C : then the extent CE , or CD , keeping one Point in C , shall cross the Ground-line in the Point B , by which , and E , you may draw the perpendiculer Line EB , which is but the converse of the former . 6. To draw a Line Parallel to another , at any distance . To the Line AB , I would have another Parallel thereto ; to the distance of AI , take AI between your Compasses , and setting one Point in one end of the Line , as at A , sweep the Ark EIF ; then set the Compasses in the other end , as at B , and sweep the Ark GDH ; then just by the Round-side of those Arks , draw a Line , which shall be the parallel-Line required . Or thus , Take BC , the measure from the Point that is to cut the Parallel-line , and one end of the given-Line , viz. B ; with this distance , set one foot in A , at the other end of the given-Line , and draw the Arch at K ; then take all AB , the given-Line , and setting one Point in C , cross the Ark at K , then C and K shall be Points to draw the Parallel-line by . Note the Figure 4. 7. To make one Angle equal to another . The Angle BAC , being given , and I would have another Angle equal unto it ; set one point of the Compasses in A , and draw the Arch CB ; then on the Line DE from the Point D , draw the like Ark EF ; then in that Ark make EF equal to CB , then draw the Line DF , it shall make the Angle EDF , equal to the Angle BAC ▪ which was required . 8. To divide a Line into any Number of parts . Let AB represent a Line to be divided into Eight parts : On one end , viz. A , draw a Line , as AD , to any Angle ; and from the other end B , draw another Line Parallel to AD , as BE , then open the Compasses to any convenient distance , and from A and B , divide the Lines AD , and BE , into eight parts ; then Lines drawn by a Ruler , laid to every division , in the Lines AD , and BE , shall divide the Line AB in the parts required . Note the Figure marked VI. This Proposition is much easier wrought by the Line of Lines on the Sector , thus ; Take AB between your Compasses , and fit it over parrally in 8 , and 8 of the Line of Lines ; then the Parallel distance between 1 and 1 , shall divide AB into 8 parts required . 9. Any three Points given , to bring them into a Circle . Let ABC be three Points to be brought into a Circle ; first set one Point on A , and open the other above half-way to C , and sweep the part of a Circle above and below the Point A , as the two Arches at D and E ; not moving the Compasses , do the like on C , as the Arks F and G ; then set the Compass-point in B , and cross those Arks in DEF and G ; then a Rule laid from D to E , and from F to G , and Lines drawn do inter-sect at H , the true Center , to bring ABC into a Circle . 10. Any two Points given in a Circle , to draw part of a Circle , which shall cut them , and the Circumference first given into two equal parts . Let A and B be two Points in a Circle , by which two Points , I would draw an Arch , which shall cut the whole Circumference into two equal parts . First , draw a Line from A , the Point remotest from the Center , through the Center , and beyond the Circumference , as AD ; then draw another Line from A , to a Point in the Circumference , perpendiculer to AD , ( and cutting the Center C ) as the Line AE : Then on the Point E , draw another Line perpendiculer to the Line AE , till it inter-sect AD at D ; then these three Points ABD brought into a Circle , or Arch , by the last Rule , shall divide the Circumference into two equal parts . Note the Figure 8 , where the first Circle is cut into two equal parts at F and G , by part of a Circle passing through the Points A and B. 11. Any Segment of a Circle given , to find the Diameter and Center of the Circle belonging to it . Let ABC be the Segment of a Circle , to which I would find a Center ; any where about the middest of the Segment , set one point of the Compasses at pleasure , as at B ; on the point B ( at any meet distance ) describe a Circle , and note where the Circle doth cross the Segment , as at D and E , then ( not stirring the Compasses ) set one point in D , and cross the Circle twice , as at F and I ; and again set one point in E , and cross the Circle twice in G and H : Lastly , by the Points GH , and FI , draw two Lines , which will meet in the point O , the center required . 12. Or else to find the Diameter , thus . Multiply the Chord ( or flat-side ) of the half-Segment , viz AK , 12 by it self ( which is called Squaring ) which makes 144 ; then divide that Product 144 by 8 , the Line KB , called a Sine , the Quotient which comes out will be found to be 18 ; then if you adde 8 the Sine , and 18 the Quotient together , it shall make 26 for the Diameter required to be found . 13. Any Segment of a Circle given , to find the Length of the Arch of the Segment . Lay the Chord of the whole Segment , and twice the Chord of half the Segment , from one Point severally ; and to the greatest extent , adde one third part of the difference between the Extents , and that sum of Extents shall be equal to the Arch. Example . 14. To draw a Helical Line from any Three Points , to several Radiusses without much gibbiosity ; useful for Architect , Shipwrights , and others . Let ABCDE be five Points , to be brought into a Helical-Line , smoothly , and even without gibbiosity or bunches , as the under-side of an Arch , or the bending of a Ship , or the like . First , between the two remote Points of 3 , as A and C , draw the Line AC , then let fall a Perpendiculer from B , to cut the Line AC at Right Angles , and produce it to F : draw the like perpendiculer-Line from the point D , to cut the Line CE at Right-Angles produced to F. I say , the Center both for the Arches AB the lesser , and BC the greater , will be found to be in the Line BF ; the like on the other-side for DE and CD , the Helical-Circle , or Arch required . But if you divide the Arch ABCDE into 24 or more parts , the several Centers of the splay-Lines are thus found ; Take the measure AG , and lay it from B , or D , or C , on the Line GF ; and those Points on GF , shall be the several Points to draw the splay-Lines of the Arch , and Key-stone by . CHAP. IV. Of the Explanation of certain Terms used in this following Book . 1. RAdius , or Sine of 90 , or Tangent of 45 , or Secant of 00 , are all one and the same thing , yet taken respectively in their proper places , and is the whole Line of Sines , or Tangents , to 45 ; or more particulary that point at the end of the Natural-Sines , on the Sector-side , and at 90 and 45 on the edge of the Rule for the Artificial Sines and Tangents , or 10 on the Line of Numbers , and 10 and 90 on the Line of Lines , and Sines , on the Quadrantal-side of the Instrument . 2. A Right Sine of any Ark , or Angle , is the measure from the beginning of the Line of Natural-Sines , to that Point on that Line of Sines , which represents the degrees and minutes contained in that Ark or Angle required . But on the Artificial-Sines we respect not any measure but the Point only . 3. The same account is used both for the Right-Tangent , and Secant also ; the Natural-Tangent taken from the beginning to the degree and minute required ; the Artificial respecting the Point only . 4. In the same manner count for the Secants , and Chords , Lines , or versed Sines . 5. A Cosine , or Sine Complement of any Ark or Angle , is the measure from the Point representing the Ark or Angle , counted from 90 , to the beginning of the Line of Sines , being in effect the Right-Sine of the Cosine of the Ark or Angle required : As for Example ; I would take out the Cosine of the Latitude of London , which is 51 , 32 ; Count 51 32 from 90 toward the beginning , and you shall find your account to end at the Right-Sine of 38 28 , which is the Complement of 51 32 ; for both put together , makes 90 , the whole Sine or Radius . But on the Artificial-Lines count backward to the Point required , without minding any distance or measure , till you come to Proportion . 6. A Lateral Sine , Tangent , or Secant , or Scale of Equal Parts , is any Sine Tangent or Secant , taken along the length of any Line , from the beginning onwards , being a term used only in operation with a Sector , or one Line and a Thred , and opposed to a parallel-Sine , Tangent , or Secant , the thing next to be explained . 7. A Parallel Sine , Tangent , or Secant , is any Sine , Tangent , or Secant , taken across from one Leg to the other of a Sector ; or from any degree and minute on one Line to a Thred drawn streight with the other hand , or any other fixed Line whatsoever , at the nearest distance . 8. The Nearest Distance to any Line , is thus taken ; When one Point of the Compasses stands in any one Point , and the Line being laid , I open or close my Compasses till the other moveable-Foot , being turned about , will but just touch or cleave the Thred . But if you are to lay the Thred to the nearest distance , then one Point of the Compasses being set fast , the other is to be turned about , and the Thred also slipped to and fro , till the Compass-point shall just cleave the Thred in the middest . 9. To adde one Sine or Tangent , to a Sine or Tangent , is to take the Right-Sine , or Tangent of any Ark or Angle between your Compasses , and setting one Point of the Compasses in the Point of the other Number , and then to see how far the other Point will extend Laterally . Example . To adde the Sine of 20 , to the Sine of 30 , take the Sine of 20 between your Compasses , and then putting one Point in 30 , the other shall reach to the Sine of 51 21 ; therefore the distance from the beginning to 51 21 , is the sum of the Sine of 30 and 20 added together . The like way is to add Tangents . 10. To Substract a Sine from a Sine , or a Tangent from a Tangent , is but to take the Lateral least Right-Sine or Tangent between your Compasses , and setting one Point in the term of the greatest turn , the other toward the beginning , and note the degree and minute that the other Point stayes in , for that is the difference or remainder . Example . Suppose I would take the Sine of 10 degrees from 25 ; Take the distance 10 between your Compasses , and setting one foot in 25 , and the other turned toward the beginning , shall reach to 14 23 , the residue or difference required . Or , you may sometimes take the distance between the greater and the less , and lay this from the beginning , shall give the remainder in distance on the Sines as before . 11. The Rectifying-Point , is a Point or Hole on the Head of the Trianguler Quadrant in the inter-secting of the hour and Azimuth-line , and the common Line to the Lines and Sines on the Head-leg ; in which Point you are , when the Rule is open , to stick a small Pin to look to the object whose Altitude above the Horizon you would have in degrees and minutes . Of Terms used in DIALLING . PLain , is that Board , Glass , or flat Superficies you intend to draw the Dial upon , either single of it self , or joyned to some other . Pole of the Plain , is an imaginary Point in the Horizon ( for all upright Dials ) directly opposite to the Plain , or in all Plains , a Point every way 90 degrees from the Plain . Declination of a Plain , is only the number of degrees and minutes , that the Pole-point of the Plain is distant from the North and South-points of the Horizon . The Perdendiculer-Line on the Plain , is a Line Square to a Horizontal-line , being part of a Circle passing through the Zenith , and Nadir , and Pole-point of the Plain . The Horizontal-line , is a Line drawn on any Plain , exactly parallel to the true Horizon of the place you dwell in . Reclination , is when a Plain beholdeth the Zenith-point over our heads : But Inclination , is , when a Plain beholdeth the Nadier ; as in a Roof of a House , the Tiled-part reclines , and the Celid-part inclines . The Meridian-line , on all Plains is the Hour-line of 12 ; but the Meridian of the Plain , is the great Circle of Azimuth perpendiculer to the Plain , bing the same with the Perpendiculer-line on the Plain , passing through the Points of Declination . The Substile-line on all Dials , is that Line wherein the Stile , Gnomon , or Cock of the Dial doth stand , usually counted from 12 , the Meridian-line , or from the Perpendiculer-line , which in all erect Dials is 12. The Stile of a Dial , is the Angle , between the common Axis of the World and the Plain , upon the Substile-line on the Plain , on all Dials . The Angle between 12 and 6 , is onely the number of degrees and minutes contained between the Hour-line of 12 , and the Hour-line of 6 a clock , on any kind of Plain ; especially those having Centers . The Inclination of Meridians , is the number of degrees and minutes , counted on the Aequinoctial , between the Meridian or Hour-line of 12 : and the Substile being the distance , between the Meridian of the place , viz. 12 a clock , and the Meridian of the Plain , but counted on the Aequinoctial ; and doth serve to make the Table of Hour-Arks at the Pole , and to prove your work . The Lines Parallel to 12 , are two Lines peculiar to this way of Dialling by the Sector , and are only two Lines drawn equidistant from , and parallel to the Hour-line of 12. The Contingent or Touch-line in this way of Dialling with Centers , is a Line drawn parallel to the Hour-line of 6 ; but in those without Centers , it is drawn alwayes perpendiculer to the Substile , and so may it be also , if you please , in those with Centers also . The Vertical Line on the Plain , is the same with the Perpendiculer-line on the Plain , being perpendiculer to the Horizontal-line . By the word Nodus , is meant a Knot or Ball , on the Axis or Stile of the Dial , to make a black-shaddow on the Dial , to trace out the Suns motion in the Heavens ; or sometimes an open or hollow-place in the Stile , to leave a light-place to do the same office . But by Apex is meant the same thing , when the Top-end , or Point of an upright Stile shall shew the Hour and Suns place , as the Spot doth in Celing-Dials , where the Hours and Quarters are all of one length , and distinguished by their tullours or greatness only . The Perpendiculer height of the Stile , is nothing else but the nearest distance from the Nodus or Apex to the Plain . The Foot of the Stile is properly right under the Nodus or Apex at the nearest distance . The Vertical-Point , is a Point only used in Recliners and Incliners , being a Point right over , or under the Apex ; and yet in the Meridian , being let fall from the Zenith , by or through the Apex or Nodus , to the Plain in the Meridian-line . The Axis of the Horizon , is only the measure from the Apex to the Vertical-point last spoken to , being the Secant of the complement of the Reclination to the Radius of the Perpendiculer height of the Stile . Erect , is when Plains are upright , as all Walls are intended to be . Direct , is when the Dial-plain beholdeth one of the Four Cardinal Points of the Horizon , as South or North , East or West , that is to say , when the Pole of the Plain , being 90 degrees every way from the Plain , doth lie precisely in one of those Four Cardinal Azimuths : Which in an Erect and Direct-Plain will be in the Horizon . Declining , and Reclining , or Inclining-Plaines , are as the upper or under-side of Roofs at any Oblique Scituation from the Cardinal Points of the Horizon . Oblique , is only a wry , slanting , crooked ; contrary to direct , right , plain , flat , or perpendiculer ; and applied variously , as to the Sphear , to Triangles , to Dial-plains , to Discourse and Conversation . Circles of Position , or rather Semi-circles making 12 Houses , are Circles , whose Pole or Meeting-point is in the Meridian and Horizon of every Country , dividing the Aequinoctial into 12 equal parts , being then called Houses , when used in Astrologie , and some times drawn on Sun-Dials . But when they are used in Astronomy , they require a more near account , as to degrees and minutes . Of certain Terms in Astronomy , and Spherical Definitions of Points and Lines in the Sphear . NOt to be curious in this matter , a Sphear may be understood to be a united Spherical Superficies , or round Body , contained under one Surface ; in the middle whereof is a Point or Center , from whence all Lines drawn to the Circumference are equal : Or you may conceive a Sphear to be an Instrument , consisting of several Rings or Circles , whereby , the sensible motion of the Heavenly Bodies are conveniently represented . For the better Explanation whereof , Astronomers have contrived thereon , viz. on the Sphear , ten imaginary Points , and ten Circles , which are usually drawn on Globes and Sphears ; besides others not usually drawn , but apprehended in the fancy , for Demonstrations-sake , in Spherical Conclusions . The ten Points are , the two Poles of the World , the two Poles of the Zodiack , the two Aequinoctial Points , the two Solstitial Points , the Zenith , and Nadir . The ten Circles are , The Horizon , the Meridian , the Aequinoctial , the Zodiack ; the two Colures , viz. that of the Equinoxes , and that of the Solstices ; the Tropick of Cancer , and the Tropick of Capricorn ; and the two Polar-Circles , viz. The Artick or North , the Antartick or South , Polar-Circle . The first six , are great Circles , cutting the Sphear into two equal parts : And the four last are lesser Circles , dividing the Sphear unequally . All which Points and Circles shall be represented by the Figure of the Analemma , from whence the Trianguler Quadrant is derived , as a general Instrument , and also by the Horizontal projection of the Sphear fitted for London , being better for the fancy to apprehend the Mystery of Dialling , one thing mainly intended in this Discourse . Of the 10 Points in the Sphear . THe two Poles of the World , are the two Points P and P in the Analemma , being directly opposite one to another ; about which two Points , the whole frame of the Heavens moveth from East to West ; one of which Poles may alwayes be seen by us , called the Artick or North-Pole ; represented in the particular Scheam by the Point P. The other being not seen , is not represented in the particular Scheam ; but the Line PEP , in the general Scheam , drawn from Pole to Pole , is called the Axis ▪ or Axeltree of the World , because the whole Sphear appears to move round about it . The Poles of the Zodiack are two Points diametrically opposite also , upon which Points the Heavens move slowly from West to East , represented by the two Points , I and K , 23 degrees and 31 minutes distant from the two former Poles , in the Analemma , and by the Point PZ in the Horizontal projection ; but the other Pole of the Zodiack cannot be represented in that particular Scheam . The Equinoctial Points , are the Points of Aries and Libra ; to which two Points , when the Sun cometh along the Ecliptick , it maketh the Dayes and Nights equal in all places ; at Aries March 10th or 11th ; to Libra about the 13th of September , where the Spring , and Autumn begins ; being represented in the Analemma by the Point ● , and in the particular Scheam by the Points E , and W. The two Solsticial Points , are represented one by the Point ♋ , and the other by the Point ♑ , in both Scheams ; to which Points when the Sun cometh along the Ecliptick , it makes the Dayes in Cancer ♋ , longest ; in Capricorn ♑ , shortest ; ♋ being about the 11th of Iune ; and ♑ about the 11th of December . The Zenith is an imaginary Point right over our heads , being every way 90 degrees distant form the Horizon ; in which Point all Azimuth Lines do meet , represented by the Points Z , in both Scheams . The Nadir is an imaginary Point under our feet , directly opposite to the Zenith , represented by the Point N in the Analemma , but not in the particular Scheam , because it is not seen at any time . Of the Circles of the Sphear . THe Horizon is twofold , viz. Rational , and Sensible : The Rational Horizon , is an imaginary great Circle of the Sphear , every where 90 degrees distant from the Zenith , and Nadir ; Points cutting , or dividing the whole Sphear into two equal parts , the one called , The upper or visible Hemisphear ; the other the lower or invinsible Hemisphear . This Rational Horizon , is distinguished also into Right , Oblique , and Parallel-Horizon . 1. The Right Horizon is when the two Poles of the World lie in the Horizon , and the Equinoctial at Right Angles to it ; which Horizon is peculiar to those that live under the Equinoctial , who have their Dayes and Nights alwayes equal , and all the Stars to Rise and Set , and the Sun to pass twice in a year by their Zenith-point , thereby making two Winters , and two Summers ; Their Winters being in Iune and December , and their Summers , in March and September . 2. The Oblique Horizon is when one Pole-point is visible , and ( the other not ) having E●evation above , and depression below the North or South part of the Horizon , according to the Latitude of the place : in which Horizon when the Sun cometh to the Equinoctial , the Dayes and Nights are only then equal ; and the nearer the Sun comes to the visible Pole , the Dayes are the longer , and the contrary ; also some Stars never set , and some never rise in that Horizon : And all Horizons but two , are in a strict sense Oblique-Horizons , viz. The Right Horizon already spoken to : And The Parallel Horizon , is that Horizon which hath the Equinoctial for its Horizon , and one of the Pole-points for its Zenith ; peculiar only to those Inhabitants under the Pole , ( if any be there . ) In which Horizon , ona half of the Sphear doth only alwayes appear , and the other half alwayes is hid ; and the Sun , for one half year , doth go round about like a Skrew , making it continual Day , and the other half year is continual Night , and cold enough ; which Circle in the Analemma is represented by the Line HES , but in the particular Scheam by the Circle NESW . The Meridian is a great Circle which passeth through the two Pole-points , the Zenith and Nadir , and the North and South-points of the Horizon , and is called Meridian , becuse when the Sun ( or Stars ) cometh to that Circle , it maketh Mid-day , or Mid-night , which is twice in every 24 hours : Also all places , North and South , have the same Meridian ; but places that lie Eastward , or Westwards , have several Meridians . Also , when the Sun or Stars come to the South , or North-part of the Meridian , their Altitudes are then highest , and lowest . And the difference of Meridians is the difference of Longitudes of Places , noted by the Circle ZHNS in the Analemma ; and NZ ♋ S in Horizontal-projection . The Equinoctial is a great Circle , every where 90 degrees distant from the two Poles of the World , dividing the Sphear into two halfs , called the North and South Hemisphear ; and is called also the Aequator , because when the Sun passeth by it twice a year , it makes the day and nights equal in all places ; noted by WAEE , and AEEAE in both . The Zodiack , or Signifer , is another great Circle that divides the Sphear and Equinoctial into two equal parts , whose Poles are the Poles of the Zodiack , being 90 degrees from it ; and it inter-sects the Equinoctial in the two Points of Aries and Libra ; and one part of it doth decline Northward , and the other Southward , 23 degrees 31 minutes , as the Poles of the Zodiack decline from the North and South-Poles of the World : The breadth of this Zodiack , or Girdle , is counted 14 or 16 degrees , to allow for the wandring of Luna , Mars , and Venus ; the middle of which breadth is the Ecliptick-Line , because all Eclipses are in , or very near in this Line . And this Circle is divided into 12 Signs , and each Sign into 30 degrees , according to their Names and Characters , ♈ Aries , ♉ Taurus , ♊ Gemini , ♋ Cancer ; , ♌ Leo , ♍ Virgo , ♎ Libra , ♏ Scorpio , ♐ Sagittarius , ♑ Capricornius , ♒ Aquarius , ♓ Pisces . 6 being Northern , and the 6 latter Southern . The two Colures are only two Meridians , or great Circles , crossing one another at Right Angles ; the one Colure passing through the Poles of the World , and the Points of Aries and Libra , there cutting the Equinoctial and Ecliptick : And the other Colure passeth by the Poles of the World also , and cuts the Ecliptick in ♋ , and ♑ , making the Four Seasons of the year ; that is , the equal Dayes , called the Equinoctial-Colure ; and the unequal Dayes , in Iune and December , called the Solsticial-Colures , represented in the Analemma by ZP ♋ NS , and PEP ; and in the particular Scheam by WPE , and NPS , the Solsticial-Colure . The lesser Circles are the Tropicks of ♋ , and ♑ ; being the Lines of the Suns motion in the longest and shortest dayes , noted in the Scheams by ♋ , ☉ , ♋ , and 6 ♋ E , and ♑ , ♑ ; and W ♑ 6 ; to which two Circles when the Sun cometh , it is on the 11th of Iune , and the 11th of December , making the Summer and Winter Solstice . The Polar Circles , are two Circles drawn about the Poles of the World , as far off as the Poles of the Zodiack are , viz. 23 degrees , 31 minutes ; That about the North-Pole is called the Artick , and that about the South the Antartick , being opposite thereunto , shewed in the Analemma by II , and KK ; and by the small Circle about P in the particular Scheam . Of the other Circle imagined , but not described on Sphears or Globes . 1. HOurs are great Circles , passing through the two Poles , and cutting the Equinoctial in 24 equal parts , as the Lines P1 , P2 , P3 , &c. in the Particular ; and P ☉ H in the Analemma ; such also are degrees of Longitude , and Meridians ; the Meridian being the hour of 12. 2. Azimuths are great Circles , passing through , or meeting in the Zenith and Nadir-points , numbred and counted on the Horizon , from the Four Cardinal Points of North and South , East and West , according to Four 90ties , or 180 degrees , or according to the 32 Rombs or Points of the Compass , as Z ☉ A , and ZE , the Azimuth of East and West , being called the prime Virtical , viz. SE , WZ . 3. Almicanters , or Circles of Altitude , are lesser Circles , all parallel to the Horizon , counted on any Azimuth from the Horizon to the Zenith , to measure the Altitude of the Sun , Moon , or Stars above the Horizon , being the Portion of some Azimuth , between the Center of the Sun , or Star ; and the Horizon , commonly called its Altitude above the Horizon , showed by A ☉ in the Analemma , and SAE in the particular Scheam . 4. Parallels of Declination , are parallels to the Equinoctial , as the Almicanters were parallel to the Horizon , as ♋ ☉ ♋ , the greatest Declination or Circle of ♋ : These parallels have the 2 Poles of the World for their Centers , and in respect of the Sun or Stars are called degrees of Declination ; but in respect of the Earth , degrees of Latitude ; being the Arch on the Meridian of any place , between the Pole and Horizon , as 4 ♋ 4 in the Particular , and HP , in the Analemma . 5. Parallels of Latitude , in respect of the Stars , are Lines drawn parallel to the Ecliptick , as the Almicanters were parallel to the Horizon ; so that the Latitude of a Star is counted from the Ecliptick toward the Poles of the Zodiack ; but the Sun being alwayes in the Ecliptick , is said to have no Latitude . 6. Degrees of Longitude , in respect of the Heavens , are measured by the degrees on the Ecliptick , from the first point of Aries forward , according to the succession of the 12 Signs of the Zodiack . But Longitude on the Earth , is counted on the Equinoctial Eastwards , from some principal Meridian on the Earth , as the Isles of Azores , or the Peak of Tenneriff , or the like . 7. Right Ascention is an Arch of the Equinoctial ( counted from the first Point of Aries ) that cometh to the Meridian with the Sun , Moon , or Stars , at any day , or time of the year , being much used in the following discourse , noted in the Analemma by EH , or the like ; but counted as afterward is shewed . 8. Oblique Ascention is an Arch of the Equinoctial , between the beginning of Aries , and that part of the Equinoctial that riseth with the Center of a Star , or any portion of the Ecliptick in an Oblique-Sphear . 9. Ascentional Difference , is the difference between the Right and Oblique Ascention , to find the Sun or Stars rising before or after 6. 10. Amplitude is an Arch of the Horizon , between the Center of the Sun and the true East-point , at the very moment of Rising , represented by ♋ F , in the particular Scheam , and GE , and FE in the Analemma : useful at Sea. 11. A Circle of Position is one of the 12 Houses in Astronomy or Astrology . 12. An Angle of Position , is the Angle made in the Center of the Sun , between his Meridian , or Hour , and some Azimuth , as the Prime , Vertical , or the Meridian , or any other Azimuth , being useful in Astronomy , and sometimes in Calculation , represented bp P ☉ Z in the Analemma . Thus much for Astronomical terms . CHAP. V. Some Uses of the Trianguler Quadrant . Use I. And first to rectifie the Rule , or make it a Trianguler Quadrant . FIrst open the Rule , and put in the loose piece into the two Mortice-holes , ( which putting together makes it a Trianguler Quadrant ) but if you do not use the loose-piece , then open it to an Angle ▪ of 60 degrees , which is thus exactly done : Measure from the Rectifying-point , to any Number on the Sines or Lines ; then keeping the Point of the Compasses still fixed in the Rectifying-point , turn the other to the Common-Line of the Hour and Azimuth-Line , that cuts the Rectifying-point , and there keep it ; then removing the Point of the Compasses from the Rectifying-point , open or close the Rule till the other Point shall touch the distance first measured in the Line of Sines or Lines , then shall you see the Lines on the Head , and Moveable-leg , to meet ; and also see quite through the Rectifying-point , to thrust a Pin quite through ; and thus is it set to an Angle of 60 degrees , without the help of the loose-piece , or to an Angle of 45 , or whatsoever else the Rule is made for . Use II. To observe the Sun or a Stars Altitude above the Horizon . Put a Pin in the Center-hole on the Head-Leg , and another in the Rectifying-point , and a third ( if you please ) in the end of the Hour-line on the Moving-leg . Then on the Pin in the Leg-center , hang a Thred and Plummet ; then if the object be low , viz. under 25 degrees high ; Look along by the two Pins in the Rectifying-point , and the Moving-leg , and see that the Plummet playeth evenly and steady , then the degrees cut by the Thred ▪ shall be the Altitude required , counting from 60 / 0 toward the Head , as the smaller Figures shew . But if the Object be above 25 degrees high , then look by the Pin in the Rectifying-point , and that on which the Plummet hangeth ; and observe as before , and the Thred shall shew the Altitude required , as the Figures before the Line sheweth ; If you have Sights , use them instead of Pins ; and by Practice learn to be accurate in this Work , the ground and foundation in every Observation ; and according to your exactness herein , is the following Work also . Note also , that this looking up toward the Sun , is only then when the Sun is in a cloud , and may be seen in the Abiss , but will not give a clear shadow : Or else you must use a piece of Red , or a Blue , or Green Glass , to darken the luster that it offend not the eyes . But if the Sun be clear and bright , then you need not look up toward it , but hold the Trianguler Quadrant so , that the shadow of the Pin in the Center may fall just on the shadow of the Pin in the Rectifying-point , and both those shadows on your finger beyond them , and the Plummet being somewhat heavy , and the Thred small and playing evenly by the Rule , then is the Observation so made , likely to be near the very truth . Note also further , That the shaking of the hand , you shall find will hinder exactness ; therefore , when you may , find some place to lean your Body , or Arm , or the Instrument against , that you may be the more steady . But the surest and best way is with a Ball-socket , and a Three-leg-staff , such as Land Surveighers use to support their Instruments withal , then you will be at liberty to move and remove it , to and fro , till the Sights or Pins , and Plummet and Thred play to exactness ; without which care and exactness , you cannot certainly and knowingly attain the Sun's or a Star's Altitude to a minute , either by this nor any other Instrument whatsoever , though they be never so truly made : Yet I dare affirm to do it , or it may be done as well by this , as by any other graduated Instrument whatsoever : The Line of degrees on this , being only two thirty degrees of a Tangent laid together ; of which , that on the in-side of the loose-piece is the largest , and consequently the best , to distinguish the minuts of a degree withal . Use III. To try if any thing be Level , or Upright . Set the Moving-leg of the Trianguler Quadrant on the thing you would have to be Level ; then if the Thred play just on 60 degrees , or the stroke by 600 , then is it Level , or else not . But to try if a thing be upright or not ▪ apply the Head-leg to the Wall or Post , and if it be upright , the Thred will play just on the common Line between the Lines and Sines on the Head-leg , and cut the stroke by 90 on the Head of the Instrument , or else not . Use IV. To find readily what Angle the Sector stands at , at any opening . First , on the Sector side , about the Head , is 180 degrees , or twice 90 graduated to every two degrees ; so that opening the Rule to any Angle , the in-side of the Moving-leg , passing about the semi-circle of the Head , sheweth the Angle of opening to one degree . But to do it more exactly , do thus : The two Lines of Sines that issue from the Center in Rules of a Foot , shut , are drawn usually just 5 degrees assunder ; or rather the two innermost Lines , on each Leg , are always just one degree from the inside , so that if you put a Center-pin in the Line of Tangents , just against the Sine of 30 , it makes the two innermost Lines that come from the Center , just 2 degrees assunder , which is easie to remember either in adding or substracting as followeth , two wayes . 1. Take the Latteral Sine of 30 , viz. the measure from the Center to 30 : the Compasses so set , set one Point in the Center-pin in the Tangents just against 30 ; and turn the other till it cut the common Line , in the Line of Sines on the other Leg , and there it shall shew what Angle the two innermost-Lines make , counting from the end toward the Head , and two degrees less is the Angle the Sector stands at , both on the in-side and out-side , the Legs being parallel ; which Number must nearly agree with what the in-side of the Leg cuts on the Head-semicircle , or there is a mistake . As thus for Example . Suppose I open the Rule at all adventures ▪ and taking the Latteral Sine of 30 from the Sines on the Sector-side , and putting one Point of the Compass in the Center on the Tangents , right against the Sine of 30 on the other Leg ( or the beginning of the Secants on the same Leg ) and turning the other Point to the Line of Sines on the other Leg , it cuts the Sine of 60 on the innermost Line that comes from the Center ; then I say , that the Lines of Sines and Tangents are just 30 degrees assunder , and the in-side or out-side of the Legs but 28 , viz. two degrees less , as a glance of your eye to the Head will plainly shew . 2. This way will serve very well for all Angles above 20 , and under 80 : But for all under 20 , and above 80 , to 120 , this is a better way ; Open the Rule to any Angle at pleasure , and take the distance parallelly ( that is , across from one Leg to the other ) between the Center-pin at 30 in the Sines , and that in the Tangents right against it , and measure it latterally from the Center , and it shall shew the Sine of half the Angle the Sines and Tangents stand at ; and one degree less is the Sine of half the Angle the Sector stands at . Example . Suppose that opening the Sector at adventures , or to the Level of any thing , I would know the Angle it stands at : I take the parallel Distance between the two Centers ; and measuring it latterally from the Center , I find it gives me the Sine of 51 degrees , viz. the half Angle the Lines stand at ; or 50 , the Angle the Rule stands at ; which doubled , is 102 for the Lines , or 100 for the Legs of the Sector , as a glance of the eye presently resolves by the inner-edge of the Moving-leg , and the divided semi-circle . 3. On the contrary , Would you set the Legs or Lines to any Angle , take the half thereof latterally , or one degree less in the half for the Legs , and make it a Parallel in the two Centers , and the Sector is so set accordingly . Example . I would set the Legs to 90 degrees , or a just Square : take out the Latteral Sine of 44 , one degree less than 45 , the half of 90 , and make it a Parallel in the two Centers abovesaid , and you shall find the Legs set just to a Square , or Right-Angle , as by looking to the Head you may nearly see . At the same time if you take Latteral 30 , and lay it from the Center , according to the first Rule , you shall see a great deficiency therein , as above is hinted . Use V. The Day of the Month being given , to find the Suns Declination , true place in the Zodiack , Right Ascention , Ascentional Difference , or Rising and Setting . 1. Lay the Thred to the Day of the Month ( in the upper Line of Months , where the length of the Dayes are increasing ; or in the lower-Line , when the Dayes are decreasing , according to the time of the year ) then in the Line of degrees you have his Declination ; wherein note , that if the Thred lie on the right hand of 600 , then the Suns Declination is Northwards ; the contrary-way is Southwards : Also on the Line of the Sun 's Right Ascention , you have his Right Ascention , in degrees and hours , ( counting one Hour for 15 degrees ) as the Months proceed from March the 10th , or Equinoctial , the Right Ascention being then 00 , and so forward to 24 hours , or 360 degrees , as the Months and Dayes proceed . Again , on the Line o● the Sun 's true place , you have the sign and degree of his place in the Ecliptick , Aries , or the Equinoctial-point being the place to begin , and then proceeding forward as the Months and Dayes go . Lastly , on the Hour-line you have the Ascentional-difference , in degrees and minutes , counting from 6 ; or the Suns Rising , counting as the morning hours proceed ; or his Setting , counting as the afternoon hours proceed . Of all which , take two or three Examples . 1. For March the 12th , lay the Thred to the Day , and extend it streight ; then on the Line of degrees , it sheweth near 1 degree , or 54 minutes Northward . 2. The Suns Right Ascention , is in time 8 minutes and better , or in degrees , 2 deg . 5 minutes . 3. The Suns Place , is 2 degrees and 16 minutes in Aries , ♈ . 4. The Ascentional Difference , is 1 degree and 10 minutes ; or the Sun riseth 4 minutes before , and sets 4 minutes after 6. Again , for May the 10th , the Thred laid thereon , cuts in the degrees , 20 deg . 9 min. for Northern Declination ; and 57 deg . 24 min. or 3 hours 52 min. Right Ascention ; and 29 37 in ♉ Taurus for his true place ; and 27 12 for difference of Ascentions , or riseth 11 minutes after 4 , and sets 49 minutes after 7. Again , on the last of October , or the 21 of Ianuary , near the Declination , is 17 22 Southwards , the Right Ascention for October 31 , is 225 , 53 , for Ianuary 21 , 314 21 : The true place for October 31 , is ♏ Scorpio , 18 deg . 22 min ; but for Ianuary 21 , ♒ Aquarius 11 , 52 ; according as the Months go to the end at ♑ , and then back again ; but the Ascentional difference , and Rising and Setting , is very near the same at both times , viz. 23 10 , and Riseth 32 minutes , and more , after 7 ; and Sets 28 minutes less after 4. Use VI. The Declination of the Sun , or a Star , given , to find his Amplitude . Take the Declination , being counted on the particular Scale of Altitudes , between your Compasses ; and with this distance , set one foot in 90 on the Azimuth-Line , the other Point applied to the same Line , shall give the Amplitude , counting from 90. Example . The Declination being 12 North , the Amplitude is 19 deg . 15 min. Northwards . Or the Declination being 20 South , the Amplitude is 34 deg . 10 min. Southwards . Use VII . The Right Ascention and Ascentional-difference being given , to find his Oblique-Ascention . When the Declination is North , then the difference between the Right Ascention , and the Ascentional-difference , is the Oblique-Ascention . But in Southern declinations , the sum of the Right Ascention , and difference of Asscentions , is the Oblique Ascention . Example . On or between the 25 and 26 of Iuly , the Oblique-Ascention is by Substraction 112 , 15 : On the 30th of October , the Oblique-Ascention is 337 , 45 by Addition . Use VIII . The Day of the Month , or Sun's Declination and Altitude being given , to find the Hour of the Day . Take the Suns Altitude , from the particular Scale of Altitudes , setting one Point of the Compasses in the Center , at the beginning of that Line ; and opening the other to the degree and minute of the Sun's Altitude , counted on that Line ; then lay the Thred on the Day of the Month ( or Declination ) and there keep it : Then carry the Compasses ( set at the former distance ) along the Line of Hours , perpendiculer to the Thred , till the other Point , being turned about , will but just touch the Thred ; the Compasses standing between the Thred and the Hour 12 , then the fixed Point in the Hour-Line shall shew the hour and minute required ; but whether it be the Fore or Afternoon , your judgment , or a second observation must determine . Example . On the first of August in the morning , at 20 degrees of Altitude , you shall find it to be just 52 minutes past 6 ; but at the same Altitude in the afternoon , it is 7 minutes past 5 at night , in the Latitude of 51 32 for London . Use IX . The Suns Declination and Altitude given , to find the Suns Azimuth from the South-part of the Horizon . First , by the 4th Use , find the Suns Declination , count the same on the particular Scale , and take the distance between your Compasses ; then lay the Thred to the Suns Altitude , counted the same way as the Southern-Declination is from 600 , toward the loose-piece ; and when need requires on the loose-piece , then carry the Compasses along the Azimuth-line , on the right-side of the Thred , that is , between the Thred and the Head , when the Declination is Northward ; and on the left-side of the Thred , that is , between the Thred and the End , when the Declination is Southward . So as the Compasses set to the Declination , as before , and one Point staying on the Azimuth-line , and the other turned about , shall but just touch the Thred at the nearest distance ; then , I say , the fixed-Point shall , in the Azimuth-line , shew the Suns-Azimuth required . Example 1. The Sun being in the Equinoctial , and having no Declination , you have nothing to take with your Compasses , but only lay the Thred to the Altitude counted from 600 toward the loose-piece , and in the Azimuth-line it cuts the Azimuth required . Example . At 25 degrees high , you shall find the Suns Azimuth to be 54 , 10 ; at 32 degrees high , you shall find 38 , 20 , the Azimuth . Again , At 20 degrees of Declination , take 20 from the particular Scale , and at 10 degrees of Altitude , lay the Thred to 10 counted as before ; then if you carry the Compasses on the right-side for North-Declination , you shall find 109 , 30 , from South ; but if you carry them on the left-side for South-Declination , you shall find 38 , 30 , from South . The rest of the Vses you shall have more amply afterwards . CHAP. VI. The Use of the Line of Numbers on the Edge , and the Line of Lines on the Quadrantal-side , or on the Sector-side , being all as one . HAving shewed the way of Numeration on the Lines , as in Chapter the first . Also to add or substract one Line or Number to or from one another , as in Chapter 4th , Explanation the 9th . I come now to work the Rules of Multiplication and Division , and the Rule of Three , direct and reverse , both by the Artificial and Natural-Lines ; and first by the Artificial , being the most easie ; and then by the Natural-lines both on the Sector and Trianguler Quadrant , being alike : and I work them together ; First , because I would avoid tautology : Secondly , because thereby is better seen the harmony between them , and which is best and speediest . Thirdly , because it is a way not yet , as I know of , gone by any other . And last of all , because one may explain the other ; the Geometrical Figure being the same with the Instrumental-work by the Natural way . Sect. I. To multiply one Number by another . 1. By the Line of Numbers on the Edge Artificially , thus : Extend the Compasses from 1 to the Multiplicator ; the same extent applied the same way from the Multiplicand , will cause the other Point to fall on the Product required . Example . Let 8 be given to be multiplied by 6 ; If you set one Point of the Compasses in 1 , ( either at the beginning , or at the middle , or at the end , it matters not which ; yet the middle 1 on the Head-leg , is for the most part the most convenient ) and open the other to 6 , ( or 8 , it matters not which , for 6 times 8 , and 8 times 6 , are alike ; ( but yet you may mind the Precept if you will ) the same Extent , laid the same way from 8 , shall reach to 48 , the Product required ; which , without these Parenthesis , is thus : The Extent from 1 to 6 , shall reach the same way from 8 to 48. Or , The Extent from 1 to 8 , shall reach the same way from 6 to 48. the Product required . By the Natural-Lines on the Sector-side , or Trianguler Quadrant with a Thred and Compasses , the work is thus ; 1. For the most part it is wrought by changing the terms from the Artificial way , as thus ; The former way was , as 1 to 6 , so is 8 to 48 ; or as 1 to 8 , so is 6 to 48 ; but by the Sector it is thus : As the Latteral 6 taken from the Center toward the end , is to the Parallel 10 & 10 , set over from 10 to 10 , at the end counted as 1 ; so is the Parallel-distance between 8 & 8 , on the Line of Lines taken a-cross from one Leg to the other , to the Latteral-distance from the Center to 48 , the Product required . Or shorter thus . As the Latteral 8 , to the Parallel 10 ; So is the Parallel 6 , to the Latteral 48. See Figure I. 2. Another way may you work without altering the terms from the Artificial way , as thus , by a double Radius ; Take the Latteral-Extent from the Center to 1 , ( or from 10 to 9 , if the beginning be defective ) make this a Parallel in 6 & 6 , then the Latteral-Extent from the Center to 8 of the 10 parts between Figure and Figure , shall reach across from 48 to 48 , as before . See Fig. II. The same work as was done by the Sector , is done by the Line of Lines , and Thred on the Quadrant-side , that if your Sector be put together as a Trianguler Quadrant , you may work any thing by it , as well as by the Sector , in this manner ; ( or by the Scale and Compass , as in the Figure I. ) and first , as above , Sector-wise . Take the Extent from the Center to 6 latterally , between your Compasses ; set one Point in 10 , and with the other lay the Thred in the nearest distance , turning the Compass-Point about , till it will but just touch the Thred , then there keep it ; then set one Point of the Compasses in 8 , and take the nearest distance to the Thred ; this distance laid latterally from the Center , shall reach to 48 the Product . Or as Latteral 6 , to Parallel 10 ; so is the Parallel 8 , to the Latteral 48 , the Product required . Or the last way by a double Radius , or a greater and a smaller Scale , as the Latteral Extent from the Center to 1 , is to the Parallel 6 , laying the Thred to the nearest distance ; so is the latteral Extent from the Center to 8 parts , less than the 1 before taken , carried parallelly along the common-line , till the other Point will but just touch the Thred , it shall on those conditions stay only at 48 , the Product required . Observe and note the Figure , by protraction , with Scale and Compass . 3. But if you have an Index and a Square , as is used in the Demonstrative Work of Plain Sayling , as you shall have afterwards , then the representation of this Natural-way will most evidently appear , as thus : From hence you may observe , That the first and third Numbers must alwayes be accounted alike , and on like Scales ; and the second and the fourth in like manner on like Scales and counting ; and the Latteral-first Number , must alwayes be less than the Parallel-second , in length or quantity , or you cannot work it ; which you must make so , either by changing the terms , or using a less Scale , to begin and end upon . Here you must except a Decimal gradation , as thus ; sometimes the same place which is called 10 in the first , may be counted 100 in the third , and the contrary ; or more or less differing in a Decimal Account . But if you would see a Figure of the Sector-way of operation , then it is thus ; Let the Line C 6 , represent one Leg of the Sector ; and the other Line C 6 , represent the other Leg of the Sector ; then take 1 out of any Scale , as 1 inch , or one tenth part of a foot , or what you please : or the distance from the Center to 1 , or 2 , on the Line of Lines between your Compasses : put this distance over in 6 & 6 , of the Line of Lines . Then is the Sector set to its due Isosocles Angle . Then take 8 parts , or rather 8 tenth parts of the former 1 , from the Scale from whence you took the first Latteral distance , and carry it parallel between the Line of Lines till it stay in like parts , which you shall find to be at 48 , the Product required . Or to get the Answer in a Latteral-line , is generally most convenient , by changing the terms ; work thus : Take the Latteral-distance from the Center to 8 , on the Line of Lines , make it a Parallel in 10 & 10 ; then the Sector being so set , take the Parallel-distance between 6 & 6 , and lay it latterally from the Center , and it shall reach to 48 , the Product required . See Fig. IIII. Thus you see that the way of the Sector-side , and of the Quadrant-side , is in a manner all one ; and the laying of the Thred , or Index to the nearest distance , is the same with setting the Legs of the Sector to their Angle ; and the taking the nearest distance from any Point or Number to the Thred , is the same with taking parallelly from Point to Point , or from Number to Number : So that having thus fully explained the Latteral and Parallel-Extent , and laying of the Thred , and setting of the Sector , the following Propositions will be more easie , and ready ; and to that purpose , these brief Marks for Latteral , Parallel , and Nearest-distance , will frequently be used ; as thus , for Latteral , thus — ; for Parallel , thus = ; for Nearest-distance , thus ‑ , or ND , thus : for the Sine of 90 , or Radius , or Tangent of 45 , thus R ; &c. In all which wayes you may see , that for the want of several Radiusses , which do properly express the unites , tens , hundreds , and thousands , and ten thousands of Numbers , there must a due and rational account , or consideration , go along with this Instrumental manner of work , else you may give an erroneous answer to the question propounded ; to prevent which , observe , that in Multiplication there must be for the most part , as many figures in the Product , as is in the Multiplicator and Multiplicand , put together ; except when the first figures of the Product , be greater than any of the first figures of the Multiplicator , or the Multiplicand , and then there is one less ; As for Example . 2 times 2 makes 4 , being only one figure , because 4 is greater than 2 ; but 2 times 5 is 10 , being two figures ; wherein 1 , the first figure , is less than 5. Again , in a bigger sum ; 52 multiplied by 23 , makes 1300 , consisting of four figures , as many as is in the Multiplier and Multiplicand put together ; but if you multiply 42 by 22 , it makes but 924 , which is but three figures ; because the first figure 9 is greater than 2 or 4 , as in the former , the first figure 1 was less than 5 or 2 : And this Rule is general , as to the number of places , or figures , in any Multiplication whatsoever ; but note , that no Instrument extant , and in ordinary use , is capable to express above 5 or 6 places : Yet with this help you may come true to 5 places , with a good Line of Numbers . As thus ; Suppose I was by a Line of Numbers to multiply 168 by 249 ; the extent from 1 to 168 , will reach the same way from 249 , to 41832 : Now by the Line of Numbers , you can only see but the 418 , and estimate at the 3 ; but the last figure 2 , I cannot see by any Line usually put on two foot Rules , therefore the 168 , and 249 being before you , say ( according to the vulgar Rules of Multiplication ) 9 times 8 is 72 ; therefore 2 must needs be the last figure ; and if you can see the former 4 , you have the Product infallibly true : if not , multiply a figure more : But by this help you shall be sure to come right alwayes to 4 figures , or places , in any Multiplication whatsoever . 4. Also , by these operations , you may plainly see , That the Line of Numbers , or Gunters-Line , as it is usually called , is the easiest and exactest for Arithmetical operations , being performed with an Extent of the Compasses only , without any opening or shutting of the Rule , or laying a Thred or Index : But in Questions of Geometry , where a lively draught or representation is required , as to the reason of the work , there the Natural-Lines are more demonstrative . In which Natural Work , Note , That the Parallel-distance must alwayes be the greatest , or you cannot work it , unless you make use of a greater , and a lesser Scale ; to which purpose , this Instrument is well furnished , with three or four Radiusses , bigger and less , both of Sines , Tangents , and Secants , and Equal-Parts , as in their due places shall be observed ; and taken notice of , in the Astronomical-Work . And note also , That if the Line of Lines were repeated to 4 Radiusses , or 400 instead of 10 , you might work right-on to 4 figures ; but then the Radiusses must be very small , or the Instrument very large . Therefore this of 10 , being the most usual , I shall make use of , and work every Question the most convenient way ; that by a frequent Practice , the young Beginners , for whom only I write , may see the Reason and Nature of the Work , and the sooner understand it . For a Conclusion of this Rule of Multiplication , take three or four Examples more , both by the Line of Numbers , and Equal-Parts also ; 1. First 15 foot , 8 tenths , multiplied 9 foot and 7 tenths , by the Line of Numbers ; the Extent of the Compasses from 1 to 9 foot , 7 tenths , shall reach the same way from 15 — 8 , to 153 , 26. By the Line of Lines , or Equal Parts . As the Latteral — 15 — 8 , to the Parallel = 10 , at the End counted as 1 ; So is the Parallel = 9 — 7 , to the Latteral — 153 — 26. Where you may observe , that the first and fourth , are measured on like Scales ; and the second and third , also on like Scales . But note ; that as you diminished the account in the third = work , counting 9 — 7 less than 10 reckoned as 1 : So likewise in the — fourth , you count 153 — 26 , which is less in Extent than the 15 — 8 , first taken Latterally ; yet is to be read as before , viz. 153 — 26 , because 9 times 15 , must needs be above 100. 2. As 1 , to 9 foot 10 inches ; So is 10 foot 9 inches , to 105 foot 6 inches ½ . To work this properly by the Line of ●umbers , you are first by the inches and ●ot-measure on the in-side of the Rule , to ●educe the inches into decimals of a foot ; as ●hus : Right against 10 inches , in the Line 〈◊〉 Foot-measure , you shall find 83 ● / ● . Also , ●ight against 9 inches , on the Foot-measure , ●ou shall find 75 ; this being done , which 〈◊〉 with a glance of your eye only , on those two Lines , then the work is thus : As 1 , to 10 75 ; so is 9 83 , to 105 ½ , or 105 foot , 6 inches : Now for the odd 6 square inches , you cannot see them on the Rule , but must find them by the help before mentioned , as thus , Having set down the 9 — 10 , & the 10 — 9 , as in the Margent , say by vulgar Arithmetick thus , Ten times 9 is 90 ; for which you must set down 7 foot and 6 inches , which is the 6 inches you could not see on the Line of Numbers , and there must needs be 105 foot , and not 10 foot and an half , and better , which is as to the right Number of Figures . But by the Line of Lines as the — 10 — 75 , from any Scale , as the Line of Lines doubled , or foot-measure , or the like , is to the = 10 , so is the = 9 — 83 , to the Latteral 105 — 6 ½ , as before , though not so quick or plain , as by the Line of Numbers . 3. As 1 , to 1528 , so is 3522 , to 5371616 , the true answer ; which indeed is to more places than possible the Rule can come to without the help last mentioned : But if the Question had been thus , with the same Figures , 15 foot 100 / 28 parts , by 35 foot 100 / 22 parts , as so many feet , or yards , and hundred parts : Then the answer would be as before , 537 foot ; and cutting off four figures for the four figures of Fractions , both in the Multiplicator and Multiplicand , viz. the 1616 , which makes near 2 inches of a foot more , or 16 / 100 parts of a yard more , which in ordinary measuring is not considerable . By the Line of Lines . 4. As — 15 — 28 , to = 1 , next the Center : So is = 35 — 22 , to 537 — 1616 / 100000 To multiply 3 pound , 6 shillings , and 3 pence , by it self ; the Product is , 10. l. — 19. s. — 5. d. — 1. f. — 7 / 10 : For the Extent from 1 , to 3 — 3125 , the Decimal number for 3 ● . — 6. s. — 3. d. shall reach from thence to 10 — 9726 , which reduced again , is as before , 10 — 19 — 5 — 2 ; as followeth . Note , That in this way of Multiplication by the Pen , works thus ; You must first multiply Pounds by Pounds , one over the other , as 3 by 3 : Then the Shillings by the Pounds cross-wise both-wayes , as the black-line sheweth . Then Pounds by Pence , as the long Prick-lines sheweth both-wayes also . Then Shillings by Shillings , as the 6 by 6. Then Shillings by Pence , both-wayes , as the short Prick-lines sheweth . Then lastly , the Pence by the Pence , as 3 by 3 ; whose true power , or denomination , is somewhat hard to conceive ▪ which is thus : First , 3 times 3 ( next the left-hand ) is 9 Pounds . Secondly , 3 times 6 , is 18 Shillings . Thirdly , 3 times 6 , is 18 shillings again . Fourthly , 3 times 3 , is 9 Pence , as the long Prick-line sheweth . Fiftly , 3 times 3 , is 9 Pence more . Sixtly , 6 times 6 is 36 , every 20 whereof is 1 Shilling ; and every 5 thereof is 3 Pence ; and every 1 is 2 Farthings and 4 / 10ths of a Farthing : So that 36 make 1 Shilling , 9 Pence , 2 Farthings , 4 / 10ths of a Farthing . Seventhly , 6 times 3 is 18 ; every 5 whereof is a Farthing , and every 1 is two tenths of a Farthing , as the short Prick-line sheweth . Eightly , 6 times 3 is 18 , or 3 Farthings and 6 tenths , as before . Ninthly and lastly ; to the right-hand , 3 times 3 is 9 ; where note , that there goes 60 to make 1 Farthing ; therefore 6 makes one tenth of a Farthing : So that here is 1 tenth and ½ : Consider the Scheam and the Decimal-work , to prove it exactly to the hundreds of millions of a Pound , and you will find it to be very near . Example . Product is 10. 19. 5. 1. 7. ½ . The same Decimally . Which Sum , being brought to Shillings , Pence , and Farthings , and tenths of Farthings , is just as aforesaid , viz. Or else find the Square of the least Denomination in 20 s. and divide the Product of the Sums being brought to that least Denomination thereby , and the Quotient shall be the Answer required . Example . 960 , the Farthings in 20 s. squared , is 921600. The sum of 3 l. — 6 s. — 3 d. in Farthings , is 3180 ; multiplied by it self is 10112400 : This Product divided by 921600 , the square of the Farthings in 20 s. makes 10 l. 896400 / ●216●● in the Quotient , which reduced , is 10 l. — 19 s. — 5 d. — 1 7 / 10. To find this Decimal Fraction is very easie thus , by the Line of Numbers ; for if 20 shillings be 1000 , what shall 6 shillings and 3 pence be ? Set one Point in 2 , representing 20 ; and the other in 1 , representing 1000 : then the same Extent laid the same way from 6 and ¼ , shall reach to 3.125 , the decimal fraction for 6 shillings and 3 pence . Or by inches and foot-measure ; for if you account every 8th of an inch a farthing , then every inch is 2 d. and 6 inches is 12 d. right against which , in the Foot-measure , is the Decimal Fraction required : So that right against 12 farthings , or 1 inch and ½ on the Line of Foot-measure , is 125 , the Decimal Fraction sought for . Or if one Pound be the Integer , or whole Number , then every 10th part is 2 shillings ; and every 5th is 1 shilling : and the inter-mediate pence and farthings is very near the 5th part ; for if you conceive a 5th part , or 50 of an hundred to contain one shilling ▪ or 48 farthings ; then one of 50 is very near one farthing , for 12 and ⅓ is just 3 d. and 25 is just 6 d. 37 ½ is just 9 d. and 50 just 12 d. So that to set the Compass-point to 3 l. 6 s. and 3 d. is to set the Point on 3.3125 , as before , which a little practice will make easie . By the Line of Lines on the Trianguler-Quadrant , or Sector . As the Latteral 3.3125 is to the Parallel 10 , So is the Parallel 3.3125 to the Latteral 10 l. — 19 s. — 5 d. — 2 farthings ferè , or 10.973 . Sect. II. To divide one Number by another . First , by the Line of Numbers the Rule is , Extend the Compasses from the Divisor to 1 , then the same extent of the Compasses , applied the same way from the Dividend , shall reach the Quotient required . Or the Extent from the Divisor to the Dividend , shall reach the same way from 1 , to the quotient required . Example the first . Let 40 be a Number given , to be divided by 5 ; here 40 is the Dividend , and 5 the Divisor ; and the answer to how many , viz. 8 is the Quotient . Extend the Compasses from 5 to 1 , the same Extent shall reach the same way from 40 to 8 the Quotient required ; or the Extent from 5 to 40 , shall reach the same way from 1 to 8 the quotient required . But by the Line of Lines , the work is thus ; As the — Latteral 40 , to the = Parallel 5 ; So is the = 10 counted as 1 to the — 8. Or so is the = 1 , to the — 8 of the smaller part . Observe the Figure with the Line AB . Or , as the — 5 , to the = 10 ; So is the — 40 , to the — 8 ; As the Line CB in the Figure doth demonstrate , being the manner of working by the Trianguler-Quadrant , the way of the Sector being the same . A second Example . Let 1668 , be divided by 19. As 19 to 1 , so is 1668 to 87 ●8 / 100 ; Or , 15 of 19 : Or , As 19 to 1668 , so is 1 to 87 7● / 100 ; Or , 15 of 19 , as before . For the Extent from 19 to 1668 , shall reach the same way from 1 to 87 — 78 / 100 ; the work by the Lines is as before . In this work of Division , for most ordinary questions , where there is not above four figures in the quotient , you may come very near with a good Line of Numbers , as that on Serpentine-lines , and the like ; but the difficulty is , to know the Number of Figures , which is thus most certainly done : Write down the Dividend , and set the Divisor under it , as in the vulgar way of Division ; and there must alwayes be as many Figures as the Dividend hath more than the Divisor ; and one more also , when the first figure of the Dividend is greater than the first figure of the Divisor ; as if 152178●365 were to be divided by 365 , then there would be 3 figures in the Quotient ; for the Divisor would be written 3 times under the Dividend , in the usual way of Division ; and those figures be 417 almost : But if 9172318 , is divided by 8231 , you will have 4 figures , viz. 1115 , being one figure more than 3 , the difference of places . In this Rule also you may see the excellency of the Artificial-Lines of Numbers , before and above the Natural-Lines . Sect. III. To two Lines or Numbers given , to find a third in continual Proportion Geometrical . By the Line of Numbers work thus : The Extent from one Number to the other , shall reach the same way from that second to a third , &c. Example . As 5 to 7 , so is 7 to 9 — 82 ; So is 9 — 82 to 13 — 76 , &c. ad infinitum . By the Trianguler-Quadrant , or Sector . As the Lateral first Number to the Parallel second , laying the thred to the nearest distance , there keep it : Then so is the Latteral second , to the Parallel third . Sect. IV. Any one side of a Geometrical Figure being given , to find all the rest , or to find a Proportion between two or more Right Lines . This Proposition is most proper to the Line of Lines , and not to the Line of Numbers ; and done thus : Take the Line given , and make it a Parallel in its respective Numbers ; the Thred so laid to the nearest distance , or the Sector so set , there keep it : then take out all the rest severally , and carry the Compasses parallelly till they stay in like parts , which shall be the Numbers required . Note the Figure . Note also , That the Line of Sines , and the Thred will readily lie on all the Angles , and be removed from Radius to Radius more nimbly than any Sector whatsoever , only by drawing the Thred streight , and observing on what degree and part it cuts being so laid . Let ABCDEFG be the Plot of a Field , whose side ED is only given to be 9 Chains ; and I would know all the rest : Take ED , make it a = in 9 ; lay the Thred to ND , or set the Sector to that gage , and there keep it : then measure every side severally , and you shall find what every one is in the same proportional parts , by carrying the Compasses parallelly , till it stay in like parts by the Sector , or ND , by the Quadrant . Sect. V. To lay down any Number of parts in a Line , to any Scale less than the the Line of Lines . Take 10 , or any other Number , out of your given Scale , or design any distance to be so much as you please , and set one Point in the same Number , on the Line of Lines ; and with the other , lay the Thred to the nearest distance , and there keep it , by noting the degree cut by ; then take out any other Number that you would have , setting one Point in that Number on the Line of Lines ; and opening the Compasses , till the other Point will but just touch the middle ●f the Thred , at ND , and that shall be the other part required ; or the length of so much , according to the first Scale given . Example . Figure I. Let AB represent a Line which is 100 parts , and I would lay down 65 , 30 , 42 , 83 parts of that 100. First , take all AB between your Compasses , and set one Point in 100 at 10 with the other , lay the Thred to ND , then take out 65 , 30 , 42 , 83 , &c. parallelly , and lay them down for the parts required , as here you see . The like work is by the Sector , making AB a = in 100 & 100 ; then take out = 30 , 42 , 65 , 83 , or any thing else for the parts required . But note , If the Line be too large for your Scale , or Line of Lines , then take half , or one third , or fourth part of the given Line ; then if you take half , you must at last turn the Compasses two times : If you take one third , then turn the Compasses three times ; which may prove a very convenient help in many cases , in Surveying and Dialling . Sect. VI. To divide a Line into any Number of Parts . Take the whole length of the Line between your Compasses , and setting one Point in the Number of Parts , you would have the Line divided into ; with the other , lay the Thred to ND , and there keep it ; then take the ND from 1 to the Thred , and that shall divide the Line into the parts required . Example . Let AB be to be divided into 7 parts : Take AB , make it a Parallel in 7 , laying the Thred to the ND , there keep it ; then the = 1 shall divide the Line into 7 parts . But if the Line were to be divided into many parts , as suppose 73 : Then first , fit the whole Line in = 73 ; then take out the = 72 , 71 , & 70 , for the odd 3 ; then the = 10 s. for every 10th division , then the = 1 for the smaller parts ; or else you shall find it almost an impossible thing , to take at once any distance , which , being turned above 50 times over , shall not at last happen to be more or less than the desired Number required . Note , That if the given Number happen to be such , that the Part will fall too near the Center ; as suppose 11 , 12 , or any Number under 30 ; then you may double , treble , or quadruple the Number , and then count 2 , 3 , or 4 , for one of the Numbers required . As for Example . Suppose I would divide a Line into 15 parts ; multiply 15 by 6 , and it makes 90 : Now in regard you have multiplied 15 by 6 , you must take the = 6 , instead of the = 1 , to divide the Line into 15 parts , between your Compasses , because the whole Line is set in = 90 , instead of = 15 ; which is 6 times as much as 15. Note also , if the Line be too big for your Scale , then take half , or a third , and make it a = in the given Line ; then take out the = 1 , and turn two or three times , to divide the Line according to your mind , when it is too large for your Scale . These two last are not to be done by the Line of Numbers , but proper for the Line of Sines only ; unless you turn your Lines to be divided into Numbers , and then work by Proportion , as thus ; As the whole Number of Parts , is to the whole Line , in any other parts ; So is 1 , to as many of those Parts as belongs to 1. Sect. VII . To find a mean Proportion , between two Lines , or Numbers given . A mean Proportion between two Lines or Numbers , is that Number , which being multiplied by it self , shall produce a Number equal to the Product of the two Numbers given , when they are multiplied the one by the other . Example . Let 4 and 9 be two Numbers , between which a Geometrical mean is required . 4 and 9 , multiplied together , make 36 : So also 6 , multiplied by it self , is 36 : Therefore 6 is a mean Proportional between 4 and 9. To find this by Arithmetick , is by finding the Square-root of 36. But by the Line of Numbers , thus ; Divide the distance between 4 and 9 into two equal parts , and the middle-point will be found to be 6 , the Geometrical mean proportional required . But to do it by the Line of Lines , do thus ; First , joyn the Lines , or Numbers , together , to get the sum of them , and also the half sum ; and substract one from the other to get the difference , and half the difference ; then count the half difference from the Center down-wards ; and note where it ends : then taking the half sum between your Compasses , lay your Thred to 00 on the loose-piece ; then , setting one Point in the half-difference , on the Line of Lines : See where , on the loose-piece , the other Point shall touch the Thred ; and mark the place , with a Bead on the Thred , or a speck of Ink , or otherwise : for the measure from thence to the Center is the mean Proportional required . Or else use this most excellent way by Geometry Draw the Line AB , and from any Scale of Equal Parts , take off 4 and 9 , and lay them from C , to A and B ; then find out the true middle between A and B , as at E ; and draw the half Circle ADB ; then on C erect a perpendiculer Line , as CD ; then if you take CD between the Compasses , and measure it on the same Scale that you took 4 and 9 from , and you shall find it to be 6 , the true mean proportional required : being only the way by the Line of Lines , as by considering the Triangle CDE will appear . To do this by the Sector , open the Line of Lines to a Right-Angle ( by 3 , 4 , & 5 , or 6 , 8 , & 10. thus : Take 10 Latterally between your Compasses , make it a Parallel in 6 and 8 , then is the Line of Lines opened to a Right-Angle ; or if your Rule be large , and your Compasses small , then take Latteral 5 , the half of 10 , and make it a Parallel in 3 and 4 , the half of 6 and 8 , and it is rectangle also : ) Then set half the difference on one Leg from the Center , then having half the sum between your Compasses , set one Point in the half-difference last counted , and turn the other Point to the other Leg , and there it shall shew the mean proportional Number required . 1. To make a Square , equal to an Oblong . Find a mean proportion between the length and the breadth of the Oblong , and that shall be the side of a Square equal to the Oblong . Example . Let the breadth of the Oblong be 4 , and the length 9 , the mean proportion will be found to be 6 ; Therefore a Square , whose side is 6 , is equal to an Oblong , whose breadth is 4 , and length 9 , of the same parts . 2. To make a Square , equal to a Triangle . Find a mean proportion between the half Base , and the whole Perpendiculer ; and that shall be the side of a Square equal to the Triangle . Example . If the half-Base of a Gable-end be 10 , and the whole Perpendiculer 11-18 ; the mean proportion between 10 and 11-18 , is 10-575 ; the side of the Square equal to that Triangle , or Gable-end required . 3. To find a Proportion between the Superfecies , though unlike to one another . First , to every Superfecies , find the side of his equal Square , whether it be Circle , Oblong , Romboides , or Triangle ; then the proportion between the sides of those Squares , shall be the Proportion one to another . Example . Suppose I have a Triangle , and a Circle , and the side of the Square , equal to the Circle , is 10 inches ; and the side of the Square , equal to the Triangle , is 15 inches : The Proportion between these two Squares , as they are Lines , is as 10 to 15 ; but as Superfecies , as 100 to 45 ; being thus found out , Take the Extent between 15 and 10 , on the Line of Numbers , and repeat it two times the same way from 100 , and it shall reach to 45 , the Proportion as Superfecies , between that Circle and Triangle , whose Squares equal were 15 and 10. 4. To make one Superfecies , equal to another Superfecies , of another shape : but like to the first Superfecies given . First find a mean proportion between the unequal sides of the given Superfecies , that you are to make one like ; and find the mean proportion also between the unequal sides of the Figure that you are to make one equal to . As thus for Example . I have a Romboides , whose base is 5 , and perpendiculer is 3 , ( and side is 3-55 ) the mean proportion between is 3-866 : Also , I have a Triangle , whose half-base is 8 , and the perpendiculer 4 , the mean proportional is 5-6552 ; and I would make another Romboides as big as the Triangle given , whos 's Area is 32 : Then by the Line of Numbers , say , As 3-866 , the one mean proportion , is to 5-6552 , the other mean proportion ; so is the Sides of the Romboides , whose like I am to make , to the sides and perpendiculer of the Romboides required , to make a Romboides equal to a Triangle given , and like to another Romboides first given . As thus for Example . As 3-866 is to 5-6552 ; so is 5 , the base of the Romboides given , to 7-30 , the base of the Romboides required . And so is 3 , the given perpendiculer of the Romboides , to 4-38 , the perpendiculer of the Romboides required : So also is 3-55 , the side of the Romboides given , to 5-19 , the side of the Romboides required : for , if you multiply 7-30 , the base thus found , by 4-38 , the perpendiculer now found , it will make a Romboides , whose Area is equal to 32 , the Area of the Triangle , that I was to make the Romboides equal to ; and making the side to be 3-55 , it will be like the first Romboides propounded . If it had been a Trapesia , or other formed Figure , it might have been resolved into Triangles , and then brought into Squares , as before : Then all them Squares added into one sum , whose Square-root is the mean proportional or side of a Square , equal to that many-sided Figure , whose like or equal is desired to be made and produced . 5. One Diameter and Content of a Circle given , to find the Content of another Circle , by having the Diameter thereof only given . The Extent from one Diameter to the other , being twice repeated the right-way from the given Area , shall reach to the Area required . If the Area's of two Circles be given , and the Diameter required ; then the half-distance on the Numbers , between the two Area's , shall reach from the one Diameter to the other . Sect. VIII . To find the Square-root of a Number . To do this by the Line of Numbers , you must first consider , whether the Figures , whereby the Number , whose Root you would have , is expressed , be even or odd figures , that is , consist of 2 , 4 , 6 , 8 , or 10 ; or 1 , 3 , 5 , 7 , or 9 figures . For if it be of even figures , then you must count the 10 at the end for the unite ; and the Root and Square are backwards toward 1. But if it consist of odd Figures , then the 1 , in the middle of the Line , is the unite ; and the Root and Square is forwards towards 10 : for the Square-root of any Number , is alwayes the mean proportional , or middle space between 1 , and the Number propounded ; counting the unite according to the Rule abovesaid : So that the Square-Root of 1728 , consisting of four figures , it is at 41 and ● / ● , counting 10 for the unite ; for the Number 42 & ● / ●● , is just in the middest between 1728 and 10. And to find the Square-root of 144 , consisting of three figures ; divide the space between the middle 1 and 144 , counted forwards , into two equal parts , and the Point shall rest at 12 , the Square-root required . To do this by the Line of Lines , or Sector . First , find out a Number , that may part the Number given evenly , or as even as may be ; then the Divisor shall be one extream , and the Quotient another extream ; the mean proportional between which two , shall be the Square-root required , working by the last Rule . Example . To find the Square-root of 144. If you divide 144 by 9 , you shall find 16 in the Quotient : Now a mean proportion between 9 the Divisor , and 16 the Quotient , is 12 the Root , found by the last Rule , viz. the 7th . Sect. IX . To find the Cubick-Root of a Number . The Cubick-root of a Number , is alwayes the first of two mean proportionals between 1 , and the Number given ; counting the unite with the following cautions : Set the Number down , and put a Point under the 1st , the 4th , the 7th , and the 10th figure ; and look how many Points you have , so many figures shall you have in the Root . Then if the last Point fall on the last Figure , then the middle 1 must be the unite , and the Root , the Square , and Cube will fall forwards toward 10. But if the last Point fall on the last but one , then the unite may be placed at either end , viz. at 1 at the beginning , or at 10 at the end ; and then the Cube will be one Radius beyond the unite , either forwards or backwards . But if it fall on the last but two , then 10 at the end of the Line must be the unite ; and the Root , the Square , and Cube will alwayes be in the same Radius , that is between 10 at the end , and the middle 1. So that by these Rules , the Cubick-root of 8 is 2 ; for putti●g a Point under 8 , being but one figure it hath but one Point , therefore but one figure in the Root : Secondly , the Point being under the last figure , the middle 1 is the unite ; then dividing the space between 1 and 8 , into three equal parts ; the first part ends at 2 , the Root required . So likewise in 1331 , there is two Points , therefore two figures in the Root ; and the last Point being under the last Figure , the middle 1 is the unite ; and the space between 1 and 1331 , being divided into three equal parts , the first part doth end at 11 , the Cubick-root of 1331. Again , for 64 there is one Point , and it falls on the last figure but one ; therefore the Root contains but one figure , and 1 at the beginning , or 10 at the end , which you please , may be the unite . But yet with this Caution , That the Cube must be in the next Radius beyond that which belongs to the unite ; so that dividing the space between 10 and 64 , beyond the middle 1 , towards the beginning , into three equal parts ; the first part falls on 4 , the Cubick-root required : Or , if you divide the space between 1 and 64 , near the 10 , into three equal parts , the first part falls on 4 also . Again , for 729 , there is but one Point ; therefore but one figure : Again , it falls on the last but 2 ; therefore 10 at the end is the unite ; and between 10 and the middle 1 backwards , you shall have both Root , Square , and Cube , for the Number required , which will be at 9 ; For if you divide the space between 10 , and 729 , into three equal parts , the first part will stay at 9 , the Cubick-root required . Note , if it be a surd Number that cannot be cubed exactly ; yet the Number of figures to be accounted as Integers is as before ; and the residue discoverable by the Line , is a Decimal Fraction . Example . For 1750 the Root resulting , is 12 04● / 1000 , or 12 , and near 5 of a 100. Thus you have a very good and ready way for this hard question in Arithmetick ▪ and will come near enough for most uses . But to perform this by the Natural-lines , at the best it is very troublesome , and cannot come to no such exactness , as by the Line of Numbers ; and therefore I shall omit it a● inconvenient . For Application or Use of this last Rule of finding the Cube-Root , observe with me as followeth : 1. Between two Numbers , or Lines given , to find two mean proportional Numbers , or Lines required . Divide the space on the Line of Numbers , between the two Numbers given , into three equal parts ; and the Numbers where the Points of the Compasses stay at each repetition , ( or turning ) shall be the two mean proportional Numbers required . Example . Let 4 and 32 , be two extream Numbers ( or the measure of two extream Lines ) between which I would have , two mean proportional Numbers ( or Lines ) required . In dividing the space on the Line of Numbers , between 4 and 32 , into three equal parts , you shall find the Compasses to stay first at 8 , secondly at 16 ; the two mean proportionals , between 8 and 32 , the two extream Numbers first given . For the Square or Product of 4 and 32 , the two extreams , 128 , is equal to the Square or Product of 8 and 16 , the two means multiplied together , being 128 also . 2. To apply it then thus for Example : If I have the solid content of a Cube to be 1728 Cubick inches , and the side thereof be 12 inches ; I would know what shall the side of the Cube be , whose solid content is 3456 , the double of 1728 ? Divide the space between 1728 , and 3456 , into three equal parts ; then , lay the same distance the Compasses stand at from 12 , the side of the Cube given , and it shall reach to 15-12 , the side of the Cube required , whose solid content is 3456 inches . Also , If I have a Shot of Iron , whose weight is 3 pound , and the diameter thereof 2 inches , and 780 parts of an inch in a 1000 ; what shall the diameter of a Shot be , whose weight is 71 pound ? One third part of the distance , on the Line of Numbers , between 3 pound , and 71 pound , shall reach from 2 inches , 780 parts , the given diameter , to 8 inches , the true diameter of a cast Iron Bullet , whose weight is 71 pound . 2 Secondly , on the contrary , if the Diameter and Content of one Globe , or Cube be given , and the Diameter of another Globe or Cube , to find the content thereof . As the diameter of the Globe , whose content is also given , is to the diameter of the Globe whose content is required ; so is the content given , to the content required ; by repeating the Extent the same way three times . Example . Suppose the capassity or content of a Globe , whose diameter is 10 inches , be 523 inches solid , and 80 parts ; what shall the content of that Globe be whose diameter is 20 inches ? the Extent from 10 to 20 , being turned three times from 523-8 , the content of a Globe of 10 inches diameter , shall reach to 4190 10 / 1 , the Cubick inches contained in a Globe of 20 inches diameter , being 8 times as much as the former . 3. The Proportion between the weights and magnitudes of several Metals , are as followeth , according to Marinus Ghetaldi . If 7 pieces of the 7 Metals , are all of one shape , and bigness , either Sphears , or Cubes , or Cillenders , or Parallelepipedons ; then their weights are in proportion as followeth , according to Marinus Ghetaldi . The Shape and Magnitudes equal , The Weights are in proportion , as , ♃ Tinn 1554 ♂ Iron 1680 ♀ Copper 1890 ☽ Silver 2030 ♄ Lead 2415 ☿ Quicksilver 2850 ☉ Gold 3990 So that if a Cillender of Tinn , whose side is one inch , weigh 1824 grains ; What shall a Cillender of Gold weigh , the height and diameter being just one inch , viz. 4682. For as 1554 , is to 3990 ; So is 1824 , to 4682 , the grains in one inch of Gold. The Shapes and Weights of the pieces of the seven several Metals being equal , then the Magnitudes of the sides are as followeth , according to Mr. Gunter . ☉ Gold 3895 ☿ Quicksilver 5433 ♄ Lead 6435 ☽ Silver 7161 ♀ Copper 8222 ♂ Iron 9250 ♃ Tinn 10000 So that if I have a Sphear of Iron , weigheth 9 pound , whose Diameter is 4 inches ; What must the Diameter of a Leaden Sphear , or Bullet , be of the same weight ? Say thus ; One third part of the space between 9250 , and 6435 , shall reach from 4 the Diameter of the Iron Bullet , to 3 inches 54 parts , the diameter of the Leaden Bullet , that weighs 9 pound . 4. So that if I have the weight and magnitude , of a body of one kind of Metal , and would know the magnitude of a body of another Metal , having the same weight : work thus ; The first of two mean proportionals , between the two Points on the Line of Numbers , representing the Numbers in the last Table , for the two Metals , shall reach the right way , from the Magnitude given , to the Magnitude required . As in the Example before , and illustrated by another thus ; Suppose a Cube of Gold , whose side is 2 inches , weigh 29000 grains ; What shall the side of a Cube of Tinn be , having the same weight ? Divide the space on the Line of Numbers , between 3895 , the Point on the Numbers for Gold ; and 10000 , the Point for Tinn ; and this extent parted into 3 equal parts , and that distance laid from 2 , the side of the Cube of Gold , shall reach to 2-74 ; the side of the Cube of Tinn required . 5. The Magnitudes of two bodies of several Metals being given , and the weight of the one , to find the weight of the other . Take the Extent between their Points , on the Line of Numbers , according to the last Table , for each several Metal ; and this Extent laid from the given weight , shall reach to the enquired weight , of the other Metal propounded . Example . If a Bullet of Iron , of 4 inches Diameter , will weigh 9 pound ; a Bullet of Lead , of the same Diameter , will weigh 13 pound . 6. A body of one Metal being given , to make another body like unto it of another Metal , and any other weight , to find the Diameters and Magnitudes thereof . First , by the 4th last past , find the Magnitude of the side , or Diameter of the Sphear , having equal weight ; and note that down , or keep it . Then find out two mean proportions , between the weights given ; and setting this distance , on the Line of Numbers , the right way , either increasing , or diminishing from the side given , shall shew the side or diameter required . Example . Suppose I have a Bullet of Iron weighing 60 pound , and the diameter thereof is 7 inches , 57 parts ; I would have a Bullet of Lead like unto it , to weigh one third part more ; what must his diameter be ? First by finding out of two mean proportionals , between the Points , on the Line of Numbers , for each Metal ; and laying it the right way from the side of the body given , it gives the diameter of the other body of equal weight , with the first given body ; as the Iron Bullet being 7 inches , 57 parts diameter , the Leaden Bullet of the same weight is but 6 inches , and 70 parts . Then find two mean proportionals between 60 , the weight given ; and 80 , one third part more : and repeat this the right way ( viz. increasing ) from 6 inches , 70 parts , and it shall shew on the Line of Numbers 7-37 , the diameter of a Leaden Bullet , that shall weigh one third part more than an Iron Bullet of 60 pound weight , viz. 80 pound . The performance of these Propositions by the Sector , without the help of a Line of Superfecies , and solids , is very troublesome to do : And if you have a right Gunters Sector , you may have also his Book of the Use thereof , but this way by the Numbers is as quick , and more certain , having your proportional Points or Numbers true ; but the way by the Pen is best . Sect. X. To divide a Line , or Number , by extream and mean proportion . To part , or divide a Line or Number , by extream and mean proportion , is to part it so , that the greatest part shall bear such proportion to the lesser , as the whole doth to the greatest part ; so that the square of the greater part , shall be equal to the Product of the whole , and the lesser part multiplied together : Or , if you add the Square of the whole , and the square of the lesser-part together , it is 3 times as much as the square of the greater-part . See Dig's Theorems . As if you would part 12 , by extream and mean proportion , the greater-part will be 7-42 , & the lesser-part 4-58 near ; and the whole is 12 : as by squaring and dividing you shall find . For the extent of the Compasses on the Line of Numbers , from 12 to 7-42 , being turned once , the same way , from 7-42 , shall reach to 4-58 , the lesser part . To find this by Arithmetick , do thus ; First , square the given Number ( that is , multiply it by it self ) then multiply the Product by 5 , and divide this Product by 4 ; then find the Square-root of the Quotient , and from it take half the given Number , the residue is the greater portion , then the greater part taken from the whole , leaves the lesser-part . By the Sector , work thus ; Open the Sector to a Right Angle , in the Line of Lines ( making Latteral 10 , a Parallel in 8 and 6 ) or else make the Latteral 90 , a Parallel Sine of 45 ( or the Latteral Sine of 45 , a Parallel Sine of 30 ) then upon both Legs count the given Number ; then take the Parallel Ex●ent from the whole Number on one Leg , to the half on the other Leg , and lay this from the Center Latterally ; and whatsoever the Point reacheth beyond the whole Number , must be added to the half Number , to make up the greater Number ; or taken from the half to make the lesser . Example . Let the Number given be 12 , which may be represented at 6 on the Line of Lines ; then the Sector standing at Right Angles , take the Parallel-distance from 3 , the half of 6 ( counted as 12 ) on one Leg , to 6 on the other Leg , and you shall find it reach to 6-71 , which doubled is 13-42 ; from which if you take 6 , the half sum , rest 7-42 for the greater part : and if you take 7-42 from 12 , there remains 4-58 , the lesser-part . But by the Line of Numbers work thus : following the Arithmetical way . Extend the Compasses from 1 to 12 , and that extent shall reach from 12 to 144 ; then next the extent from 1 to 144 , shall reach from 5 to 720 ; then the extent from 4 to 1 , shall reach from 720 to 180 : then to find the Square-root of 180 ; the half-distance between 180 and 1 , you will find to be at 13-42 , as before ; which used as abovesaid , gives the extream , and the mean proportional parts of 12 required . Another Example of 26. Extend the Compasses from 1 to 26 , and repeat the same again forward from 26 , shall reach to 676. Again , the Extent from 1 to 676 , shall reach from 5 to 3380. Lastly , the Extent from 4 to 1 , shall reach from 3380 , to 845 ; and the Square-root of 845 , is 29-07 : from which Number or Root , if you take half the given Number , viz. 13 ; then there will remain 16-07 , one extream : then 16-07 taken from 26-0 , rest 9-93 , the other extream required . For the Extent from 26 to 16-07 , will reach from 16-07 , to 9-93 . Another way by the Line of Sines , Geometrically . The best and quickest way is by the Line of Sines , thus ; Make the given Line a Parallel-Sine of 90 ; then take out the parallel-Sine of 38 degrees , 10 minutes , and that shall be the greater part . Also , take out the Parallel-Sine of 22 degrees , 27 minutes , and that shall be the lesser extream required : Or , according to Mr. Gunter , use 54 for the whole Line , 30 for the greater part , and 18 for the less . Also by consequence , having the mean , or greater part , make it a parallel-Sine of 38 degrees , 10 minutes ; then Parallel 90 shall be the whole Line , and Parallel 22 degrees , 27 minutes , shall be the lesser part . And lastly , having the least part , make it a Parallel in 22 degrees , 27 minutes ; then Parallel 90 deg . 10 min. shall be the whole Line ; and Parallel 38-10 , the greater part . The Use whereof , you shall have afterwards in the 11 th Chapter , about the cutting off the Platonical Bodies . Sect. XI . Three Lines or Numbers given , to find a Fourth , in Geometrical Proportion ; or , the Rule of Three direct . 1. In all Questions of the Rule of Three , there be three terms propounded , viz. two of Supposition , and one of Demand . 2. Also note , that two of the terms propounded , are of one denomination , ( or at least to be reduced to one denomination ) and one of another denomination . 3. Of the three termes propounded , ( in direct proportion ) that of Demand is alwayes the third term , and one of the terms of Supposition , viz. that of the same Denomination , with the term of Demand , is alwayes the first ; then the other of Supposition left , must needs be the second term in the Question . 4. In direct proportion Alwayes ; As the first term is to the second , so is the third to the fourth term required . 5. Having discovered which be the first , second , and third terms ; If the first and third term be of divers Denominations , they must be reduced to one Denomination , if it cannot be done on the Line in the operation , as many times it may ; As thus for instance : If one pound cost two shillings , what shall 30 ounces cost ? Here you see that the term of Demand , 30 ounces ▪ viz. the third term , is not directly of the same Denomination with one pound , the first term ; but is thus to be reduced to ounces : Saying ; If 16 ounces cost 2 shillings , what shall 30 ounces cost ? 3 s — 9 d. Thus the first and third terms , are brought to one Denomination : Also you see that the Demand or Question , viz. What shall 30 ounces cost ? is joyned to the third term ; and also that 16 ounces the first term , is of the same Denomination ; therefore the 3 s. must needs be the second term , and the Answer to the Question is the fourth . 6. Having thus discovered , which are the first , second , and third terms , and reduced the first and third to the like Denominations ; then the work by the Line of Numbers is alwayes thus ; As the first , to the second ; so is the third , to the fourth . Or the Extent of the Compasses upon the Line of Numbers , from the first , to the second ; shall reach the same way , from the third , to the fourth required . As 16 ounces is to 2 s ▪ so is 30 ounces to 2 s ¾ , or 9 pence . Or , As 16 ounces is to 24. d ; so is 30 ounces to 45 d ; which is 3 s ▪ — 9 d. as before . 7. But by the Line of Lines or Sector , if you will work on one Scale only , you must consider which term of the first or second is biggest ; for you must alwayes order it so , that the Parallel wor● must be the largest , ( or at least so as it may be wrought ) and as much as may be , that the fourth term may be a Latteral Extent , as the first alwayes is ; for then it is wrought the soonest , and also the exactest . Yet by this Instrument , you need not much care for these Cautions , having several Scales of Equal-Parts , to begin and end the work on , you are freed from that trouble . As thus for Example . When the second term is greater then the first , then the Work is well performed thus , two wayes . As the Latteral first 16 or else As Latteral third 30 from a lesser Scale . To the Parallel second 02 or else To Parallel first 16 So is the Latteral third 30 or else So Parallel second 2 To the Parallel fourth 3 75 / ●00 or else To Latteral fourth 3 75 / 100 by the same Scale . Or , As — second , to = first ; So = third , to — fourth ; by a less Scale also , if need be . But when the second term is less than the first , then the work is performed thus : If 50 Foot of Timber cost 40 s. what shall 20 Foot cost ? As the Latteral second 40 Or as before ; As — 3d 20 foot To the Parallel first 50 Or as before ; To = 1st 50 foot . So is the Parallel third 20 Or as before ; So = 2d 40 shill. To the Latteral fourth 16 Or as before ; To — 4th 16 shill. 8. Thus you see several wayes of working : but for Beginners , I would advise thus , briefly . First , either to observe this Rule of changing the terms , from the first to the second , viz. To take the second Latterally , and make it a Parallel in the first ; then the Parallel third gives you a Latteral Answer . Or else to work directly , as the first to the second , and so be content with a Parallel Answer , which you may alwayes do with the help of a smaller Scale , when need requires it . Note the Figures of Operation , by the Trianguler Quadrant . Sect. XII . The Rule of Three inversed . 1. The Rule of Three inversed , or the back-Rule of Three , is , when the term required , or fourth term , ought to proceed from the second term , according to the same proportion , that the first term proceeds from the third . As thus for Example . Car. Hour . Car. Hour . 20. 16. 10. 32. 1. 2. 3. 4. If 20 Carts carry 60 Square yards of Earth in 60 hours , how many Square yards of Earth shall 10 Carts carry in 16 hours ? Here it is apparent that fewer Carts must have a longer time to carry the like quantity ; therefore to the same time must less work be allotted , as in the work doth follow . Car. Hour . Car. Hour . 20. 16. 10. 32. 1. 2. 3. 4. For if you extend the Compasses from 10 to 20 , terms of like Denomination , viz. that of Carts ; the same extent applyed the contrary way , from 16 , the time required , by 20 Carts , shall reach to 32 , the time required by 10 Carts , to carry 60 Load . For Note , as in the former Rule of Three direct : Look how much the third term is greater than the first ; so much the fourth is greater than the second . And contrarily , Look how much the third term is less than the first , by so much is the fourth term less than the second . As thus in Numbers . As 2 is to 4 , so is 6 to 12 , for as 6 the third term , is thrice as much as 2 , the first term ; so is 12 , the fourth term , thrice as much as 4 , the second . And contrarily decreasing . As 12 is to 6 , so is 4 to 2 ; For as 4 is one third part of 12 , so is 2 one third part of 6. 2. But now in this Rule of Three inversed , or the back-Rule of three ; it is contrarily ordered , as thus ; Look how much the third term is greater ( or lesser ) than the first , by so much is the fourth term lesser ( or greater ) than the second . As thus in Numbers . As 5 is to 60 , so is 30 to 2 1 / ● ; that is , If 5 s. is 60 d. How many shillings is 30 pence ? The Answer is , 2 s. ½ . For as 30 is greater than 5● ; so is 2 ½ less than 60. Again , as 2 ½ is to 30 ; so is 60 to 5 , in the like manner . Pion. Dayes . Pion. Dayes . 18. 40. 15. 48. 1. 2. 3. 4. If 18 Pioneers make a Trench in 40 days , how many Pioneers is needful to perform the same in 15 dayes ? As 40 to 15 , so is 18 to 48 : Here , as the third is lesser than the first ; so is th● fourth greater than the second . Hors. Dayes . Hors. Dayes . 12. 30. 24. 15. 1. 2. 3. 4. Again , if 12 Horses eat 20 bushels of Provender , in 30 dayes ; how soon will 24 Horses eat up the like quantity of Provender ? The Answer is in 15 dayes . 3. The manner of working this Rule on the Line of Numbers , is thus . Extend the Compasses from one term to the other of like Denomination ; the same extent laid the contrary way from the other term , shall reach to the Answer required . As in the last Example ; the extent from 12 to 24 , the terms under the denomination of Horses , shall reach the contrary way from 30 to 15 , the number of dayes required . 4. Note , That by due consideration , this back-Rule may be wro●ght by the Precepts , for the direct-Rule , Thus : In all Questions of this nature , there be three terms given to find a fourth ; of which three terms , two are of one Denomination , and one of a different Denomination ; of which , the fourth must alwayes be ; which in the first Rule of the tenth Section before going , are called two termes of Supposition , and one of Demand . Now here you are to consider , That 5. When the fourth term required , ought to be greater than that of Demand ; which by reason you may certainly know ; Then say , As the lesser term of Supposition is to the greater ; So is the term of Demand , to his Answer , the fourth . Example . Men. Dayes . Men. Dayes . 80. 12. 40. 24. 1. 2. 3. 4. If 80 Men do a Work in 12 dayes , how soon may 40 Men do the like Work ? Here Reason tells me , that fewer Men must have longer time ; therefore the fourth term required must be greater . Therefore , As 40 to 80 , viz. As the lesser term of Supposition 40 , to the greater 80 ; So is 12 , the term of Demand , to 24 , the Answer required . 6. But if the required term ought to be lesser , which Reason will discover in like manner ; Then thus : As the greater term of Supposition , is to the lesser ; so is the term of Demand to the fourth term required . As 80 to 40 , so is 24 to 12 ; extending the Compasses the same way from the third to the fourth , as from the first to the second . But Note here , That you are not tyed to observe which is the first , second , or third term ; but to consider only the nature of the Question , that you may Answer accordingly ; and indeed this way will , generally , take in the direct Rule also . For alwayes in Direct Proportion , you may as well say , As the third term is to the first , so is the second to the fourth ; as to say , As the first to the second , so is the third to the fourth . Also backwards , or inversly ; As the third to the first , so is the second to the fourth ; extending the Compasses the contrary way . As 80 to 40 ; So is 12 to 24. 8. To perform this by the Sector , or general Scale and Thred , on the Quadrantal-side , you may generally observe this Rule ; Enter the second term taken Latterally , Parallelly in the first ▪ keeping the Sector , or Thred , at that Angle ; then the Parallel-third , shall give the Latteral-fourth , LATTERALLY . 9. Or else , As the Latteral-first , to the Parallel-second ; so is the Latteral-third , to the Parallel-fourth , PARALLELLY . And if the second be less than the first , make use of a smaller Scale ; or change the terms , as is shewed before ▪ Sect. XIII . The Double or Compound Rule of Three , Direct and Reverse . Having premised the way to bring the back ( or inversed ) Rule of Three , to be performed by the Rules for the Direct ; and considering that the Double and Compound Rules of Three are alike by the Line of Numbers ; I have therefore joyned them together in one Section . 1. The Compound , or Double ( Golden ) Rule of Three ; is , when more than three terms are propounded , or given . 2. The Double Rule of Three , is when five terms are propounded , and a sixt term proportional unto them is demanded . As thus ; If 6 Men spend 18 l. in three months ; How much will serve 12 Men for 6 months ? Or , again . If two Barrels of Beer serve 12 Men for 14 dayes ; How many dayes will 4 Barrels serve 24 Men ? 3. The five terms given consi●t of two parts , viz. a Supposition , and Demand ; as in the Rule of Three direct . The Supposition lies in these three Numbers first propounded , viz. If 6 Men spend 18 l. in 3 months ; and the Demand lies in the two remaining ▪ viz. How much will serve 12 Men 6 mon●●s ? Or in the other Example , viz. If 2 Barrels of Beer serve 12 Men 14 dayes , are the terms of Supposition ; and , how many dayes will 4 Barrels serve 24 Men , are the terms of Demand ? 4. The next work is to rank the three terms of Supposition , and the two of Demand , in their due and proper order , for convenience of Operation ; which may be thus : Of the three terms of Supposition , that which hath the same Denomination with the term required , pla●e in the second place ; and the other two , one above another in the first place : Thus ; 6 18 12 3   6 And then place the two terms of Demand one above another in the third place , only observing to keep the Numbers of like Denomination in the same ranks ; as 6 Men , and 12 Men in the upper rank ; and 3 Months , and 6 Months in the lower rank ; as in the Work is exprest . 5. When Questions of this nature are resolved by two single Rules , then the Analogy , or Proportion , is thus ; Operation I. As the first term , in the upper Rank , is to the second ; So is the third , in the same Rank , to a fourth . Again , Operation II. As the first term in the lower Rank , is to the fourth last found ; So is the other term in the lower Rank , to the term required . As in the first Example ; As 6 to 18 ; so is 12 to 36 a fourth . Again , as 3 to 36 ; so is 6 to 72 , the term required . Which by the Line of Numbers , is thus wrought ; Extend the Compasses from 6 to 18 ; the same extent applyed the same way from 12 , shall reach to 36. Then again , extend the Compasses from 3 to 36 , the same extent applied the same way from 6 , shall reach to 72 , the term required . By the Trianguler Quadrant , or Sector , thus ; 6. As — 18 to = 6 ; so is = 12 to — 36. Again , As — 36 to = 3 ; so is = 6 to — 72 , the term required . Or else work it Parallelly , observing the same order , as by the Line of Numbers , thus ; As — 6 to = 18 ; so is — 12 to = 36 , the fourth term . Again , As — 3 to = 36 ; so is — 6 , to = 72 , the sixt term required . The Double Rule of Three inversed . 7. In the other Example , is comprehended the double Rule of Three inverse ; which runs thus ; If two Barrels of Beer , serve 12 Men 14 dayes ; How many dayes will 4 Barrels serve 24 Men ? If you Rank the terms , according to the former Precept , they will stand thus : 2 — 14 — 4 12 24 or thus , 12 — 14 — 24 2 4 8. Which if you work according to the back-Rule , the way is thus ; Operation I. Extend the Compasses from 2 to 4 , term● of like Denomination , viz. of Barrels ; th●● same Extent applied the contrary w●y from 14 , shall reach to 7 , for a Fourth Proportional . Operation II. Again , Extend the Compasses from 12 to 7 , the fourth last found ; the same Extent shall reach the contrary way ▪ from 24 to 14 , the number of dayes required . 9. But if you would reduce this , to be wrought by two single direct Rules ; you must consider the Precept Rule , the 5th and 6th , of the Eleventh Section ; and the terms of Supposition and Demand ; and the increasing , or decreasing of the fourth term , which is required . As thus ; First , I part this into two single Rules , thus : Operation I. If 12 Men drink 2 Barrels in 14 dayes , then 24 Men may drink 2 Barrels in 7 dayes . Operation II. Again , If 2 Barrels last 24 Men 7 dayes , ● Barrels will last them 14 dayes ; the Answer to the Question required . Here by the 6th Rule , where the Number sought is to be less ; As 24 , the greater term of Men , is to 12 the less of the same Denonomination ; So is 14 to 7 , the fourth . Again . As 2 the lesser term , is to 4 the greater of the same Denomination ; so is 7 to 14 , the Answer required , by the 5th Rule of the 11th Section . Or else thus ; As 2 to 7 , so is 4 to 14 ; that is , the Extent from 2 to 7 , shall reach the same way from 4 to 14 , the term required . To work this by the Trianguler Quadrant , or Sector , the general Rule in this Section , Rule 6 and 7 , giveth sufficient direction . 10. The Rule of Three , compounded of five Numbers , is no other than the double Rule of Three ; and is , or for the most part , may be wrought by one Operation , having prepared the Numbers by Multiplication , for that purpose : Which two Multiplications by the Line of Numbers , though they are presently wrought , yet the two Rules of Three are done as soon ; so that the Compound Rule , is here of no advantage at all , therefore I might wave it ; yet because the only difficulty lies in the ordering the Question , I shall propound it , for the addition sake of another Example , which is this ; If the Carriage of 2 hundred weight , 30 miles , cost 4 s. What will the Carriage of 5 hundred weight cost for 100 miles ? The Numbers Ranked , according to the first Precept , will stand thus , as followeth . 11. Then for the Operation , multiply the two first Numbers one by the other ; as 2 times 30 is 60 , which is the first term ; and let the middle Number be the second term ; and the Product of the two last ( multiplyed together ) for the third term : Then the Numbers being so prepared , say , As 60 , the Product of the two first Numbers , is to 4 , the middle Number ; So is 500 , the Product of the two last , to 33 ● / ● , the Answer required . By the Line of Numbers , the Extent from 60 to 4 , will reach the same way from 500 to 33 ⅓ , or , thirty three shillings and four pence , the price of 5 hundred weight , carried 100 miles . Note , This Rule serves when it is performed by the Compound Rule of Three direct . 12. But if the inverse , or backer Rule of Three , be used in the work ; then Operate thus : As in this following Example , is manifest . A Merchant hath received 10 l. 10 s. for the Interest of a certain sum of Money for six Months ; and he received after the rate of 6 l. for the use of an hundred pound in a year ; the Question is , how much Mone● was Principal to 10 l. — 10 s. for 6 Months ? First , I range the Numbers , according to the order first propounded , in the 4th Rule of the 12th Section , as followeth . Then I observe diligently , whether the inverse Proportion be in the first or second Operation● or Line , as thus in this Question it is in the lower Line ; therefore after the Cross Multiplication , it is to be wrought by the single inversed Rule of Three ; but when the inverse Proportion is in the upper Line , it is wrought by the single Rule direct . Then I multiply the double terms across ; that is , the lowest on the right-hand by the uppermost on the left ; and the uppermost on the right , by the lowest on the left ; As thus : 6 by 6 , which makes 36 , to be set under 6 ; and 12 by 10-5 , or 10 l. ( which is 126 , and 10 s. ) and set it under 10 : then say by the inversed Rule , thus ▪ * As 126 to 36 , so is 100 to 350 , the Answer demanded ; So that 350 l. as Principal will yield 10 l. — 10 s. in 6 Months ; Or , the Extent from 126 to 36 , shall reach the contrary way from 100 to 350 , the Principal Money required . Which you may more readily prove by reasoning thus : 13. If 3 l. be the Interest of 100 l. in 6 Months , to how much Money shall 10 l. 10 s. be interest in 6 Months ? work thus ; The Extent of the Compasses from 3 to 100 , shall reach the same way from 10 l. 10 s. to 350 , the Principal Money answering to 10 l. — 10 s. the Answer required . By the Line of Lines , work thus ; As — 3 to = 10 , counted 100 ; So is — 10 ½ at the first 1 next the Center , to = 350. Or , As — 100 , to = 3 ; so is = 10 1 / ● to — 350. Sect. XIV . The Rule of Fellowship . 1. Rules of Plural Proportion are those , by which those Questions are resolved , which requi●e more Golden Rules than one ; and yet cannot be Resolved by the double ( Rule of Three , or ) Golden 〈…〉 was last mentioned . 2. Of these Rules there be ●ivers kinds and varieties , according to the nature of the Question propounded ; for here the terms given , are sometimes four , five , or six , or more ; and the terms required also more than one , two , or three . 3. The Rule of Fellowship , is to discover the Gain or Loss of every Partner in the Stock , by their several Stocks , and the whole gain or loss of the whole Stock . Also observe , That the Rule of Fellowship may be either single or double ; of both which in order . 4. The single Rule of Fellowship is , when the Stocks propounded are single Numbers . As thus for Example . ABC and D , representing the Names of 4 Men , put into one common Stock 100 l. to trade withal : A puts in 10 l. B puts in 20 l. C 30 l. and D 40 l ; and with this Stock , in a certain time , they gained 10 l. or 200 s ; Now the Question is , what ought each man to have of the 200 s. that may be proportionable to his particular Stock ? 5. The Rule of Operation is , first , by Addition find the total of all the particular Stocks , for the first term ; the whole gain ( or loss ) , for the second term ; and each particular Stock for a third term ; and repeating the Rule of Three as often as there be particular Stocks in the Question , you shall bring forth , or find out , as many fourths for the particular gains ( or losses ) of each particular Man required . As thus for Instance . The sum of the four Stocks are 100 l. The whole gain is 10 l. or 200 s. Then , For the Extent from 100 to 200 , shall reach from 10 to 20 , and from 20 to 40 , and from 30 to 60 , and from 40 to 80 ; the particular gains due to ABCD , which was required . 6. For proof whereof , if you add 20 , 40 , 60 , and 80 together , they make up 200 s , or 10 l ; the whole gain of the whole Stock . 7. The double Rule of Fellowship is , when the Stocks propounded are double Numbers . As thus for Example . AB and C , holds a Field in common , for which they pay 50 l. a year ; and in this Field , A had 25 Oxen went 30 dayes ; B had 15 Oxen there 40 dayes ; and C had 29 Oxen went there 40 dayes : What ought each man to pay for his part of the Rent , viz. 50 l ? Here you see the Stocks propounded are double Numbers , as of Oxen , and their dayes , or time of feeding ; as 25 & 30 , 15 & 40 , 20 & 40 , being double Numbers . 8. The Rule of Operation is thus , in the double Rule of Fellowship : Multiply the double Numbers , severally one by the other , one after another , and take the sum of their several Products , for the first term ; and the whole gain or loss , for the second term ; and the particular Products of every double Number , for the third term , one after another : This done , repeating the Rule of Three , as often as there be double Numbers , the 4th term produced from those Operations , shall be Answers to the Questions required , viz. the quantity of each mans gain or loss . Example . 25 & 30 , A's Oxen and time of feeding , multiplied , is 750 15 & 40 , B's Oxen and time of feeding , multiplied , is 600 20 & 40 , C's Oxen and dayes of feeding , multiplied , is 800 The Sum 2150 AS 215 , to 50 ; so is 750 A's Stock to 17-9 A's Rent   600 B's 13-19 B's   800 C's 18-12 C's 9. To work by the Line of Numbers , the Extent of the Compasses from 1 to 25 , shall reach the same way from 30 to 750 , the first Product of a A's double Number , or Stock . And as 1 to 15 , so is 40 to 600 , the Product of B's double Number , and Stock . And as 1 to 20 , so is 40 to 800 , the Product of C's double Number , and Stock . Which three Products added , make 2150 , the first term ; and 50 is the second term ; and 750 , 600 , and 800 , the three Products severally , the third term . Then , The Extent from 2150 to 50 , shall reach the same way from 750 to 17-45 , or 17 l. 9 s. And from 600 to 13-95 , or 13 l. — 19 s. And from 800 to 18-60 , or 18 l. = 12 s. the several Answers required ; which being added together , make up 50 l. the whole Rent to be paid among them . There be other Rules of Arithmetick , as the Rule called Allegation , Medial , and Alternate , and the Rule of Position or Falsehood ; in the working of which , are so many Cautions in ordering the Numbers , before you come to the proportional work , that it would make the Book more bulky than useful ; therefore I shall wave it , and refer you to the particular Books of Arithmetick , as that of Mr. Record , Dee , and Mellis ; or that of Mr. Wingate Natural and Artificial , having in it plenty of Examples ; and others also , as Iohnf●us , Iaggers , or Moores Arithmetick , any of which exceed the bounds I intend for this whole discourse ; I shall therefore pass on to the Rules of Practice , in several kinds , as measuring Superfecies , and Solids , and Rules of double and treble Proportion and Questions of Interest ; which are tedious by the Pen , without the help of particular Tables , and very easie by the Line of Numbers , as will fully appear in the next Chapters . CHAP. VII . The use of the Line of Numbers in measuring of any kind of Superficial Measure . THe Measure that is commonly used in this Work , is a Foot-Rule , divided into 100 parts ; or else into 12 inches , and those inches into halves , and quarters , or 8 parts ; or inches and 10 parts ▪ but in regard that the Numbers do most fitly agree to the 100 parts of a Foot , it will be convenient here to shew how to reduce them , or any other Fraction , from 12 s. to 10 s. or any other whatsoever , from one Fraction to the other , which by the Line of Numbers is quickly done ; as thus , from 12● to 10● . Reduction . Extend the Compasses from one Denominator to the other , the same Extent shall reach the same way from one Numerator to the other . Example . As 12 to 10 , so is 6 half of 12 , to 5 half of 10. Again . As 120 to 100 , so is 30 a 4th of 120 , to 25 a 4th of 100. Which two Lines of Inches , and Foot-Measure , are usually set together on Rules , for the ready way of Reduction by Occular inspection , only in this manner , as in the Figure ; And the like may be for any thing whatsoever , as Mr. Edmond Windgate hath largely shewed in his Arithmetick . Which Line being next to the Line of Numbers on your Rule , will be very plain and ready in the use of the Line of Numbers for feet and inches , or shillings and pence ; and the same Rule of Reduction , serves for all manner of Fractions : For as the Denominator of one Fraction is to the Denominator of the other , ( which in the Decimal work is alwayes a unite , with one , two , or more Cyphers ) so is the Numerator of one , to the Numerator of the other . And Note , That the operation of Decimal Numbers , and their Fractions , is no other than whole Numbers , except only the cutting off so many Figures as there is Fractions in the Multiplicator and Multiplicand , after any Multiplication ; as in the following Examples will appear . This being premised , I come next to the Work. Problem I. The breadth of an Oblong Superficies given in Foot ▪ Measure , to find how much in length makes one Foot. The Extent of the Compasses from the breadth to 1 , shall reach the same way from 1 , to the length required . Example at 7 10th broad . As 7 to 1 , so is 1 to 1 Foot and 43 parts The breadth given in inches , to find how much make a Foot. As the breadth in inches to 12 , so is 12 to the length of a Foot in inches , and 10 parts . Example . At 8 inches broad , you must have 18 inches to make a Foot ; for the Extent from 8 to 12 , shall reach the same way from 12 to 18. To work these two by the Line of Lines . By Inches . As — 1 to = 7 , so is = 1 to — 1 — 43 , the length in Foot-measure ; By Inches . As — 12 to = 8 , so is = 12 to — 18 ; Or else , By Inches . As — 8 to = 12 , so is — 12 to = 18 , the length in inches . Problem II. Having the breadth of an Oblong Superficies given in Foot-measure , to find how much is in a Foot long . This is soon wrought ; for in every Foot long there is just as much as the breadth is , either in Foot-measure or inches ; for a piece of Board half a Foot broad , and a Foot long , is just half a Foot. Problem III. Having the length and breadth in Foot-Measure , to find the Content in Feet . The Extent from 1 to the length , shall reach the same way from the length to the Content in Feet . Example . As 1 to 1 foot 50 , the breadth ; so is 11 foot , 10 parts , the length , to 16 foot and 65 parts , the Coment required . The breadth given in inches , and the length in feet , to find the Content in feet . As 12 to the breadth in inches , so is the length in feet , to the Content in feet required . Example at 9 inches broad , and 11 foot long . The Extent from 12 to 9 , shall reach the same way from 11 to 8 foot , 3 inches , or ¼ . By the Line of Lines . As — 11 , to = 12 ; So is — 9 , to — 8 ¼ . But Note , That in working this , and many such-like , it will be convenient to double your Scale in account , calling 10 at the end 20 , and every single figure as much more , as to call 12 , and 24 , &c. So that in this Operation , the work runs thus ; As — 11 taken from the Line of Lines , counting 1 for 10 as usually , To = 6 , the half of 12 reckoned double for 12 : So is = 4 ½ counted for 9 , to — 8 ¼ between the Center and 1. Or else thus ; As — 5 ½ counted for 11 , is to = 6 counted for 12 ; So is = 9 , to — 8 ¼ near the end , and as large as may be . Thus you may many times vary the manner of work to get the Answer latterally , and as large as may be on the Scale of Lines , by doubling or halfing the Numbers , or taking the whole Number of quarters , or using a less or a bigger Scale , as hath been hinted , and shall be more in places convenient , in the following Discourse , to attain exactness and ease , ●s much as may be , as time and practice will demonstrate to the willing Practitioner in these Operations . Problem IV. Having the length and breadth given in Inches to find the Content in Superficial-Square Inches . As 1 inch , to the breadth in inches ; so is the length in inches , to the Content in Superficial inches . Example , 20 inches broad , and 36 inches long . The Extent of the Compasses from 1 on the Line of Numbers to 20 , shall reach the same way from 36 to 72 , the true Number of Superficial Square inches in that Oblong . By the Line of Lines . As — 36 to = 5 , counted as 1 ; So is = 10 counted as 20 , to — 72 , at the largest Extent . For Note , The reason that the Latteral 72 and 36 , are from the same Scale in account ; and the Parallel 1 and 20 counted Decimally , are from the same Scale also , or else according to the Proportion by the Line of Numbers ; As — 1 to = 20 , So is — 36 to = 72. Here also is the same advance Decimally from 1 to 20 , as before . Problem V. Having the length and breadth given in Inches , to find the Content in Feet Superficial : As 144 to the breadth in inches , so is the length in inches , to the Content in Feet Superficial . Example at 40 broad , and 60 long . For the Extent on the Line of Numbers from 144 , the number of inches in one foot , to 40 the breadth , shall reach from 60 inches the length , to 16 foot ½ and 26 inches . To count so many inches on the Line , observe with me this way of Reduction , the 16 foot and ½ is very plainly seen . And Note , That there is 10 cuts in this place between 16 and 17 ; and 10 times 14 is 140 , which is near 144 , the inches in a foot ; so the Point of the Compasses staying at near 2 10ths beyond the half-foot , I count almost twice 14 , which is 26 inches for the Fraction above 16 foot and ½ . By the Line of Lines . As — 144 ( found between 1 and 2 near the Center ) is to = 40 ( at the figure 4 ) So is half = 60 , or the measure from the Center to 3 , to = 8-35 , which is the half of 16 foot ½ , and 26 inches : For if you had taken all 60 , it would have exceeded the whole Parallel Radius , where the Answer would have been right 16 ½ and better , but taking the half , it gives the half also . Or else work thus with a Latteral Answer . As ½ — 60 to = ½ 144 , So is all = 40 to — 16 7 / 10. Or , As all — 40 , to ½ = 144 ; So is all = 60 to — 33 and 4 / 10 the double of 16 and 7 / 10. Note , That in both these two last workings , the 144 is at 72 , which is the half of 144 ; to make the work the larger . By these the excellency of the Line of Numbers , over the Line of Lines , is evident in these kind of Proportions . And for discovering the Reason of these Proportions , read the beginning of the 6th Chapter , Section 3d. Problem VI. The length and breadth of an Oblong Superficies being given , to find the side of a Square equal to it , by the Line of Numbers . Divide the space between the length and breadth into two equal parts , and the middle Point shall be the side of the Square equal to the Oblong given , in quantity . Example . If a long Square , or Oblong , be 18 foot one way , and 12 foot the other way ; the middle Point between 18 and 12 is 14 and 7 / 10 ferè ; for 18 multiplied by 12 , makes 216 , and 14-7 multiplied by 14-7 , is near 216 also . To do this by the Line of Lines is shewed at large in the 7th Section of the 6th Chapter . Problem VII . Having the Diameter , or Circumference of a Circle , to find the Circumference , or Diameter , or Squares equal , or Inscribed , and Content . For this purpose there are certain Proportional Numbers found out , As thus ; If the Diameter of a Circle be 10 , then the Perifera , or Circumference , is 31 42 , the side of the Square equal to the Circle , is 8-862 , the side of the Square inscribed is 7-071 , and the Superficial Content is 78-54 ; so that any one of these being given , you may find out any of the rest by the Line of Numbers . Thus having the Diameter , to find the Circumference . As 10 to 31 42 , so is the given Diameter to the Circumference required . Or , As 10 to 8-862 , so is the given Diameter to the Square equal . As 10 to 7-071 , so is the given Diameter to the Square inscribed . As 10 to 30 , so is 78-54 to the Square of the Area of that Circle , whose Diameter is 30. Or , To the Diameter turning the Compasses twice . As for Example . Let the given Diameter of a Circle be 30 , The Circumference is 94 26 Diameter 10 00 The qu are equal is 26 58 ½ Circumference ●1 42 The Square within is 21 21 Square equal 8 862 The Content or Area is 707 00 Square within 7 071 The Diameter is 030 00 Area or Content 78 54 Also Note , If the Circumference be first given , then say , As 31 42 is to the Number 10 for the Diameter , or 8-862 for the Square equal ; or to 7-071 for the inscribed Square ; so is the given Circumference to the rest . But to find the Area say , As the fixed Diameter 10 , is to the given Diameter 30 ; so is the fixed Area for 10 , viz. 78-54 , to 707 — by turning the Compasses two times . Or if the Circumference be given , and I would find the Area . As 31-42 , the fixed Circumference , is to 94-26 the given Circumference ; so is 78-54 , the fixed Area , answering the fixed Circumference 707 , the Area required , turning the Compasses two times the same way . Thus by having five Centers at the five fixed Numbers ; or four Centers answering to the four fixed Numbers ; for a Circle whose Diameter is 10 , having any one of those 5 given , you may find any of the other required . Thus you have eight Problems couched in one ; therefore be the more diligent to understand it . To work these by the Line of Lines , observe the former directions , which for brevity sake I now omit . Problem VIII . The Content of a Circle being given , to find the Diameter . Divide the distance on the Line of Numbers , between the fixed Content , or the Point 78-54 , and the given Content into two equal parts ; that distance laid the same way , from the fixed Diameter , shall reach to the required Diameter . Example . The Content being 707-00 . The half distance between 78-54 , and 707 , shall reach from 10 to 30 , the Diameter required . Problem IX . The Content of a Circle being given , to find the Circumference . Divide the distance between the fixed and the given Contents or Area's into two equal parts , the distance laid from the fixed Circumference , shall reach to the required Circumference . Example . A Circle , whose Area is 707 , shall be 94-26 about . For the half distance between 78-54 , and 707-00 , shall reach from 31-42 the fixed Circumference , to 94-26 , the enquired Circumference . And from 8-862 , the fixed Square , equal to 26-58 ½ , the inquired Square Equal . And from 7-071 , the fixed Square inscribed , to 21-21 , the inquired Square Inscribed . Problem X. Certain Rules to measure several Geometrical Figures Superficially . For the Square , the long Square and Circle , hath been spoken to just before ; All other Figures are to be reduced to a Square , or to a long Square ; and then measured by Multiplication , as before . Or thus . Multiply the Diameter by it self ; and then that Product by 11 : then lastly , divide this last Product by 14 , and the Quotient shall be the Area , or Content , of the Circle required . For a Circle ( otherwise ) thus ; Multiply half the Diameter , by half the Circumference , and the Product shall be the Content required . For a Half Circle ; Multiply half the Diameter of the whole Circle , and a quarter of the whole Circumference together , and the Product shall be the Content . For a Quadrant , or a quarter of a Circle ; Multiply half the Arch , by the half Diameter , or Radius of the Circle and the Product shall be the Superficial Content . The like Rule holds for any lesser portion of a Circle , whose Point goeth to the Center , viz. to take half the Arch , and the whole Radius , and multiply them together , and the Product shall be the Content . Any Segment of a Circle given , to find the true Diameter . Square half the Chord , and divide the Product by the Sine , then add the Quotient and Sine together ; the sum is the Diameter . Chord is 24 ½ 12. Squared is 144 ; divided by 8 , gives 18 in the Quotient , which added to 8 , makes 26 for the Diameter . For any other Segment of a Circle , find the true semi-Diameter , and measure it as before ; then take out the Triangle , and the remainder is the true Content of the Segment . See Chapter 3.11 , 12. Or else thus , by the Line of Segments joyned to a Line of Numbers , in this manner . To the Segment given , find the true Diameter , by Chap. III. 11 , 12. Then having the Diameter , find out the Area , or Content of the Circle , by any of the former Rules , then the Proportion or Analogy is thus ; As the whole Diameer is to 100 on the Segments ; So is the Altitude of the Segment , whose Area is required to a 4th Number on the Line of Segments , which you must keep . Then Secondly , As 1 , to the whole Content of the whole Circle given ; So is the 4th Number , kept , counted on the Numbers , to the Area of the Segment required . If the Line of Segments is not on your Rule , then this Table annexed , will supply the defect , reasonably well thus : A Table to divide a Line of Segments , making the whole Circle 10000 parts . A. Table of Segments . Seg. par Seg. par Seg. par Seg. par . 70 1 1127 2 3484   6682   112 2 1151 4 3566   6786   147 3 1177 6 3645   6850   178 4 1201 8 3729   6935   206 5 1224 7 3810 35 7020 75 233 6 1248 2 3892   7106   258 7 1272 4 3971   7193   282 8 1296 6 4050   7281   307 9 1318 8 4131   7370   329 1 1341 8 4211 40 7460 80 350 1 1365 2 4290   7550   371 2 1388 4 4369   7642   392 3 1411 6 4448   7735   412 4 1433 8 4527   7829   431 5 1455 9 4606 45 7924 85 451 6 1478 2 4686   8022   469 7 1500 4 4766   8119   487 8 1522 6 4844   82●2   507 9 1544 8 4922   8327   524 2 1565 10 5000 50 8436 90 558 2 1673 11 5078   8552   592 4 1778 12 5156   8669   626 6 1881 13 5234   8788   657 8 1978 14 5314   8908   688 3 2076 15 5394 55 9029 95 718 2 2171 16 5473   9172   749 4 2265 17 5552   9330   779 6 2358 18 5●31   9505   808 8 2450 19 5710   97●0   083● 4 2540 20 5789 60 10000 100 0864 2 2630 21 5869       0892 4 2719 22 5950       0918 6 2807 23 6029       0948 8 2894 24 6108       0970 ● 2980 25 6190 65     1000 2 3065   6271       1027 4 3150   6355       1051 6 3214   6434       1077 8 3318   6516       1102 6 3402 30 6598 70     The Diameter of the Circle , answering to the Segment given , being found out , Say , As the whole Diameter to 100 ; so is the Altitude of the Segment to a 4th Number , which sought in the Table of Segments , or the nearest to it , gives in the parts the Number to be kept . Then again , As the whole Content of the Circle fixed , viz. 100 , is to the whole Content of the new Circle ; so is the Number kept , being the Content , of Area , of the fixed Segment , to the Area of the Segment required . Example . Let the Segment of a Circle , whose whole Area is 314-2 , and whose Diameter is 20 , and let the Altitude of the Segment be 5 , one 4th part of the whole Diameter . Then say As 2-000 , the whole Diameter given , is to 10000 : So is the Altitude of the Segment 5 , to 2500 , the 4th ; which sought in the Table of Segments , in the Parts , gives 19-50 for a 5th Number to be kept . Then again , As 1 , to 314-2 , the whole Area ; So is 19-50 , to 61-30 , the Area , or Content of the Segment required . For all manner of Triangles , multiply the longest side ( being properly called the base ) by half the perpendiculer , and the Product shall be the Content of the Triangle ; or as 2 to the base , so is the perpendiculer to the Content . For a Rhombus , being a Figure like a Quarry of Glass , containing 4 equal sides , and two pair of equal Angles : And any Figure having his opposite-sides Parallel one to another ; then the length of one side and the nearest distance between the other two opposite-sides multplied together , shall be the true Area required . For all other four-sided-figures , call'd Trapeziaes , being irregular Figures ; draw a Line from one corner to the other , which makes it two Triangles ; then multiply that Line , being the whole base of both the Triangles , by the half sum of both the Perpendiculers , and the Product shall be the Content required . Or , For all Regular Polligons , or Figures , with equal sides , the measure from the Center to the middle of one side , and the half sum , of the measure of all the sides multiplied together , shall be the true Area , or Content thereof . All other Figures whatsoever , of how many sides soever they be , may be reduced to Triangles , or to Trapeziaes , and measured as before ; which kind of Figure , Surveyors and Builders oftentimes meet withal , in their Operations . Problem XI . For the measuring of on Oval , the best way Ovals ▪ is to reduce it to a Circle thus ; Divide the distance on the Line of Numbers , between the length and the breadth of the Oval into two equal parts ; and the middle Point where the Compass stayeth on , shall be the Diameter of a Circle equal in Area to the Oval given . Example . Suppose an Oval be 10 foot long ( transverse ) and 8 foot broad ( conjugate ) ; the mean proportion , between 10 and 8 , is 8-95 : I say , that a Circle whose Diameter is 8-95 , is equal to an Oval of 8 broad , and 10 long ; And how to measure the Circle , is shewed before . Of these Figures . If the Content be 100 , then the sides of these Regular Figures are 〈…〉 , and also so in proportion , is any other quantity , of content required . Perpendiculer-Triangle , 13. 123. Trianguler-side , 15. 2. Square , its Side , 10. 0. Pantagon of five Sides , 7. 62. Hexagon of six , 6. 02. Heptagon of seven , 5. 26. Octagon of eight , 4. 55. Nonagon of nine , 4. 03. Decagon of ten , 3. 06. Half Diameter , or Radius , 5. 64. Example as thus : I would have a Triangle to contain 200 , What must the Sides be ? The half distance on the Numbers between 100 and 200 , shall reach from 15-2 to 21-5 , the side required . And from 13-123 the fixed perpendiculer for a Triangle , whose Area is 100 , to 18-6 , the perpendiculer of an equilatteral Triangle , whose Area is 200. But if the Sides be given , and you would find the Area , work thus ; The Extent from the fixed-side , to the given-side , shall reach at two turnings , from the fixed Area , to the Area required . The Extent from 15-2 , to 21-5 , shall reach , at twice repeating , from 100 to 200. Problem XII . To make an Oval equal to a Circle , having the Diameter of the Circle , and the length or breadth of the Oval given . S●t one Point of the Compasses in the Diameter of the Circle found out on the Line of Numbers , and the other Point to the Ovals length ; then turn that distance the contrary way from the same Diameter-point , and it shall reach to the breadth of the Oval required . Example . Let the Diameter of a Circle be 10 foot , I would have an Oval to contain as much as the Circle , and be 12 foot long ; the Query is , how broad must it be ? Set one Point in 10 , and the other in 12 , that Extent turned the other way from 10 , shall reach to 8-34 , the breadth of the Oval required . If you please to alter the breadth or length , you shall soon find the length or breadth accordingly . To work this by the Line of Lines , you must work by the Directions in the 7th Section of the 6th Chapter , as thus ; First , To find the Content of the Oval , joyn the length and breadth in one sum , to get the sum , the half sum , and difference , and half difference ; then open the Sector , ( or lay the Thred on 600 ) to a Right Angle ; Then count half the difference from the Center downwards , and note the place ; then take half the sum between your Compasses , and setting one Point in the half-difference , and extending the other to the other Leg , ( or perpendiculer Line ) and it shall shew a Point , whose distance from the Center is the mean proportional required ; which is the Diameter of a Circle , equal in Area , to the Oval , or Elipsis given to be measured ; as before is shewed . To make an Oval equal to a Circle . Take the guessed half-sum of the length and breadth of the Oval , and setting one Point in the Diameter of the Circle ; and on the other Leg , set at a Right Angle , the other Point shall shew half the difference , between the length and breadth of the Oval ; then if the mean proportional between them be equal to the Diameter , you have wrought right ; if not , then resolving upon the length or breadth of the Oval , take more or less , for the breadth or length accordingly : Herein also is seen the excellency of the Line of Numbers , in many operations . Problem XIII . The length and breadth of any Oblong Superficies given in Feet , to find the Content in Yards . As 9 foot ( the number of feet in one yard ) to the length in feet and parts ; So is the breadth , in feet and parts , to the Content in yards . Example at 13 Foot 6 Inches long , and 7 Foot 6 Inches broad . The Extent of the Compasses from 9 to 13 ½ the length , shall reach the same way from 7 ½ the breadth , to 11 yards and a quarter , the Content . Note , That if yo● measure by feet and hundred parts , you shall find this way exceeding ready ; the Answer being given in yards , and hundred parts of a yard . But if you have a yard divided into a 100 parts , to measure withal ; Then the Rule is thus ▪ As 1 to the length o● breadth , so is the breadth or length to the Content in yards . Example at 3 yards , 72 parts broad , and 5 yards , 82 parts long . The Extent of the Compasses on the Line of Numbers , from 1 to 3-72 , shall reach the same way from 5-82 , to 21 yards 65 parts , the Content in square yards , and 100 parts . By the Line of Lines . As — 5-82 , to = 1 at 10 the end ; So is = 3-72 , to — 21-65 yards . Or in the Example before . As — 13-6 , counting 6 ½ for 13 , is to = 9 ; So is = 7 ½ to — 11 ¼ ; as you counted at first . Problem XIV . The length and breadth of any Wall , being given in feet and 100 parts , to find how many Rods of Walling there shall be at a Brick and an half thick . First you must Note , That 272 foot and a quarter , makes one Rod , ( or so many feet is in a Rod ) . Secondly , That let the Walls be half a Brick , one Brick , two Bricks , two and a half , or three Bricks thick ; it is to be reduced to Brick and a half thick , as a standard thickness . Thirdly Note , That this reducing to a Brick and half thick , may be at the measuring , or after the casting-up , as you please , as in the Examples following will plainly appear . As thus for Instance ; A Front , or side-Wall of a House is to be measured , wherein the Celler-story Wall is 2 Bricks and a half thick ; The Shop and first Chamber-story is two bricks thicks ; the other Stories 1 Brick and a half thick ; and the Gable-ends 1 Brick thick . The nearest way to measure this Wall , I conceive is thus ; 1. The Cellar-story is 10 foot high , but being 2 bricks and a half thick , I make it 16 foot 8 inches high , by adding two thirds of 10 foot , to the 10 foot high , which is 6 foot 8 inches , in all 16 foot 8 inches . 2. The other two Stories , are supposed 22 foot ; but in regard they are two bricks thick , I add one third part of 22 foot , which is 7 foot 4 inches , to 22 ; and it makes 29 foot and 4 inches , the height of the Shop and next Story above . 3. The other two Stories being a brick and half thick , need no alteration , which suppose may be 19 foot . 4. The Gable-end , or Garret-story , if ●ny be , being but one brick thick ; you must take away one third part to bring it to a brick and a half . Also if it be a Gable-end , Note , it is a Triangle , and you must measure but half the height , and the whole breadth , to find the Content ; which here may be 5 foot . The Cellar Story , 16 — 8 Two next Stories , 29 — 4 Two next Stories , 19 — 0 The Garret , 5 — 0   70 — 0 5. Add all these sums of feet high together , and they make 70 ; then measure the breadth , which is common to every Room , the out-side going upright , which in a double House may be 36 or 40 foot . 6. Then having gotten the Dimentions right by the Line of Numbers , Say , As 272 ¼ ( the feet in one Rod ) is to 40 foot , the breadth of the House ; so is 70 foot , the whole height of every several Story , ( reduced ) to 10 Rod and 29 parts ; which 29 parts you may call a quarter of a Rod , and 10 foot and a half . The reason whereof is apparent thus : As 100 is to 272 ¼ ; so is 29 to near 79 ; of which 79-68 is a quarter of a Rod , or 25 of 100 is a quarter likewise , which by the Line of Numbers is apparently seen ; then every 10th part is 2 foot , and 72 of a hundred , which is near two and three quarters ; so that here 25 being a quarter of a Rod , there is 4 hundred parts more in 29 : Then thus ; the double of 4 is 8 , or , twice 4 is 8 , and four times three quarters is three foot more ; of which you must abate somewhat ( because 72 ¼ is not 75 , which is just three quarters ) and all put together , make ten rod , one quarter , ten foot and a half : for if you shall divide the Product of 40 , multiplied by 70 , which is 2800 by 272 ¼ , you shall find the Quotient to be 10 rod , 78 ½ , which is , as before , 10 rod , 1 quarter and 10 foot and a half . But note also by the way , That when you come to take out the deductions for the doors and windows , if any happen in a Wall of two Bricks and a half , or in two Bricks ; you must add two thirds , or one third more to the length or bredth one way ; and then casting them up severally , when they be of several lengths or breadths , you shall do no wrong to the Work-master nor Work-man : For true Arithmetick and Geometry will lie for no man , or use any kind of partiality . This I conceive is as near a way , as any such business can be performed . But if you will measure every Story severally , taking account of each Story severally in their thicknesses ; then , after it is cast up , the best way , by the Rule , to reduce it , is thus ; As 3 half bricks , for a brick and a half , is to any other number of half bricks thick , over or under 3 ; So is the Content at that rate accordingly , to his Content , at a brick and a half required . Example . 1269 foot at 5 half bricks thick is 2115 , for two thirds of 1269 , which is 846 , added to 1269 , makes 2115 ; For the Extent on the Line of Numbers , from 3 to 5 , shall reach the same way from 1269 to 2115 , the Number required to be found out . Otherwise thus ; To bring any kind of thickness , to one brick and a half thick , at one operation , by the Line of Numbers . For this purpose , you must use several Points , as so many gage Points , as in the short Table following doth appear . For half a brick , use 3-00000 For 1 brick , 1-50000 For 1 brick & a half , use 1-0000 For 2 bricks , 0-7500 For 2 bricks & a half , use 0-6000 For 3 bricks , 0-5000 For 3 bricks & a half , 0-4285 For 4 bricks , 0-3750 For 4 bricks and a half , 0-3333 For 5 bricks , 0-3000 For 5 bricks and a half , 0-2727 For 6 bricks , &c. ad infinitum . 0-2500 Example at the 6 ordinary thicknesses . Let a Wall be 30 foot long , and 10 foot high ; and let it be supposed of any of these thicknesses following , from half a bricks length , to three bricks length in thickness ; then thus in order , increasing , &c. First , at half a Foot. For ½ brick . As 3 to 30 ; so is 10 to 100 foot , at 1 ½ . For 1 brick . As 15 to 30 ; so is 10 to 200 foot , at 1 brick . For 1 ½ thick . As 10 to 30 ; so is 10 to 300 foot , at 1 ½ . For 2 bricks . As 0-75 to 30 ; so is 10 to 400 foot , at 1 ½ . For 2 ½ thick . As 0-60 to 30 ; so is 10 to 500 foot , at ½ . For 3 bricks . As 0-50 to 30 ; so is 10 to 600 foot , at 1 ½ . For 3 ½ thick . As 0-4285 to 30 ; so is 10 to 700 foot , at 1 ½ . For 4 bricks . As 0-3750 to 30 ; so is 10 to 800 foot , at 1 ½ . And so for any other thickness , as far as you please ; which Points are found thus ; The Exten● ▪ from the number of bricks , any Wall is thick to 15 ( or 1 and ½ ) shall reach the same way from 10 , or 1 , to the Gage-Point required for that Wall , or Walls of that thickness . Example . As 2 to 1 ½ ; so is 10 , to 0-750 , for 2 bricks thick , &c. Lastly , having the Number of Feet in the whole work , to find how many Rods there is . Say , If 272 ¼ , be one Rod ; what shall any other Number of Feet make in Rods ? The Extent of the Compasses from 172 ¼ , to 1 , shall reach the same way , from the Number of Feet , to the Number of Rods , and hundred Parts , or Rods , and Quarters , and Feet ; as by the 6th , last mentioned . Example . In 5269 Feet , how many Rods ? The Extent from 272 ¼ , to 1 , shall reach the same way , from 5269 , to 19 Rod , and 36 parts of a 100 ; or , 19 Rod 1 quarter , and 29 foot , and a quarter of a foot . The 19 Rod , and a quarter , is seen plainly on the Rule ; and 25 being a quarter , 36 is 11 parts more ; for which 11 parts more , I say , 2 times 11 is 22 foot , and 11 , 3 quarters of a foot is near 8 foot , which put together , makes 29 foot , as before : Or , as the Compasses stand , turn them the contrary way , from the Decimal parts , above the even quarter , and it shall reach to the odd feet above the quarter required . Example . The Extent from 272 ¼ , to 100 ; or 1 , shall reach the contrary way from 10 ½ , to 29 foot , the feet above ¼ of a Rod. 8. Observe , That the Tyling , the Roof , the Floors , and Partitions , are measured by the Square ; which is 10 foot Square every way , or 100 foot in Area . The Chimneys are usually done by a certain rate for a Chimney ; or if to be measured , thus are the height and breadths taken , &c. If a Chimney stand singly and alone , not leaning against , or in a Wall , the usual way is to girt it about ; and if the Jaumes are but a brick thick , and wrought upright over the Mantle-tree to the Floor ; then I say , girt it about for a length , and the height of the Story is the breadth , at a brick thick , because of the gathering together , to make room for the next Hearth above in the next Story . But if the Chimney-back be a Party-Wall , the Wall being first measured , then the brest and the depth of the two Jaumes is one side , and the height of the Story another side , to be multiplied together , at a brick and a half thick , or a brick thick , according as the Jaumes be , and nothing to be abated for the want between the Hearth and the Mantle-tree , because of the Wit hs and thickning for the next Hearth . For measuring the Shafts of the Chimneys . Girt with a Line , round about the least place of them , for one side ; and the height for the other side , at a brick thick , in consideration of the Wit hs , Pargitting , and Scaffolds . In measuring of Ceiling a foot broad , and the length of the Vallies is alwayes to be allowed , more than the whole Roof ; Also the length of the Rafter feet , above or beyond the Roof . When Rafters have their usual pitch , which is , when the breadth of the House is 12 foot , the Rafter is 9 foot long , which is 3 quarters of the Floors breadth , be it more or less ; then , I say , that the Content of one Floor , and half so much , is the Area of the whole Roof in Squares ; to which is to be added , the Vallies and Rafter-Feet , or Eves , in Tileing . And also a Deduction for Chimney-room , and Gutters , if any be . Which work by the Line of Numbers , is done at one Operation , thus ; As 6666 , is to the length of the House ; So is the breadth to the Content in the Roof . Example . A House 30 foot long , and 20 foot broad , is 900 foot , or 9 square . For the Extent , from 6666 , to 20 , shall reach the same way from 30 to 900. Also in measuring of the Roof , as to Carpenters work , by the Square , there is to be allowance for those Rafters in the Dormers , and Gable-ends , on which no Tiles are laid , as over-work for a particular use and convenience , more than need be in a bare Covering , or Roof . Also in measuring of Plasterers work in Partitions and Walls ; the Timbers and Quarters , are not to be deducted out of the rendring for Work only , except when the Workman finds the Work and Stuff also , then substract a 6th part for the Quarters in the rendring Work : But in Ceilings , the Summers which are seen , are alwayes abated ▪ and Doors and Windows also , unless by a due considerate ( or an unconsiderate ) bargain of running measure . Thus you have a brief account of the usual order , used among Workmen , in taking the Dimentions of a House , viz. Brick-work by the Rod ; Tileing and Carpenters-work by the Square ; Chimneys usually by the Fire ; And Plasterers and Painters-work by the Yard ; Glasiers , by the Foot. There are many other things to be taken notice of in the Carpenters Bill , as Lintels , Mantle-trees , and Tassels ; Luthern Lights , and other Lights , both Architrave and Plain Lights , Sky-lights , or Cubiloes , Modillean Cornish , and guttering Penthouse Cornish , Timber-Front-Story , Cellar-doors , and Door-cases ; the Plank and Curb at the Cellar-stairs , Dogleg-stairs , and Open-Newel-stairs , Canted-stairs , counted either by the step or pair ; together with the half Spaces on the Corners of the open Newel-stairs , the Rayles and Ballasters , small and great Cornish , Outside-work and Partitions , Ceiling Joysts , and the Ashlering , Boarded Partitions , and Chequer-work ; back-Doors , and Door-cases ; Window-boards , and Wall-timber ; Planks in the Foundation , Palcing , Penthouse-floors , and Penthouse-roof , furring the Platform , Centerings for the Chimney , Trimmers , Girders-ends , Ends of Brest-summers , and Plate ; and more the like , which will come in Accompt to be remembred and set down according as the Building is . Also , with due allowance into the Wall that way the ends of the Joysts are entred or laid in the Wall , as thus ; If it be Framing Work is only measured , then 9 Inches ought to be allowed into each Wall , that way the Joysts ends are laid ; because every Joyst , if well laid , should have 9 inches , at least , hold on the Wall. But if it be Timber , and Boarding , both to be measured , then 6 inches only is a competent allowance ; because the Timber is usually vallued at one third part more than the Boarding is . Also , As the Workman doth think on this , the Work-master may not forget to deduct for Stairs , and Chimneys also , where Work and Stuff are both measured ; though for Work only it may be very well allowed , unless the better Price make an allowance for it . Note also , That by the Line of Numbers , you may readily find the length of the Hips and Rafters , in a Roof of any largeness , at true pitch , by this following Proportion and Table . The Breadth of the House being 40 Feet , and the Ends Square , the Length and Angles are , as in the Table , at the usual tru● pitch .   f●et . 100 par ▪ Whole breadth 40 00 Half breadth 20 00 Rafter 30 00 Hip-Rafter 36 00 Diagonal Line 56 57 Half-Diagonal 28 28 Perpendiculer 22 36     deg . min. Hip Angles at Foot 38 22 at Top 51 38 on the Outside 116 12 Rafter Angles at Foot 48 10 at Top 41 50 For any other House , by the Numbers thus : as suppose 18 Foot broad . The Extent of the Compasses from 40 , the breadth in the Table , to 18 the breadth given , shall reach the same way from 30 , the Rafter in the Table , to 13-50 , the Rafter required . And from 36 , the Hip in the Table , to 16-22 , the Hip required . And from 22-36 , the Perpendiculer in the Table , to 10-06 , the Perpendiculer required . And from ●6-57 , the Diagonal in the Table , to ●5-48 , the Diagonal required . The Angles are alwayes the same in all Rooss , small or great , as in the Table , being Square and true pitch . If you would have Directions for Bevel or Taper Frames , to find the Lengths and Angles of Ra●ters and Hips , you may have it at large , in an Appendix to the Mirrour of Architecture ; or , Vincent Stamo●●i , Printed for William Fisher , at the Postern-Gate , 1669. By which Directions , and the Sector , you may find any thing that is there set down . As also , by the Trianguler Quadrant , Thred and Compasses . Note also , That having Inches and Foot-measure together , you may presently , by inspection , find the price of one Foot , having the price of the Square , and the contrary . Also , having the 12 Inches on the other Foot , divided into 85 parts ( near ) , and figured at every 8 with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10. shall represent pence and half farthings ; then at any price the Rod , you have the price of one Foot , & the contrary . As thus ; Let every Inch , represent one pound ; 〈◊〉 ●very 8th part , 2 shillings and 6 pence ; or every 10th part , 2 shillings ; because 8 half-crowns , or 10 two shillings , is 20 shillings . Example . Right against 6 Inches and a half , for 6 l. ●0 s. on this other Line , I find ● pence 3 farthings , the price of one Foot , at 6 l. 10 s. per Rod : And at 7 farthings per Foot , I find near 40 shillings , or 2 pound per Rod. Also , at 40 shillings per Square , found at 40 , on Foot-measure , is 4 pence 3 farthings 1 / ● per Foot , found just against it on the Inches . CHAP. VIII . The use of the Line of Numbers , in measuring of Land by Perches and Acres . Problem I. At any length of the Land , to find the breadth of the Acre . IN the Answering of this Question , it is not amiss , but very needful to premise , how many Square Inches , Feet ▪ Yards , Perches ; or Chains I mean a Chain of ●6 ●oot long ) is contained in a Square Acre of Land ; for which purpose , have recourse to the Table annexed , which is drawn with great care and exactness for that purpose . By which Table you may perceive , That 6272640 Square inches are contained in one Square Acre . And ( 100000 , or ) one hundred thousand Square Links of a 4 Pole Chain , make a Square Acre . And 43560 Square Feet , make a Square Acre . And 4840 Square Yards , make a Square Acre . And 1742 , 4 Square Paces , make a Square Acre . And 160 Square Perch , make a Square Acre . And 10 Square 4 Pole Chain , make one Acre . As in the Table you may see . And 3097 1 / ● Square Ells , make one Acre of Land , Statute measure . The Table .   Inch Links Feet Yards Pace Perch Chain Acre Mile Inch 1 7.92 12 36 60 198 792 7920 63360 Link 62.720 1 1.515 4.56 7.575 25 100 1000 8000 Feet 144 2.295 1 3 5 16.5 66 660 5280 Yard 1296 20.755 9 1 1.66 550 22 220 1760 Pace 3600 57.381 25 2.778 1 3.3 13.2 132 1056 Perch 39204 625 272.25 30 25 10.89 1 4 40 320 Chain 627264 10000 4356 484 174.24 16 1 10 80 Acre 6272640 100000 43560 4840 1742.4 160 10 1 8 Mile 4014489600 64000000 27878400 3097600 1115136 102400 6400 640 1 Square Inches Links Feet Yards Pace Perch Chain Acre Mile Then as the length of the Land given in Feet , Yards , Paces , Perches , or Chains , is to the number of Square Feet , Yards , Paces , Perches , or Chains in a Square Acre ; So is 1 to the breadth of the Land ( in that measure the length was given ) to make a Square Acre : See the Examples of all these measures in their order , viz. of Feet , Yards , Paces , Perches , and Chains . Suppose a piece of Land be 660 Feet long , or 220 Yards , or 132 Paces , or 40 Perches , or 10 Chains in length ▪ which several measures are all of the same quantity ; I would know how much in breadth I must have to make a Square Acre ? Extend the Compasses from the length given , viz. 660 Feet , or 220 Yards , or 132 Paces , or 40 Perches , or 10 Chains ; to 43560 , for Feet ; or to 4840 , for Yards ; or to 1742 , for Paces ; or to 160 , for Perches ; or to 10 , for Chains ; To the Number in the Table for that measure in a Square Acre ; the same Extent applyed the same way from 1 , shall reach to the Feet , Yards , Paces , Perches , or Chains required . Note the Work. 1. As 660 , to 43563 , the Feet in a Square Acre ; So is 1 , to 66 , the breadth in Feet required . 2. As 220 , the length in Yards , to 4840 , the Square Yards in a Square Acre ; So is 1 , to 22 , the breadth in Yards required . 3. As 132 , the length in Paces , to 1742 ; So is 1 , to 13-2 , the breadth in Paces sought . 4. As 40 , the length in Perches , to 1●0 ; So is 1 , to 4 , the breadth in Perches . 5. As 10 , the length in Chains , to 10 ; So is 1 , to 1 , the breadth in Chains required . 6. As 176 , the length in Elles , to 3097 ⅓ ; So is ● , to 17-6 , the breadth in Elles required . To work this by the Line of Lines , say ; 1. As the — 43560 , to = 660 ; So is = 10 , to 66 , Latterally . 2 : As the Latteral 220 , to Parallel 4840 ; So is Latteral 1 , to Parallel 22 , ( single , double , or four-fold ) . 3. As — 132 doubled , is to = 1742 likewise doubled , because it falls near the Center ; So is — 1 quadrupled , viz. 4 , to = 13-2 quadrupled , viz. 52-4 . 4. As — 160 , to = 40 ; So is = 10 for 1 , to — 4 Perch . 5. As — 10 , to = 10 ; So is = 1 to — 1 , the breadth required . If you would know how much breadth , at any length , shall make 2 , 3 , or 4 Acres ; Then say , As the length given to the quantity of one Acre in that measure , according to the Table ; So is 2 , 3 , 4 , or 5 , to the breadth required . Example at 30 Perch in length . The Extent from 30 to 160 , shall reach the same way from 4 to 21 Perch , and 34 of 100 ( or 5 Foot , 06 Inches ) the breadth of 4 Acres , at 30 Perches in length . Problem II. The length and breadth given in Perches , to find the Content in Perches , of any piece of Land. The Extent from 1 , to the breadth in Perches , shall reach the same way , from the length in Perches , to the true Content in Square Perches . Example . As 1 to 50 , so is 179 to 8950 , the Content in Square Perches . Problem III. The length and breadth being given in Perches , to find the Content in Square Acres . The Extent from 160 to the breadth in Perches , shall ●each the same way , from the length in Perches , to the Content in Square Acres . Example . As 160 to 50 , so is 179 to 5-58 Acres , or 5 A●res , 2 Rood , and 13 Perches . Problem IV. The length and breadth of a piece of Land being given in Chains , to find the Content in Acres . The Extent from 1 , to the breadth in Chains , and 100 parts , which are Links , shall reach the same way from the length in Chains and Links , to the Content in Square Acres . Example . As 1 to 5 Chains , 52 Links , the breadth ; So is 8 Chains , 72 Links , to 48 Acres , and 3960 Square Links . Problem V. Having the Base and Perpendiculer of a Triangle given in Chains or Perches , to find the Content in Acres . The Extent from 2 , if you use Chains ; or from 320 , if you measure by Perches , to the whole Base , shall reach the same way from the whole Perpendiculer , to the whole Content of the Triangle ; or if it be a Trapezia , joyn both the Perpendiculers in one sum . Example . As 2 ( for Chains ) to 3-63 , the whole Perpendiculer ; So is 11-80 , the whole Base , to 21 Acres , 42 Links , the Content of the whole Triangle . Or in Perches . As 320 to 14-55 , the Perpendiculer in Perches ; So is 47-20 , the length , or base Line , in Perches , to 21 Acres , 24 Links , the Content in Acres . Problem VI. The Area , or Content of a piece of Land given , that was measured by Statute-Perches ; to find the Content of the same piece of Land in Wood-land measure , or Customary Acres , or Irish Acres . For the better understanding of this Problem , it is necessary to describe the several kinds and quantities of Perch●s , which are spoken of by Authors , and used in several places ; together with their proportion to the Statute Perch of 16 Foot and a half square , London measure . The kinds of Perches , are first Statute-measure of 16 foot ½ to the Perch , according to the Standard at Guild-Hall , or the King's Majesties Exchequer . Secondly , Woodland-measure , a Perch whereof contains 18 Foot Square of the same London measure . Thirdly , Irish Acres , of 21 Foot to the Perch or Pole. And lastly , Three sorts of Customary , used in several places of England , of 20 , 24 , Cheshire measure , and 28 Foot square to the Perch . As for the Proportions one to another , that is , as 16 ½ , to 18 , 20 , 21 , 24 , 28 , or any the like wha●soever . But to find their difference in Squares or Scales , the Work is thus ; By the Line of Numbers , First appoint what Number in an Inch shall be the Scale for Statute measure , which I shall appoint a Scale of 30 in an Inch. Then the Extent from 16 ½ , to 18 , for Woodland measure , shall reach the contrary way from 30 , being twice repeated , to 25-2 ; so , I say , that a Scale made to 25-2 in an Inch , shall be the Scale for a Woodland Perch of 18 Foot Square ; and in proportion to that of 16 Foot ½ , at 30 parts in an Inch. Again , For Irish Acres , which are measured by a Pole of 21 Foot to a Perch , the Extent on the Line of Numbers from 21 to 16 ½ , shall reach ( being turned twice the same way ) from 30 , to 16 ½ , the quantity of the Scale for Irish Acres , to be in proportion to a Scale of 30 in an Inch for Statute-measure ; and so for the rest , or any other whatsoever , as in the following Table . 16 ½ The Scale that is to it proportionable to 30 for Statute measure is , 16 ½ is 30 — 00 Statute-Measure . 18 18 25 — 22 Woodland-Measure . 20 20 20 — 42 Customary . 21 21 18 = 50 In an Inch for Irish. 22 22 16 — 89 Customary . 24 24 14 — 20 Customary Cheshire-measure . 28 28 10 — 42 Customary . 30 30 09 — 08 Or any other . So that if you have several Scales made upon a Rule ( to draw the Plot of your Field withal ) to these Proportions ( which may be convenient enough for Difference between one another ) , then for the reducing of the quantity of Acres found by Statute-measure , to Woodland , Irish , or Customary , is no more but thus : Take the Acres , measured by Statute-measure , out of the Scale of 30 in an Inch appointed for Statute measure , and measure it in the Scale of 25-22 in an Inch for Woodland ; or by the Scale of 18-55 for Irish Acres ; or by the Scale of 16-89 for Customary ; and you shall have the quantity of Woodland , Irish , or Customary Acres required . Example . Suppose I have 30 Acres of Statute measure , how many Acres of Woodland , Irish , or Customary measure will they make ? Take 30 from the Scale of 30 in an Inch , and on the Scale of 25-22 , it shall give 25-22 , for so many Woodland Acres ; and on the Scale of 18-55 , for Irish Acres , it shall give 18-55 for so many Irish Acres ; and on the Scale of 16-89 in an Inch for Customary Acres , it shall give 16-89 for so many Customary Acres , at 22 Foot to the Perch or Pole , &c. This being thus fully premised , to work these Questions by the Line of Numbers only : the Extent of the Compasses from 1●-5 , the Feet in a Statute Perch , to 18 the Feet in a Woodland Perch ( or to 21 the Feet in an Irish Perch , or to 22 , 24 , 28 the Feet in a Customary Perch ) shall reach from 30 , the Acres in Statute measure , beng twice repeated , to 25-22 , the Acres in Woodland measure required , &c. it being a larger-Acre ▪ must nee●s be less in quantity . Which work is performed by the back-Rule of Three in a duplicated proportion . Problem VII . Having the Plot or Draught of a Field , and its Content in Acres , to find by what Scale it was Plotted ; that is , by what parts in an Inch. Suppose a Triangle , or a Parallellagram , or long Square , do contain 4 Acres and a half , which is set down in figures thus , 4-50 ; which if I should measure by a Scale of 12 in an Inch , might happen to be 2-25 Chains one way , and 1 Chain , 25 Links ●he other way ; which two sums being multiplied together , make 2-5200 , whereas it should be 4-5000 ; Therefore by th● 〈◊〉 of Numbers , to gain 〈…〉 do thus ; Divide the dista●ce between 2-5200 , and 4-5000 , into two equal parts ; that distance laid the right way from 12 , the Scale I measured by , shall reach to 16 , the Scale the Plot was made by . For Note , That if the Scale I guessed a● , gives more than I should have , then I have too many in an Inch ; but if less , I must have more in an Inch , as here , which infallibly sheweth which way , which is alwayes the same way as you divided the space , from the guessed Sum or Product , to the true Product . To this Rule may be referred the way to discover the true size of Glasiers Quarries ; the method whereof is thus : They are usually cut to , and called by 8s , 10s , 12s , 15s , 18s , and 20s in a Foot , or any other what you please ; that is to say , 8 quarries of Glass of 8s , make a Superficial Foot ; and 10 quarries of 10s , make a Foot Superficial ; and 12 of the 12s , &c. Also they are cut in a Diamond form to one sort of Angle for the Squ●re quarries ; and another for the Long quarries : The acute Angle of the Squar● quarries being 77 degrees , and 19 〈…〉 and the acute Angle of the Long quar●●es ●7 degrees and 22 minutes : The long ●as being just 6 inches long , and 4 inches broad ; and the Square 10s , 6 inches long , and 4 inches , and 80 parts of a 100 broad ▪ This being the standing Rule or Method , and those two sizes being known , I would find out any other , as 13s , or 14s , or 17s and the like . Do thus ; Divide the distance on the Line of Numbers , between the Content of some known size , and the Content of the inquired size into two equal parts ; and that distance laid the right way from the sides of the known size ( increasing for a bigger , and decreasing for a less ) shall give the reciprocal sides of the size required . Example . The Sides , Ranges , Lengths , and Breadt● of Square 10s , are as in the Table following ; and I would have the Ranges , Sides , Length , and Breadth of 14s , an unusual Size . The Content of a Square quarry of Glass called 10s , is a just 10th part of a Foot , which is 1 inch and 20 parts ; or one 10th part of a Superficial Foot , containing 12 long inches . And the Content of the size called 14 s , must be one 14th part of the same measure , or Foot Superficial , which is 0-85714 , that is 0-857 parts of one long inch in a 1000 parts . Then , by the Line of Numbers , divide the space between 1-2000 , the Content of the 10s , and 0-857 the Content of the 14s into two equal parts ; that Extent , I say , laid the same way from 3-76 , the Ranges of Square 10s , shall reach to 3-18 , the Ranges for 14s : And from 3-84 , the sides of Square 10s , to 3-25 the sides of Square 14s : And from 4-80 the breadth of Square 10s , to 4-05 the breadth of Square 14s : And from 6-00 , the length of Square 10s , to 5-07 the length of Square 14s , the requisites of the unknown Size required . And the like for any other whatsoever . The true size of Glasiers Quarries , both Long and Square , By J. B. 1660. Square Quarries 77 deg . 19 min. Long Quarries 67 deg . 22 min. Sizes Rang. Sides . bredth length Content . Content . Size . Rang. Sides . bredth length Content . Content . In. 100 In. 100 In. 100 In 100 Inc. 100 Tens . In. 100 In. 100 In. 100 In. 100 Inc. 100 Tens . 8 4 20 4 30 5 36 6 70 1 500 1 250 8 4 09 4 41 4 90 7 34 1 500 1 250 10 3 76 3 84 4 80 6 00 1 200 1 000 10 3 65 3 95 4 38 6 57 1 200 1 000 12 3 43 3 51 4 38 5 47 1 000 0 833 12 3 34 3 61 4 00 6 00 1 000 0 833 15 3 07 3 13 3 92 4 90 0 800 0 667 15 2 98 3 23 3 58 5 37 0 800 0 667 18 2 80 2 86 3 57 4 77 0 666 0 555 18 2 72 2 95 3 26 4 90 0 666 0 555 20 2 66 2 72 3 39 4 24 0 600 0 500 20 2 58 2 79 3 10 4 65 0 600 0 500 CHAP. IX . The use of the Line of Numbers in measuring of Solid measure , as Timber , Stone , or the like Solid bodies . Problem I. A piece of Timber being broader one way than the other , to find the side of a Square that shall be equal thereunto , being called , Squaring the Piece . THe side of the Square , that shall be equal to the square of the Oblong , is nothing else but a mean proportion between the length and breadth of the Oblong : As thu● ; Suppose a piece of Timber is 12 inches in depth , and 16 inches in breadth ( and 10 foot in length . ) 16 the breadth , and 12 the depth , multiplied together , make 192 ; the Square or Product of 16 and 12 multiplied . Now the square Root of 192 , which is near 13 ●59 / 1000 , is the side of a Square , equal to 12 and 16 , the depth and thickness of the piece of Timber propounded . For if you shall multiply 13-859 by 13-859 , you shall find 192-071881 , the nearest Root , you can express in 5 figures , and an indifferent true mean proportion , between 12 and 16 , the depth and breadth ; so that in fine , 13-86 , is the side of a Square , nearly equal to 12 and 16 , whereas the doubling and halfing , the old false way , gives full 14. To work this by the Line of Numbers , is thus ; Divide the distance on the Line of Numbers , between 12 and 16 , into two equal parts , and you shall find the Point to stay at 13 , and near 86 parts , the Answer required . The way of doing it , by the Line of Lines , is shewed in the VI Chapter , and 7th Proposition , either by the Sector , or Trianguler Quadrant , and therefore needs no repetition in this pla●e . Problem II. At any Breadth and Depth , or Squareness , to find how much makes a Foot of Timber . 1. If the Timber be square ( or squared ) then the way by the Line of Numbers , is thus ; First by Foot measure . Extend the Compasses from the side of the square to the middle 1 , the same Extent applyed , or turned twice the same way from 1 , shall reach to the length that makes a Foot of Timber , at that squareness . Example . Suppose a piece of Timber be 50 of a 100 ( or 6 inches ) or half a Foot Square ( which is all as one ) Extend the Compasses from 5 to 1 ( forwards ) the same Extent being turned two times , the same way from 1 , shall reach to 4 , being 4 Foot , or 40 such parts , whereof the side of the Square was 5. 2. Secondly , The same again by Inches . The Extent from 6 to 12 , shall reach , being turned two times the same way from 12 , to 48 , the number of inches in length that makes a Foot , at that Squareness ; being 48 such parts whereof the side of the Square was 6. So that , As the side of the Square , in inches , is to 12 : so is 12 to a 4th , and so is that 4th to the length of a Foot required , turning the Compasses twice , the same way as you turned from the side of the Square in inches to 12. 3. If the piece of Timber , or Stone , be not Square or Squared , Then The Extent from 1 to the depth , shall reach the same way from the breadth to a 4th Number . Again , The Extent from that 4th Number to 1 , shall reach , being turned once , the same way from 1 , to the length of a Foot in Foot-measure required . Example . Suppose a piece of Timber be 0-333 deep , and 0-750 broad in Foot-measure ; or 4 inches deep , and 9 inches broad , as with a glance of your eye on inches and foot-measure , you may see how these Numbers agree . The Extent , I say , from 1 to 0-333 , shall reach the same way from 0-750 to 2-50 . Again , I say , The Extent from 250 the 4th , to 1 , shall reach the same way , from 1 to 40 , or 4 Foot , the length required , to make a Foot at that breadth and depth . 4. By Inch-measure , 〈◊〉 find the length of a Foot in Inches . As 12 to the breadth in inches , so is the depth in inches to a 4th ; then as that 4th to 12 , so is 12 to the length in inches required . Example . The Extent from 12 to 9 the breadth , shall reach the same way from 4 the depth , to 3 for a 4th . Then the Extent from 3 the 4th to 12 , shall reach the same way from 12 to 48 , the inches in length required , to make a Foot. 5. The breadth and depth given in Inches , to find the length of a Foot of Timber , in Feet and Parts . Then say , As 1 , to the depth ; so is the breadth to a 4th . Again , As that 4th to 12 , so is 12 to the length in feet and parts . Example . The Extent from 1 to 4 , shall reach the same way from 9 to 36 , a 4th ; Then , The Extent from 36 to 12 , shall reach the same way from 12 to 4 foot , the length in feet required . The reason of this Order and Method , if you consider , you will find thus ; In the 4th way of working , you went thus ; As 12 , the inches in a foot , is to the breadth in inches ; So is the depth to 3 Foot. But in the 5th and last way you say , As 1 foot to the depth in inches ; So is the breadth to 36 inches , which is 3 foot also . But altering the Order in the beginning , alters it in the issue , though the same truth , yet in or under divers denominations ; for 48 inches , and 4 foot , are the same ; yet sometimes one way is more convenient than another . Problem III. At any Squareness , or Breadth and Depth given in Foot-measure , or Inches , to find how much Timber is in a Foot long , in Foot-measure , or feet , and 100 parts or inches . 1. If the piece of Timber be Square ( or Squared ) then work thus for Foot-measure . As 1 , to the side of the Square , so is the side of the Square to the quantity of Timber in one Foot long ; which multiplied by the length , gives the whole Content required . Example . At 50 , or half a Foot Square , how much is in a Foot long ? Extend the Compasses from 1 to 5 , the same Extent turned the same way from 5 , reaches to 25 , or a quarter of a Foot ; then if the Tree be 12 foot long , 12 quarters will make 3 foot , the Content . 2. The Side of the Square given in Inches , to find the Quantity , or Content , in a Foot. As 12 , to the side of the Square , so is the side of the Square to 3 twelve parts of a Foot Solid , or ¼ of a Foot. Or , As 1 , to the side of the Square , so is the side of the Square to 36 , 144th parts of a Foot Solid . Example . The Extent from 12 to 6 , the inches Square , shall reach the same way from 6 to 3 inches in a Foot long , which is 3 12th parts of a Foot Solid . Again , The Extent from 1 to 6 , the inches square , shall reach the same way from 6 to 36 , the number of long inches in a foot long ; or pieces of 1 inch square , and a foot long , 144 of which makes one foot of Timber . 3. But if the Piece be not square ( or squared ) then to find how much is in a Foot-long , work thus ; As 1 to the depth , so is the breadth , to the quantity in a Foot. Example 3 wayes : At 9 and 4 breadth and depth . 1. The Extent from 1 to 0-333 , shall reach the same way from 0-75 , to 0-25 , or a quarter of a Foot ; for Foot-measure . 2. The Extent from 1 to 9 , shall reach the same way from 4 to 36 , the long inches in a foot long ; for Inch-measure . 3. The Extent from 12 to 4 , shall reach the same way from 9 to 3 inches , or 3 12ths , viz. a quarter of a Foot ; for Inch-measure . Problem IV. The side of the Square , or the breadth and depth given in Inches , or Foot-measure , and the length in F●●t , to find the Quantity , or Conten● of the whole Piece , in feet and parts . 1. First , for Foot-measure ; As 1 , to the side of the Square , in Foot-measure , so is the length in Feet to a 4th , and then that 4 to the Content in feet and parts . Example . The Extent from 1 to 0-833 , the side of the Square , shall reach from 10 foot 25 parts , the length to 8-54 , and from thence to 7-11 , the Content in feet and parts required . 2. For Inch-measure , Say , As 12 to the side of the Square , in inches , so is the length in feet , to a 4th ; and then that 4th to the Content in feet and parts . Example at 10 Inches Square , and 10 Foot , 3 Inches in length . The Extent of the Compasses , on the Line of Numbers , from 12 , to 10 inches Square , shall reach the same way from 10 foot ¼ , or 3 inches , to 85-4 for a 4th ; and from thence to 7-11 , or 7 foot 1 inch , and a third part , the Content required . As by looking for 11 on the Line of Foot-measure , right against which , on the inches , is 1 inch and a quarter , and somewhat more . 3. But if the piece of Timber be not square , and you would measure it without squaring , by the first Problem ; Then say first by Foot-measure , thus ; As 1 is to the breadth , so is the depth to a 4th . Then , As 1 to the 4th , so is the length in feet to the true Content , in feet and parts . Example . Let a Timber-tree of one foot 25 , or a quarter one way , and one foot 50 the other way , and 12 foot long be measured . The Extent of the Compasses from 1 to 1-25 , shall reach the same way from 150 , to 18-74 , for a 4th . Then the Extent from 1 to 18-74 , shall reach the same way from 12 foot , the length , to 22-50 , for the Content ; viz. 22 foot and a half , the whole Content required . 4. When the breadth and depth is given in Inches , and the length in Feet , to find the Content without squaring . As 12 , to the breadth in inches ; So is the depth in inches to a 4th : Then , As 12 to that 4th , so is the length in feet and parts , to the Content in feet and parts required . Example at 15 inches deep , and 18 inches broad , and 13 foot long . Extend the Compasses on the Line of Numbers from 12 to 15 the depth ; the same Extent applied the same way from 18 the breadth , shall reach to 22-50 , for a 4th . Then the Extent , from 12 to 22-50 , the 4th , shall reach the same way from 13 foot , the length , to 24 foot 38 parts , or 4 inches and a half , as a glance of your eye to the Inches and Foot-measure will plainly shew . Thus you have the Solution of any Question that may concern proper Measuring by Foot-measure , and Inches ; using only the Center at 10 for Foot-measure , and at 12 for Inch-measure , without troubling you with 144 , or 1728 , or 41-57 , or the like , as in the little Book of the Carpenters Rule , may be seen . To work these Questions by the Line of Lines , though it may be done several ways , yet no way so soon , nor so exact , as by the Line of Numbers : Yet I shall shew now in this place , together by themselves , the Three principal Questions , viz. How much makes a Foot in quantity ; And , How much is in a Foot long ; And , By the length , breadth and depth , the Content in Feet : In the doing whereof , you must conceive the 10 principal parts to be doubled , and then 10 is called 20 ; and consequently 6 is called 12 , the Point so often used ; and 5 is called 10 , the Point used for Foot-measure . 1. To find how many Inches makes a Foot at any Squareness . As the — side of the Square , to = 12 ; So is the = side of the Square again , to a — 4th Number . Again , As = 12 , to that = 4th Number ; So is = 12 , to the — Number of Inches that goes to make a Foot of Timber . Example , at 8 Inches Square . Take the distance from the Center to 4 , accounted as 8 ; and make it a Parallel in 6 , counted as 12 ; or lay the Thred to the nearest distance , and there keep it . Then , take the nearest distance from 4 to the Thred , and that shall be a Latteral 4th . Then take the Latteral distance from the Center to 12 , according to the usual account , and make it a Parallel in the 4th last found , laying the Thred to the nearest distance , and there keep it ; then take the nearest distance from 6 , counted as 12 , to the Thred , and that shall reach Latterally from the Center to 27 Inches , the length required , to make a Foot of Timber , at 8 Inches Square . Which work I more briefly word thus , as formerly is done . As — 4 , counted as 8 , to = 6 , counted for 12 ; So is = 8 , to a — 4th . Then , As = 12 , to = 4th ; So is = 12 , to — 27 , the length in inches required . 2. If you would use Foot-measure , count the 5 in the midst for 10 , or 1 Foot ; and work all the rest as before : As thus for Example . In the same quantity , Square , exprest in Decimals : As — 0-666 , counted double , to = 5 , counted double for 10 ; So is = 0-666 , to — 22 1 / ● , for a 4th . Then , As — 1 , to = 22 1 / ● ; So is = 5 counted for 1 , to 225 , which is 2 Foot ¼ , as by the Foot-measure and Inches you may see . 3. If the Piece be not square , then say thus ; As — breadth , to = 12 ; So is the = depth , to the — 4th . Then , As — 12 , to the = 4th ; So is = 12 , to — length that goes to make 1 Foot. Example , at 9 Inches , and 4 Inches , for breadth and depth . As — 9 , to = 12 ; So is = 4 , to — 150 , for a 4th . As — 12 , to = 4th , best taken at 75 for largeness sake ; So is = 12 , to — 48 Inches . Or else thus ; As — 9 , to = 1 ; So is = 4 , to — 1-80 , a 4th . Then , As — 12 , to = 1-80 ; So is = 12 , to — 4 Foot , the length in Feet , that goes to make 1 Foot of Timber . 4. To find how much is in a Foot-long , at any Squareness . As the — side of the Square is to = 1 , counted double as before ; So is the = side of the Square to the — quantity in a Foot. Example at 6 Inches , or ( 5 ) half ● Foot Square . As — 5 , to = 1 ; so is = 5 , to — 125 for Foot-measure : Or , As — 6 , to = 12 ; so is = 6 , to — 3 , for 3 inches , or ¼ of a foot . 5. The side of the Square given in Inches , and the length in Feet , to find the Content in Feet . As — side of the Square , to = 12 ; So is the = length to a 4th . Then , As — 4th ; to = 12 ; so is = side of the Square to — Content required , in feet and parts . Example , at 9 Inches Square , and 16 Foot long . As — 9 to = 12 , so is = 16 to — 12. Again , As — 4th , viz. 12 , to = 12 ; So is = 9 , the square to 9 , the true Content of such a Piece in feet and parts required . The like Work serves for Foot-measure , using of 1. 6. The Length given in Feet , 〈◊〉 the Breadth and Depth in Inches , to find the Content in feet and parts . As — breadth , to = 12 ; So is = depth , to a 4th . Then , As — 4th , to = 12 ; So is = length , to — Content in feet . Example at 5 Inches and a half Deep , and 15 Inches Broad , and 16 Foot Long. As — 5 ½ , to = 12 ; So is = 15 , to — 69 for a 4th ( at 34 ½ ) Then , As — 69 ( or 34 ½ ) that — 4th , to = 12 ; So is = 16 , taken at 8 , to — 9 foot 2 inches ; the Content required . Thus you see the way and manner of working by the Line of Lines , either on the Quadrant , or Sector-side , for the usual Questions ; for I have neglected to give the Content of Pieces in Cube Inches , for two Reasons : First , Because it is very seldom required . Secondly , Because the Line of Numbers at most will shew but 4 figures , which is not sufficient for any Piece above 6 Foot , therefore not fit for Instrumental Work. And withal you may observe , That alwayes the Latter●l Extent first taken , must be less than the distance from the Center to the parallel Point of Entrance , which in these Examples is remedied by calling 6 12 : And also , there are so many Cautions in doubling and halfing of Numbers , to make it applicable , that without due consideration , you may soon err ; Also , the opening and shutting the Rule , and using of several Scales , makes it far inferior to the Line of Numbers , which may be easily enlarged . CHAP. X. To measure Round Timber , or Cillenders , by the Line of Numbers . Problem I. Having the Diameter of a Cillender , given in Inches , or Foot-measure , to find the length of one Foot. 1. AS the Diameter in inches , to 46-90 , ( at which Diameter one Inch makes a Foot ) ; So is 1 to a 4th , and that 4th to the length in inches . Example at 10 Inches Diameter . The Extent from 10 to 46-90 , being turned two times the same way from 1 , shall reach to 21 inches , 8 10ths , for the length of a Foot , at that Diameter , in Inches . Or rather work thus ; As the Inches Diameter , to 13-54 ; So is 12 twice , to the Inches that make a Foot of Timber . Or , The Extent from 10 , to 13-54 , turned twice the same way from 12 , shall reach to 22 Inches . Or , The same Extent being turned two times the same way from 1 , shall reach to 1-831 , which is the Decimal for 22 Inches , as by looking on Inch and Foot measure , you may plainly see . Again , 2. For the same Diameter in Foot-measure . The Extent from 0-833 ( the Decimal of 10 Inches ) to 1-128 , being turned twice the same way from 1 , shall reach to 1-83 , which is almost 22 Inches , as by comparing Inches and Foot-measure together , is plainly seen . Problem II. Having the Diameter given in Inches , or Foot-measure , to find how much is in a Foot long . 1. As 13-54 ( the Inches Diameter that make a Foot of Timber , at one Foot long ) , to the Diameter in Inches ; So is 12 to a 4th , and so is that 4th , to the quantity in a Foot long . Example at 10 Inches Diameter . The Extent from 13-54 to 10 , being repeated two times the same way from 12 , shall reach to 6 Inches ½ , or , 54 of 100 , being somewhat more than a half Foot , for the true Content of one Foot long . 2. But if the Timber is great , then it is more convenient to have the quantity of a Foot , in feet and parts . Then say , As 13-54 , is to the Diameter in Inches ; So is 1 , to a 4th , and that 4th to the quantity in a Foot , in feet and parts . Example , as before , at 10 Inches . The Extent from 13-54 to 10 , the Diameter in Inches , shall reach , being turned twice the same way from 1 , to 0-545 , the Content of a Foot long . Again at 30 inches Diameter . The Extent from 13-54 , to 30 , being turned two times the same way from 1 , shall reach to 4 foot , 93 parts ; which 4-93 multiplied by the length in feet , shall give the whole Content of the Tree . 3. To perform the same , having the Diameter given in Foot-measure , Do thus ; The Extent of the Compasses from 1-128 , ( the feet and 10ths Diameter that make a Foot , at one foot in length ) to the Diameter in Foot-measure , shall reach , being turned twice the same way from 1 , to the quantity in a Foot long . Example at 1 Foot , 50 / 100 Diameter . The Extent from 1-128 , to 1-50 , shall reach , being turned twice the same way from 1 , to 1-77 , the true quantity in one Foot long . Problem III. 1. The Diameter of any Cillender given in Inches , and the length in Feet , to find the Content in Feet . As 13-●4 , to the Diameter in Inches ; So is the length in Feet to a 4th . Then , As the length , to the 4th ; So is the 4th , to the Content in Feet required . Example at 8 Inches Diameter , and 20 Foot long . The Extent from 13-54 , to 8 , being turned twice the same way from 20 , the length , shall stay at 6-94 , or near 7 foot . 2. The Diameter and length of a Cillender given in Inches , to find the Content in Cube-inches . The Extent from 1-128 , to the Diameter in Inches , being turned twice the same way from the length in Inches , shall reach to the Content in Inches . Thus the Extent from 1-128 to 10 inches Diameter , shall reach from 24 inches , the length , to 1888 , the Content in inches . 3. The Diameter and Length given in Foot-measure , to find the Content in Feet . The Extent from 1-128 , to the Diameter , shall reach from the length , being twice repeated the same way , to the Content in feet required . Thus the Extent from 1-128 , to 1-50 , shall reach , being turned twice the same way , from 5-30 , to 9-37 , the Content in feet required . Problem IV. Having the Circumference of a Cillender given in Inches , or Foot-measure , to find the length that makes one Foot of Solid-measure . 1. First to find the Inches in length , that makes a Foot. As the Circumference in Inches , is to 134-50 , ( because at so many inches about , one of a Foot in length , is a Foot ) so is 12 to a 4th , and so is that 4th to the length of a Foot in inches . Example at 30 Inches about . The Extent from 30 to 134-50 , being turned twice the same way from 12 , shall reach to 24 inches , 13 parts ; the inches and parts that make one Foot Solid . 2. To find the length of a Foot in feet and parts . As the Circumference in Inches , to 134-50 ; So is 1 to a 4th , and that 4th to the length in feet and parts , that makes 1 Foot. For the Extent of the Compasses from 30 to 134-50 , being turned twice from 1 , the same way , shall reach to two foot , and one tenth , the length that makes one Foot Solid . 3. When the Circumference is given in Foot-measure . As the Circumference in Feet , or Feet and parts , is to 3-54 ; So is that Extent twice repeated the same way from 1 , to the length that makes a Foot Solid . Example . The Extent from 2-50 , to 3-54 , being turned two times the same way from 1 , doth reach to 2 foot , 001 , the length in Foot-measure . Problem V. The Circumference given in Inches , or Foot-measure , to find how much is in a Foot long . 1. The Circumference of a Tree , when one Foot long makes a Foot of Timber . As 3 foot , 545 parts , to the feet about ; So is 1 foot to a 4th , and that 4th to the solid Content in one foot long . Example . The Extent of the Compasses from 3-545 , to 2-50 , the feet about , shall reach , being turned twice the same way from 1 , to 0-497 , the quantity in a foot long , viz. near half a foot . 2. The Circumference given in Inches , to find the Content of one Foot in length , Solid-measure , in Inches . The Inches a Tree is about , when one 10th of a Foot in length , makes a Foot of Timber in quantity . As 134-5 , to the Inches about ; So is 12 to a 4th , and that 4th to the Content of one foot long . Example at 30 inches about . Th● Extent from 134-5 , to 30 , being turned two times from 12 , shall reach to near 6 inches for the Content of one foot long , at 30 inches about . 3. The Circumference of a Cillend●r given in Inches , to find the quantity of one Foot long in feet and inches . As 134-5 , to the Circumference ; So is 1 to a 4th , and that 4th to the quantity of one foot long in Feet and Inches . The Extent from 134-5 , to 30 , being twice repeated the same way from 1 , shall reach to 0-497 , or near half a foot , the Content of one foot long , at that Circumference , which being multiplied by the length in feet , gives the true Content of any Cillender whatsoever . Problem VI. The Circumference , and length of any Cillender given in Inches , or Feet and Inches , to find the Content . 1. The Circumference given in Inches , and the length in Feet , to find the Content in feet and parts . As 42-54 ( the Circumference in Inches , that makes 1 foot long , a Foot ) is to the Inches in Circumference ; So is the length in Feet to a 4th , and that 4th to the Content in Feet . Example . The Extent from 42-54 , to 48 the inches about , being twice repeated from 12 foot the length , shall reach to 15-28 , the Content in feet required . 2. The Circumference and length given in Feet , to find the Content in feet and parts . As 3-545 , ( because at 3 foot and a half about , and a foot in length , is a Foot ) is to the Circumference ; So is the length in Feet to a 4th , and that 4th to the Content in Foot-measure . Example . The Extent from 3-545 , to 4-0 , the Circumference , being turned two times from 12 foot the length , shall reach to 15-28 , the Content in feet required . 3. The Circumference and length given in Inches , to find the Content in Inches . As 3-545 , to the Circumference in Inches ; So is the length in Inches to a 4th : Then , As the length to that 4th ; So is the 4th , to the Content in Cube-Inches . Example . The precise Extent on a true Line of Numbers , from 3-545 , to 48 , being turned two times from 144 , the length in Inches , shall reach to 26383 , the number of Inches , in a Tree 48 inches about , and 144 inches in length . This is sufficient for the Mensuration of any solid body , in a square , or Cillender-like form , as Timber or Stone usually is , after the true quantity of a foot , or 1728 Cubical inches ; but there is a custome used in buying of Oaken-Timber , and Elm-Timber , when it is round and unsquared , to take a Line , and girt about the midst of th● Piece ; and then to double the Line 4 times , and account that 4th part of the Circumference , to be the side of the Square , equal to that Circle ; but this is well known to be less than the true measure , by a fifth part of the true Content , be it more or less . Also in measuring Elm , and Beech , and Ash , whose bark is not peeled off , as Oak usually is ; to cast away 1 inch out of the 4th part of the Circumference , which may well be allowed when the Bark is 3 quarters of an inch , or more in thickness , and the Tree about 40 inches about , or the 4th part , 10 inches ; but if the Bark is thinner , and the Tree less , then 8 inches-square ; then an inch is too much to be allowed . Also , if the Tree is greater than a foot-square , and the Bark thick , an inch is too little to be allowed , as by this Rule you may plainly see , by the 7 th Problem of Superficial-measure in the 7 th Chapter . Suppose a Tree be 48 inches about , the Diameter will be 15 ¼ , the 4th of 48 , for the square is 12. Now if I take away 1 inch ½ from the Diameter , then the Tree will be but 43 inches and 1 / ● about , whose 4th part is under 11 ; so that here I may very well abate 1 inch from the 4th part of the Line ; So consequently , if the Rind be thinner , and the Tree less , a less allowance will serve ; and if the Rind be thicker , and the Tree large , there ought to be more , as by cutting the Rind away , and then taking the true diameter , you may plainly see . This measuring by the 4th part of the Circumference , for the side of the Square , and allowance for the Bark being allowed for , as before , I say will prove to be just one 5th part over-measure . Especially considering this , That when it is hewed , and large wanes left , then the Tree is marked for more measure , sometimes by 10 foot in 60 , than there was before it was hewed ; the reason is , because when the Tree is round and unhewn , the girting it , and counting the 4th part for the side of the Square , is but very little more than the inscribed Square ; and then being hewen , and that scarce to an eight Square , and measuring with a pair of Callipers , to the extremity of that , doth not then allow the Square equal to the Circle for the side of the Square , as by the working by those several Squares , will very plainly appear , which being foretold and warned of , let those whom it concerns look to it . But this being premised , and the Parties agreeing , the difference being as 4 to 5 , the best way to measure round Timber , I conceive , is by the Diameter taken with a pair of Callipers , and the length ; which for the just and true measure is largely handled already . But if this allowance be agreed on , then the Proportion for it is thus ; As 1-526 , to the Diameter ; So is the length to a 4th , and so is that 4th to the Content in feet . Example . The Extent from 1-526 to 15-26 , shall reach , being twice repeated from 10 foot , the length , to 10 foot the Content required , being all at one Point . Or , another Example . The Extent from 1-526 , to 20 inches the Diameter , being twice repeated the same way from 10 foot , the length , shall reach to 17 foot ¼ the Content . Or , if you have the Circumference and length . Then the Extent from 48 , to the inches about , being turned twice the same way from the length in feet , shall reach to the Content required . The Extent from 48 , to 62 , the inches about , being turned twice from 10 , the same way , shall reach to 17 foot ¼ , the Content in that measure . Thus you have full and compleat Directions for the measuring of any round Timber by the Line of Numbers , by having the Diameter and length given , after any usual manner , there remains only one general and natural way , by finding the base of the middle , or one end , by the 7 th Problem of Superficial measure ; and then to multiply that base by the length , will give the true Content in feet or inches . Thus , Having found the Base of the Cillender by the 7th or 10th Problem of Superficial-measure ; then if you multiply that Base being found in square inches , by the length in inches you shall have the whole Content in Cube Inches . Example . Suppose a Cillender have 10 inches for its Diameter , then by the 7th or 10th abovesaid , you shall find the Base to be 78-54 ; then if you multiply 78-54 by 80 , the supposed length in inches , you shall find 2356-20 Cube Inches , which divided by 1728 , the inches in a Cube Foot , sheweth how many feet there is , &c. And as to the number of figures , and the fractions cutting off , you have ample Directions in the first Problem , and the third Section of the six● Chapter . Problem VII . How to measure a Pyramis , or taper Timber , or the Section of a Cone . 1. First , get the Perpendiculer length of the Pyramis or Cone , thus ; Multiply half the Diameter of the Base , AB , by it self ; then measure the side AD , and multiply that by it self ; then take the lesser Square out of the greater , and the Square root of the residue is the Perpendiculer Altitude required , viz. DB. Example . Suppose the half Diameter of the Base AC , were 10-25 , and the side DA 100 , AB 10-25 , and 10-25 multiplied together , called Squaring , makes 105 , 0625 ; DC 100 , multiplied by 100 , called Squaring , makes 10000 ; then the lesser Square 105 , 0625 , taken out of 10000 , the greater Square , the remainder is 9894 , 9375 , whose square Root found by the 8 th Problem of the sixt Chapter , is 99-475 , the true length of the Line DB , the length or height of the Cone . Then if you multiply the Area or Content of the Base AC 20-5 , which by the 7th or 10th of Superficial measure is found to be 160-08 , by 33-158 , a third part of 99-475 , the whole height makes 5308 , cutting off the Fractions for the true Content of the Cone , whose length is 99 inches , and near a half , and whose Base is 20 inches and a half Diameter . 2. Then Secondly , for the Segment or Section of a Cone , the shape or form of all round taper Timber , the truest way is thus ; By the length and difference of Diameters , find the whole length of the Cone , which for all manner of Timber as it grows this way is near enough . As thus ; As the difference of the Diameters at the the two ends , is to the length between the two ends ; So is the Diameter at the Base , to the whole length of the Cone . Example . The difference between the Diameters AC , and EF , is 13-70 , the length , AE is 66-32 . then the Extent on the Line of Numbers from 13-70 , the difference of the Diameters ; to 66-32 , the length between , shall reach the same way from 20-50 , the greater Diameter to 99 and better , the length that makes up the Cone , at that Angle o● Tapering in the Timber ; then if by the last Rule you measure it as a Cone of that length , and also measure the little end or point at his length and diameter ; and then lastly , this little Cone taken out of the great Cone , there remains the true Content of the Taper-piece that was to be measured , viz. 5246-71 , when 61-30 , the Content of the small Cone at the end , is taken out of 5308 , the Content of the whole Pyramid . 3. If this way seem too troublesome for the common use , then use this , being more brief : To the Content that is found out , by the Diameter in the midst of the Timber , and the length , add the Content of a Piece found out , by half the difference of Diameters , and one third part of the length of the whole Piece , and the sum of them two shall be the whole Content required . 4. Or else ; Divide the length of the Tree into 4 or 5 parts , and measure the middle of each part severally , and that cast up by his proper length , shall give the Content of each Piece ; then the sum of the Contents of all the Pieces put together , is the true Content of the whole Taper Piece , very near . Note , That this curiosity shall never need to be used , but when you meet with Timber much Taper , and Die-square , or on a Contest or Wager ; for according to the usual way ( and measure ) of squaring the Timber , it is well , if the measure of the Square , taken with Callipers from side to side , in the middle of the length of the Piece , will make amends for half the Timber which is wanting in the wany edges of your squared Timber , and the knots or swellings , & hollows of most round Timber , may well ballance this over-measure found by the Diameter taken in the middle of the length of the Piece . But indeed for Masts of Ships and Yards , being wrought true and smooth , where the price of a Foot is considerable , there exactness is requisite , and necessary to be used ; and thus much for Solid-measure in Squares and Cillenders . Problem VIII . To measure Globes , and roundish Figures . 1. To measure a Sphear or Globe by Arithmetick , the ancient way , is to multiply the Diameter by it self , and then that Product , to multiply by the Diameter again ; which two multiplications is called Cubing of the Diameter ; then multiply this Cube by 11 , and then divide this last Product by 21 , and the Quotient shall be the Solid Content of the Sphear , in such measure as the Diameter was . Example . Let a Sphear be to be measured , whose Diameter is 10 inches : First , 10 times 10 , is 100 ; and 10 times 100 , is 1000 ; the Cube of 10 , that multiplied by 11 , makes 11000 ; which being divided by 21 , makes 523-81 , for the Solid Content . Which by the Line of Numbers , you may work thus ; 2. The Extent from 1 , to the Diameter , shall reach the same way from the Diameter to the Square of the Diameter . Then again , The Extent from 1 , to the Square of the Diameter , shall reach the same way from the Diameter , to the Cube of the Diameter . Then , The Extent from 1 , to the Cube of the Diameter , shall reach the same way from 11 , to the Product of the Cube of the Diameter , multiplied by 11. Lastly , This Extent from 21 , to this last Product , shall reach the same way from 1 , to the Solid Content of the Sphear required . Or else more briefly thus ; 3. The Extent from 1 , to the Diameter , being turned three times the same way from 0-5238 , shall stay at the Solid Content of the Sphear , or Globe , required . Example at 12 Diameter . The Extent from 1 to 12 , being turned three times the same way from 0-5238 , shall reach to 905-143 , the Solid Content required . 3. The Diameter given , to find the Superficial Content . Square the Diameter , an● multiply that by 3-1416 , and the Product is the Superficial Content . Or , by the Line of Numbers ; The Extent from 1 , to the Diameter , being turned twice the same way from 3-141● , shall reach to the Superficial Content , of the out-side round about the Gobe , viz. at 12 Diameter , 452-44 . 4. Having the Superficial Content , to find the Diameter . The Extent from 1 to 0-3183 , shall reach the same way from the Superficial Content , to the Square of the Diameter , whose Square-root is the Diameter required . As at 452-44 , gives 144. 5. Having the Solid Content , to find the Diameter of a Globe . The Extent from 1 to 1-90986 , shall reach from the Solid Content to the Cube of the Diameter , as ●t 905-143 Solidity gives 1728 , the Cube of 12. 6. Having a Segment of a Sphear , to find the Superficial Content . The Extent from 1 , to the Chord of the half Segment , shall reach , being twice repeated , from 3-1416 , to the Superficial Content of the round part of the Segment , ABC . Example . Let the Segment be the half Sphear , ABC ; AC being 12 , then BC which is the Chord of the Peripheria , BC is 8-485 , whose Square is 72. Then , The Extent of the Compasses from 1 , to 8-485 , being turned twice the same way from 3-1416 , shall reach to 220-22 , the Superficial Content of the round part of the Segment , or half Sphear or Globe , to which if you add the Content of the Circle or Base , you have the whole Superficies round about . 7. To find the Solid Content of a Segment of a Globe . First , you must find the Diameter of tha● Sphear , of which the given Segment to b● measured is part . Thus ; Add the Square of the Altitude , and the Square of the Diameter of the Segment together , and the sum divide by the Altitude of the Segment , the Quotient shall be the whole Sphears Diameter . Then , Taking the Altitude of the Segment given , from the whole Diameter , there remains the Altitude of the other Segment . Then ; Extend the Compasses from the whole Diameter of the Sphear , to 1 ; the same Extent applied the same way from the Altitude of the given Segment , shall reach to a 4th Number , on a Line of Artificial Solid Segments joyned to the Line of Numbers , which 4th Number keep . Then , Example . Let the whole Diameter of a Sphear be 14 , then the whole Solid Content by the former Rules , you will find to be 1437 ⅓ , a Segment of that Sphear whose Altitude or Depth is 4 , the Solidity is required . Extend the Compasses from 14 , the whole Sphears Diameter , to 1 ; that Extent applied the same way from 4 , the Altitude of the Segment , shall reach to 2-86 on the Numbers , or to 19-88 , on the Line of Solid Segments joyn'd to the Line of Numbers , which 19-88 , is the 4th Number to be kept . Then secondly , The Extent from 1 to 1437 , the whole Content of the whole Sphear , shall reach the same way from 19-88 , to 284 2 / ● , the Content of the Segment required to be found . If you want the Line of Segments , the Table annexed will supply that defect : Thus ; Look for the 4th Number , found on the Line of Numbers , among the figures on the Table , and the number answering it in the first Column , is the Solid Segment , or 4th to be kept ; as here , on the Numbers , you find 2-86 ; seek 2-86 in the Table annexed , and in the first Column , you find near 20 fo● the 4th in Segments . 8. To perform the same by Arithmetick after the way set forth by Mr. Thomas Diggs , 1574. To find the Superficial Content of a Globe or Sphear . Multiply the Diameter by the Circumference , the Product shall be the Superficial Content round about the Globe . 9. Or , Multiply the Content of a Circle , having like Diameter , by 4 , the Product shall be the Superficial Content . 10. And If you multiply the Superficial Content , by a 6th part of the Diameter , the Product shall be the Solid Content of the Sphear . A Table of Segments . Num. Seg Num. Segm. Num. Segm. Num. Segm. 059 335 506 679 084 342 513 686 104 349 520 694 122 356 527 703 5 137 30 363 55 534 80 712 152 371 540 720 164 378 547 728 176 385 554 737 188 392 560 746 10 197 35 399 60 567 85 753 207 406 574 763 218 413 580 772 228 420 587 782 237 426 594 793 15 245 40 433 65 601 90 803 254 440 608 812 263 447 615 825 272 453 622 836 280 460 629 848 20 288 45 466 70 637 95 865 297 473 644 878 306 480 651 896 314 487 658 916 321 494 665 941 25 328 50 500 75 672 100 1000 11. For the Segment work thus ; Multiply the whole Superficial Content of the whole Globe , by the Altitude of the Segment , and divide the Product by the Sphears whole Diameter , the Quotient shall be the Superficial Content of the Convexity or round part of the Segment . 12. But for the Solid Content , work thus ; First , find the difference between the height of the Segment , and the half Diameter of the Sphear ; then multiply this difference ( being found by subtracting the less from the greater ) by the Superficial Content of the Base of the Segment , and the Product subtract from the Product of the Sphears semi-Diameter , and the Convex Superficies of the Segment ; then a third part of the remainder shall be the Solid Content of the Segment required . Example as before . The Sphears Diameter is 14 , the Segments Altitude is 4 , the Segments Altitude taken from 7 , the half Diameter , the remainder is 3 , which multiplied by 126 , the Superficial Content of the Base of the Segment , makes 378 ; then having multiplied 7 , the Sphears half Diameter , by the Convex Superficies of the Segment 176 , the Product is 1232 , from which number take 378 , the Product last found , and the remainder is 854 , whose third part 284 ⅔ , is the Solidity of the Segment required . There are other fragments of Sphears , as Multiformed and Irregular , Cones or Pyramids , and Solid Angles ; but the Mensuration of these I shall not trouble my self , nor the Learner with , for whom I only write , intending the Mensuration of things that may come in use only . 13. But yet to conclude this Chapter , take these Observations along with you , concerning the Proportion of a Cube , a Prisma , and a Pyramid , a Cillender , Sphear , and Cone ; whose Shapes and Proportions are as in the Figures . If a Cube be made or conceived , whose side is 12 inches , then the solidity thereof is 1728 Cube inches ; and a Prisma , having the same Base and Altitude , contains 864 Cube inches ; and a square Pyramis , of the same Base and Altitude , contains 576 Cube inches ; and a Trianguler Pyramid , as before , contains 249-6 Cube inches ; A Cillender contains 1357 5 / 7 , being the same Height and Diameter of 12 inches : A Sphear , whose Axis is 12 inches , contains 905 1 / 7 Cube inches ; and a Cone , of the same Diameter and Altitude , contains 452 4 / 6. The Superficies of the Cillender about , ( excepting the top and bottom ) is equal to the Superficial Content of a Globe . Cube 1728 Cillender 1357 5 / 7 Sphear 905 2 / 7 Prisma 864 Piramis 576 Cone 452 4 / 6 △ Tetrahed . 2496 Octahed . whose Triangle Side is 12 814-6 By the foregoing Proportions , it is evident that a Cube is double the Prisma , and treble to the Square Pyramis of equal Base and Altitude , or as 3 , 2 , 1 ; for 3 times 576 is 1728 , and 2 times 864 is 1728. Also a Cillender is 11 / 14 of a Cube ; and a Globe is 11 / ●● of a Cube , or ⅔ of a Cillender , whose Sides and Diameters are equal ; and a Cone is ● / ● of a Cillender ; so that the Proportion between the Cone , Sphear , and Cillender , is as 1 , 2 , & 3 ; for 3 times 452 4 / 6 , makes 1357 5 / 7 ; and two thirds of 452 4 / 6 makes 905 1 / 7 ; the Content of a Sphear . The Trianguler Pyramid is little more than 1 / 7 of a Cube ; so that if any one have frequent occasions for these proportions , let Centers be put in the Line of Numbers , at these proportional Numbers , and then work with those Points from the Cube and Cillender , as is directed before , for the Circumference , and Diameter , and Squares , equal and inscribed in Chap. 8. Prob. 6. So much for the measuring of regular ordinary Solids ; for the extraordinary and irregular , the best Mechanick way is by Weights or Waters to measure their Crassitudes or Solidities , either by Weight or Measure . A further improvement of the Trianguler Quadrant , as I have made it several times , with a sliding Cover on the in-side , when made hollow , to carry Ink , Pens , and Compasses ; then on the sliding Cover , and Edges , is put the Line of Numbers , according to Mr. White 's first Contrivance for manner of Operation ; but much augmented , and made easie , by John Brown. 1. THe description thereof for one side ▪ being the same with the Line of Numbers on the outter-Edge , except that the first part is sometimes ( when required for that particular purpose ) divided into 12 parts , for inches , instead of 10 that is to say , The space between the first 1 , and the middle 1 , on the Rule ; ( the space I say ) between every Figure , on the first half part , is cut into 12 parts , instead of 10 , to answer to the 12 inches in a Foot ; and the other half , as the Line of Numbers on the Edge . And in the same manner are White 's sliding Rules made , only for this particular purpose . 2. On the other side , is the Line of Numbers drawn double , the one Line to the other , for the ready measuring of solid Measure at one Operation ; the description whereof in brief is thus ; First , The divisions on the sliding-piece in hollow-Rules , or on the right-side in sliding-Rules ; when the figures of the Timber-side stand fit to read , I call the right-side , or single-side , being alwayes toward the right hand , and a single Radius . The divisions therefore on the fixed-edge of the Rule , must needs be the left-side , and is also divided to a double Radius , or one Radius twice repeated . So also in sliding-Rules , the double Radius is on the left-side also . See the Figure thereof , with right and left-side exprest upon it . For the right reading those Lines , the Method is thus ; The Figures on the right or single-side , do usually begin at 3 or 4 , and so proceed with 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , for so many inches of a Foot. Then 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , &c. for so many whole Feet . The smaller Cuts between the first Figures , from 3 inches to 1 foot , being quarters of inches ; And the small Divisions between the Figures , that represent Feet , are only every whole Inch ; The halfs , and quarters of Feet , also noted by a longer stroke ▪ as in such work is necessary and usual . 3. On the same right-side also , for more ease and readiness in the use , are noted several Gage-points ( as it were ) ; As , First , At 1 Foot is the word square . Secondly , At 1 Foot , 1 inch and ½ , is a spot ; and close to it is set t.d. for true Diameter of a round solid Cillender . Thirdly , At 1 foot 3 inches ⅜ is another spot , and near to it is set D , for the Diameter of a rough piece of Timber , according to the usual allowance for unhewed Timber , according to the fourth part of a Line girt about and counted for the side of the square . Fourthly , At 3 foot 6 inches and ½ , and near to it is set t : r : for the true Circumference of a round Cillender . Fiftly , At 4 foot just is set R , for the Circumference , according to the former allowance . Sixtly , At 1 foot 5 inches 1 / 7 f●rè , is a spot ; and close to it the letter W , as the Gage-point for a Wine-gallon . Seventhly , At near 19 inches , or 1 foot 7 inches , is another spot ; and close to it the letter a , as the Gage-point of an Ale-gallon . Eightly , At 2 foot 8 inches 8 / 10 is a spot , and close to it is set B , for the Gage-point of a Beer Barrel ; and at 2 foot 7 inches is set A , for an Ale Barrel . The Uses whereof in order follow . The Figures on the left-side , or fixed-edges , are read and counted as those on the right : For the small , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , are to represent inches , and the cuts between , quarters of inches ; Then the 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 Figures next , somewhat bigger , as to represent so many feet , and the cuts between , are whole inches : Then 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , 150 , for tens of feet , and the parts between , for single-feet , for the most part ; or else whole and half feet , as is usual . The Uses follow . Use I. A piece of Timber being not Square , ( or having its breadth and depth unequal ) to make it Square , or find the Square equal ; Set the breadth of the Piece , counted on the right-side , to the same breadth counted on the left-side ; then right against the depth found on the left-side , on the right or single-side , is the inches and quarters square required . Example , at 15 inches broad , and 9 inches thick , or deep . Set 9 inches on the right-side , to 9 on the left ; then right against 15 inches , or 1 foot 3 inches , found out on the double or left-side , on the right or single-side , is 11 inches and ⅝ the Square equal required . Also , if you set 15 to 15 , then right against 9 , found out on the left-side , on the right-side is 11 inches ⅝ , the Square equal required . Use II. The Side of the Square given , to find how much in length will make 1 Foot. Set the inches ( or feet & inches ) found out on the right-side , to 1 foot on the left ; then right against 1 foot on the right , is the inches , or the feet and inches required , to make a Foot of Timber . Example at 9 inches square . Set 9 inches , found out on the right-side , to 1 foot on the left-side ; then right against 1 , on the right-side , is 1 foot 9 inches 2 / ● on the left . If the Square be so big , that the 1 on the right falls beyond the End at the beginning , then right against 10 foot on the right-side , is on the left , the hundredth part of a foot , that makes a Foot of Timber . Example , at 4 Foot Square . Set 4 foot , found out on the right , to 1 foot on the left ; then right against 10 foot on the right-side , is 0-063 on the left-side , or against 12 foot you have 9 , 12 parts of 1 inch , the length that goes to make 1 foot of Timber required . Use III. At any ( bigness or ) Inches , or feet and inches square , to find how much is in 1 Foot long . Just as the Rule stands even , that is 1 foot on the right , against 1 foot on the left , seek the inches , or feet and inches , the Piece is square on the right or single-side ; and just against it on the left or double-side is the Answer required ; in inches , or feet and inches . Example at 19 inches square . Just against 1 foot 7 inches , or 19 inches , ( which is all one ) found out on the right-side , on the left-side is 2 foot 6 inches , the quantity of Timber in 1 foot long , at 19 inches square ; which Number of 2 foot 6 inches , multiplied by the length in feet , gives the true Content of the whole Piece of Timber required . Note , That this is a most excellent way for great Wood , and very exact . Also Note , That here , by inspection , you may square a small Number , or find the square-root of a small Number . As thus ; The square of 8 & ½ , is near 72 ; Or , The square-root of 72 , is near 8 & ½ . Use IV. The side of the Square , and the length of any Piece being given , to find the Content in feet and parts . Set the word square , or 1 foot , alwayes to the length , found out on the left-side ; then right against the inches , or feet and inches square counted on the right , on the left is the Content required . Example , at 20 foot long , and 15 inches square . Set 1 foot on the right , to 20 on the left , then right against 1 foot 3 inches on the right , is 31 foot 2 inches and ½ , the Content . Note , That if the Piece be very small , call the feet on the left-side , inches ; and the parts between 12● of inches ; then the Answer will be found on the left-side in 144● of a Foot. Example , at 2 inches square , and 30 foot long , how much is there ? Set 1 foot on the right , to 30 foot on the left ; then right against 2 foot on the right , counted as 2 inches , is 120 parts of a foot divided into 144 parts , being just 10 inches , for 10 times 12 is 120. But if it be ● great Piece of Timber , then work thus ; Set 1 foot , or the word square , to the length on the left , counting the single feet 10● of feet ; then right against the feet and inches square are the 100● of feet required . Example , at 40 foot long , and 4 foot square . Set 1 foot to 4 foot , counted as 40 , on the left ; then right against 4 foot on the right , is 640 , the true Content , increasing the 10● to 100● . Thus much for square Timber . Though there be many other wayes and manner of workings , some whereof you may find in a Book set forth under the name of The Carpenters Rule , 1666 , by I. Brown , and well known abroad already . Use V. For Round Timber . The middle Diameter of any Piece given , to find how much is in a Foot long , at true measure . Set the spot by t.d. to 1 foot on the left , then just against the inches , or feet and inches Diameter , found on the right , is the quantity of Timber in 1 foot long on the left-side required . Example , at 2 foot 9 inches Diameter . Suppose a piece of Stone-Pillar , or Garden-Roul , be two foot 9 inches Diameter , set the spot by t.d. just against 1 foot , then right against 2 foot 9 inches , found on the right , on the left is 6 foot , the quantity of solid measure in one foot long ; which being multiplied by the length in feet , gives the true Content of the whole Piece . Note , That if you would have the usual allowance , set D to 1 , instead of t.d. Use VI. The Diameter of any Piece of Timber given , to find how much in length will make one Foot. Set 1 foot on the left , to the inches Diameter counted on the right ; then right against t.d. for true measure ; or D for the usual allowance , is the Answer required , found on the left-side . Example , at 9 inches Diameter . Set 1 on the left to 9 on the right ; then just against t.d. is 2 foot 3 ½ on the left ; and right against D , is 2 foot 11 inches on the left , the length required to make a foot solid at true measure , or the usual allowance , when the 4th part of the girt about , is counted the side of the Square , equal to the round piece of Timber . Note , That for great Timber , you must set the left 1 foot , to the feet and inches Diameter as before ; but count the t.d. or D , as far beyond 12 foot , as it is placed beyond 12 inches , and you shall have the Answer in 144● of a foot . Example , at 5 foot Diameter . If you set 1 on the left to 5 foot on the right , and count so much beyond 12 foot on the right , as t.d. is beyond 12 inches , you shall find 7 ¼ , that is , 7 144● and a quarter , to make 1 foot true measure , and 9-12s and ½ for the usual allowance . But for small Timber , set 1 foot , 2 foot , &c. on the right , counted as 1 , 2 , 3 , 4 , and inches , to 1 on the left ; then right against t.d. or D , is a Number , that multiplied by 12 , is the Number of feet required . Example , at 2 inches Diameter , how much makes 1 Foot ? Set 2 foot on the right , counted as 1 inch , to 1 on the left ; then right against t.d. is 3 foot 10 inches , calling the inches feet , and the feet 10s of feet ; which 3 foot 10 inches , multiplied by 12 , make 46 foot , for the length of 1 foot of Timber at 2 inches Diameter , the thing required for true measure . Use VII . The Diameter and Length given , to find the Content . Set t.d. or D , for true measure , or usual allowance , alwayes to the length counted on the left ; then right against the inches Diameter counted on the right , on the left-side is the Content required . Example , at 6 inches Diameter , an● 30 Foot long . Set t.d. to 30 ; then right against 6 inches , counted on the right ; on the left is 5 foot 11 inches , the Content required . Note , If the Piece be small , then count every foot on the right as inches , and you have the Answer in 144s of a foot , which is easily counted by having 1 set at 12 , 2 at 24 , 3 at 36 , 4 at 48 , 5 at 60 , &c. to 12 at 144 , which little small figures are counted as inches of 12s of a foot . Example , at 2 inches Diameter , and 20 foot long . You shall find 64 , 144s ; that is , 5 in ⅓ true measure . But for great Pieces , set t.d. or D , to the length , counting 1 foot , or the left for 10 foot , then have you the Answer in 100s of feet . Example , at 5 foot Diameter , and 30 foot long . Set t.d. to 3 inches , counting 30 foot for the length ; then right against 5 foot on the right , on the left is 592 foot , the Content required . Use VIII . To measure round Timber , by having the Girt , or Circumference about , and length given . This being the same in operation with the Diameter , I shall pass it over more briefly ; which way of wording , may serve for the Square and Diameter also ; only I labour to be plain and brief . The Circumference given , to find how much in a Foot long . Set t.r. or R , for true round , or allowance to 1 foot on the left ; then against the inches about , on the left is the Answer required . Example . At 2 foot about , you will find 3 inches and 10 / 12 in a foot long true measure ; or just 3 inches at the usual allowance . The Circumference given , to find how much makes a Foot. Set the inches , or feet and inches about , to 1 foot on the left ; so is t.r. or R , to the length to make a foot . Example , at 18 inches about . As single 18 , to double 1 ; So is t.r. to 5 foot , 7 inches ½ Or , So is R , to 7 foot 2 inches , for the usual allowance . The Circumference and length , to find the Content . As t. r , or R , to the length ; So is the feet and inches about , to the Content . Example , at 3 foot about , and 30 foot long , true measure . As t.r. to 30 ; so is 3 foot , to 21 foot 5 inches , the Content . For great things , call 1 foot on the left , 10 foot , as before . For small things , call 1 foot on the right , 1 inch , as before . Use IX . To Gage small Cask by the mean Diameter and Length . Set the spot by W , for Wine-gallons , alwayes to the length of the Vessel , given in inches , counted on the left-side ; then right against the inches , or feet and inches Diameter , counted on the right ; on the left is the Content in Gallons required . Example , at 24 inches , or 2 foot mean Diameter , and 30 inches long . As the spot at W , to 30 inches counted at 30 foot ; So is 24 inches , or 2 foot , to 58 Gallons 3 quarters , the Content required . For greater Vessels , count the feet on the left for 10s s of inches in the length , and you have your desire . Example , at 60 inches long , and 38 inches mean Diameter . As W , to 6 foot on the left for 60 inches ; So is 3 foot 2 inches , or 38 inches to 295 Gallons , the Answer required . If you would have it in Ale-gallons , use the mark at a. Use X. To gage Great Brewers Vessels , round Tuns . The Diameter and length being given in feet and inches , to find the Content in Beer-Barrels , at one Operation . Set the spot at BB , to the depth of the Tun , counted on the left , in feet and inches ; then right against the mean Diameter , found out on the right , on the left is the Content in Barrels required . Example , a Tun 4 foot deep , and 10 foot mean Diameter . As the spot at BB , to 4 foot ; So is 10 to 53 Barrels , and 2 third parts . If you would have the Content in Ale-Barrels , use the mark at AB . Thus much for the Timber-side , the use of the other , or board-side , is the same with that by the Compasses before treated of , and therefore needs here no repetition , unless as to the bare manner of working with it . The sliding-Rule is only two Rules , or Pieces fitted together , with a short Grove , and Tenon , and two Braces at each end , to keep it from falling assunder ; and even so also is the sliding-Cover , and two edges on the inside of the Trianguler Quadrant ; and the Numbers graduated thereon , are cut across the middle Joynt , having the same divisions on both sides ; that is to say , on each Rule , or on the Cover and Edge of the inside of the Rule . The Reading and description is the same with that in Chap. I. Page 12 , 13 , 14 , 15 , 16 ; and the general Method in use is thus ; That side or part of the Rule , on which you count the first term in the Question , is called alwayes the first-side ; then the other must needs be the second . Then for Multiplication , thus ; As 1 , on the first-side , to the Multiplier on the second , or other-side ; So is the Multiplicand , on the first-side , where 1 was , to the Product on the second . For Division , alwayes thus ; As the Divisor found on any one side , is to 1 on the second , or other-side ; So is the Dividend on the same first-side , to the Quotient on the second . For the Rule of Three . As the first term on the first-side , to the second on the other ; For Superficial Measure , by Inches and Feet . As 12 , to the breadth in inches on the second ; So is the length in feet , to the Content on the second . For any thing else , the same Rules and Precepts you find in Chap. VII . will give you ample and plain directions . The Lines being fitted , as much as may be , to speak out the Answer to the Question , as by well considering the Figure , you may see . CHAP. XI . To make and measure the Five Regular Platonical Bodies , with their Declinations and Reclinations . 1. For the Cube , being the Foundation of all other . IT is a Square Solid Body , every way alike , and spoken of largely before , as to the Mensuration thereof , and obvious enough to every indifferent Workman , as to the making thereof , and needs no repetition in this place . 2. For the Tetrahedron . It is a Figure , comprehended of 4 equilatteral plain Triangles , or a Trianguler Pyramid , last mentioned , the best and nearest way , as I conceive of making , is thus . According to directions of Mr. Iohn Leak : On any rough Piece , make one side plain and flat , so large , as to contain the Triangle which you intend shall be one side of the Tetrahedron ; then set a Bevel to 70 degrees 31 minutes and 42 seconds ; and plain another side , to fit the former side , and the Bevel ( secundum Artem ) ; then mark this last plained side , according to the former , and cut away the residue , plaining them away just to the strokes , and fit to the Bevel formerly set , and you shall constitute the Tetrahedron required . The Superficial Content is the Area of the 4 equilatteral Triangles mentioned before , and the solid Content is found by multiplying the Area of one Triangle by one third part of the Altitude of the Pyramid , or Tetrahedron , from the midst of one Plain to the Apex , or top of the opposite Solid Angle . If the measuring the sides , Perpendiculer , and Altitude of the Tetrahedron with Compasses , Callipers , and Scale , serve not to exactness ; then proceed thus ; First , for the Perpendiculer , the Triangle being equilatteral . Multiply one side given by 13 , and divide the Product by 15 ; the Quotient is the Perpendiculer . Example . If the side of the Tetrahedron be 12 , that multiplied by 13 , gives 156 ; which divided by 15 , leaves 10-4 , for the length of the Perpendiculer in the equilatteral Triangle , whose side is 12. Then for the Perpendiculer Altitude , work thus , by the Artificial Numbers and Sines . As the Sine of 90 , to the sine of 70 deg . 31 min. 42 sec ; So is 10-4 , the Perpendiculer , to 9-80 , the perpendiculer Altitude required . Or by the Sector , work thus ; Take 12 , the side of the Tetrahedron , from ( any Scale , or ) the Line of Lines , and set the Sector to 60 degrees , by making the Latteral 12 , a Parallel 12 , then the nearest distance from 12 , to the Line of Lines , is the true Perpendiculer ; which measured on the same Line of Lines , will be found to be 10-4 , as before ; then make this 10-4 a Parallel Sine of 90 , and 90 the Sector so set , take out the Parallel-sine of 70-31-42 , and measure it on the same Scale , and it shall be 9-8 , as before . Then , Lastly , This perpendiculer Altitude being multiplied by the Area of the Base , gives a Number , whose third part is the Solid-Content of the Tetrahedron required . For 12 the side , and 5-2 the half Perpendiculer , makes 62-4 , the Superficial-Content of one Triangle , or Base ; then 62-4 , the Base , multiplied by 9-8 , the perpendiculer Altitude , gives 611-52 , and a third part of 611-52 is 203-86 , the solid Content required . The three Triangles recline from the Perpendiculer upright , 19 degr . 28 min. and 18 sec. and decline when the edge is South 60 , South East , and South West , and the opposite Plain a just North ; but if you make one South , then the other two are are North-east and North-west 60 deg . 3. For the Octahedron , Whis is a solid body , comprehended under 8 equilatteral Triangles : The way of making which , is thus ; Make a plain Parallelepipedon , or long-Cube , if the breadth both wayes be 1000 , let the length be 1-414 ; or if the length be 500000 , the breadth both wayes must be 3-53553 ; then to these Measures square it exactly ; then divide the length and breadth just in the midst , and draw Lines both wayes on all the 6 sides ; then draw the Diagonal-Lines from the midst of the length , to the midst of the breadth ; and cut by these Diagonal-Lines , and the Octahedran will appear to be truly made . For the Mensuration thereof , it is the same as in the Tetrahedron ; For , supposing the side of one of the Triangles 12 ; the Base is 144 in Content , and the Triangles Perpendiculer is 10-4 , as before : But the Perpendiculer Altitude is just half the length , viz. 8-49 ; for if the breadth be 12 , then the length must be 16-98 , whose halfs are 6 and 8-49 ; Then if you multiply 144 the Base , by 8-49 the perpendiculer Altitude , the Product will be 1222-56 , whose third part is 407-52 , the half of the Tetrahedron , and 815-04 is the whole solid Content of the Tetrahedron required , as near as we can see by Instrumental Operation ; but if you work to a Figure more , you shall find the total Area to be but 814-656 more exact . To find this Perpendiculer Altitude by the Sector , work thus ; First , The Triangles Perpendiculer being 10-4 , as before ; Take the Latteral 10-4 , from the Line of Lines , make it a Parallel in 90 , lay the Thred exactly to the nearest distance , and there keep it ; then the Parallel-distance from the Sine of 54 deg . 44 min. 45 sec. the Reclination shall be 8-486 , the true Perpendiculer Altitude required . Then if the Octahedron stand on one Triangle , you have one Horizontal Plain , and one South and North Reclining and Inclining 19 deg . 28 min. 18 sec. as the Tetrahedron was ; and two South , and two North , declining 60 , and reclining and inclining 19 deg . 28 min. 18 sec. as afore . But if it stand on a Point , then you have 4 direct or declining 45 , and reclining 54 , 44 , 45 ; and 4 incliners , inclining as much and direct , or declining , as you shall please to set them . 4. For the Dodecahedron , Which is a regular solid body , contained under ( or made up of ) 12 Pentagonal Pyramids , or Pyramids whose Base hath 5 equal sides , and the perpendiculer Altitudes of those 12 Pyramids equal to half the Dodecahedrons Altitude , standing on one side , or equal to the semi-diameter of the inscribed Sphear . To cut this Body , take any round Piece , and if the Diameter be 100000 , the length must be 0-81005 , or as 4-906 to 3-973 , then the Piece being turned round , and the two Ends flat to the former measures of Length and Diameters ( which are near according to the Sphear inscribed , and to the Circle circumscribing ) being measured by Compasses , Callipers , and Line of Lines very carefully and exactly . Then divide the Circumference of the two Ends of the Cillender into 10 equal parts , and draw Lines Perpendiculer from end to end , and plain all away between the Lines flat and smooth , so that the two Plains on both ends will become a regular ten-sided Figure . Then making the whole Diameter abovesaid , 10000 in the Line of Lines , take out 0-309 , and with this measure ( as a Radius on the Center ) at both ends describe a Circle ; and if you draw Lines , from every opposite Line of the 10 first drawn , you shall have Points in the last described circle , to draw a Pentagon by ; which is the Base of one of the 12 Pentagonal Pyramids , contained in the body . This Work is to be done at both Ends ; but be sure that the Angle of the Pentagon at one end , be opposite to a side of the Pentagon at the other end ; then these Lines drawn , the two ends are fully marked . Then to mark the 10 Sides , do thus ; Count the first length 1000 , viz. the measure from the top to the bottom , or from Center to Center ; and fit this length in 10 and 10 , of the Line of Lines ; the Sector so set , take out 0-3821 , and lay it from the two ends , and either draw , or gage Lines round about from each end ; and in the midst between the two Lines will remain 0-2358 ; then Lines drawn Diagonally on the 10 sides , will guide to the true cutting of the Dodecahedron . If you set a Bevel to 116 deg . 33 min. 54 sec. and apply it from the two ends , you may try the truth of your Work. The Declination and Reclination of all the 10 Pantagonal Plains , are as followeth . First , You have 1 North , reclining 26 deg . 34 min ; and 1 South , inclining as much . Secondly , You have 2 North declining 72 , and reclining 26 , 34 ; and 2 South , declining 72 , and inclining 26 , 34. Thirdly , You have 2 North , declining 36 , and inclining 26 , 34 ; and 2 South , declining 36 , and reclining 26 , 34 ; And 1 Horizontal Plain , and his opposite Base to stand on . As for the measuring of this Body , the Plain and Natural way is thus ; First , find the Superficial Content of the Base of one of the Pentagons , by multiplying the measure from the Center to the middle of one of the Sides , ( which is the contained Circles semi-diameter ) and half the sum of the measure of all the sides put together ; and then to multiply this Product by one third part of half the Altitude of the body , and the Product shall be the Content of one Pentagonal Pyramid , being one twelft part of the Dodecahedron ; and this last , multiplied by 12 , gives the solid Content of the Dodecahedron ; or 12 times the Superficial Content of one side , is the Superficial Content thereof . Example . Suppose the side of a Dodecahedron be 6 , then the sum of the sides measured is 30 , the contained Circles semi-diameter is 4-12 ; then 15 the half of 30 , and 4-12 multiplied together , make 61-80 ; and 12 times this , makes 741-60 , for the Superficial Content of the Dodecahedron . Then for the Solid Content , multiply 61-80 , the Superficial Content of one side by 2-233 , one 6th part of 13-392 , the whole Altitude of the body ; the Product is 137-99940 : Again , this multiplied by 12 , the number of Pyramids , makes 16●● ▪ 9928 , the Solid Content , as near as may be , in such a Decimal way of Computation . 5. For the Icosahedron , Which is a regular solid body , made up of , or contained under 20 Trianguler Pyramids , whose Base ( or one of whose Sides ) is an equilatteral Triangle ; and the perpendiculer Altitude of one of these 20 Pyramids , is equal to half the perpendiculer Altitude of the Icosahedron , from any one side , to his opposite side , or equal to the semi-diameter of the inscribed Sphear . To cut this body , take any round Piece , and if the Diameter thereof be 10000 , let the length thereof be turned flat and even to 8075 ; or if the true Round and Cillendrical Form in Diameter be 4910 , let the true length , when the ends are plain and flat , be 3964 ; then divide the Cillendrical part into 6 equal parts , and plain away all to the Lines , so that the two ends may be two 6-sided-figures ; then making 5000 , the former semi-diameter , 1000 in the Line of Lines , take out 616 , and on the Center , at each end , describe a Circle ; and by drawing Lines to each opposite Point , make a Triangle , whose circumscribing Circle may be the Circle drawn at each end ; but be sure to mark the side of one Triangle opposite to the Point of the other Triangle at the other end , as before in the Dodecahedron ; thus both the ends shall be fully and truly marked . Then making the length a Parallel in 1000 , of the Line of Lines , take out -379 , and -095 , and prick those two measures from each end , and by those Points ( draw or gage ) Lines round about , on the 6 sides . Then Diagonal Lines drawn from Point to Line , and from Line to Point round about , shews how to cut the Body at 12 cuts . Note , That if you set a Bevel to - 138-11-23 , and apply it from each end , it will guide you in the true plaining of the sides of the Icosahedron . And a Bevel set to 100 degrees , will fit , being applied from the midst of one side , to the meeting of two sides . The Reclination of the three Triangles , whose upper sides are adjacent ( or next ) to the three sides of the upper Horizontal-Triangle is 48 11.23 , from the Perpendiculer , or 41 48 37 , from the Horizontal , and when one corner stands South , the Declination of one of these 3 , viz. that opposite to the South-corner a direct North ; th' other two decline 60 degrees , one South-east , the other South-west ; the other 6 , about the corners of the Horizontal-plain , do all recline 19 deg . 28 min. 16 sec. the two that behold the South , decline 22 deg . 14 min. 29 sec. and those two that behold the North , decline 37 deg . 45 min. 51 sec. toward the East and West ; the other two remaining , recline as before , and decline one North-east , and the other North-west 82 deg . 14 min. 19 sec. The other Nine under-Plains , opposite to every one of these , decline and incline , as much as the opposite did recline and decline , as by due consideration will plainly appear . For the measuring of this body , do as you did by the Dodecahedron , find the Area of one Triangle , and multiply it by 20 , gives the Superficial Content ; and the Area of one Triangle , multiplied by one sixt part of the Altitude of the body , gives the solid Content of the Trianguler Pyramid ; and that Product multiplied by 20 , the number of Pyramids , gives the whole Solid Content of the Icosahedron . Example . Suppose the side of an Icosahedron be 12 , first square one side ( viz. 12 , which makes 144 ) ; then multiply that Square by 13 , and then divide the Product by 30 , the Quotient and his remainder is the Superficial Content of the Equilatteral Triangle , whose side is 12 ; namely , 62-400 ; or more exactly , the Square-root of 3888 , which is near 62-354 ; 20 times this , is the Superficial Content , namely , 1247-08 . Then for the Solid Capacity or Content , multiply 3-023 , the sixt part of the bodies Altitude , or one third of the Pyramids Altitude , by 62-354 , the Area of one Trianguler Base , and the Product will be 188-493229 . Lastly , this multiplied by 20 , the number of Pyramids in the Body , the Product is 3769-864380 , the true solid Content of the Icosahedron . Thus you have the way of cutting , and the Declinations and Reclinations and Measures , Superficial and Solid , of the 5 Regular Bodies , as near as by Decimal Accompt to 100 part of an Integer may be , the exact measuring whereof , requires the help of Algebra , whereof I am ignorant . The Measures of the Containing , and Contained Sphears , Circles , and Diameters , Sides and Axis's , Diagonal-lines and Altitudes of the five Regular Bodies , gathered in a Table to a Containing Sphear , whose Diameter was 10 inches ( or Integers ) found out by Geometry , according to this Scheam , taken from Mr. Tho. Diggs . Let the Line AB be 10 of some Diagonal Scale , representing the Diameter of the Containing Sphear . Which Line AB , you must divide into two parts at C , and into three parts at E ; AE being one third part , and on the Points C and E , raise two Lines Perpendiculer to AB ; and with 5 of your Diagonal Scale , on the Center C , describe the semi-Circle AFDB , and note the Points F and D , in the semi-Circle , with F and D , drawing Lines from either of them to A , and from F to B. Then , Divide AF by extream and mean Proportion ; the greater Segment being AG , ( by the 10th Problem of the 6th Chapter ) then extend the Line AF to H , making FH equal to FG , and draw the Line HB ; and from F , draw another Line Parallel to HB , cutting the Diameter in I , and from I , draw a Line Parallel to CD , as IK ; then make IL a third part of IB , and draw ML Parallel to IK ; also , draw the Line MB , and divide it into two parts at N , and into 4 parts at 5 ; then divide the 4th part , MS , by extream and mean Proportion , whose greater Segment ( or part ) let be SV ; then divide FB in 4 parts , making FO the half , and FR the quarter ; divide likewise FE in two parts , and at the middle set P : The Figure being thus made , then with your Compasses , and Diagonal Scale , you may measure all the Diameters , Sides , and Altitudes , of all the 5 Regular Bodies . As thus ; AB is in all of them , the Contained Sphears Diameter . EC , OF , RO , NC , NC , the Contained Sphears Semi-Diameter . EF , OB , OF , NB , MB , the Containing Circles Semi-Diameter . EP , OC , CO , VN , MN , the Contained Circles Semi-Diameter . FB , AF , AD , AG , KB , the Length of the Sides of each Body . EB , AF , FA , MA , AM , the Altitude of the Bodies . AD , FB , VB , SB , the Perpendiculer Line of the Bases . FB , AF , the Diagonal-Line of the Bases , as in the Table . These Measures and Proportions are for a Sphear of 10 inches Diameter . If you would have the like for any other , then say by the Line of Numbers , or Line of Lines , or Rule of Three , thus ; As the side ( Diameter or Altitude ) for 10 , as in the Table , is to the given Side , Diameter , or Altitude ; So is any other Number , in the Table , for Diameter , Side , or Altitude , to his Proportional Measure required . Example . I have a Dodecahedron , whose Side is 6 , What shall all his other Sphears , or Circles , Diameters , or Altitude be ? The Extent of the Compasses from 3-570 , the Dodecahedrons-side in the Table , to 6 the side given , shall reach from 10 , the Containing Sphears Diameter in the Table , to 16 , the Containing Sphears Diameter , for a Dodecahedron , whose side is 6 : And from 7-970 , the Contained Sphears Diameter , to 12-643 , the Contained Sphears Diameter . And so for any other whatsoever . The Table . The Names of the Bodies Tetrahedron Cube . Octahed . Dodecahed . Icosahed . Containing Sphear .           The containing Sphears Diameter , that comprehends the body in it , is for ever one of them . AB AB AB AB AB 10.000 10.000 10.000 10.000 10.000 Contained Sphear .           The contained SphearS Diameter that is contained in the body , called also Axis , is 2.332 8.1648 4 0824 7.970 7.970 EC OF RO NC NC The half thereof , is 1.666 4.0824 2 . 041● 3.983 3.985 Containing Circle .           The containing Circles Diameter , ( or the Diameter of that Circle which comprehends one side or base of the body , is 9.420 9.648 9.644 6.070 MB EF OB OF NB 6.070 The half thereof , is 4.710 4.0824 4.0824 3.035 3.035 Contained Circles .           The contained Circles Diameter , comprehended in the Base of one side , is 4.410 5.7840 3.7840 4.910 MN EP OC OC VN 3.035 The half thereof , is 2.355 2.8920 2.8920 2.455 1.5175 Sides ,           The length of one side of the Triangle Square , or Pentagon , being the base of the figure , is FB AF AD AG KB 8.1647 5.774 7.071 3.570 5.260 The half therof , 4.08235 2.887 3.5355 1.785 2.630 Altitude .           The Altitude from side , to the side opposite , or from Side to the Point opposite , EB AF AF MA MA 6.666 5.774 5.774 7.960 7.960 The half thereof , 3.3333 2.887 2.887 3.980 3.980 Perpendiculer .           The Length of the Perpendiculer Line of any one Side or Base , AD FB   VB SB 7 073 8.1647 6.123 5.485 4.556 The half thereof , 3 5365 4.0823 3.0625 2.7425 2.278 Diagonal-Line .           The Diagonal-Line , from Corner to Corner of the same Base , is ,   FB   AF   none 8.1647 none 5.774 none The half thereof is ,   4.0823   2.887   This Table was gathered from this Geometrical Figure , drawn on a Slate , by a good Diagonal Scale of 6 parts in a Foot , whereby I could very well come to the 100th part of an Integer ; and is true enough for any Mechanick Operation , for whose use I only do it , and I hope it may be as kindly accepted , as it was carefully Calculated , and offered to Publick view . CHAP. XII . The use of the Line of Numbers , in Gaging of Vessels , close or open . GAging of Vessels , is no other than the Measuring of Solid Bodies ; and the former directions for solid Measure , conveniently and aptly applied , is fully sufficient ; only observing this difference , That the result or issue of the Question is to be rendred in proper terms , according to the demand of the Question , as thus ; in measuring of Timber or Stone , the Question is , How many Feet , or Inches , is there in the Solid Body ? But in Gaging , the Question is , How many Gallons , Kilderkins , or Barrels is there in the Vessel to be measured ? For which purpose there are fit Numbers , or Gage-Points , requisite to be known , for the more speedy attaining the Answer to the Question , of which in their order , as followeth ; First , You are to remember , That the solid capacity of a Wine-Gallon , is 231 Cube Inches ; a Corn-Gallon 272 ¼ Cube inches ; an Ale or Beer-Gallon , is 282 ¼ Cube inches ; or as some say , 288 Cube inches ; So that when you have found the Content of any Vessel in Cube inches , if you divide that sum in inches , by the respective Number for the Gallons you would have , the Quotient shall be the Content in Gallons required . Problem I. To measure a Square Vessel . From hence it follows , That to measure any Square or Oblong Vessel , you must multiply the length and breadth taken in inches , and tenth parts , together ; that is to say , The one by the other ; and the Product shall be the Content of the Base in inches , superficially : Then multiply this Superficial Content of the Base , by the inches , and tenth parts deep , and the Product shall be the solid Content in Cube inches ; then divide this Product by 282 , gives the Content in Ale-Gallons in the Quotient , and the remainder , if any be , are Cube inches . But if you divide by 10161 , the Cube inches in a Beer Barrel ; or , by 9032 the Cube inches in an Ale Barrel ; the Quotient sheweth the Number of Beer or Ale Barrels , ( and the remainder Cube inches . ) Example of a Brewers Cooler . The length let be 78 inches and 1 tenth , the breadth let be 320 inches and 5 tenths , and the depth 9 inches and 5 tenths , or half an inch ; by multiplying and dividing , as above , you will find 843 Gallons , and 68 Cube inches , to be the solid Content of that Cooler ; which work is very readily done by the Line of Numbers , in this manner ; Extend the Compasses from 1 , to the breadth or length ; and the same Extent shall reach from the length or breadth to a 4th , which is the Superficial Content of the Base , or bottom , in Superficial inches . Then , The Extent from 282 ¼ , to the last Number found , shall reach the same way from the inches , and tenths deep , to the Content in Gallons . Example . The Extent from 1 , to 78-1 , shall reach the same way from 320-5 , to 25031 ; then the Extent from 282 ¼ , to 25031 , shall reach the same way from 9-5 , to 842-68 , the Solid Content in Gallons required . Indeed , you must Note , You cannot see so many Figures on the Line , as the Product of 4 figures multiplied by 3 ; yet by the Rules ( in Chap. 6. Sect. 3. ) you have directions as to the number of Figures , which here is 7 ; the two last ( next the right hand ) being Fractions , or parts of an Inch , and is therefore neglected . Again , In dividing the Product of 25031 , and 9-5 , multiplied together , which makes 6 Figures beside the Fraction by 282 , there must needs be three Figures in the Quotient , which are the Gallons : This artificial help you have , beside the present view of the Vessel , which will direct you not to call 842 Gallons , only 84 ½ , nor 8420 , as you must needs do , if you mistake as to the denomination . Again , You need not to trouble your self , to know what the 4th Number is ; but having found the Point representing it , keep the Compass-point fixed there , and open the other to 282 ¼ , where you may have a Brass Center-pin for more readiness ; but let your account go as 282 ¼ to the 4th , for methods sake , and not as the 4th to 282 ¼ ; for then you must say , so is the depth the contrary way to the Content in Gallons . All this is hinted for plainness and caution sake , in benefit to young Learners . Also Note , That if you would have had the Answer in Ale or Beer Barrels ; then instead of 282 ¼ , you must use the Point at 9032 , for Ale Barrels ; or the Point at 10161 for Beer Barrels , being the number of Cube inches in those Barrels , as 282 ¼ is the number of inches , in a Gallon of Ale or Beer . Example for the same Cooler . The Extent from 1 to 78-1 , shall reach from 320-5 , the same way to 25031 ; then , the Extent from 10161 to 25031 , shall reach the same way from 9-5 to 23 ⅓ , the true number of Beer Barrels required . Or , The Extent from 9032 , to 25031 , shall reach to 26 Barrels , and near 1 third : which is as quick and ready a way as can be for Square Vessels . Problem II. To Gage or Measure any round Tunn or Vessel . The plain and natural way for measuring of a round Tun , is this ; Measure the Diameter in inches and tenths , and set down half thereof ; Measure also , the Compass round about the inside , and set down the half of that also , in inches and tenth parts ; and multiply those two Numbers together , the Product shall be the Content of the Base , or bottom , in Superficial inches ; then this Product multiplied by the depth in inches , gives the solid Content in inches ; then lastly , this Product divided by 282 , or by 10161 , or by 9032 , gives the solid Content in Gallons , or Beer , or Ale Barrels , as before . For , half the Diameter , and half the Circumference , doth reduce the round Vessel to an Oblong Vessel , equal to that round Vessel . Which Vessel , when it is brought to a Square , by taking of half the Diameter , and half the Circumference ; then the Rule last mentioned , for Square Vessels , performs the work exactly , to Gallons , or Barrels , as you please . But when the Vessel is Taper , that is to say , the bottom and top of different Diameters , as generally they all are ; then the chief care is to come by the true Diameters , which is best done by a sliding Rule applied to the inside , whose regular equal computation is thus to be ordered ; When the Vessel is taper , and the Sides go streight , like the Segment of a Con● ; then you may add the Diameters at top and bottom together , and count the half sum for the mean Diameter of that taper Vessel , and multiply half the Diameter , and half his proportional Circumference , as before ; and multiply and divide , to get the solid Content in Gallons , or Barrels . But when the Staves are bending , as most of your close Cask are , then the readiest way to come to a mean Diameter , is thus ; Say , As 10 to 7 , or as 10 to 6 & 3 / 10 , 4 / 10 , 5 / 10 , 6 / 10 , 7 / 10 , 8 / 10 , 9 / 10 ; according as you shall find most true for several Cask : So is the difference of Diameters to a 4th Number , which is to be added to the least of the Diameters , to make up a mean Diameter . As for Example . If the Sides be round or arching , and the less Diameter be 30 inches , and the greater 40 inches ; then , As 10 to 7 ; So is 10 , the difference to 7 inches ; which makes ( being added to 30 , the least Diameter ) 37 , for a mean Diameter . But Note , It is hinted by Mr. Dary , That Vessels , usually , are between a Spheroid and a Parabolick Spindle ; then , if as 10 to 7 , be too much to add to the least Diameter ; You may say , As 10 , to 6 ½ ; Or , As 10 , to 6 6 / 10 , 7 / 10 , 8 / 10 , 9 / 10 ; So is the difference of Diameters to a 4th Number , which you must add to the least Diameter , to make a mean Diameter . Having thus gained a mean Diameter , you may work as before ; or rather thus more readily and easily , by the Line of Numbers , thus ; As the Gage-point is to the mean Diameter ; So is the Length to a 4th , and that 4th to the Content required . The Gage-point for Wine , and Oyl-gallons , at 231 Cube inches in a Gallon , is 17 — 15 The Gage-point for Ale-gallons , at 282 ¼ , is , 18 — 95 The Gage-point for Ale , or Beer-gallons , at 288 , is , 19 — 15 The Gage-point for a Corn-gallon , at 272 ¼ , is , 18 — 62 The Gage-point for a Beer Barrel , at 10161 , is , 35 — 96 The Gage-point for an Ale Barrel , at 9032 , is , 33 — 91 The Extent of the Compasses , on the Line of Numbers , from the Gage-point to the mean Diameter of a Vessel ; being turned two times the same way from the length of a Vessel , shall reach to the Content of the Vessel , in Gallons or Barrels , according to the nature of the Gage-point . Example . A mean Diameter being 30 , and the Length 40 , the Content is in Wine-gallons 123 , near . In the lesser Ale or Beer-gallons , 100-½ . In the greater Ale-gallons , at 288-098 gallons and a half . In Corn-gallons , at 272-¼ , — 104 Gallons . In Beer Barrels , by his Gage-point you will find 2-78 , or 2 three quarters : 2 — 78. In Ale Barrels , you will find 3 and 11 of a hundred : 3 — 11. And the like for any other Measure , whose Gage-point is known . Problem III. To find the Gage-point of any Measure . The Gage-point of any Solid Measure , is only the Diameter of a Circle , whose Superficial Content is equal to the Solid Content of the same Measure . As thus more plainly ; The Solid Content of a Wine Gallon is 231 Cube inches : Now if you have a Circle that contains 231 Superficial inches , the Diameter thereof will be found to be 17 inches , and 15 of a hundred ; as by the 7th Problem of the 7th Chapter , is well seen . These Directions may serve for any round Vessel , either close or open ; yet Mr. Oughtred , a very able Mathematition , hath a way accounted somewhat more exact , and consequently more tedious and troublesome to use either by the Pen or Compasses , which is this ; You must measure the Diameters at head and bung , or the top and bottom in inches and 10ths , the length also by the same measure ; then find out the Superficial Content of the Circles , answerable to those two Diameters , and take two thirds of the greatest , and one third of the least , and add them together in one sum ; which sum you must multiply by the length in inches and tens , and the Product shall be the Content in Cube inches ; which Product divided by 282 , gives Ale Gallons ; or by 231 , gives Wine Gallons , as before . By the Line of Numbers , this way is more easie and ready thus ; The Extent from 1 , to 0-5236 , a Number fit for 2 thirds , of the Circle at the bung ; So is the Square of the Diameter at the bung to a 4th . Then again ; As 1 , to 0-2618 , the half of the former Number , and fit for one third of the Circle at head ; So is the Square of the Diameter at head to a 4th . These two 4ths add together , then say ; As 231 ( for Wine , or 282 ¼ for Ale-Gallons ) , is to the sum of the two 4ths added together ; So is the length to the Content in Wine-Gallons . Example , at 18 inches at head , and 32 at bung , that old Example . The Square of 32 , is 1024 ; The Square of 18 , is 324 : Then , The Extent of the Compasses from 1 , to 0-5236 , shall reach from 1024 , the Square of 32 , to 536-4 , two thirds of the bung-Circle . Again , The Extent from 1 , to 0-2618 , shall reach from 324 , the Square of 18 , the Diameter at head , to 84-9 , the sum of 536-4 , and 84-9 , is 621-3 . Then lastly , The Extent from 231 , the 621-3 , shall reach from 40 the length , to 107-58 , or 107 Gallons and a half , and better , the Content in Wine-gallons , as briefly as can be done this way . But if you take the Diameters at head and bung , with a Line called Oughtred's Gage-line ; and set the measure found at the bung by that Line , down twice ; and the measure found at the head , found by the same Line , once , and bring them into one sum ; then multiply that sum by the length of the Vessel in inches , and 10 parts , and then the Product shall be the Content in Wine-gallons required . As if I should measure a Cask of 18 , and 32 , as before : right against 18 inches on Oughtreds-Line , you find 0-367 ; and right against 32 , you shall find 1-161 ; this last set down twice , and 0-367 once ; added , makes 2-689 ; and then this sum multiplied by 40 , makes 107-56 , being very near to the former operation , but differing about 2 Gallons , from the way set before by the mean Diameter and Gage-point , by reason of the extream swelling of the Cask ; But if this way should prove the truest in the Book of the Carpenters Rule , you have a Table to rectifie this difference , which you will very seldom have occasion to use . Note also , That this Line , called Oughtred's Gage-Line , is very excellently improved to find the Content of Great Vessels , either in the whole , or inch by inch ; which you will find at large in the Book before mentioned . Also , The use of the Lines called Diagonal-Lines , and Lines to find the emptiness of Cask , and to measure Corn-measures by , to which I shall , for the present , refer you . Problem IV. The Diameter and content of a Vessel being given , to find the length of the Vessel . Extend the Compasses from the Diameter to the Gage-point , the same Extent twice repeated from the Content , shall give the length required . Example . If the Content be 60 , and the Diameter 24 , then extend the Compasses from 24 , to 17-15 , the Gage-point for Wine ; this Extent turned twice the same way , from 60 the Content , shall reach to 30 inches , and 6 tenths , and a half , the length required . Problem V. The Length and Content of a Vessel given , to find the Diameter . Divide the space on the Line of Numbers , between the Length and the Content , into two equal parts ; the Compasses so set , shall reach the same way from the Gage-point to the Diameter of the Vessel . Example . The half distance between 31-65 the length , and 60 the Content , shall reach the same way from 17-15 the Gage-point , to 24 the Diameter required . These two last Problems may be useful for Coopers , to make Cask of any length , diameter , and quantity . Problem VI. To find what is wanting in any close Cask , at any number of inches and parts , ( the Cask lying after the usual manner , with the bung-hole uppermost ) from the bung-hole to the superficies of the Liquor given , two wayes . This Problem I shall resolve two wayes , either of which is experimented to come near the truth , and will very well serve , till a better comes to light . The One , by a Line of Segments , joyned to the Line of Numbers , as before in the measuring the Fragments of a Globe ; But , The Other , is by a way found out by Mr. Bennit , a Cooper , that hath long exercised the way of Gaging , which is by comparing a Cask known , and its quantity of emptiness , to a Cask unknown , and its inches of emptiness , as followeth . First , by the Line of Numbers , and Artificial Line of Segments , to find the quantity of Gallons that any Vessel wants of being full , at any number of Inches , from the inside of the bung-hole , to the superficies of the Liquor , which is usually called Inches dry . Extend the Compasses , on the Line of Numbers , from the inches and tenths diameter at the bung , to 100 on the Line of Segments , the same extent applied the same way from the inches and parts dry , shall reach to a 4th Number , on the Line of Artificial Segments ; which 4th Number you must keep . ( Or , if you will , you may use the inches wet , laying the same extent from the inches wet , and that also will on the Segments give a 4th Number , which you must likewise keep . ) Then secondly , As the Extent from 1 , to the whole Content of the Vessel in Wine or Ale-gallons ; So is the 4th Number kept to the Gallons of emptiness , or fullness , that it wants of being full , or the quantity of Gallons in the Vessel . Example , of a Canary-Pipe , whose Diameter at bung , is 28 inches and 7 , and full Content is Gallons 116 ½ , at 12 inches dry , or 16 inches , and 7 tenths wet . The Extent of the Compasses from 28-7 , to 100 , ( at the end of the Line of Segments ) shall reach the same way from 12 , the inches dry , to 39 ½ on the Line of Segments for a 4th ; or from 16-7 wet , to 60 2 / 10 on the Segments , for his 4th also , which two 4ths keep . Then secondly , The Extent from 1 , to 116 ½ , the whole Content in Gallons , shall rea●h from 39 , the dry 4th , on the Line of Numbers , to 46 3 / 10 , for the gallons dry or wanting : or the same extent shall reach the same way , on the Line of Numbers , from 60 2 / 10 , the 4th Number for wet , to 70 gallons , and 2 tenths in the Vessel , at 16 inches and 7 tenths wet ; which two Numbers put together , makes up 116 gallons and a half , the full Content . The like manner of working serves for any Cask whatsoever , and the nearer the Vessel wants of being half empty , the more near to the truth will your work be , and the most errour in very round and swelling Cask , when the emptiness is not above one or two inches ; but in Vessels near to Cillenders , it will give the Answer very true , and as readily as any way whatsoever . Observe also , That if you use the Segments in taking the wants , you must abate of the gallons found , till you come to the 2 thirds of the half diameter ; that is to say , the Rule sayes , there is more wanting than indeed there is ; and that somewhat considerable about the first 6 inches in a vessel of 30 inches diameter : So that I find a Table made as a mean between the Superficial and solid Segments , would do the work the truest and best of any other ; Or else , use the mean diameter and mean parts of emptiness ; found thus . Take the equaded diameter , from the diameter at the bung ; and note the difference : then half this difference taken from the inches and parts empty gives the mean emptiness ; then use the mean diameter , and mean emptiness , instead of the other , and the work is more exact . The other way of Mr. Bennits invention is thus ; First , you are to fill an ordinary Cask , of a competent magnitude , as 60 or 100 gallons , of a mean form , between a Spheriord ( or roundish form ) and a Cillenderical form ; or else fill two Casks of each form , and learn the true Content , and Diameter of that mean Vessel , or rather of both those Vessels ; and the Vessel being full , draw off with a true gallon-measure , and on the drawing off every gallon , take the exact quantity of inches and 10th parts , that the drawing off of every gallon makes in the emptiness or driness of that mean Vessel , or rather both those Vessels , at least until you have drawn off the half quantity of the Vessel , which number of gallons drawn off , and the inches and tenth parts of emptiness , or fulness , or driness or wetness , you must draw into a Table , or insert them on a Rule , making the inches as equal parts , and the gallons , and his proportional part of a gallon , the unequal parts ; then with the Line of Numbers , and this mean Table , or rather two Tables or Scales , which you may put on a Rule , as Mr. Bennit hath done , you may find out the wants of any Cask whatsoever ; either Spherioid , or Cillender-like , as followeth . This measured Cask on the Scale , or Table , for methods sake , and avoiding tautologie , I shall call the first Cask , and the Vessel or Cask , whose wants you would know , I shall call the second Cask ; then the proportion is thus . As the Diameter at the bung of the second Cask , is to the bung diameter of the first Cask ( which is always fixed ) ; So is the inches dry of the second Cask to a 4th ( on the Line of Numbers ) which 4th Number sought on the inches of your Table , or Scale , on the opposite-part of your Scale or Table , gives a 5th Number , which you must keep . Then , As the whole Content of the first Cask , is to the whole Content of the second Cask ; So is the first Number kept , to the Number of Gallons the Vessel wants of being full , at so many inches dry . Example . There is such a Scale made on purpose for Victuallers use , to measure what they want of a Barrel of Ale , being put into a Beer-barrel , which Scale I shall here use , to try this former Example by . Suppose , as before , a Canary Pipe want 12 inches of being full , and the Content 116 ½ gallons , and 28 inches and 7 tenths diameter at bung ; The Extent on the Line of Numbers from 28-7 , to 22-5 , shall reach from 12 , to 9-4 ; then just against 9 inches and 4 tenths , on that Barrel Scale , I find 14 gallons of Beer , which is 17 gallons and a half of Wine , being the 5th Number to be kept . Then the Extent from 44 , the Content of a Barrel in Wine-gallons , to 116 ½ , the Content of a Canary-Pipe in the same gallons , shall reach the same way from 17 ½ the Number kept , to 46 , and near a half , the gallons wanting at 12 inches dry , in the Canary Pipe , and 46 gallons , and 3 quarts , is the Number Mr. Bennit finds in a Canary-Pipe , by measuring at 12 inches dry . Thus you have an account of the two easie Mechanick wayes , to discover the wants of Cask , very applicable , and ready , and experimented to be Propè verum . The Gallons wanting in a Barrel , at every inch and quarter .   Beer Gall. Wine Gall       gal . pi . 100 gal . pi . 100 gal . 1000     0 0 40 0 0 49 0 0612     0 1 20 0 1 47 0 184 22   0 2 10 0 2 57 0 321   1 0 3 10 0 3 80 0 475     0 4 33 0 5 30 0 663     0 6 00 0 7 35 0 920 21   0 7 60 1 1 29 1 161   2 1 1 80 1 4 00 1 500     1 3 90 1 6 56 1 821     1 6 10 2 1 22 2 153 20   2 0 66 2 4 34 2 543   3 2 3 50 2 7 98 2 998     2 6 16 3 3 10 3 388     3 0 70 3 6 20 3 772 19   3 3 80 4 2 00 4 250   4 3 6 50 4 5 30 4 663     4 1 80 5 1 35 5 169     4 5 25 5 5 60 5 700 18   5 0 42 6 1 45 6 182   5 5 3 90 6 5 70 6 713   5 7 20 7 1 70 7 213     6 2 80 7 6 20 7 777 17   6 6 50 8 2 65 8 333   6 7 2 20 8 7 20 8 900     7 5 50 9 3 20 9 400     8 1 10 9 7 70 9 960 16   8 4 80 10 4 20 10 525   7 9 0 70 11 1 00 11 125     9 4 50 11 5 40 11 806     10 0 40 12 2 20 12 275 15   10 4 30 12 7 00 12 876   8 11 0 50 13 4 10 13 513     11 4 30 14 0 80 14 110     12 0 30 14 5 80 14 725 14   12 4 29 15 2 80 15 350   9 13 0 30 15 7 70 15 926     13 4 30 16 4 60 16 577     14 0 40 17 1 60 17 200 13   14 4 60 17 6 60 17 827   10 15 0 50 18 3 40 18 425     15 4 48 19 0 30 19 037     16 0 80 19 6 50 19 815 12   16 5 50 20 3 25 20 40●   11 17 2 20 21 1 00 21 225   17 7 90 22 0 00 21 000     18 5 49 22 6 98 22 644 11   19 2 00 23 4 31 23 391   12 19 6 16 24 1 48 24 184     20 3 00 24 7 30 24 961     20 7 40 25 4 60 25 575 10   21 3 10 26 1 36 26 170   13 21 7 40 26 6 38 26 799     22 3 00 27 3 36 27 130     22 7 00 28 2 18 28 174 9   23 3 00 28 5 18 28 648   14 23 7 30 29 2 19 29 275     24 3 70 29 7 40 29 926     24 7 40 30 3 90 30 488 8   25 3 60 31 1 00 31 125   15 25 7 50 31 5 80 31 726     26 3 30 32 2 60 32 325     26 7 00 32 7 00 32 875 7   27 3 00 33 3 80 33 475   16 27 6 40 34 0 30 34 037     28 2 20 34 4 80 34 600     28 5 80 35 0 80 35 100 6   29 1 40 35 5 34 35 668   17 29 4 80 36 1 80 36 225   30 0 40 36 6 29 36 788     30 4 10 37 2 29 37 287 5   30 7 50 37 6 54 37 820   18 31 3 00 38 2 39 38 299     31 6 10 38 6 64 38 833     32 1 80 39 2 70 39 338 4   32 5 00 39 6 00 39 752   19 32 7 80 40 1 80 40 225     33 2 10 40 4 90 40 614     33 4 80 41 0 10 41 012 3   33 7 40 41 3 65 41 457   20 34 2 00 41 6 77 41 848     34 4 30 42 1 43 42 180     34 6 20 42 4 00 42 500 2   35 0 10 42 6 70 42 840   21 35 2 00 43 0 64 43 055     35 3 60 43 2 70 43 338     35 4 80 43 4 19 43 524 1   35 6 00 43 5 42 43 678   22 35 6 80 43 6 51 43 816     35 7 40 43 7 50 43 938     36 0 00 44 0 00 44 000   CHAP. XIII . The use of the Line of Numbers , in Questions of Interest and Annuities . Problem I. At any rate of Interest per annum for a hundred pounds , to find what the Interest of any greater or lesser sum comes to in one year . EXtend the Compasses from 100 to the increase of 100 l. in one year , the same Extent shall reach from the sum propounded , to its increase for one year , at that rate propounded . Example . What is the increase or profit of 124 l. 10 s. for one year , at 6 per cent . per annum ? The Extent of the Compasses , from 100 to 6 , being laid the same way from 124 l. 10 s. ( which is at 124-5 ) shall reach to 7-47 , which is 7 l. — 9 s. — 4 d. the profit of 124· 10 s in one year . Problem II. Any sum of Money , and the rate of Interest propounded , to find what it will increase to , at any number of years , counting Interest upon Interest . The Extent of the Compasses from 100 , to the increase of 100 , being turned as many times from the sum propounded the same way , as there be years propounded , shall at last stay at the Principal and Interest required . Example . To what sum shall 143 pounds 10 shillings , amount to in 10 years , counting Interest upon Interest , at 6 per cent ? The Extent of the Compasses from 100 , to 106 , being turned 10 times from 143 ½ , shall reach to 257 l. 0 s. the sum of Principal and Interest at 10 years end . Note , That in doing this , you ought to be very precise , in taking the first Extent from 100 , to 106 ; but to cure the uncertainty thereof , you have this very good remedy : If you have a Diagonal Scale , equal to the Radius of the Line of Numbers , then use that ; if not , use the Line of Lines on the Sector-side , which should be made fit to ( or the double , or the half of ) the Radius of the Line of Numbers . As thus ; Take the Extent from the Line of Numbers , between 100 , and 106 ; this Extent measured on the Line of Lines , will be 0253058 , could you see so many Figures , but 02531 , will serve your turn very well ; which Number you must note , is the Logarithm of 106 , neglecting the Caracteristick ; then this Number multiplied by 10 , the Number of years , is 25310 ; this Extent taken from the Center , on the Line of Lines , and laid increasing from 143 ½ , shall reach to 257 l. 0 s. 0 d. the true Number of the Use and Principal of 143 l. 10 s. put out , or forborn for ten years . Problem III. A sum of Money being due at any time to come , to find what it is worth in ready Money to be paid presently , at any rate propounded . This Problem is the contrary to the last , for if you shall turn the Extent between 100 and 106 , ten times backward from 257 , it will stay at 143 ½ , the worth in ready Money . Or , to make use of the former remedy ; Multiply 0253058 , the Logarithm of 106 by 10 ; then this Extent taken and laid the decreasing way from 257 , shall reach to 143 ½ . For Note , That the Line of Lines is the Scale of equal parts , that makes the Line of Numbers , and 10 , or 7 , or 15 , or any other Number multiplied by the Logarithm of 106 , taken from that Scale of Lines all at once , is equal to so many repetitions ; and consequently more exact , because of the difficulty of taking the 10 , 12 , or 15th part of any Number whatsoever ; and observe , That so much as you err in the first , it will be 10 , 12 , or 15 , or 20 times so much at last , which may be considerable in this . Problem IV. A yearly Rent , or Annuity being forborn a certain number of years , to find what the Arrears thereof will amount unto , according to any rate propounded . First , you must find out the Principal-Money , that answers to the Rent , or Annuity in question ; then find the sum of that Principal and Use , at the end of the term given , at the rate propounded ; then the Principal taken out of this sum , both of Arrears and Principal , the Arrears do remain , which is the sum you look for . Example . Suppose a Landlord live far from his Tennant , and yet judging his Tennant honest , and able , is content to take his Rent once in every fourth year , which should be paid every year , or every quarter of the year ; and suppose the Rent be 10 l. per annum , and the rate of profit , for the forbearance , be 8 per cent . First , to find the Principal for 10 l. per annum , at the rate of 8 l. per cent . Say , If 8 l. have 100 for his Principal , what shall 10 l. have ? The Answer will be 125 ; for the Extent from 8 to 100 , shall reach from 10 , the same way , to 125 ; then by the 2d Problem of this Chapter , 125 l. forborn for four years , will come to 170 l. which is 170 l. 0 s. 0 d. from which sum , if you substract 125 l. there remains 45 l. the Arrears for 10 l. per annum forborn four years , at the rate of 8 per cent . But if you would have the profit of these Arrearages , supposing 2 l. — 10 s. the 4th part of 10 l. per annum to be paid quarterly , and to count Use upon Use at the rate abovesaid , then you will find the Principal and Arrears to be 171 l. 10 s , For if you multiply 0086 , the log . of 102 l. the Interest and Principal of 100 l. for a quarter of a year by 16 , the quarters in four years , it will be 1376 , which Number taken from the Line of Lines , and laid from 120 , on the Line of Numbers , shall reach to 171 ½ , or 171 l. 10 s. being 30 s. more than the former sum , when 150 l. the Principal is taken away , the residue Arreares is 46 l. 10 s. Or , If you turn the distance on the Numbers between 100 and 102 , 16 times from 125 , which you may help thus ; turn first 4 times , then take them 4 times in one Extent , and turn 3 times more , and you will stay at 271 ½ , the Answer required . Problem V. A yearly Rent , or Annuity propounded , to find the worth thereof in ready Money , at any rate whatsoever . First , by the 4th Problem , find the Arrears that shall be due at the end of the term , and at the rate propounded ; then by the 3d Problem , find what those Arrears are worth in ready money , which shall be the worth of the Annuity , or Rent required . Example . There is a Lease of a House or Land worth 12 l. per annum , and there is 16 years yet to come ; which Lease a man would buy , provided he may lay out his money to gain after the rate of 10 l. per cent : the question is , What is it worth ? First , by the last , if 10 l. have 100 for his Principal , What shall 12 ? the Answer is 120 ; Then by the second part of the second , 120 l. forborn 16 years , comes to 551 l. the Principal and Interest : from which sum , taking 120 l. the Principal , there remains 431 the Arrears . Then by the third Problem find what 431 due 16 years to come , is worth in ready money ; and the Answer will be at 10 in the 100 , 93 l. 14 s. Also herein observe , That if there be any Reversion of a Lease to be expired , before it may be injoyed ; then you are to find the worth of 431 l. after so many years more ; as suppose it be 5 years before the Annuity begin ; then find the worth of 431 , forborn 21 years , which will be 58 l. 4 s. Problem VI. A sum of Money is propounded , and the rate whereby a man intends to Purchase , to find what Annuity , and how many years to continue , that sum of money will buy . Take any known Annuity at pleasure , and find by the last , the value of that in ready money , then this proportion holds ; As the value found , is to the Annuity supposed ; So is the sum of money to be improved , to the Annuity required . Example . What Annuity , to continue 16 years , will 500 l. Purchase , whereby a man may gain after the rate of 10 l. per cent ? By the last Problem I find , That 93 l. 14 s. will purchase 12 l. a year , for 16 years , at 10 per cent . Therefore , The Extent of the Compasses from 93 l. 7 , to 12 l. per annum , shall reach the same from 500 , to 64 l. per annum . For such an Annuity , to continue 16 years , will 500 l. purchase , to gain 10 l. per annum , per cent . for your Money . Problem VII . Or , first rather ; Lands or Houses , sold at any certain number of years Purchase ; to find what the value of the whole will be ? The usual way of valuing Land or Houses , is by the years Purchase , and Land Fee-simple is usually vallued at 20 years Purchase ; Coppy-hold-Land , at 15 or 16 years Purchase ; and good , strong , and new Houses , at 12 , 13 , or 14 years Purchase for Fee-simple . But a Lease of 〈…〉 of 21 years about 7 years Purchase ; and a Lease of 31 years , about 8 years Purchase , rather less than more ; and a Lease of 60 , or 100 , not worth above 8 ½ years Purchase . Again , The usual profit allowed for Land in Fee-simple , is not above 5 l. in the 100 per annum , because of the certainty thereof ; for Coppy-hold Land , full 6 l. in the 100 per annum ; for the best Houses , 7 and 8 l. in the 100 Fee-simple . But in laying out Money on Leases , either of Land or Houses , Men shall hardly be savers , if they gain not 8 , 9 , or 10 in the 100 per annum , for their Money ; The reason and demonstration whereof , you may read at large in Mr. Phillips his Purchasers Pattern . Thus the number of years Purchase agreed on , ( which ought to be cleer , from Quit-rent , and Taxes , and the like ; the Rent is usually various , according to the place , and time where , and wherein , the Purchase shall happen to be ) then to find the quantity of the whole Purchase , Say , As 1 , to 20 , 18 , 15 , 14 , 12 , 10 , or 8 , the number of years Purchase , for Fee-simple , or Coppy-hold Land , or Houses Fee-simple , or Coppy-hold ; For Leases of 60 , 50 , 40 , or 30 years , or 21 years ; So is the yearly Rent to the whole value . Example . A Parcel of Land worth 10 l. per annum Fee-simple , valued at 20 years Purchase , will amount to 200 l. For , The Extent from 1 , to 20 , will reach the same way from 10 to 200 , the whole price of 20 years Purchase , at 10 l. per annum . CHAP. XIV . The Use of the Line of Numbers IN Military Questions . Problem I. Any Number of Souldiers being propounded , to order them into a Square Battel of Men ; that is , as many in Rank as in File . FInd the Square-root of the Number of Souldiers , and that shall be the Number of Men in Rank and File required . As suppose it were required to order 1770 Men , in the order abovesaid , you shall , by the 8th Probl. of the 6th Chapt. find , that the Square-root of 1770 is 42 , and 6 over , which here is not considerable . Problem II. Any number of Souldiers propounded , to order them into a double Battel of Men ; that is to say , twice as many in Rank as File . Find the Square-root of half the Number of Men , and that is the Number of Men in File , and the double the Number in Rank . As for Example . If 2603 , were so to be placed , the half of 2603 , is 1301 ; whose Square-root by the 8th of 6th , is 36 ; the number of Men in File : and 72 , the double thereof , is the number in Rank . For if you shall multiply 72 by 36 , the Product is 2592 , almost the number of Men propounded . Problem III. Any Number of Souldiers being propounded , to order them into a Quadruple Battel of Men ; viz. 4 times as many in Rank as File . Find the Square-root of a 4th part of the Number of Men , and that shall be the Number in File ; and 4 times so many the Number in Rank . So the 4th part of 2603 , is 650 ; whose Square-root is 25 ½ , and 4 times 25 is 100 , the Number in Rank . Problem IV. Any Number of Souldiers being given , together with their Distance one from another in Rank and File , to order them into a Square Battel of Ground . As suppose I would order 3000 Men so , that being 7 foot asunder in File , and 3 foot apart in Rank , the Ground whereon they stood should be Square . Extend the Compasses from 7 foot , the distance in File ; to 3 foot , the distance in Rank ; then that Extent applied the same way from 3000 , the Number of Souldiers , reaches to 1286 , whose greatest Square-root is 35-7 ; that is , 35 , the Number of Men to be placed in File . Then , If you divide 3000 , the whole Number , by 35-7 , the Quotient is 84 , the Number in Rank , to use and imploy a Square plat of ground to stand in . As 7 , to 3 ; so is 3000 , to 1286 , whose Square-root is 35-7 . Then , As ●5-7 , to 1 ; so is 3000 , to 84. Problem V. Any Number of Souldiers propounded , to order them into Rank and File , according to the ratio of any two Numbers given . This Question is all one with the former ; For , As the Number given for the distance in File , is to that for the distance in Rank ; So is the whole Number of Souldiers to a 4th , whose Square-root is the Number of Men in Rank . Then. The whole Number divided by the Number in Rank , the Quotient is the Number to be placed in File . Example . Suppose 3000 Souldiers were to be ordered in Rank and File : As 5 is to 10 , or as 5 is to 9 ; that is to say , that the Men in Rank , might be in Proportion to them in File , as 9 is to 5. Say thus ; As the Extent from 5 , to 9 ; So is 3000 , to 5400 , whose Square-root is 73 ½ , the Number of Men in Rank . Then , As 73 ½ , to 1 ; So is 3000 , to near 41 , the Number in File . Problem VI. There are 8100 Men to be ordered into a Square Body of Men , and to have so many Pikes , as to arm the main Square-Body round about , with 6 Ranks of Pikes ; the Question is , How many Ranks must be in the whole Square Battel ? And , How many Pikes and Musquets ? First , the Square-root of 8100 , is 90 , the Number of Men in , and Number of Ranks and Files ; now in regard that there must be 6 Ranks of Pikes round about the Musquetiers , there will be 12 Ranks less of them , both in Front and Flank , than in the whole Body ; therefore substracting of 12 from 90 , rest 78 , whose Square is 6084 , the number of Musquetiers ; which taken from 8100 , there remains 2061 , the number of Pikes . Problem VII . To three Numbers given , to find a fourth in a doubled Proportion . For as much as like Squar●● , are in double the Proportion of their answerable sides ; therefore you must work by their Squares , and Square-root . But by the Line of Numbers , in this manner . If a Fathom of Rope , of 6 inches compass about , weigh 6 pound , 2 ounces , ( or 6-●25 / 1000 ) what shall a Fathom of Rope of 12 inches compass weigh ? Here Note alwayes , That when the two Numbers of like denomination , which are given , are of Lines , or sides of Squares , or Diameters of Circles ; then the Extent of the Compasses upon the Line of Numbers , from one Line to the other , or from one side to the other side ; that Extent turned twice the same way from the given Area , or Content , shall reach to the other required : So here , the Extent of the Compasses from 6 to 12 , being turned two times the same way from 6-125 , shall reach to 24-50 , for 24 pound and a half , the weight required . But if the two terms given of one denomination , are of Squares , or Superficies , or Areas ; then the half distance , on the Line of Numbers , between one Area and the other , being turned the same way on the Line , from the given Line or Side , it shall reach to the Side , or Line , required . For the half-distance , between 24-50 , and 6-125 , shall reach from 12 to 6 ; or the contrary , from 6 to 12. An Example whereof , you have in the 4th and 5th Problems of the 12th Chapter : Also , in the 6th and 7th Problems of the 8th Chapter , which treates of Superficial-measure , in measuring of Land. Note also , That if you have three Lines of Numbers , viz. a Great , a Mean , and a Less ; after Mr. Windgates way ; then these Questions are wrought without doubling , or halving , and very neatly and speedily . As thus ; The Extent on the mean Line , from 24-50 , to 6-125 , the weight of the two Ropes , shall reach on the great Line , from 12 to 6 ; or from 6 to 12 , the inches in compass about of each Rope . Problem VIII . To three Numbers given , to find a fourth in a tripled Proportion . For as much as like Solids , are in a tripled Proportion to their answerable side ; the Cubes of their sides are proportional one to another ; therefore , to work these Questions by the Line of Numbers , do thus ; When the two given terms , of like denomination in the Question , are of Sides , Lines , or Diameters ; then the Extent of the Compasses , on the Line of Numbers , from one side to the other , that is , from the side , whose Cube or Solidity is also given , to the other ; the same Extent , turned three times from the given Cube , or Solidity , shall reach to the inquired Cube , or Solidity . As for Example . If a side of a Cube , being 12 inches , contain in Solidity 1728 cube inches ; How many inches is there in a Cube , whose side is 8 inches ? The Extent from 12 to 8 , being turned three times from 1728 , shall reach to 512 , the Solidity required of the Cube , whose side is 8 inches every way . Again , on the contrary . When the two terms of the same denomination , are Cubes or Solidities , then divide the space on the Line of Numbers , between the two Solidities , into three equal parts , and lay that Extent the same way , as the reason of the Question doth require , either increasing or diminishing , from the given Side or Line , and it shall reach to the inquired Side , or Line . Example . If 1728 , be the Cube of 12 , the Root or side ; what shall be the Root or side of 864 , the half of 1728. being half a foot of Timber ? The Extent between 1728 , and 864 , being divided into three parts , and that third part , laid decreasing from 12 , shall reach to 9-525 , the side or root required of half a foot of Timber , though not exactly , yet very near . Again for another Example . If an Iron Bullet , of 6 inches Diameter , weigh 30 pound ; what shall a Bullet of 7 inches Diameter weigh ? The Extent from 6 to 7 , shall reach , being turned three times , to 47-7 . Again , If a Ship , whose Burthen is 300 Tun , be 75 Foot by the Keel ; what shall that Ship be , whose Keel is 100 Feet ? The Extent between 75 and 100 , turned three times from 300 , shall reach to 713 Tun Burthen . Again , If a Ship of 29 Foot and a half at the beam , be 300 Tun Burthen , what shall a Ship of 713 Tun burthen be ? The third part of the distance between 300 and 713 , shall reach from 29 ½ to 39-35 , its measure at the beam . Again , If a Ship of 300 Tun be 13 Foot in hold , what shall a Ship of 713 Tun be in hold ? The third part between 300 and 713 , shall reach from 13 foot , to 17-35 , the Feet in the Hold of a Ship of 713 Tun. If you have a treble Line , then you may save the dividing , by taking from the little-Line , and measuring on the great-Line , and the contrary , as the nature of the Question doth require . Lastly , Know , that by adding of twelve Centers and Points , the Line may be made to speak , as it were , and so made more fit for any mans more particular occasions . A Brief Touch of the Use of the Logarithms , or Tables , of the Artificial Numbers , Sines , and Tangents . See more in Gunter's Works . IT may happen , that some may meet with this Book , that had rather use the Tables of Logarithms , from whence these Lines are framed , than the Lines on the Rule ; or out of curiosity to prove the truth of their work , for whose sakes I have added these following plain Precepts , without Examples . 1. To multiply one Number by another . Set the Logarithm of the Multiplicator , and Multiplicand , right under one another , and add them together , and the sum is the Logarithm of the Product . 2. To divide one Number by another . Set down first the Logarithm of the Dividend , and then right under it the Logarithm of the Divisor , and then substract the log ▪ of the Divisor , from the log . of the Dividend , and the remainder is the Log. of the Quotient required . 3. To find the Square-root of a Number . Half the Logarithm of the Number given , is the whole Logarithm of the Square-root of it . 4. To find the Cubick-root of a Number . One third part of the Logarithm of the given Number , is the full Logarithm of the Cubick-root of the given Number , as a third of 14313637 ; the logarithm of 27 is 0-4771212 , the Log. of 3 , the Cube-root of 27 , required . 5. To work the Rule of Three direct , or three Numbers given , to find a 4th by the Logarithms . 5. To work the Rule of Three direct , or three Numbers given , to find a 4th by the Logarithms . Set down the Logarithms of the 1 , 2 , & 3 Numbers , one right over another ; then add the logarithms of the second and third together ; and from the sum , substract the logarithms of the first , and the remainders is the logarithms of the 4th required . 6. When in common Arithmetick the second term is divided by the first , and the Quotient multiplied by the third . Then by Logarithms , Take the Logarithm of the first term , from the Logarithm of the second ; and add the difference to the log . of the third , and the sum is the log . of the 4th . 7. When in common Arithmetick the second term is divided by the first , and the third by the Quotient . Then take the log . of the second , from the log . of the first term ; and take the difference out of the log . of the third , and the remainder is the log . of the 4th term required . 8. Between two extream Numbers , to find a mean Proportional . Add the logarithms of the two extream Numbers together 〈…〉 sum is the 〈…〉 . 9. To work the Rule of Three in the Logarithms of Artificial Numbers , Sines , and Tangents . 1. When Radius is the first term . Add the Logarithms of the second and third terms together , and Radius , or a unite , in the first place , taken from the sum , there shall remain the logarithm of the 4th term required ; according to the 5th Precept . 2. When Radius is in the second place , or term . Then the first term ( and second virtually ) taken from the third , cutting off a unite in the first place for Radius , is the 4th term . 3. When Radius is in the third place . Then substract the logarithm of the second term , from the log . of the first term , cutting off a unite for Radius , and the remainder is the 4th term . 4. If Radius be none of the three terms . Then add the Logarithms of the second and third terms ; and from the sum , substract the logarithm of the first term , and the remainder is the logarithm of the 4th term . 5. Or else . Set down the Arithmetical complement of the first term , and the logarithms of the second and third term , and add all together , and the sum cutting off Radius , is the 4th term . 10. When the Number is not to be found in the Canon of Logarithms of Numbers , Sines , or Tangents ; take the next nearest , or for more exactness use the part proportional . 11. Though Numbers and Sines , or Numbers and Tangents are used together , the work is all one , as with Sines and Tangents , as to the Precept in working . 12. In using the Logarithms , great regard is to be had to the Index , or Charracteristick , to rule in the Number of places ; the Characteristick being one unite less than the Number of places to express that Number ; thus the Characteristick of 3-5932861 , the logarithm of 3920 is 3 , being one less than the Number of places in 3920 , which consists of 4 figures . CHAP. XV. The use of the Trianguler Quadrant , in Geometry , and Astronomy . Use I. The Radius of a Circle , or Line being given , to find readily , any required Sine , Tangent , or Secant , or Chord , to that Radius . And first , to do it by the Quadrantal-side . FIrst , If your Radius happen to be equal to the greater Scale of ( Altitudes or ) Sines , issuing from the Center , then the measure of any degree or minuite , from the Center toward the head , shall be the Sine , the measure from the Center-point at 600 , on the degrees , to any degree and minuit required , shall be the Tangent to the same Radius ; and the measure from the Tangent to the Center , shall be the Secant , to the same Radius . And if you have an Index , or a Bead upon your Thred , and set the Bead , when the Thred is drawn streight , to the Center at 600 on the degrees or Tangents , or to the Sine of 90 ; then if you lay the Thred to any Number of degrees and minuits , counted from 90 , and there keep it ; then the extent from the Sine of 90 , to the Bead , shall be the Chord of the Angle the Thred is laid to , to the Radius of the greater Scale of Sines , issuing from the Center . But if this happen to be too large , then the other lesser Line of Sines , issuing upwards from the Center , being about one third part of the other , hath first it self for Sines ; secondly , the degrees on the loose-piece for Tangents , counting from the Center at 60 ; thirdly , the measure from the Tangent , to the Center , for a Secant ; fourthly , the Bead and Thred , for a Chord , as before ; all at once to one Radius , clearly and distinctly , without any interruption , to 75 degrees of the Tangent , or Secant . But if any other Radius be given , then they will not be had so readily altogether , but thus in order one after another , and first for the Sine , by the Trianguler Quadrant . Take the Radius between your Compasses , set one foot in 90 , with the other lay the Thred to the nearest distance , and there keep it ; then take the nearest distance from the Sine of any Ark or Angle you would have , and that shall be the Sine of the Ark or Angle required to the given Radius . 1. But by the Sector-side work thus , being near alike , fit the given Radius in the Parallel-sine of 90 & 90 ; then take out the Parallel-sine , of the Ark or Angle required , and you have your desire . 2. Also the Sector being so set , if you take out any Parallel Tangent under 45 , you have that also to the same Radius . 3. Also , if you would have any Tangent under 76 , as the Sector stands , take out the parallel Tangent thereof , and that shall be the 4th part of the Tangent required to the same Radius , and is to be turned 4 times for that greater Radius . 4. Also , if you want a Secant under 60 degrees ; at the same Radius take out the parallel Secant of the Ark or Angle required ; and that shall be the half of the Secant required ; for note , the Secant of one degree is more than Radius ; and why I use a half , rather than a 4th part , in time you may well see . By the Artificial Numbers , Sines , and Tangents , this cannot properly be done ; only thus you may do by them , counting your given Radius ( be it great or little ) 10000 parts , you may by them find out readily how many of them parts will go to make the Sine , Tangent , or Secant , to any Number of degrees and minuts . As thus ; Take the distance from the Sine of 90 , on the Artificial Sines , to the Sine of any degree and minuit required ; and set the same distance , the same way , from 10 on the Line of Numbers , reading it as a Scale of equal parts , and that shall be the Natural Sine of the degree and minuit required . Or , If you lay a Square to the Sine given , on the Numbers , it cuts the Natural Sine required . Example . Right against the Artificial Sine of 30 , on the Line of Numbers , you find 5000 , which is the Natural Number thereof . But , if you measure this distance from 10 , in the Line of Lines , it will give the Logarithmal Sine thereof , viz. 69897. And the like for the Tangent also , under 45 , in the same manner . But for the Secants , and , for the Tangents above 45 , you must count thus ; Measure , as before from 90 , to the Co-sine of the Angle , required , for a Secant ; and from 45 , to the co-Tangent of 45 , for a Tangent ; This extent laid the contrary way from 1 , in the Numbers , shews how many Radiusses , and also how much above Radius , you must have to make up the Natural Tangent , or Secant required , in Numbers . Example . The Secant of 50 degrees , and the Tangent of 57 degrees , 16 minuts , being near alike , is 1 Radius 5556 ; for the Natural Number thereof : and this distance measured on the Line of Lines , gives Radius because above 45 , and 1919 ; more for the Artificial Tangent of 57-16 , or the Secant of 50 degrees . This have I hinted , in the first place , that thereby you might see the nature of the Lines , and the making of the Instrument , with its great convenience in the Contrivance of the Work on both sides , and the harmony , and proportion , the Natural way hath to the Artificial ; also hereby you may readily prove the truth of your Instrument , being an equilatteral Triangle , whether you use the greater or the lesser Sines ; For the measure from the Center , where the Thred is fastened , to the Center-point of Brass on the moveable-leg , and loose-piece at 60 on the degrees , ought to be equal to each Line of Sines ; and also to the Tangent of 45 on the Tangent Line : The measure from the Center to the rectifying-point on the Head , at the meeting of the Lines for the Hour and Azimuth , and the Lines for the Sines and Lines , is equal to the Tangent of twice 30 on that piece . Again , The measure from the Center , to the rectifying-point on the end of the Head-leg , shall reach from thence to 30 on the loose-piece ; and being turned twice , reaches to 060 on the loose-piece : Also , the Radius , or Tangent of 45 , turned twice from 060 on the loose-piece , shall reach to 75 , as by comparing the Natural Numbers together , will most exactly appear ; Though perhaps without this hint , it might not have beeen observed by an ordinary eye . Having been so large , and plain , in this first Use , I shall be , I hope , as plain , though far more brief in all the rest ; for if you look back to Chapt. VI. Probl. I. Sect. 3. you shall there see the full explaining of Latteral and Parallel , and Nearest-distance , and how to take them ; the mark for Latteral being thus — ; The mark for Parallel thus = ; Nearest-distance thus ND , &c. Use II. The Sine of any Ark or Angle given , to find the Radius to it . Take the Sine between your Compasses , and setting one foot of the Compasses in the given Sine ; and with the other Point lay the Thred to the nearest-distance , and there keep it ; then the nearest-distance from the Sine of 90 to the Thred , shall be the Radius required . Make the given Sine a Parallel Sine , and then take out the Parallel Radius , and you have your desire . The Artificial Sines and Tangents , are not proper for this work , further then to give the Natural Number thereof , as before ; therefore I shall only add the use of them when it is convenient in the fit place . Use III. The Radius , or any known Sine being given , to find the quantity of any other unknown Sine , to the same Radius . Take the Radius , or known Sine given , and make it a Parallel in the Sine of 90 for Radius , or in the Sine of the known Angle given , and lay the Thred to ND . Then , take the unknown Sine between your Compasses , and carry one Point along the Line of Sines , till the other foot being turned about , will but just touch the Thred ; then the place where the Compasses stayes , shall be the Sine of the unknown Angle required , to that Radius or known Sine . Make the given Radius a Parallel Radius , or the given Sine a = Sine , in the answerable Sine thereof : Then , taking the unknown Sine , carry it parallelly along the Line of Sines till it stay in like parts , which parts shall be the Numerator to the Sine required . Use IV. The Radius being given , by the Sines alone to find any Tangent or Secant to that Radius . Take the Radius between your Compasses , and set one Point in the Sine complement of the Tangent required , and lay the Thred to the ND ; then the ND from the Sine of the Tangent required , to the Thred , shall be the Tangent required : And the ND from 90 , to the Thred , shall be the Secant required . Make the given Radius a = in the co-Sine of the Tangent required ; then the = Sine ( of the inquired Ark or Angle ) shall be the Tangent required ; and = 90 shall be the Secant required to that Radius . Use V. Any Tangent or Secant being given , to find the answerable Radius ; and then any other proportionable Tangent , or Secant , by Sines only . First , if it be a Tangent that is given , take it between your Compasses , and setting one foot in the Sine thereof , lay the Thred to ND , then the = Co-sine thereof shall be Radius ; But , if it be a Secant , take it between your Compasses , and set one foot alwayes in 90 , lay the Thred to the ND , then the nearest distance from the Co-sine to the Thred ( or the = Co-sine ) shall be the Radius required . Take the given Tangent , make it a = in the Sine thereof ; then the = Co-sine thereof shall be Radius . Or , if it be a Secant given , then Take the given Secant , make it a = in 90 , then the = Co-sine thereof , shall be the Radius required . Then having gotten Radius , the 4th Use shewes how to come by any Tangent , or Secant , by the Sines only . Use VI. To lay down any Chord , to any Radius ; less then the Sine of 30 degrees . Take the given Radius ▪ set one Point in the Sine of 30 , lay the Thred to the ND ( and for your more ready setting it again , note , what degree and minuit the Thred doth stay at , on the degrees ) and there keep it . Then the ND from the Sine of half the Angle you would have , shall be the Chord of the Angle required . Take the given Radius , and make it alwayes a = in 30 , and 30 of Sines ; the = Sine of half the Chord , shall be the Chord required . Use VII . To lay down any Chord to the Radius of the whole Line of Sines . Take the Radius between your Compasses , and setting one Point in 90 of the Sines , lay the Thred to the ND , observing the place , there keep it . Then taking the = Sine of the Angle required , with it set one Point in the Line to which you would draw the Angle , as far from the Center as the Radius is ; then draw the Convexity of an Ark , and by that Convexity , and the Center , draw the Line for the Angle required . Example . Let AB be a Radius of any length , under or equal to the whole Line of Sines : Take AB between your Compasses , and setting one Point in 90 ; lay the Thred to ND , then take out the = Sine of 38 , or any other Number you please , and setting one Point in B , the end of the Radius from A the Center , and trace the Ark DC , by the Convexity of which Ark , draw the Line AC for the Angle required . Take the given Radius AB , make it a = in 90 , and 90 of Sines ; then take out = 38 , and setting one foot in B , draw the Ark DC , and draw AC for the Angle required . Or else work thus ; Take AB , the given Radius , ( having drawn the Ark BE ) and make it a = in the Co-sine of half the Angle required ; and lay the Thred to ND , ( or set the Sector ) . Then , Take the = ND , from the right-sine of the Angle required , and it shall be BE , the Chord required to be found . Note , That the contrary work finds Radius . Use VIII . To lay off any Angle by the Line of Tangents , or Secants , to prove it . Having drawn the Ground-Line , AB , at the Point B , raise a Perpendiculer , as the Line BC extended at length , then make AB , the Radius , a = Tangent in 45 and 45 ; then take out the = Tangent of the Angle required , and lay it from B to C in the Perpendiculer , and draw the Line AC for the Angle required . Also , If you take out the Secant of the Angle , as the Sector stands , and lay it twice in the Line AE , it will reach just to C , the Point required . Also Note , That if you want an Angle above 45 degrees , as the Sector stands , take the same from the small Tangent that proceeds to 75 , and turn that Extent 4 times from B , and it shall give the Point required in the Line BC. Use IX . To lay down , or protract any Angle by the Tangent of 45 only . First , make a Geometrical Square , a ABCD , and let A be the Anguler Point ; then making AB Radius , make AB a = Co-sine of the Angle you would have , and lay the Thred to the nearest distance , then the ND from the right Sine of the Angle to the Thred , shall be the Tangent required . Example . I make AB Radius a = in 50 , the co-Sine of 40 , then the = Sine of 40 shall be BE. Again , If I make AD equal to AB , the = co-Sine of 30 , viz. 60 ; and then take out the = Sine of 30 , and lay it from D to F , it shall be an Angle of 60 from AB , or 30 from D to F. But by the Sector this is more easie ; Use X. To take out readily , any Tangent above 45 , by the Tangent to 45 on the Sector-side . Take the given Radius , make it a = in the co-Tangent of the Tangent required ; then the = Tangent of 45 , shall be the Tangent required . Example . I would have a Tangent to 80 degrees ; take the given Radius , make it a = in 10 , the complement of 80 ; then the = Tangent of 45 , shall be the Tangent of 80 required . But if your Radius be so big , that you cannot enter it , then take the half , or a quarter of your Radius , and then = 45 will be the half , or the quarter of the Tangent required . Use XI . How to work Proportions , in Sines alone , by the Natural Sines . There are 4 Varieties in this Work , that include all Proportions , viz. 1. When the Sine of 90 is the first term , then the work is thus ; Lay the Thred to the second term , counted on the degrees from the Head , toward the loose-piece ; and count the third term on the Line of Sines , from the Center downwards ; and taking the nearest distance from thence to the Thred , and that distance measured from the Center downwards , on the Line of Sines , gives the 4th term required . Example . As Sine 90 , to Sine 23-30 ; So is 30 , to 11-31 . Take the Latteral second term , make it a = Sine of 90 ; then take out the = third term , and measuring it from the Center , it gives the 4th term required . 2. When the Sine of 90 is the third term , then work thus ; Take the — Sine , of the second term , from the Center downwards , and make it a = Sine in the first term , laying the Thred to ND ; then on the degrees , the Thred shall give the 4th term required . Example . As the Sine of 30 , to 23-31 ; So is the Sine of 90 , to Sine of 52-56 . But by the Sector , Take the — Sine of the second , make it a = Sine of the first term ; then take out = 90 , and measure it from the Center , and it shall give the 4th term required . Example as before . 3. When the Radius , or Sine of 90 , is in the second place , work thus ; Take — 90 from a lesser Scale , as the uppermost Sine above the Center , or the Line of Right-Ascentions , or the Azimuth-Scale , or the like ; and make it a = in the Sine of the first , laying the Thred to ND , then the = third term , taken and measured on the same Scale that 90 was taken from , shall give the 4th term required . Example . As — 90 , on the Line of Right-Ascention , is to = 30 ; So is = 20 , to 43-12 , measured on the same Line that 90 was taken from . Or else secondly , work thus ; As — 30 , to = 90 ; So is — 20 , to = 43-12 . By carrying the Compasses till it so stayes , as that the foot turned about , will but just touch the Thred , at the nearest distance . Or else thus , thirdly ; By transposing the terms , when the third is not greater than the first : thus ; As the first , to the third ; So is the second term , to the 4th : Where the Radius being brought to the third place , it is wrought by the second Rule , as before . By the Sector . Take a smaller — Sine of 90 , make it a = in 30 ; then the = Sine of 20 , taken and measured on the small Sine , gives 43-12 , as before . Again , As — 90 , to = 30 ; So is = 20 , to — 43-12 . Again , As — 30 , to = 90 ; So is = 20 , to = 43-12 : Lastly , by transposing . As — 20 , to = 30 ; So is = Radius , to — 43-12 ; as before . 4. When Radius is none of the given terms . Then when the first term is greater than the second and third , work thus ; Take the — second term , make it a = in the first , laying the Thred to the ND ; then the nearest distance , from the third term , to the Thred measured from the Center downward , give the 4th Sine required . Example . As 20 , to 12 ; so is 18 , to 10-50 . By the Quadrant . As — 12 , to = 20 ; So is = 18 , to — 10-50 . When only the second term is greater than the first , then transpose the terms , and work as before : Or else use a double Radius , which is on this Instrument very easily done , having several Radiusses . Or , Lastly , use a Parallel entrance , or answer rather , as before , which being carefully wrought , will do very well . By the Sector . The same manner of work , is as before by the Quadrant , and the setting the Sector , is all one to the laying the Thred , as will be largely seen in all the following Propositions , wrought both by the Artificial and Natural Lines , of Numbers , Sines , and Tangents , as followeth . Use XII . Having the day of the Month , or Suns place given , to find his Declination . Lay the Thred on the day of the Month in the Kalender , and in the Line of degrees , on the Moving-leg , you have his Declination , either Northward , or Southward , according to the time of the year , counting from 600 , toward the Head , for North-declination ; or toward the End , for South-declination . By the Artificial Sines and Tangents on the Edge of the Instrument . Extend the Compasses from the Sine of 90 , to the Sine of 23 degrees 31 minuts , the Suns greatest Declination : The same Extent applied the same way , from the Sine of the Suns place , or the Suns distance from the next Equinoctial-point , shall cause the Moving-point to fall , on the sine of the Suns declination ; This being the general way of working . Example . The Extent from the sine of 90 , to the sine of 23-31 , shall reach from the sine of 30 , to 11 deg . 31 min. the Suns declination , in ♉ Taurus 30 degrees from ♈ Aries , the next Equinoctial-point , and from 60 degrees , the Suns distance in ♊ Gemini 60 degrees , from ♈ 20 deg . 12 min. the Suns declination then . This being the manner of working by these Lines , by extending the Compasses from the first to the second term : I shall for the rest wave this large repetition of extending the Compasses , and render it only thus by the words of the Cannon-general in all Books ; As Sine 90 , to Sine 23-31 ; So is the Sine of 30 , to Sine 11-31 . Lay the Thred to 23-31 , on the degrees on the Moveable-piece , counted from the Head toward the End ; then count the Suns place from the next Equinoctial-point , on the Line of Sines from the Center downwards , and take the ND from thence to the Thred ; then this distance being measured from the Center downwards , shall be the sine of the Suns declination , required for that distance , from the next Equinoctial-point ; ( by the 1 st Rule abovesaid ) . By the Sector . Take — 23-31 , from the Sines , make it a = in the sine of 90 ; then the = sine of the Suns distance from the next Equinoctial-point , shall be the — sine of the Suns declination ; Example as before ( Rule the 1 st ) . Use XIII . The Suns Declination being given , to find his true place or distance from ♈ or ♎ , the two Equinoctial Points . Lay the Thred to the Declination counted in the degrees from 600 , and in the Line of the Suns place , is his true place required . Example . When the Suns declination is 12 degrees Northward , the dayes increasing , then the Sun will be 31 deg . and 23 min. from ♈ , or 1 deg . 23 min. in ♉ , his true place required . As Sine of 23-31 , the Suns greatest declination , to Sine of 90 ; So Sine of 12-00 , the Suns present declination , to Sine of Suns distance from ♈ or ♎ 31-23 . Which , by considering the time of the year , gives his true place , by looking on the Months and Line of Suns place on the Quadrantal-side . Take the — Sine of the present declination , make it a = Sine in the greatest declination , laying the Thred to ND ; and on the degrees the Thred shall give the Suns distance from ♈ , or ♎ , required . Example as before . Make — Sine of the given Suns declination , a = Sine in the Suns greatest declination , then = Sine of 90 , measured from the Center , is the = Sine of the Suns distance , from ♈ or ♎ , required ; or count 30 deg . for one sign , and the Center for the next Equinoctial-point , and 90 for the two Tropicks of Cancer , and Capricorn . ♋ . ♑ . Use XIV . The Suns place , or Day of the Month , and greatest Declination given ; to find his Right Ascention from the same Equinoctial . Lay the Thred to the day of the Month , or place given , and in the Line of the Suns Right Ascention , you have his Right Ascention in degrees , or hours and minutes , counting 4 minuts for every degree . Example . On the 9th of April , near night , the Sun being then entring ♉ , the Suns Right Ascention will be 1 hour 52 min. or 28 degrees of Right Ascention , distant from ♈ . As the Sine of 90 , to the Sine complement of the Suns greatest declination ( or C.S. ) of 23-31 , counting backwards from 90 , which will be at the Sine of 66-29′ . ) So is the Tangent of the Suns distance from the next Equinoctial-point ; to the Tangent of the Suns Right Ascention from the same Equinoctial-point . Take the — co-sine of the greatest declination from the Center downwards , being the — sine of 66-29′ . make it a = sine of 90 , laying the Thred to ND ; and note what degree and minuit it cuts , for this is fixed to this Proportion : Then take the Tangent of the Suns distance from the next Equinoctial-point , from the Center at 600 , on the degrees toward the End , and lay it on the sines , from the Center downwards , and note the Point where it stayeth , for the ND from thence to the Thred , shall be the Tangent of the Suns Right Ascention required . Note , That if the Suns distance from ♈ , or ♎ , be above 45 degrees , then the Tangents on the loose-piece , are to be used instead of the Tangents on the moveable-leg . Or , by Sines only thus ; Or , Take — Sine of the present Suns declination , make it a = in the Sine of the Suns greatest declination , and lay the Thred to ND ; then take = Co-sine of the Suns greatest declination , and make it a = in Co-sine of the Suns present declination , and lay the Thred to ND , and in the degrees it cuts the Suns Right Ascention , required . Make — Co-sine of 23-31 , viz. the right Sine of 66-29 , a = sine of 90 , then the = Tangent of the Suns distance from ♈ , or ♎ , is the = Tangent of the Suns Right Ascention from the same Point of ♈ , or ♎ ; as at 30 from ♈ , it is 28 degrees , or 1 hour and 52 minuts from ♈ , ( neer ) . Use XV. Having the Suns Right Ascention , and greatest Declination , to find the Angle of the Ecliptick and Meridian . As Sine 90 , to Sine 23-31 ; So is the Co-sine of the Suns Right Ascention , to the Co-sine of the Angle of the Ecliptick and Meridian . Lay the Thred to 23-31 , counted on the degrees from the Head ; then count the Co-sine of the Right Ascention , from the Center downward , or the Sine from 90 upwards , and take the ND from thence to the Thred , and measure it from the Center , and it shall reach to the Co-sine of the Angle required . Example . The Right Ascention being 30 degrees , or 2 hours , the Angle shall be 69-50 . Make the — right sine of 23-31 , a = sine of 90 ; then the = co-sine of 30 , viz. = 60 , shall make the — sine of 69-50 , the Angle of the Ecliptick and Meridian . Use XVI . Having the Latitude , and Declination of the Sun or Stars , to find the Suns or Stars Amplitude , at rising or Setting . Take the Suns declination , from the particular Scale of Sines , and lay it from 6 , in the hour or Azimuth-line , and it shall give the Amplitude from South , as it is figured ; or from East , or West , counting from 90 ; observing to turn the Compasses the same way from 90 or 6 , as the declination is Northward , or Southwards . Example . The Suns declination being 10 degrees Northward , the Suns Amplitude , or Line , is 106-12 , from the South , or 16-12 from the East-point . As co-sine of the Latitude , to S. 90 ; So is S. of the Suns declination , to S. of the Amplitude . Take the — Sine of the Suns declination , make it a = in the co-sine of the Latitude , and lay the Thred to the nearest distance , and on the degrees the Thred shall shew the true Amplitude required . Make the — right Sine of the Suns declination , a = in co-sine latitude , then = 90 , taken and measured from the Center , gives the Amplitude or Line . Use XVII . Having the same Amplitude , and Declination , to find the Latitude . As S. of the Suns Amplitude , to S. the Suns Declination ; So is S 90 , to Co-sine Latitude . Take the — sine of the Suns declination ; set one Point in the Sine of the Suns Amplitude , lay the Thred to ND , and on the degrees it sheweth the complement of the Latitude required . Example . The Declination being 20 degrees , and the Amplitude 33-15 , the complement of the Latitude will be 38-28 — , counting from the Head , toward the End. Make the right Sine of the Suns Declination , a = sine in the Suns Amplitude ; then the = sine of 90 , shall be the — co-sine of the Latitude required . Use XVIII . Having the Latitude , and Suns Declination , to find his Altitude at East or West , commonly called the Vertical-Circle ; or Azimuth of East or West . Take the Suns Declination from the particular Line of Sines , set one Point in 90 on the Azimuth-line , and lay the Thred to the ND , and on the degrees it sheweth the Altitude required ; counting from 600 toward the End. As S. latitude S. of 90 ; So S. of Suns declination , to S. Suns height , at East or West . Take the — sine of the Suns declination , make it a = in the sine of the latitude , and lay the Thred to ND , and on the degrees it shall shew the Suns Altitude , at East and West required . Example . Declination 10. Latitude 51-32 ; the Altitude is 12 degrees , and 50 minuts . As — S. of the Suns Declination , to = S. of Latitude ; So is the = S. of 90 , to — S. of Vertical Altitude . Use XIX . Having the Latitude , and Suns Declination , to find the time when the Sun will be due East or West . Having gotten the Altitude by the last Rule , take it from the particular Sine ; then lay the Thred to the Suns declination , counted on the degrees ; then setting one Point in the Hour-line , so as the other turned about , shall but just touch the Thred , and the Compass-point shall stay at the hour and minuit of time required . As Tangent latitude , to Sine 90 ; So is the Tangent of the Suns declination , to Co-sine of the hour . Or , As sine 90 , to Tangent Suns declination ; So is Co-tangent-latitude , to Co-sine of the hour from noon . Example . Latitude 51-32 , declination 10 , the Sun will be due East at 6-32 , and West at 5-28 . Take the — Tangent of the Latitude ( on the loose-piece , counting from 60 towards the moveable-leg ; or else from 600 , on the moving-leg , or degrees , according as the Latitude is above or under 45 degrees ) and lay it from the Center downwards , and note the Point where it ends . Then take from the same Tangent , the Tangent of the Suns declination , and setting one foot in the Point last noted , lay the Thred to ND , then the = sine of 90 , shall be the — sine of the hour from 6. Or by the Sines only work thus ; Take the — sine of the Suns declination , make it a = in sine of the latitude ; lay the Thred to ND , then take ND from the Co-sine latitude to the Thred ; then set one foot in the Co-sine of the Suns declination , lay the Thred to ND , and on the degrees it gives the hour from noon , as it is figured , or the hour from 6 , counting from the head , counting 4 minuts for every degree . Make the small Tangent of the Latitude , if above 45 , taken from the Center , a = sine of 90 ; then the — Tangent of the Suns declination , taken from the same small Tangent , and carried Parallely till it stay in like Sines , shall be the Sine of the hour from 6. Or , as before , by Sines only . Make — sine Declination , a = sine Latitude ; then take = Co-sine Latitude , and make it a = Co-sine of the Suns Declination ; then take = 90 , and lay it from the Center , it gives the Sine of the hour from 6. Use XX. Having the Latitude , and Suns Declination , to find the Ascentional Difference , or the Suns Rising and Setting , and Oblique Ascention . Lay the Thred to the Day of the Month , ( or to the Suns Declination , or true Place , or to his Right Ascention ; for the Thred being laid to any one of them , is then also laid to all the rest ) then in the Azimuth-line , it cuts the Ascentional difference , if it you count from 90 , or the Suns Rising , as you count the morning hours ; or his Setting , counting the afternoon hours . The Oblique Ascention is found out for the six Northern signs , or Summer half-year , by substracting the Suns difference of Ascentions , out of the Suns Right Ascention . But for the other Winter-half year , or six Southern signs , it is found by adding the Suns difference of Ascentions to his Right Ascention ; this sum in Winter , and the remainder as above-said in Summer , shall be the Suns Oblique Ascention required . As Co-tangent Lat. to sine 90 ; To is the Tangent of the Suns declination , to the sine of the Suns Ascentional difference . Take the — co-tangent latitude , from the loose or moveable-piece , as it is above or under 45 degrees , make it a = in sine 90 , lay the Thred to ND , then take the — Tangent of the Suns declination from the same Tangents , and carry it = till it stay in the parts , that the other foot , turned about , will but just touch the Thred , which parts shall be the Sine of the Suns Ascentional difference required . Or ●hus , by Sines only ; Make the — sine of Declination , a = Co-sine of the Latitude ; lay the Thred to ND , then take the = sine of Latitude , make it a = in Co-sine of the declination , and lay the Thred to ND , and on the degrees it shall cut the Suns Ascentional-difference required ; which being turned into time , by counting 4 minuts for every degree , and added to , or taken from 6 , gives the Suns Rising in Summer , or Winter . Make the — Co-tangent Latitude , a = sine of 90 ; then take — Tangent of the Suns declination , and carry it = till it stay in like parts , viz. the Sine of the Suns Ascentional difference required . Example otherwise ; As — sine 90 , to = Tangent 38-28 ; So is = Tangent of 23-31 , the Suns greatest declination , to the — sine of the Suns greatest Ascentional difference , 33 deg . and 12 min. Use XXI . The Latitude and Suns Declination given , to find the Suns Meridian Altitude . When the Latitude and Declination is both alike , viz. both North , or both South ; then substract the Declination out of the Latitude , or the less from the greater , and the remainder shall be the complement of the Suns Meridian Altitude . But when they be unlike , then add them together , and the sum shall be the complement of the Meridian Altitude : The contrary work serves when the complement of the Latitude and Declination is given , to find the Meridian Altitude . Lay the Thred to the Declination , counted on the degrees from 600 , the right way , toward the Head for North , and toward the End for South declination . Then , Take the nearest distance , from the Center-prick at 12 , in the Hour-line , to the Thred ; this distance measured on the Particular-line of Sines , shall shew the Suns Meridian-Altitude required . Use XXI . The Latitude , and Hour from the midnight Meridian given , to find the Angle of the Suns Position , viz. the Angle between the Hour and Azimuth-lines in the Center of the Sun. As Sine 90 , to Co-sine of the Latitude ; So is the Sine of the Hour from Midnight , to the sine of the Angle of Position . Example . As Sine 90 , to Co-sine Latitude 38-28 ; So is the Co-sine of the Hour from midnight , 120 , for which you must use 60 , to 32-34 , the Angle of Position . Take the distance from the Hour to the 90 Azimuth on the Hour-line , and measure it in the particular sines , and it shall shew the Angle of Position required . This holds in the Equinoctial . Take — Co-sine Latitude , make it a = in sine 90 ; then take out the = Co-sine of the Hour from the Meridian , and it shall be the — sine of the Suns Position . Make — Co-sine Latitude a = sine 90 ; then = Co-sine of the Hour , shall be — sine of the Suns Position . Note , The Angle of the Suns Position may be varied , and it is generally the Angle made in the Center of the Sun , by his Meridian or Hour-circle , being a Circle passing thorow the Pole of the World , and the Center of the Sun ; and any other principal Circle , as the Meridian , the Horizon , or any Azimuth , the Anguler-Point being alwayes the Center of the Sun. Use XXII . The Suns Declination given , to find the beginning and end of Twi-light , or Day-break ▪ Lay the Thred to the Declination on the degrees , but counted the contrary way , viz. South-declination toward the 〈…〉 North-declination toward the 〈◊〉 ; then take 18 degrees from the particul●● 〈◊〉 Sines for Twi-light , or 13 degrees for Day-break , or clear light ; Then carry this distance of 18 for Twi-light , or 13 for Day-break , along the Line of Hours on that side of the Thred next the End , till the other Foot , turned about , will but just touch the Thred , then shall the Point shew the time of Twi-light , or Day-break , required . Example . The Suns Declination being 12 degrees North , the Twi-light continues , till 9 hours 24 minuts ; or it begins in the morning at 38 minuts after 2 ; but the Day-break is not till 22 minuts after 3 in the morning , or 38 minuts after 8 at night , and last no longer . To work this for any other place , where the Latitude doth vary , do thus ; Find the Hour that answers to 18 degrees of Altitude , in as much Declination the contrary way , and that shall be the time of Twi-light ; or at 13 degrees for Day-break , according to the Rules in the 26th Use , where the way how is largely handled to the 33d Use , both wayes generally . Use XXIII . To find for what Latitude your Instrument is particularly made for ; Take the nearest distance from the Center on the Head-leg , to the Azimuth-line on the moveable-leg ; this distance measured on the particular Scale of Sines , shall shew the Latitude required ; or the Extent from 0 to 90 , on the Azimuth-line , shall shew the complement of the Latitude , being measured as before . Use XXIV . Having the Meridian Altitude given , to find the time of Sun Rising or Setting , true Place , or Declination . Take the Suns Meridian Altitude from the particular Scale , and setting on Point in ☉ on the Azimuth-line ; lay the Thred to the ND , and on the Hour-line it sheweth the time of Rising or Setting ; and on the degrees , the Declination ; and the rest in their respective Lines . Example . The Meridian Altitude being 50 , the Sun riseth at 5 , and sets at 7. Use XXV . The Latitude and Declination given , to find the Suns height at 6. Lay the Thred to the Day of the Month , or Declination , then take the ND from the Hour-point of 06 , and 6 to the Thred , and that distance measured on the particular Scale of Sines , shall be the Suns Altitude at 6 in Summer time , or his depression under the Horizon in the Winter time . As sine of 90 , to sine of the Suns Declination ; So is sine Latitude , to sine of the Suns Altitude at 6. Count the Suns declination on the degrees from 90 , toward the End , and there lay the Thred ; then the least distance from the sine of the Latitude to the Thred , measured from the Center downwards , shall be the sine of the Suns Altitude at 6. Make the — sine of the Declination a = sine of 90 ; then the = sine of the Latitude , shall be the — sine of the Suns height at 6. Example . Latitude 51-32 , Declination 23-31 , the height at 6 , is 18 deg . 13 min. Use XXVI . Having the Latitude , the Suns Declination and Altitude , to find the Hour of the Day . Take the Suns Altitude , from the particular Scale of Sines , between the Compasses ; then lay the Thred to the Day of the Month , or Declination ; then carry the Compasses along the Line of Hours , between the Thred and the End , till the other Point ( being turned about ) will but just touch the Thred , and then the fixed Point shall shew the true hour and min. required , in the Fore , or After-noon ; if you be in doubt which it is , then another Observation presently after , will determine it . Example . May 10th , at 30 degrees of Altitude , the hour will be 32 minuts after 7 in the Morning , or 28 minuts after 4 in the Afternoon . This Work being somewhat more difficult than the former , I shall part it thus ; 1. First , to find the Hour the Sun being in the Equinoctial . Take the — sine of the Suns Altitude , make it a = Co-sine of the Latitude ; lay the Thred to ND , and on the degrees it shall give the Hour from 12 , as it is figured , counting 15 degrees for an hour , or from 6 , counting from the Head at 90. Example . Latitude 51-30 , Altitude 20 , the hour is 8 & 12′ in the forenoon , or 3-48′ in the afternoon . The same by Artificial Sines & Tangents . As Co-sine Latitude , to sine 90 ; So is the sine of the Suns Altitude , to sine of the hour from 6. Make — S. ☉ Altitude , a = S. in ☉ Latitude ; then take out = S. 90 , and it shall be the — sine of the hour from 6. 2. The Latitude , Declination , and Altitude given , to find the Hour at any time . First by the 25th Use , find the Suns Altitude or depression at 6 ; then in Summer-time , lay this distance from the Center downwards ; and in Winter-time , lay it upwards from the Center toward the End of the Head-leg ; and note that Point for that day , or degree of Declination ; for by taking the distance from thence to the Suns Altitude , on the General Scale , you have added , or substracted the Altitude at 6 , to , or from the present Altitude . ( For by taking the distance from that noted Point , over , or under the Center , to the Suns present Altitude , you have in Summer the difference between the Suns present Altitude , and his Altitude at 6. And in Winter you have the sum of the present Altitude , and the Altitude at 6. ) This Operation is plainly hinted at , in the 4th Chapter , and 9th and 10th Section , which being understood , take the whole Operation in shorter terms , thus ; Count the Suns Declination from 90 , toward the end , and thereunto lay the Thred ; the nearest distance from the sine of the Latitude to the Thred , is the Suns height , or depression at 6 : In Winter use the sum of , in Summer the difference between , the Suns Altitude at 6 , and his present Altitude ; with this distance between your Compasses , set one Point in the co-sine of the Latitude ; lay the Thred to ND , then take the ND from 90 , to the Thred ; then set one foot in the Co-sine of the Suns declination , and lay the Thred to ND , and on the degrees it gives the hour required ; from 6 counting from 90 , or from 12 , as it is figured . Example . On April 20 , at 30 deg . 20 min. of Altitude , Latitude 51-32 , the hour will be found to be just 2 hours from 6 , or just 8. Again , On the 10th of November , at 8 deg . 25 min. high , it is just 3 hours from 6 , or 9 a clock in the forenoon , or 3 afternoon . Or somewhat differing thus ; Take the — sine of the sum , or difference , of the Suns present Altitude , and Altitude at 6 , and make it a = in the co-sine of the Latitude , and lay the Thred to the nearest distance ; then take out the = Secant of the declination beyond 90d , and make it a = sine of 90 ; and laying the Thred to the nearest distance , on the degrees it shall shew the hour from 6 required . First , by Use 25 , find the Suns height at 6 , or depression in Winter ; then by the former 2d , find the sum or difference between the Altitude at 6 , and the Suns present Altitude ; but if you have Tables of Natural Sines and Tangents ; then in Winter , add the Natural Sines of the two Altitudes together ; and in Summer , substract the lesser out of the greater , and find the Ark of difference more exactly . Then , As the Co-sine of the Latitude , to the Secant of the Declination ( counted beyond 90 , as much forward as from 90 to the Co-sine of the Suns Declination ) ; So is the Sine of the sum , or difference , to the hour from 6 ▪ required . Or else ●hus ; As the Co-sine of the Latitude , to the Sine of the sum , or difference ; So is sine of 90 , to a 4th . Then , As the Co-sine of the Suns declination , to that 4th ; So is sine 90 , to the hour from 6. By the Sector . Take the — secant of the Suns declination , make it a = in the co-sine of the Latitude ; then take out the = sine of the sum or difference , and turn it twice from the Center lattera●ly , and it shall be the sine of the hour from 6 , required . Example . April 20 , the Suns Declination is 15 degrees ; and the Suns Height at 6 , then is , 11 deg . 42 min. now the Natural sine of 11-42 , 20278 , taken from the Natural sine of 30 deg . 20 min. 50502 , the Suns present Altitude , the residue is 30224 , the sine of 17 deg . 35 min. and a half . Then , The — Secant of 15 made a = sine of 38-28 , and the Sector so set , the = sine of 17-35 ½ , turned latterally twice from the Center , shall reach to 30 , the sine of 2 hours from 6 , the hours required . Use XXVII . Having the Latitude , the Suns Declination , and Altitude , to find the Suns Azimuth . Take the Declination from the particular Scale of Sines , for the particular Latitude the Instrument is made for ; Then , count the given Altitude on the degrees from 600 toward the loose-piece , and sometimes on the loose-piece also ; and thereunto lay the Thred , then carry the Compasses , so set , along the Azimuth-line on the right-side of the Thred in Northern-declinations , and on the left-side in Southern-declinations , till the other foot , turned about , will but just touch the Thred ; then the fixed-point shall stay at the Suns true Azimuth required . Take two or three Examples . 1. First , When the Sun is in the Equinoctial and hath no Declination , then there is nothing to take between your Compasses , but just to lay the Thred to the Suns Altitude , counted from 600 on the loose-piece toward the End : then 〈◊〉 the Azimuth-line , it cuts the Azimuth from the South required . Example . At 00 degrees high , the Azimuth is 90 from South ; and at 10 degrees high , it is 77-5 ; at 20 high , the Azimuth is 62-45 ; at 30 degrees high , it is 43-30 ; at 34 degrees high , it is 32 degrees of Azimuth from South ; and at 38-28 degrees high , it is just South . 2. Secondly , at 10 degrees of Declination Northward , and 20 degrees of Altitude , take 10 degrees from the particular Scale , and lay the Thred to the Suns present Altitude , as before , and carry the Compasses on the right-side of the Thred on the Azimuth-line , till the other foot , being turned about , will but just touch it ; then shall the Point rest at 80 degrees , 42 min. of Azimuth from the South . 3. But if the Declimation be the same to the Southwards , and the Altitude also the same ; then carry the Compasses on the left-side of the Thred , on the Azimuth-line , till the other foot , turned about , will but just touch it , and you shall find the Point to stay at 41 deg . 10 min. the true Azimuth from the South required . Note , That any thing , as thick as the Rule , laid by the Rule , and the Thred drawn over , it will keep the Thred steady , till you get the nearest distance more truly . First , by the 18th Use , find the Suns Altitude in the Vertical Circle , or Circle of East and West , thus ; Take the sine of the Suns Declination , and set one foot in the sine of Latitude , lay the Thred to ND , and in the degrees you shall have the Altitude at East and West required . Which Vertical Altitude in Summer or Northern Declinations , you must substract out of the Suns present Altitude ; or take the lesser from the greater , to find a difference ; but in Winter , you must add this depression in the Vertical Circle , to the Suns present Altitude to get a sum , which must be done on a Line of Natural Sines , or by the TABLE of Natural Sines , as before , in the Hour , by laying it over or under the Center , and taking from that noted Point to the Suns present Altitude all that day . Then take the distance from the Center to the Tangent of the Suns present Altitude on the loose-piece , which is the Secant of the Suns present Altitude , and lay it from the Center on the Line of Sines , and note the place ; then take the distance from 60 , on the loose-piece , to the co-tangent of the Latitude ( by counting 10 , 20 , 30 , &c. from 60 , toward the moveable-leg ) between your Compasses ; then setting one Point on the Secant of the Suns Altitude last found , and noted on the Line of Sines ; and with the other , lay the Thred to the nearest distance , and there keep it , ( by noting what degree , day of the month , or hour & minut , or Azimuth it cuts ) . Then take the — distance on the Sines , from the sine of the Suns Vertical Altitude , to his present Altitude , for a difference in Summer ; Or , The distance from a Point made beyond the Center , ( equal to the sine of the Suns Vertical depression ) to the Suns present Altitude , for a sum in Winter . Then having this — distance of sum or difference , for Winter or Summer , between your Compasses ; carry one Point parallelly on the Line of Sines , till the other , being turned about , shall just touch the Thred at the ND , the place where the Point stayeth , shall be the Azimuth from East or West , as it is figured from the Center ; or from North or South , counting from 90. Which work in brief , may be sufficiently worded thus ; As — co-tangent of the Latitude , to the = secant of the Suns present Altitude , laying the Thred to ND ; So is the — sine of the sum , or difference , of the Suns present Altitude , & Vertical depression in Winter , or the difference between his Vertical and present Altitude in Summer ; to the = sine of the Suns Azimuth , at that Altititude and Declination . Yet again , more short . As — C.T. Lat. to = Sec. ☉ Alt. So — S. of sum or difference , to = S. ☉ Azimuth . But note , That in Latitudes under 45 , when the complements of the Latitude are too large , then work thus ; As the — co-sine of the Suns Altitude , to = Tangent of the Latitude , taken from the degrees on the moveable-leg , laying the Thred to ND , then the — sine of the sum or difference carried parallelly , shall stay at the Suns Azimuth required . If the Tangents are too small , on the Sector-side is a larger ; and if the Sines are too great , on the Head-leg there is a less . Find the Vertical Altitude by Use 18 , and the sum or difference of the present and Vertical Altitude by the Table , or Line of Natural Sines , as before shewed ; then the Canon or Proportion runs thus ; As the Co-sine of the Suns Altitude , to the Tangent of the Latitude ; So is the sine of the sum or difference , to the sine of the Azimuth , from East or West . Or , As Co-tangent Latitude , to Secant of the Suns Altitude ; So is the sine of the sum or difference , to the sine of the Azimuth . Make the — Secant of the Suns Altitude , a = Co-tangent of Latitude ; then the — sine of the sum or difference , shall be half the — sine of the Azimuth ; or being turned twice from the Center , the whole sine . Or else thus ; Make the — Tangent of the Latitude , a = Co-sine of the Suns Altitude ; then the = sine of the sum or difference , shall be the — sine of the Azimuth , measured on the Sine , equal to the Radius of the Tangents first taken . Example . In Latitude 51-32 , Declination North and South 13-15 , the Vertical Altitude or Depression being 17-01 , and the present Altitude 20 ; the Azimuth for South-declination will be found to be 31-45 , from South , the Depression at East and West being 17-01 ; and the sum of the present Altitude and Depression 39-25 . Again , For North-declination , or Summer-time , the difference between the Vertical and present Altitude , is 2-54 ; and the Azimuth from South , will be found to be 86 degrees and 15 minuts . Use XXVIII . To make a Scale , whereby to perform all th●se Propositions , by the former Rules , agreeable to the Trianguler Quadrant , being added chiefly as a Demonstration of the Instrument , and former Operations . Then making DE Radius , describe the Circle 90 EI , and divide it into 180 equal degrees ; Also ▪ draw the lesser Circle 90 F to the Radius DF , then a Rule laid to the Center D , and every one of the 180 degrees , shall divide the Tangent Lines AC , and BC , into 180 degrees ; and if you work right , you will meet with all the former Points , F , G , 45 , I , 69-54 , 60-45 , H , and E , in their true places , as first drawn . Also , Perpendiculers let fall from every degree in the Circle 90 EI to the Line DB , shall divide the Line of Sines , D 90 , to the the greater Radius ; and the like Perpendiculers from the degrees in the lesser Circle , to the Line DA , shall divide the lesser Line of Sines ; Also , the Extent from the Center D , to the Tangent of any Ark or Angle in the Line AC , counting from F , shall be the Secant to that Ark or Angle , to the lesser Radius ; and the measure from the Center D , to the Tangent of any Ark or Angle in the Line CB ( but counted from E ) shall be the Secant to that Ark or Angle , to the greater Radius . This little Instrument thus made , and a Thred fastened at D , will perform any Proposition by the Rules here inserted , and is the very making of the Trianguler Quadrant ; or you may put these Lines on a Rule as a plain Scale , and use them thus : As for Example , for the Azimuth last treated on . First draw a streight Line , as AB , representing the Line AB in the Trianguler-Quadrant ; then appoint in that Line any Point for a Center , as C ; then for this Proposition of finding the Azimuth , the Sines and Tangents being on a streight Scale , work thus ; First , to find the Suns Altitude , or Depression in the Vertical-Circle . Take the Sine of the Latitude , and lay it from C to 51-30 ; then take out the Sine of 13-15 , between your Compasses , and setting one Point in the Point 51-30 , last made in the Line AB , and strike the touch of the Arch at D , and draw the Line CD ; also , on the Line CB , lay down from C the sine of 90 out of the Scale , then the nearest distance from the Point for 90 in CB , to the Line CD , shall be the Sine of the Suns Altitude in the Vertical , in Summer or Northern declination , or his depression in Winter , viz. 17-01 . Then , as before , on the Line of Sines , find a sum for Winter , or a difference in Summer , between the Vertical and present Altitude ; Now supposing the Altitude 15 , the sum is 33-30 , or the difference is 1-58 , which you must remember . Then take the Secant of 15 , the Suns present Altitude from the Scale , lay it from C to E ; then take out the Co-tangent of the Latitude between your Compasses , set one Point in E , and strike the touch of an Ark , as at F , and draw the Line CF ; then take the sine of 33-30 , the sum , if it be Winter , or 1-58 , if it be Summer , between your Compasses , carry one Point in the Line CB , higher or lower , till the other foot , turned about , will but just touch the Line CF ; then the measure from thence to the Point C , shall be the Sine of the Azimuth required , viz. in Winter 43-50 ; and in Summer 92-30 , from the South , because the present Altitude is less than the Vertical , or East and West . But when the Co-tangent of the Latitude is too large for a Parallel entrance , then prick off first the Tangent of the Latitude , and take the Co-sine of the Suns Altitude to work in a Parallel way which will remedy the inconveniences ; Thus you see that by drawing three Lines only this work is done ; yet not so soon by far , as by the Instrument with the Thred , which represents those Lines more certainly and exactly , after the same way of Operation . To find the Suns Azimuth in Southern Declinations . As the Co-sine of the Latitude , to the Sine of the Suns present Altitude ; So is the Sine of the Latitude to a 4th sine ; which 4th sine is to be added to the Suns Amplitude , for that time , on a Line of Natural sines , and the sum observed , as a 5th . Then , As the Co-sine of the present Altitude , is to the sine of the sum last found ; So is the sine of 90 , to the sine of the Suns Azimuth , from East or West , required . For the Amplitude , work thus ; As Co-sine Lat. to S. Suns declination ; So is S. 90 , to the sine of Amplitude . Use XXIX . Having the Latitude , Suns Altitude , and Vertical Altitude , to find the Azimuth . And first for Northern-Declinations . First , find the Vertical Altitude by the former Rule , and find the difference between it and the present Altitude , by the Line of Sines : then take this difference from the general Sines between your Compasses , and setting one foot in the Co-sine of the Latitude , lay the Thred to the ND , then take the ND from the sine of the Latitude to the Thred ; having this distance , set one one foot in the Co-sine of the Suns Altitude , and lay the Thred to ND , and on the degrees it shall shew the Suns true Azimuth at that Altitude and Declination required . Example . The Suns Declination being 7 , the Virtical Altitude is 8-57 ; the Suns present Altitude being 30 , the difference or residue in Sines will be 20-13 , and the Suns Azimuth found thereby will be 60-12′ . The same by Artificial Sines and Tangents , in Summer . As Co-S. Lat. to S. of residue ; So is S. 90 , to a 4th sine . Then , As Co-S. ☉ Alt. to the 4th sine ; So is S. 90 , to S. of ☉ Azimuth , from East or West . Secondly , in Southern-Declinations , work thus ; First , find the Suns Amplitude for that Declination , thus ; Take the — sine of the Declination , make it a = in the C●-sine of the Latitude ; lay the Thred to ND , and on the degrees it gives the Suns Am●litude for that Declination , which you must remember . Then , Take the — sine of the Suns present Altitude , make it a = in the Co-sine of Latitude , lay the Thred to the ND , then take the ND from the sine of the Latitude to the Thred , and as the Compasses so stand , set one foot in the sine of the Suns Amplitude first found , and turn the other foot onward toward 90 ; then take from thence to the Center . Thus have you added the Amplitude , and last found distance together on Sines , then this added Latteral-distance , must be made a Parallel in the Suns Co-altitude , and the Thred laid to the nearest distance in the degrees , gives the Azimuth required . Example . At 15 degrees of Declination , and 10 degrees of Altitude , the Azimuth will be found to be 49 degrees 46 minuts from the South , and the Amplitude 24-30 in 51-32 of Latitude . The same , work by Artificial Sines and Tangents , in Winter . As co-S. of Lat. to S. of ☉ present Alt. So is S. of Lat. to a 4th ; which you must add to the Suns Amplitude on Natural Sines , and keep it as a sum ; Then , As co-S. of ☉ Alt. to S. of the sum ; So is S. 90 , to S. Azim . from East or West . Use XXX . Having the Latitude , the Suns Declination , his Meridian and present Altitude given , to find the Hour . Make the — Secant of the Latitude , a = in the Co-sine of the Suns declination , laying the Thred to ND ( and note the place ) ; then take the — distance on the sines , between the Suns Meridian and present Altitude , and lay it from the Center toward 90 ; then the ND from that Point to the Thred ( as before laid ) shall be the versed Sine of the Hour , measured on a Line of versed Sines , equal in Radius to the Line of Secants first taken , as the Sines above 90 are . Make the — Secant Latitude , a = Sine of Co-declination ; then the — distance between the Suns Meridian and present Altitude , laid on both Legs from the Center latterally , and the = distance between , measured on versed Sines , equal to the Secants , shall give the hour required ; as the great Line of Sines on the Sector are , by turning the Compasses twice , because the Line of Secants is half the Radius of those Sines , as at first was hinted . Example . Latitude 51-32 , Meridian Altitude 50 , present Altitude 40 , Declination North 11-30 , the Hour will be found to be 9-32 in the forenoon , or 2 h 28′ in the afternoon . Note , That if the Suns Meridian Altitude be above 90 , count the excess from 90 toward the Center , and take from thence to the present Altitude . Use XXXI . To find the Suns Azimuth , by having the Latitude , Declination , and Suns present Altitude . First , make the — Secant of the Latitude , a = Sine in the Suns Co-altitude , and lay the Thred to ND , and note the exact place where it is laid ; then find by Addition the sum of the complements of the Latitude , and Suns Altitude ; and observe whether the sum of them , be above or under 90. For when under 90 , count it from 90 toward the Center , as is usual , and take the distance from thence to the Center , and substract this on the Line of Sines , out of the declination , or take the Sine of the declination , out of this , ( and lay the residue from the Center ) ; and note the place . Or shorter thus ; If the sum of the complements of the Co-latitude and Co-altitude be under 90 , then count the same on the Line of Sines from 90 , toward the Center , and take the distance from thence to the Sine of the Suns Declination , and lay this from the Center ; for the = distance from thence to the Thred ( as first laid ) shall be the versed Sine of the Azimuth from the South required . Example . Latitude 51 32 Declination 15 00 Altitude 50 00 Co-altitude 40 0 Co-latitude 38 28 Sum 78 28 Azimuth is , 33 0 But if the sum of the complements of the Latitude and Suns Altitude be above 90 , then count from 90 to the Center as 90 , and reckon the excess above 90 , from the Center toward 90 , and take from thence to the Center , and add this distance on the Sines to the Suns declination toward 90 , and take from thence the nearest distance to the Thred , and that shall be the versed Sine of the Suns Azimuth from noon . But when the complements are under 90 , then the ND from the noted place to the Thred , shall be the versed Sine of the Azimuth required . But in Winter , when the sum of the complements are above 90 , and are counted backwards , from the Center , towards 90 ; take the — distance from thence to the Sine of the Suns declination , the lesser from the greater , and set this distance , or residue , from the Center downwards ; then the nearest distance from thence to the Thred , shall be the versed Sine of the Azimuth . But when the Latitude is less than the Suns Declination , and the same way ; then take the — distance ( on the Sines ) from the sum of the Suns Altitude , and Co-latitude , found by Addition , when under 90 , and counted from the Center to the declination , and lay that from the Center , as before is shewed . But if the sum of the Suns Altitude , and the complement of the Latitude , be above 90 ; then , having counted forwards from the Center to 90 , count the excess from 90 toward the Center , and take the — distance from thence , to the Sine of the Suns declination , and lay it from the Center , as before ; then the ND from thence to the Thred , shall give the versed Sine of the Suns Azimuth on the small Sines beyond the Center . The very same manner of Operation that serves for the General-Quadrant , serves also for the Sector , and this way being more troublesome than the rest , I shall say no more to it , but proceed to others . Use XXXII . The Suns Altitude , the Latitude , and Declination given , to find the Hour . Add the Co-latitude , Co-altitude , and Suns distance from the Elevated Pole together , for a sum ; and find the half sum , and the difference between the half sum and the Co-altitude . Then say ; As Sine 90 , to Co-sine Latitude ; So is the Sine of Suns distance from the Pole , to a 4th Sine . Again , As the 4th Sine , to the Sine of the half - sum ; So is the Sine of the difference , to the versed Sine of the Hour , if you have them on the Rule ; if not , to a 7th Sine , whose half-distance on the Sines towards 90 , gives a Sine , whose complement doubled , and turned into time , is the Hour from South required . Example , at 36 deg . 42 min. Altitude , and 23 deg . 31 min. Declination , Latitude 51-32 North. 53-18 , the Co-altitude ; 38-28 , the Co-latitude ; and 66-29 , added together , makes 158-15 for a sum ; then the half - sum is 79-07 , and the difference between 79-07 and 53-18 , is 25-49 for a difference . Then , The Extent from sine 90 , to the Sine of 38-28 , will reach the same way from the sine of 66-29 , to the sine of 34-47 , for a 4th Sine . Again , The Extent from Sine 34-47 , to Sine 79-7 , shall reach the same way from the Sine of 25-49 , the difference , to the Sine of 48-34 , a 7th Sine , right against which , on the versed Sines , is 60 , viz. 4 hours from noon . Or else , The half-distance , between Sine 48-34 , and the Sine of 90 , is the Sign of 60 degrees , whose complement , viz. 30 doubled is 60 degrees , or 4 hours in time , from noon . Use XXXIII . To find the Suns Azimuth , having the same things given , viz. Co-latitude , Co-altitude , and Suns distance from the Pole. Add , as before , the three Nunbers together , and thereby find the sum , and half - sum , and the difference between the half - sum , and the Suns distance from the Elevated Pole. Then say , As the sine of 90 , to the Co-sine of the Latitude ; So is the Co-sine of the Altitude , to a 4th sine . Again , As the Sine of the 4th , to the Sine of the half - sum ; So is the Sine of the difference , to the versed Sine of the Suns Azimuth , from South , ( or to a 7th sine , whose half-distance , toward 90 , gives a sine , whose complement doubled , is the Azimuth from South ) . Example , Latitude 51-32 , Altitude 41-53 , Declination North 13. The 3 Numbers , viz. 38-28 , 49-7 , and 77-0 , added together , makes 163-35 ; whose half is 81-47 ½ , and the difference between the half - sum , and the Suns distance from the Pole , is 4-47 ½ . Then , As sine 90 , to sine 38-28 ; So is sine 48-7 , to sine 27-36 . Then , As sine 27-36 , to sine 81-47 ; So sine 4-47 ½ , to ( V.S. of 130 , the Azimuth from the North : ) the sine of 10-15 , a 7th sine , whose half-distance toward 90 , is 25 , whose complement 65 doubled , is 130 , the Azimuth from the North , whose complement to 180 , viz. 50 , is the Azimuth from South . Having the same complements , to find the Hour , and Azimuth , by the General-Quadrant and Sector ; and first for the Azimuth . First , of the complements of the Latitude , and Suns present Altitude , by substraction find the difference . Secondly , Count this difference on the Line of Natural Sines from 90 , toward the Center , as the smaller figures are counted . Thirdly , Take the distance on the Sines , from thence to the — sine of the Suns declination . But note , That when the Latitude and Declination differ , viz. one North , and the other South , as it is with us in Winter ; you must count the Suns Declination beyond the Center , and call it the Suns distance from the Elevated Pole , and take from thence . Fourthly , Make this — distance , a = in the Co-sine of the Latitude , laying the Thred to ND , or keeping the Sector at that opening . Then , Fiftly , Take out the = sine of 90 , And Sixtly , Make it a = sine in the Suns Co-altitude , setting the Sector , or laying the Thred to the ( nearest distance ) ND . Seventhly , Take out the = sine of 90. And , Eightly , Measure it from the sine of 90 , towards ( and if need be beyond ) the Center , and it shall reach to the versed sine of the Suns Azimuth from North or South , when you count from 90 ; or from East or West , if you count from the Center , on a Line of Sines , or middle of the Line of versed Sines . Note , That if the general Sines are too big , you have a less adjoyning , whereon to begin and end the Work ; as sometime the Hour-Scale , and sometimes the Line of Right Ascentions . Example . In the Latitude of 51-32 , the Suns Declination 18-30 , the Suns Altitude 48-12 , you shall find the Suns Azimuth to be 130 from the North , or 50 from South . Secondly , for the Hour , by the same data , or things given . 1. First of the complement of the Latitude , and the Suns distance from the Elevated Pole , find the difference by Substraction . 2. Count it on the Line of Sines from 90 toward the Center , ( or beginning of the Sines ) . 3. Take the — distance from thence , to to the sine of the Suns present Altitude . 4. Make this — distance , a = in the Co-sine of the Latitude , setting the Sector , or laying the Thred to the ND , and there keep it . 5. Then take out the = sine of 90 ; And , 6. Make that a = in the Co-sine of the Suns Declination , laying the Thred to ND . 7. Then take out the = sine of 90 again . And , 8. Measure it from 90 , toward the Center , and it shall shew the versed sine of the Hour from Mid-night , or the contrary from noon ; or from 6 , if you count from the Center of the Sines , or the middle on versed Sines . Example . Latitude 51-32 , Declination North 20-14 , Altitude 50-55 , you shall find the Hour to be 150 from North , viz. 10 in the fore-noon , or 30 degrees short of South . Use XXXIV . Having the Latitude , Suns Altitude , and distance from the Elevated Pole , to find the Hour , by the Line of versed Sines , on the Sector . First , By Addition , find the sum of , and by Substraction , the difference between the complement of the Latitude , and the Suns distance from the Elevated Pole. Secondly , Count this sum and difference from the Center , or the versed Sines on the Sector , ( or the beginning of the Azimuth-Line , if you use that , or any other , which is not drawn from a Center ) and with Compasses take the — distance between them . Thirdly , Make this — distance , a = versed Sine of 180. Fourthly , Take the — distance between the versed sine of the sum , and the complement of the Suns Altitude , and carry parallelly till it stay in like versed Sines , which shall be the versed Sine of the Hour from the North Meridian , or mid-night . Or , If you take the — distance from the difference to the Co-altitude , and carry that = till it stay in like sines , it shall be the hour from noon ; counting the Center 12 at noon , the middle at 90 , the two sixes and 180 at the end , for 12 at night . Use XXXV . Having the Latitude , the Suns Altitude , and Distance from the Elevated Pole , to find his true Azimuth from South or North , by Natural versed Sines . First , Of the Co-altitude , and Co-latitude , find the sum and difference , by Addition and Substraction . Secondly , Count the sum and difference from the Center , and take the — distance between them with Compasses on the versed Sines . Thirdly , Make it a = versed sine of 180 , and so keep the Sector . Fourthly , Take the — distance , between the sum , and the Suns distance from the Pole , ( counting the Center the Elevated Pole , and 90 the Equinoctial ) and carry it = till it stay in like parts , which shall be the Azimuth from South . Or , If you take the — distance from the difference , to the Suns distance from the Pole , and carry it as before , it shall stay at the versed sine of the Azimuth , from the North part of the Horizon . These five general wayes of finding the Hour and Azimuth , are not all needful to be learned by every one , but to delight the ingenious , and to hold forth the usefulness of the Instrument , and to supply defects that at some times may happen by Excursions , and as a four-fold Testimony , to shew the harmony in several wayes of Operation ; the first particular way , and this last by versed Sines , being most easie and comprehensive of any other . Use XXXVI . To work the last without the Line of versed Sines . Note , That if for want of room , the versed Sines be set but on one Leg , then it is to be laid at the nearest distance instead of like parts , after the manner of using the Thred on the General Quadrant . Also , If you have it not at all , then the Azimuth-line for the particular Latitude ; and if that be too large , the little Line of Sines beyond the Center , will supply this defect very well thus ; First , Turn the Radius , or whole length of that Line of Sines , two times from the Center downwards , ( which in Sea-Instruments , will most conveniently stay at 30 on the large Line of Sines , or general Scale , as was hinted in the 28th Use , being just 4 times as much one as the other ) . For a Point representing 180 of versed Sines , to set the Compasses in , when you lay the Thred to ND , and to take any versed sine above 90 degrees ; this being premised , the Operation is thus : Example . Lat. 51-32 , ☉ Dist. from Pole 80 , ☉ Alt. 25 , to find the Hour ; The sum of Co-lat . 38-28 , and 80 , is 118-28 ; And the difference is 41-32 . Now in regard the sum is above 90 , count the Center 90 , 10 on the smaller Sines 100 , and 20 on the same Sines , 110 , and 28 deg . 28 min ; 118 deg . 28 min. turn this distance the other way from the Center downwards , and note that place , for the Point , representing the sum on the versed sines . Then , The — Extent between this sum , and the difference 41-32 , as the smaller figures reckon it , being taken between your Compasses , set one Point in 180 , the Point first found , and lay the Thred to ND , and there keep it ( or observe where it cuts ) , then taking the — distance between the versed sine of the difference , counted as the small figures are reckoned , and the sine of the Suns Altitude 25 , as the greater figures are reckoned from the Center toward the End ; and carrying this Extent parallelly along the greater Line of sines , till the other Point will but just touch the Thred at ND ; Then , I say , the measure from that Point to the Center , measured on the small sines , as versed sines , shall be the versed sine of the Hour required , viz. 62 from South , or 7 hours 52 minuts from mid-night . This Rule , or Use , is longer far in wording , than the Operation need be in working ; for if you shall approve of this way , the adding of two brass Center-pins will shew you the two Points most used very readily , and the Thred is sooner laid , than the Legs can be opened or shut , and the Instrument keeps its Trianguler form as it is in , during the time of Observation . Use XXXVII . Having the Latitude , Suns Declination , and Hour , to find his Altitude . This Problem being not of such use as the contrary , viz. having the Altitude , to 〈◊〉 the hour , it shall suffice to hint only two ●ayes , the most convenient . And , First by the Particular Quadrant . Lay the Thred to the Day , or Declination , then the ND from the Hour to the Thred , measured in the particular Scale of Altitudes , shall shew the Suns Altitude required . Secondly , by the versed Sines . 1. First , of the Co-latitude , and Suns distance from the Pole , find the sum and difference . 2. Take the — distance between them , and make it a = versed sine of 180 , by setting the Sector , or laying the Thred to ND . 3. Then take the = versed sine of the Hour , and lay it latterally from the sum , and it shall give the complement of the Altitude required . This work is the same , both by the Sector , or General Quadrant , as is shewed in Use ●he 36th , and is nothing else but a backward working ; but the Altitude at any Azimuth , is not so to be done . To do the same by the Natural-Sines . First , having the Latitude , and the Suns Declination , find the Suns Altitude , or Depression at 6 ; and note the Point , either below , or above , or in the Center , as is largely shewed in Use the 26th , where the Altitude is given , to find the hour in any Latitude . Then , Lay the Thred to the Hour , counted in the degrees either from 12 , or 6 ; Then , Take the ND from the Co-sine of the Suns Declination , and make it a = in the sine of 90 , laying the Thred to the ND ; then the ND from the sine complement of the Latitude to the Thred , shall reach from the noted Point , for the Suns Altitude or Depression at 6 , to the Suns Altitude required . Example . Latitude 51-32 , Declination 23-71 , a 8 or 4 , viz. 2 hours from 6 Southwards the Altitude will be found to be 36-42′ . 1. For the Altitude at 6 , at any time of the year , say ; As the sine of 90 ; to sine of the Latitude ; So is the sine of the Suns Declination , to the sine of the Suns Altitude at 6. 2. For the Suns Altitude , at any hour or quarter , in Aries or Libra , ( the Equinoctial ) . As sine 90 , to Co-sine Latitude ; So is the sine of the Suns distance from 6 in degrees , to sine of the Suns Altitude . 3. For the Suns Altitude at all other hours , or times of the year . As sine 90 , to Co-tangent Latitude ; So is sine of the Suns distance from 6 , to the Tangent of a 4th Ark , in the Tangents . Which 4th Ark being taken from the Suns distance from the Elevated Pole , then the residue is the 5th Ark ; but for hours before and after 6 , add the 4th Ark , and the Suns distance from the Pole together , to make a 5th Ark. Then say , As the Co-sine of the 4th Ark , to the sine of the Latitude ; So is the Co-sine of the residue ( or sum ) being the 5th Ark , to the sine of the Suns Altitude at that hour . Use XXXVIII . The Latitude , Suns Azimuth and Declination given , to find the Altitude , or height thereof . First , to find the Suns Altitude at all Azimuths in the Equinoctial . As sine 90 , to Co-tangent Latitude ; So is the Co-sine Azimuth from South , to the Tangent of the Suns Altitude in Aries . Or , As sine 90 , to the Co-sine of the Azimuth from South ; So is Co-tangent Lat. to the Tangent of the Suns Altitude , at that Azimuth in the Equinoctial , which you must gather into a Table for every single degree . Then , As the sine Lat. to the sine of the Suns Declination ; So is the Co-sine of the Suns Altitude in Equinoctial , to the sine of a 4th Ark. Then , When the Latitude and Declination are alike , as both North , or South ; then add the 4th Ark and the Altitude ( in the Equator ) together , and the sum is the Altitude required . But in Winter-time , when the Latitude and Declination is unlike , take the 4th Ark out of the reciprocal Altitude in the Equator , and the residue is the Suns Altitude required . Also , in all Azimuths from East and West Northwards , in Summer-time also , you must use Substraction also , and not Addition ; as the Rule before-going suggests . By the Particular Quadrant , work thus ; Take the Sun or Stars Declination from the particular Scale , and setting one Point in the Suns Azimuth , on the Azimuth Line , and with the other lay the Thred to the ND , the right way , and on the degrees the Thred cuts the Altitude required . By the General Quadrant . As the — Co-tangent Latitude , taken from the Moving-leg , or Loose-piece , to = sine of 90 , laying the Thred to ND ; So is the = Co-sine of the Suns Azimuth from South , to the — Tangent of the Suns Altitude in the Equator , at that reciprocal Azimuth . Which being remembred , or gathered into a Table together , then say ; As the — Co-sine of the Suns Altitude in the Equator , to the = sine of the Latitude , laying the Thred to the ND ; So is the = sine of the Suns Declination , to the — sine of the 4th Ark. Which 4th Ark is to be added , or substracted , as immediately before is directed , and the sum or residue , shall be the true Altitude required . Example . At 60 degrees of Azimuth from South , the Equinoctial Altitude will be found to be 21-40 , for London latitude of 51-32 ; and the 4th Ark in ♋ or ♑ is 28-16 . Then , 21-40 , the Suns Altitude at 60 in ♈ , and 28-16 , the reciprocal 4th Ark in ♋ added , makes 49-56 , the Suns Altitude at 60 degrees from the South in ♋ . The same way of working serves for the Sector , as is used for the General Quadrant , only observing to set the Sector , instead of laying the Thred to the nearest distance , as the Ingenious will soon perceive . Use XXXIX . Having the Latitude , Declination , Azimuth , and Altitude , to find the Hour . As the — Co-sine of the Suns Altitude , to = Co-sine of the Suns Declination ; So is the = sine of the Suns Azimuth , to — sine of the Hour . Or else thus ; First find the Altitude , at that Azimuth ; and then at that Altitude , and Declination , the Hour . As — Co-sine of Declination , to = sine of the Azimuth ; So is the — Co-sine Altitude , to = sine of the Hour . As Co-sine Declination , to the Sine of the Azimuth ; So is Co-sine Altitude , to Sine of the Hour . Use XL. Having the Latitude , Declination , Hour , and Altitude , to find the Azimuth . As — Co-sine of Declination , to = Co-sine of the Suns Altitude ; So is = sine of the Hour , to — sine of the Azimuth . First , find the Altitude at that Hour , and then the Azimuth at that Altitude , as before . As Co-sine of Altitude , to sine of the Hour ; So is Co-sine Declination , to sine of the Azimuth from South , or North , as the Hour is counted ; that is to say , from South , if the Hour is between 6 at morning , and 6 at night ; and from the North if the contrary ; that is to say , between 6 at night , and 6 next morning , or next to midnight . Use XLI . Having the Latitude , and the Suns Declination , to find the Suns Azimuth at 6. As the sine of 90 , to the Co-sine of the Latitude ; So is the Tangent of the Suns Declination , to Co-tangent of the Suns Azimuth from the North , at the hour of 6. First find the Suns height at 6 , and then the Suns Azimuth at that Altitude . Make the — Tangent of the Declination , a = sine of 90 , laying the Thred to ND , then the = Co-sine Latitude shall be the — Co-tangent of the Suns Azimuth from the North at 6. Use XLII . To find the Amplitude , Azimuth , Rising , Setting , and Southing of the fixed Stars , having the Latitude , Altitude , and Declination , or time of the year given . First for the Amplitude , Take the Stars Declination , out of the particular Scale of Altitudes , and measure it from 90 in the Azimuth-line ; and count the same way , and the other Point shall shew the Stars Amplitude required . Example . The Declination of the Bulls Eye , being 15-48 ; if you take 15-48 from the particular Scale , and lay it from 90 in the Azimuth-line , it will reach to 26 degrees , counting from 90 towards either end , the same as for the Sun in Use 16. But in other Latitudes , work as you do for the Sun by the Rules in the 16th Use abovesaid . For a Stars Azimuth . The work here is the same as for the Sun , thus ; Take the Stars Declination from the particular Scale of Altitudes , or Sines , between your Compasses , and lay the Thred to the Stars Altitude , counted from 600 toward the Loose-piece ; then carry the Compasses , or the right-side of the Thred , for Northern-stars ; and on the left-side for Southern-stars , along the Azimuth-line , till the other foot , being turned about , will but just touch the Thred ; then the fixed Point on the Azimuth-line shall shew the Stars Azimuth , from the South , required . Example . The Bulls Eye being 30 degrees high , shall have 77 degrees and 10 minuts of Azimuth from the South ▪ If you be in other Latitudes , use the general wayes , as for the Sun in all respects , having the same Declination that the Star hath North or South . To find the Stars Rising , or Setting . Count the Stars Declination on the degrees , as you count the Suns , North , or South , and there lay the Thred ; and in the Line of Hours is the Stars Rising , or Setting , when the Stars Right Ascention and Declination are equal . But at other times , you must reckon thus ; First , find the Suns Right Ascention , by Use 14 , and set down the complement thereof to 12 Hours , and the Stars Right-Ascention , and the hour of Rising the Thred cuts , and add them into one sum , and the sum , if under 12 , is the time of his Rising in common hours ; or if you add the hour of Setting that the Thred sheweth , it shall give his setting . Example . If you lay the Thred to 15-48 , the Declination of the Bulls Eye , in the Hour-line it cuts 4 hours 36 min. for Rising ; or 7-24 , for his Setting ; then if you work , for April the 23d , the Suns Right Ascention , then is 2-44 , and the complement thereof to 12 , is 9-16 ; and the Stars Right-Ascention is 4 hours and 16 minutes ; and the Hour cut , is 4-36 for Rising ; and the three Numbers , viz. 9-16 , the complement of the Suns Right Ascention , and 4-16 , the Stars Right Ascention , and 4-36 , the Hour of Rising the Thred cuts , being added , makes 18-8 ; from which , taking 12 , rest 6-8 , the time that the Bulls Eye Riseth on April 23 ; and if you add 7-24 , the time of Setting that the thred cuts , there comes forth 8-56 , viz. one hour and 32 min. after the Sun. To find the time of a Stars coming to South . Substract the Right Ascention of the Sun , from the Right Ascention of the Star , increased by 24 , when you cannot do without , and the remainder , if less than 12 , is the time between 12 at noon , and 12 at night ; but if the remainder be more than 12 , it is the time between mid-night , and mid-day , following . Example . The Lyons-Heart , whose Right Ascention is 9-50 , will come to the South on March 10 , at 9-48 , the Suns Right Ascention , being then only 2 minuts . By the Line of 24 hours ( say , or ) work thus ; Extend the Compasses from the Suns Right Ascention , to the Stars Right Ascention ; that distance laid the same way from 12 at the middle , or at the beginning , shall reach to the time of the Stars coming to South . To find the time of the Stars continuance above the Horizon . First , find what the Suns semi-diurnal-Ark is , having the same declination , and that doubled , is the whole time of continuance ; Or , if you shall add and substract it to , or from the time of the Stars coming to South , you shall find the time of Setting or Rising . Or else , By laying the Thred to the Stars Declination , it sheweth the Ascentional difference in this Latitude , which added in those Stars that have North declination , or substracted in Southern to 6 hours , gives the semi-diurnal Ark of the Star above the Horizon . Example . The Eye 's Ascentional-difference , is one hour and 24 minuts ; which added to 6 hours , because of Northern declination , makes 7-24 , for the semi-diurnal-Ark , or 14● 48′ , for the whole time of being above the Horizon . Note , That to work this for other Latitudes , the Suns Ascentional-difference is to be found for that Latitude you are in , and the Operation is general for all places . To find a Meridian Line by the Sun. On any flat Horizontal-Plain , set up a streight Wyre in the Center of a Circle ; or hold up a Thred or Plummet , till the shadow of the Thred cut the Center , and any where in the Circumference , which two Points you must note ; then immediately take the Suns Altitude , and find the Suns Azimuth , and count so many degrees in the Circle the right way , as the Suns Azimuth comes to , from the Points of the shadow marked in the Circumference , and draw that Line for a true Meridian-line . This Work is best done before 10 in the morning , and after two afternoon ; or in the night , by two Plumb-lines , set in a right-Line with the North-Star , at a right scituation . Use XLIII . To find the Hour of the Night by the Fixed Stars . First , find the Stars Altitude , by looking along the Fixed or Moveable-leg , to the middle of the Star , letting the Thred , with a weighty Plummet , play evenly by the degrees , between your Thumb and Fore-finger , to the end you may command the Thred , and know whether it playeth well or no by feeling . Then , Take the Altitude found , from the particular Scale of Sines , and laying the Thred over the Stars declination , which for readiness sake is marked with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , according to the Figures set to the 12 Names of the 12 Stars on the Rule ; and then carrying the Compasses as you do in finding the hour by the Sun , you shall find how much the Star wants , or is past the Meridian , which is called the Stars-Hour ; And note , That if the Star be past the South , it is an aft●rnoon hour ; if not come to the South a morning hour , which you must remember . Also , knowing the Suns Right Ascention , set one Point of the Compasses in the Suns Right Ascention , ( counted in the Line of twice 12 , or 24 hours , on the outward-leg of the fixed-piece , next to the particular Scale of Sines ) and open the other to the Stars Right Ascention , noting which way you turn the Compasses ; for the same Extent , applied the same way , from the Stars hour last found , shall shew the true hour of the night required . Example . Suppose on the 10th of Ianuary , I should observe the Altitude of the Bulls Eye to be 20 degrees ; if you take 20 degrees , the Altitude , from the particular Scale , and lay the Thred on 15-48 , the Stars declination Northward , and measure from the Hour-scale the nearest distance to the Thred , you shall find the Compass-point to stay at 6-49 on the East-side of the Meridian ; ( suppose ) Also , The Suns Right Ascention , the same day , is 8 hours and 12 minuts . Then , The Extent from 8 hours 12 minuts ( on the Line of twice 12 hours ) the Suns Right Ascention , to 4-16 , the Stars Right Ascention , shall reach the same way from 6-49 , the Stars hour , to 2-53 , the true hour . Use XLIV . To find the Hour of the Night by the Moon . First , by an Almanack , or Ephemerides , find the Moons Age , and true Place for the present time ; then , by laying the Thred on the Moons place , you may have her Right Ascention , and also the Suns Right Ascention ; and by the Moons Altitude , taken from the particular Scale , and the Thred laid over the Moons place , you find what the Moon wants , or is past coming to South , which is called the Moons hour . Then , by the Line of 24 Hours , say ; As the Suns Right Ascention , is to the Moons Right Ascention ; So is the Moons hour last found , to the true hour . Example . Suppose that on the 8th of Ianuary , about 40 min. after 3 , there is a New Moon ; then note , That the Suns true place , is the Moons true place ; and consequently , their Right Ascentions ; and the Moons Hour and Altitude is the same with the Suns . Therefore , As 8 hours 04 min. the Suns Right Ascention , is to 8-04 , the Moons Right Ascention ; So is the Moons hour at any Altitude , to the Suns true hour . Again , Suppose that on the 1st Quarter-day , the Moon being gone 90 degrees from the Sun , to find her place ; Then do thus ; Set one Point in the Moons place the Change-day , and open the other to the beginning or the end of the Line of 24 hours , Then , The same Extent applied the contrary way from 6 hours , or 7 dayes and a half , the Moons Age , shall give 28 deg . 58 min. ♈ ; to which you must add 7 degrees and 30 minuts ( the Suns place ) between , and the sum shall be the Moons true place required , viz. 6-28 degrees in ♉ . Example . If the Moon Change on the 8th day , the First Quarter being 7 dayes and a half after , will be on the 15th day later at night ; then the difference between the Sun and Moons Right Ascention , will be found to be near 6 hours ; for the Suns Right Ascention , Ianuary 15 , is 8-32 ; and the Moons Right Ascention , the same day , being about 8 degrees and a half in ♉ , is 2 hours and 28 minuts ; if you take the distance between them , on the 24 hours , it is near 6 hours ; which is the difference of time between the Moon and the Suns hour . Again , For the Full Moon ; on the 22 day , near 4 hours after noon , the Moons Age being 14 dayes ¾ ; if you add 12 hours , or 6 signs , to the Moons place a● the Change , you shall find ♋ 29-0 ; to which if you add 14-45 , the dayes between the New and Full , you shall find ♌ 13 deg . 45 min. for the Moons place ; the Suns Right Ascention the 22 day is 9 hours , and the Moons the same day at 1 afternoon , is 9 hours also ( or rather 12 difference ) so that the Suns hour and the Moons is equal ; only one is North , and the other South . Again , For the Last Quarter 22 ¼ dayes , or 18 hours added , and 22 degrees also together , makes ♏ 22 deg . 11 min. for the Moons place , by help of which , to find the Moons hour by her Altitude above the Horizon found by observation . Or , Without regarding the Sun or Moons Right Ascention , having her true Age , and Hour , Say thus ; As 12 on the Line of 24 hours , is to the Moons Age in the Line of her Age ; So is the Moons hour , to the true hour . For , The Extent from 12 in the middle , to the Moons Age under or over the middle , shall reach the same way , on the same Line , from the Moons hour , to the true hour . The like work serves to find the hour of the night by any Planets , as Saturn , Mars , or Iupiter , which are seen to shine very brave and bright in Winter evenings ; and having learned their Place by their distance from the fixed Stars , or by the Ephemerides , then their Altitude and Place will find their hour from the Meridian , and the comparing their Right Ascentions with the Suns , gives the true hour , as before , in the Fixed-Stars . Use XLV . To find the Moons Place and Declination , without the Ephemerides , somewhat near . First , observe when the Moon is in the Meridian , and then find her Altitude , and take the same from the particular Scale between your Compasses ; then set one Point in the hour 12 , and lay the Thred to ND , and on the degrees it shall shew the Moons declination ; and in the Line of the Suns Place , the Moons present Place , counting her Progress orderly from the last Change-day , or New Moon , when she was with the Sun. Otherwise thus ; Observe what Hour the Moon sheweth on any Sun-dial , at the same instance by the Fixed Stars , or other wayes , find the true Hour ; Then , The Extent from the Moons Hour , to the the true Hour , shall reach the same way from 12 , to the Moons Age , right against which is her coming to South , at which time you may find her true Altitude , and so come by her Declination . Yet again , for her Age and Place , according to Mr. Street , and Mr. Blundevil . Add the Epact , the Month , and Day of Month in one sum , counting the Months from March , by calling March the first Month , April the second , &c. then that sum , if under 30 , is the Moons Age ; but if the sum be above 30 , then substract 30 , and the remainder is the Moons Age , when the Month hath 31 dayes ; but if the Month hath but 30 , or less than 30 dayes , then substract but 29 , and the remainder is the Moons Age. Or thus ; Add to the Epact for the present year , and in Ianuary 0 , in February 2 , in March 1 , in April 2 , in May 3 , in Iune 4 , in Iuly 5 , in August 6 , in September 8 , in October 8 , in November 10 , in December 10 ; and the sum , if under 30 , or the excess above 30 , added to the day of the Month , abating 30 , if need be , gives the Moons Age that day ; but substracted from 30 , leaves the day of her Change in that Month , or from the 〈…〉 ●onth . Example . July 10. 1668. The Epact that year is 26 , and the Number for Iuly is 5 , the Excess above 30 , is 1 ; which added to any day of the Month as to 10 , gives 11 , for the Moons Age , Iuly 10. 1668. Then for the Moons Place . Multiply the Moons Age by 4 , and the Product divided by 10 , the Quotient giveth the signs ; and the remainder multiplied by 3 , gives the degrees , which you must add to the Suns place that day , to find out the Moons place for that day of her Age. Example . On Iuly 10. 1668 , the Moons Age is 11 , which multiplied by 4 , makes 44 ; and 44 divided by 10 , gives 4 signs in the Quotient ; and 4 , the remainder , multiplied by 3 , makes 12 degrees more ; which added to Cancer , 29 degrees , the Suns place on the 10th day of Iuly , makes 11 degrees in Sagittarius , the Moons place the same day , propè verum . Or rather by the Rule thus , on the Line of 24 hours by particular Scale , having the Moons place , to find her Age by the Line of 24 hours . The Extent from the Suns true place , to the Moons true place , shall reach the same way , from 0 day , to the day of her Age. Or contrarily , having the Moons true Age , to find her true Place . The Extent from 0 day old , to the Moons true Age , shall reach the same way from the Suns true Place to the Moons . Or , having the Moons true Place at the New Moon , to find her Place any day of her Age after . The Extent from ♈ , to the Moons true Place at the Change , shall reach the same way , from the day of her true Age , to her true Place , adding as many degrees to the Number found , as the Moon is dayes old . Then , Having her Place , and Age , it is easie to find the Moons Hour , and then her true Hour ; but I fear I spend herein too much time on an uncertain subject . Use XLVI . The Right Ascention and Declination of any Star , with the Suns Right Ascention , and the Hour of the Night given , to find the Altitude and Azimuth of that Star , and thereby to know the Star , if you knew it not before . Set one Point of the Compasses in the Stars Right Ascention , found in the Line of twice 12 hours ; and open the other to the Suns Right Ascention , found in the same Line ; then this Extent shall reach , in the same Line , from the true hour of the Night , to the Stars hour from the Meridian ; then laying the Thred to the Stars Declination , the ND from the Stars hour , in the Line of hours , to the Thred , measured on the particular Scale of Altitudes , gives the Stars Altitude ; then by his Declination and Altitude , you may soon find his Azimuth , by Use 27. And if the Instrument be neatly fixed to a Foot , to set North and South , and turn to any Azimuth and Altitude , you may find any Star , at any time convenient and visible . Use XLVII . The Altitude and Azimuth of any Star being given , to find his Declination . Lay the Thred to the Altitude on the degrees , counted from 600 toward the end , then setting one Point on the Stars Azimuth , counted in the Azimuth Line , and take the ND from thence to the Thred ; which distance measured from the beginning of the particular Scale of Altitudes , shall give the Declination . If the Compasses stand on the right-side of the Thred , then the Declination is North ; if on the left , it is South ; according as you work for the Suns Azimuth in a particular Latitude . Use XLVIII . The Altitude and Declination of any Star , with the Right Ascention of the Sun , and the true Hour of the Night given , to find the Right Ascention of that Star. First , by the 43d Use , find the Stars Hour , viz. How many hours and minuts it wants of coming to , or is past the Meridian ; then the Extent of the Compasses ( on the Line of 24 hours on the Head-leg ) from the Stars hour to the true hour , shall reach the same way from the Suns Right Ascention , to the Stars Right Ascention , on the Line of twice 12 , or 24 hours . Use XLIX . To find when any Fixed-Star cometh to South , by the Line of twice 12 , or 24 hours . In Use 42 , Section 4 , you have the way by Substraction , with its Cautions : But by the Line of twice 12 , or 24 hours , work thus ; Count the Suns Right Ascention on that Line , and take the distance from thence to the next 12 backward , viz. that at ♈ , at the beginning of the Line , when the Suns Right Ascention is under 12 hours ; or , to the next 12 in the middle of the Rule at ♎ , when the Suns Right Ascention is above 12 hours , ( which is nothing but a rejecting 12 for more conveniency ) . Then , The same Extent laid the same way from the Stars Right Ascention , shall reach to the Stars coming to South . Or , The Extent from the Sun , to the Stars Right Ascention , shall reach the same way from 12 , to the Stars coming to South . Example , for the Lyons-Heart , August 20. The Suns Right Ascention the 20th of August , is 10 hours 36 minuts ; the Right Ascention of the Lions-Heart , is 9 hours and 50 min. Therefore , The Extent from 10 hours 35 min. to the beginning , shall reach the same way from 9 hours 50 min. ( by borrowing 12 hours ) because the Suns Right Ascention is more than the Stars ) to 11 hours 13 min. of the next day , viz. at a quarter past 11 ; or , at 11 hours and 13 min. the same day ; where you may observe , that the remainder being above 12 , if you add 24 hours , the time of Southing is between mid-night , and mid-day next following . Use L. To find what two dayes in the year are of equal length , and the Suns Rising and Setting . Lay the Thred on any one day in the upper Line of Months and Dayes , and at the same time the Thred cuts in the lower-Line of Months the day that is answerable to it in length , rising , setting , and declination , and other requisites . Example . The 1st of April , and the 21 of August , are dayes of equal length ; and the Suns Rising and Setting is the same on both those dayes ; only in the upper-Line , the dayes are increasing in length , and in the lower-Line they are decreasing . Use LI. To find how many degrees the Sun is under the Horizon at any Hour , the Declination and Hour being given . Count the Suns Declination on the degrees , the contrary way , viz. for North Declination , count from 600 toward the end ; and count for Southern Declination toward the Head , and thereunto lay the Thred ; then take the nearest distance from the hour given to the Thred ; this distance measured in the particular Scale of Altitudes , shall shew the Suns Depression under the Horizon at that hour . Example . January the 10th at 8 at Night , how many degrees is the Sun under the Horizon . On that Day and Hour , the Suns Declination is about 20 degrees South ; then if I lay the Thred to 20 degrees of Declination North , and take the nearest distance from 8 to the Thred , that distance , I say , measured in the particular Scale , gives 34 degrees and 9 min. for the Suns Depression under the Horizon of 8 afternoon . To do this in other Latitudes , you are to find the Suns Altitude at 8 in Northern Declination , by Use 37. CHAP. XVII . The use of the Trianguler Quadrant , in finding of Heights and Distances , accessable or inaccessable . Use I. To find an Altitude at one Station . FIrst , The Trianguler Quadrant being rectified , and fixed to a Ball and Socket and three-legged-staff , being necessary in these Operations to perform them exactly , especially for Distances ; look up to the object as you would to a Star ; and observe what degree and minut the Thred cuts , and set it down : Also , observe the place where you stand at the time of Observation , and the distance from your Eye to the ground , and the place on the object that is level with your eye also ; as the playing of the Thred and Plummet will plainly shew . Also , you must have the measure from the place where you stood observing , to the Point exactly right under the object , whose height you would have in Feet , Yards , Perch , or what you please , to Integers , and Fractions in Decimals , if it may be . Also Note , That in all Right-Angle-Triangles , one Acute Angle is alwayes the complement of the other ; so that observing or finding one by Observation , by consequence you have the other , by taking that from 90. These things being premised , the Operation followes , by the Artificial Numbers , Sines and Tangents , and also by the Natural . Note also by the way , That in regard the complement of the Angle observed is frequently used , if you count the degrees the contrary way , that is to say from the Head , you shall have the complement required ; as hath been oftentimes hinted before . Then , As the sine of the Angle , opposite to the measured side , is to the measured side , counted on the Numbers ; So is the sine of the Angle found , to the Altitude or Height required on Numbers . Example at one station . Standing at C , I look up to B the object , whose Height is required , and I find the Thred to fall on 41 degrees and 45 minuts ; but if you count from the Head , it is 48-15 , the complement thereof , as in the Figure you see . Also , the measure from C to A , is found to be 218 foot . Then , As the sine of 48-15 , the Angle at B , being the complement of the Angle at C , is to 218 on the Line of Numbers ; So is the sine of the Angle at C , 41-45 , to 195 the Altitude of AB the height required , found on the Line of Numbers . A second Example standing at D. But if I were standing at D , 129 foot and a half from A , and would find the height AB , the complement of the Angle at D , that is to say , the Angle at B is 33-30 . This being prepared , then say ; As the sine of 33-30 , the Angle at B , to the measured-side DA , 129 ½ counted on the Numbers ; So is 56-30 , the sine of the Angle at D , to 195 , the Altitude required , AB , and 5 foot more , the usual height of the eye from the Level to the ground , makes 200 , the whole height required . To work this by the Trianguler Quadrant , say thus ; As — 129 ½ , taken from any Scale , is to the = sine of 33 deg . 30 min. laying the Thred to the nearest distance ; So is the = sine of 56-30 , the Angle at D , to the — measure of 195 on the Scale you took 129 ½ from . The like manner of work is by the Sector , as thus , in the foregoing Example . As 218 , taken from the Line of Lines , to the = sine of 48 deg . 15 min. So is the = sine of 41-45 , to 195 on the Line of Lines latterally . And yet further , So is the = sine of 90 , to 291 , the Line CB. Use II. To find an Altitude at two stations . But if you cannot come to measure to the foot of the object , then you must observe at two places . As thus for Example . First , as before , find the Angle at D , or rather the complement thereof , viz. 33-30 ; then go further backward in a right Line with the object and first station , any competent Number of feet , as suppose 88 ½ to C ; there also observe the Altitude or Complement , viz. the Angle ABC , 48-15 . Then , Find the difference between 48-15 , and 33-30 , and it is 14-45 . Then , As the sine of the difference last found , viz. the Angle CBD , 14-45 , to 88 ½ , on the Line of Numbers ; So is the sine of the Angle at C , 41-45 , to the measure of the side DB , 233 , on the Line of Numbers . Again , for the second Operation . As the sine of 90 , the Angle at A , to the Hypothenusa DB , 233 ; So is the sine of 56-30 , the Angle at D , to 195 , the Altitude required . The same by the Trianguler Quadrant , or Sector . As — 88 ½ , the measured distance CD , to the = sine of 14-45 , CBD ; So is = sine of 41-45 , to the — measure of 233 , the opposite-side DB. Again , As — 233 , taken from the Line of Lines , to = sine of 90 ; So is the = sine of 56-30 , the Angle at D , to — 195 , on the Line of Lines , the height required . Use III. Another way to save one Operation from IC . First , observe the complement of the Angle at D , and also the complement of the Angle at C ; then count these two complements on the Line of Natural Tangents , on the loose-piece , or moving-leg , and take the distance between them , and measure it on the same Tangent-line from the beginning thereof , and note what Tangent the Compass-point stayeth at , and count that for the first term , in degrees and minuts . Then , As the Tangent of this first term , to the measured distance CD , 88 ½ , on the Line of Numbers ; So is the Tangent of 45 , to the Altitude required . Thus in our Example ; The distance measured is 88 ½ , the two complements 33-30 , and 48-15 ; the distance between them makes the Tangent of 24-34 , to be used as a first term . Then , As the Tangent of ●4-34 , the first term last found , to 88 ½ on the Numbers ; So is the Tangent of 45 , to 195 , ferè , on the Numbers , the height required . But if the distance from D or C , to A , the foot of the Object , were required , then the manner of Calculation runs thus ; As the Tangent of the difference of the Co-tangents first found , 24-34 , is to the distance between D and C 88 ½ ; So is the Co-tangent of the greater Ark 48-15 , to the greater distance CA 218. Or , So is the Co-tangent of the lesser Ark 33-30 , to the lesser distance DA , 129 ½ . But if the Hypothenusaes be required , then reason thus ; As the Tangent of the difference first found is 24-34 , to the distance between the stations D and C , 88 ½ ; So is the Secant of the Angle at B the greater , viz. 48-15 , counted beyond 90 , to CB 291. Or , So is the Secant of 33-30 , the lesser Angle at B , to 233 the lesser distance DB , the Hypothenusa required . To work these two last by the Trianguler Quadrant . First , prick off the Tangents and Secants to be used parallelly , from the loose-piece , on the greater general Scale ; and note those Points for your present use . As thus ; The Tangent of 24-34 , taken from the loose-piece from 60 , counted as 00 will reach to the sine of 10-40 , on the general Scale . Secondly , The Secant of 33-30 , being the measure from the Tangent of 33-30 , on the loose-piece ( counting from 60 ) to the Center , will reach on the general Scale from the Center , to 28-50 . Thirdly , The measure from the Tangent of 48-15 , on the loose-piece , to the Center , being the Secant of 48-15 , will reach from the Center to 32-5 , on the general Scale . This being prepared , the work is thus ; As — distance between the two stations , to = Tangent , of the first term , at 10-40 ; So is = Tangent of 45 , to the Altitude required . Again , for the Distance . As — distance between the two stations , to the = Tangent of the first term ; So is the = Tangent of the greater Angles complement , at 26-36 , to the greatest distance CA 218. Or , So is the = Tangent of the lesser Angles complement , at 15-25 , to the lesser distance DA , 129 ½ ; Or , So is the = Secant of the greater Angles complement , at 32-5● to the greater Hypothenusa CB , 291. Or , So is the = Secant of the lesser Angles complement , at 28-50 , to the lesser Hypothenusa DB , 233. Use IV. Another way for Altitudes , by the Line of Shadows , either accessable or unaccessable , by one or two stations . If this way be desired , it may be put on this , as well as any other Quadrants . Then the use is thus ; Figure II. Suppose that AB be the height of a Tree , or other Object to be found ; go so far back from it , as suppose to C , till looking up by the two Pins put for sights , the Thred falls on 45 degrees on the Quadrant , or on 1 on the Line of Shadows ; then , I say , that the height AB , is equal to the distance CA , more by the height of your eye from the ground . But if you go further back still to D , till the Thred falls on 2 on the Line of Shadows ; that is to say , at 26 deg . 34 min. the Altitude will be but half the distance from A ; but if you remove to E , the Thred falling on 3 on the Shadows , the Altitude will be but one third part of the distance EA . From hence you may observe , that observing at C , and at D , where the Thred falls on 1 , and on 2 , the distance between C and D , is equal to the Altitude ; so likewise at D and at E , and so by consequence at 1 ½ and 2 ½ and 3 ½ , or any other equal parts . This is an excellent easie way . The like will be if you observe at D and C , looking up to F , where the Altitude AF is twice the distance AC . Use V. Another way , by the Line of Shadows , at one station . Measure any distance , as feet , yards , or the like from any object ; as suppose from A to D were 200 foot , and looking up to B , the Thred cuts the stroke by 2 on the Line of Shadows . Then by the Line of Numbers , say ; As 2 , the parts cut , is to 1 ; So is 200 , the distance measured , to 100 the height . Or , Suppose I measured any other uneven Number , and the Thred fall between 00 on the Loose-piece , and 1 on the Shadows , commonly called contrary Shadow . The Rule is alwayes thus ; As the parts cut by the Thred , are to 1 ; So is the measured distance , to the height required , being less than the measured distance . But when the Thred falls between 1 and 90 at the Head , called right Shadow ; then the Rule goes thus ; As 1 , to the parts cut by the Thred ; So is the measured distance , to the height , being alwayes more than the measured distance from the foot of the object , to the station . Use VI. Another way by the Line of Shadows , and the Sun shining . When the Sun shineth , find his Altitude , and also as the Thred lies , see what division on the Line of Shadows is cut by the Thred , and then straightway measure the shadows length on the ground ; and if the Sun be under 45 degrees high , the shadow is longer than the length of that object which causeth the shadow ; but if the Sun be above 45 degrees high , then the object is longer than the shadow ; and the Operation is thus by the Line of Numbers , only with a pair of Compasses . The Height of the Sun being under 45 , say ; As the parts cut by the Thred on the Shadows , is to 1 ; So is the Shadow measured , to the height required . The Height of the Sun being above 45 , say ; As 1 , to the parts cut by the Thred on the Line of Shadows ; So is the measure of the shadow , to the height in the same parts . Use VII . To find an inaccessable Altitude , by the Quadrat and Shadows , otherwise . Observe the Altitude at both stations , and count the observed Altitudes at both stations , on the Quadrat or Shadows , according as it happens to be either above or under 45 degrees ; and take the lesser out of the greater , noting the remainder for the first term ; and the Divisor to divide the distance between the stations , increased with Cyphers , if need be ; and the Quotient is the Answer required . But by the Line of Numbers , work thus ; The Extent from the difference to 1 , shall reach the same way from the measured distance , to the height required . Example . Figure II. Let ABCDE represent the Object and three Stations ; let the Line AC represent the Altitude ; the Point B one station , 50 foot from A ; D another station , 100 foot from A , or 50 from B ; and E another station , 73 foot from D , or 173 foot from A ; all which measures you need not know before , but only BD , and DE ; Also , the Angle at B , 63-27 , and his complement , counting the other way , being the Angle at C 26 degrees 33 minuts ; the Angle at D 45 , and his complement so also ; the Angle at E 30 , and his complement 60. Now mind the Operation by either of these , First lay the Thred on 26-33 , and in the Quadrat it cuts 50 ; lay the Thred on 45 , and in the Shadows , or Quadrat , it cuts 100 , or 1 ; or , if you lay the Thred to 60 , then in the shadows it cuts 173. The difference between 173 , and 100 , is 73. Then , As 73 , the difference in Tangents between the two observations , is to the distance in feet 73 ; So is Radius 100 , or the side of the Quadrat , to 100 , the hight required . Again , for the two nearest Observations , whose difference of Tangents , is 50. As 50 , the difference in Tangents , to 50 foot the measured distance ▪ So is 100 , the side of the Quadrat , to 100 the height . Again , lastly by the observations at B & E , the difference of Tangents being 123. As 123 , the difference in Tangents , to 123 , the measured distance ; So is 100 , the Radius or side of the Quadrat , to 100 , the height required . Or , In the first Figure , the Angles at the top being 33-30 , and 48-15 ; and the measured distance 88 foot and a half , the difference in Tangents will be 45-8 . Then , As 45-8 , to 100 , the side of the Quadrat ; So is 88 ½ , the measured distance to 194 , the Altitude required . This way is general for any Station , though both of right shadow , or both of contrary , or mixt of right and contrary , and done by the Line of Numbers , or by Multiplication and Division . Also Note , That you may find this difference in Tangents or Secants , by the Natural Tangents , or Natural Secants on the Sector , and the Scale of equal parts belonging to them . Thus ; Take the distance between the compleplement of the two observations , on the greater or lesser Line of Tangents , ( as is most convenient ) and measure this distance in the Line of Lines , or equal parts equal to that Radius ; and that shall be the difference in Tangents required . The like for the Secants . Also , By the Artificial Numbers , Sines , and Tangents , you may come by this differences in Tangents , or Secants , very well thus ; Just right against the Tangent of the Co-altitude , counted on the Line of Tangents , in the Line of Numbers , is one Number ; and against the Tangent of the complement of the other Angle , is the other Number ; only with this Caution , That if the Tangent be above 45 , then take the distance from 45 to the Tangent , as it is counted backward , with Compasses , and set the same the increasing way from 1 , on the Numbers , to the other Number required ; then the lesser taken from the greater , leaves the difference in Tangents that was required . In the same manner , the Sines counted from 90 , and laid the contrary way from 1 increasing , will give the difference in Secants , to measure the Ba●● , and Hypothenusa by Numbers only . Use VIII . Another pretty way by Scale and Compass , without Arithmetick , from T. S. Then draw the Line , that the Thred maketh , on the Board ; Then measure from your standing , to the foot of the Object , and take the number of feet , or yards from any Scale , and lay it from the right Angle on the other Line , and raise a Perpendiculer from thence to the Plumb-line made by the Thred , and that shall be the Altitude required , being measured on the same Scale . Example . Let ABGD represent the Boards end , or Trencher , and on that , let AB be one streight Line , and AG another Perpendiculer to it ; in the Point A , knock in one Pin ; and in B , or any where toward the end , another ; On the Pin at A , hang a Thred and Plummet ; and standing at I , any convenient station , look up by the two Pins at B and A , till they bourn in a right Line with the Point H , the object whose height is to be measured ; then the Plummet playing well and even , make a Point just therein , and draw the Line AD , as the Thred shewed . Then , having measured the distance from G the foot of the Object , to I the station , take it from any first Scale , and lay it from A to G ; then on the Point G , raise a Perpendiculer to AG , till it intersect the Plumb-line AD ; then , I say , the distance CD , measured on the same Scale you took AC from , shall be equal to the Altitude GH , which was required . Use IX . The same work at two stations . But if you cannot come to measure from I , the first station to G ; then measure from I to K ; and having observed at I , and drawn the Plumb-line AD , take the measure between I and K , the two stations , from any fit Scale of equal parts , and lay it on the Line AC , from A to C , viz. 79 parts and in the Point C , knock another Pin , and hang the Thred and Plummet thereon , and observe carefully where this last Plumb-line doth cross the other , as suppose at E ; then from E , let fall a Perpendiculer to the Line AC , which Line AC shall be the height GH required ; ( or thus , the nearest distance from E to AC is the height required ) viz. 120 of the same parts that IK is 79 ; Note the Figure , and behold that ACFE , the small Figure on the Board is like and proportional to AA , GH , the greater Figure . Other wayes there be , as by a Bowl of Water , or a Glass , or a Plash of Water , or a Square ; but these set down , are as convenient and ready as any whatsoever ; As in the next Figure you may see the way by the Glass , and Square . As thus ; Let C represent a Glass , a Bowl , or Plash of Water , wherein the Eye , at A , sees the picture or reflection of the Object E. Then , by the Line of Numbers ; As CB , the measure from your foot to the Glass , is to AB , the height from your eye , to the ground at your foot ; So is the measure from C to D , to the height DE. See Figure VI. Again , to find a distance by the Square , that is not over-long . Let C represent the upper-corner of a Square , hung on a staff at F ; then the one part of the Square directed to E , and the other to A. The Proportion will hold , by the Line of Numbers . As FA 11-37 , to FC 50 ; So is FC 50 , to FE 220. That is , So many times as you find AF in FC ; So many times is FC in FE , and the like . Note , That you must conceive AFE to be the Ground , or Base-line in this Operation by the Square ; C being the top of an upright Staff , 5 foot long , called 50 for Fraction sake . Use X. To find a Distance not approachable by the Trianguler Quadrant . First , I plant my Trianguler Quadrant , set upon a three legged Staff and Ball socket , right over the place A ; and then bring the Index with two sights in it , laid or fastened to the Center of the Trianguler Quadrant , right over the Lines of Sines , and Lines cutting 90 at the Head ; the Index and sights so placed , hold it there , and bring it and the Instrument together , till you see the mark at C , through the two sights , by help of the Ball-socket , and then there keep it ; then remove the Index only to 0-60 on the loose-piece , which makes a right Angle ; and set up a mark in that Line , at any convenient distance ; as suppose at B , 102 foot from A ; then remove the Instrument to B , and laying the Index on the Center , and 0-60 on the loose-piece , direct the sights to A , the first station , by help of a mark left there on purpose ; Then remove the sights till you see the mark at C , and note exactly on what degree the Index falleth , as here on 60 , counting from 060 on the loose-piece ; or on 30 , counting from the Head , which is the Angles at B , and at C. Then by the Artificial Numbers , Sines and Tangents on the edge , say ; As the sine of 30 , the Angle at C , to 102 , the measured distance counted on the Numbers ; So is the sine of 60 , the Angle at B , to 117 , on the Numbers , the distance required . So also is 90 , the Angle at A , to 206 , the distance from B to C. Or , by the Lines and Sines on the Quadrant-side , as it lies , thus ; As the — measure of 102 , taken from any Scale , as the Line of Lines doubling , to the = sine of 30 , laying the Index , or a Thred , to the nearest distance ; So is the = sine of 60 , to 117 , measured latterally on the same Line of Lines . And , So is the = sine of 90 , to 206 , the distance from B to C. So also , If you observe at B , and at D only , you must be sure to set your Instrument at one station , at the same scituation ▪ as at the other , as a looking back from station to station will do it , and the same way of work will serve . For , As the Sine of 20 , to 110 ; So is the Sine of 40 , to 206. And , So is the Sine of 120 , to the Line DC 278 , &c. Use XI . To find a Breadth and a Distance at any two Stations . Let AB be two marks , as two corners of a House or Wall , and let the breadth between them be demanded , and their distance from C and D , the two stations ; First , set up two marks at the two stations , then setting up the Instrument at C , set the fiducial Line on the Rule to D , the other mark ; then direct the sights exactly to B , and to A ; observe the Angles DCB 45 , and DCA 113-0 , as in the Figure . Secondly , Remove the Instrument to D , the other station , and set the fiducial-Line of the Quadrant ( viz. the Line of Lines and Sines ) directly to C ; then fix it there , and remove the Index and sights to A , and to B , to get the Angles CDA 42-30 , and CDB 109-0 ; Then observe , that the 3 Angles , of every Triangle , being equal to 180 degrees ; having got the Angles at C 113 , and the Angle at D 42-30 , by consequence , as you take 155 , the sum of the Angles at C and D , out of 180 , then there remains 24-30 , the Angle at A. So also , Taking 109 and 45 from 180 , rests 26 , the Angle at B ; then also , taking 45 , the Angle BCD , out of 113 , the Angle DCA , rests 68 degrees ; the Angle BCA , in like manner , taking 42-30 from 109 , the Angles at D , rests 66-30 , the Angle ADB ; and let the distance measured , between the two stations , be 100 , viz. CD . These things thus prepared by the Artificial Numbers , Sines and Tangents on the edge , Say , As the Sine of 24-30 , the Angle at A , to 100 , on the Numbers , the measured side CD ; So is the Sine of the Angle at D 42-30 , to 164 , on the Numbers , the side CA. So is the Sine of 113 , the Angle ACD , to 222 , on the Numbers , the distance from C to B. Also , for the other Triangle , at the other Station D. As the Sine of 26 , the Angle CBD , to 100 , on the Numbers , the measured distance CD ; So is the Sine of 45 , to 161 , on the Line of Numbers , the distance from D to B ; So is the Sine of 109 , the Angle CDB , to 216 , on the Numbers , the distance from D to A. Then , having the Sides DB 161 , and AD 222 , and ADB the Angle included 66-30 , to find the Angles DAB , or ABD , use this Proportion . As the sum of the two sides given , is to the difference between the two sides ; So is the Tangent of half the sum of the two Angles sought , to the Tangent of half their difference . Example . 222 , and 161 , make 383 for a sum ; and 161 , taken from 222 , rest 61 for a difference . Again , 66-30 , taken from 180 , rest 113-30 , for a sum of the Angles sought , whose half 56-45 , is the third Number in the proportion . As 383 , the sum of the two known sides , is to 061 , the difference on the Numbers ; So is the Tangent of 56-45 , the half - sum of the two Angles sought , to the Tangent of half the difference 13-40 ; which half - difference , 13-40 , added to 56-45 , makes 70-25 , the greater Angle required at B , viz. ABD . Then also , If you take 13-40 , from 56-45 , the half - sum of the Angles inquired , rest 43-05 , the Angle BAB ; the like may you do with the other Triangle ABC , being needless in our Proposition . Thus having found the Angles , and one side , the Sines of the Angles , as proportional to their opposite sides . As the Sine of 44-33 , the Angle ABC , is to the side AC 146 , on the Numbers ; So is the Sine of 68 , the Angle a● C , to 217 , the distance between the marks required . Or , As Sine 43-05 , the Angle at A , to 161 ; So is Sine of 66-30 , the Angle at D , to 217 , the distance between the marks required . Also note , That if this manner of Calculation be tedious or difficult , then on a Slate , or sheet of Paper , you may do it by protraction , by the Line of Lines and Chords , or half Sines , very near the matter with care ; Thus : Draw CD the Station-Line , or measured-distance ; and make AD 100 , from any fit Scale . Then , on C and D draw a Circle , and on that Circle lay off from C and D the Angles , found by observation , and draw those Lines , and where they cross one another , as at A and B , draw the Line AB : those Lines and Angles measured on the same Scales and Chords , shall be the Sides , breadth , and distances required ; as you see in the Figure . Use XII . Another way for a long Distance . Let C be your standing place to set your Instrument , and let E be the mark afar off , whose distance from you C would know : first , move in a right Line between C and E to A , any number of Yards or Perches , as suppose 50 Perch , and set a mark at A ; Then move in a Perpendiculer-Line to CE , from A to B any distance , and there set up a mark at B , as suppose 66 Perches from A. Then come back again to C , and remove in a Perpendiculer-Line to CE , till you see the mark set up at B , and the enquired point at the distance E in a Right-Line ; and note that place at D , getting the exact distance thereof from C , suppose 76. Then substract the measured distance AB from the measured distance CD , and note the difference 10. Then , by the Line of Numbers , or by the Rule of Three , say , As the Difference between AB and BC , 10 , is to the distance between A & C 50 : So is the measured distance CD 76 , to the distance CE 380. Or , So is the measured distance AB 66 , to the distance AE 330 , the distance required . Note , That you must be careful and exact in measuring the Distances AC , AB , & CD , and the Answer will be the more exact accordingly . Use XIII . To find an Altitude of a House or Tower , by knowing part of it . If you divide the inside-edge of the Loose-peice into inches , or any equal parts , such as the nearest distance from the Rectifying-Point to that inside-edge may be 1000 , and for this use two small sliding sights may be convenient : Then the use is thus for any Angle under 30 degrees ; Fix the Instrument to the Ball-socket and Staff , and turn it toward the Object , causing the Plummet to play on 30 degrees ; for then the Loose-piece is perpendiculer . Then one pin or sight set in the Rectifying-Point , slip on a sight along the inner-edge of the Loose-piece , till you see the Object at the upper part of the Altitude , and another sight at the lower part of the Altitude known ; and observe the precise distance in parts between the two sights , on the Loose-piece ; Or , the several parts cut by the Index at each Observation : Then , As the distance between the two sights , is to the distance between the remotest sight from the middle of the Loose-piece ; So is the height of the known part , to the whole height required above the level of the eye . Example . Let CI represent the Altitude of a Pyramid on the Tower of a Steeple 30 foot high , and , standing at B , I would know the height of IA from the level of the eye upward . Fix the Trianguler Quadrant on the Staff and Ball-socket , with the Head-Center at B , with the Plummet playing on 30 degrees , and the Loose-piece perpendiculer : Then slip two sights on the Loose-piece , one in a Right-Line to C , the other to I ; and note the parts between , and the parts the furthest sight cuts , from the middle stroak on the Loose-piece , from whence the parts are numbred , which in our Example let be 500 , the sight of H , and the sight at G to cut 359 ; then the distance between the sights will be 143 , and the remotest from the middle of the Loose-piece to be 500 ; and the known Altitude , being part of the whole , to be 30 foot . Then , by the Line of Numbers , say , As 143 , the distance betwixt the sights at G & H , to 500 the remotest sight from the level or middle , viz. FH : So is 30 foot , part of the Altitude known , CI , to 105 , the whole Altitude unknown , AC . Or , So is 75 , the height of the lower part , to 105 the whole height AC . Or , As 143 , the distance between the sights , to IC the part of the height known 30 : So is 357 , the parts cut between F and G , to 75 the height AI unknown , &c. Use XIV . Having the Height , to find a Distance . Let CA be the Altitude given , and AB the distance required . Then I standing at C , observe the Angle CAB , by setting the end of the Head-leg to my eye , and the Head-end downwards , and set down , as the Thread cuts , numbring both wayes , for the Angle at C and at B his complement . Then say , As the Angle at B , 30 deg . 40 minutes , counted on the Sines , to 105 the height of the Tower : So is 59 deg . 20 min. the Angle at C on the Sines , to 176 the distance required on the Numbers . Also note by the way , That if you take an Altitude at two stations , as suppose at E and at B ; if the Angle observed at B , be found to be the half of the Angle at E ; as here in Figure VIII , the Angle at E , being 61-20 , and the Angle at B 30-40 , the just half thereof ; then , I say , that the distance between the two stations , is equal to the Hypothenusa EC , at the first station , viz. EB is equal to EC ; which being observed , say ; As the sine of 90 , to 120 , on the Numbers . So is 61-20 on the Sines , to 105 , the height required on the Numbers . A further proof hereof , take in this following Figure IX . Let AB be a breadth of a Wall , or Fort , not to be approached unto ; then by the degrees on the in-side of the loose-piece , to find that breadth one way , is thus ; Put two Pins into the two holes in the Head and Moving-leg , ( or set the sights there in large Instruments ) ; then move nearer or further from the objects , till your eye , fixed at the rectifying Point , can but just see the marks A and B by the two Pins in each Leg , which will only be at the mark C , at an Angle of 60 degrees ; for so the Rule is made to that Angle : then the Instrument being still fixed at C , look backward in a right Line from the middle of the loose-piece , and rectifying Point toward D , putting up a mark either in , or over , or beyond the Point D ; and also be sure to leave a mark at C , the first place of observation ▪ Then remove the sights to 15 degrees , the half of 30 , counting from the middle , and go back in a right Line from C , toward D , till you can just see the marks by the two sights set at 15 degrees each way ; for then , I say , that the measure between the two stations , C and D , shall be exactly equal both to AB , the breadth required , and also to CB , or CA , the Hypothenusaes ; then , having the sides CB , and CD , and the Angles BCE , and CBE , and BDC , it is easie to find all the other Sides and Angles , by the Rules before rehearsed , by the Lines of Artificial Nmmbers and Sines . For , As the Sine of 15 degrees , the Angle at D , viz. BDC , to 108 on the Numbers ; So also is the Angle at B , viz. DBC 15 , to 108 on the Sines and Numbers . So also is the Sine of 150 , the Angle at C , viz. DCB , to BD 208 ½ on the Numbers . Note also , That if the Angles of 60 and 30 be inconvenient , then you may make use of 52 and 26 , or 48 and 24 , or 40 and 20 , or any other , and the half thereof ; and then the measured distance , and the Hypothenusa BC , at the nearest station , will alwayes be equal ; but not equal to the breadth at any other Angle , except 30 and 60 , as in the Figure . But having the Angles , and those Sides , you may soon find all the others by the Artificial Numbers , Sines and Tangents , by the former directions . The End of the First Part. The Table or Index of the things contained in this Book . TRianguler Quadrant , why so called , Page 2 The Lines on the ou●ter-Edg , N. T. S. VS , Page 2 The Line on the inner-edge , I. F. 112 , Page 3 The Lines on the Sector-side , L.S.T. Sec. Page 3 Lesser Sines , Tangents , and Secants , Page 5 The Lines on the Quadrant-side , Page 6 The 180 degrees of a semi-Circle variously accounted , as use and occasion requires , Page 7 60 Degrees on the Loose-piece , as a fore-Staff for Sea-Observation , Page 7 The Line of right Ascentions , Page 8 The Line of the Suns true Place , ibid. The Months and Dayes , ibid. The Hour and Azimuth-line for a Particuler Latitude , Page 9 Natural versed Sines , ibid. Lines and Sines , or the general Scale of Altitudes for all Latitudes , Page 10 The particular Scale of Altitudes , or Sines , for one Latitude only , Page 11 The Degreees of a whole Circle , 12 Signs , 12 Inches , or 24 Hours , and Moons Age , ibid. The Appurtenances to this Instrument , ibid. Numeration on Decimal-lines , Page 12 Three Examples thereof , Page 13 Numeration on Sexagenary Circular-lines , with Examples thereof . Page 17 How Right Sines , Versed Sines and Chords , are counted on the Rule , Page 20 Of a Circle , Diameter , Chord , Right Sine , Sine Complement , or Co-sine , Versed Sine , Tangent , Secant , what it is , Page 23 , 24 Two good Notes or Observations , Page 25 Of the division of a Circle , ibid. What a Radius is , Page 26 What an Angle , a Triangle , Acute , Right , or Obtuse ; Plain , or spherical Angle is , Page 26 , 27 Parallel-lines , and Perpendiculer-lines , what they are , ibid. The usual Names of Triangles , ibid. Of four sided Figures , and many sided , Page 28 Terms in Arithmetick , as Multiplicator , Product , Quotient , &c. what they mean , Page 29 Geometrical Propositions , Page 31 To draw a right Line , ibid. To raise a Perpendiculer on any line , ibid. To let fall a Perpendiculer any where , Page 33 To draw Parallel-lines , Page 34 To make one Angle equal to another , Page 35 To divide a Line into any number of parts , ib. To bring any 3 Points into a Circle , Page 36 To cut any two Points in a Circle , and the Circle into two equal parts , Page 37 A Segment of a Circle given , to find the Center and Diameter , Page 38 A Segment of a Circle given , to find the length of the Arch , Page 39 To draw a Helical-line , and to find the Centers , of the Splayes , of Eliptical arches , and Key-stones , Page 41 To draw an Oval , ibid. Explanation of Terms particularly belonging to this Instrument . Radius , how taken , Page 41 Right Sines , how taken and counted , Page 42 Tangents , Secants and Chords , how taken , ib. Sine complement , or co-sine Tangent , complement or co-tangent , how taken and counted on this Instrument , ibid. Latteral Sine and Tangent , Page 43 Parallel Sine and Tangent , ibid. Nearest Distance , what it means , ibid. Addition on Lines , ibid. Substraction on Lines . Page 44 Of Terms used in Dialling . Plain , and Pole of the Plain , Page 45 Declination , Reclination , and Inclination of a Plain , what it is , Page 46 What the Perpendiculer-line , and Horizontal-line of a Plain are , ibid. Meridian-line , Substile-line , and Stile-line , Angle of 12 and 6 , and the Inclination of Meridians , what they are , Page 47 Parallels and Contingent-lines , what , Page 48 Vertical-line and Point , what , ibid. Nodus Apex and foot of the Stile , what , ibid. Axis of the Horizon , what , Page 49 Erect , Direct , what , ibid. Declining , Reclining , or Inclining , what , ibid. Circles of Position , what , ibid. Of Terms in Astronomy . What a Sphear is , Page 50 Of ten Points , and ten Circles of the Sphear , Page 51 The 2 Poles of the World or Equinoctial , ibid. The 2 Poles of the Zodiack , Page 52 The 2 Equinoctial-points , ibid. The 2 Solstitial-points , Page 53 The Zenith and Nadir , Page 54 The Horizon , the Meridian , the Equinoctial , the Zodiack , the 2 Colures , the 2 Tropicks , and 2 Polar Circles , Page 55 , 56 , 58 Hours , Azimuths , Almicanters , Declination , Latitude , Longitude , Right Ascention , Page 59 , 60 Oblique Ascention , Difference of Ascentions , Amplitude , Circles , and Angles of Position , what they are , Page 61 , 62 To rectifie the Trianguler Quadrant , Page 63 To observe or find the Suns Altitude , Page 64 To try if any thing be level , or upright , Page 66 To find what Angle the Sector stands at , at any opening ; or to set the Sector to any Angle required , Page 67 , 68 The day of the Month given , to find the Suns Declination , true Place , Right Ascention , or Rising and Setting , by inspection only , Page 71 To find the Suns Amplitude , and difference of Ascentions , and Oblique Ascention , Page 73 To find the Hour of the Day , Page 74 To find the Suns Azimuth , Page 75 The use of the Line of Numbers , and the use of the Line of Lines , both on the Trianguler Quadrant and Sector , one after another , in most Examples . To multiply one Number by another , Page 78 A help to Multiply truly , Page 85 A crabbed Question of Multiplication , Page 90 Precepts of Reduction , Page 94 To divide one Number by another , Page 95 A Caution in Division . Page 97 To 2 Lines or Numbers given , to find a 3d in Geometrical proportion , Page 98 Any one side of a Figure being given , to find all the rest ; or to find a proportion between two or more Lines or Numbers , Page 99 To lay down any number of parts on a Line to any Radius , Page 100 To divide a line into any number of parts , Page 102 To find a Geometrical mean proportion between two Lines or Numbers , three wayes , Page 104 To make a Square equal to an Oblong , Page 107 Or to a Triangle , ibid. To find a Proportion between unlike Superficies , Page 108 To make one Superficies like another Superficies , and equal to a third , Page 109 The Diameter and Content of a Circle being given , to find the Content of another Circle by having his Diameter , Page 111 To find the Square-root of a Number , ibid. To find the Cube-root of a Number , Page 113 To find two mean Proportionals between two Lines or Numbers given , Page 116 The Diameter and Content of a Globe being given , to find the Content of another Globe , whose Diameter also is given , Page 118 The proportion between the Weights and Magnitudes of Metals , Page 119 The Weight and Magnitude of a body of one kind of Metal being given , to find the Magnitude of a body of another Metal of equal weight , Page 121 The magnitudes of two bodies of several Metals , having the weight of one given , to find the weight of the other , Page 122 The weight and magnitude of one body of any Metal being given , and another body like unto the former , is to be made of any other Metal , to find the diameters or magnitudes of it , Page 123 To divide a Line , or Number , by extream and mean proportion , Page 124 Three Lines or Numbers given , to find a fourth in Geometrical proportion , Page 128 The nature & reason of the Golden Rule , Page 129 The Rule of Three inversed , with several Cautions and Examples , Page 132 The double and compound Rule of Three Direct and Reverse , with Examples , Page 139 The Rule of Fellowship with Examples , Page 148 The use of the Line of Numbers in Superficial measure , and the parts on the Rule , Page 154 The breadth given in Foot-measure , to find the length of one Foot , Page 156 The bredth given in Inches , to find how much in length makes one Foot , ibid. The bredth given , to find how much is in a Foot-long , Page 157 Having the length and bredth given in Foot-measure , to find the Content in Feet , ibid. Having the bredth given in Inches , and length in Feet , to find the Content in Feet , Page 158 Having the length & bredth given in Inches , to find the content in superficial Inches , Page 160 Having the length & bredth given in Inches , to find the Content in Feet superficial , Page 161 The length and bredth of an Oblong given , to find the side of a Square equal to it , Page 163 The Diameter of a Circle given , to find the Circumference , Square , equal Square , inscribed and Content , Page 164 The Content of a Circle given , to find the Diameter or Circumference , Page 166 , 167 Certain Rules to measure several figures , Page 108 A Segment of a Circle given , to find the true Diameter and Area thereof , Page 169 A Table to divide the Line of Segments , Page 170 The use of it in part , Page 171 The measuring of Triangles , Tapeziaes , Romboides , Poligons , and Ovals , Page 172 , 173 A Table of the Proportion between the Sides and Area's of regular Poligons , and the use thereof for any other , Page 174 , 175 To make an Oval equal to a Circle , and the contrary , two wayes , Page 175 , 176 The length and bredth of any Oblong Superficies given in Feet , to find the Content in Yards , Page 177 The length and bredth given in feet and parts , to find the Content in Rods ▪ Page 179 The nearest way to measure a party Wall , Page 180 To multiply and reduce any length , bredth , or thickness of a Wall to one Brick and a half at one Operation , Page 183 Examples at six several thicknesses , Page 184 To find the Gage-points for this reducing , Page 185 At one opening of the Compasses , to find how many Rods , Quarters , and Feet in any sum under 10 Rods , Page 186 The usual and readiest equal wayes to measure Tileing and Chimnyes , Page 187 Of Plaisterers-work , or Painters-work , Page 188 Of particulars of work , usually mentioned in a Carpenters-Bill , with Cautions , Page 189 , 190 At any bredth of a House , to find the Rafters , and Hip-rafters , length and Angles , by the Line of Numbers readily , Page 191 The price of one Foot being given , to find the price of a Rod , or a Square of Brick-work , or Flooring , by inspection , Page 193 At any length of a Land given , to find how much in bredth makes one Acre , Page 194 A useful Table in measuring Land , and the use thereof in several Examples , Page 196 , 197 The length and bredth given in Perches , to find the Content in Squares , Perches , Poles , or Rods , Page 200 The length and bredth in Perches , to find the Content in Acres , ibid. The length and bredth given in Chains , to find the content in square Acres , Quarters , and Links , Page 201 To measure a Triangle at once , without halfing the Base or Area , ibid. To reduce Statute-measure , or Acres , to Customary , and the contrary , ibid. A Table to make Scales to do it by measuring or inspection , with Examples , Page 204 Knowing the content of a piece of Land plotted out , to find by what Scale it was done , Page 206 The same Rule applied to the measuring of Glaziers Quarries , Page 208 A Table of all the usual sizes of quarries , Page 210 The bredth & depth of any solid body being given , to find the side of the square equal , Page 211 The bredth and depth , or square equal given , to find how much in length makes one foot solid , four manner of wayes , according to the wordin● of the question , Page 212 The bredth an● depth , or the side of the square of any Solid given , to find how much is in a Foot long solid measure , three wayes , according to the wording the question , Page 219 , 220 The bredth depth and length of any solid body given , to find the solid Content , four wayes , according to the wording the question , Page 221 , 222 , 223. The 3 last Probl. wrought by the Sector , Page 226 The Diameter of a Cillender given , to find how much in length makes 1 foot , 4 wayes , Page 230 The diameter of a Cillender given , to find how much is in a foot long , 3 wayes , Page 232 The diameter of a Cillender with the length given , to find the Content 3 wayes , Page 233 The Circumference given , to find a foot , 3 wayes , Page 234 The Circumference , to find how much in a foot , 3 wayes , Page 236 The Circumference and Length , to find the Content 3 wayes , Page 238 The customs & allowances in measuring round Timber , as Oak or Elm , & the like , Page 240 The use of 2 Points for that allowance , Page 242 To measure a round Pyramid or Steeple , ibid. A nicity in measuring round Timber , stated , Page 246 To measure Globes , and Segments of Globes , both superficially round about , and with th● solidity several wayes , by Arithmetick ▪ and the Line of Numbers , and solid Segments ; with a small Table of solid Segments , Page 252 , 253 The Experimented Proportions , between a Cube , a Cillender , a Sphear , a Cone , a Prism , a Square and Trianguler Pyramid , Page 257 The use of the sliding ( cover or ) Rule , Page 259 The description , Page 260 The Gage-points , and places of them , Page 261 The Uses , to square a Piece , to find how much in length will make 1 foot of square Timber , Page 263 To find how much is in a foot long , Page 264 The square and length given , to find the Content , Page 265 The Diameter of round Timber given , to find how much is in a foot long , Page 267 To find how much in length makes 1 foot , Page 268 Diameter and length given , to find the Content , Page 269 The Circumference given , to find how much is in a foot long , Page 271 The Circumference given , to find how much makes a foot , ibid. The Circumference and length given , to find the Content , Page 272 To Gage round Cask by the Rule or Square , counting 6 Foot for a Barrel of Beer , or one Foot for 6 Gallons , or one Foot for 7 Gallons and a half of Wine measure , Page 273 The diameter and length of a Cask given , to find the Content in Wine-gallons , or Ale-gallons , ibid. To Gage Brewers great round Tuns , and to have the Content in Barrels at one work , Page 274 The use of the other-side in superficial measure , Golden Rule , and Division , Page 275 To make and measure the 5 regular bodies , with the Declination and Reclination of every side , at any scituation of them . The Cube , Page 277 The Tetrahedron , Page 279 The Octahedron , Page 281 The Dodecahedron , Page 283 The Icosahedron , Page 286 A Figure and a Table of all the Sides and Angles , Page 294 Gaging by the Line of Numbers , Page 295 To Gage great square Vessels , and round Vessels , Page 297 — Artificially and Naturally , with Examples , Page 300 To find the mean Diameter , and Gage-point , Page 303 To find the Contents of Cask otherwise , ibid. The Content and mean Diameter given , to find the length of the Cask , & contrary , Page 308 To find the wants & nullage , two wayes , Page 311 A Table of the Wants in a Beer Barrel , in Beer and Wine Gallons , at any Inches , wet or dry , Page 317 The use of the Line of Numbers in Interest , and several Examples thereof , many wayes useful , Page 324 The use of the Line in Military questions , Page 332 The use of the Line in solid Proportions , as the weights and measures of Rope , and Burthen of Ships , Page 336 The way to use the Logarithmal Tables , Page 340 The use of the Rule in Geometry & Astronomy , in 50 Propositions , or Uses , by the perticular Scale or Quadrant , the general Scale or Quadrant , the Sector and Artificial Numbers , Sines , and Tangents , Page 345 , to 448 The use of the Trianguler Quadrant , in finding of Heights and Distances , accessable or inaccessable , in 14 Uses , Page 449 , to 483. FINIS . Notes, typically marginal, from the original text Notes for div A29761-e3430 Radius . Right-Sine . Tangent . Secant , Chord . Cosine . Lateral-Sine . Parallel . Nearest-Distance . Addition on Lines . Substraction on Lines . Rectifying Point . Plain . Pole of the Plain . Declination . Perpendiculer-line on the Plain . Horizontal-line . Reclination and Inclination . Meridian-line . Substile . Stile . Angle between 12 and 6. Inclination of Meridians Parallels ▪ Contingent . Vertical-line . Nodus or Apex . Perpendiculer height of the Stile . Foot of the Stile . Virtical-point . Axis of the Horizon . Erect . Direct . Declining Reclining or Inclining-plains Oblique . Circles of Position . Poles . Poles of the Zodiack . Equinoctial-Points . Solsticial-Points . Zenith . Nadir . Horizon . 1. Zodiack 4. Colures . 5. & 6. Tropicks . 7. & 8. Polar-Circles . 9. & 10. Hours . Azimuths Almicanters . Declination . Latitude . Longitude . Right-Ascention . Oblique-Ascention . Ascentional Difference . Amplitude . Circles and Angles of Position . * In both these the inversed Proportion is in th lower line Circle . Half-Circle . Quadrant , or the quarter . Lesser-parts . Segments . Triangles . Rhombus . Trapeziaes . Regular-Polligons . By th' Trianguler-Quadrant . Sines . Tangent . Secant . Chord . Sines . Tangents . Secants . Chords . By the Sector-side . Sine . Tangent . Tan. to 76. Secant . By the Lines on the Edge . Sines . Tangent . Secants & Tangents beyond 45 degrees . Artificial-logarithms The proof of the truth of the Instrument . Quadrant . Sector . Quadr. Sector . Quadrant . Sector ▪ Quadr. Sector . Quadr. Sector . Quadr. Sector . Sector . Sector . Quadr ▪ Sector . Quadr. Sector . Quadr. Quadr. By the Quadr. particularly . Quad. Generally . Quadr· Particularly . Artificial-S . & T. Quad. Generally . Sector . Particular Quadr. Artificial-S . & T. Quad. Generally . Sector ▪ Art. Sine . Quadr. Sector· Partic. Quadr. Art. Sines . Quadr. generally . Sector . Art. Sine . Quadr. Sector Partic. Q. Artific . S. Gen. Quad. Sector . Part. Q. Artificial S. & T. Gen. Quad. Sector . Partic. Q. Oblique-Ascention . Artificial S. & Tan. G. Quad. Sector . Artificial-S . & T. Partic. Q. Gen. Quad Sector . Partic. Q. Particular Quadr. Particular Quadrant . Particular Quadrant . Artificial-S . & T. Gen. Quad. Sector . Particular Quadrant ▪ Gen. Quad. Sector . Gen. Quad By Artificial Sines & Tang. Particular Quadrant . General-Quadr . By the Artificial-Sines and Tangents . By the Sector . By Artifl . S. & T. General-Quadr ▪ Gen. Quad , By the Sector . Gen. Q●ad Sector . Artificial S. & Tat. Artificial S. & T. By the General-Quadrant & Sector . Gen. Quad. By Artificial Sines & Tang. By Artifi . S. & T. By the Sector . General-Quadr . Or , Sector . Particular Quadrant . By the Artificial-Sines and Tangents . General-Quadr . Particular Quadrant . Artificial-S . & T. Artificial-S . & T. Partic. Q. Gen. Quad Particular Quadrant . Fig. I. Fig. I. Fig. I. Fig. II. Fig. V ▪ Fig. VI. Fig. IV. Fig. VII . Fig. VIII . 
A34005	----	The sector on a quadrant, or A treatise containing the description and use of four several quadrants two small ones and two great ones, each rendred many wayes, both general and particular. Each of them accomodated for dyalling; for the resolving of all proportions instrumentally; and for the ready finding the hour and azimuth universally in the equal limbe. Of great use to seamen and practitioners in the mathematicks. Written by John Collins accountant philomath. Also An appendix touching reflected dyalling from a glass placed at any reclination. Collins, John, 1625-1683. 1659 Approx. 798 KB of XML-encoded text transcribed from 209 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2007-01 (EEBO-TCP Phase 1). A34005 Wing C5382 ESTC R32501 99899660 99899660 66044 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A34005) Transcribed from: (Early English Books Online ; image set 66044) Images scanned from microfilm: (Early English books, 1641-1700 ; 1524:9 or 2400:6) The sector on a quadrant, or A treatise containing the description and use of four several quadrants two small ones and two great ones, each rendred many wayes, both general and particular. Each of them accomodated for dyalling; for the resolving of all proportions instrumentally; and for the ready finding the hour and azimuth universally in the equal limbe. Of great use to seamen and practitioners in the mathematicks. Written by John Collins accountant philomath. Also An appendix touching reflected dyalling from a glass placed at any reclination. Collins, John, 1625-1683. Lyon, John, professor of mathematics. Appendix touching reflective dialling. Sutton, Henry, mathematical instrument maker. [16], 284; [2], 54, [10], 26 p., [6] leaves of plates (some folded) : ill. (woodcuts) printed by J.M. for George Hurlock at Magnus Corner, Thomas Pierrepont, at the Sun in Pauls Church-yard; William Fisher, at the Postern near Tower-Hill, book-sellers; and Henry Sutton, mathematical instrument-maker, at his house in Thred-needle street, behind the Exchange. With paper prints of each quadrant, either loose or pasted upon boards; to be sold at the respective places aforesaid, London : 1659. The quadrants described were made and engraved by Henry Sutton, who also calculated some of the tables and drew the projections. "The description and vses of a great universal quadrant" has separate title page dated 1658; pagination and register are continuous. "The description and uses of a general quadrant, with the horizontal projection, upon it inverted" has separate title page dated 1658; pagination, and register are separate. "An appendix touching reflective dialling" by John Lyon has separate title page dated 1658; pagination is separate; register is continuous. A reissue, with cancel title page, of the 1658 edition having "printed by J. Macock" in imprint (Wing C5381). In this issue, the 2 contents leaves, bound after p. 275 in the original issue, are bound with the preliminaries, following title page and "To the reader" (A2). The catchword "The" on the verso of the second contents leaf does not match the first word of the following page (a1). "The description and uses of a general quadrant" filmed separately as Wing C5371 on UMI microfilm set "Early English books, 1641-1700", reel 2400. Reproduction of originals in: Harvard University Library; Henry E. Huntington Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematical instruments -- Early works to 1800. Astronomy -- Early works to 1800. Navigation -- Early works to 1800. Dialing -- Early works to 1800. 2005-07 TCP Assigned for keying and markup 2006-03 SPi Global Keyed and coded from ProQuest page images 2006-06 Derek Lee Sampled and proofread 2006-06 Derek Lee Text and markup reviewed and edited 2006-09 pfs Batch review (QC) and XML conversion THE SECTOR ON A QUADRANT , OR A Treatise containing the Description and Use of four several QUADRANTS ; Two small ones and two great ones , each rendred many wayes , both general and particular . Each of them Accomodated for Dyalling ; for the Resolving of all Proportions Instrumentally ; And for the ready finding the Hour and Azimuth Universally in the equal Limbe . Of great use to Seamen and Practitioners in the MATHEMATICKS . Written by JOHN COLLINS Accountant Philomath . Also an Appendix touching Reflected Dyalling from a glass placed at any Reclination . London , Printed by J.M. for George Hurlock at Magnus Corner , Thomas Pierrepont , at the Sun in Pauls Church-yard ; William Fisher , at the Postern near Tower-Hill , Book-sellers ; And Henry Sutton , Mathematical Instrument-Maker , at his House in Thred-needle street , behind the Exchange . With Paper Prints of each Quadrant , either loose or pasted upon boards ; to be sold at the respective places aforesaid . 1659. To the Reader . Courteous Reader , THou hast in this Treatise , the Description and Uses of three several Quadrants , presented to thy View and Acceptance ; and here I am to give thee an account of their Occasion and Original . Being in conference with my loving friend M. Thomas Harvie , he told me , that he had often drawn a Quadrant upon Paper pastboard , &c. derived by himself , and never done by any man before , as to his knowledge , from the Stereographick Projection , which for a particular Latitude , would give the Hour in the equal Limb , and would also perform the Azimuth very well ; and but that it was so particular , was very desirous to have one made in Brass for his own use by an Instrument Maker : whereto replying , that with the access of some other Lines to be used with Compasses , it might be rendred general for finding both the Hour and the Azimuth in the equal Limb : He thereupon intimated his desires to M. Sutton , promising within a fortnight after their conference , to draw up full directions for the making thereof . But M. Sutton having very good practise and experience in drawing Projections , speedily found out the drawing of that Projection , either in a Quadrant or a Semicircle , without the assistance of the promised directions , and accordingly , hath drawn the shape of it for all Latitudes , and also found how the Horizontal Projection might be inverted and contrived into a Quadrant without any confusion , by reason of a reverted tail , and let me further add , that he hath taken much pains in calculating Tables for the accurate making of these and other Instruments , in their construction more difficult then any that ever were before ; and the said M. Sutton conceiving that it would be an advancement to their Trade in general , besides satisfactory to the desires of the studious in the Mathematiques , to have the uses of a good Quadrant published , prevailed with me , in regard M. Harvey was not at leisure ( though willing his Quadrant should be made publique ) to write two or three sheets of the use of it , which I intended to have given M. Sutton ( who very well understood the use as well as the making ) to be published in his own name ; whereto he being unwilling , and finding that therein many of the uses of one Quadrant , much less of more could not be comprized , at his earnest request , I wrote what is here digested , succisivè & horis Antelucanis , having little leisure for that purpose , and all this performed before the Instruments were cut , wherefore the description given of them , may not so nearly agree with the Instruments , as if they had been first made , nor possibly some of the examples about finding the hour of the night by the Stars ; which examples were fitted from Tables of present right Ascension , whereas the Quadrant is fitted to serve the better for the future , the difference notwithstanding will be but small . And thus hoping thou wilt cover my failings with the mantle of love , and kindly accept of my endeavours , tending to the publique advancement and increase of knowledge , I still remain a Wellwiller desirous thereof . John Collins . depiction of a quadrant How the Projections on both the Quadrants may be Demonstrated . TO satisfie the inquisitive Reader herein , I shall only in this Edition quote such Latine Authors and Propositions as will evince the truth thereof , the performance whereof in English , is hereafter intended by my loving friend M. Thomas Harvie , in an elaborate Treatise , concerning all the Projections , with their Demonstration and Application , who is accomplished with singular knowledge in that kind , as in general in the Mathematiques . Now the Demonstration of these two Projections is as much included in the Demonstration of the Stereographick Projection , which by Aguilonius in his 6 Book is largely insisted upon , as a peculiar question in Trigonometry , is included in a general Case , and both the Projections on these Quadrants being derived from the grounds of the said general Projection , are necessarily involved in one and the same Demonstration . Stofler in his Astrolabe supposeth the eye in the South Pole Stereographically projecting upon the Plain of the Equator those Circles between the North Pole Horizon and Troprick of Capricorn , neglecting that part under the Horizon . But the Projection on the Quadrant , considered as it may be derived from his Astrolabe , supposeth the eye in the same Position , and makes use of one half of the Projection of the other part of the Circles intercepted between the Horizon and Tropick of Capricorn , namely , of that space between the Tropick below our Horizon , only changing the names of Cancer for Capricorn in their use , and using the depressed Parallels to the Horizon , instead of the Parallels of Altitude ; so that the Azimuths of the Quadrant made by this inversion , are no other then the Azimuths of Stoflers Projection continued below the Tropick of Capricorn where he breaks them off , and the rule he prescribes to draw the parallel of 18d of Depression for the Twylight serves to draw that , and all the other parallels of Altitude in this Quadrant . In like manner the Horizontal Projection supposeth the eye in the Nadir projecting upon the Plain of the Horizon . That part of the Sphere intercepted between the two Tropicks , neglecting that part thereof under the Horizon . But the Projection on the other great Quadrant , considered as derived therefrom , supposeth the eye there projecting that part of the Sphere which is there neglected with the like change of denomination ; and the Parallels of Declination are no other then the continuance of the said Parallels of the Horizontal Projection round to the Midnight Meridian , and the Hour circles the continuance of the said hours , only the Index of Altitudes is fitted to the Depressed Parallels of the Horizon , in stead of the Parallels of Altitude . Now it is evident , either from the Sphere or Analemma , that that part of either of these Projections which falls under the Horizon , will supply the use of that which hapned above , admitting only a change of denomination ; for in the Horizontal Projection , that Parallel of Declination which was called the Winter Tropick , being no other then the same Circle continued about , now in its use and denomination , becomes the Summer Tropick ; and the reason is , because what ever Altitude the Sun hath in any Sign upon any Hour or Azimuth reckoned from the Noon Meridian , he hath the like Depression on the like Hour and Azimuth in the opposite Signe counted from the Midnight meridian . The terms of Noon and Midnight Meridian are afterwards used in relation to some general Proportions : By the Hour in general is meant the Angle between the Meridian of the Sun or Stars , and the Meridian of the place : By the Hour counted from Noon Meridian , is meant the said Angle counted from that part of the Meridian of the place which falls above the Elevated Pole , continued towards the Depressed Pole : and by the Midnight Meridian , the opposite thereto under the Elevated Pole , continued as before . By the Azimuth counted from the Midnight Meridian , is meant an Angle at the Zenith between the Suns Vertical or Azimuthal circle , and the Meridian of the place , measured by the Horizon , counted from the Intersection of the Horizon with the Meridian under the Elevated Pole ; and by the Azimuth counted from the Noon Meridian , is meant the Complement of the said Angle to a Semicircle , counted from the opposite Intersection of the Horizon with the former Meridian continued above the Elevated , and towards the Depressed Pole , according to which acceptions , the general Proportions are fitted for finding it either way in both Hemispheres , without any restriction to North or South . A more immediate account of these Projections . HItherto we have accommodated our Discourse , to shew how these Projections are derived from Stoflers Astrolabe , and from the Horizontal Projection , of which neither Stofler ( as to my knowledge ) for I have only seen his 8 Book ) nor the learned M. Oughtred , give no peculiar Demonstration , as being particular examples of a general case , largely ( as such ) insisted on ; and this we have done for the accommodation of Instrument makers , to whom this Derivation may seem most suitable ; whereas such a deduction is not at all necessary to the Demonstration of the Projections so derived . For in the Projection derived from the Inversion of Stofler , let the eye be supposed to be placed in the North Pole , projecting upon the Plain of the Equinoctial , such Circles in the Sphere , as are described in the Quadrant between the two Tropicks , a quarter of which Projection will be the same with that on the Quadrant , namely , one of those quarters between the South part of the Meridian and hour of six , which will leave out all the outward part of the Almicanters between it and the Tropick of Cancer , and in stead thereof , there is taken in such a like part of the depressed Parallels to the Horizon between the same Hour of six , and Tropick of Capricorn , which is the reverted tail ; for the Parallels of Depression have the same respect to the Tropick of Capricorn , that the Parallels of Altitude have to the Tropick of Cancer , and will work the same in effect . In like manner , the Eye in the other Projection may be supposed in the Zenith , Stereographically Projecting upon the Plain of the Horizon , that part only of the space between the Tropicks , which falls without the Projection of the Horizontal Circle , save only the reverted tail , which is the Projection of so much of the Parallels of South Declination , as is intercepted between the prime Vertical Circle and the Horizon , and is taken in to serve in stead of that part of the Parallels of North Declination , which will fall without the Quadrant . In any of these Positions of the Eye , all Circles passing through the same , will be projected in right lines by 91 Prop. 6 Book of Aguilonius , such are the Azimuths on the Horizontal Quadrant , and the Hours on the other Quadrant , represented by the thred lying over any Ark in the Limb , so also in this latter Quadrant is the Parallel of Altitude equal to the Latitude of the place , a right line . All Circles parallel to the Horizon and Equinoctial , will be projected in concentrick Circles by 94 Prop. 6. Aguilonius , such are the Parallels of Altitude in the Horizontal Quadrant , and the Parallels of Declination in the other Quadrants , represented by the Bead , when it is rectified to the Index of Altitudes in the one , and to the Ecliptick in the other , carried in a circular trace from one side of the Quadrant to the other . All other Circles in the Sphere , whatsoever and howsoever scituated , being projected according to the supposed position of the Plain and Eye , will be represented by Excentrick Circles . by 96 Prop. 6. Aguilonius , and the hours in the Horizontal Projection will ( if they be produced ) meet with the projected Pole points , so also the Azimuths in the other Projection , which by the like parity of Reason may be denominated The Equinoctial Projection , will ( being produced ) meet with the projected points of the Zenith and Nadir ; and how in particular to project and divide any Circle however scituated in the Sphere , is abundantly shewn in the 6th Book of the aforesaid Author , and amplified with many examples , though none of them agreeing with the particular Draughts of these Quadrants , yet if put in practise according to the proposed Scituation of the Eye , will be found to agree with the prescribed Directions for the making of these Quadrants . See also Clavius his Book of the Astrolabe , Guido Ubaldus his Theorick of the general Planispheres , and M. Oughtreds 2d Scheme B in his late Trigonometry in English . An Appendix to the Description of the Small Quadrant . SInce the Printing of the sheet B , we have thought fit to vary a little from the Description there given of the Small Quadrant . The Dyalling Scale of Hours described in page 9 , near the beginning , which I say in Page 191 may be omitted , is accordingly left out , and instead of it , a line of Versed Sines of 90d put on , the uses whereof are handled in the great Equinoctial Quadrant . Also there is two Scales added to the small Quadrant more then was described ; namely , the Scale of Entrance , the same that was placed upon the Horizontal Quadrant , with a Sine of 51d 32′ put through the whole Limb serving to give the Altitude at six , which the thred will intersect , if it be laid over the Declination in the Limb ; but enough of the uses of these Scales is said in the Horizontal Quadrant . Lastly , Those that like it best , instead of having on the Small Quadrant one loose fitted Scale for the Hour , and another for the Azimuth , may have the Hour-Scale only divided into two parts , serving to give the Hour and Azimuth for the Sun , and all the Stars in the Hemisphere , the one part for South Declinations , the other for North Declinations , in imitation of the Diagonal Scale . An Advertisement . ALL manner of Mathematical Instruments , either for Sea or Land , are exactly made in Wood or Brass , by Henry Sutton , in Thredneedle-street , near Christophers Church , or by William Sutton in Upper Shadwel , a little beyond the Church . Pag. Li. Errors to be thus Corrected . 8 33 apply supply 19 25 76 d 54′ 79 d 54′ 20 7 48 d 45 d 23 3 50 d 41′ 50 d 41′ 23 11 58′ 53′ 24 25 a Letter Character . the Character plus + 28 is lesse if lesse 55 14 difference of one of differences of 55 17 of the Leggs from the Leggs of one of the Leggs The affections in page 57 line 16 , 17. wanting Braces , are expressed at large page 140 , 141. also the last affection in that page having a mistake of lesser for greater in the middle brace is reprinted in page 138 60 14 The last term of the 4th Proportion should be the Sine of the angle sought , and not the Cosecant . 92 29 a Leg and its adjacent angle The Hipotenusal and its adjacent angle 30 to find the other angle by 4 to find the side opposite there to by 8 Case Case . See page 138. 103 34 Acquimultiplex Aequimultiplex 35 therefore by therefore by 18 Prop. 7. Euclid . 121 18 Hour 3¼ Altitude 41 d 31′ Altitude for the hour 3¼ is 43 d 31′ 158 1 Line of Line of Sines 159 21 of 90 d at 90 d 164 ●8 any some 174 15 in the Limb in the lesser Sines 181 25 between as also between 27 would find it would in the other Hemisphere find it 184 21 the common as the common 189 1● either it may be found either 192 24 As the second including side As the Sine of the 2d including side 207 1 great Scale great Quadrant 209 7 60 parts 60 equal parts The Angle C in the Scheam page 52 is wrong cut , and should be 113d 22′ , See it in page 156. Page 98 a wrong Scheam printed , the true one is in page 93. Page 102 in the under Triangle the Angle D should be 108d 37′ See page 201. The first ten lines of calculation p. 53 are somwhat misplac'd , & should stand thus .     D●ffs Log ms     B C 126 Legs 169   2,2278867   A C 194   101   2,0042214 Sum with double Radius A B 270 Base 25 1,3979400 24,232208●   Sum 590           half sum 295   Logarith . 2 4698220 3 8677620 former Rectangle       Residue 20,3644461   The half is the tangent of 56 d , 41′ t.       10,1822230   Which Ark doubled is       113 , 22 A C B the Angle sought .   A Table shewing the Contents of the Book . AN account of the original of the Projections , and their Demonstration . In the Preface . The Description and making of the small Quadrant : Page 1. The Vse of the perpetual Almanack . 12 The Vses of the Projection on the small Quadrant . 15 Of the Stars inscribed thereon . 24 Of the Projection drawn in a Semicircle . 33 Of the Stars and quadrant of Ascensions drawn on the back-side of the quadrant . 34 The Vses of the Quadrat and Shaddows . 35 Divers ways to measure the altitude of a Tower or object . 36 Affections of Plain Triangles 45 Proportions for the Cases of right angled plain Triangles 46 Proportions for the Cases of Oblique Plain Triangles 49 Doubtful Cases manifested , and mistakes about them rectified 50 Four several Proportions for finding an angle when three sides are given , not requiring the help of Perpendiculars 52 Affections of Sphoerical Triangles 56 Six several Proportions suited to each of the 16 Cases of right angled Sphoerical Triangles , with all necessary caution 60 Directions for varying of Proportions . 72 Some Cases of oblique Sphoerical Triangles resolved without the help of Perpendiculars 76 , 83 When some of the Oblique Cases will be doubtful , and when not 80 , 81 Six of those Cases demonstrated to be doubtful 86 , 87 Perpendiculars needless in any of those Cases , and not used 88 Some new Proportions applyed to the Tables 88 And mistakes about Calculating the distances of places noted 89 And Proportions esteemed improper to the artificial Tables applyed for that purpose 96 Proportions both Instrumental and others for finding a Sphoerical angle when 3 sides are given 100 , 101 Those Instrumental Proportions how demonstrated and derived from other Proportions in use for the Tables . 103 New Proportions for finding the Hour and Azimuth demonstrated from the Analemma . 108. And applyed to the Calculating of a Table of Hours . 117 A Table Calculated thereby , showing the Suns altitudes on all hours for some Declinations useful for the trial and construction of some Instruments . 121 A Table of the Suns Altitudes for each 5d of Azimuth fitted to the same Declinations . 122 To Calculate a Table of Azimuths to all Altitudes . 124 And a Table of Altitudes to all Azimuths . 128 And the Proportions used laid down from the Analemma . 132 Affections of some doubtful Cases of Sphoerical Triangles determined . 138 The ground of working Proportions on a quadrant . 144 Proportions is equal Parts resolved instrumentally . 146 , 147 All Proportions in Tangents alone resolveable by the tangent line on the quadrant , or by a tangent of 45d. 148 Proportions in Sines and Tangents wrought on the quadrant . 151 Proportions in equal parts and tangents resolved by the quadrant . 152 Proportions in equal Parts and Sines so resolved , being useful in Navigation . 156 , 157 Of a line of Sines , and how from it or the Limb to take off a Sine , Tangent or Secant . 158 Proportions in Sines wrought on the quadrant . 161 How to proportion out lines to any Radius from lines inscribed in the Limbe . 167 To operate Proportions in Sines and Tangents on the quadrant . 168 A general way for finding the Hour . 173 As also of the Azimuth . 174 Particular Scales fitted for finding the same . 177 Another general way for finding the Hour . 181 And for finding the Suns Altitudes on all Hours . 183 Another general way for finding the Azimuth . 183 And of distances of places by the quadrant . 188 How from a quadrant to take off Chords ▪ 190 The Description of the Diagonal Scale . 193 , 194 The said Scale fitted for the ready finding the Hour and Azimuth for all parts of England , Wales , Ireland , and the uses thereof . 195 , 196 The description and uses of the particular Scales in Dyalling . 197 Also the uses of other Scales fitted for finding the Hour and Azimuth near noon . 200 Hitherto the Uses of the small Quadrant . THe Description of the great Equinoctial Quadrant . 2 The Vses of a line of Versed Sines from the Center . 208 And of Versed Sines in the Limb. 214 The Vses of a fitted particular Scale , with the Scale of entrance , for finding the Hour and Azimuth in the equal Limb. 216 As also for finding the same in the Versed Sines of 90d in the Limb. 220 , 221 And of the Diagonal Scale therewith . 222 , 223 New general Proportions for finding the Hour and Azimuth . 225 Divers Proportions demonstrated from the Analemma . 227 Several general Proportions for finding the Hour applyed . 230 And several Proportions applyed for finding the Altitudes on all Hours . 233 General Proportions for finding the Azimuth 237 A new single Proportion in Sines for Calculating a table of hours . 243 The Stars how put on and the hour found by them . 244 , 245 The Construction of the Graduated Circle . 248 The line of Latitudes demonstrated . 249 Vses of the graduated Circle and Diameter in resolving of Proportions , and in finding the Hour and Azimuth universally . 253 Vses of other Lines within the said Circle . 261 To prick down a Horizontal Dyal in a Square or a Triangle by the Dyalling lines on the Quadrants . 262 Proportions for upright Decliners . 265 To prick down upright Decliners in a Square or rather right angled Parralellogram , is likewise in a Triangle from the Substile . 268 To prick them down in a Triangle or Paralellogram from the Meridian . 272 An Advertisement about observing of altitudes . 275 The description and making of the Horizontal quadrant . Page 1 The perpetual Almanack in another form 12 A tide Table with the use of the Epacts in finding the Moons age . 11 The uses of the Horizontal Projection . 18 The uses of the particular curved line and fitted Scales thereto . 20 Some Proportions demonstrated from the Analemma . 26 A particular Scale for finding the Hour and Azimuth fitted thereto , with its uses . 27 The uses of the Scale of entrance , being another particular fitted Scale for finding the Hour and Azimuth in the Limb. 32 A Chord taken off from the equal Limb and the Hour and Azimuth found universally thereby . 34 A new general Proportion for the Azimuth to find it in the equal Limb. 36 Another for the Converse of the fourth Axiom . 38 The Stars how inscribed , and the hour of the night found by them . 39 Some uses of the Dyalling Lines . 44 , 45 The use of the Line of Superficies . 46 , 47 And of the Line of Solids . 48 And of the line of inscribed Bodies and other Sector Lines . 49 , 50 A Table of the Latitudes of the most eminent places in England , Scotland , Wales , and Ireland . 52 , 53 A Table of the right Ascensions and Declinations of 54 of the most eminent fixed Stars . 53 , 54 Lastly , A Table of the Suns Declination and right Ascension for the year 1666 , with a Table of equation to make it serve sooner or longer . An Errour Page 248 , Line 30 and 31 for so is L N to L F read so is L N to N F. Of the Lines on the foreside of the QUADRANT . ON the right edge from the Center is placed a Line of equal parts , of 5 inches in length , divided into 100 equal parts . On the left edge a Line of Tangents , continued to two Radii , or to 63d 26 m the Radius whereof is 2½ inches . These two Lines make a right Angle in the Center , and between them include the Projection , which is no other then a fourth part of Stoflers particular Astrolabe inverted . Next above this Projection , towards the Center , is put on in the Quadrant of a Circle , the Suns declinations . And above that in four other Quadrants of Circles , the days of the Moneth , respecting the four seasons of the year . Underneath the Projection , towards the Limbe is put on , in one half of a Quadrant , one of the sides of the Geometrical Quadrat , and in the other half the Line of shadows . All which is bounded in by the equal Limbe . There stands moreover on the very edges of the Quadrant , two Dyalling Scales , which do not proceed from the Center ; that on the right edge is called the Line of Latitudes ; and that on the left edge the Scale of Hours ( equal in length to the Sines ) which is no other then a double Tangent , or two Lines of Tangents to 45d each set together in the middle , and so might , if there were need , be continued , ad infinitum . The Construction and making of such of these Lines as are not commonly described in other Treatises . To inscribe the Line of Declinations , there will be given the Suns declination to find his right Ascension , which is the Ark of the Limb , that by help of a Ruler , moving on the Center of the Quadrant , and laid over the same , will in-scribe the Declination proposed . The Canon to find the Suns right Ascension from the nearest Equinoctial point , correspondent to the Declination proposed , is As the Radius To the Cotangent of the Suns greatest Declination : So the Tangent of the Declination given : To the Sine of the Suns right Ascension . The four Quadrants for the days of the moneth are likewise to be graduated from the Limb , by help of a Table of the Suns right Ascensions , made for each day in the year . The Geometrical Quadrat is inscribed in half the Quadrant of a Circle , by finding in the Table of natural Tangents , what Arches answer to every equal Division of the Radius , and so to be graduated from the Limb ; so 300 sought in the Tangents gives the Ark of 16d 42 m of the Limb against which 3 of the Quadrat is to be graduated . The Line of shaddows is no other then the continuance of the Quadrat beyond the Radius , and so the making after the same manner ; thus having the length of the shadow assigned , annex the Ciphers of the Radius thereto , and seek in he natural Tangents , what Ark corresponds thereto ; thus the shadow being assigned thrice as long as the Gnomon , I seek 3000 in the natural Tangents , the Ark answering thereto , is 71d 34′ which being counted from the left edge of the Quadrant , towards the right in the Limb , the Line of shadows may from thence be graduated ; the Complement of this Ark is the Suns Altitude , answering to that length of the right shadow , being 18d 26′ . The Canon to make the Line of Latitudes , will be As the Radius to the Chord of 90d so the Tangents of each respective degree of the Line of Latitudes . To the Tangents of ohter Arks : The natural Sines of which Arks are the numbers that from a Diagonal Scale of equal parts shall graduate the Divisions of the Line of Latitudes to any Radius . To draw the Projection . Those Lines that cross each other , are Arches of Circles , whose Centers fall in two streight Lines . Of the Paralels of Altitude . All those Arks whose Aspect denotes them to be drawn from the right edge of the Quadrant towards the left , are called Paralels of Altitude , and their Centers fall in the right edge of the Quadrant , continued both beyond the Center and Limb so far as is needful . To find the Intersections of the Paralels of Altitude , with the Meridian , that is , Points therein limiting the Semi-diameters of the Paralells . Assume any Point in the right edge of the Quadrant , ( which is called the Meridian Line ) near the Limbe to be the Tropick of Cancer ; the distance of this Point from the Center of the Quadrant , must represent the Tangent of 56d 46′ which is half the Suns greatest Declination more then the Radius ; the distance of the Equator from the Center , shall be equal to the Radius of this Tangent . For the finding the Intersections of the other Paralells of Altitude , it will be best to make a Line of Semi-tangents to the same Radius , that is to number each degree of this Tangent with the double Ark , and so every half degree will become a whole one : Out of this Line of Semi-tangents prick off from the Center of the Quadrant 66d 29′ the Complement of the Suns greatest Declination , which will find the Intersection of the Tropick of Capricorn with the Meridian . Now to fit the Projection to any particular Latitude : Out of the said Line of Semi-tangents from the Center of the Quadrant , prick off the Latitude of the place , and it will find a point in the Meridian Line , where the Horizon , or Paralell of 00d of Altitude will intersect the Meridian ; this Point is called the Horizontal Point , and serves for finding the Centers of all the Paralells . To the Latitude of the place add each degree of Altitude successively till you have included the greatest Meridian Altitude ; these compound Arks are such as being prickt from the Center of the Quadrant out of the Line of Semi-tangents will find points in the Meridian Line , limiting the Semi-diameters of the paralells of Altitude . Above the Horizon , and between the Circle that bounds the Projection falls a portion thereof called the Reverted Tail , which otherwise would if it had not been there reverted , have excurred the limits of a Quadrant . To find the Intersections for those Paralells of Altitude , substract successively each degree out of the Latitude of the place , and the remaining Arks prick from the Center out of the Line of Semi-tangents : The use of this Tail being to find the hour and Azimuth before or after 6 in the Summer time only , it need be continued no further above the Horizon then the Ark of the Suns greatest Altitude at 6 , which at London is 18d 12′ . To finde the Centers of the Paralells of Altitude . These are to be discovered by help of a Line of natural Tangents , not numbred with the double Arks , whose Radius must be equal to the distance of the Equator from the Center of the Quadrant , or which is all one to 90d of the Line of Semi-tangents : Out of this Line of Tangents prick off beyond the Center of the Quadrant the Complement of the Latitude , the distance between the Point thereby found , and the Horizontal Point is the Semi-diameter wherewith the Horizon is to be drawn . To find the Centers of the rest of the Paralells . To the Complement of the Latitude add each degree of Altitude successively till you have included the greatest Meridian Altitudes ; The Tangents of these Arks prick beyond the Center , the distance from the Points so discovered to the Horizontal Point , are the Semi-diameters of the Paralells of Altitude ; the extreamities of which Semi-diameters being limited in the Meridian Line ; these extents thence prict , finds their Centers . Some of these Compound Arks will exceed 90 degrees , as generally where any Meridian Altitude is greater then the Latitude . In this case substract those Arkes from 180d and prick the Tangents of the remaining Arks from the Center of the Quadrant on the Meridian Line continued beyond the Limbe , and then as before the distances between those Points and the Horizontal Point , are the Semi-diameters of those Paralells , whose Extremities are limited in the Meridian Line . To find the Centers of the Paralells of the reverted Tail. From the Complement of the Latitude substract each degree of Altitude in order , till you have included the greatest Altitude of 6 the Tangent of the remaining Arks prick from the Center of the Quadrant , and you will find such Points the distances between which and the Horizontal Point are the Semi-diameters of those Paralells . To find the Centers and Semi-diameters of the Azimuths . All those Portions of Arks which issue from the top of the Projection towards the Limb are called Azimuths , the Centers of them all fall upon that Paralell of Altitude which is equal to the Latitude of the place whereto the Projection is fitted , which will always be a streight Tangent Line . Out of the former Line of Tangents , whose Radius is equal to the distance of the Equator from the Center of the Quadrant , prick down the Latitude of the place on the Meridian Line , and thereto perpendicularly erect the Line for finding the Centers of the Azimuths , which must be continued through and beyond the Projection . Out of the said Line of Tangents and beyond the Center prick down the Tangent of half the Complement of the Latitude at London 19d 14 m and it will discover a Point which is called the Zenith Point , because in it all the Azimuths do meet ; The distance between this Point and the Point where the Center Line of the Azimuths intersects the Meridian make the Radius of a Tangent , out of which Tangent prick down each degree successively , both within and beyond the Projection on the Line of Centers , and you have the Centers for all the Azimuths ; where note , that the Centers of all Azimuths which exceed 90d will fall within the Projection , and of all others without , the distances of these respective Points from the Zenith Point , are the Semi-diameters of the Azimuths , with which extents let them be respectively drawn . To draw the Summer and Winter Ecliptick and to divide them . The Summer Ecliptick is drawn from the Point of the Equator in the left edge of the Quadrant to the Tropick of Cancer , and the Winter thence to the Tropick of Capricorn out of a Line of Tangents to the Radius equal to the distance of the Equator , from the Center prick down the Tangent of 23d 31′ the Suns greatest declination from the Center of the Quadrant on the Meridian Line towards the Limbe , and you shall discover the Center of the Summer Ecliptick with the same extent , being the Semi-diameter thereof , set one foot down at the Tropick of Capricorn , and the other will fall beyond the Center of the Quadrant on the right edge , and discovers the Center for drawing the Winter Ecliptick ; to divide them use this Canon . As the Radius to Tangent of the Suns distance from the nearest Equinoctial Point : So the Cosine of the Suns greatest Declination : To the Tangent of the Suns right Ascension , which must be counted in the Limbe , and from it the Suns true place graduated on both the Eclipticks . To draw the two Horizons , and to divide them . One of the Horizons is the Paralel of 00d of Altitude , which being intersected by the Azimuth Circles , is thereby divided into the degrees of the Suns Amplitude ; this is the upper Horizon , and the drawing hereof was shewed already . The other Horizon is but this inverted , and the Divisions transferred from that , the Center of it is found by pricking the Tangent of the Complement of the Latitude on the Meridian Line from the Center of the Quadrant , the distance of the Equator being Radius . But it may be also done from the Limbe by the Proportion following . As Radius , to Tangent of the Latitude ; So the Tangent of the Suns greatest Declination , to the sine of the greatest Ascensional difference ( which converted into Time , gives the time of the Suns rising or setting before or after 6 ) by which Ark of the Limbe the Horizon is limitted ; Then to divide it say As the Radius , to the Tangent of the assigned Amplitude : So is the Sine of the Latitude : To the Tangent of the Ascensional difference agreeing thereto , which counted in the Limb , from it the Amplitudes may be divided on both the Horizons ; and note , if these Amplitudes be not coincident with those the Azimuths have designed , then are the said Azimuths drawn false . To inscribe the Stars on the Projection . Such only , and no other as fall between the two Tropicks , may be there put on . Set one foot of the Compasses in the Center of the Quadrant , and extend the other to that place of either of the Eclipticks , as corresponds to the given declination of the Star , and therewith sweep an occult Ark : I say then that a Thread from the Center of the Quadrant laid over the Limb to the Stars right Ascension where it intersects , the former occult Ark is the place where the proposed Star must be graduated . Of the Almanack . There is also graved in a Rectangular Square , or Oblong , a perpetual Almanack , which may stand either on the foreside or back of the Quadrant , as room shall best permit . On the Backside of the Quadrant there is , 1. On the right edge a Line of Signs issuing from the Center , the Radius whereof is in length 5 inches . 2. On the left edge a Line of Chords issuing from the Center . 3. On the edges of the Quadrant there are also two Scales for the more ready finding the Hour and Azimuths in one Latitude ; the Hour Scale is no other then 62d of a Line of Sines , whose Radius is made equal to half the Secant of the Latitude being fitted for London ) to the common Radius of the Sines ; the prickt Line of Declination annexed to it , and also continued beyond the other end of it , to the Suns greatest Declination is also a portion of a Line of Sines , the Radius whereof is equal to the Sine of the Latitude taken out of the other part of the Scale , or which is all one the Sine of the Suns greatest declination is made equal to the Sine of the greatest Altitude at the hour of 6 taken out of the other part of the Scale , which at London is 18d 12 m 4. The Azimuth Scale is also 62d of a Line of Sines , whose Radius is made equal to half the Tangent of the Latitude to the common Radius of the Sines , the Line of the Declination annexed to it , and continued beyond it : To the Suns greatest Declination is also a portion of a Line of Sines of such a length whereof the Sine of the Latitude is equal to the Radius of the Sines of the other part of this fitted Scale ; or which is all one , the length of the Suns greatest Declination is made equal to the Suns greatest Vertical Altitude , which in this Latitude is 30d 39′ of the other Sine or Line of Altitudes . The Limbe is numbred both with degrees and time , from the right edge towards the left . Between the Limbe and the Center are put on in Circles , the Scales following . 1 , A Line of Versed Sines to 180 degrees . 2. A Line of Secants to 60d the graduations whereof begin against 30d of the Limbe , to apply which Vacancy , and for other good uses , there is put on a Line of 90 Sines , ending where the former graduations begin ; this is called the lesser Sines . 3. A Line of Tangents graduated to 63d 26′ 4. A Line of Versed Sines to 60d through the whole Limbe , called the Versed Sines quadrupled , because the Radius hereof is quadruple to the Radius of the former Versed Sines . 5. A Line of double Tangents , or Scale of hours , being the same Dyalling Scale as was described on the foreside . 6. A Tangent of 45d or three hours through the whole Limbe for Dyalling , which may also be numbred by the Ark doubled to serve for a Projection Tangent , alias a Semi-tangent . 7. In another Quadrant of a Circle may be inscribed a portion of a Versed Sine to eight times the Radius encreased , of that of 180d called the Occupled Versed Sine , and at the end of this from the other edge , another portion of a Versed Sine to 12 times the Radius encreased may be put on . 8. Lastly , above all these is the Scale of Hours or Nocturnal with Stars names graved within it towards the Center ; this is divided into 12 equal hours and their parts , and the Stars are put on from their right Ascensions , only with their declination figured against them . All the Lines put on in Quadrants of Circles must be inscribed from the Limbe by help of Tables , carefully made for that purpose ; an instance shall be given how the Line of Versed Sines to 180d was inscribed , and after the same manner that was put on , must all the rest : Imagine a Line of Versed Sines to 180d to stand upon the left edge of a Quadrant from the Center with the whole length thereof upon the Center sweep the Arch of a Circle , and then suppose Lines drawn through each graduation or degree thereof continued parralel to the right edge till they intersect the Arch formerly swept which shall be divided in such manner as the Line of Versed Sines on this Quadrant is done . But to do this by Calculation , A Table of natural Versed Sines must first be made , which for all Arks under 90d are found by substracting the Sine Complement from the Radius , so the Sine of 20d is 34202 which substracted from the Radius rests 65798 , which is the Versed Sine of 70d : And for all Arks above 90d are got by adding the Sine of the Arks excess above 90d unto the Radius : thus the Versed Sine of 110d is found by adding the Sine of 20d to the Radius , which will make 134202 for the Versed Sine of the Said Ark. This Table , or the like of another kind , being thus prepared , the proportion for inscribing of it will hold . As the length of the Line supposed to be posited on the left edge , Is to the Radius , So is any part of that length To the Sine of an Arch , which sought in the Tables , gives the Arch of the Limbe against which the degree of the Line proposed must be graduated . But in regard the Versed Sine of 180d is equal to the double of the Radius ; the Table for inscribing it will be easily made by halfing the Versed Sine proposed , and seeking that half in the Table of natural Sines , so the half of the Versed Sine of 70d is 32899 which sought in the Table of natural Sines , gives 19d 13′ fore of the Limb against which the Versed Sine aforesaid is to be graduated , and so the half of the Versed Sine of 110d is 67101 which answers to 42d 9′ of the sines or Limb. So likewise the Table for putting on the lesser Sines was made by halfeing the natural Sines , and then seeking what Arks corresponded thereto in the natural Sines aforesaid ; those that think these Lines to many may very well want the Versed Sines so oft repeated ; And they that will admit of a Radius of 6 or 7 inches , may have the Line of Lines Superficies and Solids , put on in the Limb on the foreside , and the Segments Quadrature , Equated Bodies , Mettalls , and inscribed Bodies , or other Lines at pleasure put on upon the backside , as hath been already done upon some Quadrants . Now to the Use . The Vses of the PROJECTION . Of the Almanack . BEfore the Projection can be used , the day of the Moneth , the Suns place or Declination must be known ; but these are commonly given by the knowledge thereof : Now this Almanack will as much help to the obtaining hereof , as any other common Almanack . It consists of a Rectangular Oblong , or long Square divided into 7 Colums in the breadth to represent 7 days of the week , accounting the Lords day first ; and length ways into 9 Columns , the two uppermost represent the months of the year , accounting March the first , the five middlemost the respective days of each Month , and the two undermost some certain leap years , posited in such Columns , as that thereby may be known by Inspection , what day of the Week the first of March happened upon in the said Leap years ; the contrivance hereof owns its original from my Worthy Friend Mr. Michael Darie , for the due placing of the Months over the Columns of days , take the following Rule in his own words . First having March assign'd to lead the round , The rest o' th Months are easily after found ; If that you take the complement in days To 35 of a plac'd Month always , And count it from its place with due Progression It shews you where the next Month takes possession . Thus placing the Month of March first , then if I would place April , or the second Month , March having 31 days , the Complement thereof to 35 is 4 then counting four Columns from the place of March , it falls upon the 5 Column , where the figure 2 is placed for the 2d Moneth , then April being placed ; if I would place May I take 30 , the number of days in April , from 35 there rests 5 , and counting 5 Columns from the place of April where it ends , which is in the 3d Column , the figure 3 is placed for the 3d Moneth or Moneth of May. The next thing to be known is on what day of the Week the first day of March falleth upon , which is continually to be remembred in using the Almanack . This for some Leap years to come , may be known by counting in what Column the said Leap year is graved , thus in Anno 1660 , the first of March falls upon a Thursday , because 60 is graved in the 5 Column , that being the fift day of the Week : But for a general Rule take it in these words . To the number two add the year of our Lord , and a fourth part thereof , neglecting the odd remainder , when there is any ; the Amount divide by 7 the remainder , when the Division is finished , shews the number of Direction , or day of the week , on which the first day of March falleth , accounting the Lords day the first ; but if nothing remain , it falls on a Saturday .   2   Example for the year 1657 The even fourth thereof 414   2073 ( 296 quotient . 7 ) 1 remaining .   By this Rule there will be found to remain one for the year of our Lord 1657 whence it follows that the first day of March fell on the first day of the Week , alias , the Lords day in that year ; so in Anno 1658 , there remains 2 for Munday : in 1659 , rests 3 for Tuesday ; in 1660 rests 5 for Thursday ; so that hence it may be observed , that every 4 years the first of March proceeds 5 days : Upon which supposition the former Rule is built ; say then As 4 to 5 , or as 1 to 1¼ so is the year of the Lord propounded , to the number of days , the first of March hath proceeded in all that Tract , caused by the odd day in each year , and the Access of the days for the Leap years ; this number divided by 7 , the remainder shews the fractionate part of a Week above whole ones , which the said day hath proceeded , which wil not agree with the day of the Week the first of March falls upon , according to common tradition , unless the number two be added thereto , which argues that the first of March , as we now account the days of the week fell upon Munday , or the second day of the week in the year of our Lords Nativity : This is only for Illustration of the former Rule , being to shew that the adding of the even fourth part of the year of our Lord thereto , works the proportion of 4 to 5. The Vse of this Almanack is to know for ever on what day of the Week any day of the Month falls upon . Remembring on what day of the Week the first day of March fell upon in the year propounded ( which doth then begin in the use of this Almanack , and not sooner or later , as upon New-years day , or Quarter day ) all the figures representing the days of the Month do also represent the same day of the week in the respective Months under which they stand ; and the converse , the Moneth being assigned , all the figures that stand as days under it , inform you what days of the said Month the Week day shall be the same , as it was upon the first day of March , and then by a due Progression it will be easie to find upon what day of the Moneth any day of the week falleth , as well as by a common Almanack , without the trouble of always one , and sometimes two Dominical Letters quite shunned in this Almanack , by beginning the year the first of March , and so the odd day for Leap year is introduced between the end of the old , and the beginning of the new-year . Example . In Anno 1657. looking for the figure 10 in the Column for Months , for the Month of December ; under it I find 6 , 13 , 20 , 27 , now the first of March being the Lords day , I conclude also that these respective days in December , were likewise on the Lords day ; and from hence collect , that Christmas day , which is always the 25 of that Month , happened on a Friday . Vses of the Projection . THis Projection is no other then a fourth part of Stoflers particular Astrolabe , fitted for the Latitude of London inverted , that is , the Summer Tropick and Altitudes , &c. turned downwards towards the Limb , whereas in his Astrolabe they were placed upwards , towards the Center ; thus the Quadrant thereof made , is rendred most useful and accurate when there is most occasion for it ; before the projection can be used , the Bead must be rectified , and because the Thread and Bead may stretch , there may be two Beads , the one set to some Circle concentrick to the Limb , to keep the other at a certainty in stretching , and the other to be rectified for use . To rectifie the Bead. LAy the Thread over the day of the Month in its proper Circle , and if the season wherein the Quadrant is to be used , be in the Winter half year , set the Bead by removing it to the Winter Ecliptick ; but in Summer let it be set to the lower or Summer Ecliptick , and then it is fitted for use , One Caution in rectifying the Bead is to be given ; and that is in Summer time if it be required to find the hour and Azimuth of the Sun by the Projection , before the hour of 6 in the morning , or after it in the evening , or which is all one , when the Sun hath less Altitude then he hath at 6 of the clock ; then must the Bead be rectified to the Winter Ecliptick , and the Parralels above the Horizon in the Reverted Tail , are those which will come in vse . To find what Altitude the Sun shall have at 6 of the clock in the Summer half year . This will be easily performed by bringing the Bead that is rectified to the Summer Ecliptick to the left edge of the Quadrant , and-there among the Paralels of Altitude it shews what Altitude the Sun shall have at 6 of the clock : It also among the Azimuths shews what Azimuth the Sun shall have at the hour of 6. Example , So when the Sun hath 17 degrees of North Declination , as about the 27 of April , his Altitude at the hour of 6 will be found to be 13d 14 m and his Azimuth from the Meridian 79d 14 m whence I may conclude if his observed Altitude be less upon the same day , and the Hour and Azimuth sought , the Bead must be set to the Winter Ecliptick , and the Operation performed in the reverted Tail. Here it may be noted also that the exactest way of rectifying the Bead , will be either from a Table of the Suns Declination , laying the Thread over the same in the graduated Circle , or from his true place , laying it over the same in the proper Ecliptick , or from his right Ascension counted in the Limb. Or Lastly from his Meridian Altitude on the right edge of the Quadrant , for these do mutually give each other the Bead , being rectified to the respective Ecliptick as before . for Example . To find the Suns Declination . The Thread laid over the day of the Moneth , intersects it upon that Circle whereon it is graduated , which in the Summer half year is to be accounted on this side the Equinoctial , North , and in the Winter-half year , South ; so laying the Thread over the 27th . day of April , it intersects the Circle of Declination at 17 degrees , and so much was the Suns Declination . To find the Suns true place . The Thread lying as before , shews it on the respective Ecliptick , So the Thread lying over the 17 of April , will cut the Summer Ecliptick , in 17d 7 m of Taurus ; or in 12d 53 m of Leo , which agrees to the 26 day of July , or thereabouts , the Thread intersecting both these days at once ; and the opposite points of the Ecliptick hereto , are 17d● m in Scorpio , about the 20 of October ; and 12d 53 m of Aquarius , about the 22d of January , all shewed at once by the Threads position . To find the Suns right Ascension . Lay the Thread over the day of the Month as before , and it intersects it in the equal Limb ; whence taking it in degrees and minutes of the Equator , whilst the Sun is departing from the Equator towards the Tropicks , it must be counted as the graduations of the Limb , from the left edge towards the right ; but when the Sun is returning from the right edge towards the left ; the right Ascension thus found , must be estimated according to the season of the year . From June 11 to Sept. 13 It must have 90 degrees added to it . Sep. 13 to Dec. 11 It must have 180 degrees added to it . Dec. 11 to Mar. 10 It must have 270 degrees added to it . But in finding the Hour of the night by the Quadrant , we need no more then 12 hours of Ascension , for either Sun or Star , and the Limb is accordingly numbred from the left edge towards the right , from 1 to 6 in a smaller figure , and thence back again to 12 , and the other figures are the Complements of these to 12. so that when the Sun is departing from the Equator towards the Tropicks ; his right Ascension is always less then 6 hours , and the Complement of it more ; but when he is returning from the Tropicks towards the Equator , it is always more then 6 hours , and the Complement of it less ; the odd minutes are to be taken from the Limb , where each degree being divided into 4 parts , each part signifies a Minute of time , and to know whether the Sun doth depart from , or return towards the Equator , is very visible , by the progress and regress of the days of the month , as they are denominated on the Quadrant . Example . So the Thread laid over 17d of Declination , which will be about the 27 April The Suns right Ascension will be 44d 37 m In time 2 h 58′ 26 July The Suns right Ascension will be 135 23 In time 9 2 20 October The Suns right Ascension will be 224 37 In time 2 58 22 January The Suns right Ascension will be 315 23 In time 9 2 But here the latter 12 hours are omitted . Such Propositions as require the use of the Bead , are , To find the Suns Amplitude , or Coast of rising and setting from the true East or West . Bring the Bead , being rectified to either of the Eclipticks , it matters not which , to either of the Horizons , and the Thread will intersect the Amplitude sought , upon both alike : Example ; The Suns Declination being 17 North , or South , the Suns Amplitude , will be found to be 28● 2m. The Amplitude before found for the Summer half year , is to be accounted from East or West Northwards ; and in the Winter half year from thence Southwards . To find the time of the Suns rising or setting . The Thread lying in the same Position , as in the former Proposition , intersects the Ascensional difference in the Limb , which may there be counted either in degrees or Time. Example . So the Bead lying upon the Horizon , being rectified to 17● of Declination , the Thread intersects the Limb at 22d 38 m , which is 1 h 30 m of time , and so it shews the time of Suns rising in Summer , or setting in Winter , to be at half an hour past 4 ; and his rising in Winter , and setting in Summer , to be at half an hour past 7. To find the length of the Day or Night . The time of the Suns rising and setting are one of them ; the Complement of the other to 12 hours ; so that one of them being known , the other will be found by Substraction ; the time of Suns setting is equal to half the length of the day ; and this doubled gives the whole length of the day ; in reference to the Suns abode above the Horizon , the time of setting converted into degrees , is also called the Semi-diurnal Ark ; the time of Sun rising ( so converted is called the Semi-nocturnal Ark ) doubled gives the whole length of the Night ; so upon the 27th day of April , the Sun having 17d of Declination , the length of the day is 15 hours , and the length of the night 9 hours . To find the Suns Altitude on all Hours ; or at any time proposed . In Summer time , if the hour proposed be before 6 in the morning , or after it in the evening ▪ lay the Thread to the hour in the Limb , the Bead being first rectified to the Winter Ecliptick , and amongst the Paralels of Altitude above the upper Horizon , it shews the Altitude sought . Example . So the Sun having 16d of declination Northwards , as about the 24th of April , laying the Thread over the Declination , I set the Bead to the Winter Ecliptick , and if it were required to find what Altitude the Sun shall have at 36 minutes past 6 in the afternoon , lay the Thread over the same in the Limb , and the Bead among the Parralels of Altitude will fall upon 7d , At all other times the Operation is alike ; the Bead being rectified to that Ecliptick that is proper to the season of the year : Lay the Thread over the proposed hour in the Limb , and the Bead amongst the Parralels of Altitude , sheweth the Altitude sought . Example . So if it were required the same day to find what Altitude the Sun should have at 19 m past 2 in the afternoon ; Lay the Thread in the Limb over the time given , and the Bead among the Parralels of Altitude will fall upon 45d for the Altitude sought . To finde the Suns Altitude on all Azimuths . IN the Summer half year , if the Azimuth propounded be more Northward then the Azimuth of the Sun shall have at the hour of 6 ; The Bead must be rectified to the Winter Ecliptick , and brought to the Azimuth proposed above the upper Horizon , and there among the Parralels of Altitude , it sheweth the Altitude sought . So about the 24th of April , when the Suns Declination is 16d his Azimuth at 6 of the clock will be found to be 76d 54 m from the South ; Then if it were required to find the Suns Altitude upon an Azimuth more remote , as upon 107d from the South , laying the Thread over the Declination , I set the Bead to the Winter Ecliptick , and afterwards carrying it to the Azimuth proposed among the Parralels of Altitude above the upper Horizon , it falleth upon 7d for the Suns Altitude sought . In all other Cases bring the Bead rectified to the Ecliptick proper to the season of the year , to the Azimuth proposed ; and among the Parralels of Altitude it sheweth the Altitude sought ; So far the same day , I set the Bead to the Summer Ecliptick , and if it were required to know what Altitude the Sun shall have when his Azimuth is 50d 48′ from the Meridian carry the Bead to the said Azimuth , and among the Parralels of Altitude it will fall upon 45d for the Altitude sought . The Hour of the night Proposed to find the Suns Depression under the Horizon . IMagine the Sun to have as much Declination on the other side the Equinoctial , as he hath on the side proposed ; and this Case will be co-incident with the former of finding the Suns Altitude for any time proposed ; the reason whereof is because the Sun is always so much below the Horizon at any hour of the night , as his opposite Point in the Ecliptick is above the Horizon at the like hour of the Day . Such Propositions as depend upon the knowledge of the Suns Altititude , are to find the Hour of the Day , and the Azimuth ( or true Coast ) of the Sun. THe Suns Altitude is taken by holding the Quadrant steady , and letting the Sun Beams to pass through both the Sights at once , and the Thread hanging at liberty shews it in the equal Limb , if this be thought unsteady , the Quadrant may rest upon some Concave Dish or Pot , into which the Plummet may have room to play ; but for greate Quadrants there are commonly Pedistalls made . The Altitude supposed to find the Hour of the Day , and the Azimuth of the Sun in Winter . REctifie the Bead to the Winter Ecliptick , and carry it along amongst the Parralels of Altitude till it cut or intersect that Parralel of Altitude on which the Sun was observed , and the Thread in the Limb sheweth the hour of the Day , and the Bead amongst the Azimuths sheweth the Azimuth of the Sun. Example . So about the 18 of October , when the Suns Declination is 13d 20′ South if his observed Altitude were 18d the true time of the day would be found to be either 36 minutes after 9 or 24 minutes past 2 and his Azimuth would be 37 degrees from the South . To finde the Hour of the Day , and the Azimuth of the Sun at any time in the Summer half year . IT was before intimated , That if the question were put when the Sun hath less Altitude then he hath at the hour of 6 of the clock , that then the Operation must be performed among those Parralels above the upper Horizon , in the reverted Tail , the Bead being rectified to the Winter Ecliptick ; and that it might be known what Altitude the Sun shall have at 6 of the clock , by bringing the Bead rectified to the Summer Ecliptick , to the left edge of the Quadrant . So admitting the Sun to have 16d of North Declination , which will be about the 24 April , I might finde his Altitude at 6 of the Clock by bringing the Bead rectified to the Summer Ecliptick to the left edge of the Quadrant ; to be 12d 28 m whence I conclude , if his Altitude be less , the Bead must be rectified to the Winter Ecliptick , and be brought to those Parralels above the upper Horizon ; and it may be noted , that the Suns Altitude at 6 is always less then his declination . Example . Admit the 24th of April aforesaid the Suns observed Altitude were 7d laying the Thread over the Suns Declination , or the day of the month ; I rectifie the Bead to the Winter Ecliptick , and bring it to the said Parralel of Altitude above the upper Horizon ; and the Thread intersects the Limb at 9d 3 m shewing the hour of the day to be 24 minutes past 5 in the morning , or 36′ past 6 in the evening , and the Bead amongst the Azimuths shews the Azimuth or Coast of the Sun to be 107d from the South . Another Example . But admitting the Sun to have more Altitude then he hath at the hour of 6 , the Operation notwithstanding differs not from the former , but only in rectifying the Bead , which must be set to the Summer Ecliptick , and then carried to the Parralel of the Suns observed Altitude , and the Thread will intersect the Limb at the true time of the day , and the Bead amongst the Azimuths sheweth the true Coast of the Sun. So upon the 24th of April aforesaid , the Suns observed Altitude being 45d , I bring the Bead rectified to the Summer Ecliptick , to the said Parralel of Altitude , and the Thread intersects the Limb at 55d 15 m shewing the hour to be either 41 m past 9 in the morning , or 19 m past 2 in the ofternoon ; to be known which by the increasing or decreasing of the Altitude , and the Bead amongst the Azimuths shews the Azimuth or true Coast of the Sun to be 50d 40● from the South : Another Example . Admit when the Sun hath 19d 13 m of North Declination which will be about the 6th of May , his observed Altitude were 56d the Bead being set to the Summer Ecliptick , and brought to that Parralel of Altitude amongst the Azimuths shews the Suns true Coast to be 23d from the South , Eastwards in the forenoon , and Westwards in the afternoon , and the Thread in the Limb sheweth the true time of the day to be either 7′ past 11 in the forenoon , or 5● m past 12. The Depression of the Sun supposed to find the true time of the night with us , or the hour of the day to our Antipodes ; As also the true Coast of the Sun upon that Depression . THis Proposition may be of use to know when the Twilight begins or ends , which is always held to be when the Sun hath 18d of Depression under the Horizon , to perform this , Imagine as much Declination on the contrary side the Equinoctial , as the Declination given , and find the time of the Day , as if the Suns Altitude were 18d So when the Suns Declination is 16d North , as about the 24th of April , laying the Thread over it I rectifie the Bead to the Winter Ecliptick , and bringing it to the Parralel of 18d the Thread in the Limb shew the Twilight to begin at 54 m past 1 in the morning , and ends at 6′ past 10 at night , and the Azimuth of the Sun to be 28d 58′ which in this Case is to be accounted from the North. But if the Suns greatest Depression at night be less then 18d as that it may be in any Latitude where the Meridian Altitude at any time in Winter or the opposite Signe is less then 18d there is no dark night which in our Latitude of London will be from the 12th of May to the 11th of July . Of the Stars graduated on the PROJECTION . SUch Stars as are between the two Tropicks only , are there inscribed , and such haue many things common in their Motion with the Sun when he hath the like Declination , as the same Amplitude , Semidiurnal , Arke , Meridian , Altitude , Ascentional difference , &c. These Stars have Letters set to them to direct to the Circle of Ascensions on the back of the Quadrant , where the quantity of their right Ascension , is expressed from one of the Equinoctial points ; those that have more Ascension then 12 hours from the point of Aries , are known by the Character plus + set to them ; many more Stars might be there inserted , but if they have more then 23d 31′ of Declination , the Propositions to be wrought concerning them are to be performed with Compasses , by the general Lines on the Quadrant . To find the true time of the Day or Night when any Star commeth to the Meridian . In the performing of this Proposition we must make use of the Suns whole right Ascension in time , which how that might be known hath been already treated of , as also of the Stars whole right Ascension , which may be had from the Circle of Ascensions on the back of the Quadrant if 12 hours be added to the right Ascension of a Star taken thence that hath a Letter Character † affixed to it . Substract the Suns whole right Ascension from the Stars whole right Ascension , encreased by 24 hours when Substraction cannot be made without it , the remainder is less then 12 shews the time of the afternoon or night when the Star will be upon the Meridian ; but if there remain more then 12 , reject 12 out of it and the residue shews the time of the next morning when that Star will be upon the Meridian . Example . The 23d of December the Suns whole right Ascension is 18 hours 53′ which substracted from 4 ho : 16′ the right Ascension of the Bulls eye encreased by 24 there remains 9 h 23′ for the time of that Stars comming to the Meridian , and being substracted from 6 ho : 30′ the right Ascension of the great Dogg , there rests 11 ho : 37′ for the time of that Stars coming to the Meridian at night . This Proposition is of good use to Sea-men , who have occasion to observe the Latitude by the Meridian Altitude of a Star , that they may know when will be a fit time for observation . In finding the time of the Night by the Stars , we use but 12 ho : of right Ascension , nor no more in finding the time of their rising or setting , so that when it is found whether it be morning or evening is left to judgement , and may be known by comparing it with the former Proposition , if there be need so to do . To find the Declination of any of these Stars . This is engraven or annexed to the Stars names , yet it may be found on the Projection , by rectifying a Bead to the proposed Star , and bringing the Thread and Bead to that Ecliptick it wil intersect ; and in the same Position the Thread will intersect the said Stars declination in the Quadrant of Declinations ; if the Bead meet with the Summer Ecliptick the Declination is North , if with the Winter South . To find the Amplitude and Ascensional difference of any of the Stars on the Projection . BRing the Bead rectified to the Star to either of the Horizons , the Thread being kept in its due Extent , and where it intersects the same it shews that Stars Amplitude which varies not , and is Northward if the Star have North declination , otherwise Southwards , the Thread likewise intersecting the Limb , sheweth the Stars Ascensional difference . Example . So the Bead being rectified to the Bulls eye , and brought to the lower Horizon , shews the Amplitude of that Star to be 25d 54′ Northwards because the Star hath North Declination ; And the Thread lyeth over 20′ 49′ of the Limb which is this Stars Ascensional difference , which in Time is 1 ho 23 m The Thread in the Limb lyeth over 4 ho 37 m from midnight for the Stars hour of rising , and over 7 ho 23 m from the Meridian for the Stars hour of setting always in this Latitude which with the Amplitude varies not , except with a very small allowance in many years . To find a Stars Diurnal Ark , or the Time of its continuance above the Horizon . When the Star hath North Declination add the Ascensional difference of the Star before found in Time to 6 hours , the Sum is half the time . South Declination Subtract . the Ascensional difference of the Star before found in Time from 6 hours , the Residue is half the time . Of that Stars continuance above the Horizon , which doubled , shews the whole time , the Complement wherof to 24 ho is the time of that Stars durance under the Horizon . Example . So the Ascensional difference of the Bulls eye being in time 1 ho : 23 added to 6 hours , and the Sum doubled makes 14 hours 46 m for the Stars Diurnal Ark or abode above the Horizon , the residue whereof from 24 is 9 ho : 14 m for the time of its durance under the Horizon . To find the true time of the Day or night , when the Star riseth or setteth . THe Stars hour of rising or setting found as before , being no other but the Ascensional difference of the Star added to , or substracted from 6 hours ; which the Thread sheweth in the Limb the Bead being rectified to a Star , and brought to that Horizon it will intersect ; is not the true time of the night ; but by help thereof that may be come by ; this we have denominated to be the Stars hour , and is no other but the Stars horary distance from the Meridian it was last upon ; If a Star have North Declination the Stars hour of rising must be reckoned to be before 6 and the time of its setting after 6 South Declination the Stars hour of rising must be reckoned to be after 6 and the time of its setting before 6 Now the time of the Stars rising or setting found by this and the former Propositions must be turned into common time by this Rule . To the Complement of the Suns Ascension add the Stars Ascension , and the Stars hour from the Meridian it was last upon , the Amount if less then 12 shews the the time of Stars rising or setting accordingly ; but if it be more then 12 reject 12 as oft as may be , and the remaind-sheweth it . So upon the 23d of December for the time of the Bulls eye rising .   h m The Complement of the Suns Ascension found by the foreside of the Quadrant is — 5 7 And the said Stars Ascension on the backside is — 4 16 The Stars hour of rising is — 4 37 14 hours . From which 12 rejected rests 2 hours for the time of that Stars rising , which I conclude to be at 2 in the afternoon , because that Star was found to come to the Meridian at 23 m past 9 at night , the like Operation must be used to get the time of that Stars setting , which will be found to be at 4 ho 46 m past in the morning .   h m Complement ☉ Ascension — 5 7 Stars Ascension — 4 16 Stars hour of setting — 7 23 16 h. 46′ To find what Altitude and Azimuth a Star that hath North Declination shall have when it is 6 hours of Time from the Meridian . REctifie the Bead to the Star , and bring the Bead and Thread to the left edge of the Quadrant , and there among the Parralels of Altitude and Azimuths it sheweth what Altitude and Azimuth the Star shall have . Example . So the Bead being set to the Bulls eye , and brought to the left edge of the Quadrant it will be found to have 12′ 17′ Altitude , and 80d 3′ Azimuth from the South , when it is 6 hours of time from the Meridian , which Proposition is afterwards used to know to which Ecliptick in some Cases to rectifie the Bead as hath likewise been intimated before . The Azimuth of a Star proposed , To find what time of the Night the Star shall be upon that Azimuth , and what Altitude it shall then have . SUpposing the Azimuth proposed to be nearer the South Meridian then that Azimuth the Star shal have when it is 6 hours from the Meridian : Bring the Bead rectified to the Star , to the proposed Azimuth , and among the Parralels of Altitude it shews that Stars Altitude , and the Thread in the Limb shews that Stars hour to be turned into common time to attain the true time sought . Example . If the question were What Altitude the Bulls eye shall have when his Azimuth is 62d 48′ from South , this being less Azimuth then he hath at 6 hours from the Meridian , the rectified Bead being brought to the Azimuth sheweth among the Parralels the Altitude to be 39d and the Stars hour shewn by the Thread in the Limb is either 8 ho : 56′ or 3 ho : 4′ from the Meridian ; then if upon the 23 of December you would know at what time the Star shall have this Altitude on this Azimuth , Change the Stars hour into common time by the former Rule . Decemb. 23 Complement of ☉ Ascension 5 h 7′ 5 h 7′   Stars Ascension — 4 16 4 16   Stars hour — 8 56 3 4     18 19 12 27 And you will find it to be at 19′ past 6 in the evening , or at 27 m past midnight . For Stars of South Declination being they have no Altitude above the Horizon at 6 ho : distance from the Meridian , the operation will be the same , void of Caution . But for Stars of North Declination when the proposed Azimuth is more remote from the South Meridian then the Azimuth of that Star 6 ho from the Meridian , another Bead must be rectified to the Winter Ecliptick , and carried to the Azimuth proposed above the upper Horizon , where amongst the Parralels it shews the Altitude sought ; and the Thread in the Limb sheweth the Stars hour to be converted into common time . Example . The Azimuth of the Bulls eye being 107d 53′ from South , which is more then the Azimuth of 6 hours , the other Bead set to the Winter Ecliptick , and carried to that Azimuth in the Tail , shews the Altitude to be 6d and the Stars hour to be 5 ho : 18′ Or 6 ho : 42′ which converted into common time , as upon the 23d of December , will be either 41 m past 2 in the afternoon , or 5 m past 4 in the morning following . h ' h ' December 23 Complement Suns Ascension 5 7 5 7   Stars Ascension — 4 16 4 16   Stars hour — 5 18 6 42 Rejecting 12 the Total is — 2 41 Or 4 5 The Hour of the night proposed to find what Altitude and Azimuth any of the Stars on the Projection that are above the Horizon shall have at that time . FIrst turn common time into the Stars hour , the Rule to do it is , To the Complement of the Stars Ascension add the Suns Ascension , and the time of the night proposed , the Aggregate if less then 12 is the Stars hour ; if more reject 12 as oft as may be , and the remainder is the Stars hour sought . So the 23 of December , at 8 a Clock 59 minutes past at night what shall be the Horarie distance of the great Dogg from the Meridian Complement of great Doggs h ' Ascension — 5 30 Suns Ascension — 6 53 Time of the night — 8 59 The Sum is , 12 rejected — 9 22 Then for Stars of South Declination , rectifie the Bead to the Star proposed , and lay the Thread over the Stars hour in the Limb , and the Bead amongst the Parralels and Azimuths , shews the Altitude and Azimuth of the Star sought . Example . So the Bead being rectified to the great Dogg , and the Thread laid over 9 ho 22′ in the Limb , the Bead will shew the Altitude of that Star at that time of the night to be 14d and its Azimuth 39d from the South . The Operation is the same for Stars of North Declination when the Stars hour found as before is not more remote from the South Meridian then 6 hours on either side . But if it be more then 6 ho distance from the Meridian as before 6 after its rising , or after it before its setting , then as before suggested , one Bead must be rectified to the Star , and brought to the Summer Ecliptick , where the Thread being duly extended , another must be set to the Winter Ecliptick , and afterwards the Thread laid over the Stars hour in the Limb , this latter Bead will shew the Stars Azimuth and Parralel of Altitude in the reverted Tail above the upper Horizon . Example . So upon the 23 of December , I would know what Azimuth and Altitude the Bulls eye shall have at 4 a Clock 5 minutes past the morning following . Time proposed 4 h 5′ Complement of Bulls eye Ascension — 7 44 Suns Asce●sion 23 of December — 6 53 12 rejected rests — 6h 42′ Proceed then and lay the Thread over 42′ past 6 and the Bead among the other Paralels in the Tail sheweth the Stars Altitude to be 6d and its Azimuth from the Meridian 107d 53′ These two Propositions have a good tendency in them to discover such Stars as are upon the Projection if you know them not , but supposing them known the Proposition of chiefest use is By having the Altitude of a Star given to find out the true Time of the night , and the Azimuth of that Star. If the Stars observed Altitude be less then its Altitude at 6 ho : distance from the Meridian ; Bring the Bead , rectified to the Star , to the Summer Ecliptick and set another Bead to the Winter Ecliptick , Then carry it to the Parralell of Altitude above the upper Horizon in the Reverted taile and there it will shew the Azimuth of that Star ; and the thread in the Limbe shews the houre . Example . So if the observed Altitude of the Bulls eye were 6d its Azimuth would be found to be 107d 53′ from the South , and its hour 42′ past 6 from the Meridian the true time would be found to be 5 minutes past 4 in the morning the 24 of December .   ho :   Complement of ☉ Ascension the 23 of December — 5 7 Stars hour — 6 42 Stars Ascension — 4 16   4 5 But for Stars that have South declination or north , when their Altitude is more then their Altitude being 6 hours from the Meridian , this trouble of rectifying two Beads is shunned ; in this Case only bring the Bead that is rectified to the Star to the Parralel of Altitude , and there among the Azimuths it will shew the Stars Azimuth , and the Thread in the Limb intersects the Stars hour sought . Example . December 11th Bulls eye Altitude 39● Azimuth from the South 62d 48 Hours from the Meridian — 8 h 56′ Complement of ☉ Ascension — 6 00 Ascension of Bulls eye — 4 16 The true time of the night was 12′ past 7 of the Clock 7 12 Another Example . The great Doggs observed Altitude being 14d his Azimuth from the South would be 39d. h m And the Stars hour from the Meridian — 9 22 Stars Ascension — 6 30 If this Observation were upon the 31 of December , the Complement of the Suns Ascension would be — 4 30   8 22 And the true time of the night 22 minutes past eight of the Clock . For varieties sake there is also added to the Book a Draught of the Projection for the Latitude of the Barbados ; in the use whereof the Reader may observe that every day when the Sun comes to the Meridian between the Zenith and the Elevated Pole , he will upon divers Azimuths in the forenoon ( as also in the afternoon ) have two several Altitudes , and so be twice before noon , and twice afternoon , at several times of the day , upon one and the same Azimuth , viz. only upon such as lye between the Suns Coast of rising and setting , and his remotest Azimuth from the Meridian , which causeth the going forward and backward of the shaddow ; but of this more hereafter , when I come to treat of Calculating the Suns Altitude on all Azimuths ; It may also be observed that the Sun for the most part in those Latitudes hath no Vertical Altitude or Depression , and so comes not to the East or West . Moreover there is added a Draught of this Projection for the Latitude of Greenland , in the use whereof it may be observed that the Sun , a good part of the Summer half year comes not to the Horizon , and so neither riseth nor sets . And that no convenient Way that this Projection can be made should be omitted , there is also one drawn in a Semi-Circle for our own Latitude , which in the use will be more facile then a Quadrant , there being no trouble before or after six in the Summer time , with rectifying another Bead to perform the Operation in the reverted Taile , neither doth the Drawing hereof occupy near the Breadth , as in a Quadrant , and so besides the ease in the use is more exact in the performance ; there being no other Rule required for rectifying the Bead , but to lay the Thread over the day of the Month , and to set the Bead to that Ecliptick the Thread intersects . A Semi-Circle is an Instrument commonly used in Surveigh , and then it requires a large Center-hole ; however this Projection may be drawn on a Semi-Circle for Surveigh , but when used at home there must a moveable round Bit of brass be contrived to stop up that great Center-hole , in which must be a small Center-hole for a Thread and Plummet to be fastned , as for a Quadrant and some have been so fitted . The Reader will meet with variety of Lines and furniture in this Book to be put in the Limb , or on other parts of the Semicircle , as he best liketh . The Projection for the Barbados & Greenland , are drawn by the same Rules delivered in the Description of the Quadrant , and so also is the Summer part of this Semi-Circle , and the Winter part by the same Rules that were given for drawing the Reverted Taile . Of the Quadrant of Ascensions . The turning of the Stars hour into the Suns hour and the the converse may be also done by Compasses upon the Quadrant of Ascensions on the back side . To turn the Stars Hour into common time , called the Suns hour . THe Arithmetical Rule formerly given is nothing but an abridgment of the Rule delivered by Mr. Gunter , and others , and the work to be done by Compasses , differeth somewhat from it , though it produce the same Conclusion which is : To get the difference between the Ascension of the Sun and the Star by substracting the less from the greater ; this remainder is to be added to the Stars hour , when the Star is before , or hath more Ascension then the Sun , but otherwise to be substracted from it , and the Sum or remainder is the true time sought . To do this with Compasses , take the distance between the Star and the Suns Ascension , and set the Suns foot to the observed hour of the Star from the Meridian it was last upon , letting the other foot fall the same way it stood before , and it sheweth the time sought , if it doth not fall off the Quadrant . If it doth , the work will be to finde how much it doth excur , and this may be done by bringing it to the end beyond which it falleth , letting the other foot fall inward , the distance then between the place where it now falleth , and where it stood before , which was at the Stars hour , is equal to the said excursion , which being taken , and measured on the other end of the Scale , shews the time sought . This trouble may be prevented in all Cases , by having 12 hours more repeated after the first 12 , or 6 hours more may serve turn if the whole 18 hours be also double numbred , and Stars names being set to the Additional hours , possibly the Suns Ascension and Star do not both fall in the same 12 hours , yet notwithstanding the distance is to be taken in the same 12 hours between the quantity of the Suns Ascension and the Stars , and to proceed therewith as before , and the Compasses will never excur ; in the numbring of these hours , after 12 are numbred they are to begin again , and are numbred as before , and not with 13 , 14 , &c. And this trouble may be shunned when there is but 12 hours by assuming any hour to be the Stars hour , with such condition that the other foot may fall upon the Line ; and the said assumed hour representing the Stars hour ; count from it the time duly in order , till you fall upon the other foot of the Compasses , and you will obtain the true time sought . To turn common time , or the Suns hour into the Stars hour . THis is the Converse of the former ; take the distance between the Star and the quantity of the Suns Ascension , and set the Star foot to the Suns hour , letting the other fall the same way it stood before , and it shews the time sought . Of the Quadrat and Shaddows . Both these as was shewed in the Description of the Quadrant , are no other then a Table of natural Tangents to the Arks of the Limb and may supply the use of such a Cannon , though not with so much exactness , all the part of the Quadrate are to be estimated less then the Radius , till you come against 45● of the Limb , where is set the figure of 1 , and afterwards amongst the shadows is to be accounted more then the Radius , and so where the Tangent is in length 2 Radii as against 63d 26 m of the Limb. 3 Radii as against 71 34 of the Limb. 4 Radii as against 75 58 of the Limb. 5 Radii as against 78 42 of the Limb. are set the figures 2 , 3 , 4 , 5 , and because they are of good use to be repeated on the other side of the Radius in the Quadrat , there they are not figured , but have only full points set to them , falling against the like Arks of the Limb from the right edge towards the left , as they did in the shadows from the left edge towards the right . To find a hight at one Observation . LEt A B represent a Tower , whose Altitude you would take , go so far back from it that looking through the sights of the Quadrant , the Thread may hang upon 45 degrees of the Limb , or upon 1 , or the first prick of the Quadrat , and the distance from the foot of the Tower will be equal to the height of the Tower above the eye , which accordingly measure , and thereto add the height of the eye above the ground , and you will have the Altitude of the Tower. So if I should stand at D and find the Thread to hang over 45d of the Limb , I might conclude the distance between my Station and the Tower to be equal to the height of the Tower above my eye , and thence measuring it find to be 96 yards , so much would be the height of the Tower above the eye . If I remove farther in till the Thread hang upon the second point of the Quadrat , then will the Altitude of the Tower above the level of the eye be third point of the Quadrat , then will the Altitude of the Tower above the level of the eye be fourth point of the Quadrat , then will the Altitude of the Tower above the level of the eye be fifth point of the Quadrat , then will the Altitude of the Tower above the level of the eye be twice thrice four times as much as the distance from the Tower is to the Station . five times as much as the distance from the Tower is to the Station . So removing to C ; I find the Thread to hang upon the second point of the Quadrat , and measuring the distance of that Station from the Tower , I find it to be 48 yards , whence I may conclude the Tower is twice as high above my eye , and that would be 96 yards . So if I should remove so much back that the Thread should hang upon 2 of the shaddows &c. as E the distance between 3 of the shaddows &c. at F the distance between 4 of the shaddows &c. 5 of the shaddows &c. the Station and the foot of the Tower would be twice thrice four times five times as much as the height of the Tower above the eye , and consequently if I should measure the distance between D and E where it hung upon 1 and 2 of the Shaddows , or between E and F , where it hung upon 2 and 3 of the shaddows , &c I should find it to be equal to the Altitude ; but other ways of doing it when inaccessible will afterwards follow . A Second way at one Station . WIth any dimension whatsoever of a competent length , measure off from the foot of the Object , whether Tower or Tree , just 10 or 100 , &c. of the said Dimensions , as suppose from B to K , I measure of an hundred yards ; there look through the sights of the Quadrant to the Top of the Object at A , and what parts the Thread hangs upon in the Quadrat or Shadows , shews the Altitude of the Object in the said measured parts , and so at the said Station at K the Thread will hang upon 96 parts , shewing the Altitude A B to be 96 yards above the Level of the eye , and so if any other parts were measured off , they are to be multiplyed by the Tangent of the Altitude , or parts cut by the Thread , rejecting the Ciphers of the Radius , as in the next Proposition . A third way by a Station at Random . TAke any Station at Random as at L , and looking through the sights observe upon what parts of the Quadrat or Shadows the Thread falls upon , and then measure the distance between the Station and the foot of the object , and the Proportion will hold . As the Radius To the Tangent of the Altitude , or to the parts cut in the Quadrat or shadows , So is the distance between the Station and the Object To the height of the Object above the eye . So standing at L , the Thread hung upon 30d 58 m of the Limb , as also upon 600 of the Quadrat , the Tangent of the said Ark and the measured distance L B was 160 yards , now then to work the former Proportion , multiply the distance by the parts of the Quadrat , and from the right hand of the product cut off three places , and you have the Altitude sought . In this Work the Radius or Tangent of 45d is assumed to be 1000. To measure part of an Altitude , as suppose from a window in a Tower to the top of the Tower , may be inferred from what hath been already said ; first get the top of the Tower by some of the former ways , and then the height of the window which substract from the former Altitude and the remainder is the desired distance between the window and the top of the Tower. The former Proportion may also be inverted for finding of a distance by the height , or apprehending the Tower to lye flat on the ground , and so the height to be changed into a distance and the distance into a height , the same Rules will serve only the height of a Tower being measured , and from the top looking to the Object through the sights of the Quadrant , what Angle the Thread hangs upon is to be accounted from the right edge of the Quadrant towards the left ; but in taking out the Tangent of this Ark , after I have observed it , the Thread must be laid over the like Ark of the Quadrant from the left edge towards the right , and from the Quadrat or Shaddows the Tangent taken out by the Intersection of the Thread , and so to measure part of a distance must be done by getting the distances of both places first , and then substract the lesser from the greater . To find the Altitude of any Perpendicular , by the length of its shadow . THis will be like the first Proposition , with the Quadrant take the Altitude of the Sun , if in so doing the Thread hang over the 1st pricks in the Quadrat , the length of the Shaddow is equal to the height of the Object Tree , or Perpendicular 2d pricks in the Quadrat , the length of the Shaddow is double the height of the Object Tree , or Perpendicular 3d pricks in the Quadrat , the length of the Shaddow is triple the height of the Object Tree , or Perpendicular 4th pricks in the Quadrat , the length of the Shaddow is four times the height of the Object Tree , or Perpendicular 5th pricks in the Quadrat , the length of the Shaddow is five times the height of the Object Tree , or Perpendicular whatever it be ; But if it hang 1 in the shaddows 2 in the shaddows 3 in the shaddows 4 in the shaddows 5 in the shaddows the highth of the object is equal to the length of the double the length of the triple the length of the four times the length of the five times the length of the Shaddow which may happen where the Sun hath much Altitude , as in small Latitudes , and so the length of the Shadow being forthwith measured , the height of the Gnomon may be easily attained . If the Thread in observing the Altitude hang on any odd parts of the Quadrant or Shadows , the Proportion will hold as before . As the Radius To the length of the Shadow , So the Tangent of the Suns Altitude , or the parts cut by the Thread To the height of the Gnomon to be wrought as in the third Proposition ; if the length of the Gnomon , and the length of its shadow were given ; without a Quadrant we might obtain the Suns Altitude , for As the length of the Gnomon Is to the Radius So is the Length of its Shadow To the Tangent of the Complement of the Suns Altitude . And the height of the Sun , and the length of the Gnomon assigned ; We may find the length of the Shadow by inverting the Proportion aforesaid . As the Radius To the length of the Gnomon So the Cotangent of the Suns Altitude To the Length of the Shadow . To find an innaccessible Height at two Stations . To work this with the Pen , Out of the Line of Sines take the Sine of 70d with Compasses , and measure it on the equal parts , where admit it reach to 94 parts , Then multiply the said number by the distance 102 , 1 and the Product will be 95 , 974 from which cutting off three figures to the right hand the residue being 95 , 974 , is the Altitude sought feré , but should be 96 caused by omitting some fractionate parts in the distance which we would not trouble the Reader withall . Another more general way by any two Stations taken at randome . ADmit the first Station to be as before at G where the observed Altitude of the Object was 70d and from thence at pleasure I remove to H , where observing again I find the Object to appear at 48d 29 m of Altitude , and the measured distance between G and H to be ●0 yards , a general Proportion to come by the Altitude in this case will hold . As the difference of the Cotangents of the Arks cut at either Station . Is to the Distance between the two Stations , So is the Radius To the Altitude of the Object or Tower. To save the substracting of the two Arks from 90d to get their Complements , I might have accounted them when they were observed from the right edge of the Quadrant towards the left , and have found them to have been 20d and 41d 31 m ; to work this Proportion lay the Thread over these two Arks in the Limb from the left edge towards the right , and take out their Tangents out of the Quadrat and Shadows , then substract the less from the greater , the remainder is the first tearm of the Proportion , being the Divisor in the Rule of three to be wrought by annexing the Ciphers of the Radius to the Distance ; or as Multiplication in Decimalls ; and then dividing by the first tearm the Quotient shews the Altitude sought , 41d 31′ Tangent from the Shadows is — 885 20d Tangent from the Quadrat is — 364 Difference — 521 By which if I divide 50000 — the distance encreased by annexing the Ciphers of the Radius thereto , the Quotient will be 96 fere the Altitude sought . This may be performed otherwise without the Pen , as shall afterwards be shewn ; If the distance from the Stations either to the foot or the top of the Tower be desired , the Proportions to Calculate them will be As the difference of the Cotangents of the Arks cut at either Station Is to the distance between those Stations , So is the Cotangent of the greater Arke to the lesser lesser Arke to the greater distance from the Station to the foot of the Tower : And so is the Cosecant of the greater Ark to the lesser distance lesser Ark to the greater distance from the top of the Tower to the eye , Or having first obtained the Height , we may shun these Secants , for As the Sine of the Ark of the Towers observed Altitude at the first Station . Is to the height of the Tower or Object above the eye , So is the Radius To the distance between the Eye and the Top of the Tower ; and by the same Proportion using the Ark at the second Station , the distance thence between the Eye and the Top of the Tower may be likewise found . If any one desire to shun this Proportion in a difference , as perhaps wanting natural Tables , it may be done at two Operations : The first to get the distance from the eye to the Top of the Tower at first Station , in regard the difference of two Arks is equal to the difference of their Complements , it will hold As the Sine of the Ark of difference between the Angles observed at each Station Is to the distance between the two Stations , So is the Sine of the Angle observed at the furthest Station , To the distance between the eye and the Top of the Object at the first Station , which being had , it then holds , As the Radius To the said Distance , So the Sine of the Angle observed at the first Station , To the Altitude of the Object . The rest of the Lines on the Quadrant are either for working of Proportions , or for Protractions , and Dyalling depending thereon ; wherefore I thought fit to reduce all the common Cases of Plain and Sphoerical Triangles to setled Cannons , and to let them precede their Application to this or other Instruments , which shall be endeavoured ; it being the cheif aim of this Book to render Calculation facil , in shunning that measure of Triangular knowledge hitherto required , and to keep a check upon it . Some Affections of Plain TRIANGLES . THat any two sides are greater then the third . That in every Triangle the greater side subtendeth the greater Angle , and the Converse . That the three Angles of every Triangle are equal to two right Angles , or to 180. That any side being continued , the outward Angle is equal to the two inward Opposite ones . That a Plain right angled Triangle hath but one right Angle , which is equal to both the other Angles , and therefore those two Angles are necessarily the Complements one of another . By the Complement of an Arch is meant the residue of that Arch taken from 90d unless it be expressed the Complement of it to 180d , to save the rehearsal of these long Words , The Tangent of the Complement , The Secant of the Complement ; others are substituted , namely the Cotangent and the Cosecant . An Obutse angular Triangle hath but one Obtuse angle , an Acutangular Triangle hath none . If an Angle of a Triangle be greater then the rest , it is Obtuse , if less Acute . The Side subtending the right Angle is called the Hipotenusal and the other two the Sides or Leggs . 1. To find a side . Given the Hipotenusal , and one of the Acute Angles , and consequently both . As the Radius , To the Sine of the Angle opposite to the side sought : So is the Hipotenusal , To the side sought . 2. To find a Side . Given the Hipotenusal and the other side . As the Hipotenusal To the given side : So is the Radius to the Sine of the Angle opposite to the given side . Then take the Complement of that Angle for the other Angle . As the Radius , to the Hipotenusal : So is the Sine of the Angle opposite , To the side sought , To the side sought 3. To find a Side . Given a Side , and one Acute Angle , and by consequence both As the Radius , To Tangent of the Angle opposite to the side sought : So is the given side , To the side sought . Or , As the Sine of the Angle opposite to the given side Is to the given side : So is the sine of the other Angle To the Side sought . 4. To find the Hipotenusal . Given one of the Sides , and an Acute Angle consequently both , As the Sine of the Angle opposite to the given side , Is to the Radius : So is the given Side , To the Hypothenusal Or , As the Radius , To Secant of the Angle adjacent to the given Side : So is the given side , To the Hipotenusal . 5. To find the Hipotenusal . Given both the Sides . As one of the given Sides . To the other given Side : So is the Radius , To the Tangent of the Angle opposite to the other side , Then take the Complement of the Angle found for the other Angle . Then , As the Sine of the Angle opposite to one of the given Leggs , or Sides , Is to the given Sides : So is the Radius , To the Hipotenusal . 6. To find an Angle . Given the Hipotenusal , and one of the Sides . As the Hipotenusal , Is to the given Side : So is the Radius , To the Sine of the Angle opposite to the given side ; The Complement of the angle found , is the other Angle . 7. To find an Angle . Given both the Sides . As one of the given sides , To the other given Side : So is the Radius , To the Tangent of the angle opposite to this other Side . The Complement of the angle found is the other Angle . Because the Sum of the Squares of the Leggs of a right angled Triangle is equal to the Square of the Hipotenusal by 47 1 Euclid . Therefore the 2d and 5th Case may be otherwise performed . 2. Case , To find a Side or Legg . Given the Hipotenusal , and other Legg . To do it by Logarithmes , Add the Logarithm of the Sum of the Hipotenusa and Legg , to the Logarithm of their difference , the half Sum is the Logarithm of the Side sought . Which in natural Numbers is to extract the Square root of the Product of the Sum and difference of the two Numbers given , namely , the Hipotenusal and Legg , which said root is equal to the side sought . 5. Case , When both the Sides are given to find the Hipotenusal . To do it by Logarithmes , Substract the Logarithm of the lesser side from the doubled Logarithm of the greater , and to the absolute number answering to the remaining Logarithm add the lesser side the half sum of the Logarithmes of the Sum thus composed , and of the lesser side , is the Logarithm of the Hypotenusa sought . Which work was devised , that the Proposition might be performed in Logarithmes ; The same Operation by natural numbers , would be to Divide the Square of the greater of the Sides by the lesser , and to the Quotient to add the lesser Side , then multiply that Sum by the lesser Side , and extract the Square root of the Product for the Hipotenusal sought . Cases of Oblique Plain Triangles . 1. Data To find an Angle , TWo sides with an Angle opposite to one of them to find the Angle Opposed to the other side . As one of the Sides , To the Sine of its opposite Angle : So the other side ; to the Sine of the angle opposed thereto . If the Angle given be Obtuse , the Side opposite to it will be greater then either of the rest , and the other two Angles shall be Acute ; But if the given Angle be Acute , it will be doubtful whether the angle opposed to the greater side be Acute or Obtuse , yet a true Sine of the 4 ●h Proportional . The Sine found will give the angle opposite to the other given side , if it be Acute , and it will always be Acute when the given angle is Obtuse , But if it be fore known to be Obtuse , the Arch of the Sine found substract from a Semicircle , and there will remain the Angle sought . In this Case the quality or affection of the angle sought must be given , and fore-known , for otherwise it is impossible to give any other then a double answer , the Acute angle found , or its Complement to 180d , yet some in this Case have prescribed Rules to know it , supposing the third Side given which is not , if it were then in any Plain Triangle it would hold , That if the Square of any Side be equal to the Sum of the Squares of the other two Sides , the angle it subtends is a right angle , if less Acute , if greater Obtuse ; but if three Sides are given we may with as little trouble by following Proportions come by the quantity of any angle , as by this Rule to know the affection of it . Two Sides with an Angle opposite to one of them , to find the third Side . In this Case as in the former , the affection of the angle opposite to the other given side , must be fore-known , or else the answer may be double or doubtful . In the Triangle annexed there is given the sides A B and A C with the angle A B C to find the angle opposite to the other given side , which may be either the angle at C or D , and the third side , which may be either B C , or B D , the reason hereof is because two of the given tearms the Side A B , and the angle at B remain the same in both Triangles , and the other given side may be either A C or A D equal to it , the one falling as much without as the other doth within the Perpendicular A E. The quality of the angle opposite to the other given side being known ; by the former Case get the quantity , and then having two angles the Complement of their Sum to 180d is equal to the third angle , the Case will be to find the said side , by having choice of the other sides , and their opposite angles as follows . 2. Two Angles with a Side opposite to one of them given to find the Side opposite to the other . As the Sine of the angle opposed to the given Side , Is to the given Side ; So is the Sine of the angle opposite to the side sought ▪ To the side sought . Ricciolus in the Trigonomerical part of his late Almagestum Novum , suggests in the resolution of this Case , that if the side opposite to the Obtuse angle be sought , it cannot be found under three Operations ; as first to get the quantity of the Perpendicular falling from the Obtuse angle on its Opposite side , and then the quantity of the two Segments of the side , on which it falleth , and so add them together to obtain the side sought ; But this is a mistake , and if it were true by the like reason a side opposite to an Acute angle , could not be found without the like trouble ; for in the Triangle above , As the Sine of the angle at B Is to its Opposite side A C or A D , So is the Sine of the angle B C A , or B D A , To its opposite Side B A , where the Reader may perceive that the same side hath opposite to it both an Acute and an Obtuse angle , the one the Complement of the other to 180d , the same Sine being common to both , for the Acute angle A C E is the Complement of the Obtuse angle B C A ; but the angle at C is equal to the angle at D being subtended by equal Sides , and the Proportion of the Sines of Angles to their opposite Sides is already demonstrated in every Book of Trigonometry . 3. Two Sides with the angle comprehended to find either of the other Angles . Substract the angle given from 180d , and there remains the Sum of the two other Angles , then As the Sum of the Sides given , To tangent of the half sum of the unknown Angles : So is the difference of the said Sides , To Tangent of half the difference of the unknown Angles . If this half difference be added to half sum of the angles , it makes the greater , if substracted from it , it leaves the lesser Angle 4. Two Sides with the Angle comprehended , to find the third side . By the former Proposition one of the Angles must be found , and then As the Sine of the angle found , is to its Opposite side : So is the Sine of the angle given , To the Side opposed thereto , If two angles with a Side opposite to one of them be given , to find the side opposite to the third Angle is no different Case from the former , because the third angle is by consequence given , being the Complement of the two given angles to 180d , and the like if two angles with the side between them were given . 5. Three Sides to find an Angle . The Side subtending the Angle sought is called the Base . As the Rectangle or Product of the half Sum of the three sides , and of the difference of the Base therefrom , Is to the Square of the Radius : So is the Rectangle of the difference of the Leggs , or conteining Sides therefrom , to the Square of the Tangent , to half the Angle sought . And by changing the third tearm into the place of the first . As the Rectangle of the Differences of the Leggs from the half Sum of the 3 sides , Is to the Square of the Radius : So is the Rectangle of the said half sum , and of the difference of the Base there from , To the Square of the Tangent of an angle , which doubled is the Complement of the angle sought to 180d , Or the Complement of this angle doubled is the angle sought . To Operate the former Proportion by the Tables . From the half Sum of the three Sides substract each Side severally then from the Sum of the Logarithmes of the Square of the Radius ( which in Logarithms is the Radius doubled ) and of the differences of the Sides containing the angle sought . Substract the Sum of the Logarithms of the half sum of the three sides , and of the difference of the Base therefrom the half of the remainder is the Logarithm of the Tangent of half the angle sought Example . In the Triangle A B C let the three sides be given to find the Obtuse angle at C. Differences from the half sum . B C 126 Leggs — 101 — 2,0043214 A C 194 Leggs — 169 — 2,2278867 A B 270 Base — 25 Logarithms — —   — 1,3979400 Sum with double Radius 24,2322081 Sum 590     half sum 295 — 2,4698220       — 3,8677620     3,8677620 —     — 20,3644461 Tangent of 56d 41′ which doubled is 113d 22′ the angle sought . 10,1822230 half For the 2d Proportion for finding an Angle the Operat on varies ; to the former sum add the double Radius , and from the Aggregate substract the latter sum , the half of the Relique is the Logarithm of the Cotangent of half the Angle sought . Tangent of 33d 19′ the Complement whereof is 56d 41′ which doubled makes 113d 22′ as before The Tables here made use of are Mr. Gellibrands . It may be also found in Sines . As the Rectangle of the Containing Sides or Leggs Is to the Square of the Radius : So the Rectangle of the differences of the said Leggs from half Sum of the three Sides , To the Square of the Sine of half the Angle sought . Or , We may find the Square of the Cosine of half the Angle sought by this Proportion . As the Rectangle of the containing Sides , Is to the Square of the Radius : So the Rectangle of the half sum of the three Sides , and of the difference of the Base there from , To the Square of the Cosine of half the Angle sought , viz. To the Square of the Sine of such an Ark , as being doubled is the Complement of the Angle sought to 180d , or the Complement of the found Ark doubled is the angle sought . All these Proportions hold also in Spherical Triangles if instead of the bare names of Sides you say the Sines of those Sides , and this Case may also be resolved by a Perpendicular let-fall ; but the Reader need not trouble himself with name nor thing in no Case of Plain or Sphoerical Triangles , but when two of the given Sides or Angles are equal . If the Square of the Radius and the Square of the Sine of an Ark be both divided by Radius the Quotients will be the Radius , and the half Versed Sine of twice that Arch , and in the same Proportion that the two tearms propounded , as will afterwards be shewed , upon this Consideration may any of these Compound Proportions be reduced to single Tearms for Instrumental Operations , by placing the two Tearms of the first Rectangle , as two Divisors in two single Rules of three , and the tearms of the other Rectangle as middle tearms . An Example in that Proportion for finding an Angle in the Sineés : As one of the Leggs or including Sides , Is to the difference thereof from the half sum of the three Sides : So is the Difference of the other Legg therefrom , To a fourth Number . Again . As the other Legg , To that fourth Number : So is the Radius , To the half Versed Sine of the Angle sought ; And so is the Diameter or Versed Sine of 180d Or Secant of 60d To the Versed Sine of the Angle sought . And upon this Consideration , that the fourth Tearm in every direct Proportion bears such Proportion to the first Tearm , as the Rectangle or Product of the two middle Tearms doth to the Square of the first Tearm , as may be demonstrated from 1 Prop. 6. Euclid . We may place the Radius or Diameter in the first place of the first Proportion , and still have the half or the whole Versed Sine in the last place as before , therefore As the Radius , To one of the given Leggs : So the other given Legg , To a fourth Number , which will bear such Proportion to the Radius , as the Rectangle of the two given Leggs doth to the Square of the Radius , then it holds , As that fourth number , To the Radius : So the Rectangle of the difference of 〈…〉 the Leggs from the half sum of the three sides ; To the Square of the Sine of half the Angle sought , and omitting the Raidius , it will hold ; As that fourth Number , To the differences of the Leggs from the Leggs from the half sum of the three Sides : So is the difference of the other Legg therefrom , To the half Versed Sine of the Angle sought . And if the Diameter had been in the first place , then would the whole Versed Sine have been in the last . This Case when 3 Sides are given to find an Angle is commonly resolved by a Perpendicular let-fall , it shall be only supposed and the Cannon will hold , As the Base or greater Side , To the sum of the other Sides : So is the difference of the other Sides , To a fourth Number , which taken out of the Base , half the remainder is the lesser Segment , and the said half added to this fourth Number is the greater Segment . Again . As the greater of the other Sides , To Radius : So is the greater Segment , To the Cosine of the Angle adjacent thereto , Or , As the Lesser of the other Sides , To Radius : So is the lesser Segment , To the Cosine of its adjacent Angle : This for either of the Angles adjacent to the Base or greatest side but for the angle opposite to the greatest side , which may be sometimes Obtuse , sometimes Acute ; not to multiply Directions , the Reader is remitted to the former Cannons , or to find both the Angles at the Base first , and by consequence the third angle is also given , being the residue of the sum of the former from a Semicircle . In right angled Spherical Triangles . By the affection of the Angles to know the Affection of the Hipotenusal , and the Converse . If one of the angles at the Hipotenusal be a right angle The Hipotenusal will be A Quadrant . If both be of the same kind , The Hipotenusal will be lesser then a Quadr. If of a different kind , The Hipotenusal will be greater then a Quadr. By the Hipotenusal to find the affection of the Leggs ▪ and the Converse : If the Hipotenusal be A Quadrant one of the Legs wil be a Quadrant If the Hipotenusal be Less then A Quadrant The Leggs , and their Opposit Angles wil be less then Quadrant If the Hipotenusal be Greater , then A Quadrant One Legg will be greater , & the other less then the Quadrant : The Leggs of a right angled Sphoerical Triangle are of the same Affection as their Opposite Angles and the Converse : If a Legg be a Quadrant the Angle opposite thereto will be a Quadr. If a Legg be less then a Quadrant the Angle opposite thereto will be Acute If a Legg be greater then a Quadrant the Angle opposite thereto will be Obtuse . All these Affections are Demonstrated in Snellius ; I shall add some more from Clavius de Astrolabio , and the Lord Nepair . The three Sides of every Sphoerical Triangle are less then a Whole Circle . In an Oblique angular Triangle , If two Acute Angles be equal the Sides opposite to them shall be lesser then Quadrants . Obtuse Angles be equal the Sides opposite to them shall be greater then Quadrants . Reg. 10 ▪ 11 , 4 If an Oblique angular Triangle , if two Acute Angles be unequal the side opposite to the lesser shall be lesser then a Quadrant Obtuse Angles be unequal the side opposite to the greater shall be greater then a Quadrant Reg. 12 , 13.4 . An Acute angular Triangle hath all its Angles Acute , and each side less then a Quadrant . Two sides of any Spherical Triangle are greater then the third . If a Sphoerical Triangle be both right angled and Quadrantal , the sides thereof are equal to their Opposite Angles . If it hath three right Angles , the three sides of it are Quadrants . If it have two right Angles , the two sides subtending them are Quadrants and the contrary , and if it have one right angle , and one side a Quadrant , it hath two right angles , and two Quadrental sides . Any side of a Sphoerical Triangle being continued , if the other sides together are equal to A Semicircle the outward angle wil be equal to the inward opposit angle on the side continued . lesser then A Semicircle the outward angle wil be lesser then the inward opposit angle on the side continued . greater then A Semicircle the outward angle wil be greater then the inward opposit angle on the side continued . If any Sphoerical Triangle have two Sides equal to lesser then greater then a Semicircle , the two angles at the Base or third Side will be equal to lesser then greater then two right Angles . In every right angled Spherical Triangle having no Quadrantal Side , the angle Opposite to that Side that is less then a Quadrant is Acute , and greater then the said Side ; But that angle which is Opposite to the Side , that is greater then a Quadrant , is Obtuse ; and less then the said Side . In every right angled Spherical Triangle all the three angles are less then 4 right angles , that is the two Oblique angles are less then 3 right angles , or 270d. In a right angled equicrural Triangle , if the two equal angles be Acute , either of them will be greater then 45d , but if Obtuse less then 135d. In every right angled Sphoerical Triangle either of the Oblique angles is greater then the Complement of the other , but less then the difference of the same Complement from a Semicircle . Two angles of any Sphoerical Triangle are greater then the difference between the third angle and a Semicircle , and therefore any side being continued , the outward angle is less then the two inward opposite angles . The sum of the three Angles of a Sphoerical Triangle is greater then two right angles , but less then 6. In Spherical Triangles , that angle which of all the rest is nearest in quantity to a Quadrant , and the side subtending it are doubtful , Whether they be of the same , or of a different affection , unless foreknown , or found by Calculation ; But the other two more Oblique angles are each of them of the same kind as their Opposite Sides , which Mr Norwood thus propounds , Two Angles of a Spherical Triangle , shall be of the same affection as their Opposite Sides , and to this purpose , If any Side of a Triangle be nearer to a Quadrant then its opposite Angle , two Angles of that Triangle ( not universally any two ) shall be of the same kind , and the third greater then a Quadrant . But if any Angle of a Triangle be nearer to a Quadrant then its opposite side , two Sides of that Triangle ( not universally any two ) shall be of the same kind , and the third less then a Quadrant . In any Sphoerical Triangle , if one of the angles be substracted from a Semicircle , and the residue so found substracted from a Whole Circle , the Ark found by this latter Substraction , will be greater then the Sum of the other two Angles . In every Sphoerical Triangle the difference between the sum of two angles howsoever taken , and a whole Circle or 4 right angles is greater then the difference between the other Angle and a Semicircle ; The demonstration of most of these Affections are in Clavins his Comment on Theodosius , or in his Book de Astrolabio , where shewing how to project in Plano all the Cases of Sphoerical Triangles , and so to measure the sides and Angles , he delivers these Theorems to prevent such Fictitious Triangles as cannot exist in the Sphere . The 16 Cases of right Angled Sphoerical Triangles , Translated from Clavius de Astrolabio . 1. To find an Angle . Given the Hipotenusal and side opposite to the Angle sought . As the Sine of the Hipotenusal , To Radius : So the Sine of the given side To the Sine of the Angle sought . Or , As Radius , To sine of the Hipotenusal : So the Cosecant of the side , To the Cosecant of the Angle . As Radius , To sine of the side : So Cosecant of the Hipotenusal , To sine of the angle . As Cosecant of the side , To Radius : So Cosecant of the Hipotenusal , To Cosecant of the angle . As Cosecant Hipotenusal , To Radius : So Cosecant of the side , To Cosecant of the angle . As the Sine of the side to Radius : So sine of the Hipotenusal , To Cosecant of the Angle . The Angle found will be Acute if the Side given be less then a Quadrant , Obtuse if greater . 2. To find an Angle . Given the Hipotenusal and side adjacent to the Angle . As Radius , To Cotangent Hipotenusal : So tangent of the side , To Cosine of the angle . As tangent Hipotenusal . To Radius , So tangent of the Side . To Co-sine of the angle . As the Cotangent of the side , To Radius : So Cotangent Hipotenuse , To Cosine of the angle . As Radius , To Cotangent side : So tangent Hipotenusa , To Secant of the angle . As Cotangent Hipotenusal : To Radius : So Cotangent side , To Secant of the angle . As the tangent of the side , To Radius : So tangent Hipot : To Secant of the angle . The Angle found will be Acute , if both the Hipotenusal and the given side be greater , or less then a Quadrant , but Obtuse if one of them be greater , and the other less . 3. To find an Angle . Given the Hipotenusal , and either of the Oblique Angles As the Radius , To Cosine Hip : ∷ So Tangent of the given angle : To the Cotang of sought angle As the Cotangent of the given angle : To Radius ∷ So Cosine of : the Hipot : To the Cotang of the sought angle . As the Cosine of the Hipot : To Radius ∷ So Cotang : of the given angle : To the tang . of the ang : sought As the Radius , To Secant Hip : ∷ So Cotangent of the given angle : To the tang of the angle sought As Secant of the Hipotenusal : To Radius ∷ So the tangent of the angle given : To the Cotan : of the angle sought As the tangent of the given angle : To Radius ∷ So Secant of the Hipotenusal : To tangent of the angle sought . The angle found will be Acute , if the Hipotenusal be less then a Quadrant , and the given angle Acute ; or if the Hipotenusal be greater then a Quadrant , and the given angle Obtuse ; And the said angle will be Obtuse , if the Hipotenusal be less then a Quadrant , and the given angle Obtuse ; Of if the Hipotenusal be greater then a Quadrant , and the given angle Acute . 4. To find an Angle . Given the Side opposite to the Angle sought , and the other Oblique Angle . As the Radius : To sine of the angle given ∷ So is the cosine of the given side : To the Cosine of the angle sought As the Radius : To Cosecant of the given angle ∷ So Secant of the side given : To Secant of the angle sought As the sine of the given angle : To Radius ∷ So the Secant of the given side : To the Secant of the angle As the Cosine of the given side : To Radius ∷ So the Cosecant of the given angle : To the Secant of the angle sought As the Cosecant of the given angle : To Radius : So Cosine of the given side : To the Cosine of the angle sought As the Secant of the given side : To Radius ∷ So sine of the given angle : To the Cosine of the angle sought The angle found will be Acute , if the side given be less then a Quadrant , Obtuse if greater . 5. To find an Angle . Given a side adjacent to the angle sought , and the other Oblique angle , if it be foreknown whether the angle sought be Acute or Obtuse , or whether the Base or other side not given be greater , or lesser then a Quadrant . As the Cosine of the given side : To Radius ∷ So Cosine of the given angle : To the sine of the angle sought As the Radius : To Secant of the given side ∷ So Cosine of the given angle : To Sine of the angle sought As the Radius : To Secant of the given angle ∷ So Cosine of the given side : To Cosecant of sought angle As the Cosine of the given angle : To Radius ∷ So Cosine of the given side : To the Cosecant of the angle sought As the Secant of the given side : To Radius ∷ So is the Secant of the given angle : To the Cosecant of the angle sought As the Secant of the given angle : To Radius ∷ So is the Secant of the given side : To the Sine of the angle sought The angle found will be Acute , if the side not given be less then a Quadrant , Obtuse if greater . In like manner if the Hipotenusal be less then a Quadrant , and the given angle Acute ; Or if the Hipotenusal be greater then a Quadrant , and the given angle Obtuse , the angle found will be Acute . But if the Hipotenusal be less then a Quadrant , and the given angle Obtuse , or if the Hipotenusal be greater then a Quadrant , and the given angle Acute , the angle found will be Obtuse , 6. To find an Angle . Given both the Leggs . As Radius : To sine of the side adjacent to sought angle ∷ So Cotang of the side opposite to the angle sought : To Cotang of the angle sought As the sine of the side adjacent to sought angle : To Radius ∷ So the tangent of the side opposite to the sought angle : To the tang of the angle sought : As the Tangent of the side opposite to sought angle : To Radius ∷ So the sine of the side adjacent to the sought angle : To Cotang of the angle sought : As the Radius : To Cosecant of the side adjacent to sought angle ∷ So the tangent of the side opposite thereto : To the tang of the angle sought : As the Cosecant of the side adjacent to the angle sought To Radius ∷ So is the Cotang of the side opposite to the angle sought : To Cotang of the sought angle : As the Cotangent side opposite to the sought angle : To Radius ∷ So Cosecant of the side adjacent to the angle sought : To the tang of the angle sought The Angle found will be Acute , if the side opposite to the angle sought be less then a Quadrant , but Obtuse if greater . 7 , To find a Side or Legg . Given the Hipotenusal and the other Legg . As the Cosine of the given side : To Radius ∷ So Cosine of the Hipotenusal : To Cosine of the side sought As Radius : To Secant of the side ∷ So Cosine of the Hipotenusal : To Cosine of the side sought As Cosine Hipotenusal : To Radius ∷ So Cosine of the given side : To Secant of the side sought As Radius : To Secant of the Hipotenusal ∷ So Cosine of the given side : To Secant of the side sought As the Secant of the Hipotenusal : To Radius ∷ So Secant of the given side : To Cosine of the side sought As the Secant of the given side : To Radius ∷ So Secant of the Hipotenusal : To the Secant of the side sought The side sought will be less then a Quadrant , if both the Hipotenusal and given sides be less then Quadrants , but greater then a Quadrant if either the Hipotenusal be greater , and the given side less , Or if the Hipotenusal be less , and the given side greater . 8. To find a Side . Given the Hipotenusal , and an Angle opposite to the side fought . As the Radius : To sine of the Hipotenusal ∷ So sine of the given angle : To the sine of the side sought As Radius : To Cosecant of the Hipotenusal ∷ So the Cosecant of the given angle : To Cosecant of the side sought As the sine of the Hipotenusal : To Radius ∷ So Cosecant of the given angle : To Cosecant of the side sought As the Cosecant of the given angle ▪ To Radius ∷ So the sine of the Hipotenusal : To the sine of the side sought As the sine of the given angle : To Radius ∷ So the Cosecant of the Hipotenusal : To the Cosecant of the side sought The side found will be less then a Quadrant , if the Angle opposite thereto be Acute , but greater if Obtuse . 9. To find a Side . Given the Hipotenusal and Angle adjacent to the Side sought As the Radius : To Cosine of the given angle ∷ So tangent of the Hipotenusal : To the tangent of side sought As the Cosine of the given angle : To Radius ∷ So Cotangent of the Hipotenusal : To Cotang of the side sought As the Cotangent of Hipotenusal : To Radius ∷ So the Cosine of the given angle : To tangent of the side sought As the Radius : To Secant of the angle ∷ So Cotangent of the Hipotenusal : To Cotang of the side sought As the Secant of the given angle : To Radius ∷ So tangent of the Hipotenusal : To the tangent of the side sought As the tangent of the Hipotenusal : To Radius ∷ So Secant of the given angle : To Cotang : of the side sought The Side sought will be less then a Quadrant , if the Hipotenusal be less then a Quadrant , and the given angle Acute : Or if the Hipotenusal be greater then a Quadrant , and the given angle Obtuse ; But it will be greater then a Quadrant , if the Hipotenusal be less then a Quadrant , and the given angle Obtuse : Or if the Hipotenusal be greater then a Quadrant , and the given angle Acute . 10. To find a Side . Given a Side , and an Angle adjacent to the sought side . Provided it be foreknown whether the side sought be greater or less then a Quadrant , or whether the other angle not given be Acute or Obtuse ; or finally whether the Hipotenusal be greater or less then a Quadrant . As the Radius : To the Cotangent of the given angle ∷ So tangent of the given side : To the Sine of the side sought As the tangent of the given angle : To Radius ∷ So tangent of the given side : To the sine of the side sought As the Cotang of the given side : To Radius ∷ So the Cotang of the given angle : To the sine of the side sought As Radius : To Cotangent of the given side : So tangent of the given angle : To Cosecant of the side sought As the tangent of the given side : To Radius ∷ So tangent of the given angle : To the Cosecant of the side sought As the Cotangent of the given angle : To Radius ∷ So Cotang of the given side : To the Cosecant of the side sought The Side found will be less then a Quadrant , if the angle opposite thereto and not given be Acute , but greater if it be Obtuse ; In like manner it will be less , if the Hipotenusal be less then a Quadrant , and the side given also less then a Quadrant : Or if the Hipotenusal be less then a Quadrant , and the given side greater , the side found will be greater then a Quadrant ; Lastly , if both the Hipotenusal , and the side given be greater then a Quadrant , the side found will be less then a Quadrant , but greater if the Hipotenusal be greater then a Quadrant , and the given side less . 11. To find a Side . Given a Side , and an Angle opposite to the Side sought As Radius : To sine given side ∷ So tangent given angle : To tangent of the side sought As the sine of the given side : To Radius ∷ So Cotangent of the given angle : To Cotangent of the side sought As the Cotangent of the given angle : To Radius ∷ So sine of the given side : To the tangent of the side sought As the tangent of the given angle : To Radius ∷ So Cosecant of the given side : To Cotangent of the side sought As the Radius : To Cosecant of the given side ∷ So Cotang of the given angle : To Cotangent of the side sought As the Cosecant of the given side : To Radius ∷ So tangent of the given angle : To tangent of the side sought The side found will be less then a Quadrant , if the given Angle opposite thereto be Acute , but greater if Obtuse . 12. To find a Side . Given both the Oblique Angles . As the sine of the angle adjacent to side sought : Is to Radius ∷ So Cosine of the angle opposite to side sought : To Cosine of the side sought As the Radius : To Secant of the angle opposite to side sought ∷ So sine of the angle adjacent to the side sought : To Secant of the side sought As Radius : To Cosecant of the angle adjacent to side sought ∷ So Cosine of the angle opposite to side sought : To Cosine of the side sought As the Cosine of the angle opposite to side sought : To Radius ∷ So side of the angle adjacent to the side sought : To Secant of the side sought As the Secant of the angle opposite to side sought : To Radius ∷ So Cosecant of the angle adjacent to the side sought : To Cosine of the side sought As Cosecant of the angle adjacent to side sought : To Radius ∷ So Secant of the angle opposite to side sought : To Secant of the side sought The side found will be less then a Quadrant , if the given angle Opposite thereto be Acute , but greater if Obtuse . 13. To find the Hipotenusal . Given a side and an Angle adjacent thereto . As the Radius : To Cosine of the given angle ∷ So Cotangent of the given side To Cotang of the Hipotenusal As the Cosine of the given angle : To Radius ∷ So tangent of the given side : To tangent of the Hipotenusal As the tangent of the given side To Radius ∷ So Cosine given angle : To Cotangent of the Hipotenusal As Radius : To Secant of the given angle ∷ So tangent of the given side ; To the tangent of the Hipotenusal As the Secant of the given angle : To Radius ∷ So Cotang of given side : To the Cotang of the Hipotenusal As the Cotangent of given side : To Radius ∷ So Secant of the given angle : To tangent of the Hipotenusal The Hipotenusal found will be less then a Quadrant , if the given side be less then a Quadrant , and the angle given adjacent thereto Acute ; As also if the given side be greater then a Quadrant , and the given angle adjacent thereto be Obtuse . But it will be greater then a Quadrant , if the given side be greater then a Quadrant , and the given Angle adjacent thereto Acute ; As also when the given side is less then a Quadrant , and the given Angle Obtuse . 14. To find the Hipotenusal Given a Side , and an angle Opposite thereto . If it be fore-known whether the Hipotenusal be greater or less then a Quadrant , or whether the other angle not given be Acute or Obtuse ; Or lastly , whether the other side not given , be greater or less then a Quadrant . As the sine of the given angle : To Radius ∷ So sine of the given side : To the sine of the Hipotenusal As the Radius : To Cosecant of angle given ∷ So sine of the given side : To the sine of the Hopotenusal As Radius : To Cosecant given side ∷ So sine of the given angle : To Cosecant of the Hipotenusal As the sine of the given side : To Radius ∷ So sine of the given angle : To Cosecant of the Hipotenusal As Cosecant of given side : To Radius ∷ So Cosecant of the given angle : To sine of the Hipotenusal As Cosecant of the given angle : To Radius ∷ So the Cosecant of the given side : To Cosecant of the Hipotenusal The Hipotenusal found will be less then a Quadrant , if both the Oblique Angles be Acute or Obtuse , or if both the sides be greater or less then Quadrants . It will also be greater then a Quadrant , if one of the Oblique Angles be Acute , and the other Obtuse , or if one of the sides be less and the other greater then a Quadrant : 15. To find the Hipotenusal , Given both the sides , distinguished by the names of first and second . As Radius : To Cosine 1st side ∷ So Cosine 2d side : To Cosine of the Hipotenusal As Radius : To Secant 1st side ∷ So Secant 2d side : To Secant of the Hipotenusal As Secant 1st side : To Radius ∷ So Cosine of the 2d side : To Cosine Hipotenusal As Secant 2d side : To Radius ∷ So Cosine 1st side : To Cosine of the Hipotenusal As Cosine 1st side : To Radius ∷ So Secant 2d side : To Secant of the Hipotenusal As Cosine 2d side : To Radius ∷ So Secant 1 side : To the Secant of the Hipotenusal The Hipotenusal found will be less then a Quadrant , if both the Sides are less or greater ; But otherwise , it will be greater , if one be less and the other greater . 16. To find the Hipotenusal . Given both the Oblique Angles , distinguisht by the names of the first and second . As Radius : To Cotangent 1st angle ∷ So Cotangent 2d angle : To Cosine of the Hipot : As the tangent 1st angle : To Radius ∷ So Cotangent 2d angle : To Cosine of Hipoten : As tangent of 2d angle : To Radius ∷ So Cotangent 1st angle : To Cosine Hipotenusal As the Radius : To tangent 2d angle ∷ So tangent 1st angle : To Secant of the Hipot : As Cotangent 2d angle : To Radius : So tangent 1st angle : To Secant of the Hipotenus ; As the Cotangent 1st angle : To Radius ∷ So Tangent 2d angle : To Secant Hipotenus : The Hipotenusal found will be less then a Quadrant , if both the Oblique angles be Acute or Obtuse , but greater if one of them be Acute , and the other Obtuse . I shall not spend time to shew Examples of all these Cases , but shall onely instance in an Example or two . In the Right angled Sphoerical Triangle P S N , let the side P N represent the Poles height , the side S P the Complement of the Suns declination , the side S N the Suns Amplitude of rising from the North Meridian , the Angle S P N the time of Suns rising from Midnight , and the angle P S N the angle of the Suns Position ; and in it let there be given the side P N 51d 32′ , and the side P S the Complement of the Suns Declination to find the angle S P N the time of the Suns rising ; then in this Case there is given the Hipotenusal , and the side adjacent to the angle sought to find the said angle ; and this is the 2d Case , whence the Proportion taken is , As Radius , To Cotangent of the Hipotenusal : So the Tangent of the given side , To the Cosine of the angle sought ; and so the Proportion to find the time of Suns rising will be As the Radius , To the tangent of the Suns declination : So the tangent of the Latitude , To the Sine of the time of Suns rising before 6 in Summer or after it in Winter , the Complement whereof is the time of its rising from Midnight . Tangent 13d Suns declination — 936336 Tangent 51d 32′ the Latitude — 1,009922 Sine 16d 53′ — , 946328 the Compl : of which Ark is 73 , 6 , which converted into time shews that the Sun riseth in our Latitude when he hath 13d of North Declination at 52● and a half past 4 in the morning ferè . By the same Data we may find the Side S N the Suns Amplitude of rising or setting , and this will agree with the 7 ●h Case ; for here is given the Hipotenusal , and one of the Leggs to find the other Leg the Proportion will be , As the Cosine of the given side : To Radius : So the Cosine of the Hipotenusal , To the Cosine of the side sought ; that is in this Cass , As the Cosine of the Latitude , To Radius : So is the Sine of the Suns Declination , To the Sine of his Amplitude from the East or West . Example , Logme . Sine 13d † Radius is — 1,935208 Sine 38d 28′ — 979383 Sine 21d 12′ — 955825 the Complement wherof , viz 68d 48′ is the side S N sought , and this Proportion is of good use to obtain the Variation of the Compass at Sea by the Suns Coast of rising ; More Examples need not be given , the Reader may try over all the Cases by the Calculated Triangle annexed . Some may say here are more Proportions then needs , especially seeing there are no Logarithmical Tables of Secants ; but Alterna amant Camenae , they have not hitherto been published in English ; the Instruments to be treated of will have Secants ; besides in some Cannons there are Tables of the Arithmetical Complements of the Logarithmical Sines and Cosines , which augmented by Radius , are the Logarithmical Secants of the Complements of those Arks to which they do belong ; and for Instruments , especially Quadrants , a Proportion having Tangents or Secants many times cannot be Operated on the Quadrant without changing the Proportion , by reason those Scales cannot be wholy brought on , being infinite ; Now the chief Grounds for varying Proportions , are built upon a few Theorems . 1. That the Rectangle or Product of a Tangent , and its Complement is equal to the Square of the Radius , or which is all one , that the Radius is a mean Proportional between the Tangent of an Arch , and the Tangent of its Complement , that is , As the Tangent of an Arch , To Radius : So Radius , To tangent of that Arks Complement , And by Inversion . As the Cotangent of an Arch , To Radius : So Radius , To tangent of that Arch , that is , As the 4 h tearm to 3d , So second to first . 2. That the Radius is a mean Proportional between the Sine of an Arch , and the Secant of that Arks Complement . That is , As the Sine of an Arch , To Radius : So is the Radius , To Secant of that Arks Complement , and the Converse . 3. That the Rectangles of all Tangents and their Complements , being respectively equal to the Square of the Radius , are Reciprocally Proportional , That is , As the Tangent of an Arch or Angle : Is to the Tangent of another Arch or Angle : So is the tangent of the Complement of the latter Arch : To the tangent of the Complement of the former , And by varying the Second Tearm into the place of the Third , we may compare the Tangent of one Ark to the Cotangent of another , &c. that is , As the tangent of an Ark or Angle : Is to the Cotangent of another Ark ∷ So is the tangent of this latter Ark To the Cotangent of the former . 4. That the Sines of Arches , and the Secants of their Complements are reciprocally proportional , that is , As the Sine of an Arch : To the Sine of another Arch or Angle : So is the Cosecant of the latter Arch , To the Cosecant of the former , And by changing the 2 and 3 Tearms , a Sine may be compared with a Secant . Now hence to be directed to vary Proportions , observe that if 4 Tearms or Numbers are Proportional , it is not material which of the two middle Terms be in the second or third place ; for instance if it be , As 2 to 4 ∷ So is 3 to 6 : It will also hold , As 2 to 3 ∷ So 4 to 6. Secondly , that when 4 Tearms are in direct Proportion , if a question be put concerning a fifth Tearm not ingredient in the Proportion , it is not material whether the two former , or the two latter Tearms be taken : As if it should be demanded ; When 2 yards of Linnen cost 4sh . What shall 8 yards ? Answer , 16. It might as well be said , If 3 cost 6 , What 8 ? Answer , 16. Hence then in any Proportion , if the two first Tearms be , As the Tangent of an ark , To Radius , to bring the Radius into the first place , it may be said , As the Radius , Is to the Cotangent of that Ark , because there is the same Proportion between these two latter Tearms , as between two former ; Now in all the former Theorems , the two latter Tearms consist either of the parts , or of the Complements of the parts of the two former , whence it will not be difficult to vary any Proportion propounded . 1. From whence it will follow , that a Proportion wholly in Tangents may be changed into their Complements without altering the Order of the Tearms , and the Converse . If it were As Tangent 10d , To tang 20d : So tang 52d , To tan 69d 15′ It would also be , As tang 80d , To tan 70d : So tan 38d , To tan 20d 45′ 2. That if the two latter Tearms of any Proportion being Tangents are only changed into their Complements , it infers a Transportation of the first Tearm into the second place . That is in the first Example , As Tang 20d , To Tangent 10d : So Tangent 38d , To Tangent 20d 45′ . 3. That if the two former Tearms of a Proportion being Tangents are changed into their Complements , it likewise infers a changing of the third Tearm into the place of the fourth . And then if the fourth Tearm be sought , it will hold , As the second Tearm , To the first : So is the third Tearm , as at first propounded to the fourth . In the first Example , as tangent 70d to tangent 80d : So tang ●2 to tang 69d 15′ . 4. That a Proportion wholly in Secants may be changed into a Proportion wholly in Sines , without altering the Order of the places , only by taking their Complements , and the Converse . If it were , As Secant of 80 To Secant 70d ∷ So Secant 60d To Secant 10d It would also hold in Sines , As the Sine 10 to Sine 20d ∷ So the Sine of 30d To Sine 80d 5. That if the two latter Tearms being Secants , should be changed into Sines , and the Converse , if they were Sines to be turned into Secants , it will be done only by taking their Complements , but then must the second and first Tearms change places one with another . If the Proportion were , As Sine 12d to Sine 42d ∷ So is the Secant of 36d to Secant of 75d 26′ . It would also hold , As Sine 42d to Sine 12d ∷ So Sine of 54d to the Sine of 14d 34′ . 6. That if the two former Tearms of a Proportion in Secants , should be changed into Sines and the Converse ; this would infer a changing of the fourth Term of that Proportion into the place of the Third : But the third Tearm not being that which is sought : The Rule to do it , would be to imagine the two first Tearms to change places , and then to take their Complements . If the Proportion were : As Secant of 39d to Secant of 75d 26′ So is the Sine of 12d to the Sine of 42d. It would also hold , As Sine 14d 34′ , to Sine of 52d. So is the Sine of 12d , to the Sine of 42d. 7 Two Tearms whether the former or latter in any Proportion being as a Sine to a Tangent , may be varied . For , As the Tangent of an Arch , To the Sine of another Arch : So is the Cosecant of the latter Arch , To the Cotangent of the former . And by transposing the Order of the Tearm . As a Sine , To a Tangent : So the Cotangent of the latter Arch , To the Cosecant of the former . This will be afterwards used in working Proportions on the Instrument , and there Instances shall be given of it . 8. Lastly , Observe that if 4 Tearms or Numbers are Proportional , their Order may be so transposed , that each of those Tearms may be the last in Proportion ; and so of any 4 Proportional Tearms , if there be given , the other that is unknown may be found , Thus , As first to second ∷ So third to fourth . As second to the first ∷ So the fourth to the third : As the third to the fourth : , So the first to the second . As the fourth , To the third ∷ So the second , To the first . Cases of Oblique Sphoerical Triangles . 1. TWo Sides together less then a Semicircle with the Angle comprehended given to find one of the other Angles . At two Operations they may be both found by a Proportion demonstrated in the late Trigonometry of the Learned Mr. Oughtred . As the Sine of half the sum of the sides , To Cotangent of half the contained angle : So the sine of half the difference of the sides , To the Tangent of half the difference of the other angles . Again , As the Cosine of half the sum of the sides , To Cotangent of half the contained angle : So the Cosine of half the difference of the sides , To the Tangent of half the sum of the other angles . Add the half difference to the half sum , and you have the greater Angle ; but substracted from it , and there remains the lesser angle . If the sum of the two given Sides exceeds a Semicircle , the Opposite Triangle , must be resolved instead of that propounded . Here note that evey Sphoerical Triangle hath opposite to each angular Point , another Triangle , having the side that subtends the said Angle common to both , and the angle opposite thereto equal , the other parts of it are the Complements of the several parts of the former to a Semicircle . So if in the Triangle B C D there were given the sides B C , and C D with their contained Angle B C D to find the Angle C B D because these two sides are greater then a Semicircle , resolve the opposite Triangle C A D , in which there will be given C A , which may be the Complement of the Latitude 38d 28″ , and C D the Complement of the Altitude 83d with the angle A C D , the Suns Azimuth from the North 73d to find the angle C A D the hour from Noon . C A 38d : 28′ Sides , C D 83 : 00 Sides , 121 : 28′ sum 60 : 44 half sum Logarithms Logarith . 44 : 32 difference 22 : 16 difference , Sine 957854 Cosine — 996634 36 : 30 half the Angle 53 : 30 Complement , Tang : 1013079 Idem — 1013079 1970933 2009713 Sine of 60d 44′ half sum 994069 Cosine 968919 Tang : of 30d 24′ ½ — 976864 tan 68d 39′ 1040794 Sum 99 : 3 hour , in Time 36′ before 6 in the morning , or as much after it in the afternoon , difference 38d 15′ Angle of ☉ position . 2 , Two Angles together less then a Semicircle with the side between them , alias , the Interjacent side , To find one of the other sides . This is but the Converse of the former to be performed at two Operations to get them both , and the Proportion thence applyed by changing the sides into Angles . As the Sine of the half sum of the angles , To the Sine of half their difference : So is the Tangent of half the interjacent side , To the Tangent of half the difference of the other sides . Again . As the Cosine of the half sum of the angles , To the Tangent of half the interjacent side : So the Cosine of half their difference , To the Tangent of the half sum of the other sides . If half the difference of the sides be added to half the sum of the sides , it makes the greater side ; but substracted from it , leaves the lesser . If the Sum of the two given Angles exceeds a Semicircle , then , as in the former Case , resolve the Opposite Triangle . So in the Triangle Z P ☉ if there were given the angle ☉ Z P , the Suns Azimuth from the North 63d 54′ , and the hour from Z P ☉ 105● in time 5 in the morning , or 7 in the evening , and the Complement of the Latitude Z P 38d 28′ , to find the Complement of the Altitude Z ☉ 80d 31′ , or the Complement of the Declination ☉ P 66● 29′ , two Operations finds both , and neither with less . Example . Angle Z P ☉ — 105d , 00d P Z ☉ — 63 : 54 difference — 41d 6′ half difference — 20 : 33 Sine — 954533 Cosine 997144 half the side Z P — 19 : 14 Tang — 954268 Idem 954268 Sum of the 〈…〉 — 168 : 54 1908801 1951412 half sum — 84 : 27 Sine — 999796 Cosine 898549 Tangent — 7 : 1′ 909005 ta 73● 30′ 1052863 73 : 30 Sum — 80 31 the greater side Z ☉ Difference — 66 29 the lesser side ☉ P 3. Two sides with an Angle opposite to one of them given , To find the Angle opposite to the other , its Affection being fore-known . As the Sine of the side opposite to the angle given Is to the Sine of its Opposite angle : So is the sine of the side opposite to the angle sought , To the sine of its opposite angle . Here note , that the same Sine is common to an Arch , and to its Complement to 180 , if the Angle sought be foreknown to be Obtuse , substract the Arch found from 180● and there remains the angle sought . Example . So in the former Triangle , if there were given the side ☉ P 66d 29′ the Complement of the Declination with its opposite angle P Z ☉ 63d 54′ , the Suns Azimuth from the North , and the side Z ☉ , the Complement of the ☉ Altitude 80 31′ , the Angle Z P ☉ the hour from Noon would be found to be 105. Sine 63d 54′ — 995329 80 31 — 999402 1994731 Sine 66d 29′ — 9906234 Sine 75d — 998497 The Complement of 75d is the angle sought , being 105d , and so much is the hour from Noon . In some Cases the Affection of the angle sought cannot be determined from what is given ; Such Cases are , When the given Angle is Acute , and the opposite Side less then a Quadrant , and the adjacent or other Side greater then the opposite Side , and its Complement to a Semicircle also greater then the opposite Side . Also when the given Angle is Obtuse , and the opposite Side greater then a Quadrant , and also greater then the other side , and greater then the Complement of the said other Side to a Semicircle . In all other Cases the Affection of the Angle sought may be determined from what is given ; in these it cannot without the help of the third side ( or something else given ) Where Cases are thus doubtful , there can be but a double answer , and both true ; wherefore find the Acute Angle and its Complement to 180d and the like answer give in Case 4 ●h , 5 ●h , 6th , 7th and 8th following . 4 , Two Angles with a Side opposite to one of them being given , To find the Side opposite to the other , its Affection being foreknown . As the Sine of the angle opposite to the given side , Is to the Sine of the given Side : So is the Sine of the angle opposite to the side sought , To the Sine of the side sought . If the side sought be foreknown to be Obtuse , the Complement of the Ark found to 180 will be the side sought . Example . So in the former Triangle , if there were given the angle at Z the Suns Azimuth from the North 63d 54′ , and the Complement of the Suns Declination ☉ P 66d 29′ with the hour from Noon Z P ☉ to find the Side Z ☉ the Complement of the Suns Altitude , it would be found to be 80d 31′ , and the Altitude it self 9d 29′ . Sine 66d 29′ — 996234 Sine 105 that is of 75d is — 998497 1994731 Sine 63d 54′ is — 995329 Sine 80d 31′ — 999402 In some Cases the Affection of the Side sought cannot be determined from what is given ; Such Cases are , When the given Angle is Acute , and the opposite Side less then a Quadrant , and the other Angle greater then the former Angle , and its Complement to a Semicircle also greater then the said former Angle . Also when the given Angle is Obtuse , and the opposite Side greater then a Quadrant , the other Angle being less then this Angle , and its Complement to a Semicircle also less then this Angle : What Snellius hath spoke concerning these doubts , is in some Cases false , in others impertinent , however I conceive not that Learned Author mistakes , but the Supervisors after his death . In all other Cases the determination is certain , as may be hereafafter shewed . 5. Two sides with an Angle opposite to one of them being given , To find the third side , the kind of the angle opposite to the other side being foreknown . First find the Angle opposite to the other side by 3d Case , and then you have two Sides and their opposite Angles . To find the third side by the Inverse of either of the Proportions used in the 2d Case , the former will be , As the Sine of half the difference of the angles given , To tangent of half the difference of the sides given : So is the sine of half the sum of those angles , To the tangent of half the side required . In the latter Case , if the sum of the given Angles exceed a Semicircle , the opposite Triangle must be resolved . Example . If in the former Triangle there were given the side ☉ P , the Complement of the Declination 66d 29′ and angle ☉ Z P , the Azimuth from the North 63d 54′ with the side Z P , the Complement of the Latitude 38d 28′ , to find the side ☉ Z , the Complement of the Suns Altitude on the Proposed Azimuth : The first Operation will be to find the Suns angle of Position Z ☉ P 37d 32′ , which is always Acute when the Sun or Stars do not come to the Meridian between the Zenith and the elevated Pole. The said angle being found by the former Directions , we proceed to the second Operation . Sides 66 29 difference 28d 1′ half 14d 00′ 30′ Tang — 939705 Sides 38 28 difference 28d 1′ half 14d 00′ 30′ Tang — 939705 Angles 63 54 Sum 101d 26 , half 50d 43 Sine — 988875 Angles 37 32 Sum 101d 26 , half 50d 43 Sine — 988875 1928580 26 22 difference , half 13d 11′ Sine — 935806 Tangent of 40d 15′ 30″ — 992774 doubled is 80d 31′ the side sought being the Complement of the Suns Altitude . 6. Two sides with an angle opposite to one of them being given , To find the angle included , or between them , the species of the opposite to the other side being foreknown . First find the angle Opposite to the other side by 3d Case , and then we have two angles and their opposite sides to find the other angle , by the Inverse of either of the Proportions used in the first Case , the former will be , As the sine of halfe the difference of the sides , To the Tangent of halfe the difference of the angles : So is the sine of halfe the sum of the sides . To the Cotangent of half the angle required ; That is , to the Tangent of an Ark , whose Complement is half the angle inquired . If the sum of the given sides be more then a Semicircle , in the resolution of this latter Case resolve the Opposite Triangle . Example . In the former Triangle given ☉ P Comple : Declination 66d 29′ Z P Comple : Latitude — 38 28 Angle ☉ Z P the Azimuth — 63 54 To find the hour Z P ☉ — 105 The first operation wil find the angle of Position as before 37 d 32′ The second Operation . half difference of the given angles 13 d 11 m Tangent — 936966 half sum of the side — 52 , 28′ , 30″ Sine — 989931 1926897 half difference of the sides 14 d 00′ 30″ Sine — 938393 Tangent 37 d 30′ — 988504 Comple : is 52 30 doubled makes 105 d , the Angle sought . 7. Two Angles with a side opposite to one of them being given . To find the third Angle , the kind of the side opposite to the other Angle being foreknown . First find the side opposite to the other Angle by 4th Case , And then we have two angles , and their opposite sides to find the third angle ; by transposing the order of either of the Proportions used in the first Case , the latter will be , As the Cosine of halfe the difference of the sides , To the Tangent of halfe the sum of the angles : So the Cosine of halfe the sum of the sides , To the Cotangent of half the contained angle . Example . In the Triangle Z ☉ P Data angle ☉ — 37 d 32′ Angle P — 105 00 Side ☉ Z — 80 31 To find the angle Z — 63 54 The first Operation will find Z P — 38 28 The second Operation . half sum of the angles — 71 d 16′ Tangent — 1046963 half sum of the sides — 59 d 29′ 30″ Sine Compl : — 970558 2017521 half difference of the sides 21 d 1′ 30″ Cosine — 997007 Tangent 58d 3′ — 1020514 Compl : 31d 57′ doubled is 63● 54′ the angle sought . 8. Two angles with a side Opposite to one of them being given , To find the Interjacent side , the kind of the side opposite to the other angle being fore known . First find the side opposite to the other angle by 4 Case , And then you have two sides , and their opposite angle given to find the 3 side by , tranposing the Order of either of the Proportions used in the 2d Case , the latter will be , As the Cosine of halfe the difference of the two angles , To the tangent of halfe the sum of the two sides : So the Cosine of halfe the sum of the two given angles , To the Tangent of halfe the third side . Example . In the former Triangle given the Hour angle at P 105d 00 Azimuth angle at Z 63 54 Compl Altitude Side Z ☉ 80 31 To find the Compl. of the Latitude the side Z P 38 28 The first Operation will find the side P ☉ 66● 29′ Second Operation . half the sum of the two sides 73d 30′ Tangent 1052839 half the sum of the two angles 84 27 Cosine 898549 1951388 half the difference of the two angles 20d 33′ Cosine 997144 Tangent of 19d 14′ — 954244 Doubled is 38 28 the side sought These 6 last precedent Cases may be called the Doubtful Cases , because that three given terms are not sufficient Data to find one single answer without the quality of a fourth , which is demonstrated by Clavius , in Theodosium , and seeing it passes without due caution in our English Books , I shall insert it from him : LEt A D and A C be two equal sides including the angle D AC , and both of them less or greater then a Quadrant . Draw through the Points C and D , the arch of a great Circle C D , continue it , and draw thereunto another Arch or Side from A , namely A B , neither through the Poles of the Arch C D , nor through the Poles of the Arch A D , so that the angles B and B A D may not be right angles , nor the angle A D B , if then each of these sides A D A C be less then a Quadrant , the two angles C , and A D C will be Acute ; and if these Arks be greater respectively then a Quadrant , the two angles C and A D C , will be Obtuse , whence it comes to pass that the angle A D B is Obtuse , when the angle A D C is Acute , and the contrary : Now forasmuch as the sides A C and A D are equal to each other , the other Data , viz. the side A B , and the angle at Bare common to both , for in each Triangle A B D , and A B C there is given two sides with the angle at B opposite to one of them ; Now this is not sufficient Data to find the angle opposite to the other side , which may be either the acute angle at C , or the Obtuse angle ADB the Complement thereof to a Semicircle : Nor to find the third side , which may be either B D , or the whole side B C , nor the angle included , which may be either B A D , or B A C , therefore in these 3 Cases we have required the quality of the angle opposite to the other given side A B , and though it be not so much observed ; in the other Trigonometry , by Perpendiculars let fall , without the knowledge of the said angle it could not be determined whether the Perpendicular would fall with in or without the Triangle , nor whether the angle found in the first Case be the thing sought , or its Complement to 180● , nor whether the angles or Segments found by 1st and 2d Operation in the other Cases are to be added together , or substracted from each other , to obtain the side or angle sought . So also two angles with a side opposite to one of them , are not sufficient Data to obtain a fourth thing in the said Triangle , without the affection of the side opposite to the other given angle . LEt A B and A C be two unequal sides containing the angle B A C both together equal to a Semicircle , one being greater , the other less then a Quadrant Draw through the Points B and C , the arch of a great Circle B C , continue it , and draw thereto from A another side AD ; but not through the Poles of A C , nor through the Poles of B C , so that the angles D and C A D may not be right angles , nor the angle A C D a right angle ; for if it were a right angle , the angle A B C whereto it is equal , should be also a right angle , and so the two sides A B and A C , by reason of their right angles at B and C should be equal , and be Quadrants contrary to the Supposition ; Now the angles A C D and A B C being equal , which is thus proved : Suppose the two sides A B and B D to be continued to a Semicircle at E , then will the said angle be equal to its opposite angle at B , the side A C by supposition is equal to the side A E , the Complement of the side A B to a Semicircle , but equal sides subtend equal angles , therefore the angle at C is equal to the angle at B or at E , which being admitted retaining the side A D and angle at D , we have another angle opposite thereto , either C or B , which are equal and common to both Triangles , and so if the side opposite to the given angle at D were sought , a double answer should be given , either the side A C , or the other side A B its Complement to 180 , and the interjacent side might be C D or B D , and the third angle the lesser angle C A D , or the greater B A D , which is not commonly animadverted . Two Sides with the Angle comprehended , to find the third Side . That the former Cases might be resolved without the help of Perpendiculars , hath been long since hinted by Mr Gunter , Mr Speidel , and Mr Gellibrand , but so obscurely that I suppose little notice was taken thereof ; but this Case hath not hitherto been resolved by any man , to my knowledge , under two Operations with a Perpendicular let fall , working by Logarithms , unless by Multiplication and Division in the natural Numbers , which being the onely Case left wherein we are to use Perpendiculars , I shall shew how to shun both , with the joynt use of the Natural and Logarithmical Tables , by a novel Proportion of my own , and illustrate the usefulness thereof by some Examples . Two Sides with the Angle comprehended , to find 3d Side . As the Cube of the Radius , To the Rectangle of the Sines of the comprehending sides : So is the Square of the Sine of half the angle contained , To half the difference of the Versed sines of the third side , and of the Ark of difference between the two including sides , Which half difference doubled , and added to the Versed Sine of the difference of the Leggs or containing sides , gives the Versed Sine of the side sought . And if you will make the third Tearm the Square of the Sine of half the Complement of the contained angle to 180d , you will find the half difference of the Versed Sines of the third side , and of the sum of the two including sides to be doubled and substracted from the Versed Sine of the said sum . But to apply the former to Logarithms . Double the Logarithmical Sine of half the angle given , & thereto adde the Logarithms of the sines of the containing Sides , & from the left hand of the Sum , Substract 3 for the Cube of the Radius , so rests the Logarithm of half the difference of those two Versed Sines above . And if instead of the second Tearm be taken into the Proportion , the double of the Rectangle of the Sines of the containing Sides ; that is , if the Logarithm of the Number 2 be added to the Logm of the other middle Tearms , you will have the Logarithm of the whole Difference in the last place ; having found it , take the Number that stands against it , either in the Natural Sines or Tangents , and accordingly add it to the Natural Versed Sine of the Difference of the Leggs , and the summe is the natural Versed Sine of the side sought . This is the Inverse of the 4 h Axiom , used when 3 sides are given to find an angle , and will be of great use to Calculate the Distances of Stars by having their Declinations and right Ascensions , or Longitudes and Latitudes given , by means whereof the Altitudes of two of them , or of the Sun with the difference of time , or Azimuth , being observed at any time off the Meridian , the Latitude may be found , as also for Calculating the distances of places in the Arch of a great Circle , all of them Propositions of good use in Navigation ; as for the latter it hath hitherto been delivered in our English Books doubtfully , erroneously , or not sufficiently for all Cases , the Rules delivered being only true in some Cases , and doubtful in most , not determining whether the side sought be greater or less then a Quadrant . The Reader may observe how necessary it is to have such Tables , as have the natural Sines and Versed Sines , &c. standing against the Logarithmical Sines , for this and other following Proportions discovered by my self for the easie calculating a Table of hours and Azimuths to all Altitudes , as also a Tables of Altitudes to all hours ; but as yet there are none such made as have the Versed Sines , but will in due time be added to Mr. Gellibrands Tables ; in the interim it may be noted , that the Residue of the Natural Sine of an Ark from Radius called its Arithmetical Complement , is the Versed Sine of that Arks Complement ; thus the natural Sine of 40d is 6427876 substracted from Radius , rests 3572124 , the Versed Sine of 50 d. And for Arks above 90 d we need no natural Versed Sines , because the natural Sine of any Arks excess above 90 d added to the Radius is equal to the Versed Sine of the said Ark , thus the Sine of 40 d augmented by the Radius is equal to the Versed Sine of 130 d and is 16427876 Example of this Case . In the Triangle ☉ Z P let there be given the side ☉ Z , the Complement of the Altitude 70 d 53′ and the side Z P the Complement of the Latitude 38 d 28 n with the angle ☉ Z P 145 d the Suns Azimuth from the North , to find the side ☉ P , the Suns distance from the Elevated Pole. Sine 38 d 28 m — 97938317 Sine 70 53 — 99753646 Sine 72 30 Log m dobled 199588390 Natural Sine against 97280353 it doubled is 10691964 Natural V Sine of 32 d 25 m the difference of the sides — 1558280 The Versed Sine of 103 d the — 12250244 side sought , and therefore the Sun hath 13 d of South declination . Another Example of this Case for Calculating the Suns Altitude on all hours . As the Cube of the Radius , To the double of the Rectangle of the Cosines , both of the Latitude , and of the Suns declination . So is the Square of the Sine of half the hour from noon , To the difference of the Sines of the Suns Meridian Altitude , and of the Altitude sought . This Canon will finde two Altitudes at one Operation , and will have very little trouble in it , the double Rectangle , that is the second tearm of the Proportion , being fixed for that Declination . Add the Logarithms of the Number two , and of the Cosines of the Declination and Latitude together the sum may be called the fixed Logarithm . Double the Logarithm of the Sine of half the hour from noon , and add it to the fixed Logarithm the sum rejecting 3 towards the left hand , for the Cube of the Radius is the Logarithm of the difference : Take the natural Sine that stands against it , and substract it from the natural Sine of the Meridian Altitude , both for the Winter and Summer Declination , and there remains the natural Sines of the Altitudes sought . If this difference cannot be substracted from the Sine of the Meridian Altitude , it argues the Sun hath no Altitude above the Horizon in this Case substract that from this , and there will remain the Natural Sine of the Suns Altitude for the like hour from midnight in Summer . Example . Let it be required to Calculate the Suns Altitude when he hath 23d 31 m both of North and South Declination for our Latitude of London at 2 and 5 a Clock in the afternoon , or which is all one for the hours of 10 and 7 in the morning . Sine 38d 28 m Compl Latitude — 97938317 Sine of 66 29 m Compl Declination — 99623428 Logarithm of Number 2 is — 03010300 Fixed Number — 200572045 Logm of Sine of 15d , doubled — 188259924 Nearest natural Sine against it , 761900 — 88831969 61d 59 m Summer Meridian Altitude Natural Sine — 8828110 Substract — 761900 the difference before found Rests — 8066210 the natural Sine of 53d 46′ the Summer Altitude for the hours of 10 and 2 14d 57 m Winter Meridian Altitude Nat Sine — 2579760 Substract the former difference — 761900 Rests the Natural Sine of 10d 27 m the — 1817860 Winter Altitude for the hours of two and ten . The same day for the Altitude of 5 and 7. Fixed number — 200572045 Sine of 37d 30 m Logm doubled — 195688942 Natural Sine against it 4226183 — 96260987 Winter Meridian Altitude , as before Sine 2579760 Rests — 1646423 the natural Sine of 9d 29 m Summer Altitude for 5 in the morning , or 7 in the evening . Natural Sine . Summer Meridian Altitude , as before — 8828110 The former difference — 4226183 Rests the Natural Sine of 27d 24 m — 4601927 The Summer Altitude for 7 in the morning , or 5 in the afternoon . The former Case may also be performed at two Operations by help of a Perpendicular supposed , without the help of Natural Tables . 1. If both Sides are equal , As the Radius , To the sine of the Common side : So the Sine of half the Angle , To the Sine of half the side sought . 2. If one of the sides be a Quadrant , this by continving the other side to a Quadran ( as shall afterwards be shewed ) wil become a Case of right angled Sphoericala Triangles , in which besides the right angle , instead of the quadrantal side , there will be given a Legg , and its adjacent angle to find the other angle by 4 Case of right angled Sphoerical Triangles ; and so if the angle included were 90d it would be a Case of right angled Sphoerical Triangles , in which besides the right angle , there would be given both the Leggs or Sides to find the Hipotenusal . 3. In all other Cases one or both of the including Sides being less then Quadrants , it will hold , As the Radius , To the Cosine of the angle included : So the tangent of the lesser side , To the tangent of a fourth Ark , If the angle included , be less then 90d substract the 4 ●h Ark from the other side ; but if it be more from the other sides Complement to 180d , The remainder is called the Residual Ark. Then , As the Cosine of the 4th Ark , To the Cosine of the Ark remaining : So the Cosine of the lesser side , To the Cosine of the side sought . The side sought may be greater then a Quadrant , and so be doubtfull , but we may determine , That when the Leggs are of the same kind , and the angle comprehended Acute , the side sought is less then a Quadrant . And when the Leggs or containing Sides are of a different kind , and the angle comprehended Obtuse , the side sought is greater then a Quadrant . Or it may be determined from the affection of the Residual Ark in all Cases . When the contained angle is acute , and the residual Ark more then 90d , or when the said angle is Obtuse , and the residual Ark less then a Quadrant , the side sought is greater then a Quadrant , in all other Cases less . Example . In the Triangle ☉ Z P , let there be given Z P , and ☉ Z with the angle at Z , to find the side ☉ P , the Suns distance from the Elevated-Pole . angle included 145d Logm Or , 35 Compl 55d Sine — 99133645 Tangent of 38 28′ lesser side — 99000865 Tangent 33d , 3′ — 98134510 Compl ☉ P to 180d is 109 7 The Ark remaining or differ : 76d 4 m Cosine — 93816434 Lesser side — 51 32 Cosine — 98937452 192753886 Ark found — 33d 3 m Cosine — 99233450 Sine 13 — 93520436 The Complement hereof 77 d should be the side fought , but because the angle was Obtuse , and the residual Ark less then a Quadrant , the side sought is greater , and therefore 103 d the Complement hereof is the side sought . This Case & the Converse of it being the next Case , I have thus setled to apply the to Logarithmical Tables only , in Case the natural ones were wanting , being all the other Cases are thereto fitted ; and as the trouble about the Cadence of a Perpendicular is here shunned , without so much as the name of it ; so may it be done in all the rest of the Oblique Cases , which I had so fitted up for my own use ; but forbear to trouble the Reader with them , apprehending these to be better , and that he would not willingly Calculate for a portion of an angle , or a Segment of a Side , in order to the finding out the thing sought , when with as little trouble he may come by it , and yet Calculate always either for a side or an angle , one of the six principal parts of the Triangle . Otherwise for Instruments . As the Diameter , To the difference of the Versed Sines of the sum , and of the difference of any two sides , including an Angle . Or , As the Cosecant of one of the including Sides , Is to the Sine of the other side : So is the Versed Sine of the angle included . To the difference of the Versed Sines of the Ark of difference between the two including Sides , and of the third side sought , Which difference added to the Versed Sine of the difference of the Leggs , makes the Versed Sine of the side sought . And so is the Versed Sine of the contained angles Complement to 180d To the difference of the Versed Sines of the sum of the Leggs , and of the side sought , which substracted from the Versed Sine of the said sum , there remains the Versed sine of the side sought , Here note , that the same Versed Sine is common to an Ark greater then 180 d , and to its Complement to 360 d , So the Versed Sine of 200 d is also the Versed Sine of 160 d. The Proportions delivered for Instruments having such Tables as before hinted , will not be so unsuitable to the Logarithms as commonly reputed . Example for Calculating the distance of two places in the Arch of a great Circle , otherwise then according to the general Cannon before delivered . As the Secant of one of the Latitudes , To the Cosine of the other , So the Versed Sine of the difference of Longitude , To the difference of the Versed Sines of these two Arks , The one the Ark of distance sought , the other the Ark of difference between both Latitudes , when in the same Hemisphere , or the sum of both Latitudes when in different Hemispheres , which difference added to the Versed Sine of this latter Ark , the sum is the versed Sine of the distance , By turning the Substraction to be made of the first Tearm into an Addition , the two first Tearms of the Proportion will be , As the Square of the Radius , To the Rectangle of the Cosines of both the Latitudes : Then for the third Tearm being the difference of Longitude , take the natural Versed Sine thereof , and seek that Number in the natural Tangents , and that Logarith Tangent that stands against it take into the Proportion instead of the Logarithm of the Versed Sine proposed . Admit it were required to find the Distance between London and Bantam , in the Arch of a great Circle . Logme Bantam Longitude 140 d Latitude 5 d 40′ South Cosine 9,9978725 London Longitude 25 , 50 Latitude 51 , 32 North Cosine 9,7938317 — difference of Long 114 d 10′ Nat V Sine 14093923 equal to the natural Tangent of 54 d 38′ ½ nearest Logm 10,1489900 Natural Sine 8723538 against it — 2 9940694 Nat Versed Sine of 57 d 12′ the sum of both Latitudes 4582918 — Sum — 13306456 the natural Versed Sine of 109 d 18′ 30″ the Ark of distance sought . And if to the said difference , namely — 8723538 Be added the natural Versed Sine of the difference of both Latitudes , namely the V Sine of 45 d 52′ — 3036695 — The sum being the natural V Sine of 100 d 8′ 30″ is — 11760233 the distance of two places , having the same Latitudes , and difference of Longitude , but are both in the same Hemisphere . Here note , that no two places can have above 180 d difference of Longitude , therefore in differencing the two Longitudes if the remainder be more take its Complement to 360 d. The Complements of these two distances , namely 70 d 41′ 30″ and 79 d 51′ 30″ are the distances of two places of the same Latitudes considered as in different Hemispheres , their difference of Longitude being 65 d 50′ the Complement of the former , and two places in a such Position compared with their former Positions may be apprehended to be Diametrically opposite upon the Globe , as thus , Bantam having 5d 40′ South Latitude , let another place have as much North Latitude , the difference of Longitude between them 180 d and consequently so much their distance ; now whatever be the distance between Bantam and the third place , the Complement of it to 180 d shall be the distance between the two other places . 10. Two angles with the Interjacent side given . To find the 3d angle , the proportion derived from the former Case by changing the angles into sides , and holds without any such change supposed is , As the Cube of the Radius , To the double of the Rectangle of the Sines of the two given angles : So is the Square of the Sine of half the given side , To the difference of the Versed Sines of these two Arks , the one is the angle sought , the other the Ark of difference between one of the including angles , and the Complement of the other to a Semicircle , which difference added to the Versed Sine of this Ark gives the Versed Sine of the angle sought . How to work this by Tables need not be shewed after the Logm of the difference is got , if it be less then the Radius , it may be sought either in the Sines or Tangents , and the natural Sine or Tangent that stands against it and comes nearest taken ; but when it exceeds the Radius always seek it in the Tangents , and take the natural Tangent that stands against it , which difference so found , is to be added to the Versed Sine of the difference of the Leggs to obtain the Versed Sine of the angle sought . Otherwise for Tables the common way by a supposed Perpendicular 1. If both the angles are equal , As the Radius . To the Sine of the angle given : So the Cosine of half the given Side , To the Cosine of half the angle sought . In all other Cases not belonging to right angled Triangles if one or both of the given angles be Acute , it holds , As the Radius , To Cosine of the interjacent side : So the Tangent of the lesser angle , To the Tangent of a 4● h Ark. If the interjacent side be more then 90d substract the 4● h Ark from the other angle ; but if less then 90d , substract the 4● h Ark from the other angles Complement to 180d , noting the residual Ark. Then , As the Cosine of the 4th Ark , To the Cosine of the Ark remainmaining : So the Cosine of the lesser angle , To the Cosine of the angle sought . When the interjacent side is less then a Quadrant , and the residual Ark more , or when the interjacent side is greater then a Quadrant , and the residual Ark less , the angle sought is Obtuse , in all other Cases Acute . In the Triangle ☉ Z P let there be given The angle of Position at ☉ — 21d 28′ The hour from noon angle at P — 33 47 And the side ☉ P the Suns distante from the elevated Pole — 103 00 To find his Azimuth the angle ☉ Z P Sine 13 d the Complement of the interjacent side — 93520880 Tangent 21 d 28′ the lesser angle — 95946561 Tangent of 5 d 3′ — 89467441 The other angle — 33 47 The difference being the residual ark 28 44 Cosine — 99429335 Lesser angle — 21 28 Cosine — 99687773 199117108 Ark first found — 5d 3′ Cosine — 99983109 Sine 55 d — 99133999 The Complement whereof 35 d in this Case is not the angle sought , but the residue hereof from a Semicircle 145 d is the angle sought being Obtuse , because the interjacent side is greater then a Quadrant , and the residual Ark less ; the residual Ark in Operation if greater then a Quadrant , take its Complement to 180 d , because there are no Sines to Arks above a Quadrant , and then the Complement of this Ark to 90d is the Complement of the residual Ark the Sine whereof must be taken for the Cosine of the residual Arke . Otherwise for Instruments . As the Diameter , To the difference of the Versed Sines of the sum and difference of the two including angles , Or , As the Cosecant of one of those angles , Is to the Sine of the other , So the Versed Sine of the interjacent side , To the difference of the Versed Sine of an Ark left by substracting one of the including angles from the Complement of the other to a Semicircle , and of the angle sought , which difference added to the Versed Sine of the said Ark , gives the Versed Sine of the angle sought , And so is the Versed Sine of the interjacent sides Complement to 180 d , To the difference of the Versed Sines of an Ark made by adding one of the including angles to the Complement of the other to a Semicircle , and of the angle sought , which substracted from the Versed Sine of the said Ark , leaves the versed sine of the angle sought . 11. Three Sides to find an Angle . The two sides including the angle sought are called Leggs , and the third side the Base . As the Rectangle or Product of the Sines of the half sum of the three sides and of the difference of the Base therefrom . Is to the Square of the Radius : So is the Rectangle of the sines of the differences of the Leggs from the said half sum , To the Square of the Tangent of half the angle sought . And by changing the third Tearm into the place of the first , As the Rectangle of the Sines of the differences of the Leggs from the half sum of the 3 sides , Is to the Square of the Radius : So the Rectangle of the Sines of the half sum of the three sides , and of the difference of the Base therefrom , To the Square of the Tangent of an Ark , whose Complement doubled is the angle sought , or this Ark doubled is the Complement of the angle sought to 180 d , or it might be expressed , To the Square of the Cotangent of half the angle sought . Otherways in Sines . As the Rectangle of the Sines of the containing Sides or Leggs , Is to the Square of the Radius ; So the Rectangle of the Sines of the differences of the Leggs from the half sum of the three sides , To the Square of the Sine of half the angle sought . Or the Cosine may be found . As the Rectangle of the Sines of the containing sides , Is to the Square of the Radius : So the Rectangle of the Sines of the half sum of the 3 sides , and of the difference of the Base therefrom , To the Square of the Cosine of half the angle sought . These two latter Proportions are demonstrated in the Treatises of the Lord Napier , Mr Oughtred , Mr Norwood , and are those from whence I shall educe the Demonstrations of the rest . To work the third Proportion that finds the Square of the Sine of half the angle . To the Arithmetical Complements of the Logarithms of the sines of the containing Sides or Leggs add the Logarithmical Sines of the differences of the said Leggs from the half sum of the three Sides , the half sum of these four Numbers will be the Logarithm of the sine of half the angle sought . In the Triangle ☉ Z P , Data , the three Sides to find the angle a P the hour from noon . 80 d 31′ Base 66 29 Leggs — Ar comp , 0376572 38 28 Leggs — Ar comp , 2061683 Sum-185 , 28 difference of the Leggs half-92 44 from half sum — 26d 15′ Sine 9,6457058 54 , 16 Sine 9,9094190 Sum — 19,7989503 Sine of 52 d 30′ half — 9,8994751 doubled 105 , the angle at P sought . The Arithmetical Complement of a Logarithm , is the residue of that Logarithm from the next bigger Number , consisting of an Unite and Ciphers . Otherwise for Instruments . As the difference of the Versed Sines of the sum , and of the difference of any two sides including an angle , Is to the Diameter , Or , As the sine of one of the said sides , To the Secant of the Complement of the other . So is the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides , To the Versed Sine of the angle sought . And so is the difference of the Versed Sines of the third , and of the sum of the two including sides , To the versed Sine of the sought angles Complement to 180d. 12. Three Angles to find a Side . The work here for the Canon or Tables , will be by changing the Angles into Sides , the general Rule for changing all the parts of a Triangle , is to draw a new Triangle , and let the angles be wrot against their Opposite sides , and these against those , only taking the Complements of the greatest Angle , and greatest side opposed thereto to 180 d , this for most convenience that the sides or angles of the new framed Triangle may not be too large , and so cause recourse to the Opposite Triangle , otherwise the Complements of any side and its opposite angle to 180 d ▪ might as well have been taken . But for this Case , seeing there are only angles to be changed into sides , take the Complement of the greatest angle to 180 d and proceed as if there were three sides given to find an angle . But the Proportion in Versed Sines , &c. without any such change will be , As the difference of the Versed Sines of the sum , and of the difference of any two angles adjacent to the side sought . Is to the Diameter , Or , As the Sine of one of the said angles , Is to the Cosecant of the other : So is the difference of the Versed Sines of the third or Opposite angle , and of an Ark left by substracting one of the including angles from the Complement of the other to a Semicircle , To the Versed Sine of the side sought . And so is the difference of the Versed Sines of the third angle , and of an Ark made by adding one of the including Angles to the Complement of the other to a Semicircle . To the Versed Sine of the sought sides Complement to 180d . Thus having finished the Cases , it is to be intimated that the Proportions here used in Versed Sines are variously demonstrated in diverse Writers , but in most the latter part for finding the Complement of an angle to 180d , is quite omitted , those that have demonstrated the former part , do it in these tearms following . As the Rectangle of the sines of the containing sides , Is to the Square of the Radius : So is the difference of the Versed Sines of the Base , or third Side , and of the Ark of difference between the two including sides , To the Versed Sine of the angle sought , which the Reader may see in Lansberg , Regiomantanus , Snellius , Pitiscus , and the learned Clavius , who makes 15 Cases , and twice as many Scheams , to demonstrate this part of it . I shall only shew how it may be inferred from the common Proportions in use fitted to the Tables demonstrated by the Lord Napier , Mr Oughtred , Mr. Norwood . We have two Proportions delivered in Rectangles and Squares the one for finding an angle , the other to find its Complement to 180d. The two first tearms are the Proportion between the Rectangle of the Sines of the containing sides , and the Square of the Radius ; these two tearms being divided by the Sine of one of those sides , the Quotient will be the Sine of the other , if the same Divisor divide the Square of the Radius , the Quotient will be the Secant of the Complement of the Ark belonging to the Divisor , because , As the Sine of an Ark , To Radius , So is the Radius , To the Secant of that Arks Complement ; But if any common Divisor divide any two Tearms of a Proportion , the Dividends will be Acquimultiplex to the Quotients ; and therefore by the Quotients will bear such Proportion each to other as the Dividends , and therefore it holds , As the Rectangle of the Sines of the containing sides , Is to the Square of the Radius : So is the Sine of one of those sides , To the Secant of the Complement of the other . Again , for the third Tearm , to find an angle it is proposed . So is the Restangle of the Sines of the differences of the Leggs from the half sum of the three sides . Or which is all one , So is the Rectangle of the Sines of the half sum , and half difference of the Base or third side , and of the Ark of difference between the two including sides , To the Square of the Sine of half the angle sought , And so to find the Complement of an angle to 180 d. So is the Rectangle of the Sines of the half sum of the three sides , and of the difference of the third side or Base therefrom , Or which is equivalent thereto , So is the Rectangle of the Sines of the half sum , and half difference of the Base or third side , and of the sum of the two including sides , To the Square of the Sine of an Ark , which doubled is the Complement of the angle sought to 180 d , or the Complement of that Arch to a Quadrant doubled , is the angle sought . The former of these two expressions of the third Tearm of the Proportion , as being the more facil for memory is now retained ; but the latter , ( formerly used , and now rejected ) agrees best with the Proportion , as applyed to Versed Sines , for the inferring whereof note , that such Proportion , As the difference of two Versed Sines beareth to another Versed Sine , the same Proportion doth the half difference of those Versed Sines , bear to half the Versed Sine of that other Arch : But that is the same that the Rectangle of the Sines of the half sum and half difference of any two Arks doth bear to the Square of the Sine of half that other Arch , which will be thus inferred , because if the said Rectangle and Square be both divided by Radius , the two Quotients will be the half difference of the versed Sines of the two Arks proposed , and half the versed Sine of the 4 ●h Arch. That the Sines of the half sum and half difference of any two Arks are mean Proportionals between the Radius and the half difference of the Versed Sines of those Arks is demonstrated in Mr Gellibrands Trigonometry in Octavo , that is , As the Radius , To the Sine of half the sum of any two arks : So is the sine of half the difference of those two arks , To half the difference of the versed sines of those two arks , and therefore the said Rectangle divided by Radius , the Quotient is half the difference of the versed sines of the two Arks. And that the Sine of any Arch is a mean Proportional between the Radius and half the versed Sine of twice that Arch , That is , As the Radius , Is to the sine of an Arch : So the sine of that Arch , To half the versed sine of twice that Arch , and therefore the Square of the sine of any Arch divided by Radius , the Quotient is the half versed sine of twice that Arch ; whence the Rule to make a Cannon of whole Logarithmical versed sines is to take half the arch proposed , and to the Logarithm thereof doubled , or twice wrot down , to add the Logarithm of the number two , and from the sum to substract the Radius . We have before inferred , that As the Rectangle of the sines of the containing sides , Is to the Square of the Radius : So is the sine of one of those sides , To the Secant of the Complement of the other , and that by dividing those two Plains by one of those sides ; but if we divide the said two Plains , viz. the Rectangle of the sines of the containing sides , and the Square of the Radius , by the Radius as a common Divisor , the latter Quotient will be the Radius , and the former the half difference of the versed sines of those Arks whereof the two containing sides are the half sum and the half difference ; but those Arks are found by adding the half difference to the half sum to get the greater , and substracting it therefrom to get the lesser ; Which is no other then to get the sum and difference of the two containing sides , it therefore holds , As the Rectangle of the sines of the containing sides , Is to the Square of the Radius , Or , As the sine of one of those sides , To the Secant of the Complement of the other : So is the half difference of the versed sines of the sum and difference of those two sides to the Radius , And by consequence so is the whole difference to the Diameter , and this being admitted the whole Proportion in all its parts may be inferred from Mr Daries Book of the Uses of a Quadrant , where he demonstrates , That , As the difference of the versed Sines of the sum and difference of any two sides including an angle , Is to the Diameter : So is the difference of the versed sines of the third side , and of the Ark of difference between the two including sides , To the versed sine of the angle sought , in that Scheme it lyes , As M S , To G H : So is M P , To H C. And I further add , As M S , To G H : as before , So is P S , To G C. that is , retaining the two first Tearms of the Proportion , the same it holds for the third and fourth Tearm . So is the difference of the versed sines of the third side , and of the sum of the two including sides , To the versed sine of the sought angles Complement to 180d. Now from these Proportions thus Demonstrated , are inferred those others that give the answer in the Squares of Tangents , in order whereto observe , That if 4 Numbers are Proportional , their Squares are also Proportional ( quamvis non in eadem rations ) so that any three of those Squares being given , the Square of the 4th will be found by direct Proportion , and the Proportion for making a Table of Natural Tangents from the Tables of natural sines is , As the Cosine of an Ark , To the sine of the said Ark : So is the Radius , To the Tangent of the said Ark. It will therefore hold by 22 Prop. of 6 h Book of Euclid , As the Square of the Cosine of an Ark , Is to the Square of its sine : So is the Square of the Radius , To the Square of its Tangent , Now from the two Demonstrated Proportions for the Tables , the two first Tearms are common to both , and therefore there is the like Proportion between the two latter Tearms of the first Proportion , and the two latter in the second , as between the two latter , and the two former in each Proportion : Now because the latter Proportion finds the Square of the Cosine , and the former the Square of the Sine of the same Ark , it is inferred that the third tearm in the latter Proportion , bears such Proportion to the third Tearm in the former Proportion , as the Square of the Cosine of an Ark , doth to the Square of its Sine , which is the same that the Square of the Radius bears to the Square of the Tangent of the said Ark , it therefore holds when three sides of a Spherical Triangle , are given to find an angle . As the Rectangle of the Sines of the half sum of the three sides , and of the difference of the Base therefrom , Is to the Rectangle of the Sines of the differences of the Leggs therefrom : So is the Square of the Radius , To the Square of the Tangent of half the angle sought , and by changing the 2d Tearm into the place of the first . As the Rectangle of the sines of the differences of the Leggs from the half sum of the 3 sides , Is to the Rectangle of the sines of the half sum of the three sides , and of the difference of the Base therefrom : So is the Square of the Radius , To the Square of the Cotangent of half the angle sought . These Proportions are published in order to their Application to the Serpentine Line , which will be accomodated for the sudden operating of any of them ; the Axioms to be remembred are not many , the Reader will meet with their Demonstration and Application in Mr Newtons Trigonometry now in the Press , and said to be near finished : The four Proportions in plain Triangles , when three sides are given to find an angle without the Cadence of Perpendiculars are demonstrated in the 27 Section of the late Miscellanies of Francis van Schooten . The Construction of diverse Instruments will require a Table of the Suns Altitudes to the Hour and Azimuth assigned ; And for the Acurate bounding in of the Lines , it may be a Table of Hours and Azimuths to any Altitude assigned ; for the easie Calculating whereof , I am desired for the ease and benefit of the Trade , to render this part of Calculation as facil as I can , and therefore shall handle it the more largely . To Calculate a Table of Hours to all Altitudes in all Latitudes . The 1. Proportion shall be to find the Suns Altitude in Summer , or Depression in Winter at the hour of 6. As the Radius , To the sine of the Latitude : So the sine of the Declination , To the sine of the Altitude or Depression sought This remains fixed for all that day the Suns Declination supposed not to vary , and then it holds , As the Cosine of the Declination , To the Secant of the Latitude : So in Summer is the difference in Winter the sum of the sines of the Suns Altitude proposed , and of his Altitude or Depression at 6 To the sine of the hour from 6 towards noon in Winter , and in Summer also , when the given Altitude is greater then the Altitude of 6 , but when it is less towards midnight . This Proportion also holds for Calculating the Horary distance of any Star from the Meridian . In like manner to Calculate the Azimuth . As the sine of the Latitude , To sine of the Declination : So is the Radius , To the side of the Suns Altitude or Depression in the prime Vertical , that is , being East or West . This remains fixed for one day . Then , As the Cosine of the Altitude , To the Tangent of the Latitude : So in Summer is the difference , and in Winter the sum of the sines of the Suns Altitude proposed , and of his Vertical Altitude or Depression , To the sine of the Azimuth towards noon Meridian in Winter and in Summer also , when the given Altitude is greater then the Vertical Altitude or Depression , but when it is less towards Midnight Meridian . This Proportion is general either for Sun or Stars , when the Declination is less then the Latitude of the place ; But when it is more , say as before , As the sine of the Latitude , To the sine of the Declination : So is the Radius , To a fourth we may call it a Secant . Again . As Cosine Altitude , To the Tangent of the Latitude : So in declinations towards the Depressed Pole is the sum ; but towards the Elevated Pole the difference of this Secant , and of the sine of the Sun or Stars Altitude , To the sine of the Azimuth from the Vertical towards the noon Meridian . Before Application be made , the latter part of these Proportions being of my own peculiar Invention , and of very great use both for Calculation , and Instrumentally , it will be necessary to demonstrate the same . For the Hour from the Analemma . Having in the Scheme annexed drawn the Equator and Horizon , the two prickt Lines passing through the Center , as also the Prime Vertical and Axis , the two streight Lines passing through the same . Let I X and L M represent two Parralells of Declination on each side the Equator , and O X a Parralel of the Suns Altitude in Summer , and P Q of his Depression in Winter , at the hour of 6 , because these Parralells pass through the Intersection of the Parralells of Declination with the Axis . Let R S be a Parralel of Altitude after 6 , and T V a Parralel of Altitude before it ; from the Intersections of these Parralells of Altitude with the Parralels of Declination let fall Perpendiculars on the Parralells of the Suns Altitude or Depression at 6 , and then we shall have divers right Lined right angled Triangles Constituted in which we shall make use of the Proportion of the sines of angles to their opposite sides an Axiom of common demonstration . In the Triangle A F E , As the sine of the angle at F the Radius , To its Opposite side A E , the sine of the Declination : So the sine of the Latitude the angle at E to A F , the sine of the Suns Altitude at 6. Again in the two Opposite Triangles A B C , the smaller before the greater after 6. As the Cosine of the Latitude the sine of the angle at A , To its Opposite side B , C , the difference of the sines of the Suns Altitude at 6 , and of his proposed Altitude : So is the Radius sine of the angle at B , To C A , the sine of the hour from 6 in the Parralel of Declination in the lower Triangle before , in the upper after 6. So in the Winter or lower Triangle A B D C. As Cosine of the Latitude sine of the angle at A , To B C , the sum of the sines of the Suns Depression at 6 B D , and of his given Altitude D C : So is the Radius the sine of the angle at B , To A C , the sine of the hour from 6 towards noon in the Parralel of Declination , The sine of the hour thus found in a Parralel , is to be reduced by another Analogy to the common Radius , and that will be , As the Radius of the Parralel I A , the Cosine of the Declination , Is to the common Radius E AE : So is any other sine in that Parralel . To the sine of the said Arch to the common Radius . Now it rests to be proved that both these Analogies may be reduced into one , and that will be done by bringing the Rectangle of the two middle Tearms of the first Proportion with the first Tearm under them as an improper Fraction to be placed as a single Tearm in the second Proportion , being in value the answer found in the Parralel , and then we have the Rule of three to Operate as it were in whole Numbers and mixt . The Proportion will run , As the Cosine of the Declination , To Radius : So the said Improper Fraction ▪ To the Answer . and so proceeding according to the Rules of Arithmetick . The Divisor will be the Rectangle of the Cosine of the Declination , and of the Cosine of the Latitude , one of the middle Tearms would be the Square of the Radius , and the other the former sum or difference . Now if any two Tearms of a Proportion be divided by a common Divisor , the Dividends being Equimultiplex to the Quotients , the Quotients bear the same mutual Proportion as the Dividends by 18● Propos . 7 Euclid . So in this instance if the Rectangle of the Cosines both of the Latitude and of the Declination be divided by one of those Tearms , the Quotient will be the other , and if the Square of the Radius be divided by the Sine of an Arch , the Quotient will be the Secant of that Arks Complement ; So in the present Example , if the former Rectangle be divided by the Cosine of the Latitude ▪ the Quotient is the Cosine of the Declination , if the Square of the Radius be divided by the same Divisor , the Quotient is the Secant of the Latitude , likewise if both those Plains were divided by the Cosine of the Declination , the Quotients would be the Cosine of the Latitude , and the Secant of the Declination , it therefore holds , As the Cosine of the Declination , To the Secant of the Latitude , Or , As the Cosine of the Latitude , To the Secant of the Declination : So is the former sum or difference of sines , To the sine of the hour from 6 , which was to be proved . Corrollarie . As the Radius , To the sine of an Arch in a lesser Circle or Pararlell : So is the Secant of that Parralell , To the sine of the said Arch , to the common Radius . Hence may be observed a general Canon for the double or compound Rule of three , divide the Tearms into two single Rules , by placing two Tearms of like Denomination in each Rule , and the other remaining Tearm may in most Cases be put among either of these two Tearms of like Denomination , and then by arguing whether like require like , or unlike , the Divisor in each single Rule , may be discovered , and then it will hold in all Cases , As the Rectangle or Product of the two Divisors , Is to the product of any two of the other Tearms : So is the other Tearm left , To the Number sought , For the Azimuth . Having drawn the Horizon and Axis , the two prickt Lines , the Vertical Circle Z N , and the Equinoctial Ae Ae , the Parralels of Declination I K and L M , draw T V a Parralel of lesser Altitude then that in the Vertical , and R S a Parralel of greater Altitude ; Draw also P Q a Parralel of Depression equal to the Vertical Altitude , in the point C aboue the Center the Point A being as much below it b , eing the point where the Parralel of Declination intersects the Vertical Circle , and from the point C in the lesser parralel of Altitude , let fall the perpendicular C B on the parralel of Depression P Q , by this means there will be Constituted divers right lined , right angled Triangles , and through those Points where the parralel of Declination , and parralels of Altitude intersect , are drawn Elipses prickt from the Zenith to represent the Azimuths , and in the three several Triangles thus Constituted , the side A B measureth the quantity of the Azimuth in the parralel of Altitude , and B C in the two upper Triangles is the difference of the sines of the Suns proposed Altitude , and of his Altitude in the prime Vertical : But in the lower Triangle the sum of them , it then holds by the Proportion of the sines of Angles to their opposite sides . In the two upper Triangles , As the Cosine of the Latitude , the sine of the angle at A , To its opposite side B C , the difference of the sines of the Suns Vertical , and of his proposed Altitude : So is the sine of the Latitude , that is the sine of the angle at C , To its opposite side B A , the sine of the Azimuth from the East and West , And the like in the lower Triangle , onely there the third Tearm B C , is the sum of the sines of the Suns Vertical Depression , and of his given Altitude : Such Proportion as as the Cosine of an Ark doth bear to the sine of an Ark , doth the Radius bear to the Tangent of the said Ark , this being the Canon by which the natural Tangents are made from the natural sines , and therefore we may change the former Proportion , and instead thereof say , As the Radius , To the Tangent of the Latitude : So the said sum or difference of Sines , To the Sine of the Azimuth in the Parralel of Altitude : The answer falling in a Parralel or lesser Circle is to be reduced to the common Radius by another Analogy , and that is As the Cosine of the Altitude ( the Radius of the parralel ) To the Radius : So any sine in the said parralel , To the like sine in the common Radius . Now it is to be proved that both these Proportions may be brought into one , and that will be as before , by making an improper Fraction whose Numerator shall be the Rectangle of the two middle Tearms of the former Proportion , the first Tearm , viz. the Radius being the Denominator , and placing this as the third Tearm in the second Proportion , and then those that understand how to operate the Rule of three in whole Numbers and mixt , will find their Divisor to be the Rectangle or Product of the Cosine of the Altitude , and of the Radius , and the Dividend the Product of the three other Tearms , namely , of the Tangent of the Latitude , the Radius , and the former sum or difference of sines , whence it holds , As the Rectangle of the Cosine of the Altitude , and of the Radius , Is to the Rectangle of the Tangent of the Latitude , and of the Radius : So is the former sum or difference of sines , To the sine of the Azimuth . The Reader may presently espy that the two former Tearms of this Proportion may be freed from the Radius by dividing them both thereby , and the Quotients will be the Cosine of the Altitude , and the Tangent of the Latitude , It therefore holds , As the Cosine of the Altitude , To the Tangent of the Latitude : So in Summer is the difference , in Winter the sum of the sines of the Suns Vertical and proposed Altitude , To the sine of the Azimuth from the Vertical . This is general either for Sun or Stars , when their Declination is less then the Latitude of the place ; but when it is more , the Case doth but little vary . In the Scheme annexed fitted to the Latitude of the Barbados having drawn H H the Horizon , P P the Axis , Ae Ae the Equator , Z A the Vertical draw two parralells of Declination F R , K A continued till they intersect the Vertical prolonged , draw the parralel of Altitude B ☉ , and parralel thereto from the Point A draw A E , Then doth the latter part of the Proportion lye as evident as before , In the right angled Triangle C G F right angled at G , As the sine of the Latitude the angle at F , To its Opposite side C G the sine of the Declination , So the Radius the angle at G , To the Secant C Z F. Again in Summer . As the Cosine of the Latitude the angle at ☉ , To its opposite side D Z F , the difference between the former Secant and the sine of the Altitude : So is the sine of the Latitude , the angle at F , To its opposite side D ☉ , the sine of the Azimuth from the Vertical in the Parralel of Altitude . In Winter , As Cosine Latitude angle at A , To B E the sum of the former Secant equal to E M , and of the sine of the Altitude M B : So is the sine of the Latitude the angle B , To A E , equal to B D the sine of the Azimuth in a Parallel as before , to be reduced to the common Radius . From this Schem may be observed the reason why the Sun in those Latitudes upon some Azimuths hath two Altitudes , because the Parralel of his Declination F R intersects , and passeth through the Azimuth , namely , the prickt Ellipsis in the two points S , ☉ . I now proceed to the Vse in Calculating a Table of Hours . For those that have occasion to Calculate a Table of Hours to any assigned Altitude and parralel of Declination , it will be the readiest way to write down all the moveable Tearms first , as the natural sines of the several Altitudes in a ruled sheet of Paper , and then upon a peice of Card to write down the natural sine of the Suns Altitude at 6 and removing to every Altitude , get the sum or difference accordingly , which being had , seek the same in the natural sines , and write down the Log m that stands against it , then upon the other end of the piece of Card get the sum of the Arithmetical Complements of the Logarithmical Cosine of the Declination , and of the Logarithmica Cosine of the Latitude , and add this fixed Number to the Logme before wrote down ; by removing the Card to every one of them , and the sum is the Logme of the sine of the Hour from 6 , if the Logmes be well proportioned out to the differences which may be sufficiently done by guess . Example . Comp Latitude 38d 28′ Ar Comp — 0,2061683 Comp Delinat 66 , 29 Ar Comp — 0,0376572 fixed Number — ,2438255 Let the Altitude be 36● 42′ Nat Sine 5976251 Natural sine Altitude at 6 — 3124174 difference — 2852077 Log against 9 4550441 Sine of 30d the hour from 6 towards noon — 9,6988696 Another Example . N S Altitude at 6 — 3124174 Let the Altitude be 13d 46′ N S 2379684 difference — 744490 Logm — 8,8715646 The former fixed Number — 0,2438255 Sine of 7d 30′ the hour from 6 towards midnight because the Altitude is less then the Altitude of 6. 9,1153901 This method of Calculation will dispatch much faster then the common Canon , when three sides are given to find an angle ; the Azimuth may in like manner be Calculated , but will be more troublesome not having so many fixt Tearms in it , and having got the hour , the Azimuth will be easily found ; in this Case we have two sides and an angle opposite to one of them given , to find the angle opposite to the other , and the Proportion , will hold , As the Cosine of the Altitude , To the sine of the hour from the Meridian : So the Cosine of the Declination , To the sine of the Azimuth from the Meridian . And in this Case the three sides being given , we may determine the affection of any of the angles . If the Sun , or Stars have declination towards the depressed Pole , the Azimuth is always Obtuse , and the hour and angle of position-Acute . If the Sun , &c. have declination towards the Elevated Pole , but less then the Latitude of the place the angle of Position is always acute , the hour before 6 obtuse , the hour and Azimuth between the Altitude of 6 , and the Vertical Altitude both acute , afterwards the hour acute , and the Azimuth obtuse . But when the Sun or Stars come to the Meridian between the Zenith and the Elevated Pole , as when their declination is greater then the Latitude of the place , the Azimuth is always acute , the hour before 6 obtuse , afterwards acute . The angle of position from the time of rising to the remotest Azimuth from the Meridian is acute , afterwards obtuse . Another General Proportion for the Hour . As the Radius , To the Tangent of the Latitude : So the Tangent of the Suns declination , To the sine of the hour of rising from six : Again . To the Rectangle of the Cosine of the Latitude , and of the Cosine of the declination , Is to the Square of the Radius : So is the sine of the Altitude , To the difference of the Versed sines of the Semidiurnal Ark , and of the hour sought . Having got the Logarithm of this difference , take the natural number out of the Sines or Tangents that stands against it , accordingly as the Logme is sought , and in Winter add it to the natural sine of the hour of rising from 6 , the sum is the natural sine of the hour from 6 towards noon . In Summer get the difference between this fourth and the sine of rising from 6 , the said difference is the natural sine of the hour from 6 towards noon , when the Number found by the Proportion is greater then the sine of rising , towards midnight when less . The Canon is the same without Variation as well for South declinations as for North , and therefore we may by help thereof find two hours to the same Altitude . Example . Comp Lat 38d 28′ Ar Comp Sine — , 2061683 Comp declin 77d Ar Comp — , 0112761 fixed number — , 2174444 Let the Altitude be 14d 38′ Sine — 9,4024889 Natural sine against it — 41660 , 00 — 9,6199333 Nat sine of rising from 6 — 29058 , 79 Sum — 70718 , 79 N sine of 45d the hour from six in Winter . Difference — 12601 , 21 Sine of 7d 14′ the hour from 6 in Summer towards noon to the former Altitude , and like declination towards Elevated Pole. Another Example for the same Latitude and Declination . Logme Let the Altitude be 20d 25′ Sine — 9,5426321 The former fixed Logme — 2174444 Natural sine against it — 5754811 Sum — 9,7600765 Natural sine of rising — 2905879 Sum — 9,7600765 Sum — 8660690 sine 60d the ho from 6 in Wint Difference — 2848932 sine 16d 33′ the hour from 6 in Summer towards noon ; And thus may two hours be found at one operation for all Altitudes less then the Winter Meridian Altitude , to be converted into usual Time by allowing 15d to an hour , and 4● to a degree . To Calculate a Table of the Suns Altitudes on all Hours . As the Secant of the Latitude , To the Cosine of the Declination , Or which is all one , As the Square of the Radius : To the Rectangle of the Cosines , both of the Latitude and of the Declination : So is the sine of the hour from 6. To a fourth , namely , in Summer the difference of the sines of the Suns Altitude at 6 , and of the Altitude sought , in Winter the sum of the sines of the Suns Depression at 6 , and of the Altitude sought . Having wrote down the Logarithmical sines of the hour from 6 on the Paper , at one end of a piece of Card may be wrote down the sum of the Logarithmical Cosines of the Latitude and Declination , and add the same to the sine of the hour rejecting the double Radius , and take the natural sine that stands against the sum sought in the Logarithmical sines ; having this natural sine , get the sum and difference of it , and of the natural of the Suns Altitude at 6 , the sum is the natural sine of the Altitude for Summer declinations , and the difference for Winter Declinations when the sine of the Suns Altitude is the lesser : But when it is the greater , the said difference is the natural sine of the Altitude for hours beyond 6 towards midnight . Example . Log m Complement Latitude 38d 28′ Sine — 9,7938317 Compl Declination 66 29 Sine — 9,9623428 fixed Logme — 19,7561745 Let the hour be 30d that is 2 hours before and after six in Summer sine 9,6989700 Sum — 9,4551445 Natural sine against it — 2851308 Nat sine of the Altitude at 6 — 3124174 Sum — 5975482 Sine of 36d 42′ being the two Altitudes for 4 and 8 in the morning or afternoon , in Summer . Difference — 272866 Sine of 1 34 being the two Altitudes for 4 and 8 in the morning or afternoon , in Summer . Another Example . Let the hour be 45d from six Sine — 9,8494850 the former fixed Logme — 19,7561745 N Sine against it — 4032791 Sum — 9,6056595 N Sine of Altitude at 6 — 3124174 Sum — 9,6056595 Sum — 7156965 Sine of 45d 42′ being the two Altitudes for the hours of 9 or 3 in Summer or Winter for Declination 23d 31′ both towards the Elevated and Depressed Pole. Difference — 908617 Sine — 5 13 being the two Altitudes for the hours of 9 or 3 in Summer or Winter for Declination 23d 31′ both towards the Elevated and Depressed Pole. By the former Canon was the following Table of Altitudes calculated , and that with much celerity beyond any other way , it will not be amiss to Calculate the Suns Altitude at 6 by the natural Tables only , however the Logarithms will accurately discover the natural sine of at , if duly Proportioned by the differences . A Table of the Suns Altitudes for each Hour and quarter for the Latitude of London .   North.   South .   Declination . 23d , 31 13 Equator . 13d 23 31   XII . 61d , 59 51 , 28 38 , 28. 25 , 28 14 , 57 XII . . 61 , 49 51 , 21 38 , 22 25 , 20 14 , 52   * 61 , 23 50 , 59 38 , 4 25 , 8 14 , 39   . 60 , 40 50 , 24 37 , 36 24 , 43 14 , 18   I. 59 , 42 49 , 36 36 56 24 , 10 13 , 48 XI .   58 , 29 48 , 35 36 , 5 23 , 26 13 , 9     57 , 4 47 , 24 35 , 4 22 , 34 12 , 23     55 , 29 46 , 1 33 , 55 21 , 33 11 , 29   II. 53 , 45 44 , 3● 32 , 36 20 , 25 10 , 28 X.   51 , 53 42 , 51 31 , 8 19 , 8 9 , 19     49 , 54 41 , 4 29 , 34 17 , 44 8 , 3     47 , 51 39 , 1● 27 , 53 16 , 14 6 41   III. 45 , 42 37 1● 26 , 6 14 , 38 5 13 IX .   43 , 31 35 , 9 24 , 12 12 , 55 3 , 39     41 , 16 33 , 2 22 , 15 11 , 7 1 , 59     38 , 59 30 , 52 20 , 13 9 , 15 0 , 15   IIII. 36 , 42 28 , 37 18. 7 7 , 17     VIII .   34 , 23 26 , 22 15 , 58 5 , 17         32 , 4 24 , 5 13 , 46 3 , 12         29 , 43 21 , 46 11 , 32 1 , 4       V. 27 , 23 19. 27 9 , 16     VII .     25 , 4 17 , 7 6 , 58             22 , 46 14 , 47 4 , 39             20 , 28 12 , 27 2 , 20           VI. 18 , 13 10 , 9 00 , 00 VI.         15 , 58 7 , 51                 13 , 46 5 , 34                 11 , 37 3 , 20               VII . 9 , 30 1 , 7 V.             7 , 25     Declination . Ascensionall difference .         5 , 24             3 , 27     13d   16d , 54′       VIII . 1 , 34 IIII. 23 , 31 33 , 12 R A A Table of the Suns Altitudes for every 5 degrees of Azimuth from the Meridian for the Latitude of London . North. South . Declination . 2● d 31 13d Equator . 13● 23d , 31′ Mer Alt 61 , 59 51 , 28 38d 28′ 25d , 28′ 14d , 57   5 61 , 55 51 , 23 38 , 21 25 , 21 14 , 49   10 61 , 42 51 , 7 38 , 2 24 , 57 14 , 22   15 61 , 21 50 , 40 37 , 30 24 , 20 13 , 39   20 60 , 51 50 , 3 36 , 44 23 , 27 12 , 39   25 60 , 11 49 , 14 35 , 45 22 , 16 11 , 19   30 59 21 48 , 13 34 , 32 20 , 51 9 , 43   35 58 , 20 46 , 59 33 , 3 19 , 7 7 , 46   40 57 , 7 45 , 31 31 , 19 17 , 7 5 , 31   45 55 , 43 43 , 50 29 , 19 14 , 50 2 , 57   50 54 , 3 41 , 53 27 , 3 12 , 13 0 , 03   55 52 , 7 39 , 39 24 , 30 9 , 21       60 49 , 56 37 , 9 21 , 40 6 , 11       65 47 , 27 34 , 22 18 , 34 2 , 46       70 44 , 39 31 , 18 15 , 12           75 41 , 34 27 , 58 11 , 37           80 38 , 10 24 , 23 7 , 51           85 34 , 32 20 , 38 3 , 59           90 30 39 16 , 42               95 26 , 34 12 , 40               100 22 , 28 8 , 41               105 18 , 20 4 , 44 Declinat Amplitude .       110 14 , 15 0 , 54 13d ,   21d , 12′       115 10 , 19     23 , 31′ 39d , 54′       120 6 , 36                   125 3 , 7                 Many Tables may want the naturall Tables standing against the Logarithmicall ; therefore the method of Calculation by the Logarithmicall Tables onely , is not to be omitted , albeit we wave the common Proportions , when three Sides are given to find an Angle . A general Proportion derived from the book of the honorable Baron of Marchist●n , which may bee wrought on a Serpentine Line without the use of Versed Sines , or finding the half distance between the 7th Tearme and the Radius , not encumbred with Rectangles , Squares , or Differences of Sines , or Versed Sines . Three Sides to find an Angle ; two of them or all three being lesse then Quadrants . By a supposed perpendicular , which need not to be named . As the Tangent of half the greater of the containing sides , To the Tangent of the half sum of the other sides : So is the Tangent of half their difference , To the tangent of a fourth Arke . If this Arke be greater then the half of the first assumed side ; namely , then the Arke of the first Tearme , in the Proportion , the ( Supposed perpendicular falls without ) Angles opposite to the two other Sides are of a different Affection , the greatest side subtending the Obtuse angle , and the lesser the Accute . If the Angle Opposite to the greater of the other Sides be sought ▪ Take the difference ; if to the lesser , the Sum of the 4th Arke , and of half the containing Side , which half is the first Tearme in the Proportion , Then , As the Radius , To the Cotangent of the other Containing Side : So the Tangent of the said Sum or difference , To the Cosine of the Angle sought . The first Tearme above needs no Restraint , but when one of the Containing Sides is greater then a quadrant : If the 4th Arke be less then the half of the first assumed Side , the Perpendicular falls within , in this Case , the two Angles , opposite to the two other sides may be found , being both Acute . Get the Sum and difference of half the first assumed Side , and of the 4● Arke , the Sum is the greater Segment , and the Difference or residue , the lesser Segment ; the Perpendicular alwayes falling on the side assumed , first into the Proportion ; Then , As the Radius , To the Cotangent of the lesser of the other Containing Sides ; So is the Tangent of the lesser Segment . To the Cosine of the Angle sought . As the Radius , To the Cotangent of the greater of the other Containing Sides ; So is the Tangent of the greater Segment . To the Cosine of the Angle sought . From these general Directions is derived this Canon for Calculating the Azimuth ; As the Tangent of half the Complement of the Altitude ; To the Tangent of the half sum of the Sun or Stars distance from this elevated pole , and of the Complement of the Latitude : So is the Tangent of half their difference , To the Tangent of a 4 h Arke . If this Ark be less then half the Complement of the Altitude , the Azimuth is Acute ; if more obtuse , in both Cases , get the difference of these two Arkes , if there be no difference , the Azimuth is 90d from the Meridian ; Then , As the Radius , To the Tangent of the Latitude ; So the Tangent of the said Arke of difference , To the Sine of the Azimuth from the prime Verticall . This when the Sun or Stars do not come to the Meridian between the Zenith and the elevated pole : but when they do , Let the sum of the 4th Ark , and of half the Complement of the Altitude , be the third Tearme in the latter Proportion . This is a ready Way to Calculate a Table of Azimuths ; two Tearms in each Proportion being fixed for one Declination ; and the Azimuth being known , the Hour may be found by a single Operation , As the Cosine of the Declination , Is to the Sine of the Azimuth , from the Meridian : So is the Cosine of the Altitude , To the Sine of the Hour , from the Meridian . Example , for 13d of North Declination . 77d Complement of Declination the Polar distance . 38. 28 Complement of Latitude at London . 115 , 28 Sum : half sum 57d , 44′ Tangent . 10,1997231 . 38 , 32 difference : half diff 19 , 16 Tangent 9 ▪ 5434594 fixed for that declination — 19,7432225 Altitude 4d , 44′ Comp. 85d 16′ half 42d 38′ Tang. 9,9640811 Tangent of — 31. 1 — 9,7791414 difference — 11 37 Tangent 9,3129675 Tangent of 51d 32′ the Latitude — 10,0999135 Sine of 15d , the Azimuth 9,4128810 from East or West Northwards , because the Ark found by 1st Operation was less then half the Complement of the Altitude . Another Example for the Altitude 34d 22′   Log ● The fixed Number — 19,7432225 Compl. Altitude 55● 38′ half 27 , 49● Tangent 9,7223147 Tangent of — 46 , 23 — 10,0209078 Difference — 18 , 34 Tangent 9,5261966 Tangent of the Latitude — 10,0969135 The Sine of 25d the Azimuth from — 9,6261101 East to West Southwards because the first Ark was more then half the Coaltitude . The hours to these two Azimuths will be found by the latter Proportion to be — 98d , 54′ 50 , 9 from Noon . To Calculate the Suns Altitude on all Hours and Azimuths . The first operation shall be to find such an Ark as may remain fixed in one Latitude to serve to all Declinations in both Cases : So that but one Operation more need be required . The Proportion to find it is As the Radius to the Cotangent of the Latitude , So is the Sine of any Hour from 6 , or Azimuth from the Vertical To the Tangent of a fourth Ark. This 4th Arke ( if the Azimuth be accounted from the Vertical , that is , from the points of East or West towards noon Meridian ) Is the Altitude that the Sun shall have , being in the Equinoctial , upon that Azumuth , and so one of the Quesita . If the hour from 6 be accounted upward on the Equinoctial , this 4 ●h Ark is the ark or portion of the hour Circle , between the Equinoctial and Horizon . This Ark for Hours and Azimuths beyond 6 , or the Vertical , towards the midnight Meridian , is the Depression under the Horizon , according to the Denominations already given it . For the Altitudes on all Hours . When the Sun is in the Equinoctial , As the Radius , is to the Cosine of the Latitude : So is the Sine of the Hour from 6 To the Sine of his Altitude . In all other Cases , If the Hour from noon be more then 6 Substract the Equinoctial Arks that Correspond to such parts of time as you would Calculate Altitudes for ; out of the Suns distance from the Depressed Pole , and then it will hold , If the Hour from noon be less then 6 Substract the Equinoctial Arks that Correspond to such parts of time as you would Calculate Altitudes for ; out of the Suns distance from the Elevated Pole , and then it will hold , As the Cosine of the Ark found by the first common Proportion , Is to the Cosine of the Ark remaining ; So is the Sine of the Latitude , To the Sine of the Altitude sought ▪ For the more speedy Calculating a Table of the Suns Altitudes for this Latitude to any Declination , there is added a Table of these Equinoctial Arks for every hour and quarter , as also for every 5d of Azimuth ; The use whereof shall be illustrated by an Example or two , observing by the way , that the same Ark belongs to two hours alike remote on each side from six , as also like Arks to two Azimuths equally remote on each side the Vertical . Hours on each side six , Azimuths on each side the Vertical .   d ′ VI 0 , 00 fixed Arks       5 , 55   5d 3 , 59   8 , 49   10 7 , 51 VII 11 , 37 V 15 11 , 37   14 , 19         16 , 55   0 15 , 12   19 , 22   25 18 , 34 VIII 21 , 40 IIII 30 21 , 40   23 , 49         25 , 48   35 24 , 30   27 , 39   40 27 , 3 IX 29 , 19 III 45 29 , 19   30 , 52         32 , 13   50 31 , 19   33 , 27   55 33 , 3 X 34 , 32 II 60 34 , 32   35 , 28         36 , 17   65 35 , 45   36 , 57   70 36 , 44 XI 37 , 30 I 75 37 , 30   37 , 55½         38 , 13½   80 38 , 2   38 , 24½   85 38 , 21 XII 38 , 28   90 38 , 28 Equinoctial Altitudes or Depressions . Example . Admit it were required to Calculate the Suns Altitude for 2 or 10 of the Clock , when his declination is 23d 31′ North. Suns distance from the Elevated Pole — 66 : 29 fixed Ark for 2 or 10 is — 34 : 32 Residue — 31 : 57 Complement of the Residue 58d 3′ Sine — 9,9286571 Sine of — 51 32 the Latitude — 9,8937452 19,8224023 Sine of — 55 28 Com first ark 9,9158200 Sine of — 53 45 the — 9,9065823 Altitude sought — 9,9065823 So if it were required to Calculate the Suns Altitude for the hour ▪ of 5 in the morning when his declination is 20d North ☉ distance from depressed Pole — 110d fixed Ark — 11 : 37 The Residue is 81d , 33′ Or — 98 23 Sine of 8d , 23′ the Compl of the Residue — 9,1637434 Sine of the Latitude — 9,8937452 19,0574886 Sine of 78d 23′ the Compl fixed ark — 9,9910119 Sine of 6d 42′ the Altitude sought — 9,0664767 For the Altitudes on all Azimuths . As the Sine of the Latitude , Is to the Cosine of the Equinoctial Altitude : So is the Sine of the Declination , To the Sine of a fourth Ark. Get the sum and difference of the Equinoctial Altitude , and of this fourth Ark the sum is the Summer Altitude for Azimuths from the Vertical towards noon Meridian . The difference when this 4th Ark is lesser greater then the Equinoctial Altitude is the Winter Summer Altitude for Azimuths from the Vertical towards Noon Midnight Meridian Example . Let it be required to Calculate Altitudes for the Suns Azimuth 30d from the Meridian , that is 60d from the Vertical for Declination 23d 31′ both North and South . This will be speedily done , add the Logme of the Sine of the Declination to the Arithmetical Complement of the Logme of the Sine of the Latitude , this number varies not for that declination , and to the Amount add the Logme of the Cosine of the Equinoctial Altitude , the sum rejecting the Radius is the Logarithmical Sine of the 4th Ark. Sine Latitude Ar Comp — 0,1062548 Sine declination — 9,6009901 Sine of 55d 28′ the Complement of Equinoctial Altitude — 9,7072449 9,9158200 Sine — 24d 49′ — 9,6230649 Equinoct Altitude — 34 32 Sum — 59 21 Summer Difference — 9 43 Winter Altitude for that Azimuth . Another Example : The former Number — 9,7072449 Sine 78d 23′ Comp of Equinoct 9,9910119 Altitude for 15d of Azimuth — Sine of 29d 57′ — 9,6982568 Eq Altitude 11 37 Difference — 18 20 Sum — 41 34 The Summer Altitudes for 15d Azimuth each way from the Vertical to that declination . The Suns Declination 20d North to find his Altitudes for 15d Azimuth on each side the Vertical . Sine of 51d 32′ the Latitude Ar Com — 0,1062548 Sine of 20d the declination — 9,5340517 Sine of 78d 23′ Comp Eq Altitude — 9,9910119 Sine of 25d 20′ — 9,6313184 Sine of — 25d 20′ Eq Altitude — 11 37 Difference — 13 43 Sum — 36 57 The Summer Altitudes for 15d Azimuth on each side the Vertical to that Declination . By the Arithmetical Complement of a Number is meant a residue which makes that first Number equal to the other : And so if from a number or numbers given another Number is to be substracted , and instead thereof a third number added , the totall shall be so much encreased more then it should by the sum made of the number to be substracted , and of that was added : That is to say , in this last Example instead of substracting the Sine of the Latitude from another sum we added the residue thereof , being taken from Radius thereto , and so increased the Total too much each time by the Radius , which is easily rejected . If the Sun or Stars come to the Meridian between the Zenith and the Elevated Pole , as when their Declination is more then the Latitude of the place , the former Rule of Calculation varies not only the sum of the Equinoctial Altitude , alias , the fixed Ark , and of the Ark found by the second Proportion will be more then 90● In this Case the Complement of it to 180 d is the Altitude sought . A double Advertisement . The Declination towards the Elevated Pole supposed more then the Latitude of the place . If the Complement of the Declination be more then the Latitude of the place also , as in this case it always is for the Sun ; the Sun or such Stars shall have two Altitudes on every Azimuth between the Coast of rising or setting , and the remotest Azimuth from the Meridian ; To find what Azimuths those shall be , As the Cosine of the Latitude , To Radius ; So is the Sine of the Declination , To the Sine of the Amplitude , And So is the Cosine of the Declination , To the Sine of the remotest Azimuth from the Meridian . Between the Azimuth of rising , and the remotest Azimuth , the angle of Position is Acute , afterwards Obtuse . The Sun upon the remotest Azimuth , the angle of Position being a right angle , will have but one Altitude to find it . As the Sine of the Declination . To Radius ; So is the Sine of the Latitude , To the Sine of that Altitude . Example . In North Latitude 13d of Barbados . Declination 20 d North , the Suns Amplitude or Coast of rising 69 d 23′ from the North , or 20 d 33′ from the East Northwards , and his remotest Azimuth from the North Meridian 74 d 4●′ . His two Altitudes upon the Azimuth of 74 d from the Meridian 27 d 27′ the lesser and 52 d 27′ the greater , the fixed Ark found by the first Operation , being — 50 d 3′ And by the second Operation the Ark found is — 77 30 Difference being the lesser Altitude is — 27 27 The sum 127d 33′ the Comp to 180 d being the greater Alt is 52 27 Altitude on the remotest Azimuth — 41 07 Upon Azimuths nearer the Meridian then the Coast of rising or setting , it need not be hinted that there will be but one Altitude . The Proportion from the 5th Case of Oblique Sphoerical Triangles to find the Suns Altitude on all Azimuths would be As the Cosine of the Declination , To the Sine of the Azimuth from the Meridian : So is the Cosine of the Latitude , To the Sine of the angle of Position . In such Cases when it will be acute or obtuse is already defined , and where two Altitudes are required it will be both , and being accordingly so made , the Proportion to find the Altitudes would be , As the sine of half the difference of the Azimuth and angle of Position , To the Tangent of half the difference of the Polar distance and Colatitude So the Sine of half the sum of the Azimuth and angle of Position , To the Tangent of half the Complement of the Altitude . The Azimuth being an angle always accounted from the Midnight Meridian ; but the former Proportion derived from the other Trigonometry in this Case is more speedy . Such Stars as have more Declination then the Complement of the Latitude never rise nor set , if their declination be also more then the Latitude of the place , they will have two Altitudes upon every Azimuth , except the remotest from the Meridian , and the Calculation the same as before . Example for the Latitude of London . The middlemost in the great Bears Rump , declination 56d 45′ , The remotest Azimuth will be 61 d 49′ from the Meridian , and the Altitude thereto 69 d 26′ . If that Star have 30 d of Azimuth from the Meridian , The first ark will be 34 d 22′ difference being the lesser Alt 27● 7′ The second ark — 61 39 difference being the lesser Alt 27● 7′ Sum — 96 11 Comp the greater Altitude — 83 49 This will be very evident on a Globe for having rectified it to the Latitude extend a Thread from the Zenith over the Azimuth in the Horizon , then turn the Globe round , and such Stars as have a more utmost remote Azimuth from the Meridian , and do not rise or set will pass twice under the Thread , the Azimuth Latitude and Declination being assigned if it were required to know the time when the Star shall be twice on the same Azimuth it may be found without finding the Altitudes first by 6 ●h Case of Oblique Spherical Triangles , As before get the angle of Position . As the Sine of half the difference of the Complement of the Latitude , and of the Complement of the declination . To the Sine of half their sum : So the Tangent of half the difference of the Azimuth from the Meridian , and of the angle of Position , To the Cotangent of half the hour from the Meridian , to be converted into common time if it relate to Stars : In each Proportion there is two fixed Tearms . The Illustration how these two fixed Arks are obtained , is evident from the Analemma in the Scheme annexed . AE E represents the Equator . H K the Horison , P T the Axis , Z N the Prime , Vertical S D a Parralel of North declination , ☉ another of South , declination , P S ☉ and P A N the Arks of two hours Circles between six and noon , and P D T another before 6. In the Triangle E L B right angled , there is given the angle at E , the Complement of the Latitude , the side E L the hour from 6 , with the right angle at L to find the side L B the ark of the hour Circle contained between the Equinoctial and the Horzion by 11th Case of right angled Sphoerical Triangles , the Proportion will be , As AE E Radius , To AE H Cotangent Latitude : So E L Sine of the hour from 6 , To L B The Tangent of the said Ark. From Z draw the Arches Z S and Z ☉ ( being Ellipses ) through the Points where the Hour Circle , and Parralels of Declination intersect , and they represent the Complements of the Altitudes sought , and let fall a Perpendicular from Z to R , then will P R be equal to L B , because the Proportion above is the same that would Calculate R P ; Which substracted from P S or P ☉ the Suns distance in Summer or Winter from the Elevated Pole rests R S or R ☉ : Now in an Oblique Spherical Triangle reduced to two right angled Triangles by the Demission of a Perpendicular , the Cosines of the Bases are in direct Proportion to the Cosines of the Hipotenusals a consequence derived from the general Axiom of the Lord Napier , and therefore it holds , As the Cosine of the Ark R P , or B L , that we call the common fixed ark , Is to the Sine of the Latitude , the Complement of Z P : So the Cosine of the Ark remaining , that is the Complement of R S or S O , To the Sine of the Altitude , the Complement of Z S or Z ☉ : This for all hours under 90● from the Meridian , but for those before or after 6 in the Summer it may be observed in the opposite Triangles E A N and E A I , counting the hour E A each way from 6 that the Ark of the hour Circle A N equal to A I , as much as it is above the Horizon in Winter , so much is it below the same in Summer , and the Suns distance from the Elevated Pole then equal to his distance from the Depressed Pole now : and the Zenith distance then equal to the Nader distance now , as is evident in the Triangle T N D. So that in this Case the Sun is only supposed to have Winter instead of Summer Declination , and the Rule for Calculating his Altitude the same as for Winter Altitudes . In like manner for the Azimuth . In the Schem following H Q represents the Horizon , AE F the Equator , S G a Parralel of Summer declination , and another passing through M of Winter declination , Z S L and Z N A two Azimuths between the Vertical and Noon Meridian , Z K I N another between it and the Midnight Meridian , from the Points S M I let fall the Perpendiculars S D , M O , I G representing the ☉ Declination in the Ellipses of several hour Circles ; So will L S , M L , A K , represent the Altitudes of these 3 Azimuths respectively , according to the proper Declination , for the finding whereof there is given in the Triangle E L B the side E L the Azimuth from the Vertical the angle B E L , the Complement of the Latitude and the right angle , and the Proportion by 11th Case of right angled Sphoerical Triangle is As AE H the Radius , To H AE the Cotangent of the Latitude : So E L the Sine of the Azimuth from the Vertical , To L B the Tangent of the Equinoctial Altitude to that Azimuth . Thut we see the first Proportions common to both , this Case issuing from the 2d Axiom of Pitiscus , that in many right angled Sphoerical Triangles having the same Acute angle at the Base , the Sines of the Bases and Tangents of the Perpendiculars are proportional . For the second Operation to find B S. Though the Analogy is derivable from the general Proposition of the Lord Napier , yet here I shall take it from Ptitiscus the 1 Axiom . That in many right angled Sphoerical Triangles , having the same acute angle at the Base the Sines of the Perpendiculars and Hipothenusals are in direct Proportion . Therefore in the Triangle AE Z B , and D S B it will hold , As AE Z the Sine of the Latitude , To Z B the Cosine of the Equinoctial Altitude , So is D S , the Sine of the Declination , To the Sine of B S , which is equal to B M : See 29th Prop. of 3d book Regiomontanus , I prove it thus , The opposite angles at B are equal , and the angles at D and O are equal , and the side D S , equal to M O , it will then be evinced by Proportion , As the Sine of the angle at B , To its opposite side M O or D S : So is the Radius , that is the angle at O or D , To its opposite side S B , or M B. This equality being admitted , if unto L B we add B S , the sum is L S the Altitude for Summer Declination , if from B L we take B M equal to B S , the remainder M L is the Altitude for the like Declination towards the Depressed Pole , being the Winter Altitude of that Azimuth . But for Azimuths above 90 d from the Meridian it may be observed in the two Opposite Triangles E A N , and E A I , counting the Azimuth E A each way from the Vertical its Equinoctial Altitude A N in the Winter is equal to its Equinoctial Depression A I in the Summer , and is to be found by the 1 Proportion . The second Proportion varies not . As the Sine of the Latitude N F equal to AE Z , Is to N I equal to Z B the Cosine of the Equinoctial Altitude or Depression : So is the Sine of the Declination , I G equal to D S : To Sine I K , from which taking A I , the Equinoctial Depression rests A K , the Altitude sought . To Calculate a Table of the Suns Altitude for all Azimuths and hours under the Equinoctial . This will be two Cases of a Quadrantal Sphoerical Triangle . 1. For the Altitudes on all Azimuths . There would be given the side A B a Quadrant , the angle at B the Azimuth from the Meridian , and the side A D the Complement of the Suns Declination . If the side B D be continued to a Quadrant , the angle at C will be a right angled , besides which in the Triangle A D C , there would be given A D as before the Complement of the Suns Declination , and A C the measure of the angle at B to find D C the Suns Altitude being the Complement of B D , and so having the Hipotenusal , and one of the Leggs of a right angled Sphoerical Triangle , by the 7th Case we may find the other Legg , the Proportion sutable to this question would be As the Sine of the Azimuth from East or West , Is to the Radius : So is the Sine of the Declination , To the Cosine of the Altitude sought . 2. For the Altitudes on all Hours . There would be given the side A B a Quadrant , A D the Complement of the Suns declination with the contained angle B A D the hour from noon , to find the side B D the Complement of the Suns Altitude . Here again if B D be continued to a Quadrant , the angle at C is a right angle , the side A D remains common , the angle D A C is the Complement of the Angle B A D , See Page 57 where it is delivered , That if a Sphoerical Triangle have one right angle , and one side a Quadrant , it hath two right angles , and two Quadrantal sides , and therefore the angle B A C is a right angle ; this is coincident with the 8th Case of right angled Spherical Triangles , the Proportion thereof is , As the Radius , Is to the Cosine of the Declination , So is the Sine of the hour from six , To the Sine of the Altitude , sought . Affections of Sphoerical Triangles . BEcause the last Affection in page 57 is not Braced in the beginning , and a mistake of lesser for greater , in the last Brace but one , I thought fit to recite it at large . Any side of a Sphoerical Triangle being continued , if the other sides together are equal to a Semicircle , the outward angle on the side continued shall be equal to the inward angle on the said side opposite thereto . If the sides are less then a Semicircle , the outward angle will be greater then the inward opposite angle ; But if the said sides are together greater then a Semicircle , the outward angle will be less then the inward opposite angle . In the Triangle annexed , if the sides A B and A C together are equal to a Semicircle , then is the angle A C D equal to the angle A B C. If less then a Semicircle then is the said angle greater then the angle at B. But if they be greater , then is the said angle A C D less then the angle at B. By reason of the first Affection in page 58 ( which wants a Brace in the first Line ) after the words two sides ) We require in the first , second , and other Cases of Oblique angled Sphoerical Triangles , the sum of the two sides or angles given , to be less then a Semicircle . Before I finish the Trigonometrical part , I think it not amiss to give a Determination of the certain Cases about Opposite sides and Angles in Sphoerical Triangles , having before shewn which are the doubtful , and the rather because that this was never yet spoke to . Two sides with an Angle opposite to one of them , to determine the Affection of the Angle opposite to the other . 1. If the given angle be Acute , and the opposite side less then a Quadrant , and the adjacent side less then the former side . The angle it subtends is acute because subtended by a lesser side , for in all Sphoerical Triangles the lesser side subtends the lesser angle and the Converse . 2. If the given angle be acute , and the opposite side less then a Quadrant , and the other side greater then the former side : This is a doubtfull Case , if it be less then a Quadrant it may subtend either an Acute or an Obtuse angle , and so it may also do if it be greater then a Quadrant , yet we may determine , That when the given angle is acute , and the opposite side less then a Quadrant , but greater then the Complement of the adjacent side to a Semicircle ( which it cannot be unless the adjacent side be greater then a Quadrant ) the angle opposite thereto will be obtuse . 3. The given angle Acute , and the opposite side greater then a Quadrant , and the other side greater . If two sides be greater then Quadrants , if one of them subtends an Acute angle , the other must subtend an Obtuse angle , by the 1st . Affection in pag 58. 4. The given angle Acute , and the opposite side greater then a Quadrant ; The other side cannot be lesser then the former , by what was now spoken . 5. If the given angle be Obtuse , and the opposite side less then Quadrant , the other side less subtends an Acute angle . 6. If the given angle be Obtuse , and the opposite side less then a Quadrant ; The other side greater then the former side must of necessity be also greater then a Quadrant , otherwise two sides less then Quadrants should subtend two Obtuse angles , contrary to the first Affection in p 58. 7. If the given angle be Obtuse , and the Opposite side greater then a Quadrant ; If the other side be greater then the former it will subtend a more Obtuse angle . 8. If the given angle be Obtuse , and the Opposite side greater then a Quadrant , If the other side be less then the former , whether it be lesser or greater then a Quadrant it may either subtend an Acute or Obtuse angle . But we may determine That when the given angle is Obtuse , and the Opposite side greater then a Quadrant ; If the Complement of the adjacent side to a Semicircle be greater then the said opposite side , the angle subtended by the said adjacent side is Acute . Two Angles with a side Opposite to one of them , to determine the Affection of the side opposite to the other . 1. If the given angle be Acute , and the opposite side less then a Quadrant ; If the other angle be less , the side opposite thereto shall be less then a Quadrant because it subtends a lesser angle . 2. If the given angle be Acute , and the Opposite side less then a Quadrant , If the other angle be greater the Case is ambiguous , yet we may determine , If the given angle be Acute , and the opposite side lesser then a Quadrant , if the Complement of the other angle to a Semicircle be less then the Acute angle ( which it cannot be but when the latter angle is Obtuse ) the side subtending it shall be greater then a Quadrant . 3. If the given angle be Acute , and the opposite side grnater then a Quadrant , if the other angle be lesser . Then by 12 of 4 book of Regiomontanus , if two Acute angles be unequal , the side opposite to the lesser of them shall be less then a Quadrant . 4. If the given angle be Acute , and the opposite side greater then a Quadrant , If the other angle be greater , It must of necessity be Obtuse , because otherways two Acute unequal angles , the side opposite to the lesser of them should not be lesser then a Quadrant , contraty the former place of Regiomontanus . 5. If the given angle be Obtuse , and the opposite side lesser then a Quadrant , If the other angle be lesser , It must of necessity be Acute , and the side subtending it less then a Quadrant , otherways two sides less then Quadrants should subtend two Obtuse angles , contrary to 1st Affection in page 58. 6. If the given angle be Obtuse , and the opposite side less then a Quadrant , If the other angle be greater , By 13 Prop of 4 h of Regiomontanus , if a Triangle have two Obtuse unequal angles , the side opposite to the greater of them shall be greater then a Quadrant . 7. If the given angle be Obtuse , and the opposite side greater then a Quadrant , If the other angle be greater or more obtuse then the former ; it is subtended by a greater side . 8. If the given angle be Obtuse , and the opposite side greater then a Quadrant , If the other angle be less then the former , the Case is ambiguous ; yet we may determine , That when the given angle is Obtuse , and the opposite side greater then a Quadrant , if the other angle be less then the Complement of the said Obtuse angle to a Semicircle , the side subtending it shall be less then a Quadrant . The former Cases that are still , and alwais will be doubtful , may be determined when three sides are given . A Sphoerical Triangle having two sides less then Quadrants and one greater , will always have one Obtuse angle opposite to that greater side , and both the other angles Acute . A Sphoerical Triangle having three sides less then Quadrants , can have but one obtuse angle ( and many times none ) and that obtuse angle shall be subtended by the greatest side ; But whether the greatest side subtend an Acute or Obtuse angle cannot be known , unless given or found by Calculation , and that may be found several ways . First by help of the Leggs or Sides including the angle sought by 15● Case of right angled Sphoerical Triangles . As the Radius , To the Cosine of one of those Leggs ; So is the Cosine of the other Legg , To the Sine of a fourth Arch. If the third side be greater then the Complement of the fourth Arch to 90d , the angle included is Obtuse , if equal to it a right angle , if less an Acute angle . Secondly , By help of one of the Leggs and the Base or Side subtending the angle sought by 7th Case of right angled Sphoerical Triangles . As the Cosine of the adjacent side , being one of the lesser sides , Is to the Radius : So the Cosine of the opposite side , To the sine of a fourth Arch. If the third side be greater then the Complement of the 4th Arch to 90d , the angle subtended is Acute , if equal to it a right angle , if less an Obtuse angle . All other Cases need no determination , if a Triangle have two sides given bigger then Quadrants , make recourse to the opposite Triangle and it will agree to these Cases . If three angles were given to determine the Affection of the sides if they were all Acute the three sides subtending them will be all less then Quadrants . But observe that though a Triangle that hath but one side greater then a Quadrant , can and shall always have but one Obtuse angle , yet a Triangle that hath but one Obtuse angle may frequently have two sides greater then Quadrants . In this and all other Cases let the angles be changed into sides , and the former Rules will serue , I should have added a brief Application of all the Axioms that are necessary to be remembred , and have reduced the Oblique Cases to setled Proportions ( with the Cadence of Perpendiculars only to shew how they arise ) whereby they will be rendred very facil ; as also the Demonstration of the Affections , which may be hereafter added to some other Treatise to be bound with this Book . Of working Proportions by the Lines on the Quadrant . BEfore I come to shew how all Proportions may in some measure be performed upon the Lines of this Quadrant , it is to be intimated in general , That the working of a Proportion upon a single natural Line , was the useful invention of the late learned Mathematician , Mr Samuel Foster , and published after his decease as his ; In the use of his Scale , a Book called Posthuma Fosteri , as also by Mr Stirrup in a Treatise of Dyalling ; in which Books , though it be there prescribed , and from thence may be learned , yet I acknowledge I received some light concerning it , from some Manuscripts lent me by Mr Foster , in his life time to Transcribe for his and my own use touching Instrumental Applications ; Yet withal be it here intimated , that there are no ways used upon this Quadrant for the obtaining the Hour and Azimuth with Compasses , and the Converse of the 4th Axiom , but what are wholly my own , and altogether novel , though not worth the owning , for Instrumental Conclusions not being so exact as the Tables are of small esteem with the learned as in Mr Wingates Preface to the Posthuma ; besides the taking off of any Line from the Limb to any Radius ; the Explanation of the reason of Proportions so wrought , the supply of many Defects , and the inscribing of Lines in the Limb , I have not seen any thing of Mr Fosters , or of any other mans , tending thereto . Of the Line of equal Parts . THis Line issueth from the Center of the Quadrant on the right edge of the foreside , and will serve for Mensurations , Protractions and Proportional work . The ground of working Proportions by single natural Lines , is built upon the following grounds . That Equiangled Plain Triangles have the sides about their equal angles Proportional , and this work hath its whole dependance on the likeness of two equiangled Plain Right angled Triangles ; as in the figure annexed , let A B represent a Line of equal parts , Sines or natural Tangents issuing from the Center of the Quadrant supposed at A , and let A C represent the Thread , and the Lines B C , E D making right angles with the Line A C , or with the Thread , the nearest distances to it from the Points B and E. I say then that this Scheme doth represent a Proportion of the greater to the less , and the Converse of the less to the greater . First of the greater to the less , and then it lies , As A B to B C : So A E to A D , whence observe that the length of the second Tearm B C must be taken out of the common Scale A B , and one foot of that extent entred at B the first Tearm , the Thread must be laid to the other foot at C , according to the nearest distance then the nearest distance , from the Point E to the Thread that is from the third Tearm called Lateral entrance , being measured in the Scale A B , gives the quantity of the 4 ●h Proportional . Secondly of the less to the greater . And then it lies , As B C to A B : So E D to A E , Or , As E D to A E : So B C to A B , by which it appears that the first Tearm B C must be taken out of the common Scale , and entred one foot at the second Tearm at B , and the Thread laid to the other at C according to nearest distance then the third Tearm E D must be taken out of the common Scale and entred between the Thread and the Scale , so that one foot may rest upon the Line , as at E , and the other turned about may but just touch the Thread , as at D , so is the distance from the Center to E the quantity of the 4th Proportional ; and this is called Parralel entrance , because the extent E D is entred Parralel to the extent B C : To avoid Circumlocution , it is here suggested , that in the following Treatise , we use these expressions to lay the Thread to the other foot , whereby is meant to lay it so according to nearest distance , that the said foot turned about may but just touch the Thread , and so to enter an extent between the Thread and the Scale is to enter it so that one foot resting upon the Scale , the other turned about may but just touch the Thread . Another chief ground in order to working Proportions by help of Lines in the Limb is , That in any Proportion wherin the Radius is not ingredient the Radius may be introduced by working of two Proportions in each of which the Radius shall be included , and that is done by finding two such midle tearms ( one whereof shall always be the Radius ) as shall make a Rectangle or Product equal to the Rectangle or Product of the two middle Tearms proposed , to find which the Proportion will be . As the Radius , To one of the middle Tearms : So the other middle Tearm , To a fourth , I say then , that the Radius and this fourth Tearm making a Product or Rectangle equal to the Product of the two middle Tearms , these may be assumed into the Proportion instead of those , and the answer or fourth Tearm will be the same without Variation , and therefore holds , As the first Tearm of the Proportion , To the Radius : So the fourth found as above , To the Tearm sought . Or , As the first Tearm of the Proportion , Is to the fourth found as above : So is the Radius , To the Tearm sought ; and here observe , that by changing the places of the second and third Tearm , many times a Parralel entrance may be changed into a Lateral , which is more expedite and certain then the other , having thus laid the foundation of working any Proportion , I now come to Examples . 1. To work Proportions in equal parts alone . If the first Tearm be greater then the second , take the second Tearm out of the Scale , and enter one foot of that extent at the first Tearm , laying the Thread to the other foot , then the nearest distance from the third Tearm to the Thread gives the 4th Proportional sought , to be measured in the Scale from the Center . If the first Tearm be less then the second , still as before keep the greatest Tearm on the Scale , and enter the first Tearm upon it , laying the Thread to the other foot , then enter the third Tearm taken out of the Scale between the Thread and the Scale and it finds the 4th Proportional . Example . Admit the Sun shining , I should measure the length of the Shaddow of a Perpendicular Staff and find it to be 5 yards , the length of the Staff being 4 yards , and at the same time the length of the Shaddow of a Chimny , the Altitude whereof is demanded , and find it to be 22½ yards , the Proportion then to acquire the Altititude would be , As the length of the shaddow of the Staff , To the length of the Staff : So the length of the shadow of the Chimney , To the height thereof , that is As 5 to 4 : So 22 ‑ 5 to 18 yards the Altitude or height of the Chimney sought , Enter 4 or the great divisions upon 5 , laying the Thread to the other foot , then the nearest distance from 22 ‑ 5 to the Thread measured will be 18 , and in this latter part each greater division must be understood to be divided into ten parts . And so if the Sun do not shine , the Altitude might be obtained by removing till the Top of a Staff of known height above the eye upon a level ground be brought into the same Visual Line with the Top of the Chimney , and then it holds , As the distance between the Eye and the Staff , To the height of the Staff above the eye : So the distance between the Eye and the Chimney . To the height of the Chimney above the Eye . Some do this by a Looking Glass , others by a Bowl of Water , by going back till they can see the top of the object therein , and then the former Proportion serves , mutatis mutandis . But Proportions in equal parts will be easily wrought by the Pen , the chief use therefore of this Line will be for Protraction , Mensuration , and to divide a Line of lesser length then the Radius of the Quadrant Proportionally into the like parts the Scale is divided , which may be readily done , and so any Proportional part taken off , to do it Enter the length of the Line proposed at the end of the Scale at 10 , and to the other foot lay the Thread the nearest distances from the several parts of the graduated Scale to the Thread shall be the like Proportional parts to the length of the Line proposed , the Proportion thus wrought is , As the length of the graduated Scale , To any lesser length : So the parts of the Scale , To the Proportional like parts to that other length . Of the Line of Tangents on the left edge of the Quadrant . THe chief Uses of this Scale will be to operate Proportions either in Tangents alone or jointly , either with Sines or equal parts , to prick down Dyals , and to proportion out a Tangent to any lesser Radius ▪ To work Proportions in Tangents alone . 1. Of the greater to the less . Enter the second Tearm taken out of the Scale upon the first , laying the Thread to the other foot , then the nearest distance from the third Tearm to the Thread being taken out and measured from the Center shews the 4th Proportional . But if the Proportion be of the less to the greater , Enter the first Tearm taken out of the Scale upon the second , and lay the Thread to the other foot , then enter the third Tearm taken out of the Scale between the Thread and the Scale , and the foot of Compasses will shew the 4 Proportional . Example . Of the greater to the less , As the Tangent of 50d , To the Tangent of 20d : So the Tangent of 30 , To the Tangent of 10d. To work this take the Tangent of 20● in the Compasses , and entring one foot of that extent at 50d , lay the Thread to the other , according to the nearest distance , then will the nearest distance from the Tangent of 30d to the Thread being measured on the Line of Tangents from the Center be the Tangent of 10d the fourth Proportional . By inverting the Order of the Tearms , it will be , Of the less to the greater . As the Tangent of 20d , To the Tangent of 50d : So the Tangent of 10d , To the Tangent of 30d , to be wrought by a Parralel entrance . This Scale of Tangents is continued but to two Radii , or 63d 26′ whereas in many Cases the Tearms given or sought may out-reach the length of the Scale , in such Cases the Propprtion must be changed according to such Directions as are given for varying of Proportions at the end of the 16 Cases of right angled Sphoerical Triangles . In two Cases all the Rules delivered for varying of Proportions will not so vary a Proportion as that it may be wrought on this Line of Tangents . First when the first Tearm is greater then 63d 26′ the length of the Scale , and the rwo middle Tearms each less then 26d 34′ the Complement of the Scale wanting ; In this Case if any two Tearms of the Proportion be varied according to the Rules for varying of Proportions , there will be either in the given Tearms or Answer such a Tangent as shall exceed the length of the Scale , but it may be remedied by a double Proportion by the reason before delivered for introducing the Radius into a Proportion wherein it is not ingredient . As the Radius , To the Tangent of one of the middle Tearms : So the Tangent of the other middle Tearm , To a fourth Tangent : Again . As the Radius , To that fourth Tangent : So is the Cotangent of the first Tearm , To the Tangent of the fourth Ark sought . The Radius may be otherways introduced into a Proportion then here is done , but not conducing to this present purpose , and therefore not mentioned till there be use of it , which will be upon the backside of a great Quadrant of a different contrivance from this , upon which this trouble with the Tangents will be shunned . An Example for this Case . As the Tangent of 65d , To Tangent of 24d : So the Tangent of 20d , To what Tangent ? the Proportion will find 4d , 10′ . Divided into two Proportions will be , As Radius , To Tangent 24d : So Tangent of 20d , To a fourth , the quantity whereof need not be measured . Again . As Radius , To that fourth : So the Tangent of 25d , the Complement of the first Tearm , To the Tangent of 4d 26′ , the fourth Tangent sought . Operation . First enter the Tangent of 24d on the Radius or Tangent of 45d laying the Thread to the other foot , then take the nearest distance to it from 20d , and enter that extent at 45d , laying the Thread to the other foot , then will the nearest distance from 25d , to the Thread if measured from the Center be the Tangent of 5d 26′ sought . The second Case is when the first Tearm of the Proportion is less then the Complement of the Scale wanting , and the two middle Tearms greater then the length of the Scale . This ariseth from the former , for if the Tearms given were the Complements of those in the former Example , they would be agreeable to this Case , and so no further direction is needful about them , for the Tangent sought would be the Complement of that there found , namely 84d 34′ . Hence it may be observed , that a Table of natural Tangents only to 45d , or a Line of natural Tangents only to 45d may serve to operate any Proportion in Tangents whatsoever . To Proportion on out a Tangent to any Radius . Enter the length of the Radius proposed upon the Tangent of 45d and to the other foot of the Compasses lay the Thread according to the nearest distance , then if the respective nearest distances from each degree of the Tangents to the Thread be taken out they shal be Tangents to the assigned Radius : Because the Tangents run but to 63d 26′ whereas there may be occasion in some declining Dyalls to use them to 75d though seldom further ; to supply this defect , they may be supposed to break off at 60 and be supplied in a Line by themselves not issuing from the Center , or only pricks or full-points made at each quarter of an hour , for the 5th hour , that is , from 60d to 75d , and so these distances prickt again from the Center as here is done , either one way or other , the Proportion will hold , As the common Radius of the Tangents , Is to any other Assigned Radius : So is the difference of any two Tangents to the common Radius . To their Proportional difference in that Assigned Radius , And so having Proportioned out the first four hours , the 5th hour may be likewise Proportioned out and pricked forward in one continued streight Line from the end of the 4th hour . To work Proportions in Sines and Tangents by help of the Limb and Line of Tangents issuing from the Center . THough this work may be better done on the backside where the Tangents lye in the Limb , and the Sines issue from the Center , and where also there is a Secant meet for the varying of some Proportions that may excur , yet they may be also performed here supposing the Radius introduced into any Proportion wherein it is not ingredient , the two middle Tearms not being of the same kind as both Tangents or both Sines . To find the 4th Proportional if it be a Sine . Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the Tangent being the other middle Tearm take the nearest distance to it , then entring this extent upon the first Tearm being a Tangent lay the Thread to the other foot , and in the equal Limb it shews the Sine sought . So if the Example were , As the Tangent of 50d , To the Sine of 40d : So is the Tangent of 36d , To a Sine , the 4th Proportion would be found to be the Sine of 23d , 4′ . If a Tangent be sought . Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the Tangent being the other middle Tearm , take the nearest distance to it , then lay the Thread to the first Tearm in the Limb , and the former extent entred between the Scale and the Thread finds the Tangent , being the 4th Proportional sought . If the Example were , As the Sine of 40d , To the Tangent of 50d : So is the Sine of 23d 4′ To a fourth a Tangent , it would be found to be the Tangent of 36. These Directions presuppose the varying of the Proportion , as to the two Tangents , when either of them will excur the length of the Scale , of which more when I come to treat of the joint use of the Sines and Tangents on the backside . If both the middle Tearms be Sines , the 1st Operation will be wholly on the Line of Sines on the backside , by introducing the Radius , and the second upon the Line of Tangents on the foreside , likewise , if both the middle Tearms were Tangents , the first Operation would be on the Line of Tangents on the foreside , and the second on the Line of Sines on the backside ; but this is likewise pretermitted for the present , for such Cases will seldom be reducible to practise . To work Proportions in equal Parts and Tangents . Because the Lines to perform this work do both issue from the Center , the Radius need not be introduced in this Case ; but here it must be known whether the first or second Tearm of the Proportion taken out of its proper Scale be the longer of the two , and accordingly the work to be performed on the Scale of the longer Tearm , which shall be illustrated only by a few Examples , the ground of what can be said being already laid down . Example . To find the Suns Altitude , the length of the Gnomon or Perpendicular being assigned , and the length of its Shadow measured . As the length of the Shadow . Is to the Radius : So is the length of the Gnomon , To the Tangent of the Suns Altitude . Example . If the length of the Shaddow were 8 foot , and the length of the Gnomon but 5 foot , because 8 of the greater divisions of the equal parts are longer then the Tangent of 45d take the said Tangent or Radius , and enter it at 8 , laying the Thread to the other foot , then the nearest distance from 5 of the equal parts to the Thread measured on the Tangents sheweth 32d for the Suns Altitude sought . So the distance from a Tower and its Altitude being observed the Proportion , to get the height of the Tower is , As the Radius , To the measured distance : So the Tangent of the Altitude . To the height of the Tower. So in the Example in Page 38 , the measured distance K B was 100 yards , and the Altitude 43d 50′ to find the height of the Tower take the Tangent of 45 d , and enter it on 10 at the end of the equal Scale , laying the Thread to the other foot , then take the Tangent of 43 d 50′ , and enter it between the Scale and the Thread , and the Compasses will rest at 96 the height of the Tower in yards , sometimes each grand division of the equal Scale must represent but one sometimes 10 , and sometimes 100 , as in Case L B , and the Altitude thereto were given to find the height assigned . Another Example . As the Tangent of 60d , Is to 50 : So is the Tangent of 40d , To 24 ‑ 2 as before . Take 50 equal parts , and enter it upon the Tangent of 60 d laying the Thread to the other foot , then the nearest distance from the Tangent of 40 d to the Thread measured on the equal parts from the Center will be 24 ‑ 2 as before . Otherwise . Enter the Tangent of 40 d upon the Tangent of 60 d laying the Thread to the other foot , then enter 50 equal parts down the Line of Tangents from the Center , and the nearest distance from the termination to the Thread measured in the equal parts will be 24 ‑ 2 as before . If both the Tangents in any Proportion be too long , they may be changed into their Complements if one of them may and the other may not be so changed without excursion , then the Proportion may be wrought by the Pen , taking the Tangents out of the Quadrat and Shaddows , or it may be made two Proportions by introducing the Radius as before shewed ; it will not be needful to speak more to this , only one Example for obtaining the Altitude of a Tower at two Stations . As the difference of the Cotangents of the Arks cut at either Stations : Is to Radius : So the distance between those Stations : To the Altitude of the Tower. In the Diagram for this Case the Complements of the angles observed at the two Stations , viz. at G were 20 d , at H 41 d 31′ . Take the distance on the Line of Tangents between these two Arks , and because equal parts are sought , and the said Extent less then 50 , the measured distance changing the second Tearm of the Proportion into the place of the third , Enter the said Extent upon 50 in the equal parts , laying the Thread to the other foot , then if the Tangent of 45 d be entred between the Scale and the Thread , the Compasses will rest upon 96 for the Altitude sought . To work Proportions in equal Parts and Sines by help of the Limb. TO suppose both the middle Tearms to be either equal Parts or Sines , will not be practical , yet may be performed as before hinted , without introducing the Radius , if it be not ingredient , because both these Lines issue from the Center , and may also be performed by the Pen by measuring the Sines one the Line of equal parts , as was instanced in page 41. But supposing the middle Tearms of a different kind . 1. If a Sine be sought , Operate by introducing the Radius . Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the other middle Tearm being equal parts , take the nearest distance to it , one foot of this extent enter at the first Tearm , and the Thread laid to the other foot cuts the Limb at the Ark sought . If the Ark sought be above 70 d this work may better be performed with the Line of equal parts and Sines jointly , as issuing from the Center . 2. If a Number be sought ▪ Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the other middle Tearm being equal parts take the nearest distance to it , Then lay the Thread to the first Tearm in the Limb being a sine , and enter the former extent between the Scale and the Thread , and the foot of the Compasses will on the Line of equal parts shew the fourth Proportional . The Proportion for finding the Altitude of a Tower at one Station by the measured distance , may also be wrought in in equal parts and Sines . For , As the Cosine of the Ark at first Station , To the measured distance thereof from the Tower : So is the Sine of the said Ark , To the Altitude of the Tower. In that former Scheme , the measured distance B H is 85 , and the angle observed at H 48 d 29′ Wherefore I lay the Thread to the Sine of the said Ark in the Limb , counted from the right edge , and from the measured distance in the equal parts take the nearest extent to the Thread , then laying the Thread to the Cosine of the said Ark in the Limb , and entring the former extent between the Thread and the Scale , I shall find the foot of the Compasses to fall upon 96 the Altitude sought . So also in the Triangle A C B , if there were given the side A C 194 , the measured distance between two Stations on the Wall of a Town besieged , and the observed angles at A 25 d 22′ , at C 113 d 22′ , if B were a Battery we might by this work find the distance of it from either A or C , for having two angles given all the three are given , it therefore holds , As the Sine of the angle ot B 41d 16′ , To its opposite side A C 194 , So the Sine of the angle at C 66d , 38′ the Complement , To its Opposite side B A 270 , the distance of the Battery from A Such Proportions as have the Radius in them will be more easily wrought , we shall give some few Examples in Use in Navigation . 1. To find how many Miles or Leagues in each Parralel of Latitude answer to one degree of Longitude . As the Radius , To the Cosine of the Latitude . So the number of Miles in a degree in the Equinoctial , To the Number of Miles in the Parralel . So in 51 d 32′ of Latitude if 60 Miles answer to a degree in the Equinoctial 37 ‑ 3 Miles shall answer to one degree in this Parralel . This is wrought by laying the Thread to 51 d 32′ in the Limb from the left edge towards the right , then take the nearest distance to it from 60 in the equal parts which measured from the Center will be found to reach to 37 ‑ 3 as before . The reason of this facil Operation is because the nearest distance from the end of the Line of equal parts to the Thread is equal to the Cosine of the Latitude , the Scale it self being equal to the Radius , and therefore needs not be taken out of a Scale of Sines and entred upon the first Tearm the Radius as in other Proportions in Sines of of the greater to the less , when wrought upon a single Line only issuing from the Center , where the second Tearm must be taken out of a Scale , and entred upon the first Tearm . 2. The Course and Distance given to find the difference of Latitude in Leagues or Miles . As the Radius , To the Cosine of the Rumb from the Meridian : So the Distance sailed , To the difference of Latitude in like parts . Example . A Ship sailed S W by W , that is on a Rumb 56 d 15′ from the Meridian 60 Miles , the difference of Latitude in Miles will be found to be 33 ‑ 3 the Operation being all one with the former , Lay the Thread to the Rumb in the Limb , and from 60 take the nearest distance to it , which measured in the Scale of equal parts will be found as before , 3. The Course and Distance given to find the Departure from the Meridian , alias the Variation . As the Radius , To the Sine of the Rumb from the Meridian : So the distance Sailed , To the Departure from the Meridian . In the former Example to find the Departure from the Meridian , Lay the Thread to the Rumb counted from the right edge towards the left , that is to 56d 15′ so counted , and from 60 in the equal parts being the Miles Sailed , take the nearest distance to it ; this extent measured in the said Scale will be found to be 49 ‑ 9 Miles , and so if the converse of this were to be wrought , it is evident that the Miles of Departure must be taken out of the Scale of equal parts and entred Parralelly between the Scale and the Thread lying over the Rumb . Many more Examples and Propositions might be illustrated , but these are sufficient , those that would use a Quadrant for this purpose may have the Rumbs traced out or prickt upon the Limb : Now we repair to the backside of the Quadrant . Of the Line of on the right Edge of the Backside . THe Uses of this Line are manifold in Dyalling in drawing Projections in working Proportions , &c. 1. To take of a Proportional Sine to any lesser Radius then the side of the Quadrant , or which is all one , to divide any Line shorter in length then the whole Line of Sines in such manner as the same is divided . Enter the length of the Line proposed at 90 d the end of the Scale of Sines , and to the other foot lay the Thread according to nearest Distance , or measure the length of the Line proposed on the Line of Sines from the Center , and observe to what Sine it is equal , then lay the Thread over the like Arch in the Limb , and the nearest distances to it from each degree of the Line of Sines shall be the Proportional parts sought . And if the Thread be laid over 30 d of the Limbe the nearest distances to it will be Sines to half the Common Radius . 2. From a Line of Sines to take off a Tangent , the Proportion to do it is , As the Cosine of an Arch , To the Radius of the Line proposed : So the Sine of the said Arch , To the Tangent of the said Arch. Enter the Radius of the Tangent proposed at the Cosine of the given Arch , and to the other foot lay the Thread then from the Sine of that Arch take the nearest distance to the Thread , this extent is the length of the Tangent sought ; thus to get the Tangent of 20 d enter the Radius proposed at the Sine of 70 d , then take the nearest distance to the Thread from the Sine of 20 d , this extent is the Tangent of the said Arch in reference to the limited Radius . Otherways by the Limb. Lay the Thread to the Sine of that Arch counted from the right edge whereto you would take out a Tangent , and enter the Radius proposed down the Line of Sines from the Center and take the nearest distance to the Thread then lay the Thread to the like Arch from the left edge , and enter the extent between the Scale and the Thread , the distance of the Foot of the Compasses from the Center shall be the length of the Tangent required . 3. From the Sines to take off a Secant . The Proportion to do it is , At the Cosine of the Arch proposed , To Radius of the Line proposed So the Sine of 90d , the common Radius , To the length of the Secant of that Arch , to the limitted Radius . By the Limb , Lay the Thread to that Arch in the Limb counted from left edge whereto you would take out a Secant , then enter the Limitted Radius between the Scale and the Thread and the distance of the foot of the Compasses from the Center shall be the length of the Secant sought , and the Converse if a Secant and its Radius be given to find the Ark thereto enter the Secant of 90 d then enter the Radius of it between the Thread and the Sines , and the Compasses shews the Ark thereto , if counted from 90 d towards the Center . Otherways . Enter the Radius of the Line you would devide into Secants at the Cosine of that Arch whereto you would take out a Secant , and to the other foot lay the Thread then the nearest distance to the Thread from the Sine of 90 d is the length of the Secant sought : Thus to get the Secant of 20 d enter the Radius limited in the Sine of 70 d then the nearest distance from 90 d to the Thread , is the length of the Secant sought . And here it may be noted , that if you would have the whole length of the Line of Sines to represent the Secant sought , then the Cosine of that Arch which it represents shall be the Radius to it ; so the whole Line of Sines representing a Secant of 70 d , the length of the Sine of 20 d shall be the Radius thereto . It may also be observed , that no Tangent or Secant can be taken of at once larger then the Radius of the Quadrant , nor no Radius entred longer then that is , and that if the Radius entred be in Length ½ ⅓ ¼ of the Sines Tangents to 63 h 26′ 71 34 75 58 Secants to 60d●′ 70 32 75 32 may be taken off by help of the Line of Sines . And here it may be observed , That if the Tangent and Secant of any Arch be added in one streight Line or otherwise in Numbers , the Amount shall be equal to the Tangent of such Ark as shall bisect the remaining part of the Quadrant , as is demonstrated in Pitiscus & Sn●llius . Whence it follows , That if we have a Tangent and Secant no further then to 60 d each , yet a Tangent by the joint use of both Lines may from them be prickt down to 75 d : Wherefore at anty time to lengthen the Tangents double the Arch proposed , and out of the Amount reject 90 d ; The Tangent and Secant of the remainder connected in one streight Sine shall be the Tangent of the Arch sought . Thus to get the Tangent of 70 d the double is 140 d whence 90 d rejected rests 50 d ; the Tangent and Secant of 50 d joined in one streight Line shall be the Tangent of such an Arch as bisects the remaining part of the Quadrant , namely of 70 d. It may also be observed , That the Tangent of an Ark , and the Tangent of half its Complement is equal to the Secant of that Arch as is obvious in drawing of any Projection . A Chord may also be taken off from the Line of Sines , but more facilly by the Line of Chords on the left edge of the Quadrant , and is therefore pretermitted . To work Proportions in Sines alone . Frst , Without the help of the Limb or lesser Sines without introducing the Radius , but upon this Line alone independently . There will be two Cases , 1. If the first Tearm be greater then the second , the entrance is lateral ; Enter the second Tearm upon the first , laying the Thread to the other foot . Then from the third Tearm in the Scale take the nearest distance to the Thread , and measure that Extent from the Center , and it shews the Tearm sought , and so if it were , As Sine 30d , To Sine 10d : So Sine 80d , the fourth Proportional would be found to be 20d. In giving Examples to illustrate the matter , I shall make use of that noted Canon for making the Tables , As the Semiradius , or Sine of 30d , To the Sine of any Arch : So the Cosine of that Arch , To the Sine of that Arch doubled . But when the first Tearm is less then the second , Enter the length of the first Tearm upon the second , laying the Thread to the other foot of that Extent , then enter the third Tearm Parralelly between the Scale and the Thread , and it shews the fourth Proportional sought . So if it were , As the Sine of 10d , To the Sine of 30d : So the Sine of 20d , To a fourth , the 4th Proportional would be found to be 80 d. Another general way will be to do it by help of the Limb , by introducing the Radius in such Proportions wherein it is not Lay the Thread to one of the middle Tearms in the Limbe , and from the other middle Tearm on the Line of Sines take the nearest distance to it , then enter one foot of that extent at the first Tearm on the Line of Sines , and lay the Thread to the other foot , and in the Limbe it shews the 4th Proportional sought . Example . If the three Proportionals were , As Sine 55 d , To Sine 70 d : So Sine 30 d , To a fourth , the fourth Proportional would be found to be 35 d. But if the first Tearm of the Proportion be either a small Ark or the answer above 70 d , the latter part of this general direction for more certainty may be turned into a Parralel entrance , that is to say instead of entring the Extent taken from one of the middle Tearms in the Sines to the Thread laid over the other middle Tearm in the Limb , and entring it at the first Tearm in the Sines finding the Answer in the Limb lay , the Thread to the first Tearm in the Limb , and find the Answer in the Line of Sines by , entring the former extent parralelly between the Scale and the Thread . What hath been spoken concerning the Limb may also be performed by the Line of lesser Sines in the Limb by the same Directions . So if it were , As the Sine of 5d , To Sine 30d : So the Sine of 10d To a fourth , the 4th Proportional would be found to be the Sine of 85d , and the Operation best performed by the joint use of the Line of Sines , and the lesser Sines by making the latter entrance a Parralel entrance . When the Radius is in the third place of a Proportion in Sines of a greater to a less ▪ the Operation is but half so much as when it is not ingredient . Example . As the Cosine of the Latitude , To the Sine of the Declination , So the Radius , To the Sine of the Suns Amplitude . If the Suns declination were 13d , to find his Amplitude in our Latitude for London , take the Sine of 13d and enter one foot of it on the Sine of 38d 28′ and to the other foot lay the Thread , and in the Limb it shews the Amplitude sought to be 21d 12′ . By changing the places of the two middle Tearms , this Example will be turned into a Parralel entrance . Lay the Thread to the Complement of the Latitude in the Limb , and enter the Sine of the Declination between it and the Scale , and you will find the same Ark in the Sines for the Amplitude sought , as was before found in the Limb. Such Proportions of the greater to the less wherein the Radius is not ingredient , that have two fixed or constant Tearms , may be most readily performed by the single Line of Sines without the help of the Limb. An Example for finding the Suns Amplitude . As the Cosine of the Latitude , To the Sine of the Suns greatest declination : So the Sine of the Suns distance from the next Equinoctial Point , To the Sine of the Suns Amplitude . Because the two first Tearms of this Proportion are fixed , the Amplitude answerable to every degree of the Suns place may be found without removing the Thread ; To do it enter the Sine of the Suns greatest Declination 23d 31′ , at the Sine of the Latitudes Complement , and to the other foot lay the Thread , where keep it without alteration , then for every degree of the Suns place counted in the Sines take the nearest distance to the Thread , and measure those extents down the Line of Sines from the Center , and you will find the correspondent Amplitudes . Example . So when the Sun enters ♉ ♍ ♏ ♓ , his Equinoctial distance being 30 d , the Amplitude will be 18 d 41′ , and when he enters ♊ ♌ ♐ ♒ Equinox distance 60 d , the Amplitude will be 33 d 42′ ; and when he enters ♋ ♑ the greatest Amplitude will be 39d 50′ , his distance from the nearest Equinoctial Point being 90 d. But for such Proportions in which there is not two fixed Tearms , the best way to Operate them will be by the joint help of the Limbe and Line of Sines . An Example for finding the Time of the day the Suns Azimuth Declination and Altitude being given . By the Suns Azimuth is meant the angle thereof from the midnight part of the Meridian , the Proportion is As the Cosine of the Declination , To the Sine of the Azimuth : So the Cosine of the Suns Altitude . To the Sine of the hour from the Meridian . Example . So when the Sun hath 18 d 37′ North Declination , if his Azimuth be 69 d from the Meridian , and the Altitude 39 d , the hour will be found to be 49 d 58′ from Noon . So if there were given the Hour , the Declination and Altitude by transposing the Order of the former Proportion , it will hold to find the Azimuth , As the Cosine of the Suns Altitude , To the Sine of the hour from the Meridian : So the Cosine of the Suns Declination , To the Sine of the Azimuth from the Meridian . Commonly in both these Cases the Latitude is also known , and the Affection is to be determined according to Rules formerly given . A Proportion wholly in Secants we have shewed before may be changed wholly into Sines ; but the like mutual conversion of the Sines into Tangents is not yet known , however it may be done in 〈◊〉 of the 16 Cases wherein the Radius is ingredient , for instance , let the Proportion be to find the time of Sun rising . As Radius , To Tangent of Latitude : So the Tangent of the Declination , To the Sine of the hour from 6. Instead of the two first Tearms it may be , As the Cosine of the Latitude , To the Sine of the Latitude , then instead of the Tangent of the Declination say , So is the Sine hereof to a fourth . Again , As the Cosine of the Declination , To that fourth : So Radius , To the Sine of the hour from six : This being derived from the Analemm● by resolving a Triangle , one side whereof is the Arch of a lesser Circle . If a Quadrant want Tangents or Secants in the Limb , but may admit of a Sine from the Center , the Tangent and Secant of the Latitude , &c , may be taken out by what hath been asserted , to half the common Radius , and marked on the Limb , and the Quadrant thereby fitted to perform most of the Propositions of the Sphoere in one Latitude , and how to supply the Defect of a Line of Versed Sines in the Limb shall afterwards be shewne . What hath been spoken concerning a Line of Sines graduated on a Quadrant from the Center , may by help of the equal Limb be performed without it . 1. A Proportional Sine may be taken off to any diminutive Radius . By the Definition of Sines the right Sine of an Arch is a Line falling from the end of that Arch Perpendicularly to the Radius drawn to the other end of the said Arch ; So the Line H K falling Perpendicularly on the Radius F G shall be the Sine of the Arch H G , and by the same Definition the Line G I falling perpendicularly on the Radius F H shall also be the Sine of the said Arch , and whether the Radius be bigger or lesser , this Definition is common , but the Line G I on a Quadrant represents the nearest distance from the Radius to the Thread , therefore a Sine may be taken off from the Limb to any Diminutive Radius , to perform which , Enter the length or Radius proposed down the streight Line that comes from the Center of the Quadrant , and limits the Limb ; observe where the Compasses rests , this I call the fixed Point , because the Compasses must be set down at it , at every taking off , then to take off the Sine of any Arch to that Radius , lay the Thread over the Arch counted in the Limb from the said edge of the Quadrant , and take the nearest distance to it for the length of the Sine sought : But to take out Sines to the Radius of the graduated Limb set down one foot at the Ark in the Limb , and take the nearest distance to the two edge Lines of the Limb , the one shall be the Sine , the other Co-sine of the said Ark. 2. A Proportion in Sines alone may be wrought by help of the Limbe . Take out one of the middle Tearms by the former Prop. and entring it down the right edge from the Center , take the nearest distance to the Thread laid over the other middle Tearm in the Limbe , counted from right edge , then lay the Thread to the first Tearm in the Limb , and enter that extent between the right edge Line and the Thread , the distance of the foot of the Compasses from the Center , is the length of the Sine sought , to be measured in the Limb by entring one foot of that Extent in it : So that the other turned about may but just touch one of the edge or side Lines of the Limb issuing from the Center , or enter that Extent at the concurrence of the Limbe with the said Line , and lay the Thread to the other foot according to the nearest distance , and in the Limbe it shews the Ark sought : Whence may be observed how to prick of an angle by Sines instead of Chords . From this and some other following Propositions I assert the Hour and Azimuth may be found generally by the sole help of the Limb of a Quadrant without Protraction . How from the Lines inscribed in the Limbe to take off a Sine , Tangent , Secant and Versed Sine to any Radius , if less then half the common Radius of the Quadrant . IT hath been asserted , that a Sine may be taken off from the Limb , and by consequence any other Line there put on ; for by being carried thither they are converted into Sines , and put on in the same manner , for by the Definition of Sines , if Lines were carryed Parralel to the right edge of the Quadrant from the equal degrees of the Limb to the left edge they would there constitute a Line of Sines and the Converse . To find the fixed Point enter the Radius proposed twice down the Line of Sines from the Center , or which is all one , Lay the Thread over 30 d of the Limb counted from the right edge towards the left , and enter the limitted Radius between the Thread and the Scale ; so that one foot turned about may just touch the Thread , and the other resting on the Line of Sines , shews the fixed Point , at which if the Compasses be always set down , and the Thread laid over any Ark in the Tangent , Secant or lesser Sines , the nearest distances from the said Point to the Thread shall be the Sine , Tangent , Secant , of the said Ark to the limitted Radius . But for such Lines as are put on to the common Radius , as the Tangent of 45 d , &c. the Radius is to be entred but once down from the Center to find the fixed Point . Of the Line of Secants . This Line singly considered is of small use , but junctim with other Lines of great use for the general finding the Hour and Azimuth : Mr Foster makes use of it in his Posthuma to graduate the Meridian Line of a Mercators Chart , which is done by the perpetual addition of Secants , and the like may be done from this Line lying in the Limb but a better way wil be to do it from a well graduated Meridian Line by doubling or folding the edge of the Chard thereto , and so graduate it by the Pen. Of the Line of Tangents . The joint use of this Line with the Line of Sines is to work Proportions in Sines and Tangents , in any Proportion wrought by help of Lines in the Limb wherein the Radius is not ingredient , the Radius must be introduced according to the general Direction . If the two middle Tearms be Sines there must be one Proportion wrought wholly on the Line of Sines on the Backside , and another on the Line of Tangents on the foreside ; but such Cases are not usual : But if the two middle Tearms be Tangents , the first Operation must be on the line of Tangents on the foreside , and the latter on the line of Sines on this backside , unless the Radius be ingredient . A general Direction to work Proportions when the middle Tearms are of a different Species . If a Sine be sought , Lay the Thread to the Tangent in the Limb being one of the middle Terms , and from the Sine being another of the middle Terms take the nearest distance to it , then lay the Thread to the other Tangent in the Limb , being the first Tearm , and enter the former extent between the Scale and the Thread , and the foot of the Compasses on the Line of Sines will shew the fourth Proportional . Example . If the Proportion were , As the Tangent of 30 d , To the Sine of 25d So is the Tangent of 20 d , To the Sine of 15 d 27′ . Lay the Thread over the Tangent of 20 d in the Limb , and from the Sine of 25 d take the nearest distance to it , then lay the Thread to the Tangent of 30 d , and the former extent so entred that one foot resting on the Sines , the other foot turned about may but just touch the Thread , and the resting foot will shew 15 d 27′ for the Sine sought . 2. If a Tangent be sought . Lay the Thread to the Tangent being one of the middle Tearms , and from the other middle Tearm being a Sine take the nearest distance thereto , then Enter one foot of that extent at the first Tearm being a Sine , and the Thread laid to the other foot shews the fourth Proportional in the Line of Tangents in the Limb. Example . So if the Proportion were , As the Sine of 25 d , To the Tangent of 30 d : So is the Sine of 32 d , To a Tangent , the fourth Proportional would be found to be the Tangent of 35 d 54′ . If the answer fall near the end of the Scale of Tangents , the latter entrance may be made by laying the Thread to the first Tearm in the Limb , and by a Parralel entrance an Ark found on the Line of Sines , then if the Thread be laid over the like Ark in the Limb it will intersect the Tangent sought . These Directions presuppose the varying of the Proportion when the Tangens excur the length of the Scale , according to the Directions in the Trigonometrical part ; but as before suggested , those Directions are insufficient when one of the Tearms or Tangents are less then the Complement of the Scale wanting , and the other greater then the length of the Scale , for two such Arks cannot be changed into their Complements without still incurring the same inconvenience ; in this Case only change the greater Tearm , which may be done by help of the Line of Secants , for , As the Tangent of an Arch , To the Sine of another Arch ; So is the Cosecant of the latter Arch , To the Cotangent of the former . And by Transposing the Order of the Tearms . As a Sine , To a Tangent : So the Cotangent of the latter Arch , To the Cosecant of the former . Example . If the Proportion were , As the Sine of 8d , To the Tangent of 25d So is the Sine of 60d , To the Tangent of 71d : Here we might foreknow by the nature of the Tearms that the Tangent sought would be large or finde by tryal that it cannot be wrought upon the Quadrant : We may therefore vary it thus , As the Tangent of 25d To the Sine of 8d : So the Secant of 30d , To the Tangent of 19d , the Complement of 71d , the Arch sought . Lay the Thread over 8d in the lesser Sines , and set down one foot of the Compasses at the Sine of the same Arch the Thread lyes over in the Limb ; and take the nearest distance to the Thread laid over the Secant of 30 , then lay the Thread to the Tangent of 25d , and enter the former extent between the Thread and the Line of Sines , and the distance of the foot of the Compasses from the Center measured on the Tangents on the foreside sheweth 19d. But a more general Caution in this Case without the help of the Secants , would be by altring the larger Tangent into its Complement by introducing the Radius , and operating the Proportion on the greater Tangent of 45d. If the Proportion were , As the Tangent of 70d , To the Sine of 60d : So the Tangent of 25d , To the Sine of 8d 27′ . By introducing the Radius at two Operations it would be easily wrought , As Radius , To Tangent 25d , So Sine 60d , To a fourth , Again , As the Radius , To the Tangent of 20 d : So that fourth , To the Sine sought . So the former Example wherein a Tangent is sought may be likewise varied . As Radius , To Tangent 25d : So Sine 60d , To a fourth , Again , As that fourth , To the Radius : So is the Sine of 8d , To the Cotangent of the Arch sought , namely to the Tangent of 19d as before . Two Proportions with the Radius in each are as suddenly done as one without the Radius . Operation . Lay the Thread over the Tangent of 25d in the greater Tangents , and from the Sine of 60d take the nearest distance to it , enter that extent at 90 , or the end of the Line of Sines , laying the Thread to the other foot according to the nearest distance , then enter the Sine of 8 parralelly between the Scale and the Thread and the distance of the foot of the Compasses from the Center is the Tangent of the Complement of the Ark sought to be measured in the greater Tangents by setting down one foot at 90d , and the Thread laid to the other , according the nearest-will lye over the Tangent of 19 d. An Example with the Radius ingredient and a Sine sought , Data , Latitude , and Declination , to find the time when the Sun shall be East or West . As the Radius , To the Cotangent of the Latitude : So the Tangent of the Declination , To the Sine of the hour from 6. To be wrought by the help of the lesser Tangents . When the Radius comes first and two Tangents in the middle , change the largest Ark into its Complement to bring it into the first place , and the Radius into the second ; then take out the Tangent of the other middle Ark , either from the foreside from the Scale , or out of the Limb by setting one foot at the Sine of 90 d , and taking the nearest distance to the Thread laid over the Tangent given , then laying the Thread to the Tangent of the first Ark , enter the former extent between the Scale and the Thread , and the foot of the Compasses will shew the Sine sought . Otherways the two middle Tearms being Tangents , as also when the first Tearm and one of the middle Tearms is a Tangent , change the Radius and one of those Tangents into Sines . For , As the Radius , To the Tangent of any Ark : So is the Cosine of the said Ark , To the Sine thereof . And , As the Tangent of any Ark , To Radius : So is the Sine of that Ark , To the Cosine thereof . And so the former Proportion changed will be , As the Sine of the Latitude , To the Cosine of the Latitude : So the Tangent of the Declination , To the Sine of the hour from six , When the Sun shall be East or West . Example . If the Declination were 23d 30′ North , in our Latititude of London 51d 32′ to find the Sine sought , Lay the Thread to the Tangent of the Declination in the Limb , and from the Complement of the Latitude in the Sines take the nearest distance to it , then lay the Thread to the Sine of the Latitude in the lesser Sines and enter the former extent between the Thread and the Scale and the foot of the Compasses sheweth the answer in degrees , if the Thread be laid to the Ark found in the Limb it there sheweth it in Time ; So in this Example the time sought is 20d 14′ , or in Time 1 h 17 h before 6 in the morning or after it in in the Evening . If the Latitude and Declination were given , To find the Suns Azimuth at the Hour of 6. As the Radius , To Cosine of the Latitude : So the Tangent of the Suns Declination , To the Tangent of his Azimuth from the Vertical . In this Case a Tangent being the 4th Tearm sought , the Operation is very facil . Lay the Thread to the Tangent of the Declination in the lesser Tangents , and from the Cosine of the Latitude take the nearest distance to it , and either measure that extent on the Tangents on the foreside , or set one foot of that extent upon the Sine of 90d , and to the other lay the Thread and it will intersect the Tangent sought in the Limb : So in our Latitude when the Sun hath 23d 30′ of declination , his Azimuth at the hour of 6 will be 15 d 9′ from the East or West . Another Example , So if the Suns distance from the nearest Equinoctial Point were 60 d , his right Ascension would be found to be 57 d 48′ . The Proportion to perform this Proposition is , As the Radius , To the Cosine of the Suns greatest Declination : So the Tangent of the Suns distance from the next Equinoctial Point , To the Tangent of the Suns right Ascension , or when the Tangents are large , As the Cosine of the Suns greatest declination , To Radius : So the Cotangent of the Suns distance from the Equinoctial Point . To the Cotangent of his right Ascension . By what hath been said it appears that the working Proportions by the natural Lines is more troublesome then by the Logarithmical , however this trouble wil be shunned in the use of the great Quadrant by help of the Circle on the backside . I now come to shew how the Hour of the Day , and the Azimuth of the Sun may be found universally by the Lines on the Quadrant , which is the principal thing intended . The first Operation for the Hour will be to find what Altitude or Depression the Sun shall have at the hour of 6. The Proportion to find it is , As the Radius , To the Sine of the Latitude : So the sine of the Suns Declination , To the sine of the Altitude sought . Example . So in Latitude 51 d 32′ , the Suns declination being 23 d 31′ , To find his Altitude or Depression at 6 , Lay the Thread to the Sine of the Latitude in the Limb , and from the sine of the Suns Declination take the nearest distance to it , which extent measured from the Center will be found to be 18 d 12′ . This remains fixed for one Day , and therefore must be recorded , or have a mark set to it . Afterwards the Proportion is , As the Cosine of the Declination , To the Secant of the Latitude , Or , As the Cosine of the Latitude , To the Secant of the Declination : So in Summer is the difference , but in Winter the Sum of the sines of the Suns proposed or observed Altitude , and of his Altitude or Depression at 6 , To the Sine of the hour from 6 towards Noon in Winter , as also in Summer when the Altitude is more then the Altitude of 6 , otherways towards Midnight . To Operate this . In Winter to the sine of the Suns Depression at 6 , add the sine of the Altitude proposed , by setting down the extent hereof outward at the end of the former extent ; in Summer take the distance between the sine of the Suns Altitude , and the sine of his Altitude at 6 , and enter either of these extents twice down the Line of sines from the Center , then lay the Thread to the Secant being one of the middle Tearms , and take the nearest distance to it . Lastly , enter one foot of this extent at the first Tearm , being a Sine , and to the other foot lay the Thread , and in the equal Limb it shews the hour from 6 , which is accordingly numbred with hours . But when the Hour is neer Noon , the answer may be found in the Line of Sines with more certainty by laying the Thread to the first Tearm in the Limb , and entring the latter extent Parralelly between the Scale and the Thread . Otherways . Enter the aforesaid sum or difference of sines once down the Line of Sines from the Center , and laying the Thread to the Secant , being one of the middle Tearms , take the nearest distance to it , then lay the Thread to the first Ark in the lesser sines , and enter the former extent between the Thread and the Scale , and the foot on the Compasses on the Line sheweth the Sine of the Hour . Example . If the Altitude were 45d 42′ , take the distance between it and the sine of 18 d 12′ before found enter this extent twice down the Line of sines from the Center , and laying the Thread over the Secant of 51 d 32′ take the nearest to it , then entring one foot of this extent at 66 d 29′ in the Line of Sines the Thread being laid to the other according to nearest distance will lye over 45 d in the Limb shewing the hour to be either 9 in the morning , or 3 in the afternoon , and so it will be found also in the latter Operation by entring the first extent once down the sines , and taking the distance to the Thread lying over the Secant of the Latitude , and then laying the Thread to 66 d 29′ in the 〈◊〉 , and entring that extent between the Scale and the Thread . To find the Suns Azimuth The first Operation will be to get the Suns Altitude in the Vertical Circle , that is , being East ar West . As the sixe of the Latitude , To Radius : So is the Sine of the Declination , To the sine of the Altitud . So in our Latitude of London , when the Sun hath 23d 31′ of declination , his Vertical Altitude in Summer will be found to be 30d 39′ and so much is the Depression when he hath as much South declination . This found either by a Parralel entrance on the Line of Sines by laying the Thread to the sine of the Latitude in the Limb , and entring the sine of the Declination between the Scale and the Thread , or by a Lateral entrance in the Limbe changing the Radius into the third place , and then enter the sine of the Declination on the Sine of the Latitude , laying the Thread to the other foot , and in the Limb it shewes the Altitude sought ; having found this Ark let it be recorded or have a mark set to it , because it remains fixed for one Day , afterwards the Proportion to be wrought is , As the Cosine of the Altitude , To the Tangent of the Latitude : So in Summer is the difference in Winter the sum of the Sines of the Suns Altitude , and of his Vertical Altitude or Depression : To the Sine of the Azimuth from the East or West towards noon Meridian in Winter as also in Summer , when the given Altitude is more then the Vertical Altitude , but if less towards the Midnight Meridian . This Proportion may be wrought divers ways on the Quadrant after the same manner as the former , I shall therefore illustrate it by some Examples . Declination 13d , Latititude 51d 32′ , Vertical Altitude 16d 42′ , Proposed Altitude in Summer — 41d , 53′ . Proposed Altitude in Winter — 12 13 Enter the aforesaid sum or difference of Sines twice down from the Center of the Quadrant , and take the nearest distance to the Thread being laid over the Tangent of the Latitude , this extent set down at the Cosine of the Altitude , and lay the Thread to the other foot and in the Limbe it shews the Azimuth sought . So in this Example the Azimuth will be found to be 40d both in Summer and Winter from East or West towards Noon Meridian . Otherways . Enter the aforesaid sum or difference of sines but down from the Center , and take the nearest distance to the Thread laid over the Tangent of the Latitude , then lay the Thread to the Complement of the Altitude in the lesser sines , and enter the former extent between the Scale and the Thread , and the answer will be given in the Line of Sines , supposing the declination unchanged , if the Altitude were 9d 21′ both for the Winter and the Summer Example , the Azimuth at London would be 9d 22′ from the East or West Northwards in Summer , and 35 d Southwards in Winter . Hitherto we suppose the Latitude not to exceed the length of the Tangents , whether it doth or not this Proportion may be otherways wrought by changing the two first Tearms of it ; Instead of the Co-sine of the Altitude to the Tangent of the Latitude , we may say , As the Cotangent of the Latitude , To the Secant of the Altitude : So when the Sun hath 23 d 31′ of North Declination in our Latitude , and his Altitude 57 d 7′ , take the distance between the sine thereof and the sine of 30 d 39′ the Altitude of East , and enter it once down from the Center , and take the nearest distance to the Thread laid over the Secant of the Altitude , viz. 57 d 7′ , then lay the Thread to 38 d 28′ in the Tangents , and enter the former extent between the Scale and the Thread , and the Compasses on the Line of Sines will rest at 50 d for the Azimuth from East or West Southwards , because the Altitude was more then the Vertical Altitude . Otherways without the Secant in all Cases by help of the greater Tangent of 45d . Enter the aforesaid Sum or difference of the Sines once down from the Center and lay the Thread to the Tangent or Cotangent of the Latitude in the greater Tangents , and take the nearest distance to it . Then for Latitudes under 45d enter the former extent at the Complement of the Altitude in the Line of Sines , and find the answer in the Limb by laying the Thread to the other foot , or if it be more convenient make a Parralel entrance of it , and find the answer in the Sines as before hinted . But for Latitudes above 45 d , first find a fourth by entring the sum or difference of sines between the Scale and the Thread , and then it will hold , As the first Tearm , To that fourth : So Radius , To the Sine of the Azimuth , and may be either a Lateral or Parralel entrance , according as it falls out , and as the Radius is put either in the second or third place , in all these Directions the introducement of the Radius is supposed according to to the general Advertisement . The finding of the Amplitude this way presupposeth the Vertical Altitude known , and then the Proportion derived from the Analemma , not from the 16 Cases is , As the Radius , To the Tangent of the Latitude , So the Sine of the Vertical Altitude , To the Sine of the Amplitude : So also to find the time of Sun rising . As the Cosine of the Declination , To Secant of the Latitude : So the sine of the Suns Altitude at 6 , To the Sine of the hour of rising from six . To find the Suns Azimuth at six of the clock otherways then by the 16 Cases . As the Cosine of the Suns Altitude at 6 , To Tangent of the Latitude , So is the difference of the sines of the Suns Altitude at 6 , and of his Vertical Altitude , To the sine of the Azimuth from the Vertical . To find the time when the Sun shall be due East or West . As Cosine of the Declination , To Secant of the Latitude : So the difference of the Sines of the Suns Altitude at 6 , and of his Vertical Altitude , To the Sine of the hour from 6 , When the Sun shall be due East or West . These Proportions derived from the Analemma , are general both for the Sun and Stars in all Latitudes ; but when the Declination either of Sun or Stars exceed the Latitude of the place , this Proportion for finding the Azimuth cannot be at some times conveniently performed on a Quadrant , but must be supplyed from another Proportion , whereof more hereafter . Of the Hour and Azimuth Scales on the Edges of the Quadrant . These Scales are fitted for the more ready finding the Hour and Azimuth in one Latitude , being only to facilitate the former general Way . The Labour saved hereby is twofold , first the Suns declination is graved against the Suns Altitude of 6 in the Hour scale , and the said Declinations continued at the other end of the said hour Scale to give the quantity of the Suns Depression in Winter equal to his Altitude in Summer ; and secondly they are of a fitted length as was shewed in the Description of the Quadrant , and thereby half the trouble by introducing the Radius shunned . The Vse of the Azimuth Scale . The Altitude and Declination of the Sun given to find his Azimuth . Take the distance between the Suns Altitude in the Scale , and his Declination in Summer time in that Scale that stands adjoyning to the side ; in Winter in that Scale that is continued the other way beyond the beginning , and laying the Thread to the Complement of the Suns Altitude in the lesser sines , which is double numbred , enter this extent between the Scale and the Thread parralelly , and the foot of the Compasses sheweth in the Line of Sines the Azimuth accordingly , Declination 23 d 31′ , Altitude 47 d 27′ , the Azimuth thereto would be 25 d from East or West in Summer , and if the Altitude were 9 d 43′ in Winter the Azimuth thereto would be 30 d either way from the Meridian . And so when the Sun hath no Altitude , lay the Thread over 90 d in the lesser Sines and enter the extent from the beginning of the Azimuth Scale to the Declination , and you will finde the Amplitude which to this Declination will be 39 d 50′ . The Vses of the Hour Scale . To find the Hour of the Day . TAke the distance between the Suns Altitude in the hour Scale , and his Declination proper to the season of the year , then laying the Thread to the Complement of the Suns Declination in the lesser sines enter the former extent between the Scale and the Thread and the foot of the Compasses sheweth the sine of the hour . Example . If the Declination were 13 d North , and the Altitude 37d 13′ take the distance between it in the Scale and 13 d in the prickt Line , then laying the Thread to 77 d in the lesser sine enter that extent between the Scale and the Thread , and the resting foot will shew 45 d for the hour from 6 , that is either 9 in the forenoon , or 3 in the afternoon . The Converse of the former Proposition will be to find the Suns Altitude on all Hours . The Thread lying over the Complement of the Suns Declination in the lesser sines from the sine of the hour , take the nearest distance to it , then set down one foot of that extent in the hour Scale at the Declination , and the other will reach to the Altitude . Example . At London , for these Scales are fitted thereto , I would find the Suns Altitude at the hours of 5 and at 7 in the morning in Summer when the Sun hath 23 d 31′ of Declination . Here laying the Thread to 23 d 31′ the Suns declination from the end of the lesser Sines being double numbred , from the sine of 15 d , taking the nearest distance to it , set down one foot of this extent at 23 d 31′ the declination it reaches downwards to 9 d 30′ , and upwards to 27 d 23′ the Suns Altitude at 5 and 7 a clock in the morning in Summer . Another Example . Let it be required to find the Suns Altitudes at the hours of 10 or 2 when his declination is 23d 31′ both North and South . The Thread lying as before over the lesser sines take the nearest distance to it from 60d in the sines , the said extent set down at 23d 31′ in the prickt Line reaches to 53 d 44′ for the Summer Altitude , and being set down at 23d 31′ on the other or lower continued Line reaches to 10d 28′ for the Winter Altitude . The Hour may be also sound in the Versed sines by help of this fitted hour Scale , Take the distance between the Suns Altitude , admit 36d 42′ , and his Meridian Altitude to that Declination 61 d 59′ , and enter one foot of this extent at the sine of 66 d 29′ , and laying the Thread to the other foot according to nearest distance and it will lye over the hours of 8 in the morning , or 4 in the afternoon in the Versed sines in the Limb , and thereby also may the time of Suns rising be found by taking the distance from 0 to the Meridian Altitude and entring it at the Cosine of the Declination as before and the Converse will find the Suns Altitudes on all hours by taking the distance from the Co-sine of the Declination to the Thread laid over the Versed sine of the hour from Noon , and the said Extent will reach from the Mridian Altitude in the fitted Scale to the Altitude sought . To find the time of Sun rising or setting , Lay the Thread over the Complement of the declination as before , in the lesser sines , and enter the extent between the ☉ Altitude , which is nothing that is from the beginning of the Hour Scale to the Declination between the Scale and the Thread and the foot of the Compasses shews it in the Line of sines , which may be converted into Time by help of the Limb. If these Scales be continued further in length as also the Declinations they will after the same manner find the Stars hour for any Star whatsoever to be converted into common Time , as in the uses of the Projection , as also the Azimuth of any Star that hath less declination then the place hath Latitude , but of this more in the next Quadrant . In Dyalling there will be often use of natural sines , whereas these Scales are continued but to 62 d , if therefore it be desired to take out any sine to the same Radius , the rest of the Scale wanting may be easily supplyed , for the difference of the sines of any two Arks equidistant from 60 d is equal to the sine of their distance . Thus the sine of 20 d is equal to the difference of the sines of 40 d and 80 d Arks of like distance from 60 d on each side , and so may be added either to 40 d forward , or the other way from the end of the Scale . In finding the Hour and Azimuth by these Scales , not in the Versed sines , the Directions altogether prescribe a Parralel entrance , but if the Extent from the Altitude to the Declination be entred at the Cosine of the Altitude or of the Declination in the Line of sines according as the Case is , and the Thread laid to the other foot , the Hour and Azimuth may be found in the lesser sines by a Lateral entrance . Or if the said Extent be doubled and entred as before hinted , the answer will be found in the equal Limb. Example to find the Suns Azimuth . Declination 23 d 31′ North. Altitude — 41 : 34 Having taken the distance between these two Tearms in the Azimuth Scale and doubled it , enter one foot in the Line of sines at 48 d 26′ , the Complement of the Altitude , and laying the Thread to the other according to nearest distance it will lye over 15 d of the equal Limb for the Suns Azimuth from the East or West Southwards . The Vse of the Versed Sin 's in the Limbe . It may be noted in the former general Proportion , I have used the word Azimuth from Noon or Midnight Meridian , though not so proper , because they are more universal and common to both Hemispheres , other expressions besides their Verbosity would be full of Caution for the following Proportion in our Northern Hemispere , without the Tropick that finds it from the South between the Tropick of Cancer and the Equinoctial , when the Sun comes to the Meridian between the Zenith and the Elevated Pole would find it from the North , wherefore it is fit to be retained . A general Proportion for finding the Hour . As the Cosine of the Declination , To the Secant of the Latitude : Or , As the Cosine of the Latitude , To the Secant of the Declination : So is the difference of the Sines of the Suns Altitude proposed , and of his Meridian Altitude , To the Versed Sine of the hour from Noon● And So is the sum of the sines of the Suns proposed Altitude , and of his Midnight Depression , To the Versed sine of the hour from Midnight : And So is the sine of the Suns Meridian Altitude , To the Versed sine of the Semidiurnal Ark : And So is the sine of the Suns Midnight Depression , To the Versed sine of the Seminocturnal Ark. The Operation will be like the former , I shall therefore onely illustrate it by one Example , the Meridian Altitude is got in Winter by differencing , in Summer by adding the Declination to the Complement of the Latitude , if the sum exceed 90 d the Complement thereof to 180 d is the Meridian Altitude . An Example for finding the Hour from Noon . Declination — 23d 31′ North the 11th June . Comp. Latitude — 38 28 London .   61 59 Meridian Altitude . Proposed Altitude — 36 42 , take the distance between the sines of these two Arks , and enter it once down the Line of sines from the Center , and take the distance to the Thread laid over the Secant , then enter one foot of that extent at the sine being the first Tearm , and to the other lay the Thread , and in the Versed sines in the Limb it will lye over the Versed Sine of the hour from Noon . In this Example , if the Thread be laid over the Secant of 51d 32′ the extent must be entred at the sine of 66d 29′ 23 31 the extent must be entred at the sine of 38 28 either way the answer will fall upon 60 d of the Versed sine shewing the Hour to be either 8 in the forenoon , or 4 in the afternoon . If the hour fall near noon , then the extent of the Compasses may be Quadrupled and entred as before , and look for the answer in the Versed Sines Quadrupled : Or before the distance be took to the Thread the extent of difference may be entred four times down from the Center . The Converse of this Proposition will be to find the Suns Altitude on all Hours universally . As the Secant of the Latitude , To Cosine Declination , Or , As the Secant of the Declination , To Cosine Latitude : So the Versed sine of the hour from Noon , To the difference of the sines of the Suns Meridian Altitude , and of his Altitude sought , to be substracted from the sine of the Meridian Altitude , and there will remain the sine of the Altitude sought . So in Latitude of London , if the Suns Declination were 13 d 00′ , and the hour from noon 75 d , that is either 7 in the morning , or 5 in the afternoon . Lay the Thread over the Versed sine of the hour from noon , namely , 75 d , and from the sine of 77 d the Complement of the Declination , take the nearest distance to it , then lay the Thread to the Secant of the Latitude , and enter the former extent between the Scale and the Thread , and you will find a sine equal to the difference sought , which sine take between the Compasses and setting down one foot at the sine of 51 d 28′ the Meridian Altitude , the other foot turned towards the Center will fall upon the sine of 19 d 27′ the Altiude sought . A General Proportion for the Azimuth . Get the Remainder or Difference between these two Arks , the Suns Altitude and the Complement of the Latitude by Substracting the less from the greater , and then the Proportion will hold , As the Cosine of the Latitude , Is to the Secant of the Altitude , Or , As the Cosine of the Altitude , To the Secant of the Latitude : So is the sum of the sines of the Suns Declination ▪ and of the aforesaid Remainder , To the Versed Sine of the Azimuth from the Noon Meridian in Summer only when the Suns Altitude is less then the Complement of the Latitude . In all other Cases , So is the difference of the said sines , To the Versed sine of the Azimuth , as before from Noon Meridian . Example . The 11th of June aforesaid , the ☉ having 23 d 31′ of North declination , his Altitude was observed to be 18 d 20′ , which substracted from 38 d 28′ the remainder is 20 d 8′ , take out the sine thereof , and set down one foot at the sine of 23 d 31′ , and set the other forwards towards 90 d , then take the nearest to the Thread laid over the Secant of the Latitude 51 d 32′ , enter one foot of this Extent at the Complement of the Altitude by reckoning the Altitude it self from 90 d towards the Center , and the Thread laid to the other foot cuts the Line of Versed sines at 105 d the Azimuth from the South . The same day when the Altitude was more then the Colatitude , suppose 60 d 11′ the Remainder will be found to be 21 d 43′ , take the distance between the sine thereof and of 23 d 31′ , and because the Extent is but small enter it four times down the Line of sines from the Center , and take the nearest distance to the Thread laid over the Secant of the Latitude , which entred at the Cosine of the Altitude , the Thread laid to the other foot shews 25 d in the Quadrupled Versed Sines for the Azimuth from the South . The Proportion hence derived for the Amplitude . As the Cosine of the Latitude , To Secant of the Declination , &c. as before . So in Summer is the sum in Winter , the difference of the sines of the Suns Declination , and of the Complement of the Latitude , To the Versed Sine of the Amplitude from Noon Meridian . The Proportion for the Azimuth will be better exprest by making the difference to be a difference of Versed sines . How the Versed sines in the Limbe may be spared in Case a Quadrant want them . If a Quadrant can only admit of a Line of sines from the Center , the common Quadrant of Mr Gunters very well may , on the right edge above the Margent for the Numbers of the Azimuths , it may be easily fitted for any or many Latitudes by setting Marks or Pricks to the Tangent and Secant of the Latitudes in the Limbe , which may be taken out by help of the Limb , Line of Sines , or by Protraction , and either of these general Proportions wrought upon it , or those which follow , if it be observed that whensoever the Thread lyes over the Versed sine of any Ark in the Limbe , it also at the same Time lyeth over a Sine equal to half that Versed sine to the common Radius : Now because the sine of 30d doubled is equal to the Radius , let it be observed whether the sine cut by the Thread be greater or less then 30 deg . When it is less let the Line of Sines represent the former half of a Line of Versed Sines , and take the sine of the Ark the Thread lay over , and enter it twice forward from the end of the Scale towards the Center , and you will obtain the Versed sine of the angle sought . When it is more take the distance between the sine of 30 d and the said sine , and letting the Line of sines represent the latter half of a Line of Versed sines , enter the said distance twice from the Center , and you will obtain the Versed sine of the Arch sought , namely , the sine of an Arch , whereto 90 d must be added . Three sides to find an Angle , a general Proportion . As the Sine of one of the Sides including an angle , Is to the Secant of the Complement of the other including side : So is the difference of the Versed Sines of the third side , and of the Ark of difference betwen the two including sides , To the Versed Sine of the Angle sought . And So is the difference of the Versed sines of the third side , and of the sum of the two including sides , To the Versed sine of the sought angles Complement to 180d . To repeat the Converse when two sides and the angle comprehended are given to find the third side were needless . If one of the containing sides be greater then a Quadrant , instead of it in referrence to the two first Tearms of the Proportion , take the Complement thereof to 180d for the reputed side , but in differencing or summing the two containing sides alter it not : And further note , that the same Versed sine is common to an Ark less then a Semicircle , and to its Complement to 360 d. The Operation of this Proportion will be wholly like the former , so that there needs no direction but only how to take out a difference of two Versed sines to the common Radius , seeing this Quadrant of so small a Radius is not capable of such a Line from the Center . And here note that the difference of two Versed sines less then a Quadrant , is equal to the difference of the natural sines of the Complements of those Arks. And the difference of two Versed sines greater then a Quadrant is equal to the difference of the natural sines of the excess of those Arks above 90 d. And by consequence the difference of the Versed sines of two Arks the one less , the other greater then a Quadrant is equal to the sum of the natural sines of the lesser Arks Complement to 90d , and the greater Arks excess above it . And so a difference of Versed sines may be taken out of the Line of natural sines considered as such . Or the Line of sines may be considered sometimes to represent the former half of a Line of Versed Sines as it is numbred with the small figures by its Complements from the end of it to 90d at the Center , and sometimes the latter half of it , and then the graduations of it as Sines must be considered as numbred from 90 d to 180 d at the end of it , and so a difference to be taken out of it by taking the distance between the two Tearms , which if the two Arks fall the one to be greater the other less then 90 d will be a sum of two sines , as before hinted , and in this Case the sine of the greater Arks excess above 90 d to be set down outwards if it may be , at the Versed sine of the lesser Ark , or which is all one , at the sine of that Arks Complement , and the distance from the Exterior foot of the Compasses to the Center will be equal to the difference of the Versed sines of the Arks proposed . To measure a difference of Versed sines to the common Radius . In this Case also the Line of sines must sometimes represent the former , sometimes the latter half of a Line of Versed sines , and then one foot of the difference applyed to one Ark , the other will fall in many Cases upon the Ark sought , each Proportion variously exprest , so that possibly either one or the other will serve in all Cases . But if one of the feet of the Compasses falls beyond the Center of the Quadrant : To find how much it falls beyond it , bring the said foot to the Center , and let the other fall backward on the Line , then will the distance between the said other foot , where it now falleth , and the place where it stood before be equal to the excess of the former foot beyond the Center , which accordingly thence measured helps you to the Arch sought , and its Complement both at once with due regard to the representation of the Line ; this should be well observed , for it will be of use on other Instruments . A difference of Versed sines thus taken out to the Common Radius must be entred but once down from the Center . To take out a Difference of Versed sines to half the common Radius . Count both the Arks proposed on the Versed sines in the Limbe , and find what Arks of the equal Limbe answer thereto , then out of the Line of sines take the distance between the said Arks , and you have the extent required , which being but half so large as it should be is to be entred twice down from the Center . To measure a difference of Versed sines to half the common Radius . The Versed sines are largest at that end numbred with 180d , count the given Ark from thence , and laying the Thread over the equal Limb find what Ark answers thereto , then setting down the Compasses at the like Ark in the Line of sines from the end of it towards the Center mind upon what Ark it falls , the Thread laid to the like Ark in the Limb sheweth on the Versed sines the Ark sought . To save the labour of drawing a Triangle , I shall deliver the Proportion for the Azimuth derived from the general Proportion , As the Cosine of the Latitude , To the Secant of the Altitude , Or , As the Cosine of the Altitude , To the Secant of the Latitude : So is the difference of the Versed sines of the Sun or Stars distance from the Elevated Pole , and of the Ark of difference between the Latitude and Altititude . To the Versed sine of the Azimuth sought ; as it falls in the Sphoere that is from the Midnight Meridian . And So is the difference of the Versed sines of the Polar distance , and of the Ark of difference between a Semicircle and the sum of the Latitude and Altitude , To the Versed Sine of the Azimuth from Noon Meridian . A Canon derived from the Inverse of the general Proportion to finde the Distance of places in the Ark of a great Circle . As the Secant of one of the Latitudes , To the Cosine of the other . So the Versed sine of the difference of Longitude , To the difference of the Versed sines of the Ark of distance sought , and of the Ark of difference between both Latitudes , when in the same Hemisphere , or the Ark of the sum of both Latitudes when in both Hemispheres , which difference added to the Versed sine of the said Ark gives the Versed sine of the Ark of distance sought . And So is the Versed sine of the Complement of the difference of Longtitude to 180 d. To the difference of the Versed sines of the Ark of distance sought , and of an Ark being the sum of the Complements of both Latitudes when in one Hemisphere ; Or the sum of the lesser Latitude encreased by 90d , and of the Complement of the greater Latitude when in different Hemispheres , which difference substracted from the Versed sine of the said Ark , there will remain the Versed sine of the Ark of distance sought . This Proportion is to be wrought after the same manner as we found the Suns Altitudes on all hours universally , and the difference to be measured in the Line of sines as representing the former half of a Line of Versed sines , according to the Directions given for measuring of a difference of Versed sines to the common Radius or Radius of the Quadrant . By altring the two first Tearms of the Proportion above , we may work this Proposition by positive entrance . As the Radius , To the Cosine of one of the Latitudes : So the Cosine of the other Latitude , To a fourth . Again . As the Radius , To the Versed sine , as above expressed in both parts , So is that fourth , To the difference as above expressed . An Example for finding the distance between London and Bantum in the Arch of a great Circle the same that was proposed in Page 96. Bantam Longitude 140d Latitude 5d 40′ South . London Longitude 25 , 50′ Latitude 51 32 North. difference of Longitude 114 : 10 Sum 57 12 Lay the Thread to 5d 40′ in the Limb counted from left edge and from 38d 28′ in the sines the Complement of our Latitude take the nearest distance to it , then lay the Thread to 114d 10′ in the Versed sines and entring the former extent down the Line of sines from the Center take the nearest distance to it , then laying the Thread over 57d 12′ in the Versed sine , it cuts the Limbe at 13d 15′ from the right edge , at the like Ark set down one foot of the former extent in the Line of sines , and the other will reach to the Sine of 41d 42′ , then laying the Thread over the like Ark in the Limb , it will intersect the Versed sines at 109d 18′ the Ark of distance sought to be converted into Leagues or Miles according to the number of Leagues or Miles that answer to a degree in each several Country . Thus when we have two sides with the angle comprehended to find the third side , either to half or the whole common Radius without a Line of natural Versed sines from the Center , or by the Proportions in page 93 , or a third way , which I pretermit to the great Quadrant ; and thus the Reader may perceive this small Quadrant to be many ways both Universall and particular , which are of sudden performance though tedious in expression . Three sides to find an angle . Each of the Proportions in Rectangles and Squares before delivered for the Tables , may as before suggested be reduced to single Tearms , an instance shall be given in that which finds the Square of the sine of half the Angle sought . Add the three sides of the Triangle together , and from the half sum substract each of the sides including the angle sought , then it will hold , As the sine of one of the Comprehending sides , ( rather the greater that the entrance may be Lateral . ) Is to the sine of the difference of the same side from the half sum : So is the sine of the difference of the other comprehending side , To a fourth sine , Again . As the Sine of the other comprehending side , Is to that fourth sine : So is the Radius , To half the Versed sine of the angle sought , And So is the Diameter , To the whole Versed sine . To work this on the Quadrant . Upon the first Tearm in the Line of sines being the greatest containing side , enter the extent of the second , and to the other foot lay the Thread , then from the third Tearm in the sines take the nearest distance to it , Which extent enter at the sine of the first Tearm in the second Proportion , and to the other foot lay the Thread , and it will cut the Versed sine at the angle sought . Having shewed how all Proportions may be performed upon the Quadrant : I now proceed to the rest of the Lines . Of the Line of Chords on the left edge of the Backside . I shall not at present speak any thing as to the use thereof , that is intended to be done in a Treatise of the general Scale ; the principal use of a Line of Chords is to prick off readily the quantity of any Arch of a Circle , to do which take the Chord of 60d , and draw a Circle with that Extent then any Arch being to be prict off it is to be taken out of the Line of Chords , and to be transferred into the Circle swept , this supposeth the Radius not to vary . But to do it to any Radius that is lesser take the Semidiameter of the Circle , and enter it at the Chord of 60d laying the Thread to the other foot , and the nearest distances to the Thread will be Chords to the Semidiameter assigned , and the Converse will measure the Chord of any Arch by a Parralel entrance . A Chord may be taken off though no Line of Chords be graduated . The Bead wheresoever it be set carryed from one edge of the Quadrant to the other , the Thread being extended doth describe the Quadrant of a Circle , if therefore extending the Thread down one edge of the Quadrant you set the Bead to the distance of the Radius ( or Semidiameter of the Circle swept ) from the Center , and at it set down one foot of the Compasses , and lay the Thread kept at a certainty in stretching over the Limb to any Arch , & then open the Compasses to the Distance of the Bead , you shall take out the Chord of the said Arch. A Chord may very conveniently be taken off from any Circle swept Concentrick to the Limbe , and divers such there are upon both sides of this Quadrant ; Sweep a Circle of the like Radius on Paper as that on the Quadrant , and then setting one foot to the Intersection of the Concentrick Circle with the right edge , the Thread being laid over any Arch whatsoever in the Limbe , take the distance to the Intersection of the Thread with the Concentrick Circle , and transform the said Extent into the Circle drawn upon Paper . Of the Versed Sines Augmented . These are to be used with fitted Scales thereto , to stand upon a loose Ruler for the ready and more Exact finding the Hour and Azimuth near Noon , or at other times , and shall be treated of in the use of the Diagonal Scale . Of the Line of Latitudes and Scale of Hours on the spare edges of the foreside . The use of these Scales are for the ready pricking down of any Diall that hath a Center in an Equicrural Triangle from the Substile , as shall be shewed in the Use of the great Quadrant , though the Schems be fitted to small Scales . The Scale of Hours standing on the Edge on the foreside may very well be supplyed from thence to another Radius , as shall in due Time be shewed , though it do not proceed from the Center , and therefore may be spared out of the Limbe on the Backside . The Description of the Diagonal Scale . THe particular Scales handled in Page 181 would find the hour and Azimuth in the equal Limbe without doubling the Extent , if laying the Thread over the Cosine of the Declination in the lesser sines when the hour is sought ; or over the Cosine of the Altitude when the Azimuth is sought , it be minded what Ark of the Limbe the Thread intersects , and then make the entrance of one foot of the Extent at the like Ark in the sines laying the Thread to the other foot according to nearest distance . But because these Scales are more convenient being twice as long , there is accordingly a Diagonal Scale fitted to serve for our English Region , and may be accommodated to any 5 or 6 degrees of Latitude , and placed conveniently on any Instrument for Surveigh to give the hour and Azimuth in the Limbe of the Instrument or on the frame of the plain Table . And here I am to intimate that either the hour or Azimuth Scale before described on the small Quadrant , will serve to finde both the Hour and Azimuth , as conveniently as either ; the foundation whereof is , that the same Proportion demonstrated from the Analemma that finds the hour from six being applyed to other sides of the Triangle will also find the Azimuth from the East or West , an instance whereof I give in the Use of a great Quadrant for finding the Azimuth when the Declination of the Sun or Stars exceeds the Latitude of the place . By the like parity of reason the Proportion that found the Azimuth is a general Proportion to find an angle , when the threee sides are given the Canon will be , As the Cosine of one of the including sides , Is to the Radius : So is the Cosine of the side Opposite to the angle sought , To a fourth a Sine or Secant . Again . As the second including side , Is to the Cotangent of the first including side : So when any one of the sides is greater then a Quadrant is the sum , but when all less the difference between the 4th abovesaid , and the Cosine of the second Includer , To the Cosine of the angle sought . Wee do suppose but one of the three sides given to be greater then a Quadrant , if there be any such it subtends an Obtuse angle , and both the other sides being less then Quadrants subtend Acute Angles . When the 4th Sine is less then the Cosine of the second Includer , the Angle sought is Obtuse , other ways Acute . Hence the peculiar Proportion educed for the hour will be , As the Sine of the Latitude , Is to the Radius : So is the Sine of the Altitude , To a 4th a Sine or Secant . Again . As the Cosine of the Declination , Is to the Tangent of the Latitude , So in Summer is the difference , but in Winter the sum of the sines of the Declination of the Sun or Stars , and of the 4th Sine , To the Sine of the hour from six . When the 4th Ark is less then the Declination the hour is Obtuse , when greater Acute , and in Winter always Acute : But of this Proportion I make no use it being liable in some Cases to Excursion , and will not hold backward to find the Suns Altitudes the hour being assigned . This Diagonal Scale is made after the same manner as the Hour Scale described in the small Quadrant , being but several Lines of sines the greater whereof are made equal to the Secant of the Latitude , whereto they are fitted , the Radius of which Secant is 5 Inches long . The lesser Sines continued the other way , their respective Radii are made equal to the sine of the Latitude of that greater sine whereto they are continued . The parralel Lines fitted to the respective Latitudes are not to be equidistant one from another ; but having determined the distance between the two Extream Latitudes , to which they are fitted for the the larger sine it will hold , As the difference of the Secants of the two extream Latitudes , It to the distance between the Lines fitted thereto : So is the difference of the Secants of the lesser extream Latitude , and any other intermediate Latitude , To the distance thereof from the lesser extream . And so for the lesser sine continued the other way , having placed the two Extreams under the two former Extreams , to place the imtermediate Lines , the Canon would be , As the difference of the sines of the two extream Latitudes , Is to the distance between the Lines fitted thereto : So is the difference of the sines of the lesser extream Latitude , and of any other intermediate Latitude , To the distance thereof from the lesser Extream . Having fitted the distances of the greater sine , streight Lines drawn through the two extream sines shall divide the intermediate Parralels also into Lines of sines , proper to the Latitudes to which they are fitted : Now for the lesser sines they are continued the other way at the ends of the former Parralells , the Line proper to each Latitude should be divided into a Line of sines , whose Radius should be equal to the sine of the Latitude of the other sine whereto it is fitted ; and so Lines traced through each degree to the Extreams ; but by reason of the small distance of these Lines , the difference is so exceeding small , that it may not be scrupled to draw Lines Diagonal wise from each degree of the two outward extream Sines , for being drawn true , they will not be perceived to be any other then streight Lines . Whereas these Lines by reason of the latter Proportion should not fall absolutely to be drawn , at the ends of the former Lines whereto they are fitted , and then they would not be so fit for the purpose , yet the difference being as we said , so insensible that it cannot be scaled , they are notwithstanding there placed and crossed with Diagonals drawn through each degree of the Extreams . The Vses of the Diagonal Scale . 1. To find the time of Sun rising or setting . In the Parralel proper to the Latitude take out the Suns Declination out of the lesser continued sines , and enter one foot of this extent at the Complement of the Declination in the Line of sines , and in the equal Limb the Thread being laid to the other foot will shew the time sought . In the Latitude of York , namely 54d , if the Sun have 20d of Declination Northward he rises at 4 and sets at 8 Southward he rises at 8 and sets at 4 2. To find the Hour of the Day or Night for South Declination . In the Parralel proper to the Latitude account the Declination in the lesser continued sine , and the Altitude in the greater sine , and take their distance , which extent apply as before to the Cosine of the Declination in the Line of sines on the Quadrant , and laying the Thread to the other foot according to nearest distance it shews the time sought in the equal Limbe . Thus in the Latitude of York when the Sun hath 20d of South declination , his Altitude being 5d , the hour from noon will be found 45 minutes past 8 in the morning , or 15 minutes past 3 in the afternoon feré . For North Declination . The Declination must be taken out of the lesser sine in the proper Parralel , and turned upward on the greater sine and there it shews the Altitude at six for the Sun or any Stars in the Northern Hemispere , the distance between which Point and the given Altitude must be entred as before at the Cosine of the declination , laying the thread to the other foot and it shews the hour in the Limb from six towards noon or midnight , according as the Sun or Stars Altitude was greater or lesser then its Altitude at six . So in the Latitude of York , when the Sun hath 20d of North declination if his Altitude be 40d , the hour will be 46 minutes past 8 in the morning , or 14 minutes past 3 in the afternoon . 4. The Converse of the former Proposition will be to find the Altitude of the Sun at any hour of the day , or of any Star at any hour of the night . I need not insist on this , having shewn the manner of it on the small quadrant , only for these Scales use the Limb instead of the lesser sines , for Stars the time of the night must first be turned into the Stars hour , and then the Work the same as for the Sun. 5. To find the Amplitude of ehe Sun or Stars . Take out the Declination out of the greater sine in the Parralel proper to the Latitude , and measure it on the Line of sines on the lesser Quadrant , and it shews the Amplitude sought . So in the Latitude of York 54d when the Sun hath 20d of Declination , his Amplitude will be 35d 35′ . 6. To find the Azimuth for the Sun or any Stars in the Hemisphere . For South Declination . Account the Altitude in the lesser sine continued in the proper Parralel , and the Declination in the greater sine , and take their distance enter one foot of this extent at the Cosine of the Altitude on the Quadrant , and lay the Thread to the other according to nearest distance , and in the Limbe it shews the Azimuth from East or West Southwards . So in the Latitude of York , when the Sun hath 20d of South Declination , his Altitude being 5d , the Azimuth will be found to be 44d 47′ to the Southwards of the East or West . For North Declination . Account the Altitude in the lesser sine continued , and apply it upward on the greater sine and it finds a Point thereon , from whence take the distance to the declination in the said greater sine in the Parralel proper to the Latitude of the place , and enter one foot of this Extent at the Cosine of the Altitude on the Line of sines , and the Thread being laid to the other foot according to nearest distance , shews the Azimuth in the Limbe from East or West . So in the Latitude of York , when the Sun hath 20d of North Declination , and 40d of Altitude , his Azimuth will be 23d 16′ to the Southwards of the East or West . When the Hour or Azimuth falls near Noon , for more certainty you may lay the Thread to the Complement of the Declination for the Hour , or the Complement of the Altitude for the Azimuth , in the Limbe , and enter the respective extents Parralelly between the Thread and the Sines , and find the answer in the sines . We might have fitted one Scale on the quadrant to give both the houre and Azimuth in the Equall Limb by a Lateral entrance , and have enlarged upon many more Propositions , which shall be handled in the great Quadrants . Mr Sutton was willing to add a Backside to this Scale , and therefore hath put on particular Scales of his own for giving the requisites of an upright Decliner in this Latitude , which he hath often made upon Rulers for Carpenters and other Artificers and Diallists , and whereof he was willing to afford them a Print ; whereto I have added other Scales for giving the Hour and Azimuth near Noon . On the Backside are drawn these Lines , A large Dyalling Scale of 6 hours or double Tangents , with a Line of Latitudes fitted thereto . A large Chord . A Line for the Substiles distance from the Meridian . A Line for the Stiles height . A Line for the angle of 12 and 6. A Line for the inclination of Meridians . All these Scales relate to Dyalling . An Azimuth Scale being two Lines of natural sines of the same Radius set together at O , and thence numbred with Declinations , this Scale must be made of the same sine that the hour Scale following is made of continued from O one way to 38d 28′ , and the other way to 23d 31′ or further at pleasure ; but numbred from the beginning which is at the end of that 38d 28′ the Complement of the Latitude with 10d 20′ , &c. up to 60d. The Hour Scale is no other then a Line of sines with the declinations set against the Meridian Altitudes in the Latitude of London , the Radius of which sine is equal in length to the Dyalling Scale of hours . Of the Vses of these Scales . The Line of Hours and Latitudes is general for pricking down all Dialls with Centers as will afterwards be shewed in the Use of the great Quadrant , and by help of the Scale of Hours may the Diameter of a such a Circle be graduated as is placed in on the back of the great Quadrant , and the Line of Latitudes will serve as a Chord to divide the upper Quadrant , and the Hour Scale or Line of Sines will serve as a Chord to divide a Semicircle , whose Diameter is equal to the Scale of Hours into 90 equal parts and their Subdivisions , and hereby may Proportions in sines and Tangents , or Tangents alone be wrought by Protraction , and so the necessary Arks in Dyalling found generally as is done by Mr Foster in the three last Schems of his Posthuma , this will easily be understood if the use of the Circle on this Quadrant be well apprehended . The particular Scales give the requisite Arks of upright Decliners in this Latitude by inspection , for count the plaines Declination in the Line of Chords , and a Square laid over it intersects all those Arks or to be found by applying the Declination taken out of the Chords with Compasses to every other Line . Example . So if an upright Plain decline 35d from the Meridian . The Substiles distance from the Meridian will be — 24d 30′ The Stiles height — 30 38 The Inclinations of Meridians — 41 49′ The angle of 12 and 6 — 54 10 These particular Scales also resolve some of the Cases of right angled Sphoerical Triangles , relating to the Motion of the Sun or Stars thus , Of the Line of the Stiles height . Account the Declination in the Line for the Stiles height , and against it in the Chord stands the Amplitude of the Sun or Stars from the Meridian . Example for Amplitude . So when the Sun hath 18d of Declination , his Amplitude will be 67d 13′ from the Meridian , and 29d 47′ from the Vertical . The reason hereof is because the two first fixed Tearms of the Proportion that Calculate the Stiles height are the Radius and the Co-sine of the Latitude , and the two first Tearms that Calculate the Amplitude are the Cosine of the Latitude and the Radius , and therefore must as well serve in this Case as in that . On this Stile Line may be found the Suns Altitudes on all hours , when he is in the Equinoctial by applying the hour from six taken from the Chords to the other end of the Stile Line . Of the Substiler Line . Hereby we may find the time of Sun rising and setting , take the Declination out of the Substilar Line and measure it on the Line of Chords . Example . So when the Sun hath 18 of North Declination , the Ascensional difference is 24d 9′ in time 1 hour 36½ minutes , and so much the Sun rises and sets from six . Hereby may be also found the Equinoctial Altitudes to every Azimuth . Of the Line for the Angle of 12 and 6. Hereby we may find the time when the Sun will be due East or West . Account the Complement of the Declination in this Scale , and against it in the Chords stands the hour from six . Example . So when the Sun hath 18d of North Declination , he will be East or West at 7 in the morning , or 5 in the afternon . By these Scales the requisites of an East or West Reclining or Inclining Diall in this Latitude may be found . 1. The Substiles distance from the Meridian . Account the Complement of the Reclination Inclination in the Chords , and against it in the Line for 12 and 6 stands the Complement of the angle sought . 2. For the Stiles height . Apply the Reclination in the lesser sines on the Diagonal Scale in the Parralel proper to the Latitude to the greater sine and it shewes the Ark sought . 3. For the Inclination of Meridians . This may be also found on the Diagonal Scale when the Substiles distance is not more then the Latitude , By Accounting the Substiles distance on the greater sine , and applying it to the lesser . 4. For the Angle of 12 and six . Account the Complement of the Reclination in the Chords , and against it in the Substilar Line is the Complement of the angle sought . So if an East or West Plain Recline or Incline 35d. The Substiles distance from the Meridian will be — 45d 52′ The Stiles height — 26 41 The Inclination of Meridians — 66 27 And the angle of 12 and 6 — 56 55 Of the Hour and Azimuth Scales . This Scale is fitted to find the Hour from Noon in the Versed sine : augmented , and the Proportion to be wrought by it the same as delivered in the use of the small Quadrant . As the Cosine of the Declination , Is to the Secant of the Latitude : So is the difference of the sines of the Suns proposed and Meridian Altitude , To the Versed sine of the hour from Noon . And of this one Proportion we make two by introducing the Radius . As the Radius , is to the Secant of the Latitude : So is the former distance , To a fourth . By fitting the Radius of the sines equal in length to the Secant of the Latiude ; this first Proportion is removed for the said difference of sines taken out of this fitted Scale is the 4th Proportional , the Proportion that remains to be wrought upon the Quadrant is , As the Cosine of the Declination , Is to the difference of the sines taken out of this fitted Scale : So is the Radius , To the Versed sine of the hour from Noon . By this means if in the same Proportion as we increase the length of the fitted Scale , we also increase the versed sines lying in the Limb , we may find the hour and Azimuth near noon with certainty if the Altitude be well given : These Scales in their Use presuppose the Hour and Azimuth of the Sun to be nearer the noon Meridian then 60d. Operation to find the Hour . Take the distance between the Altitude and the Declination proper to the season of the year out of the Hour Scale , and enter one foot of this Extent at the Cosine of the Declination in the Line of sines , and laying the Thread to the other foot according to nearest distance , it shews the hour from noon in the Versed sines Quadrupled . Example . When the Sun hath 23d 31′ of North Declination , and 60d of Altitude , the hour from noon will be 13d 58′ to be Converted into time . When the hour is found to be less then 40d from Noon , the former extent may be doubled and entred as before , and it shews the hour in the Versed sines Octupled . And when the hour is less then 30d from Noon , the former extent may be tripled and entred as before , and after this manner it is possible to make the whole Limb give the hour next Noon , the Versed Sine Duodecupled , lies on the other side of the Quadrant ; and in this case , an Ark must first be found in the Limb , and the Thread laid over the said Ark , counted from the other edge , will intersect the said Versed Sine at the Ark sought . To find the Suns Azimuth : TAke the distance in the Azimuth Scale , between the Altitude and the Declination , proper to the season of the year , and entring it at the Cosine of the Altitude , laying the Thread to the other foot , according to nearest distance , it will shew the Azimuth in the Versed Sines quadrupled ; or , when the Azimuth is near Noon , according to the former restrictions for the hour , the extent may be doubled , or tripled , and the answer found in the Versed Sines Octupled , or Duodecupled , as was done for the hour . Example . So when the Sun hath 23d 31′ of North Declination , his Altitude being 60d. The Azimuth will be found to be 26d 21′ from the South . By the like-reason , when we found the Hour and Azimuth in the equal Limb by the Diagonal Scale , if those extents had been doubled , the Hour and Azimuth near six , or the Vertical , might have been found in a line of Sines of 30d , put thorow the whole Limb , but that we thought needless . FINIS . THE DESCRPITION AND VSES Of a Great Universal Quadrant : With a Quarter of Stofters particular Projection upon it , Inverted . Contrived and Written by John Collins Accomptant , and Student in the MATHEMATIqUES . LONDON , Printed in the Year , 1658. The DESCRIPTION Of the Great Quadrant . IT hath been hinted before , that though the former contrivance may serve for a small Quadrant , yet there might be a better for a great one . The Description of the Fore-side . On the right edge from the Center is placed a line of Sines . On the left edge from the Center , a line of Versed Sines to 180d. The Limb , the same as in the small Quadrant . Between the Limb and the Center are placed in Circles , a Line of Versed Sines to 180d , another through the whole Limb to 90d. The Line of lesser Sines and Secants . The line of Tangents . The Quadrat and Shaddows . Above them , the Projection , with the Declinations , Days of the moneth , and Almanack . On the left edge is placed the fitted Hour , and Azimuth Scale . Within the Projection abutting against the Sines , is placed a little Scale , called The Scale of Entrance , being graduated to 62d , and is no other but a small line of Sines numbred by the Complements . At the end of the Secant is put on the Versed Sines doubled , that is , to twice the Radius of the Quadrant , and at the end of the Tangents tripled , to some few degrees , to give the Hour and Azimuth near Noon more exactly . The Description of the Back-side . On the right edge from the Center , is placed a Line of equal part being 10 inches precise , decimally subdivided . On the out-side next the edge , is placed a large Chord to 60● , equal in length to the Radius of the Line of Sines . On the left edge is placed a Line of Tangents issuing from the Center , continued to 63d 26′ , and again continued apart from 60d , to 75d The equal Limb. Within it a Quadrant of Ascensions , divided into 24 equal hours and its parts , with Stars affixed , and Letters graved , to refer to their Names . Between it and the Center is placed a Circle , whereof there is but three Quadrants graduated . The Diameter of this Circle is no other then the Dyalling Scale of 6 hours , or double Tangents divided into 90d. Two Quadrants , or the half of this Circle beneath the Diameter , is divided into 90 equal parts or degrees . The upper divided Quadrant , is called the Quadrant of Latitudes . From the extremity of the said Quadrant , and Perpendicular to the Diameter , is graduated a Line of Proportional Sines ; M Foster call it the Line Sol. Diagonal-wise , from one extremity of the Quadrant of Latitude to the other , is graduated a line of Sines ; that end numbred with ●0 d , that is next the Diameter , being of the same Radius with the Tangents . Opposite and parallel thereto from 45d of the Semicircle , to the other extremity of the Diameter , is placed a Line of Sines equal to the former . Diagonal-wise , from the beginning of the Line Sol , to the end of the Diameter , is graduated a Line of 60 Chords . From the beginning of the Diameter , but below it , towards 45d of the Semicircle , is graduated the Projection Tangent , alias , a Semi-tangent , to 90d , being of the same Radius with the Tangents . The other Quadrant of this Circle being only a void Line , there passeth through it from the Center , a Tangent of 45d , for Dyalling , divided into 3 hours , with its quarters and minutes . Below the Diameter is void space left , to graduate any Table at pleasure , and a Line of Chords may be there placed . Most of these Lines , and the Projection , have been already treated upon in the use of the small Quadrant , those that are added , shall here be spoke to . Of the Line of Versed Sines , on the left Edge , issuing from the Center . THis Line , and the uses of it , were invented by the learned Mathematician , M. Samuel Foster , of Gresham Colledge , deceased , from whom I received the uses of it , applyed to a Sector ; I shall , and have added the Proportions to be wrought upon it , and in that , and other respects , diversifie from what I received ; wherein I shall not be tedious , because there are other ways to follow , since found out by my self . The chief uses of it are , to resolve the two cases of the fourth Axiom of Spherical Trigonometry ; as , when three sides are given to find an Angle , or two sides with the Angle comprehended , to find the third side , which are the cases that find the Hour and Azimuth generally , and the Suns Altitudes on all hours . For the Hour , the learned Author thought meet to add a Zodiaque of the Suns-place annexed to it , both in the use of his Sector , as also in the use of his Scale , published since his death , entituled Posthuma Fosteri , that the Suns place being given , which for Instrumental use might be obtained , by knowing on what day of each moneth the Sun enters into any Signe , and allowing a degree for every days motion , come by it prope verum , and being sought in the annexed Zodiaque ( which is no other then two lines of 90d. Sines , each made equal to the Sine of 23d 31′ the Suns greatest Declination ) just against it stands the Suns Declination , if accounted in the Versed Sine , from 90d each way ; but this for want of room , and because the Declination is more easily given by help of the day of the moneth , I thought fit to omit , the rather , because it may also be taken from the Table of Declinations . But from hence I first observed , that if the two first terms of a Proportion were fixed , if two natural Lines proper to those terms , were fitted of an equal length , and posited together , if any third term be given , to find a fourth in the same proportion , it would be given by inspection , as standing against the third ; but if the Lines stand asunder , or a difference be the third term , application must be made from one Line to the other with Compasses , as in the same Scale there is also fitted a Line of 60 parts , equal in length to the Radius of a small Sine , serving to give the Miles in every several Latitude , answerable to one degree of Longitude . Three sides given to find an Angle , the Proportion , As the difference of the Versed Sines of the Sum , and difference of any two Sides including an Angle , Is to the Diameter , So is the difference of the Versed Sines of the third side , And of the Ark of difference between the two including Sides , To the Versed Sine of the Angle sought ; And so is the difference of the Versed Sines of the third side And of the sum of the two including sides , To the Versed Sine of the sought Angles , Complement to a Semicircle . Corollary . And seeing there is such proportion between the latter terms of the fore-going Proportion , as between the former , omitting the two first terms , it also holds , As the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides Is to the Versed Sine of the Angle sought , So is the difference of the Versed Sines of the third side , And of the sum of the two including sides , To the Versed Sine of the sought Angles , Complement to 180d. And this is the Proportion M. Foster makes use of in his Scale , page 25 and 27. to find the Hour and Azimuth by Protraction , as also in page 68. in Dyalling , when three sides are given to find an Angle , by constituting two right angled equi-angled plain Triangles , the legs whereof consist of the 4 terms of this Proportion . But in that Protraction work , the first and third terms of the Proportion are given together , with the sum of the second and fourth terms , to find out the said terms respectively . The Proportion for the Hour . As the difference of the Versed Sines of the Sum , and difference of the Complement of the Latitude , and of the Sun or Stars distance from the Elevated Pole , Is to the Diameter or Versed Sine of 180d , So is the difference of the Versed Sines of the Complement of the Altitude , and of the Ark of difference between the Complement of the Latitude , and of the Polar distance , To the Versed Sine of the Hour from Noon . And if the latter clause of the third term be the Sum of the Co-latitude and Polar distance , the Proportion will find the Versed Sine of the hour from midnight , And if the sum of any two Arks exceed a Semicircle , take its Complement to 360d , for the same Versed Sine is common to both . When the Declination is towards the Elevated Pole , the Polar distance is the Complement of it to 90d ; and when towards the Depressed Pole , the Polar distance is equal to the Sum of 90d , and of the Declination added together . Example . Let the Suns Declination be 15d 46′ North , Complement , — 74d 14′ The Complement of the Latitude , — 38 28 Sum — 112 : 42 Difference — 35 : 46 And let the Altitude be 20d , Complement — 70 : 00 Operation . Take the distance between the Versed Sines of 35d 46′ , and of 112d 42′ , and entring one foot of that extent at the end of the Versed Scale at 180d , lay the thred to the other foot , according to nearest distance , then take the distance between the Versed Sines of 35d 46′ , and 70d , and entring that extent parallelly , between the Thred and the Scale , and the other foot will rest upon the Versed Sine of 77d 32′ , the quantity of the Hour from the Meridian being either 50′ past 6 in the morning , or 10′ past 5 in the afternoon . The Reader may observe in this work , that the thred lies over a Star , by entring the first extent ; as also , that there is the same Star graduated at 35d 46′ of the Versed Sine , and this no other then the Bulls eye , having 15d 46′ of North Declination , for which Star in this Latitude , there needs be no summing or differencing of Arks , in regard the Stars declination varies not : So to find that Stars hour at any time , having any other Altitude , only lay the thred over that Star in the Quadrant , and take the distance between the Star in the Scale , and the Complement of its Altitude , and enter that extent parallelly between the Thred and the Scale , and it finds the Stars hour from the Meridian : Thus when that Star hath 39d of Altitude , its hour from the Meridian will be found to be 45d 54′ , in time , 3 hours 3½′ , which to get the true time of the night , must be turned into the Suns hour by help of the Nocturnal on the Back-side : But admitting the Suns Declination and Altitude to have been the same with the Stars , the true time of the day thus found , would have been 56½′ past 9 in the morning , or 3½′ past 3 in the afternoon ; and thus the Reader may have what Stars he pleases put on of any Declination , and for any Latitude ; and they may be put on at such a distance from the Center , that the distance from it to the Star , may be a Chord to be measured in the Limb , to give the Stars Ascensional difference , or the like conclusion : And thus the thred being once laid , and the former point found for one example to the Suns Declination , neither of them varies that day ; which is a ready general way for finding the time of the day for the Sun. To find the Semidiurnal , and Seminocturnal Arks. SUppose the Sun to have no Altitude , and the Complement of it to be 90d , and then work by the former precept , and you will find the Semidiurnal Ark from the beginning of the Line , and the Seminocturnal Ark from the end of the Line , which doubled , and turned into time , shews the length of the Day and Night , and the difference between 90d , and either of those Arks is the Ascensional difference , or time of rising and setting from 6. To find the Azimuth generally . The Proportions for this purpose have been delivered before , from which it may be observed , that there are no two terms fixed , and therefore to every Altitude , the containing sides of the Triangle , namely , the Complements both of the Altitude and Latitude must be summed and differenced , when the Proposition is to be performed on this Line solely , and the Operation will be after the same manner , as for the hour , namely , with a Parallel entrance : and this is all I shall say of the Authors general way ; and of any other that he used , I never heard of ; those ways that follow , being of my own supply . By help of this Line to work a Proportion in Sines alone , wherein the Radius leads . As the Radius Is to the Sine of any Ark , So is the Sine of any other Ark To the Sine of a fourth Ark. This fourth Sine , as I have said before , is demonstrated by M. Gellibrand , to be equal to half the difference of the Versed Sines of the Sum , and difference of the two middle terms of the Proportion . Operation . Let the Proportion be , As the Radius Is to the Sine of — 40d So is the Sine of — 27 To a fourth Sine Sum — 67 Difference — 13 Take the distance between the Versed Sines of the said sum and difference , and measure it on the Line of Sines from the Center , and it will reach to 17d , the fourth Sine sought . By help of this Line may the Divisions of the line Sol , or Proportional Sines , be graduated to any Radius less then half the Radius of the Quadrant , the Canon is , As the Versed Sine of any Ark added to a Quadrant , Is to the Radius , or length of the Line Sol , So is the Versed Sine of that Arks Complement to 90d To that length which pricked backward from the end of the Radius of the said Line , shall graduate the Arch proposed . Example . Suppose you would graduate 20 of the Line Sol , enter the Radius of the said Line upon the Versed Sine of 110d , laying the thred to the other foot ; and from the Versed Sine of 70d , take the nearest distance to the thred , which prick from the end of the Line Sol , towards the beginning , and it shall graduate the said 20d. This Line Sol is made use of by M. Foster in his Scale for Dyalling . The Line of Versed Sines was placed on the left edge of the foreside of the Quadrant , for the ready taking out the difference of the Versed Sines of any two Arks , and to measure a difference of two Versed Sines upon it , which are the chief uses I shall make of it ; whereas to Operate singly upon it , it would be more convenient for the hand to have it lie on the right edge of the Quadrant . An example for finding the Azimuth generally , by help of Versed Sines in the Limb , and of other Lines on the Quadrant . I shall rehearse the Proportion , As the Cosine of the Latitude is to the Secant of the Altitude , Or , As the Cosine of the Altitude is to the Secant of the Latitude , So is the difference of the Versed Sines of the Suns distance from the Elevated Pole , and of the Ark of difference between the Latititude and Altitude , To the Versed Sine of the Azimuth from the midnight meridian . And making the latter clause of the third term the Complement of the Sum of the Latitude and Altitude to a Semicircle , the Proportion will find the versed Sine of the Azimuth from the noon Meridian . Example . Altitude , — 51d 32′ Latitude — 34 : 32 Complement 55d 2● Difference 17 : 00 ☉ distance from elevated Pole , 66 : 29 Operation in the first Terms of the Proportion . On the Line of Versed Sines , take the distance between 17d , and 66d 29′ , and entring it twice down the line of Sines , from the Center , take the nearest distance to the thread laid over the Secant of 51d 32′ , the given Altitude , and entring one foot of this Extent at the Sine of 55d 28′ the Complement of the Latitude , lay the thred to the other foot , according to nearest distance , and in the line of Versed Sines in the Limb , it will lie over 95d , for the Suns Azimuth from the midnight meridian . And the Suns declination supposed the same , he shall have the like Azimuth from the North , in our Latitude of London , when his Altitude is 34d 32′ , for the sides of the Triangle are the same . Another Example . To find it in the versed Sine of 90d Latitude — 47d 27′ Altitude — 51 : 32 Sum — 98 : 59 Complement — 81 : 1 Polar distance — 66 : 29 Take the distance in the Line of Sines , as representing the former half of a Line of Versed Sines , between these two Arks counted towards the Center , viz. 66d 29′ , and 81d 01′ , and enter this extent twice down the Line of Sines from the Center , and take the nearest distance to the thred lying over the Secant of the Latitude 47d 27′ , then enter one foot of this extent at 51d 32′ counted from the end of the Sines towards the Center , laying the thred to the other foot , according to nearest distance , and in the Versed Sine of 90d , it shews the Azimuth to be 65d from the South in this our Northern Hemispere . Of the fitted Particular Scale , and the Line of Entrance thereto belonging . THis Scale serves to find both the Hour and Azimuth in the Latitude of London , to which it is fitted , in the equal Limb , by a Lateral or positive Entrance , it consists of two Lines of Sines . The greater is 62d of a Sine , as large as can stand upon the Quadrant , the Radius of the lesser Sine is made equal to 51d 32′ of this greater , being fitted to the Latitude : The Scale of Entrance standing within the Projection , and abutting on the Line of Sines , is no other but a portion of a Line of Sines , whose Radius is made equal to 38d 28′ of the greater Sine of the fitted Scale ; and this Scale of Entrance is numbred by its Complements up to 62d , as much as is the Suns-greatest meridian Altitude in this Latitude . The ground of this Scale is derived from the Diagonal Scale , the length whereof bears such Proportion to the Line of Sines whereto it is fitted , as the Secant of the particular Latitude doth to the Radius , which is the same that the Radius bears to the Cosine of the Latitude , and consequently , making the Line of Sines to represent the fitted Scale , the Radius of that Sine whereto it is fitted , must be equal to the Cosine of the Latitude : and so we needed no particular Scale , but this would remove the particular Scale , or Scale of Entrance , nearer the Center , and would not have been so ready as this fitted Scale ; however , hence I might educe a general method for finding the hour and Azimuth in the Limb , without Tangents or Secants . The first Work would be to proportion out a Sine to a lesser Radius , which would find the point of Entrance , the next would be to finde the Altitude , or Depression , at 6. the third would be to enter the sum , or difference of the Sines of the Altitude , or depression at 6 at the point of Entrance , and to lay the thred to the other foot ; but I shall demonstrate it from other grounds . 1. To find the time of Sun Rising , or Setting . Take the Declination from the lesser Sine , and enter it at the Declination in the Scale of Entrance , laying the thred to the other foot , according to nearest distance , and it shews the time of Rising or setting in the equal Limb. So when the Sun hath 13d of South Declination , he riseth at 8′ past 7 in the morning f●re , and sets at 52′ past 4 in the afternoon . 2. To find the true time of the day . In Summer , or Northwardly Declination , take the distance between the Altitude in the greater Sine , and the Declination in the lesser Sine . In Winter , take the Declination in the lesser Sine , and with your Compasses add it to the Altitude in the greater Sine . These extents enter at the Declination in the Scale of Entrance , and lay the thred to the other foot , according to nearest distance , and in the equal Limb , it will lye over the true time of the day . In Summer , when the Declination in the fitted Scale is above the Altitude , the hour is found from 6 towards midnight , when below it , towards Noon . Example . When the Sun hath 13d of North Declination , his Altitude being 39d 10′ will be a quarter past 9 in the morning , or 3 quarters past 2 in the afternoon ; and when he hath the same South Declination , his Altitude being 16d 14′ the time of the day will be found the same . The Converse will find the Suns Altitudes on all hours by this fitted Scale , which I shall handle the general way . 3. To find the Amplitude . Take the Declination from the greater Sine , and enter it at the beginning of the Scale of Entrance , laying the thred to the other foot , according to nearest distance , and it shews it in the Limb. When the Sun hath 13d of Declination , his Amplitude will be 21d 12′ . 4. To find the Azimuth of the Sun. In Summer ▪ take the distance between the Altitude in the lesser Sine , and the Declination in the greater . In Winter , or South Declinations , take the Declination from the greater Sine , and add it to the Altitude in the lesser Sine with your Compasses . These Extents , enter at the Altitude in the Scale of Entrance , and lay the thred to the other foot , according to nearest distance ; and in the equal Limb , it shews the Azimuth from the East or West . In Summer , when the Altitude falls below the Declination , the Azimuth is found from the East or West , Northwards ; when above it , Southwards . So when the Sun hath 13d of North Declination , his Altitude being 43d 50′ the Azimuth will be found to be 45d from East or West , Southwards ; and when he hath the same South Declination , his Altitude being 14d 50′ he shall have the same Azimuth . These Scales are fitted to give the Altitude at six , and the Vertical Altitude by Inspection . Against the Declination in the greater lesser Sine stands the Vertical Altitude or Depression , Altitude or Depression at six . When the Hour or Azimuth falls near Noon , mind against what Arch of the Line of Sines the point of Entrance falls , the thred may be laid to the like Arch in the Limb , and the respective extents entred parallelly between the Scale and the thred , and the answer found in the Line of Sines . But we have a better remedy by help of the Versed Sine of 90d put thorow the whole Limb. The joynt use of the Fitted Scale , with the Versed Sine of 90d in the Limb. IN the following Propositions , I shall make no use of the lesser Sine of the Fitted Scale . Get the Summer and Winter Meridian Altitude , by summing and differencing the Declination , and the Complement of the Latitude , which may be done with Compasses in the equal Limb , by applying the Chord of the Declination both ways from the Co-latitude . To find the Hour of the Day in Winter . Take the distance between the Meridian Altitude , and the given Altitude , out of the greater Sine of the fitted Scale , and as before , enter it at the Declination in the Scale of Entrance , laying the thred to the other foot , according to nearest distance , and in the Versed Sine of 90d it shews the hour from Noon . So if the Sun have 13d of South Declination , the Meridian Altitude is 25d 28′ , if the given Altitude be 17d 44′ the time of the day will be half an hour past 9 in the morning , or as much after 2 in the afternoon . To find the Hour of the Day in Summer . Take the distance between the Summer Meridian Altitude , and the proposed Altitude , and if this extent be less then the distance of the Declination in the Scale of Entrance from the Center , enter it at the Declination in the said Scale , and laying the thred to the other foot , it will in the Versed Sine of 90d shew the Hour from Noon . If the Sun have 23d 31′ of North Declination , his Meridian Altitude will be 61 59′ , if his given Altitude be 47d 51′ , the time of the day will be a quarter past 9 in the morning , or three quarters of an hour past 2 in the afternoon . If the Extent be larger then the distance of the point of Entrance ▪ to wit , the distance of the Declination in the Scale of Entrance from the Center , the hour must be found from midnight . In this case , with your Compasses add the Sine of the Winter Meridian Altitude , taken from the greater Sine of the fitted Scale , to the Sine of the Altitude in the said Scale , and enter the said whole extent at the point of Entrance , as before ; and in the Versed Sine of 90d , the thred will shew the hour from midnight . When the Sun hath 23 31′ of North Declination , if his Altitude be 5d 24′ , the time of the day will be half an hour past 4 in the morning , or half an hour past 7 in the evening , the Winter Meridian Altitude to this Declination being 14d 57′ . When the hour in these examples falls near Noon , the extent of the Compasses may be doubled , or tripled , and an Ark first found in the Limb , then if the thred be laid over the like Ark from the other edge , it will accordingly in the Versed Sines doubled or tripled , shew the time sought ; and the like may be done for the Azimuth . To find the Azimuth of the Sun in Winter : Get the Ark of difference between the Suns Altitude , and the Complement of the Latitude , and in the greater Sine of the fitted Scale , take the distance between the said Ark , and the Suns Declination , and enter one foot of this Extent at the Altitude in the Scale of Entrance , laying the thred to the other foot , and in the Versed Sine of 90d , it shews the Azimuth from Noon Meridian . Example . Colatitude , — 38d 28′ Altitude , — 12 : 13 Ark of Difference — 26 : 15 Declination — 13 : 00 The Azimuth to this example , will be 50d from the South . In Summer , get the Ark of difference between the Altitude , and the Complement of the Latitude , then when the Suns Altitude is the lesser of the two , take the sum , but when the greater , the difference of the Sines of the Suns Declination , and of the said Ark , and enter it at the Altitude on the Scale of Entrance , and you will find the Azimuth from the noon Meridian , as before ; but when either of those extents are larger then the distance between the point of Entrance and the Center , the Azimuth must be found from the midnight Meridian . In this case , take the difference , that is , the distance of the Sines of the Suns Declination , and of the Ark ▪ being the sum of the Altitude and Colatitude , out of the greater Sine of the fitted Scale , and enter it at the Altitude in the Scale of Entrance , laying the thred to the other foot , and in the Versed Sine it shews the Azimuth from the North. Example for finding the Azimuth from the North. Colatitude — 38d 28′ Altitude — 14 : 15 Sum — 52 : 43 Declination , — 23 : 31 The Azimuth to this example , will be found to be 70d from the North. Of the joynt use of the Diagonal Scale , with the Line of Sines on this Quadrant . If the respective extents that found the Hour and Azimuth in the Limb on the small Quadrant , be doubled , and applyed here to the Line of Sines issuing from the Center , which in this case becomes the Scale of Entrance , the Hour and Azimuth will be also found in the equal Limb of this Quadrant , for all those respective Latitudes to which the Diagonal Scale is accommodated . Of the Hour and Azimuth Scales on the Back-side thereof . THose Scales were fitted to the Versed Sines quadrupled on that small Quadrant , and consequently , are fitted to the Versed Sine of 90d , and the Line of Sines on this Quadrant , which is just double the Radius of that Quadrant . Those Scales are peculiarly fitted for the Latitude of London , and thereby we may alwaies find the Hour and Azimuth in the Versed Sine of 90 , without the trouble of summing or differencing of Arks. 1. By the Hour Scale , to find the Hour of the Day . Take the distance between the Declination , proper to the season of the year , and the Altitude , and entering one foot of that extent at the Complement of the Declination in the Sines , lay the thred to the other foot , according to nearest distance , and it shews the hour from Noon . Example . When the Sun hath 13d of North Declination , his Altitude being 47● 24′ , the Hour will be 30′ past 10 in the morning , or as much past 1 in the afternoon . In Summer , when this extent is greater then the Cosine of the Declination , and that it will be , when the Sun hath less Altitude then he hath at 6. The Declination is graduated against the Meridian Altitudes . In this case , add the Sine of the Altitude given , to the Sine of the Meridian Altitude in Winter , to that Declination , with your Compasses , and enter that whole extent at the Declination counted in the Line of Sines from 90d laying the thred to the other foot , according to nearest distance , and in the Versed Sine of 90d , it will shew the hour from midnight . Declination , — 23d 31′ North , The hour will be found either 4 in the morning , or 8 at night . Altitude — 1 : 34 The hour will be found either 4 in the morning , or 8 at night . 2. By the Azimuth Scale , to find the Azimuth of the Sun. Take the distance between the Declination proper to the season of the year , and the Altitude , and entering one foot of this extent at the Complement of the Altitude in the Lines of Sines issuing from the Center , to the other lay the thred according to nearest distance , and it shews the Azimuth from the noon Meridian in the Versed Sine of 90d. Declination — 23d 31′ North , The Azimuth hereto will be found 65d from the South . Altitude — 47 : 27 The Azimuth hereto will be found 65d from the South . In Summer , when this extent is greater then the Cosine of the Altitude , and that it will be , when the Sun hath less Altitude then he hath in the Vertical , the Azimuth must be found from the midnight Meridian . In this case , because the Azimuth Scale is not continued far enough , the sum of the Altitude and Colatitude must be gotten , and the distance taken between the said Ark and the Declination , counted in the hour-Scale as a Sine , and that extent entred at the Altitude counted from 90d in the Line of Sines , and the thred laid to the other foot , will shew the Azimuth from the North in the Versed Sine of 90d in the Limb. Colatitude , — 38● 28′ The Azimuth to this example will be 65d from the North. Altitude — 10 : 19 The Azimuth to this example will be 65d from the North. Sum — 48 : 47 The Azimuth to this example will be 65d from the North. Declination — 23 : 31 The Azimuth to this example will be 65d from the North. North. General Proportions . It now remains to be shewed , how the Hour , and Azimuth , &c. may be found generally , either in the equal Limb , or in the Versed Sine of 90d , and that without the help of Tangents or Secants , and possibly with more convenience then with them . In page 55. I have asserted , that the fourth term in any direct Proportion , bears such Proportion to the first term , as the Rectangle of the two middle terms doth to the square of the first term . And in page 105. That the Sine of any Arch bears such proportion to the Secant of the Complement of another Ark , as the Rectangle of the Sines of both those Arks , doth to the Square of the Radius . Whence it follows , That , As the Radius , Is to the Sine of one of the sides including an Angle , So is the Sine of the other containing side , To a fourth Sine . I say then , that this fourth Sine bears such Proportion to the Radius , as the Sine of one of those including sides , doth to the secant of the Complement of the other . And therefore , when three sides are given to find an Angle , it will hold , As the Radius , Is to the Sine of one of those including sides , So is the Sine of the other including side , To a fourth sine . Again , As that fourth Sine , Is to the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides , So is the Radius , To the Versed Sine of the Angle sought . And as that fourth Sine , Is to the difference of the Versed Sines of the third side , and of the sum of the two including sides , So is the Radius , To the Versed Sine of the sought Angles , Complement to 180d , or a Semicircle . Thus we are freed from a Secant in the two first terms of 3 several Proportions that find 〈…〉 Hour and Azimuth : All which I shall further confirm from the Analemma , and then proceed to Application , in the Scheme annexed . Proportions in the Analemma . UPon the Center C , draw a Circle , and let N C be the Axis of the Horizon , and E P the Axis of the World , AE C the Equator , ☉ F and Z Y two Parallels of Declination on each side the Equator , alike equidistant , S G the parallel of Altitude at 6 , and D E F the parallel of Depression at 6 ; draw a parallel of Altitude less then the Altitude at 6 V R , and another greater M N continued ; also a parallel of Depression less then the Depression at 6 W P , and another greater X Y , and there will be constituted diverse right lined , right angled Triangles , relating to the motion of the Sun or Stars , in which it will hold . As Radius is to the Cosine of the Declination , So is the Cosine of the Latitude To a fourth . Namely the difference of the Sines of the Meridian Altitude , and Altitude at 6 in Summer , equal to the Sum of the Sines of the Meridian Altitude and Depression at 6 in Winter , which is equal to the sum of the Sines of the midnight Depression and Altitude at 6 in Summer . A : B ☉ ∷ B : A ☉ . Again the same , As D : S E ∷ E : D S Again the same — G : B F ∷ B : F G. As that fourth is to the Radius , So is the Sine of the Meridian Altitude , To the Versed Sine of the Semidiurnal Ark A ☉ : ☉ B ∷ ☉ I : ☉ K. The two first terms are common to all the rest of the following Proportions . And so is the Sine of the midnight Depression , To the Versed Sine of the Seminocturnal Ark. G F : B F ∷ F L to K F. And so is the Sine of the Altitude , To the difference of the Versed Sines of the Semidiurnal Ark and Hour sought from Noon , ☉ A : B ☉ ∷ I M : K N. And so is the Sine of the Depression , To the difference of the Versed Sines of the Seminocturnal Ark , and of the hour from Midnight , F G : F B ∷ P L : K O And so is the difference of the Sines of the Suns Meridian , and given Altitude , To the Versed Sine of the hour from Noon , ☉ A : ☉ B ∷ ☉ M : ☉ N. If the Sun have Depression , So is the sum of the Sines of the Suns Meridian Altitude , and proposed Depression , To the Versed Sine of the hour from Noon , A ☉ : ☉ B ∷ ☉ Q : ☉ O. And so is the difference of the Sines of the Midnight and propose Depression , To the Versed Sine of the hour from Midnight , F G : F B ∷ P F : O F. But supposing the Sun to have Altitude , retaining still the two first terms , it holds . And so is the sum of the Sines of the Midnight Depression and given Altitude , To the Versed Sine of the hour from Midnight . F G : F B ∷ F R : F S. And so is the Sine of the Altitude , or Depression at six , To the Sine of the Ascensional difference , A ☉ : ☉ B ∷ A I : B-K. In Summer , if the Sun have Altitude , So is the difference of the Sines of the Altitude at six , and of the given Altitude , To the Sine of the hour from six , towards Noon , if the given Altitude be greater then the Altitude at six , otherwise towards Midnight . A ☉ : ☉ B ∷ A M : B N. Also A ☉ : ☉ B ∷ A T : B S. If he have Depression , So is the sum of the Sines of the Altitude at six , and the given Depression , To the Sine of the Hour from six , towards Midnight . A ☉ : ☉ B ∷ A Q : B O. In Winter , if the Sun have Altitude : So is the the sum of the Sines of his Depression at 6 , and of his given Altitude , To the Sine of the hour from 6 toward Noon , S D : S E ∷ D V : Z E. If he have Depression . So is the difference of the Sines of his Depression at 6 , and of has given Depression . To the Sine of the hour from 6 towards Noon , when the Depression is less then the Depression at 6 , otherways towards Midnight . S D : S E ∷ W D : Q E. S D : S E ∷ D X : E Y. When two terms of a Proportion happen in the common Radius , and two in a Parallel , there needs no Reduction . In Latitudes nearer the Poles then the Polar Circles , the Semidiurnal Arks , when the Declination is towards the Elevated Pole , will be more then the Diameters of their Parallels ; in that case , the difference , is the difference of the Versed Sine of the Hour , and of the fourth Proportional , found by the Proportion that finds the Semidiurnal Ark. General Proportions for the Hour . The Proportion selected for the Hour is , As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Declination , To a fourth : Namely , the difference of the Sines of the Meridian Altitude , and of the Altitude at 6. Again , 1. As that fourth , Is to the Radius , So in Summer , is the difference ; but in VVinter , the sum of the Sines of the Suns Altitude or Depression at 6 , To the Sine of the Hour from 6 towards Noon or Midnight , according as the Altitude or Depression is greater or less then the Altitude or Depression at 6. 2. And so is the difference of the Sines of the Meridian , and proposed Altitude , To the Versed Sine of the Hour from Noon : And so is the sum of the Sines of the Midnight Depression , and given Altitude , To the Versed Sine of the Hour from Midnight . 3. And so is the Sine of the Altitude , To the difference of the Versed Sines of the Semidiurnal Ark , and of the Hour sought . By the first Proportion , the hour may be found generally , either in the equal Limb , or Line of Sines . By the second Proportion , it may be found generally , either in the Versed Sines of 90d , or 180d. By the third Proportion , it may be found in the Line of Versed Sines issuing from the Center in many cases . I shall add a brief Application of all three ways . The first Work will be to find the point of Entrance . Example , For the Latitude of Nottingham , 53d. Lay the thred to the Declination , admit 20 in the Limb , counted from the left edge , and from the Latitude in the Line of Sines , counted towards the Center from 90d ; take the nearest distance to the thred , the said extent measured from the Center , will fall upon 34 25′ , and there will be the point of Entrance ; let it be recorded , or have a mark set to it . If the Suns Declination be North , the Meridian Altitude in that Latitude , will be 57d , the said extent will reach from the Sine thereof , to the Sine of the Suns Altitude , or Depression at 6 , to that Declination , namely , to 15d 51′ : which may also be found without the Meridian Altitude , by taking the distance from 20d in the Sines , to the thred laid over the Arch 53d , counted from the right Edge , and by measuring that extent from the Center , the point thus found , I call the Sine point . Thirdly , If the respective distances between the Sine point , and the Sine of the given Altitude , be taken and entred upon the point of Entrance , laying the thred to the other foot , according to nearest distance , the hour may be found all day for that Declination , when it is North in the equal Limb. Example , For the Latitude of Nottingham , to the former Declination , being North. When the Sun hath 11d 31′ 20 17 of Altitude , the Hour in each Case will be found half an hour from 6 , to the lesser Altitude beyond it , towards Midnight ; to the greater , towards Noon . And when the Altitude is 38d 19′ , the time of the day will be either half an hour past 8 in the morning , or half an hour past 3 in the Afternoon . An Example for the Latitude of Nottingham , when the Declination is as much South . Let the Altitude be 10d 6′ , In this case add the Sine thereof to the Sine of 15d 51′ , the whole extent will be equal to the Sine of 26d 39′ ; Enter this Extent upon the point of Entrance at 34d 25′ laying the thred to the other foot , according to nearest distance , and the time of the day found in the Limb , will be either half an hour past 9 in the morning , or half an hour past 2 in the afternoon . An Example for working the second Proportion . The Summer Meridian Altitude is 57d , if the given Altitude be 46d 11′ , take the distance between the Sines of these two Arks , and entring this extent upon the point of Entrance , lay the thred to the other foot , according to nearest distance , it will in the Versed Sine of 90d , shew the Hour from Noon to be 37d 30′ , that is , either half an hour past 10 in the morning , or half an hour past 1 in the Afternoon . And when the Hour falls near Noon , we may double or triple the extent of the Compasses , and find an Ark in the Limb , which if counted from the other edge , and the thred laid over it ▪ will give answer in the Versed Sines doubled or tripled accordingly . A third Example . If the Altitude were 3d 15′ , in this case the distance between it and the Meridian Altitude being greater then the distance of the point of Entrance from the Center , the hour must be found from Midnight ; add the Sine thereof to the Sine of 17d , the Winter Meridian Altitude , the whole extent will be equal to the Sine of 20 25′ ; Enter the said extent upon the point of Entrance ▪ as before , and in the Versed Sine of 90d , the hour will be found to be either half an hour past 4 in the Morning , or half an hour past 7 in the Evening . Examples for working the third Proportion . Take the Sine of 30d , and enter it upon the point of Entrance , laying the thred to the other foot , according to nearest distance , and there keep it ; then take the nearest distance to it from the Sine of 57 , the Meridian Altitude ; and the said Extent prick upon the Line of Versed Sines on the left edge , and it will reach to 118● 54′ , set a mark to it . Lastly , the nearest distance from the Sine of each respective Altitude to the thred , being pricked from the said mark , will reach to the Versed Sine of the hour from Noon , for North Declinations . So when the Sun hath 24d 48′ of Altitude , the Hour from 7 : 17 Noon will be found to be — 75d 105 A Winter Example for that Declination . The nearest distance from the Sine of 17d , the Winter Meridian Altitude , to the thred , will reach to the Versed Sine of 61d 6′ , the Complement of the former to a Semicircle , at which set a mark ; then if the Altitude were — 12d 30′ 14 : 26 the nearest distances to the thred prickt from the latter mark , would shew the hours to these Altitudes to be 2 hours 1 ½ hour from Noon This last Proportion in some cases will be inconvenient , being liable to excursion in Latitudes more Northwardly . Two sides with the Angle comprehended , to find the third side . As the Radius , Is to the Sine of one of the Includers , So is the Sine of the other Includer , To a fourth . Again , As the Radius , Is to the Versed Sine of the Angle included , So is that fourth , To the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides , And so is the Versed Sine of the Included Angles Complement to 180. To the difference of the Versed Sines of the third side , and of the sum of the two including sides . Another Proportion for finding it in Sines , elsewhere delivered . By the former Proportion , having the advantage both of lesser and greater Versed Sines , we may find the side sought , either in the line of Sines , or in the line of Versed Sines on the the left edge ▪ issuing from the Center . The Converse of the Proportion that found the Hour , will find the Suns Altitudes on all Hours . As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Declination , To a fourth Sine . Namely , The difference of the Sines of the Suns Meridian Altitude , and of his Altitude at 6 in Summer , but the sum of the Sines of his Depression at 6 , and Winter Meridian Altitude , hereby we may obtain the point of Entrance and Altitude , or Depression at 6 , as before , and let them be recorded , then it holds , As the Radius , Is to the Sine of the Hour from 6 , So is that fourth Sine , To the difference of the Sines of the Suns Altitude at 6 , and of his Altitude sought ; But in Winter , To the sum of the Sines of his Depression at 6 , and of the Altitude sought . Hereby we may find two Altitudes at a time . Lay the thred to the Hour in the Limb , and from the point of Entrance , take the nearest distance to it , the said Extent being set down at the Altitude at 6 , shall reach upward to the greater Altitude , and downward , to the lesser Altitude . Example . Admit the hour to be 5 and 7 in the morning , the Altitudes thereto for 20 North Declination for the Latitude of Nottingham , will be found to be 7d 17′ , and 24● 48′ If the Hour be more remote from 6 then the time of Rising , we may find a Winter Altitude to as much South Declination , and a Summer Altitude , to the said North Declination . Thus if the Hour be 45d from 6 , that is either 9 in the morning , or 3 in the afternoon , the nearest distance from the point of Entrance to the thred , will reach from the Sine of 15d 51′ , the Altitude at 6 upwards , to the Sine of 42d 18′ , the Summer Altitude to that Declination : But downwards , it reaches beyond the Center : In this case measure , that extent from the Center , and take the distance between the inward foot of the Compasses , and the Altitude at 6 ▪ which measured on the Sines , will be found to be 7d 17′ for the Winter Altitude to that Hour . So if the hour were 60d from 6 , that is either 10 , or 2 , the Summer Altitude would be found to be 49d 42′ , and the Winter Altitude 12d 30′ . And this may be found in the Versed Sines on the left edge , accounted as a Sine each way from the middle , if use be made of the lesser Sines , instead of the Limb , in finding the point of Entrance , as also , in laying it to the Sine of each hour from 6 , in which case the Compasses will alwaies find two Altitudes at once ; for when they fall beyond the midst of the said Line , it shews the Winter Altitudes counted from thence towards the end of the said Versed Sines . Having found the fourth Sine , which gives the point of Entrance as before , the Altitudes on all hours may be found by the Versed Sines of 90d in the Limb , the Proportion will be , As the Radius , Is to the Versed Sine of the Hour from Noon , So is the fourth abovesaid , To the difference of the Sines of the Meridian Altitude , and of the Altitude sought . But for hours beyond 6 , the Proportion will be , As the Radius , Is to the Versed Sine of the Hour from Midnight , So is the fourth abovesaid , To the sum of the Sines of the Suns Depression at Midnight ( equal to his Winter Meridian Altitude , ) and of his Altitude sought , Hereby also we may find two Altitudes at once . Operation . Lay the thred to the Versed Sine of the Hour from Noon , and from the point of Entrance at 34d 25′ , take the nearest distance to it , the said Extent shall reach from the Summer Meridian Altitude , accounted in the Sines to the Altitude sought , also from the Winter Meridian Altitude , to the Altitude sought . Example . Latitude of Nottingham is 53d , Complement — 37● Suns Declination , — 20 Sum being the Summer Meridian Altitude — 57d Difference being Winter Meridian Altitude 17 If it were required to find the Altitudes for the hours of 11 and 1 The Extents so taken out will find the Summer Altitudes to be — 55● 00′ And the Winter Altitudes to the same hours and Declination — 15● 51● 10 and 2 The Extents so taken out will find the Summer Altitudes to be — 49 , 42 And the Winter Altitudes to the same hours and Declination — 12 , 30 9 and 3 The Extents so taken out will find the Summer Altitudes to be — 42 , 18 And the Winter Altitudes to the same hours and Declination — 7 , 17 8 and 4 The Extents so taken out will find the Summer Altitudes to be — 33 , 47 And the Winter Altitudes to the same hours and Declination — 00 , 32 But for hours more remote from the Meridian then 6 , as admit for 5 in the morning , or 7 at night , which is 75d from the North Meridian ; lay the thred to the said Ark in the Versed Sine of 90d , and the distance from the point of Entrance to it , shall reach from the Sine of 57d , the Meridian Altitude , to the Sine of 24d 48′ , the Summer Altitude for the Hour 75d from Noon , and if that Extent be pricked from the Winter Meridian Altitude , it will reach beyond the Center , in which case , enter that Extent upon the Line of Sines , and take the distance between the point of limitation and 17d , which will ( being measured ) be found to be the Sine of 7d 17′ , the Altitude belonging to the hour 105 from Noon . In like manner , the Altitudes for the hours 97d 30 from Noon that is 82d 30′ from Midnight , will be 11d 31′ and for the like hours from Noon 20d 17′ 112 , 30 from Noon that is 67 , 30 from Midnight , will be 3 : 15 and for the like hours from Noon 29 : 19 In like manner , it might have been found in the Versed Sines issuing from the Center , if in finding the point of Entrance , and in laying the thred to the Versed Sine of the Hour , we make use of the lesser Sines , and of the Versed Sine of 180d in the Limb. For the Azimuth . Two of the former Proportions may be conveniently applied to other sides , for finding the Azimuth universally . As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Altitude , To a fourth Sine . Get the sum of the Altitude and Colatitude ; or , which is all one , the sum of the Latitude and Colatitude ; and if it exceeds a Quadrant , take its Complement to a Semicircle : This fourth Sine is equal to the difference of the Sines of this Compound Ark , and of another Ark to be thereby found , called the latter Ark. Then it holds , As the fourth Sine , Is to the Radius , So in Summer is the difference , but in Winter , the sum of the Sines of this latter Ark , and of the given Declination , To the Sine of the Azimuth from the Vertical . When the latter Ark is more then the Declination , the Azimuth will be found from the Vertical towards the Noon Meridian , otherwise towards the Midnight Meridian , and in winter , always towards the Noon Meridian . For such Stars as come to the Meridian between the Zenith and the elevated Pole , the fourth Ark will never exceed the Stars declination , and their Azimuth will be alwaies found from the Vertical towards the Meridian they come to , above the Horizon . Example for the Latitude of Nottingham . Complement of the Latitude is — 37d Altitude is 40● — 40 Sum — 77 Let the Declination be 20d North. To find the point of Entrance , take the nearest distance to the thred laid over 50 in the Limb , counted from right edge from the Sine of 37d , the said Extent measured from the Center , falls upon the Sine of 27d 26′ , and there will be the point of Entrance ; the said Extent prickt from 77● in the Sines , will reach to the Sine of 30 51′ , where the Sine point falls . Lastly , The distance between the Sine point , and the Sine of 20d being entred at the point of Entrance , and the thred laid to the other foot , the Azimuth will be found in the equal Limb to be 21d 48′ from the East or West Southwards , because the Sine point fell beyond the Declination . Another Example for that Latitude , the Declination being 20d South Altitude . 12d 30′ The point of Entrance will fall at the Sine of 36● The Sine point may be found without summing or differencing of Arks , by taking the nearest distance from the Sine of the Latitude , to the thred laid over the Altitude , counted in the Limb from the right edge ; which Extent being added to the Sine of 20d the Declination , the whole Extent will be equal to the Sine of 31d , this being entred on the point of Entrance , and the thred laid to the other foot , the Azimuth will be found to be 61d 14′ from the East or West Southwards . A third Example for the Latitude of London , 51d 32′ . Let it be required to find the Azimuth of the middlemost Star in the great Bears tail , Declination is 56d 45′ , let the Altitude be 44d 58′ . The nearest distance from the Sine of 38d 28′ to the thred laid over the Altitude counted from the right edge , will find the point of Entrance to be at the Sine of — 26d 6′ . The nearest distance from the Sine of 51d 32′ to the thred laid over the Altitude , counted from the right edge , need not be known , but the distance between that Extent , and the Sine of 56d 45′ , the Stars Declination being entred on the point of Entrance , will find the Azimuth of that Star , by laying the thred to the other foot , to be 40d from the East or West Northwards . Thus we find it the general way , and so it will also be found by the fitted particular Scale ; for the Hour , the point of Entrance , and Sine point , vary not till the Declination change ; but for the Azimuth , they vary to every Altitude . To find the Azimuth in the Versed Sines . As the fourth , found by the former Proportion ; namely , where the point of Entrance hapned , Is to the Radius , So is the difference of the Versed Sines of the Polar distance , and of the Ark of Difference between the Altitude and the Latitude , To the Versed Sine of the Azimuth from Midnight Meridian . This finds the Angle it self in the Sphere . And so is the difference of the Versed Sines of the Polar distance , and of the Ark of residue of the sum of the Latitude and Altitude taken from a Semicircle . To the Versed Sine of the Azimuth from Noon Meridian . This finds the Complement of the Angle in the Sphere to a Semicircle . The Proportion to find it from Midnight Meridian , the third term being express'd in Sines , will be thus . Get the sum of the Altitude and Colatitude , and when it exceeds a Quadrant , take its Complement to a Semicircle , the Ark thus found , is called the Compound Ark. Then it holds , As the fourth found before . Is to the Radius , So in Summer Declinations , is the difference , but in Winter Declinations , the sum of the Sines of the Suns or Stars declination , and of the compound Ark , To the Versed Sine of the Azimuth from the Midnight Meridian of the place . Use this Proportion alwaies for the Sun or Stars , when they come to the Meridian between the Zenith and elevated Pole. And to find it from the Noon Meridian , Get the difference between the Altitude and Colatitude , and then it holds , As the fourth Sine found before , Is to the Radius , So is the sum of the Sines of the said Ark of Difference , and of the Suns Declination , To the Versed Sine of the Azimuth from the Noon Meridian , in Summer only , when the Suns Altitude is less then the Colatitude . In all other cases , So is the difference of the said Sines , To the Versed Sine of the Azimuth , as before , from Noon Meridian . If by the former Proportion it be required to find the Azimuth in the Versed Sine of 90d , a difference of Versed Sines taken out of the Line of Versed Sines on the left edge must be doubled , and being taken out of the Line of Sines , as sometimes representing the former , sometimes the latter half of a Versed Sine , needs not be doubled . Example : Latitude of Nottingham — 53d Altitude of the Sun — 4 Ark of difference — 49 ☉ Declination 20 North , the Polar distance is — 70 The Point of entrance will fall at the Sine of 36d 54′ And the difference of the Versed Sines of 49● and 70● , equal to the distance between the Sines of 41● and 20 being entred at the Point of entrance , and the Tbread laid to the other foot will lye over 61d 30′ of the Versed Sine of 90● , and so much is the Suns Azimuth from the North. Another Example for finding it from the South when the Altitude is more then the Colatitude . Altitude — 47d Colatitude — 37 of Nottingham . difference — 10 The Point of entrance will fall at the sine of 24d 14′ found by taking the nearest extent from sine of 37d to the Thread lying over 43d of the Limb the Coaltitude . Then the distance between the sines of 10d , the Ark of difference as above , and the sine of 20 the Suns North Declination being entred at the Point of entrance , and the Thread laid to the other foot , will shew 53d 55′ in the Versed sine of 90d for the Suns Azimuth from the South . A third Example when the Altitude is less then the Colatitude in Summer . Complement Latitude 37d of Nottingham . Altitude — 34 difference — 3 The Point of entrance will fall at the sine of 29d 55′ , and the sum of the sines of 3d , and of 20d the Suns declination supposed North , is equal to the sine of 23d 13′ : Which Extent entred at 29d 55′ , the Point of entrance , and the Thread laid to the other foot according to nearest distance , it will intersect the Versed sine of 90d at the Ark of 77d 57′ , and so much is the Suns Azimuth from the South . And if there were no Versed sines in the Limbe , find an Ark of the equal Limbe , and enter the sine of the said Ark down the Line of sines from the other end , and you may obtain the Versed sine of the Ark sought . More Examples need not be insisted upon , having found the Point of entrance , the distance between the Versed sines of the Base or side subtending the angle sought , and of the Ark of difference between the two including sides , being taken out of the streight Line of Versed Lines on the left edge , and entred at the Point of entrance , laying the Thread to the other foot shews in the Versed Sine of 180d in the Limb the angle sought ; and if the said distance or Extent be doubled , and there entred it shews the angle sought in the Versed Sine of 90d , when the Angle is less then a Quadrant , when more , the distance between the Versed Sines of the Base and the sum of the Legs , will find the Complement of the angle sought to a Semicircle without doubling in the Versed Sine of 180d in the Limb , with doubling in the Versed Sine of 90d. Lastly , Three sides , viz. all less then Quadrants , or one of them greater , generally to find an angle in the equal Limb , the Proportion will be , As the Radius , Is to the Cosine of one of the including sides : So is the Cosine of the other Includer , To a fourth Sine . Again , As the Sine of one of the Includers , To the Cosecant of the other : So when any one of the sides is greater then a Quadrant is the sum , but when all less , the difference of the fourth Sine , and of the Cosine of the third side , To the Cosine of the angle sought . If any of the three sides be greater then a Quadrant , it subtends an Obtuse angle , the other angles being Acute ; But when they are all less then Quadrants , if the 4th Sine be less then the Cosine of the third side , the angle sought is Acute , if equal thereto , it is a right angle , if greater an Obtuse angle . From the Proportion that finds the Hour from six , we may educe a single Proportion applyable to the Logarithms without natural Tables for Calculating the Hour of the day to all Altitudes , By turning the third Tearm , being a difference of Sines or Versed Sines into a Rectangle , and freeing it from affection . The two first Proportions to be wrought are fixed for one Declination ; The first will be to find the Suns Altitude or Depression at six . The second will be to find half the difference of the Sines of the Suns Meridian Altitude , and Altitude sought , &c. as before defined , the Proportion to find it is , As the Secant of 60d , To the Cosine of the Declination : So is the Cosine of the Latitude , To the Sine of a fourth Arch. Lastly , To find the Hour . Get the sum and difference of half the Suns Zenith distance at the hour of six ; and of half his Zenith distance to any other proposed Altitude or Depression . Then , As the Sine of the fourth Arch , Is to the Sine of the sum : So is the Sine of the difference , To the Sine of the hour from six towards Noon or Midnight , according as the Altitude or Depression was greater or lesser then the Altitude or Depression at six . Observing that the Sine of an Arch greater then a Quadrant , is the Sine of that Arks Complement to a Semicircle . Of the Stars placed upon the Quadrant below the Projection . ALL the Stars placed upon the Projection are such as fall between the Tropicks and the Hour may be found by them with the Projection , as in the Use of the small Quadrant : Which may also be found by the fitted particular Scale , not only for Stars within the Tropicks , but for all others without , when their Altitude is less then 62d , and likewise their Azimuth may be thereby found when their Declination is not more then 62d. For other Stars without the Tropicks , they may be put on below the Projection any where in such an angle that the Thread laid over the Star shall shew an Ark in the Limb , at which in the Sines the Point of entrance will always fall ; And again , the same Star is to be graved at its Altitude or Depression at six in the Sines , and then to find the Stars hour in that Latitude whereto they are fitted , will always for Northern Stars be to take the distance in the Line of Sines between the Star and its given Altitude , and to enter that Extent at the Point of entrance , laying the Thread to the other foot according to nearest distance , and it gives the Stars hour in the equal Limb from six , which may also be found in the Sines by a Parrallel entrance , laying the Thread over the Star. Example . Let the Altitude of the last in the end of the great Bears Tail be 63d , take the distance between it and the Star which is graved at 37d 30′ of the Sines , the said Extent entred at the Sine of 23d , the Ark of the Limb the Thread intersects when it lies over the said Star , and by laying the Thread to the other foot you will find that Stars hour to be 46d 11′ from six towards Noon Meridian , if the Altitude increase , and in finding the true time of the night , the Stars hour must be always reckoned from the Meridian it was last upon ; in this Example it will be 5 minutes past 9 feré . Of the Quadrant of Ascensions on the backside , This Quadrant is divided into 24 Hours with their quarters and subdivisions , and serves to give the right Ascension of a Star , as in the small Quadrant to be cast up by the Pen. It also serves to find the true Hour of the night with Compasses . First having found the Stars hour , take the distance on the Quadrant of Ascensions in the same 12 hours between the Star and the Suns Ascension ( given by the foreside of the Quadrant ) the said Extent shall reach from the Stars hour to the true hour of the night , and the foot of the Compasses always fall upon the Quadrant ; Which Extent must be applyed the same way it was taken , the Suns foot to the Stars hour . Example . If upon the 30th of December the last in the end of the Bears Tail were found to be 9 hours 05′ past the Meridian it was last upon , the true time sought would be 16 minutes past 3 in the morning . Another Example for the Bulls Eye . Admit the Altitude of that Star be 39 d , that Stars hour as we found it by the Line of Versed Sines was 3 ho 3′ from the Meridian , if the Altitude increase , then that Stars hour from the Meridian it was last upon was 57 minutes past 8 — 8 h : 57′ If this Observation were upon the 23d of October , the Complement of the Suns Ascension would be — 9 : 30 The Ascension of that Star is — 4 : 16 The true time of the night would be forty — 10 : 43 three minutes past ten . The distance between the Star and the Suns Ascension being applyed the same way , by setting the Sun foot at the Stars hour will shew the true time sought . When the Star is past the Meridian , having the same Altitude , the Stars hour will be 3′ past 3 , and the true time sought , will be 49′ past 4 in the next morning . The Geometrical Construction of Mr Fosters Circle . THe Circle on the Back side of the Quadrant , whereof one quarter is only a void Line , is derived from M. Foster's Treatise of a Quadrant , by him published in An●o 1638. the foundation and use whereof being concealed , I shall therefore endeavour to explain it . Upon the Center H describe a circle , and draw the Diameter A C , passing through the Center , and perpendicularly thereto , upon the point C , erect a Line of Sines C I , whose Radius shall be equal to the Diameter A C , let 90d of the Sine end at I ; I say then , if from the point A , through each degree of that Line of Sines , there be streight lines drawn , intersecting the Quadrant of the circle C G , as a line from the point D doth intersect it at B the Quadrant C G , which the Author calls the upper Quadrant , or Quadrant of Latitudes shall be constituted , and if C I be continued as a Secant , by the same reason the whole Semicircle C G A may be occupied ; hence it will be necessary to educe a ground of calculation for the accurate dividing of the said Quadrant , and that will be easie ; for A C being Radius , the Sine C D doth also represent the Tangent of the Angle at A , therefore seek the natural Sine of the Ark C D in the Table of Natural Tangents , and the Ark corresponding thereto , will give the quantity of the Angle D A C , then because the point A falls in the circumference of the Circle , where an Angle is but half so much as it is at the Center , by 31 Prop. 3. Euc. double the Angle found , and from a Quadrant divided into 90 equal parts , and their subdivisions , by help of a Table so made , may the Quadrant of Latitudes be accurately divided : but the Author made his Table in page 5. without doubling , to be graduated from a Quadrant divided into 45 equal parts . Again , If upon the Center C , with a pair of Compasses , each degree of the line of Sines be transferred into the Semicircle C G A it shall divide it into 90 equal parts ; the reason whereof is plain , because the Sine of an Arch is half the chord of twice that Arch , and therefore the Sines being made to twice the Radius of this circle , shall being transferred into it , become chords of the like Arch , to divide a Semicircle into 90 equal parts . Again , upon the point A , erect a line of Tangents of the same Radius with the former Sine , which we may suppose to be infinitely continued , here we use a portion of it A E. If from the point C , the other extremity of the Diameter lines be drawn , cutting the lower Semicircle ( as a line drawn from E intersects it at F ) through each degree of the said Tangent , the said lower Semicircle shall be divided into 90 equal parts ; the reason is evident a line of Tangents from the Center shall divide a Quadrant into 90 equal parts , and because an Angle in the circumference is but half so much as it is in the Center , being transferred thither , a whole Semicircle shall be filled with no more parts . The chief use of this Circle , is to operate Proportions in Tangents alone , or in Sines and Tangents joyntly , built upon this foundation , that equiangled plain Triangles have their sides Proportional . In streight lines , it will be evident from the point D to E , draw a streight line intersecting the Diameter at L , and then it lies as C L to C D ; so is A L to A E : it is also true in a Circle , provided it be evinced , that the points B L F fall in a streight line . Hereof I have a Geometrical Demonstration , which would require more Schemes , which by reason of its length and difficulty , I thought fit at present not to insert , possibly an easier may be found hereafter : As also , an Algebraick Demonstration , by the Right Honourable , the Lord Brunkard , whereby after many Algebraick inferences it is euinced , that as L K is to K B ∷ so is L N to ● F : whence it will follow , that the points B , L , F , are in a right line . If a Ruler be laid from 45d of the Semicircle , to every degree of the Quadrant of Latitudes , it will constitute upon the Diameter , the graduations of the Line Sol , whereby Proportions in Sines might be operated without the other supply . From the same Scheme also follows the construction of the streight line of Latitudes , from the point G , at 90● of the Quadrant of Latitudes , draw a streight Line to C , and transfer each degree of the Quadrant of Latitudes with Compasses , one foot resting upon C into the said streight line , and it shall be constituted . To Calculate it . The Line of Latitudes C G bears such Proportion to C A as the Chord of 90d doth to the Diameter , which is the same that the Sine of 45d bears to the Radius , or which is all one , that the Radius bears to the Secant of 45 d , which Secant is equal to the Chord of 90 d ; from the Diagram the nature of the Line of Latitudes may be discovered . Any two Lines being drawn to make a right angle , if any Ark of the Line of Latitudes be pricked off in one of those Lines retaining a constant Hipotenusal A C , called the Line of Hours , equal to the Diameter of that Circle from whence the Line of Latitudes is constituted , if the said Hipotenusal from the Point formerly pricked off , be made the Hipotenusal to the Legs of the right angle formerly pricked off , the said Legs or sides including the right angle shall bear such Proportion one to another , as the Radius doth to the sine of the Ark so prickt off ; and this is evident from the Schem , for such Proportion as A C bears to C D , doth A B bear to B C , for the angle at A is Common to both Triangles , and the angle at B in the circumference is a right angle , and consequently the angle A C B will be equal to the angle A D C , and the Legs A C to C D bears such Proportion by construction , as the Radius doth to the Sine of an Ark , and the same Proportion doth A B bear to B C , in all cases retaining one and the same Hypotenusal A C , the Proportion therefore lies evident . As the Radius , the sine of the angle at B , To its opposite side A C , the Secant of 45d : So is the sine of the angle at A , To its opposite side B C sought . Now the quantity of the angle at A was found by seeking the natural Sine of the Ark proposed in the Table of natural Tangents ; and having found what Ark answers thereto , the Sine of the said Ark is to become the third Tearm in the Proportion . But the Cannon prescribed in the Description of the small Quadrant is more expedite then this , which Mr Sutton had from Mr Dary long since , for whom , and by whose directions he made a Quadrant with the Line Sol , and two Parrallel Lines of Sines upon it , as is here added to the backside of this Quadrant . Of the Line of Hours , alias , the Diameter or Proportional Tangent . This Scale is no other then two Lines of natural Tangents to 45 d , each set together at the Center , and from thence beginning and continued to each end of the Diameter , and from one end thereof numbred with 90 d to the other end . This Line may fitly be called a Proportional Tangent , for whersoever any Ark is assumed in it to be a Tangent , the remaining part of the Diameter is the Radius to the said Tangent . So in the former Schem , if C L be the Tangent of any Ark , the Radius thereto shall be A L. In the Schem annexed , let A B be the Radius of a Line of Tangents equal to C D , and also parralel thereto , and from the Point B to C draw the Line B C , and let it be required to divide the same into a Line of Proportional Tangents : I say , Lines drawn from the Point D to every degree of the Tangent , A B shall divide one half of it as required from the similitude of two right angled equiangled plain Triangles , which will have their sides Proportional , it will therefore hold , As C F , To C D : So F B , To B E , and the Converse , As the second Tearm C D , To the fourth B E : So is the first C F , To the third F B , and therefore C F bears such Proportion to F B , as C D doth to B E , which is the same that the Radius bears to the Tangent of the Ark proposed . If it be doubted whether the Diameter wil be a double Tangent or the Line here described such a Line , a Proportion shall be given to find by Experience or Calculation , what Line it will be ; for there is given the Radius C D , and the Tangent B E , the two first Tearms of the Proportion , with the Line C B the sum of the third and fourth Tearms , to find out the said Tearms respectively ; and it will hold by compounding the Proportion , As the sum of the first and second Tearm , Is to the second Tearm : So is the sum of the third and fourth Tearm , To the fourth Tearm , that is , As C D + B E , Is to B E : So is C F + F B = C B , To F B , see 18 Prop. of 5 of Euclid , or page 18 of the English Clavis Mathematicae , of the famous and learned Mr Oughtred . After the same manner is the Line Sol , or Proportional Sines made , that being also such a Line , that any Ark being assumed in it to be a Sine , the distance from that Ark to the other end of the Diameter , shall be the Radius thereto . A Demonstration to prove that the Line of Hours and Latitudes will jointly prick off the hour Di●tances in the same angles as if they were Calculated and prickt off by Chords . Draw the two Lines A B and C B crossing one another at right angles at B , and prick off B C the quantity of any Ark out of the line of Latitudes , and then fit in the Scale of Hours ; so that one end of it meeting with the Point C , the other may meet with the other Leg of the right angle at A , from whence draw A E parralel to B C ; So A B being become Radius , B C is the Sine of the Arch first prickt down from the line of Latitudes ; from the Point B through any Point in the line of Proportional Tangents , at L draw the Line B L E , and upon B with the Radius B A draw the Arch A D , which measureth the Angle A B E to the same Radius : I say , there will then be a Proportion wrought , and the said Arch measureth the quantity of the fourth Proportional , the Proportion will be , As the Radius , To the Sine of the Ark prickt down from the Line of Latitudes : So is any Tangent accounted in the Scale , beginning at A , To the Tangent of the fourth Proportional ; in the Schem it lies evident in the two opposite Triangles L C B and L A E , by construction equiangled and consequently their sides Proportional . Assuming A L to be the Tangent of any Ark , L C becomes the Radius , according to the prescribed construction of that Line , it then lies evident , As L C the Radius , To C B the Sine of any Ark , So is L A , the Tangent of any Ark , To A E , the Tangent of the fourth Proportional . Namely , of the Angle A B E , and therefore it pricks down the Hour-lines of a Dyal most readily and accurately : the Proportion in pricking from the Substile being alwaies , As the Radius , To the Sine of the Stiles height , So the Tangent of the Angle at the Pole , To the Tangent of the Hour-line from the Substile . Uses of the Graduated Circle . To work Proportions in Tangents alone . In any Proportion wherein the Radius is not ingredient , it is supposed to be introduced by a double Operation , and the Poportion will be , As the first term , To the second , So the Radius to a fourth . Again , As the Radius is to that fourth , So is the third Term given , To the fourth Proportional sought . In illustrating the matter , I shall make use of that Theorem● for varying of Proportions , that the Tangents of Arches , and the Tangents of their Complements are in reciprocal Proportions . As Tangent 23d , to Tangent 35d , So Tangent 55d to the Tangent of 67d. In working of this Proportion , the last term may be found to the equal Semicircle , or on the Diameter . 1. In the Semicircle . Extend the thred through 23d on the Diameter , and through 3● in the Semicircle , and where it intersects the Circle on the opposite side , there hold one end of it , then extend the other part of it over 55 in the Diameter , and in the Semicircle , it will intersect 67d for the term sought . 2. On the Diameter . Extend the thred over 23d in the Semicircle , and 35d on the Diameter , and where it intersects the void circular line on the opposite side , there hold it , then laying the other end of it over 55 d in the Semicircle , and it will cut 67 d on the Diameter . If the Radius had been one of the terms in the Proportion , the operation would have been the same , if the Tangent of 45 d had been taken in stead of it . To work Proportions in Sines and Tangents joyntly . 1. If a Sine be sought , the middle terms being of a different species . Extend the thred through the first term on the Diameter , being a Tangent , and through the Sine , being one of the middle terms , counted in the unequal Quadrant , and where it intersects the Opposite side of the Circle hold it , then extend the thred over the Tangent , being the other middle term counted on the Diameter , and it will intersect the graduated Quadrant at the Sine sought . Example . If the Proportion were as the Tangent of 14d to the Sine of 29d So is the Tangent of 20d to a Sine , the fourth Proportional would be found to be the Sine of 45d. 2. If a Tangent be sought , the middle terms being of several kinds , Extend the thred through the Sine in the upper Quadrant , being the first term , and through the Tangent on the Diameter , being one of the other middle terms , holding it at the Intersection of the Circle on the opposite side , then lay the thred to the other middle term in the upper Quadrant , and on the Diameter , it shews the Tangent sought . Example . If the Suns Amplitude and Vertical Altitude were given , the Proportion from the Analemma to find the Latitude would be , As the Sine of the Amplitude to Radius , So is the Sine of the Vertical Altitude , To the Cotangent of the Latitude Let the Amplitude be — 39d 54′ And the Suns Altitude being East or West — 30 39′ Extend the thred through 39 54′ , the Amplitude counted in the upper Quadrant , and through 45d on the Diameter , holding it at the intersection with the Circle on the Opposite side , then lay the thred over 30d 39′ , the Vertical Altitude , and it will intersect the Diameter at 38d 28′ , the Complement of the Latitude sought . But Proportions derived from the 16 cases of right angled Spherical Triangles , having the Radius ingredient , will be wrought without any motion of the thred . An Example for finding the Suns Azimuth at the Hour of 6. As the Radius to the Cosine of the Latitude , So the Tangent of the Declination , To the Tangent of the Azimuth , from the Vertical towards Midnight Meridian . Extend the thred over the Complement of the Latitude in the upper Quadrant , and over the Declination in the Semicircle , and on the Diameter . it shews the Azimuth sought . So when the Sun hath 15d of Declination , his Azimuth shall be 9d 28′ from the Vertical at the hour of 6 in our Latitude of London . Another Example to find the time when the Sun will be due East or West . Extend the thred over the Latitude in the Semicircle , and over the Declination on the Diameter , and in the Quadrant of Latitudes it shews the Ark sought . The Proportion wrought , is , As the Radius to the Cotangent of the Latitude , So is the Tangent of the Declination , To the Sine of the Hour from 6. Example . So when the Sun hath 15● of North Declination , in our Latitude of London , the Hour will be found 12d 18′ from 6 in time 49⅕′ past 6 in the morning , or before it in the afternoon . Another Example to find the Time of Sun rising . As the Cotangent of the Latitude , to Radius , So is the Tangent of the Declination , To the Sine of the Hour from 6 before or after it . Lay the thred to the Complement of the Latitude in the Semicircle , and over the Declination on the Diameter , and in the Quadrant of Latitudes , it shews the time sought in degrees , to be converted into common time , by allowing 15● to an hour , and 4′ to a degree . So in the Latitude of London , 51d 32′ when the Sun hath 15● of Declination , the ascensional difference or time of rising from 6 , will be 19d 42′ , to be converted into common time , as before . By what hath been said , it appears , that the Hour and Azimuth may be found generally by help of this Circle and Diameter . For the performance whereof , we must have recourse to the Proportions delivered in page 123. whereby we may alwaies find the two Angle adjacent to the side on which the Perpendicular falleth , which may be any side at pleasure ; for after the first Proportion wholly in Tangents is wrought , to find either of those Angles , will be agreeable to the second case of right angled Spherical Triangles , wherein there will be given the Hypotenusal , and one of the Legs , to find the adjacent Angle , only it must be suggested , that when the two sides that subtend the Angle sought , are together greater then a Semicircle , recourse must be had to the Opposite Triangle , if both those Angles are required to be found by this Trigonometry , otherwise one of them , and the third Angle may be found by those directions , by letting fall the perpendicular on another side , provided the sum of the sides subtending those Angles be not also greater then a Semicircle ; or , having first found one Angle , the rest may be found by Proportions in Sines only . IN the Triangle ☉ Z P , if it were required to find the angles at Z and ☉ , because the sum of the sides ☉ P and Z P are less then a Semicircle they might be both found by making the half of the Base ☉ Z the first Tearm in the Proportion , and then because the angles ☉ Z are of a different affection , the Perpendicular would fal without on the side ☉ Z continued towards B , as would be evinced by the Proportiod , for the fourth Ark discovered , would be found greater then the half of ☉ Z ; hence we derived the Cannon in page 124 , for finding the Azimuth ; Whereby might also be found the angle of Position at ☉ ; so if it were required to find the angles at ☉ and P , the sides ☉ Z and Z P being less then a Semicircle the Perpendicular would fall within from Z on the side ☉ P , as would also be discovered by the Proportion , for the fourth Ark would be found less then the half of ☉ P. But if it were required to find both the angles at Z and P , in this Case we must resolve the Opposite Triangle Z B P , because the sum of the sides ☉ Z and ☉ P are together greater then a Semicircle , and this being the most difficult Case , we shall make our present Example . The Proportion will be , As the Tangent of half Z P , Is to the Tangent of the half sum of Z B and P B : So is the Tangent of half their difference , To a fourth Tangent . That is , As Tangent 19d 14′ , Is to the Tangent of 86d 30′ , : So is the Tangent of 9d 30′ , To a fourth . Operation . Extend the Thread through 19 d 14′ on the Semicircle , and 9 d 30′ on the Diameter , and hold it at the Intersection on the opposite side the Semicircle , then lay the Thread to 86 d 30′ in the Semicircle , and it shews 82 d 44′ on the Diameter for the fourth Ark sought . Because this Ark is greater then the half of Z P , we may conclude that the Perpendicular B A falls without on the side Z P continued to A. fourth Ark — 82d 44′ half of Z P is — 19 14 sum — 101 58 is Z A difference — 63 30 is P A Then in the right angled Triangle B P A , right angled at , A we have P A and B P the Hypotenusal , to , find the angle B P A , equal to the angle ☉ Z P. The Proportion is As the Radius , Is to the Tangent of 13d , the Complement of B P : So is the Tangent of P A 63d 30′ , To the Cosine of the angle at P. Extend the Thread through 13 d on the Diameter , and through 63 d 30′ in the Semicircle counted from the other end , and in the upper Quadrant , it shews 27 d 35′ for the Complement of the angle sought . And letting this Example be to find the Hour and Azimuth in our Latitude of London , so much is the hour from six in Winter when the Sun hath 13 d of South Declination , and 6 d of Altitude , in time 1 ho 50⅓ minutes past six in the morning , or as much before it in the afternoon . To find the Azimuth . Again , in the Triangle Z A B right angled at A , there is given the Leg or Side Z A 101 d 58′ , and the Hipotenusal Z B 96 d , to find the angle B Z P ; here noting that the Cosine or Cotangent of an Ark greater then a Quadrant is the Sine or Tangent of that Arks excess above 90 d , and the Sine or Tangent of an Ark greater then a Quadrant , the Sine or Tangent of that Arks Complement to 180 d , it will hold , As the Radius , To the Tangent of 6d : So is the Tangent 78d 2′ , To the Sine of 29 d 44′ , found by extending the Thread through 78 d 2′ on the Semicircle , counted from the other end , alias , in the small figures , and in the Quadaant it will intersect 29 d 44′ ; now by the second Case of right angled Sphoerical Triangles , the angle A Z B will be Acute , wherefore the angle ☉ Z B is 119 d 44′ the Suns Azimuth from the North , the Complement being 60 d 16′ is the angle A Z B , and so much is the Azimuth from the South . To work Proportions in Sines alone . THat this Circle might be capacitated to try any Case of Sphoerical Triangles , there are added Lines to it , namely , the Line Sol falling perpendicularly on the Diameter from the end of the Quadrant of Latitudes , whereto belongs the two Parrallel Lines of Sines in the opposite Quadrants , the upermost being extended cross the Quadrant of Latitudes . The Proportion not having the Radius ingredient , and being of the greater to the less . Account the first Tearm in the line Sol , and the second in the upper Sine extending the Thread through them , and where it intersects the opposite Parrallel hold it ; then lay the Thread to the third Tearm in the line Sol , and it will intersect the fourth Proportional on the upper Parrallel . As the Sine of 30d , To the sine of any Arch : So is the Cosine of that Arch , To the sine of the double Arch and the Converse . By trying this Canon , the use of these Lines will be suddenly attained . Example . As the sine of 30d , To the sine of 20d : So is the sine of 70d , To the sine of 40d. But if it be of the less to the greater , the answer must be found on the Line Sol. Account the first Tearm on the upper Sine , and the second in the Line Sol , and hold the Thread at the Intersection of the opposite Parrallel , then lay the Thread to the third Tearm on the uper Parrallel , and on the line Sol it will intersect the fourth Proportional if it be less then the Radius . But Proportions having the Radius ingredient , will be wrought without any Motion of the Thread . As the Cosine of the Latitude , To Radius : So is the sine of the Declination , To the sine of the Amplitude . So in our Latitude of London , when the Declination is 20d 12′ the Amplitude will be found to be 33d 42′ . Extend the Thread through 38 d 28′ on the line Sol. and through the Declination in the upper Sine , and it will intersect the opposite Parrallel Sine at 33 d 42′ , the Amplitude sought . The use of the Semi-Tangent and Chords are passed by at present . The line Sol is of use in Dyalling , as in Mr Fosters Posthuma , page 70 and 71 , where it is required to divide a Circle into 12 equal parts for the hours , and each part into 4 subdivisions for the quarters , and into such parts may the equal Semicircle be divided ; that if it were required to divide a Circle of like Radius into such parts , it might be readily done by this . Of the Line of Hours on the right edge of the foreside of the Quadrant . This is the very same Scale that is in the Diameter on the Backside , only there it was divided into degrees , and here into time , and placed on the outermost edge ; there needs no line of Latitudes be fitted thereto , for those Extents may be taken off as Chords from the Quadrant of Latitudes , by help of these Scales thus placed on the outward edges of the Quadrants may the hour-lines of Dyals be prickt down without Compasses . To Draw a Horizontal Dyal . FIrst draw the line C E , for the Hour-line of 12 , and cross it with the Perpendicular A B , then out of a Scale or Quadrant of Latitudes set of C B and C A , each equal to the Stiles height , or Latitude of the place , then place the Scale of 6 hours on the edge of the Quadrant , whereto the Line of Latitudes was fitted , one extremity of it at A , and move the Quadrant about , till the other end or extremity of it will meet with the Meridian line C E ; then in regard the said Scale of Hours stands on the very brink or outward most edge of the Quadrant , with a Pin , Pen , or the end of a black-lead pen , make marks or points upon the Paper or Dyal against each hour ( and the like for the quarters , and other lesser parts ) of the graduated Scale , and from those marks draw lines into the Center , and they shall be the hour-lines required , without drawing any other lines on the Plain , the Scale of Hours on the Quadrant is here represented by the lines A E , and E B , the hour lines above the Center , are drawn by continuing them out through the Center . And those that have Paper prints of this line , may make them serve for this purpose , without pricking down the hour points by Compasses , by doubling the paper at the very edge or extremity of the Scale of Hours . Otherwise to prick down the said Dial without the Line of Latitudes and Scale of hours in a right angled Parallellogram . Having drawn C E the Meridian line , and crossed it with the perpendicular C A B , and determining C E to be the Radius of any length , take out the Sine of the Latitude to the same Radius , and prick it from C to A and B , and setting one foot at E , with the said Extent sweep the touch of an Arch at D and F , then take the length of the Radius C E , and setting down one foot at B , sweep the touch of an Ark at D , intersecting the former , also setting down the Compasses at A , make the like Arch at F , and through the points of Intersection , draw the streight lines A F , B D , and F E D , and they will make a right angled Parallellogram , the sides whereof will be Tangent lines . To draw the Hour-lines : Make E F , or E D Radius , and proportion out the Tangents of 15d and prick them down from E to 1 and 11 and draw lines 30 and prick them down from E to 2 and 10 and draw lines through the points thus found , and through the points F and D , and there will be 3 hours drawn on each side the Meridian line . Again , make A F or B D Radius , and proportion out the Tangent of 15d , and prick it down from A to 5 , and from B to 7. Also proportion out the Tangent of 30d , and prick it down from A to 4 , and from B to 8 , and draw lines into the Center , and so the Hour-lines are finished , and for those that fall above the 6 of clock line , they are only the opposite hours continued , after the like manner are the halfs and quarters to be prickt down . Lastly , By chords prick off the Stiles height equal to the Latitude of the place , and let it be placed to its due elevation over the Meridian line . Of Vpright Decliners . DIvers Arks for such plains are to be calculated , and may be found on the Circle before described . 1. The Substiles distance from the Meridian . By the Substilar line is meant , a line over which the Stile or cock of the Dyal directly hangeth in its nearest distance from the Plain , by some termed the line of deflexion , and is the Ark of the plain between the Meridian of the Plain , and the Meridian of the place . The distance thereof from the Hour-line of 12 , is to be found by this Proportion . As the Radius , To the Sine of the Plains Declination , So the Cotangent of the Latitude , To the Tangent of the Substile from the Meridian . 2. For the Angle of 12 and 6. An Ark used when the Hour-lines are pricked down from the Meridian line in a Triangle or Parallellogram , ( and not from the Substile , ) without collecting Angles at the Pole. As the Radius , Is to the Sine of the Plains Declination , So is the Tangent of the Latitude , To the Tangent of an Ark , the Complement whereof is the Angle of 12 and 6. 3. Inclination of Meridians . Is an Ark of the Equinoctial , between the Meridian of the plain , and the Meridian of the place , or it is an Angle or space of time elapsed between the passage of the shaddow of the Stile from the Substilar line into the Meridian line , by some termed the Plains difference of Longitude ; and not improperly , for it shews in what Longitude from the Meridian where the Plain is ; the said Plain would become a Horizontal Dyal , and the Stiles height shews the Latitude , this Ark is used in calculating hour distances by the Tables and in pricking down Dyals by the Line of Latitudes , and hours from the Substile . As the Radius , Is to the Sine of the Latitude , So the Cotangent of the Plains Declination , To the Cotangent of the Inclination of Meridians . Or , As the Sine of the Latitude to Radius , So is the Tangent of the Plains Declination , To the Tangent of Inclination of Meridians . 4. The Stiles height above the Substile . As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Plains Declination , To the Sine of the Stiles height . Or the Substiles distance being known , As the Radius , To the Sine of the Substiles distance from the Meridian , So is the Cotangent of the Declination , To the Tangent of the Stiles height . Or , The Inclination of Meridians being known . As the Radius , To the Cosine of the Inclination of Meridians , So is the Cotangent of the Latitude , To the Tangent of the Stiles height . 5. Lastly , For the distances of the Hour-lines from the Substilar Line . As the Radius , Is to the Sine of the Stiles height above the Plain , So is the Tangent of the Angle at the Pole , To the Tangent of the Hours distance from the Substilar Line . By the Angle at the Pole , is meant the Ark of difference between the Ark called the Inclination of Meridians , and the distance of any hour from the Meridian , for all hours on the same side the Substile falls , and the sum of these two Arks for all hours on the other side the Substile . These Proportions are sufficient for all Plains to find the like Arks , without having any more , if the manner of referring Declining Reclining Inclining Plains to a new Latitude , and a new Declination in which they shall stand as upright Plains , be but well explained , for East or West Reclining Inclining Plains , their new Latitude is the Complement of their old Latitude , and their new Declination , is the Complement of their Reclination Inclination , which I count always from the Zenith , and upon such a supposition , taking their new Latitude and Declination , those that will try , shall find that these Proportions will calculate all the Arks necessary to such Dials . So if an Upright Plain decline 25d in our Latitude of London from the Meridian . The Substiles distance from the Meridian is — 18d 34′ The Angle of 12 and 6 is — 62 : 00 The Inclination of Meridians is — 30 : 47 The Stiles height is — 34 : 19 To Delineate the same Dial from the Substile by the Line of Latitudes , and Scale of hours in an Equicrutal Triangle . To Draw an Vpright Decliner . An Vpright South Plain for the Latitude of London , Declining 25d Eastwards . TO prick down this Dial by the line of Latitudes , and Scale of Hours in an Isoceles Triangle . Draw C 12 the Meridian Line perpendicular to the Horizontal line of the Plain , and with a line of Chords , make the Angle F C 12 , equal to the Substiles distance from the Meridian , and draw the line F C for the Substile ; Draw the line B A perpendicular thereto , and passing through the Center at C , and out of the line of Latitudes on the other Quadrants , or out of the Quadrant of Latitudes on this Quadrant , set off B C and C A each equal to the Stiles height , then fit in the Scale of 6 hours , proper to those Latitudes , so that one Extremity meeting at A , the other may meet with the Substilar line at F. Then get the difference between 30d 47′ , the inclination of Meridians , and 30d the next hours distance lesser then the said Ark , the difference is 47′ in time , 3′ nearest then fitting in the Scale of hours as was prescribed . Count upon the said Scale , Hour . Min.   0 3 from F to 10 1 3 11 2 3 12 3 3 1 4 3 2 5 3 3 And make points at the terminations with a pin or pen , & draw lines from those points into the Center at C , & they shall be the true hour-lines required on this side the Substile . Again , Fitting in the Scale of Hours from B to F , count from that end at B the former Arks of time . Ho Min   00 , 03 from B to 4 1 , 3 5 2 , 3 6 3 , 3 7 4 , 3 8 5 , 3 9 And make Points at the Terminations , through which draw Lines into the Center , and they shall be the hour Lines required on the other side the Substile . The like must be done for the halfs and quarters , getting the difference between the half hour next lesser ( in this Example 22d 30′ ) under the Ark called the inclination of Meridians , the difference is 1d 17′ in time 33′ nearest to be continually augmented an hour at a time , and so prickt off as before was done for the whole hours . By three facil Proportions , may be found the Stiles height , the Inclination of Meridians , and the Substiles distance from the Plains perpendicular , for all Plains Declining , Reclining , or Inclining , which are sufficient to prick off the Dyal after the manner here described , which must be referred to another place . If the Scale of hours reach above the Plain , as at B , so that B C cannot be pricked down , then may an Angle be prickt off with Chords on the upper side the Substile , equal to the Angle F C A , on the under side , and thereby the Scale of hours laid in its true situation , having first found the point F on the under side . To prick down the former Dyal in a Rectangular ☉ blong , or long square Figure from the Substile . Having set off the Substilar F C , assume any distance in it , as at F to be the Radius , and through the fame at right Angles , draw the line E F D , then having made F C any distance Radius , take out the Sine of the Stiles height to the same Radius , and entring it at the end of the Scale of three hours , make it the Radius of a Tangent , and proportion out Tangents to 3′ and set them off from F to 10 1 hour 3 and set them off from F to G 2 3 and set them off from F to H Again , Take out the Tangents of the Complement of the first Ark , increasing it each time by the augmentation of an hour , namely 57′ and prick them from F to I and from the points 1 ho. 57 and prick them from F to K and from the points 2 57 and prick them from F to E and from the points thus found , draw lines into the Center . Then for the other sides of the Square , make C F the Radius of the Dyalling Tangent of 3 hours , and proportion out Tangents to the former Arks , namely , 3′ and prick them from B to P Also to the latter Arks , 57′ and prick them from A to — N 1 ho. 3 and prick them from B to O Also to the latter Arks. 1 h. 57 and prick them from A to — M 2 3 and prick them from B to L Also to the latter Arks. 2 57 and prick them from A to — D and draw lines from these terminations into the Center , and the Hour-lines are finished ; after the same manner must the halfs and quarters be finished . And how this trouble in Proportioning out the Tangents may be shunned without drawing any lines on the Plain , but the hour-lines , may be spoke to hereafter , whereby this way of Dyalling , and those that follow , will be rendred more commodious . Lastly , the Stile may be prickt off with Chords , or take B C , and setting one foot in F , with that Extent sweep the touch of an occult Arch , and from C , draw a line just touching the outward extremity of the said Arch , and it shall prick off the Angle of the Stiles height above the Substile . To prick off the former Dyal in an Oblique Parallellogram , or Scalenon alias unequal sided Triangle from the Meridian . First , In an Oblique Parallellogram . DRaw CE the Meridian line and with 60d of a line of Chords , draw the prickt Arch , and therein from K , contrary to the Coast of Declination , prick off 62d , the angle of 12 and 6 , and draw the line C D for the said hour line continued on the other side the Center , and out of a line of Sines , make C E equall to 65d the Complement of the Declination ; then take out the sine of 38d 28′ the Complement of the Latitude , and enter it in the line D C , so that one foot resting at D , the other turned about , may but just touch the Meridian line , the point D being thus found , make C F equall to C D , and with the sides C F and C E make the Parallellogram D G F H , namely , F H and G D equal to C E : and E G and E H equal to D C. And where these distances ( sweeping occult arches therewith ) intersect will find the points H and G limiting the Angles of the Parallellogram . Then making E H or C D Radius , proportion out the Tangents of 15d and prick them down from E to 1 and 11 and 30 and prick them down from E to 2 and 10 and draw lines into the Center through those points , and the angular points of the Parallellogram at H and G , and there will be 6 hours drawn , besides the Meridian line or hour line of 12. Then making D G Radius , proportion out the Tangent of 15d , and prick it down from D upwards to 5 , and downward to 7 , also proportion out the tangent of 30d and prick it from D to 8 , and from F to 4 , and draw lines into the Center , and so the hour lines are finished ; after the same manner are the halfs and quarters to be proportioned out and pricked down : and if this Work is to be done upon the Plain it selfe , the Parallel F H will excur above the plain , in that case , because the Parallel distance of F H from the Meridian , is equal to the parallel distance of D G the space G. 8. may be set from H to 4 , and so all the hour lines prickt down . To prick down this Dyal in a Scalenon , or unequal sided triangle from the Meridian , from E to D draw the streight line D E , and from the same point draw another to F , and each of them ( the former hour lines being first drawn ) shall thereby be divided into a line of double tangents , or scale of 6 hours , such a one as is in the Diameter of the Circle on this quadrant , or on the right edge of the foreside ; and therefore by helpe of either of them lines , if it were required to prick down the Dyal , it might be done by Proportioning them out , take the extent D E , and prick it from one extremity of the Diameter in the Semicircle on the quadrant , and from the point of Termination draw a line with black Lead to the other extremity , ( which will easily rub out again either with bread or leather parings ) and take the nearest distance from 15 of the Diameter to the said line , and the said extents 30 of the Diameter to the said line , and the said extents 45 of the Diameter to the said line , and the said extents shall reach from , E to 11 and from D to 7 shall reach from , E to 10 and from D to 8 shall reach from , E to 9 and from D to 9 and the like must be done for the line E F , entring that in the Semicirle as before ; or without drawing lines on the quadrant , if a hole be drilled at one end of the Diameter , and a thred fitted into it , lay the thred over the point in the Diameter , and take the nearest distances thereto . Lastly , from a line of Chords , prick off the substilar line , and the stiles height as we before found it . This way of Dyalling in a Parallellogram , was first invented by John Ferrereus a Spaniard , long since , and afterwards largely handled by Clavius , who demonstrates it , and shews how to fit it into all plains whatsoever , albeit they decline , recline , or incline , without referring them to a new Latitude ; the Triangular way is also built upon the same Demonstration , and is already published by Mr Foster in his Posthuma , for it is no other then Dyalling in a Parallellogram , if the Meridian line C E be continued upwards , and C E set off upwards , and lines drawn from the point , so found to D and E , shall constitute a Parallellogram . An Advertisement about observing of Altitudes . IMagine a line drawn from the beginning of the line Sol , to the end of the Diameter , and therein suppose a pair of sights placed with a thred and bullet hanging from the begining of the said line , as from a Center ; I say the line wherein the sights are placed , makes a right angle with the line of sines on the other side of Sol , and so may represent a quadrant , the equal Limbe whereof is either represented by the 90d of the equal Semicirle , or by the 90d of the Diameter and thereby an Altitude may be taken . Now to make an Isoceles equicrural , or equal legged triangle made of three streight Rulers , the longest whereof will be the Base or Hipotenusal line ; thus to serve for a quadrant to take Altitudes withal , will be much cheaper , and more certain in Wood , then the great Arched wooden framed quadrants . Moreover , the said Diameter line supplies all the uses of the Limbe , from it may be taken off Sines , Tangents , or Secants , as was done from the Limbe ; and therein the Hour and Azimuth , found generally by helpe of the line of Sines on the left edge , as is largely shewed in the uses of this quadrant , besides its uses in Dyalling , onely when such an Instrument is made apart , it will be more convenient to have the line of Sines to be set on the right edge , and the Diameter numbred also by its Complements ; this Diameter or double Tangent , or Hipotenusal line being first divided on , all the other lines may from it , by the same Tables that serve to graduate them from the equal Limbe be likewise inscribed : and here let me put a period to the uses of this quadrant . Gloria Deo. The Description of an Universal small Pocket Quadrant . THis quadrant hath only one face . On the right edge from the Center is placed a line of sines divided into degrees and half degrees up to 60 d. afterwards into whole degrees to 80 d. On the left edge issueth a line of 10 equal parts , from the Center being precise 4 inches long , each part being divided into 10 subdivisions and each subdivision into halfs . These two lines make a right angle at the Center , and between them include a Projection of the Sphere for the Latitude of London . Above the Projection are put on in quadrants of Circles a line of Declinations 4 quadrants for the dayes of the moneth , above them the names of 5 Stars with their right Ascensions graved against them , and a general Almanack . Beneath the Projection are put on in quadrants of Circles a particular sine and secant , so called , because it is particular to the Latitude of London . Below that the quadrat , and shadowes . Below that a line of Tangents to 45 d. Last of all the equal Limbe . On the left edge is placed the Dialling scale of hours 4 Inches long , outwardmost on the right edge a line of Latitudes fitted thereto . Within the line of sines close abutting thereto is placed a small scale called the scale of entrance beginning against 52 d. 35 min. of the sines numbred to 60 d. The line of sines that issueth from the Center should for a particular use have been continued longer to wit to a secant of 28 d. because this could not be admitted , the said secant is placed outward at the end of the scale of entrance towards the Limbe , and as much of the sine as was needful placed at its due distance , at the other end of the scale of entrance . Of the uses of the said quadrant . THe Almanack hath been largely spoke to in pag. 12 , and 13 , also again in the uses of the Horizontal quadrant pag. 11 , 12. The quadrat and shadowes from pag. 35 to 44. The line of Latitudes and scale of hours pag. 250. Again , from page to 262 to 274 , also the line of sines , equal parts , and Tangents , in other parts of the Booke . The use of the Projection . THis projection is only fitted for finding the hour in the limb , and not the Azimuth , all the Circular lines on it are parallels of Altitude or Depression except the Ecliptick and Horizon , the Ecliptick , is known by the Characters of the signes , and the Horizon lyeth beneath it , being numbred with 10 , 20 , 30 , 40. The parallels of Altitude are the Winter parallels of Stofler's Astrolable , and are numbred from the Horizon upwards towards the Center , the parallels of Depression which supply the use of Stoflers Summer Altitudes are numbred downwards from the top of the Projection towards the limb . To find the time of Sun rising , and his Amplitude . LAy the thread over the day of the moneth , and set the Bead to the Ecliptick , then carry the thread and Bead to the Horizon , and the thread in the limb , shewes the time of rising , and the Bead on the Horizon the quantity of the Amplitude . Example . So on the second of August the Suns Declination being 15 d ▪ his Amplitude will be 24 d. 35 min. and the time of rising 41′ past 4 in the morning . To find the Hour of the Day . HAving taken the Suns Altitude and rectified the Bead as before shewed , if the Sun have South Declination bring the Bead to that parallel of Altitude on which the Suns height was observed , amongst those parallels that are numbred upward towards the Center , and the thread in the limb sheweth the time of the day . Eample . So when the Sun hath 15 d. of South Declination , as about 28th of January , if his Altitude be 15 deg . the time of the day will be 39 m. past 2 in the afternoon , or 21 m. past 9 in the morning . But in the Summer half year bring the Bead to lye on those parallels that are numbred downwards to the limb , and the thread sheweth therein the time of the day sought . Example . If on the second of August his Declination being 15 d. his Altitude were 40 d. the true time of the day would be 8 m. past 9 in the morning , or 52 m. past 2 in the afternoon . If the Bead will not meet with the Altitude given amongst those parallels that run donwnwards towards the right edge , then it must be brought to those parallels that lye below the Horizon downward towards the left edge , and the thread in the Limb shewes the time of the day before six in the morning or after it in the evening in Summer . Example . When the Sun hath 15 d. of North Declination as on the second of August if his Altitude be 5 d. the time of the Day will be 44 m. before 6 in the morning , or after it in the evening . Of the general lines on this quadrant . THe line of sines on the right edge is general for finding either the hour or the Azimuth in the equal limb , or in the said line of sines , as I have largely shewed in page 231 for finding the hour , also for finding the Suns Altitudes on all hours , as in page 234 , for finding the Azimuth from page 237 to pag. 2.9 . Though this quadrant hath neither secants nor versed sines as the rest have , yet both may be easily supplyed , let it be required to work this Proportion . As the Co-sine of the declination , Is to the secant of the Latitude , So is the difference of the sines of the Suns Meridian and given Altitude , To the versed sine of the hour from noon , before or after six the hour may be found from midnight by the proportions in page 230. Let the Radius of the sines be assumed to represent the secant of the Latitude , the Radius to that secant will be the cosine of the Latitude , then lay the thread to the complement of the Declination in the limb counted from right edge , and take the nearest distance to it . I say that extent shall be the cosine of the Declination to the Radius of the Secant enter this at 90 d. of the line of sines laying the thread to the other foot according to nearest distance , then in the sines take the distance between the Meridian Altitude and the given Altitude , and enter that extent so upon the sines that one foot resting thereon , the other turned about may just touch the thread the distance between the resting foot and the Center is equal to the versed sine of the ark sought and being measured from the end of the line of sines towards the Center shewes the ark sought . Example . When the Suns Declination is 15 d. North if his Altitude were 35 d. 21 m. the time of the day would be found 45 d. from noon , that is 9 in the morning or 3 in the afternoon . Of finding the Azimuth generally . THough this may be found either by the sines alone in the equal limbe as before mentioned , or by versed sines as was instanced for the hour , see also page 239 , 240 , 241 , yet where the Sun hath vertical Altitude or Depression , as in places without the Tropicks towards either of the Poles , it may be found most easily in the equal limb by the joynt help of sines and tangents by the proportions in page 175. First , find the vertical Altitude as is shewed in page 174. Then for Latitudes under 45 d. Enter in Summer Declinations the difference , but in Winter Declinations the sum , of the sines of the vertical Altitude , and of the proposed Altitude once done the line of sines from the Center , and laying the thread over the Tangent of the Latitude take the nearest distance to it , then enter that Extent at the complement of the Altitude in the Sines , and lay the thread to the other foot , and in the limb it shewes the Azimuth from the East or West . Example . For the Latitude of Rome to witt 42 d. If the Sun have 15 d. of North Declination his vertical Altitude is 22 d. 45 m. If his given Altitude ●e 40 d. the Azimuth of the Sun will be 17 d. 33 m. to the Southward of the West . If his declination were as much South and his proposed Altitude 18 d. his Azimuth would be 41 d. 10 m. to the Southwards of the East or West . For Latitudes above 45. If we assume the Rad. of the quadrant to be the tangent of the Latit . the Rad. to that Tang. shall be the co-tangent of the Latit . wherefore lay the thread to the complement of the Latitude in the line of Tangents in the limb , and from the complement of the Altitude in the Sines take the nearest distance to it , I say that extent shall be cosine of the Altitude to the lesser Radius which measure from the Center , and it finds the Point of entrance whereon enter the former Sum or difference of sines as before directed , and you will find the Azimuth in the equal limb . Or if you would find the answer in the sines , enter the first extent at 90 d. laying the thread to the other foot , then enter the Sum or difference of the Sines of the vertical and given Altitude , so between the scale and the thread , that one foot turned about may but just touch the thread , the other resting on the sines , and you will find the sine of the Azimuth sought . Example . For the Latitude of Edinburg 55 d. 56 m. If the Sun have 15 d. of Declination , his vertical Altitude or depression is 18 d. 14 m. the Declination being North , if his proposed Altitude were 35 d. the Azimuth of the Sun would be 28 d. to the South-wards of the East or West . But if the Declination were as much South , and the Altitude 10 d. the Azimuth thereto would be 46 d. 58 m. to the South-wards of the East or West . The first Operation also works a Proportion to witt . As the Radius Is to the cotangent of the Latitude . So is the cosine of the Altitude , To a fourth sine . I say this 4th sine beares such Proportion to the Radius as the cosine of the Altitude doth to the tangent of the Latitude , for the 4th term of every direct Proportion beares such Proportion to the first terms thereof , as the Rectangle of the two middle terms doth to the square of the first term . But as the rectangle of the co-tangent of the Latitude and of the cosine of the Altitude is to the square of the Radius , So is the cosine of the Altitude is to the tangent of the Latitude , or which is all one , So is the co-tangent of the Latitude , To the secant of the Altitude , as may be found by a common division of the rectangle , and square of the Radius by either of the terms of the said rectangle , by help of which notion I first found out the particular scales upon this quadrant . All Proportions in sines and Tangents may be resolved by the sine of 90 d. and the Tangent of 45 d. on this quadrant if what hath been now wrote , and the varying of Proportions be understood , as in page 72 to 74 it is delivered . Because the Projection is not fitted for finding the Azimuth there are added two particular scales to this quadrant , namely , the particular sine in the limb , and the scale of entrance abutting on the sines fitted for the Latitude of London . Lay the thread to the day of the moneth , and it shewes the Suns Declination in the scales proper thereto . Then count the Declination in the Limbe laying the thread thereto , and in the particular sine , it shewes the Suns Altitude or Depression being East or West . To find the Suns Azimuth . FOr North Declinations take the distance between the sines of the vertical Altitude and given Altitude , but for South Declinations adde with your compass the sine of the given Altitude to the sine of the vertical Altitude , enter the extent thus found , at the Altitude in the scale of entrance laying the thread to the other foot according to nearest distance , and in the equal limb it shewes the Azimuth sought from the East or West , or it may be found in the sines by laying the thread to that arch in the limb that the Altitude in the scale of entrance stands against in the sines , and entring the former extent paralelly between the thread and the sines . Example . So when the Sun hath 13 d. of Declination his vertical Altitude or Depression is 16 d. 42 m. If the Declination were North and his Altitude 8 d. 41 m. his Azimuth would be 10 d. to the North-wards of the East or West . But if it were South and his Altitude 12 d. 13 m. the Azimuth would be 40d to the South-wards of the East or West . By the same particular scales the hour may be also found . To find the time of Sun rising or setting . TAke the sine of the Declination , and enter it at the Declination in the Scale of entrance and it shewes the time sought in the equal lim●e from six . Example . When the hath 10 d. of Declination the Ascensional difference is 49 m. which added to , or substracted from six shewes the time of rising and setting . To find the hour of the day for South Declination . IN taking the Altitude , mind what Ark in the particular Sine the thread cut , adde the Sine of that Ark to the Sine of the Declination , and enter that extent at the Declination in the Scale of entrance laying the thread to the other foot according to nearest distance , and in the equal limb it shewes the hour from six . So if the Declination were 13 d. South and the Suns Altitude 14 d. 38 m. the thread in the particular Sine would cut 18 d. 49 m. and true time of the day would be 9 in the morning or 3 in the morning . To find the time of the day for North Declination . HAving observed what Ark the thread in taking the Altitude hung over in the Particular Sine take the distance between the Sine of the said Arke , and the Sine of the Declination and entring that extent at the Declination in the Scale of entrance the thread in the limbe shewes the hour from six . Example . If the Suns Declination were 23 d. 31 m. North , and his Altitude 39 d. the Arch in the particular Sine would be 53 d. 32 m. and the time of the day would be about 3 quarters past 3 in the afternoon , or a quarter past 8 in the morning . When the Altitude is more then the Latitude the thread will hang over a Secant in the particular Scale , this happens not till the Sun have more then 13 d. of North Declination , in this case take the distance between the Secant before the beginning of the Scale of entrance , and the Sine of the Declination at the end of the same and enter it as before . Example . The Suns declination being 23 d. 31 m. North if his Altitude were 55 d. 29 m. the thread in the particular Scale would hang over the Secant of 18 d. 11 m. and the true time of the day would be a quarter past 10 in the morning , or 3 quarters past 1 in the afternoon . The Proportions here used are expressed in Page 193. The Stars hour is to be found by the projection by rectifying the Bead to the Sar and then proceed as in finding the Suns hour , afterwards the ●ue time of the night is to be found as in page 32. ERRATA . In the Treatise of the Horizontal quadrant , pag. 43 line 6 for the 3 January read the 30th . In the Reflex Dialling pag. 5 , adde to the last line these words , As Kircher sheweth in his Ars Anaclastica . FINIS . THE DESCRIPTION AND USES OF A GENERAL QUADRANT , WITH THE HORIZONTAL PROJECTION , UPON IT INVERTED . Written and Published By JOHN COLLINS Accountant , and Student in the Mathematicks . LONDON , Printed Anno M. DC . LVIII . The Description OF THE HORIZONTAL QUADRANT . THis Denomination is attributed to it because it is derived from the Horizontal projection inverted . Of the Fore-side . On the right edge is a Line of natural Sines . On the left edge a Line of Versed-Sines . Both these Lines issue from the Center where they concurre and make a right Angle , and between them and the Circular Lines in the Limb is the Projection included , which consists of divers portions and Arkes of Circles . Of the Parallels of Declination . THese are portions of Circles that crosse the quadrant obliquely from the left edge , towards the right . To describe them . OBserve that the left edge of the quadrant is called the Meriridian Line , and that every Degree or Parallel of the Suns Declination if continued about would crosse the Meridian in two opposite points , the one below the Center towards the Limbe , and the other above , and beyond the Center of the quadrant , the distance between these two points is the Diameter of the said Parallel , and the Semidiameters would be the Center points . It will be necessary in the first place , to limit the outwardmost Parallel of Declination , which may be done in the Meridian Line at any point assumed . The distance of this assumed point from the Center in any Latitude , must represent the Tangent of a compound Arke , made by adding halfe the greatest Meridian Altitude to 45 Deg. which for London must be the Tangent of 76 Degr. And to the Radius of this Tangent must the following work be fitted . In like manner , the Semidiameters of all other Parallels that fall below the Center , are limited by pricking downe the Tangents of Arkes , framed by adding halfe the Meridian Altitude suitable to each Declination continually to 45 Degr. Now to limit the Semidiameters above or beyond the Center onely prick off the respective Tangents of half the Suns mid-night Depression from the Center the other way , retaining the former Radius , by this meanes there will be found two respective points limiting the Diameters of each Parallel , which had , the Centers will be easily found falling in the middle of each Diameter . But to doe this Arithmetically , first , find the Arke compounded of halfe the Suns meridian Altitude , and 45 Degr. as before , and to the Tangent thereof , adde the Tangent of halfe the Suns mid-night depression , observing that the Suns mid-night depression in Winter-Summer , is equal to his Meridian Altitude in Summer-Winter , his declination being alike in quantity , though in different Hemispheres , the halfe summe of these two Tangents are the respective Semidiameters sought , and being prickt in the meridian line either way from the former points limiting the Diameters , will find the Centers . Or without limiting those Points for the Diameters : first , get the Difference between the Tangents of those Arkes that limit them on either side , and the halfe summe above-said , the said difference prickt from the Center of the quadrant in the meridian line finds the respective Centers of those Parallels , the said halfe summes being the respective Semidiameters wherewith they are to be described . Of the Line or Index of Altitudes . THis is no other then a single prickt line standing next the Meridian line , or left edge of the quadrant , to which the Bead must be continually rectified , when either the houre or Azimuth is found by help of the projection . To graduate it . ADde halfe the Altitudes respectively whereto the Index is to be fitted to 45 Degr. and prick downe the Tangents of these compound arkes from the Center . Example . To graduate the Index for 40 Degr. of Altitude , the halfe thereof is 20 , which added to 45 Degr. makes 65 Degr. which taken from a Tangent to the former Radius , and prickt from the Center , gives the point where the Index is to be graduated with 40 Degrees . Hence it is evident that where the divisions of the Index begin marked ( 0 ) the distance of that point from the Center is equal to the common Radius of the Tangents . Because this quadrant ( as all natural projections ) hath a reverted taile , the graduations of the Index are continued above the Hozontal point ( 0 ) towards the Center to 30 Degr. 40′ as much as is the Sunnes greatest Vertical Altitude in this Latitude , and the graduations of the Index are set off from the Center by pricking downe the Tangents of the arkes of difference between half the proposed Altitude , and 45 Deg. thus to graduate 20 deg . of the Index the halfe thereof is 10 Degrees , which taken from 45 Degrees , the residue is 35 Degrees , the Tangent thereof prickt from the Center gives the point where the Index is to be graduated with 20 Degrees . Of the houre Circles . THese are knowne by the numbers set to them by crossing the Parallels of Declination , and by issuing from the upper part of the quadrant towards the Limbe . To describe them . LEt it be noted that they all meet in a point in the Meridian Line below the Center of the quadrant : the distance whereof from the Center is equal to the Tangent of halfe the Complement of the Latitude taken out of the common Radius , which at London will be the Tangent of 19 Deg. 14′ . The former point which may be called the Pole-point , limits their Semidiameters , to find the Centers prick off the Tangent of the Latitude and through the termination raise a line Perpendicular to the Meridian line , the distance from the Pole-point being equal to the Secant of the Latitude , must be made Radius . And the Tangents of 15 Degrees , 30 Degrees &c. prickt off on the former raised line , gives the respective Centers of the houre Circles , the distances whereof from the Pole point are the Semidiameters wherewith those houre Circles are to be drawne . Of the reverted Tail. THis needs no Rule to describe it , being made by the continuing of the parallels of Declination to the right edge of the quadrant and the houre Circles up to the Winter Tropick or parallel of Declination neerest the Center , however the quantity of it may be knowne by setting one foot of a paire of Compasses in the Center of the quadrant , and the other extend to 00 Degrees of Altitude in the Index ; an Arch with that extent swept over the quadrant as much as it cuts off will be the Reverted Taile , and so much would be the Radius . Of a Quadrant made , of this Projection not inverted . BY what hath been said it will be evident to the judicious that this inversion is no other then the continance of the extents of one quarter of the Horizontal projection . Which otherwise could not with convenience be brought upon a quadrant . Hence it may be observed that . Having assigned the Radius , a quadrant made of the Horizontal Projection without inversion , to know how big a Radius it will require when inverted the proportion will hold . AS the Radius , is to the distance of the intersection of the Aequinoctial point with the Horizon from the Center equall to the Radius of the said Projection when not inverted , in any common measure . So is the Tangent of an Arke compounded of 45 Degrees , and of half the Suns greatest Meridian Altitude . To the distance between the Center and the out-ward Tropick next the Limbe in the said known measure when inverted , whence it followes that between the Tropicks this projection cannot be inverted , but the reverted taile will be but small , and may be drawne with convenience without inversion . Of the Curved Line and Scales belonging to it . BEyond the middle of the Projection stands a Curved or bending Line , numbred from the O or cypher both wayes , one way to 60 Degrees , but divided to 62 Degrees , the other way to 20 Degr. but divided to 23 Deg. 30′ . The Invention of this Line ownes Mr. Dary for the Author thereof , the Use of it being to find the houre or Azimuth in that particular latitude whereto it is fitted by the extension of a threed over it , and the lines belonging to it . The lines belonging to it are two , the one a Line of Altitudes , and Declinations standing on the left edge of the quadrant , being no other but a line of Sines continued both wayes , from the beginning one way to 62 Degrees , the other way to 23. Degrees 30′ . The other line thereto belonging is 130 Deg. of a line of Versed Sines , which stands next without the Projection being parallel to the left edge of the quadrant . To dravv the Curve . DRaw two lines of Versed Sines , it matters not whether of the same Radius or no , nor how posited ; provided they be parallel , let each of them be numbred as a Sine both ways , from the middle at ( 0 ) and so each of them will containe two lines of Sines , to the right end of the uppermost set C , to the left end D , and to the right end of the undermost set A , and to the left end B. First , Note that there is a certaine point in the Curve where the Graduations will begin both upwards and downwards , this is called the Aequinoctial point ; to find it , lay a ruler from A to the Complement of the Latitude counted from ( 0 ) in the upper Scale towards D , and draw a line from A to it , then count it the other way towards C , viz. 38 Degrees 28′ . for the Co-latitude of London , and lay a ruler over it , and the point B , and where it intersects the line before drawn , is the Aequinoctial point to be graduated . Then to graduate the Division on each side of it , requires onely the making in effect of a Table of Meridian altitudes to every degree of Declination ( which because the Curve will also serve for the Azimuth in which case the graduations of the Curve , which in finding the houre were accounted Declinations must be accounted Altitudes ) must be continued to 62 Degrees for this Latitude , and further also if it be intended that the Curve shall find a Stars houre that hath more declination . To make this Table . GEt the Summe and difference of the Complement of the latitude and of the Degrees intended to be graduated , and if the summe exceed 90 Degrees , take its complement to 180 degrees instead of it : being thus prepared the Curve will be readily made . To graduate the under part of the Curve . Account the summe in the upper line from O towards D , and from the point A in the under line draw a line to it . Account the difference in the upper lfne when the degree proposed to be graduated is lesse then the complement of the Latitude from O towards C : but when it is more towards D , and from the point B lay a Ruler over it , and where the Ruler intersects , the line formerly drawn is the point where the degree proposed is to be graduated . Example . Let it be required to find the point where 60 deg . of the Curve is to be graduated . Arke proposed 60 deg . Co-latitude 38 : 28   98 : 28 Summe 81 : 32 Difference 21   32 Count 81 deg . 32′ in the upper line from O towards D , and from the point A draw a line to it . Count the difference 21 degrees 32′ from O towards D , because the co-latitude is lesse then the arke proposed , and lay a Ruler over it , and the point B , and where it intersects the former line is the point where 60 deg . of the Curve is to be graduated on the lower side . Another Example . Let it be proposed to graduate the same way , The arke of 30 degr . 30 degr . Co-latitude 38 : 28 Summe 68 : 28 Difference 8 : 28 Count 68 deg . 28′ from O towards D , and from the point A draw a line to it . Again in the said upper line , count 8 deg . 28′ upwards from O towards C , & from the point B lay a ruler over it , & where it intersects the line last drawn is the point where 30 d. of the curve is to be graduated . To graduate the upper part of the curve requires no other directions , the same arkes serve , if the account be but made the other way , and in accounting the summe the ruler laid over B. in the lower line instead of A , and in counting the difference over A , instead of B , neither is there any Scheme given hereof , the Practitioner need onely let the upper line be the line of altitudes on the left edge of the quadrant continued out to 90 deg . at each end , and to that end next the Center set C , and to the other end D. So likewise let that end of the Versed Scale next the right edge of the quadrant be continued to 180 deg . whereto set A , and at the other end B , and then if these directions be observed , and the same distance and position of the lines retained , it will not be difficult to constitute a Curve in all respects agreeing with that on the fore-side of the quadrant . Of the houre and Azimuth Scale on the right edge of the Quadrant . THis Scale stands outwardmost on the right edge of the quadrant , and consists of two lines , the one a line of 90 sines made equal to the cosine of the Latitude , namely , to the sine of 38 deg . 28′ , and continued the other way to 40 deg . like a Versed sine . The annexed line being the other part of this Scale , is a line of natural Tangents beginning where the former sine began , the Tangent of 38 deg . 28′ being made equal to the sine of 90 deg . this Tangent is continued each way with the sine ; towards the Limbe of the quadrant it should have been continued to 62 deg . but that could not be without excursion , wherefore it is broken off at 40 degrees , and the residue of it graduated below , and next under the Versed sine belonging to the Curve that runnes crosse the quadrant being continued but to halfe the former Radius . Of the Almanack . NExt below the former line stands the Almanack in a regular ob-long with moneths names graved on each side of it . Below the Almanack stands the quadrat● , and shadowes in two Arkes of circles terminating against 45 deg . of the Limbe , below them a line of 90 sines in a Circle equal to 51 deg . 32′ of the Limbe broken off below the streight line , and the rest continued above it . Below these are put on in Circles a line of Tangents to 60 degrees . Also a line of Secants to 60 deg . with a line of lesser sines ending against 30 deg . of the Limbe ( counted from the right edge ) where the graduations of the Secant begins . Last of all the equal Limbe . Prickt with the pricks of the quadrat . Abutting upon the line of sines , and within the Projection stands a portion of a small sine numbred with its Complements beginning against 38 deg . 28′ of the line of sines , this Scale is called the Scale of entrance . Upon the Projection are placed divers Stars , how they are inscribed shall be afterwards shewne . The description of the Back-side . Put on in quarters or Quadrants of Circles . 1 THe equal Limbe divided into degrees , as also into houres and halves , and the quarters prickt to serve for a Nocturnal . 2 A line of Equal parts . 3 A line of Superficies or Squares . 4 A line of Solids or Cubes . 5 A Tangent of 45 degrees double divided to serve for a Dyalling Tangent , and a Semitangent for projections . 6 The line Sol , aliàs a line of Proportional Sines . 7 A Tangent of 51 degrees 32′ through the whole Limbe . 8 A line of Declinations for the Sun to 23 deg . 31′ . Foure quadrants with the days of the Moneth . 9 10 11 12 13 The Suns true place , with the Charecters of the 12 Seignes . 14 The line of Segments , with a Chord before they begin . 15 The line of Metals and Equated bodies . 16 The line of Quadrature . 17 The line of Inscribed bodies . 18 A line of 12 houres of Ascension with Stars names , Declinations , and Ascensional differences . Above all these a Table to know the Epact , and what day of the Weeke , the first day of March , hapned upon , by Inspection continued to the yeare 1700. All these between the Limbe and the Center . ON the right edge a line of equal parts from the Center decimally sub-divided , being a line of 10 inches ; also a Dyalling Tangent or Scale of 6 houres , the whole length of the quadrant not issuing from the Center . On the left edge a Tangent of 63 deg . 26′ from the Center . Also a Scale of Latitudes fitted to the former Scale of houres not issuing from the Center , and below it a small Chord . The Vses of the Quadrant . Lords-day 1657 63 68 74 ☉ 85 91 96 anno 25 1 26 3   4 11 6 epact Monday 58 ☽ 69 75 80 86 ☽ 97 anno 6   7 14 9 15   17 epact Tuesday 59 64 70 ♂ 81 87 92 98 anno 17 12 18   20 26 22 28 epact Wenesday ☿ 65 71 76 82 ☿ 93 99 anno   23 29 25 1   3 9 epact Thursday 60 66 ♃ 77 83 88 94 ♃ anno 28 4   6 12 7 14   epact Friday 61 67 72 78 ♀ 89 95 700 anno 9 15 11 17   18 25 20 epact Saturday 62 ♄ 73 79 84 90 ♄ 701 anno 20   22 28 23 29   1 epact Dayes the same as the first of March. March 1 8 15 22 29 November August 2 9 16 23 30 August May 3 10 17 24 31 Jnuary October 4 11 18 25 0 October April 5 12 19 26 00 July Septem . 6 13 20 27 00 December June 7 14 21 28 00 February Perpetual Almanack . Of the Vses of the Projection . BEfore this Projection can be used , the Suns declination is required , & by consequence the day of the moneth for the ready finding thereof there is repeated the same table that stands on the Back-side of this quadrant in each ruled space , the uppermost figure signifies the yeare of the Lord , and the column it is placed in sheweth upon what day of the Weeke the first day of March hapned upon in that yeare , and the undermost figure in the said ruled space sheweth what was the Epact for that yeare and this continued to the yeare 1701 inclusive . Example . Looking for the yeare 1660 I find the figure 60 standing in Thursday Column , whence I may conclude that the first day of March that yeare will be Thursday , and under it stands 28 for the Epact that yeare . Of the Almanack . HAving as before found what day of the Weeke the first day of March hapned upon , repaire to the Moneth you are in , and those figures that stand against it shewes you what dayes of the said moneth the Weeke day shall be , the same as it was the first day of March. Example For the yeare 1660 , having found that the first day of March hapned upon a Thursday , looke into the column against June , and February , you will find that the 7th , 14th , 21th and 28th dayes of those Moneths were Thursdayes , whence it might be concluded if need were that the quarter day or 24th day of June that yeare hapneth on the Lords day . Of the Epact . THe Epact is a number carried on in account from yeare to yeare towards a new change , and is 11 dayes , and some odde time besides , caused by reason of the Moons motion , which changeth 12 times in a yeare Solar , and runnes also this 11 dayes more towards a new change , the use of it serves to find the Moones age , and thereby the time of high Water . To know the Moons age . ADde to the day of the Moneth the Epact , and so many days more , as are moneths from March to the moneth you are in , including both moneths , the summe ( if lesse then 30 ) is the Moones age , if more , subtract 30 ; and the residue in the Moons age ( prope verum . ) Example . The Epact for the year 1658 is 6 , and let it be required to know the Moons age the 28 of July , being the fift moneth from March both inclusive 6 28 5 The summe of these three numbers is 39 Whence rejecting 30 , the remainder is 9 for the Moons age sought . The former Rule serves when the Moneth hath 31 dayes , but if the Moneth hath but 30 Dayes or lesse , take away but 29 and the residue is her ages To find the time of the Moones comming to South . MUltiply the Moones age by 4 , and divide by 5 , the quotient shewes it , every Unit that remaines is in value twelve minutes of time , and because when the Moon is at the full , or 15 dayes , old shee comes to South at the houre of 12 at midnight , for ease in multiplication and Division when her age exceedes 15 dayes reject 15 from it . Example , So when the Moon is 8 dayes old , she comes to South at 24 minutes past six of the clock , which being knowne , her rising or setting may be rudely guessed at to be six houres more or lesse before her being South , and her setting as much after , but in regard of the varying of her declination no general certaine rule for the memory can be given . Here it may be noted that the first 15 dayes of the Moones age she commeth to the Meridian after the Sun , being to the Eastward of him , and the later 15 dayes , she comes to the Meridian before the Sun , being to the Westward of him . To find the time of high Water . TO the time of the Moones comming to South , adde the time of high water on the change day , proper to the place to which the question is suited , the summe shewes the time of high waters For Example , There is added in a Table of the time of high Water at London , which any one may cast up by memory according to these Rules , it is to be noted , that Spring Tides , high winds , and the Moon in her quarters causes some variation from the time here expressed . Moones age Moon South Tide London Dayes . Ho. mi. Ho. Mi 0 15 12 — 3 00 1 16 12 : 48 3 : 48 2 17 1 : 36 4 : 36 3 18 2 : 24 5 : 24 4 19 3 : 12 6 : 12 5 20 4   7 00 6 21 4 : 48 7 : 48 7 22 5 : 36 8 : 36 8 23 6 : 24 9 : 24 9 24 7 : 12 10 : 12 10 25 8 : 00 11 : 00 11 26 8 : 48 11 : 48 12 27 9 : 36 12 : 36 13 28 10 : 24 1 : 24 14 29 11 : 12 2 : 12 This Rule may in some measure satisfie and serve for vulgar use for such as have occasion to go by water , and but that there was spare roome to grave on the Epacts nothing at all should have been said thereof . A Table shewing the houres and Minutes to be added to the time of the Moons comming to South for the places following being the time of high Water on the change day .   H. m. Quinborough , Southampton , Portsmouth , Isle of Wight , Beachie , the Spits , Kentish Knocke , half tide at Dunkirke . 00 :  00 Rochester , Maulden , Aberdeen , Redban , West end of the Nowre , Black taile . 00 :  45 Gravesend , Downes , Rumney , Silly half tide , Blackness , Ramkins , Semhead . 1 :  30 Dundee , St. Andrewes , Lixborne , St. Lucas , Bel Isle ; Holy Isle . 2 :  15 London , Tinmouth , Hartlepoole , Whitby , Amsterdam , Gascoigne , Brittaine , Galizia . 3 :  00 Barwick , Flamborough head , Bridlington bay , Ostend , Flushing , Bourdeaux , Fountnesse . 3 :  45 Scarborough quarter tide , Lawrenas , Mountsbay , Severne , King sale , Corke-haven , Baltamoor , Dungarvan , Calice , Creeke , Bloy seven Isles . 4 :  30 Falmouth , Foy , Humber , Moonles , New-castle , Dartmouth , Torbay , Caldy Garnesey , St. Mallowes , Abrowrath , Lizard . 5 :  15 Plymouth , Weymouth , Hull , Lin , Lundy , Antwerpe , Holmes of Bristol , St. Davids head , Concalo , Saint Malo. 6 :  00 Bristol , foulnes at the Start. 6 :  45 Milford , Bridg-water , Exwater , Lands end , Waterford , Cape cleer , Abermorick Texel . 7 :  20 Portland , Peterperpont , Harflew , Hague , St. Magnus Sound , Dublin , Lambay , Mackuels Castle . 8 :  15 Poole , S. Helen , Man Isle , Catnes , Orkney , Faire Isles , Dunbar , Kildien , Basse Islands , the Casquers , Deepe at halfe tide . 9 :   Needles , Oxford , Laysto , South and North Fore-lands . 9 :  45 Yarmouth , Dover , Harwich , in the frith Bullen , Saint John de luce , Calice road . 10 :  30 Rye , Winchelsea , Gorend , Rivers mouth of Thames , Faire Isle Rhodes . 11 :  15 To find the Epact for ever . IN Order hereto , first , find out the Prime Number divide the yeare of the Lord by 19 the residue after the Division is finished being augmented by an Unit is the Prime sought , and if nothing remaine the Prime is an Unit. To find the Epact . MUltiply the Prime by 11 , the product is the Epact sought if lesse then 30 , but if it be more , the residue of the Product divided by 30 is the Epact sought , there note that the Prime changeth the first of January , and the Epact the first of March. Otherwise . Having once obtained the Epact adde 11 so it the Summe if lesse then 30 is the Epact for the next yeare if more reject 30 , and the residue is the Epact sought . Caution . When the Epact is found to be 29 for any yeare , the next yeare following it will be 11 and not 10 , as the Rule would suggest . A Table of the Epacts belonging to the respective Primes . Pr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Ep. 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 The Prime number called the Golden Number , is the number of 19 years in which space the Moone makes all variety of her changes , as if she change on a certain day of the month on a certain yeare she shall not change the same day of the moneth again till 19 yeares after : and then it doth not happen upon the same houre of the day , yet the difference doth not cause one dayes variation in 300 yeares , as is observed by Mr. Philips . The Vses of the Quadrant . WIthout rectifying the Bead nothing can be performed by this Projection , except finding the Suns Meridian Altitude being shewn upon the Index , by the intersection of the Parallel of declination therewith . Also the time when the Sun will be due East or West . TRace the Parallel of Declination to the right edge of the Projection , and the houre it there intersects ( in most cases to be duly estimated ) shewes the time sought , thus when the Sun hath 21 deg . of North declination , we shall find that he will be due East or West , about three quarters of an houre past 4 in the afternoon , or a quarter past 7 in the morning . The declination is to be found on the Back-side of the quadrant by laying the thread over the day of the moneth . To rectifie the Bead. LAy the thread upon the graduated Index , and set the Bead to the observed or given Altitude , and when the Altitude is nothing or when the Sun is in the Horizon set the Bead to the Cypher on the graduated Index , which afterwards being carried without stretching to the parallel of Declination the threed in the Limbe shewes the Amplitude or Azimuth , and the Bead amongst the houres shewes the true time of the day . Example . Upon the 24th of April the Suns declination will be found to 16 deg . North. Now to find his Amplitude and the time of his rising , laying the threed over the graduated Index , set the Bead to the beginning of the graduations of the Index , and bring it without stretching to the parallel of declination above being 16 d ▪ and the threed in the limbe will lye over 26 deg . 18′ for the Suns Amplitude or Coast of rising to the Northward of the East , and the Bead amongst the houres sheweth 24 minutes past 4 for the time of Sun rising . Which doubled gives the length of the night 8 houres 49 min. In like manner the time of setting doubled gives the length of the day . The same to find the houre and Azimuth let the given Altitude be 45 degrees . HAving rectified the Bead to the said Altitude on the Index and brought it to the intersect , the parallel of declination the thread lyes over 50 degrees 48′ . For the Suns Azimuth from the South . And the Bead among the houres shewes the time of the day to be 41 minutes past 9 in the morning , or 19 minutes past two in the afternoon . Another Example wherein the operation will be upon the Reverted taile . Let the altitude be 3 deg . 30′ And the declination 16 deg . North as before . TO know when to rectify the Bead to the upper or neather Altitude will be no matter of difficulty , for if the Bead being set to the neather Altitude will not meet with the parallel of declination , then set it to the upper Altitude , and it will meet with Winter parallel of like declination , which in this case supplyes the turn . So in this Example , the Bead being set to the upper Altitude of 3 deg . 30′ and carried to the Winter parallel of declination . The thread in the Limbe will fall upon 68 deg . 28′ for the Suns Azimuth from the North , and the Bead among the houres shewes the time of the day to be either 5 in the morning or 7 at night . Another Example . Admit the Sun have 20 degr . of North Declination ( as about the 9th of May ) and his observed altitude were 56 deg . 20′ having rectified the Bead thereto , and brought it to intersect the parallel of 20 deg . among the houres it shewes the time of the day to be 11 in the morning or 1 in the afternoon , and the Azimuth of the Sun to be 26 deg . from the South The Vses of the Projection . TO find the Suns Altitude on all houres or Azimuths will be but the converse of what is already said , therefore one Example shall serve . When the Sun hath 45 deg . of Azimuth from the South . And his Declination 13 deg . Northwards . Lay the threed over 45 deg . in the Limbe , and where the threed intersects the Parallel of Declination thereto remove the Bead which carried to the Index without stretching , shewes 43 deg . 50′ for the Altitude sought . Likewise to the same Declination if it were required to find the Suns Altitude for the houres of 2 or 10. Lay the threed over the intersection of the houre proposed with the parallel of Declination , and thereto set the bead which carried to the Index shewes the Altitude sought namely 44 deg . 31′ . The same Altitude also belongs to that Azimuth the threed in the former Position lay over in the Limbe . This Projection is of worst performance early in the morning or late in the evening , about which time Mr. Daries Curve is of best performance whereto we now addresse our selves . Of the curved line and Scales thereto fitted . This as we have said before was the ingenious invention of M. Michael Dary derived from the proportionalty of two like equiangled plain Triangles accommodated to the latitude of London , for the ready working of these two Proportions . 1. For the Houre . As the Cosine of the Latitude , is to the secant of the Declination , So is the difference between the sine of the Suns proposed and Meridian Altitude . To the versed sine of the houre from noone , and the converse , and so is the sine of the Suns Meridian Altitude , to the versed sine of the semidiurnal Arke . 2. For the Azimuth . The Curve is fitted to find it from the South and not from the North , and the Proportion wrought upon it will be . As the cosine of the Latitude , is to the Secant of the Altitude . So is the difference of the versed sines of the Suns ( or Stars ) distance from the elevated Pole , and of the summe of the Complements both of the Latitude and Altitude , to the versed sine of the Azimuth from the noon Meridian . Which will not hold backward to find the Altitude on all Azimuths , because the altitude is a term involved , both in the second and third termes of the former proportion . If the third terme of the former Proportion had not been a difference of Sines , or Versed sines , the Curved line would have been a straight-line , and the third term always counted from one point , which though in the use it may seem to be so here , yet in effect the third term for the houre is always counted from the Meridian altitude . Here observe that the threed lying over 12 or the end of the Versed Scale , and over the Suns meridian altitude in the line of altitudes , it will also upon the curve shew the Suns declination , which by construction is so framed that if the distance from that point to the meridian altitude , be made the cosine of Latitude , the distance of the said point from the end of the versed Scale numbred with 12 shall be the secant of the declination to the same Radius , being both in one straight-line by the former constitution of the threed , and instead of the threed you may imagine a line drawn over the quadrant , then by placing the threed as hereafter directed it will with this line & the fitted scales constitute two equiangled plaine triangles , upon which basis the whole work is built . In the three first Proportions following relating to time , the Altitude must alwayes be counted upwards from O in the line of Altitude , and the Declination in the Curve upwards in Summer , downwards in Winter . 1 To find the time of the Suns rising and setting by the Curve . WE have before intimated that the suns Declination is to be found on the back of the quadrant , having found it , lay one part of the thread over 0 deg . in the Line of Altitude , and extending it , lay the other part of it over the Suns Declination counted from O in the Curve , and the thread upon the Versed scale shewes the time of Suns rising and setting , which being as much from six towards noon in Winter as towards mid-night in Summer , the quantity of Declination supposed alike both wayes on each side the Equinoctial , the thread may be layd either way from O in the Curve to the Declination . Example . When the Sun hath 20 deg . of Declination , the thread being laid over 20 deg . in the Curve and O in the Altitude on the left edge shewes that the Sun riseth setteth 1 houre 49′ before after six in the Summer and riseth setteth as much after before six in the Winter . 2 The Altitude and Declination of the Sun being given to find the houre of the day . COunt the Altitude from O in the Scale of Altitudes towards the Center , and thereto lay the thread , then count the Declination from O in the Curve , if North upwards towards the Center , if South downwards towards the Limbe . And lay the thread extended over it , and in the Versed Scale it shewes the time of the day sought . Example . The Altitude being 24 d. 46′ and the Declination 20 d. North counting that upwards in the Scale of Altitudes , and this upward in the curve , and extending the through thread , it will intersect the Versed Scale at 7 and 5 , shewing the houre to be either 7 in the morning , or 5 in the afternoon . Another Example for finding when twliight begins . Let the Suns Declination be 13 deg . North , the Depression supposed 18 degr . under the Horizon . In stead of the case propounded , suppose the Sun to have 13 deg . of South Declination , and Altitude 18 deg . above the Horizon accordingly extending the thread through 18 in the Altitudes counted upward from O in the line of Altitudes and through 13 deg . counted downward in the Curve from O , and upon the Versed Scale , the thread will shew that the Twilight begins at 28 minutes past 2 in the morning , and at 32 min. past 9 at night . 3 The Converse of the last Proposition is to find the Suns altitudes on all houres . EXtend the thread over the houre proposed in the versed Scale and also over the Declination in the Curve counted upward if North , downward if South . And in the Scale of Altitudes it shewes the Altitude sought . Example . If the Sun have 13 deg . of North Declination his Altitude for the houre of 7 in the morning , or 5 in the after-noon will be found to be 19 deg . 27′ . In the following Propositions the altitude must alwayes be counted from O in the Curve downwards , and the Declination in the line of altitudes , if North downward , if South upwards . 4 To find the Suns Amplitude or coast of rising and setting . Example . If the Sun had 20 deg . of Declination the thread being laid to O in the Curve , and to 20 in the line of altitudes or Declinations , either upwards or downwards the thread will lye 33 deg . 21′ from 90 in the Versed Scale , for the quantity of the Suns Coast of rising or setting from the true East or West in Winter Southward , in Summer Norhward . 5 The Suns altitude and Declination being proposed to find his Azimuth . COunt the altitude from O in the Curve downward , and the declination in the Winter upon the line of Declinations from O upwards , in Summer downwards , and the thread extended sheweth the Azimuth sought , on the Versed Scale . Example . So when the Sun hath 18 deg . 37′ of North Declination , as as about 19 July , if his altitude were 39 deg . the Suns Azimuth would be found to be 69 deg . from the South . 6 The Converse of the former Proposition will be to find the Suns Altitude on all Azimuths . THe Instrument will perform this Proposition though the Porportion for finding the Azimuth cannot be inverted . Lay the thread to the azimuth in the Versed Scale , and to the Declination in the Scale on the left edge , and upon the Curve it will intersect the altitude sought . Example . If the Sun had 16 deg . 13′ of South Declination , as about the 27th of October , if his Azimuth were 39 deg . from the South the altitude agreeable thereto would be found to be 14 deg . These Uses being understood ▪ if the houre and altitude or the azimuth and altitude were given to find the Declination , the manner of performance cannot lurke . Of the Houre and Azimuth Scale on the right Edge of the Quadrant . THis Scale being added by my selfe , and derived from Proportions in the Analemma , I shall first lay them down , and then apply them . In the former Scheme draw F C V the Horizon , Z C the Axis of the Horizon , C P the axis of the Spheare G C continued to N the Equator , O L a parallel of North , and E I a parallel of South Declination , W X a parallel of winter altitude , S L a parallel of altitude lesse then the Complement of the latitude , N Z P a parallel of greater altitude , and from the points E and B. let fall the perpendiculars E F and B H , and from the points B G and N let fall the perpendiculars B G , G M , and N O which will be the sines of the Suns declination , by this meanes there will be divers right lined right angled plaine Triangles constituted from whence are educed , the Proportions following to calculate the Suns houre or Azimuth . Note , first , that T V is the Versed sine of the Semidiurnal arke in Summer , and E I in Winter , and Y V the sine of the houre of rising before six in Summer , equal to the distance of I from the Axis continued in Winter , which may be found in the Triangle C Y V , but the Proportion is . As the Cotangent of the latitude , To Radius . So the Tangent of the Suns Declination , To the sine of his ascentional difference , being the time of his rising from six , thus we may attaine the Semidiurnal arke . Then for the houre in the Triangle B H I it holds . As the Cosine of the latitude , to the sine of the altitude . So is the Secant of the Declination . To the difference of the Versed sines of the Semidiurnals arke , and of the houre sought . In the Triangle B H I it leys . As the Cosine of the Latitude the sine of the angle at I. To its opposite side B H the sine of the altitude . So is the Radius or the angle at H. To B I h difference of the Versed sine of the Semidiurnal arke , and of the houre sought , in the parallel of declination and by consequence , so is the secant of the Declination , to the said difference in the Common Radius as we have else where noted , if this difference be subtracted from the Versed sine of the semidiurnal arke there will remaine E B the versed sine of the houre from noon , the like holds , if perpendiculars be let fall from any other parallel of Declination , from the same Scheme it also followes . As the Cosine of the Latitude , Is to the secant of the Declination . So is the sine of the Meridian Altitude . To the versed sine of the semidiurnal arke . Here observe the like Proportion between the two latter terms , as between the two former which may be of use on a Sector . If the Scheme be considered not as fitted to a peculiar question for finding the houre , but as having three sides to find an angle , it will be found upon such a consideration in relation to the change of sides , that the Proportion for the Azimuth following is no other then the same Proportion applyed , to other sides of the Triangle , and so we need have no other trouble to come by a Proportion for the Azimuth , but it also followes from the same Scheme . In the Triangles C D A and C K G , and C Z N the first operation will be to find A D , and G K , and N ● in all which the Proportion will hold . As the Radius to the Tangent of the Latitude . Or as the Cotangent of the Latitude to Radius . So is the Tangent of the Altitude , to the said respective quantities , which when the Altitude is lesse then the Complement of the Latitude , are the sins of the Suns Azimuth from the Vertical belonging to the proposed Altitudes when the Sun is in the Equinoctial , or hath no declination . The next proportion will be . As the Cosine of the Latitude , Is to the Secant of the Altitude . So is the Sine of the declination . To the difference sought being a 4 Proportional . Hereby we may find A B in the Winter Triangle A G B which added to A D , the summe is the sine of the Azimuth from the Vertical consequently W B , is the Versed sine of the Azimuth , from the noon Meridian . Also we find G L in the Summer triangle L M G , when the Altitude is lesse then the Complement of the Latitude , which added to S G the summe S L is the Versed sine of the Azimuth from the South . Likewise we may find N R in the Triangle R O N , and by subtracting it from N Z , there will remaine R Z , and consequently Q R the versed sine of the Azimuth from the Meridian in Summer when the Altitude is greater then the Co-latitude . And for Stars that come to the Meridian between the Zenith , and the Elevated Pole , we may find N c , in the Triangle N c d where it holds , as the sine of the Angle at N , the complement of the Latitude , to its opposite sides c d , the prickt line , the sine of the Declination : so is the Radius to N c , the parallel of altitude the Azimuth sought . The latter Proportion lyes so evident , it need not be spoken to , if what was said before for the houre be regarded , and the former Proportion lyes . As the Cosine of the Latitude , the sine of the Angle at A. To its Opposite side D C , the sine of the altitude . So is the sine of the Latitude , the angle at C. To its opposite side A D in the parallel of altitude . And in stead of the Cosine , and sine of the Latitude . We may take the Radius , and the Tangent of the Latitude . Another Analogy will be required to reduce it to the common Radius . As the Cosine of the Altitude to Radius . So the fourth before found in a parallel . To the like quantity to the Common Radius . These Analogies or Proportions being reduced into one , by multiplying the termes of each Proportion , and then freed from needlesse affection will produce the Proportion at first delivered . The Vses of the said Scale . WE have before noted , that if two termes of a Proportion be fixed , and naturall lines thereto fitted of an equal length , that if any third term be sought in the former line , the fourth term will be found in the other line by inspection , as standing against the third . So here , in this Scale which consists of two lines , the one an annexed Tangent , the other a line of Sines continued both Wayes , the Radius of the Sines being first fittted , the Tangent annexed must be of such a Radius , as that 38 deg . 28′ , of it may be equall in length to the Radius of the Sine to which it is adjoyned , and then looking for the Declination in the Tangent just against it stands the time of rising , from six or ascentional difference , or the Semidiurnal arke , if the same be accounted from the other end as a Versed Sine . So if the Suns Altitude be given , and accounted in the Tangent , just against it stands the Suns Azimuth , when he is in the Equinoctial upon the like altitude , and thus the point N will be found in the Tangent at the altitude , when it is more then the Colatitude . 1 An Example for finding the time of the Sun rising . If the Declination be 13 deg . looke for it in the annexed Tangent , and just against it in the houre Scale stands 16 deg . 53′ the ascentional difference in time 1 houre 7½ min. shewing that the Sun riseth so much before , and setteth so much after 6 in Summer , and in Winter riseth so much after , and setteth before 6 , for this arke may be found on either side of six where the declination begins each way . 2 To find the time of the day . To perform this Proposition wee divide the other Proportion into two , by introducing the Radius in the Middle . As the Radius is to the Secant of the Declination . So is the sine of the altitude to a fourth . Again . As the Cosine of the Latitude to Radius . So the fourth before found . To the difference of the Versed Sines of the Semidiurnal arke , and of the houre sought . The former of these Proportions must be wrought upon the quadrant , the latter is removed by fitting the Radius of the Sines that gives the answer , equal in length to the Cosine of the latitude . Wherefore to find the time of the day , lay the thread to the Secant of the declination in the limbe , and from the sine of the altitude take the nearest distance to it , and because the Secant is made , but to halfe the Common Radius , set downe one foot of this extent at the Declination in the annexed Tangent , and enter the said extent twice forward , and it will shew the time of the Day . Example . Let the Declination be supposed 23 deg . 31′ North , and the Altitude 38 deg . 59′ the nearest distance from the Sine thereof , to the thread laid over the Secant of 3● deg . 31′ will reach being turned twice over from 3● d. 31′ in the annexed Tangent neerest the Center to 33 deg . 45′ in the Sines , aliàs to 56 d. 15′ counted as a Versed Sine shewing the time of the day to be a quarter past 8 in the morning , or three quarters past three in the afternoon . 3 To find the Suns Altitude on all houres . Take the distance between the houre and the Declination in the fitted Scale , and enter it downe , the line of Sines from the Center , then laying the thread over the Cosine of the Declination in the Limbe , the nearest distance to it shall be the sine of the Altitude sought . Example . Thus whee the Sun hath 13 deg . of South Declination , count it in that part of the annexed Tangent nearest the Limbe , if then it were required to find the Suns Altit . for the houres of 10 or 2 by the former Prescriptions the Altitude would be found 10 d. 25′ 4 To find the Suns Amplitude . Take the Sine of the Declination from the line of the Sines , and apply it to the fitted Scale where the annexed Tangent begins , and either way it will reach to the Sine of the Amplitude . Example . So when the Sun hath 15 deg : of Declination his Amplitude will be found to be 24 deg . 35′ . 5 To find the Azimuth or true Coast of the Sun. Here we likewise introduce the Radius in the latter Proportion . 1 In Winter lay the thread to the Secant of the Altitude in the Limbe , and from the sine of the Declination , take the nearest distance to it , the said extent enter twice forward from the Altitude in the annexed Tangent , and it will reach to the Versed Sine of the Azimuth from the South . Example . So when the Sun hath 15 deg . of South Declination , if his Altitude be 15 deg . the nearest distance from the sine thereof to the thread laid over the Secant of 15 degrees , shall reach in the fitted Scale from the annexed Tangent of 15 deg . being twice repeated forward to the Versed sine of 39 deg . 50′ for the Suns Azimuth from the South . 2 In Summer when the Altitude is lesse then 40 deg . enter the former extent from the sine of the Declination to the thread laid over the Secant of the Altitude twice backward from the Altitude in the annexed Tangent , and it will reach to the Versed sine of the Azimuth from the South . Example . So if the Sun have 15 deg . of North Declination , and his Altitude be 30 deg . the prescribed extent doubled shall reach from the annexed Tangent of 30 deg . to the Versed sine of 75 deg . 44′ for the Suns Azimuth from the South . 3 In Summer when the Altitude is more then 40 deg . and lesse then 60 deg . apply the extent from the sine of the Declination to the thread , laid over the Secant of the Altitude once to the Discontinued Tangent placed a Crosse the quadrant from the Altitude backwards minding how farre it reaches , just against the like arke in the annexed Tangent stands the Versed sine of the Azimuth from the South . 4 When the Altitude is more then 60 deg . this fitted Scale is of worst performance , however the defect of the Secant might be supplyed by Varying the Proportion . 6 To find the Suns Altitude on all Azimuths . JUst against the Azimuth proposed stands the Suns altitude in the Equator suitable thereto , which was the first Arke found by Calculation when we treated of this subject , and the second arke is to be found by a Proportion in sines wrought upon the quadrant . This quadrant is also particularly fitted for giving the houre , and Azimuth in the equal limbe . The sine of 90 deg . made equal to the sine if 51 deg . 32′ gives the altitude of the Sun or Stars at six , for if the thread be laid over the Declination counted in the said sine , it shewes the Altitude sought in the limbe , so when the Sun hath 13 deg . of Declination his Altitude or Depression at 6 is 10 deg . 9′ . It also gives the Vertical Altitude if the Declination be counted in the limbe , seeke what arke it cuts in that particular sine , when the Sun hath 13 deg . of Declination , his Vertical Altitude or Depression is 16 deg . 42′ . To find the houre of the Day . HAving found the Altitude of the Sun or Stars at six , take the distance between the sine thereof in the line Sines , and the Altitude given , and entring one foot of that extent at the Declination in the Scale of entrance laying the thread to the other foot according to nearest distance , it will shew the houre from six in the limbe . Example . When the Sun hath 13 deg . of Declination his Altitude , or Depression at six will be 10 deg . 9′ if the Declination be North , and the Altitude of the Sun be 24 deg . 5′ the time of the day will be halfe an houre past 7 in the morning , or as much past 4 in the afternoon . In winter when the Sun hath South Declination as also for such Stars as have South Declination , the sine of their Altitude must be added to the sine of their Depression at six , and that whole extent entred as before . When the Sun hath the same South Declination , if his Altitude be 11 deg . 7′ the time of the day will be half an houre past 8 in the morning , or 30 min. past 3 in the afternoon . To find the Azimuth of the Sun or Stars . LAy the thread over their Altitude in the particular sine fitted to the Latitude , and in the equal Limbe it shewes a fourth Arke . When the Declination is North , take the distance in the line of Sines between that fourth Arke and the Declination , and enter one foot of that extent at the Altitude in the Scale of entrance , laying the thread to the other foot , and in the equal Limbe it shewes the Azimuth from the East or West . Example . When the Altitude is 44 deg . 39′ the Arch found in the equal Limbe will be 33 deg . 20′ then if the Declination be 23 deg 31′ North , the distance in the line of sines between it and the said Arke being entred at 44 deg . 39′ in the Scale of entrance the thread being laid to the other foot will shew the Azimuth to be 20 deg . from the East or West . But if the Declination be South , adde with your Compasses the sine thereof to the sine of the fourth Arke , and enter that whole extent as before , and the thread will shew the Azimuth in the equal limbe . Example . When the Altitude is 12d . 13′ the fourth Arch will be found to be 9 degrees 32 minutes , then admit the Declination to be 13 degrees South , whereto adding the Sine of the fourth Arke , the whole will be equall to the sine of 22 deg . 41 minutes , and this whole extent being entred at 12 deg . 13′ in the Scale of entrance lay the thread to the other foot according to nearest distance , and it will intersect the equal . Limbe at 40 deg . and so much is the Suns Azimuth from the East or West . Because the Scale of entrance could not be continued by reason of the Projection , the residue of it is put on an little Line neare the Amanack the use whereof is to lay the thread to the Altitude in it when the Azimuth is sought , and in the Limbe it shewes at what Arke of the Sines the point of entrance will happen which may likewise be found by pricking downe the Co-altitude on the line of Sines out of the fitted houre Scale on the right edge . How to find the houre and Azimuth generally in the equal limb either with or without Tangents or Secants hath been also shewed , and how that those two points for any Latitude might be there prickt and might be taken off , either from the Limbe , or from a line of Sines , or best of all by Tables , for halfe the natural Tangent of the Latitude of London , is equal to the sine of 〈◊〉 39 deg . And half the Secant thereof equal to the sine of 〈◊〉 53d . 30 Against which Arkes of the Limbe the Tangent and Secant of the Latitude are graduated , but of this enough hath been said in the Description of the small quadrant . Of the Quadrat and Shadowes . THe use thereof is the same as in the small quadrant onely if the thread hang over any degree of the Limb lesse then 45d . to take out the Tangent thereof out of the quadrat count the Arch from the right edge of the quadrant towards the left , and lay the thread over it , the pricks are repeated in the Limbe to save this trouble for those eminent parts . Of the equal Limbe . WE have before shewed that a Sine , Tangent and Secant may be taken off from it , and that having a Sine or Secant with the Radius thereof the correspondent Arke thereto might be found , & that a Chord might be taken off from Concentrick Circles or by helpe of a Bead , but if both be wanting enter the Semidiameter or Radius whereto you would take out a Chord twice downe the right edge from the Center , and laying the thread over halfe the and laying the thread over halfe the Arch proposed , take the nearest distance to it , and thus may a chord be taken out to any number of degrees lesse then a Semicircle . It hath been asserted also that the houre and Azimuth might be found generally without Protraction by the sole helpe of the Limb with Compasses and a thread . Example for finding the houre . THe first work will be to find the point of entrance take out the Cosine of the Latitude by taking the nearest distance to the thread laid over the said Arke from the concurrence of the Limbe with the right edge , and enter it down the right edge line and take the nearest distance to the thread laid over the complement of the Declination counted from the right edge , this extent entred down the right edge finds the point of entrance , let it be noted with a mark . Next to find the sine point take out the sine of the Declin . & enter it dowh the right edge , & from the point of termination , take the nearest distance to the thread laid over the ark of the Latit . counted from the right edge , this extent enter from the Center and it finds the sine point , let it be noted with a marke . Thirdly , take out the sine of the Altitude & in Winter add it in lenght to the sine point , in Summer enter it from the Center & take the distance between it & the sine point which extent entred upon the point of entrance , if the thread be laid to the other foot shewes the the houre from 6 in the equal limb before or after it , as the Sine of the Altitude fell short or beyond the sine point . Example . In the latitude of 39 d. the Sun having 23d . 31′ of North Declination , and Altitude 51 deg . 32′ the houre will be found to be 33 deg . 45′ from six towards noon . Note the point of entrance and sine point Vary not , till the Declination Vary . After the same manner may the Azimuth be found in the limb , by proportions delivered in the other great quadrant . Also both or any angle when three sides are given may be found by the last general Proportion in the small quadrant which finds the halfe Versed sine of the Arke fought , which would be too tedious to insist upon & are more proper to be Protracted with a line of Chords . To find the Azimuth universally . THe Proportion used on the smal quadrant for finding it in the equal limbe ( wherein the first Operation for the Vertical Altitude was fixed for one day , ) by reason of its Excursions will not serve on a quadrant , for the Sun or Stars when they come to the Meridian between the Zenith and the elevated Pole , but the Proportion there used for finding the houre applyed to other sides will serve for the Azimuth Universally , and that is As the Radius , Is to the sine of the Latitude , So is the sine of the Altitude , To a fourth sine . Again . As the Cosine of the Altitude , Is the Secant of the Latitude . Or , As the Cosine of the Latitude , Is the Secant of the Altitude . So In Declinations towards the Elevated Pole is the difference , but towards the Depressed Pole the summe of the fourth sine , and of the sine sine of the Declination . To the sine of the Azimuth from the Vertical . In Declinations towards the Depressed Pole , the Azimuth is alwayes obtuse , towards the elevated Pole if the Declination be more then the fourth Arch it is acute , if lesse obtuse . Example for the Latitude of the Barbados 13 deg . Altitude 27 deg . 27′ . Declination 20 deg . North. Lay the thread to 27 deg . 27′ in the Limbe , and from the sine of 13 deg . tahe nearest distance to it which enter on the line of Sines from the Center , and take the distance between the limited point , and the sine of 20 deg . the Declination , this latter extent enter twice downe the line of the Sines from the Center , and take the nearest distance to the thread laid over the Secant of 27 deg . 27′ this extent enter at the sine of 77 deg . the Complement of the Latitude , and laying the thread to the other foot it will lye over 16 deg . in the equal Limbe , the Suns Azimuth to the Northwards of the East or West . Otherwaies . Another Example for the same Latitude and Declination , the Altitude being 52 deg . 27′ lay the thread to it in the Limbe , and take the nearest distance to it from the sine of 13 deg . as before , and enter it downe the line of sines from the Center , and from the point of the limitation take the distance to the sine of 20 deg . the Suns Declination , this latter extent enter once downe the line of sines from the Center , and take the nearest distance to the Thread laid over the Secant of the Altitude 52 deg . 27′ then lay the thread to 77 deg . the Complement of the Latitude in the lesser sines , and enter the former extent between the Scale and the thread , and the foot of the Compasses sheweth 16 deg . as before , for the Suns Azimuth to the Northward of the Vertical , that the Sun may have the same Azimuth , upon two several Altitudes hath been spoken to before , and how to do this without Secants hath been shewne . Two sides with the Angle comprehended to find the third side . DIvers wayes have been shewed for doing of this before , I shall adde one more requiring no Versed sines nor Tangents . 1 If both the sides be lesser then quadrants , and the Angle at liberty . Or , 2 If one of the sides be greater then a quadrant , and the Angle included acute , it will hold . As the Radius , To the Cosine of one of the including sines . So is the Cosine of the other , To a fourth sine . Again . As the Cosecant of one of the including Sides ● is the Sine of the other , So is the Cosine of the angle included , To a seventh Sine . The difference between the fourth and the seventh Sine , is the Cosine of the Side sought . 1 In the first case if the angle given be obtuse , and the seventh Sine greater then the fourth Sine , the Side sought is greater then a quadrant in other cases lesse . If in the second case the seventh Sine be lesse then the fourth , the side sought is greater then a quadrant in other cases lesse . In this second case when one of the includers is greater then a quadrant , and the angle obtuse resolve the opposite Triangle by the former Rules , or the summe of the fourth and seventh Sine shall be the Cosine of the side sought in this case greater then a quadrant . We have before noted that the Cosine of an Arke greater then a quadrant is the Sine of that Arkes excesse above 90 deg . this no other then the converse of the Proportion for the houre demonstrated from the Analemma , in the Triangle O Z P. Let there be given the Sides O P 113 deg . 31′ the side Z P 38 deg . 28′ and the angle comprehended Z P O 75 to find the Side O Z. Operation . Lay the thread to 51 deg . 32′ in the Limbe , and from 13 deg . 31′ in the Sines take the nearest distance to it which measured from the Center will reach to the sine of 18 deg . 12 minutes the fourth Sine . Again , laying the thread to 23 deg . 31′ in the Limbe , from the Sine of 15 deg . take the nearest distance to it , then lay the thread to the Secant of 51 deg . 32′ and enter the said extent between the Scale and the thread , the distance between the resting foot , and the Sine of 18 deg . 12 minutes before found measured from the Center is equal to the Sine of 9 deg . 32′ being the Cosine of the side sought which in this instance because the seventh Sine is lesse then the fourth sine is greater then a quadrant , and consequently must have 90 deg . added thereto , therefore the side O Z is 99 deg . 28 minutes if the question had been put in this Latitude what depression the Sun should have had under the Horizon at the houres of 5 or 7 in the Winter Tropick it would have been found 9 deg . 28′ and this is such a Triangle as hath but one obtuse Angle yet two sides greater then quadrants , and how to shunne a Secant , and a parallel entrance hath been shewed els-where . Of the Stars on the Projection , and in other places of the fore-side of the quadrant . SUch only are placed on the Projection as fall between the Tropicks being put an according to their true Declinations , and in that respect might have stood any where in the parallel of Declination , but in regard we shall also find the time of the night by them with Compasses , they are also put on in a certain Angle from the right edge of the quadrant , to find the quantity of the Angle for Stars of Northerly declination , get the difference of the Sines of the Stars Altitude six houres from the Meridian , and of its Meridian Altitude , and find to the Sine of what Arch the said difference is equal , against that Arch in the Limbe , let the Star be graduated in its proper declination , but for Stars of Southwardly Declination , get the summe of the Sines of their Depression at six and of their Meridian Altitude , and find what Arke in the Sines corresponds thereto as before . We have put on no Stars of Southwardly Declination that will fall beyond the Winter Tropick , but some of Northerly Declination falling without the Summer Tropick , are put on that are placed without the Projection towards the Limbe . All these Stars must be graduated against the line of Sines at their respective Altitudes or Depressions at the Stars houre of Six from the Meridian , and must have the same letter set to them in both places , as also upon the quadrant of 12 houres of Ascension on the Back-side where they are put on according to their true Ascension with their Declinations and Ascensional differences graved against them with the former Letter , and such of them as have more then 12 houres of right Ascension have the Character plus ✚ affixed , denoting that if there be 12 houres of Ascension added to that Ascension they stand against , the summe is their whole true right Ascension . To find the quantity of a Stars houre from the Meridian by the Projection . SEt the Bead upon the Index of Altitude to the Stars observed Altitude , and bring it to the parallel of Declination the Star is graved in , so will it shew among the houre lines , that Stars houre from the Meridian , and the thread in the Limbe will shew the Stars Azimuth . Example . Admit the Altitude of Arcturus be 52 deg . the houre of that Star from midnight , if the Altitude increase will be 7′ past 10 ferè , and the Azimuth of that Star will be 47 deg . 43′ to the Eastwards of the South . The houre and Azimuth of any Star within the Tropicks , may be also found by the fitted Scale on the right edge of the quadrant , or by the Curve , after the same manner as for the Sun , using the Stars Declination as was done for the Suns , or in the equall limb as we shewed for the Sun , which may well serve for most of the Stars in the Hemisphere . Otherwise with Compasses according to the late suggested placing of them . To find the houre of any Star from the Meridian that hath North Declination . TAke the distance between the Star point in the line of Sines , and it s observed Altitude , and laying the thread over the Star where it is graved on or below the Projection , enter the former extent paralelly between the thread and the Scale , and it shewes the Stars houre from six in the sines towards noone , if the Altitude fell beyond the Star point , otherwise towards midnight . Example . For the Goat Star let its Altitude be 40 deg . and past the Meridian , the houre of that Star will be 44′ from six , for the Compasses fall upon the sine of 11 deg . 4′ the houre is towards noon Meridian , because the Altitude is greater then 34 deg . the point where the Star is graved , the thread lying over the Star intersects , the Limbe at 25 deg . 47′ if the distance between the Star , and its Altitude be entred at the sine of that Arke , and the thread laid to the other foot , the houre will be found in the equal Limbe the same as before . For Stars of Southwardly Declination . BEcause the Star point cannot fall the other way beyond the Center of the quadrant , therefore the distance between the Star point , and the Center must be increasing by adding the sine of the Stars Altitude thereto , which will fall more outwards towards the Limbe , and then that whole extent is to be entred as before . Example . The Virgins Spike hath 9 deg 19′ of South Declination the Depression of that Star at six will be found by help of the particular sine to be 7 deg . 17′ and at that Arke in the sines the Star is graved , if the Altitude of that Star were 20 deg . the sine thereof added to the Star will be equal to the sine of 29 deg . 6′ this whole extent entred at the sine of 37 deg . 52′ the Arke of the Limbe against which the Star is graved , and the thread laid to the other foot , the houre of that Star if the Altitude increase will be 19′ past 9. To find the true time of the right . THis must be done by turning the Stars houre into the Suns houre or common time , either by the Pen as hath been shewed before , which may be also conveniently performed by the back of this quadrant , for the thread lying over the day of the moneth sheweth the Complement of the Suns Ascension in the Limbe . Or with Compasses on the said quadrant of Ascensions . THe thread lying over the day of the moneth , take the distance between it and the Star on the said quadrant , the said extent being applyed , the same way as it was taken the Suns foot to the Stars houre shall reach from the Stars houre to the true houre of the night , and if one of the feet of the Compasses fall off the quadrant , a double remedy is els-where prescribed . Example . If on the 12th of January the houre of the Goat Star was 16′ past 5 from the Meridian , the true time sought would be 49′ past 1 in the morning . Example . If upon the third of January , the houre of the Virgins Spike , were observed to be 19′ past 9 , the true time sought would be 45′ past 2 in the morning . To find the time of a Stars rising and setting . THe Ascentional difference is graved against the Star , the Virgins Spike hath 48′ of Ascentional difference , that is to say , that Stars houre of rising is at 48′ past 6 , and setting at 12′ past 5 , And the true time of that Stars rising upon the third of January , will be at 22′ past 10 at night , and of its setting at 47′ past 8 in the morning , found by the former directions . Of the rest of the lines on the back of this quadrant . THey are either such as relate to the motion of the Sun or Stars , or to Dialling , or such as are derived from Mr Gunters Sector . The Tangent of 51 deg . 32′ put through the whole Limbe is peculiarly fitted to the Latitude of London , and will serve to find the time when the Sun will be East or West , as also for any of the Stars that have lesse Declination then the place hath Latitude . Lay the thread to the Declination counted in the said Tangent , and in the Limbe it shewes the houre from 6 if reckoned from the right edge . Example . When the Sun hath 15 deg . of North Declination the time of his being East or West will be 12 deg . 17′ in time about 49′ before or after six , ferè . The Suns place is given in the Ecliptick line by laying the thread over the day of the moneth in the quadrant of Ascensions , of which see page 16 & 17 of the small quadrant . Of the lines relating to Dialling . SUch are the Line of Latitudes , and Scale of houres , of which before , and the line Sol in the Limbe , of which I shall say nothing at present , it is onely placed there in readinesse to take off any Arke from it , according to the accustomed manner of taking off lines from the Limbe to any assigned Radius . The requisite Arkes of an upright Decliner will be given by the particular lines on the Quadrant for the Latitude without the trouble of Proportionall worke . 1 The substiles distance from the Meridian . ACcount the Plaines declination as a sine in the fitted hour Scale on the right edge of the fore-side , and just against it in the annexed Tangent , stands the substiles distance from the meridian . If an upright Plaine decline 30 deg . the substiles distance will be 21 deg . 41 minutes . 2 The Stiles height . Count the Complement of the Plaines Declination in the said fitted houre scale as a sine and apply it with Compasses to the line of sines issuing from the Center , for the former Plaine the stiles height will be found 32 deg . 37′ . 3 The Inclination of Meridians . Account the stilts height in the annexed tangent of the fitted hour Scale , and just against it in the sine stands the Complement of the Inclination of meridians which for the former plaine will be found to be 36 deg . 25′ 4 The Angle of 12 and 6. Account the Plaines Declination in the Limbe on the Backside from the right edge , and lay the thread over it , and in the particular Tangent it shewes the Angle between the Horizon and six 32 deg . 9′ in this Example the Complement whereof is the Angle of 12 and 6 , namely 57 deg . 51 min. Also the requisite Arkes of a direct East or West , reclining or inclining Dial may be found after the same manner for this Latit . 1 The substiles distance . ACcount the Plaines Reinclination in the Limbe on the Backside from the left edge , and in there lay the thread , and in the particular Tangent it shewes the Arke sought . So if an East or West plain recline or incline 60 deg . the substiles distance will be found to be 32 deg . 12′ . 2 The stiles height . Account the Reinclination in the particular Sine on the foreside and in the Limbe it shewes the stiles height , which for the former Example will be found to be 42 deg . 41′ . 3 The inclination of Meridians . The Proportion is , As the Sine of the Latitude , to Radius . So is the sine of the substiles distance . To the sine of the inclination of Meridians , when the substiles distance is lesse then the Latitude of the place it may be found in the particular sine on the foreside , by the intersection of the thread , and for this Example will be 42 deg . 53′ . 4 The Angle of 12 and 6. Account the Complement of the Reinclination in the peculiar hour Scale as a sine , and just against it in the annexed Tangent stands the Complement of the Angle sought , in this Example the Angle of 12 and 6 is 68 deg . 20′ . In other Latitudes the Operations must be performed by Proportional worke with the Compasses . Of the Lines derived from Mr. Gunters Sector . Such are the Lines of superficies Solids , &c. Of the Line of Superficies or Squares . THe chiefe uses of this Line joyntly with the Line of Lines in the Limbe , is when a square number is given to find the Root thereof , or a Root given to find the square number thereto , these Lines placed on a quadrant will perform this some what better then a Sector , because it is given by the Intersection of the thread without Compasses , the properties of the quadrant casting these lines large where on a Sector they would be narrow . To find the square Root of a number . The Root being given to find the Square Number of that Root . IN extracting the square Root pricks must be set under the first third , fift , and seventh figure , and so forward and as many pricks as fall to be under the square number given , so many figures shall be in the Root , and accordingly the line of lines , and superficies must vary in the number they represent , I am very unwilling to spend any time about these kind of Lines , as being of small performance , and by my self and almost by all men accounted meere toyes . If a number be given in the superficies , the thread in the lines sheweth the Root of it , and the contrary , if a number be given in the lines the thread laid over it intersects the Square thereof . The performance thereof by these lines is so deficient that I shall give no Example of it . When a number is given to find the square thereof , if not to large the Reader may correct the last figure of it by multiplying it in his memory . To three numbers given to find a fourth in a Duplicated Proportion . That is to worke a Proportion between Numbers and Squares . Example . If the Diameter of a Circle whose Area is 154 be 14 , what shall the Diameter of that Circle be whose Area is 616. Example . Lay the thread over 616 in the superficies , and from 14 in the equal parts , take the nearest distance to it , then lay the thread to 154 in the superficies , and enter the former extent between the thread and the Scale , and the foot of the Compasses will rest upon 28 the diameter sought . To find a Proportion between two or more like superficies . ADmit there be two Circles , and I would know what Proportion their Areas bear to each other , in this case the proper use of a Line of superficies would be to have it on a ruler , and to measure the lengths of their like sides , for Circles the lengths of their Diameters upon it , and then I say , the numbers found on the superficies beare such Proportion each to other as the Areas or superficial contents , and for small quantities may be done on the quadrant by entring downe the larger extent of the Compasses on the Line of Lines from the Center , and mind the point of limitation , enter then the other extent on the point of limitation , and lay the thread to the other foot , find what number it cuts in the superficies , and the greater shall beare such Proportion to the lesser as 100 , &c. the length of the whole line doth to the parts cut . The Proportion that two superficies beare each to other is the same that the squares of their like sides , and therefore their sides may be measured either in foot or inch measure , and then the Squares taken out as before shewed . The line of superficies serves for the reducing of Plots to any proportion . ADmit a Plot of a piece of ground being cast up containes 364 Acres , and it were required to draw another Plot which being cast up by the same Scale should containe but a quarter so much , and let one side of the said Plot be 60 inches , against 60 in the lines , the square of it will be found to be 3600 , and the fourth part hereof would be 900 , which account in the superficies and you will find the Square Root of it to be 30 , and so many inches must be the like side of the lesser Plot if being cast up by the same Scale it should containe but ¼ of what it did before . If the line of Superficies were on a streight ruler , then to perform such a Proposition as this , would be to measure therewith the side of the Plot given , minding what number it reaches to in the Superficies , the fourth part of the said Number being reckoned on the Superficies , and thence taken shall be the length of the side in the Proportion required . Of the Line of Solids . IF a number be duly estimated in the said line , and the thread laid over it , it will in the line of lines shew the cube Root of that number , and the converse the Root being assigned , the Cube may be found , but by reason of the sorry performance of these Lines I shall spend no time about it , if this line be placed on a loose Ruler , and the like sides of two like Solids be measured therewith , those Solids shall beare such Proportion in their contents each to other as the measured lengths on the Solids . Three Numbers being given to find the fourth in a Duplicated Proportion . Example . IF a Bullet of 4 inches Diameter weigh 9 pound , what shall a Bullet of 8 inches Diameter weigh ? Answer 72 pounds . In this case let the whole line of Solids represent 100 , alwayes the Solid content whether given or sought , must be accounted in the line of Solids , and the Sides or Diameters in the Equall parts . Lay the thread to 9 in the line of Solids , and from 8 in the inches take the nearest distance to it , enter one foot of that extent at 4 in the inches , and lay the thread to the other foot : and it will lye over 72 in the Solids for the weight of the Bullet sought . An Example of the Converse . If a Bullet whose Diameter is 4 Inches weigh 9 pound , another Bullet whose weight is 40 pound , what shall be the Diameter of it . Lay the thread to 40 in the Solids , and from 4 Inches in the lines take the nearest distance to it . Then lay the thread to 9 in the Solids , and enter the said extent at the equal Scale , so that the other foot turned about may but just touch the thread , and it it will rest at 6½ Inches nearest , which is the Diameter sought . Of the Line of inscribed Bodies . This Line hath these letters set to it . D Signifying the Sides of a Dodecahedron S Signifying the Sides of a _____ I Signifying the Sides of a Icosahedron C Signifying the Sides of a Cube O Signifying the Sides of a Octohedron T Signifying the Sides of a Tetrahedron And the Letter S Signifieth the Semidiameter of a Sphere , the use whereof are to find the Sides of the five Regular Bodies that may be inscribed in a Sphere . Example . A joyner being to cut the 5 Regular Bodies desires to know the lengths of the sides of the said 5 Regular Bodies that may be inscribed in a Sphere where Diameter is 6 inches . Lay the thread over S ▪ and take 3 inches out of the line of equal parts or Inches , and enter that extent so that one foot resting on the said Scale of inches , the other turned about may but just touch the thread , the resting point thus found , I call the point of entrance , from the said point take the nearest distances to the thread laid over the Letters .   Inch. Dec. parts D And measure those Extents on the Line of Inches , and you will find them to reach to 2.13 I 3.15 C 3.45 O 4.23 T 4.86 Which are the Dimensions of the respective sides of those Bodies to which the Letters belong . The uses of the Lines of quadrature , Segments , Mettals and Equated Bodies , I leave to the Disquisition of the Reader , when he shall have occasion to put them in practice , which I think will be seldome or never ; and wherein the assistance of the Pen will be more commendable . These lines were added to this quadrant to fill up spare room , and to shew that what ever can be done on the Sector , may be performed by them on a quadrant . A TABLE Of the Latitude of the most eminent Places in England , Wales , Scotland and Ireland .   d. m. Bedford 52 8 Barwick 55 54 Bristol 51 27 Buckingham 52   Cambridge 52 12 Canterbury 51 17 Carlisle 55   Chichester 50 48 Chester 53 16 Colchester 51 58 Derby 52 58 Dorchester 50 40 Durham 54 50 Exceter 50 43 Gilford 51 12 Gloucester 51 53 Hartford 51 49 Hereford 52 7 Huntington 52 19 Ipswich 52 8 Kendal 54 23 Lancaster 54 10 Leicester 52 40 Lincolne 53 14 London 51 32 Northampton 52 14 Norwich 52 42 Nottingham 53   Oxford 51 46 Reading 51 28 Salisbury 51 4 Shrewsbery 52 47 Stafford 52 52 Stamford 52 38 Truero 50 30 Warwick 52 20 Winchester 51 3 Worcester 52 14 Yorke . 53 58 WALES d. m. Anglezey 53 28 Barmouth 52 50 Brecknock 52 1 Cardigan 52 12 Carmarthen 51 56 Carnarvan 53 16 Denbigh 53 13 Flint 53 17 Llandaffe 51 35 Monmouth 51 51 Montgomeroy 51 56 Pembrooke 51 46 Radnor 52 19 St. David 52 00 The ISLANDS . d. m Garnzey 49 30 Jersey 49 12 Lundy 51 22 Man 54 24 Portland 50 33 Wight Isle . 50 39 SCOTLAND . d. m. Aberdean 57 32 Dunblain 56 21 Dunkel 56 48 Edinburgh 55 56 Glascow 55 52 Kintaile 57 44 Orkney Isle 60 6 St. Andrewes 56 39 Skirassin 58 36 Sterling . 56 12 IRELAND . d. m. Autrim 54 38 Arglas 54 10 Armach 54 14 Caterlagh 52 41 Clare 52 34 Corke 51 53 Droghedah 53 38 Dublin 53 13 Dundalk 53 52 Galloway 53 2 Youghal 51 53 Kenny 52 27 Kildare 53 00 Kings towne 53 8 Knock fergus 54 37 Kynsale 51 41 Lymerick 52 30 Queens towne 52 52 Waterford 52 9 Wexford . 52 18 A Table of the right Ascensions and Declinations of some of the most principal fixed Stars for some yeares to come .   R. Ascension . Declination . Magnitude   H m D. m.   Pole Star 00 31 87 34 N 2 Andromedas Girdle 00 50 33 50 N 2 Whales Belly 01 35 12 S 3 Rams head 1 48 21 49 N 3 Whales mouth 2 44 2 42 N 2 Medusas head 2 46 39 35 N 3 Perseus right side 2 59 48 33 N 2 Buls eye 4 16 15 46 N 1 Goat 4 52 45 37 N 1 Orions left foot 4 58 8 38 S 1 Orions left shoulder 5 6 5 59 N 3 First , in Orions girdle 5 15 00 35 S 3 Second , in Orions girdle 5 19 1 27 S 3 Third , in Orions girdle 5 23 2 9 S 3 Orions right shoulder 5 36 7 18 N 2 The Wagoner 5 39 44 56 N 2 Bright foot of the Twins 6 18 16 39 N 3 Great Dog 6 30 16 13 S 1 Castor or Apollo 7 12 32 30 N 2 The little Dog 7 22 6 6 N 2 Pollux or Hercules 7 24 28 48 N 2 Hidra's heart 9 10 7 10 S 1 Lions heart 9 50 13 39 N 1 Lions Neck 9 50 21 41 N 3 Great Beares rump 10 40 58 43 N 2 Lions back 11 30 22 4 N 2 Lions tail 11 31 16 30 N 1 The Virgins girdle 12 38 5 20 N 3 First in the great Bears taile next the rump 12 38 57 51 N 2 Vindemiatrix 12 44 15 51 N 3 Virgins Spike 13 7 9 19 S 1 Middlemost in the Great Beares tail 13 10 56 45 N 2 Last in the end of the Great Beares tail 13 34 51 05 N 2 Arcturus 14 00 21 03 N 1 South Ballance 14 32 14 33 S 2 Brightest in the Crown 15 24 27 43 N 3 North Ballance 14 58 08 03 S 3 Serpentaries left hand 15 56 02 46 S 3 Scorpions heart 16 08 25 35 S 1 Serpentaries left knee 16 18 09 46 S 3 Serpentaries right knee 16 49 15 12 S 3 Hercules head 16 59 14 51 N 3 Serpentaries head 17 19 12 52 N 3 Dragons head 17 48 51 36 N 3 Brightest in the Harp 18 25 38 30 N 1 Eagle or Vultures heart 19 34 08 00 N 2 Upper horn of Capricorn 19 58 13 32 S 3 Swans tail 20 30 44 05 N 2 Left shoulder of Aquarius 21 13 07 02 S 3 Pegasus mouth 21 27 08 19 N 3 Right shoulder of Aquarius 21 48 01 58 S 3 Fomahant 22 39 31 17 S 1 Pegasus upper Wing , or Marchab 22 48 13 21 N 2 Pegasus Lower Wing . 23 55 33 25 N 2 Mr. Sutton knowing that some of the Tables of Declination and Right Ascension in our English Books are antiquated and removed forward , took the pains to Calculate a new Table of Right Ascensions and Declinations to serve for the future , in regard I was not at leisure to accomplish it ; which followeth . Dayes . January ☉ R A. ☉ Decl. H. M. D. M. 1 19 35 21 46 2 19 39 21 36 3 19 43 21 25 4 19 47 21 14 5 19 51 21 03 6 19 56 20 52 7 20 00 20 40 8 20 04 20 27 9 20 09 20 15 10 20 13 20 01 11 20 17 19 48 12 20 22 19 34 13 20 26 19 20 14 20 30 19 05 15 20 34 18 50 16 20 38 18 35 17 20 42 18 19 18 20 46 18 03 19 20 50 17 47 20 20 54 17. 30 21 20 58 17 13 22 21 03 16 56 23 21 07 16 39 24 21 11 16 21 25 21 15 16 03 26 21 19 15 44 27 21 23 15 26 28 21 27 15 07 29 21 31 14 48 30 21 35 14 28 31 21 38 14 09 Dayes . February ☉ R A. ☉ Decl. H. M. D. M. 1 21 42 13 49 2 21 46 13 29 3 21 50 13 08 4 21 54 12 48 5 21 58 12 28 6 22 02 12 06 7 22 06 11 45 8 22 10 11 24 9 22 14 11 03 10 22 17 10 41 11 22 21 10 19 12 22 25 9 57 13 22 29 9 35 14 22 33 9 13 15 22 36 8 51 16 22 40 8 26 17 22 44 8 06 18 22 48 7 43 19 22 52 7 20 20 22 55 6 57 21 22 59 6 34 22 23 03 6 11 23 23 06 5 48 24 23 10 5 24 25 23 13 5 01 26 23 17 4 37 27 23 21 4 14 28 23 25 3 51 29         30         31         Dayes . March ☉ R. A. ☉ Decl. H. M. D. M. 1 23 28 3 27 2 23 32 3 03 3 23 36 2 39 4 23 39 2 16 5 23 43 1 52 6 23 46 1 29 7 23 50 1 05 8 23 53 0 41 9 23 57 0 18 10 0 01 North 6 11 0 05 0 30 12 0 08 0 53 13 0 12 1 17 14 0 15 1 41 15 0 19 2 04 16 0 23 2 28 17 0 26 2 51 18 0 30 3 15 19 0 33 3 38 20 0 37 4 01 21 0 41 4 24 22 0 44 4 48 23 0 48 5 11 24 0 52 5 34 25 0 55 5 57 26 0 59 6 19 27 1 03 6 42 28 1 06 7 04 29 1 10 7 27 30 1 14 7 49 31 1 17 8 11 Dayes . April . ☉ R. A. ☉ Decl. H. M. D. M. 1 1 21 8 33 2 1 25 8 55 3 1 29 9 17 4 1 33 9 38 5 1 36 9 51 6 1 40 10 21 7 1 44 10 42 8 1 47 11 03 9 1 51 11 24 10 1 54 11 44 11 1 58 12 05 12 2 02 12 24 13 2 06 12 45 14 2 10 13 04 15 2 13 13 24 16 2 17 13 43 17 2 21 14 02 18 2 25 14 21 19 2 29 14 40 20 2 32 14 58 21 2 36 15 16 22 2 40 15 34 23 2 44 15 52 24 2 48 16 09 25 2 51 16 27 26 2 55 16 43 27 2 59 17 00 28 3 03 17 16 29 3 07 17 32 30 3 10 17 48 31         Dayes . May ☉ R. A. ☉ Decl. H. M. D. M. 1 3 14 18 03 2 3 18 18 18 3 3 22 18 33 4 3 26 18 48 5 3 30 19 02 6 3 34 19 16 7 3 38 19 29 8 3 42 19 42 9 3 46 19 55 10 3 50 20 08 11 3 54 20 20 12 3 58 20 32 13 4 02 20 44 14 4 06 20 55 15 4 10 21 05 16 4 14 21 16 17 4 18 21 26 18 4 22 21 36 19 4 26 21 45 20 4 30 21 54 21 4 34 22 02 22 4 38 22 11 23 4 42 22 19 24 4 46 22 26 25 4 50 22 33 26 4 54 22 40 27 4 58 22 46 28 5 02 22 52 29 5 06 22 57 30 5 11 23 02 31 5 15 23 07 Dayes . June . ☉ R. A. ☉ Decl. H. M. D. M. 1 5 19 23 11 2 5 23 23 15 3 5 27 23 19 4 5 31 23 22 5 5 36 23 24 6 5 40 23 26 7 5 44 23 28 8 5 48 23 29 9 5 52 23 30 10 5 56 23 31 11 6 00 23 31½ 12 6 04 23 31 13 6 08 23 30 14 6 12 23 29 15 6 17 23 28 16 6 21 23 26 17 6 25 23 24 18 6 29 23 21 19 6 33 23 18 20 6 38 23 14 21 6 42 23 11 22 6 46 23 06 23 6 50 23 01 24 6 54 22 56 25 6 58 22 51 26 7 02 22 45 27 7 06 22 39 28 7 10 22 32 29 7 14 22 25 30 7 19 22 17 31         Dayes . July ☉ R. A. ☉ Decl. H. M. D. M. 1 7 23 22 09 2 7 27 22 01 3 7 31 21 52 4 7 35 21 43 5 7 39 21 34 6 7 43 21 24 7 7 47 21 14 8 7 51 21 04 9 7 55 20 53 10 7 59 20 42 11 8 03 20 30 12 8 07 20 18 13 8 11 20 06 14 8 15 19 54 15 8 19 19 41 16 8 23 19 28 17 8 27 19 14 18 8 31 19 00 19 8 35 18 46 20 8 39 18 32 21 8 43 18 17 22 8 47 18 02 23 8 51 17 46 24 8 55 17 31 25 8 58 17 15 26 9 02 16 59 27 9 06 16 42 28 9 10 16 25 29 9 14 16 08 30 9 17 15 51 31 9 21 15 33 Dayes . August ☉ R. A. ☉ Decl. H. M. D. M. 1 9 25 15 16 2 9 29 14 58 3 9 33 14 39 4 9 37 14 21 5 9 40 14 02 6 9 44 13 43 7 9 48 13 24 8 9 51 13 04 9 9 55 12 45 10 9 58 12 25 11 10 02 12 05 12 10 06 11 45 13 10 10 11 25 14 10 14 11 04 15 10 17 10 43 16 10 21 10 22 17 10 25 10 01 18 10 28 9 40 19 10 32 9 18 20 10 35 8 57 21 10 39 8 35 22 10 43 8 14 23 10 46 7 52 24 10 50 7 30 25 10 53 7 07 26 10 57 6 45 27 11 01 6 22 28 11 04 6 00 29 11 08 5 37 30 11 11 5 14 31 11 15 4 51 Dayes . September ☉ R. A. ☉ Decl. H. M. D. M. 1 11 19 4 28 2 11 23 4 6 3 11 26 3 42 4 11 30 3 19 5 11 33 2 56 6 11 37 2 33 7 11 41 2 10 8 11 44 1 46 9 11 48 1 23 10 11 51 0 59 11 11 55 0 3● 12 11 59 0 12 13 12 02 South 11 14 12 06 0 35 15 12 09 0 58 16 12 13 1 22 17 12 17 1 46 18 12 20 2 09 19 12 24 2 33 20 12 27 2 56 21 12 31 3 19 22 12 35 3 43 23 12 38 4 06 24 12 42 4 30 25 12 45 4 53 26 12 49 5 16 27 12 53 5 39 28 12 57 6 02 29 13 01 6 26 30 13 04 6 49 31         Dayes . October ☉ R. A. ☉ Decl. H. M. D. M. 1 13 08 7 11 2 13 12 7 34 3 13 15 7 57 4 13 19 8 19 5 13 22 8 42 6 13 26 9 04 7 13 30 9 26 8 13 34 9 48 9 13 38 10 10 10 13 41 10 31 11 13 45 10 53 12 13 49 11 14 13 13 53 11 36 14 13 57 11 57 15 14 00 12 18 16 14 04 12 38 17 14 08 12 59 18 14 12 13 19 19 14 16 13 39 20 14 20 13 59 21 14 24 14 19 22 14 28 14 38 23 14 32 14 57 24 14 36 15 16 25 14 39 15 35 26 14 43 15 5● 27 14 47 16 1● 28 14 51 16 29 29 14 55 16 47 30 14 59 17 04 31 15 03 17 21 Dayes . November ☉ R. A. ☉ Decl. H. M. D. M. 1 15 07 17 38 2 15 11 17 54 3 15 15 18 10 4 15 19 18 26 5 15 23 18 41 6 15 27 18 56 7 15 31 19 11 8 15 36 19 26 9 15 40 19 40 10 15 45 19 53 11 15 49 20 07 12 15 53 20 19 13 15 58 20 32 14 16 02 20 44 15 16 07 20 56 16 16 11 21 08 17 16 15 21 19 18 16 19 21 29 19 16 23 21 39 20 16 28 21 49 21 16 32 21 58 22 16 36 22 08 23 16 40 22 16 24 16 44 22 24 25 16 49 22 32 26 16 53 22 39 27 16 57 22 46 28 17 02 22 52 29 17 06 22 58 30 17 11 23 03 31         Dayes . December . ☉ R. A. ☉ Decl. H. M. D. M. 1 17 15 23 08 2 17 20 23 13 3 17 25 23 17 4 17 29 23 20 5 17 34 23 23 6 17 38 23 26 7 17 42 23 28 8 17 47 23 29 9 17 51 23 30 10 17 56 23 31 11 18 00 23 31½ 12 18 05 23 31 13 18 09 23 30 14 18 14 23 29 15 18 19 23 27 16 18 24 23 25 17 18 28 23 22 18 18 33 23 19 19 18 37 23 15 20 18 41 23 11 21 18 45 23 07 22 18 49 23 02 23 18 54 22 56 24 18 58 22 50 25 19 03 22 43 26 19 07 22 36 27 19 11 22 29 28 19 16 22 21 29 19 20 22 13 30 19 25 22 04 31 19 30 21 55 A Rectifying Table for the Suns Declination .   Years Years Years   1657 1661 1665 1669 1673 1659 1663 1667 1671 1675 1660 1664 1668 1672 1676 Moneths min. min. min. January 3 s 2 a 5 a 4 s 3 a 7 a 5 s 4 a 9 a February 5 s 5 a 10 a 5 s 5 a 11 a 6 s 5 a 11 a March 6 s 5 a 13 s 5 a 5 s 12 a 5 a 5 s 12 a April 5 a 5 s 11 a 5 a 5 s 10 a 4 a 4 s 9 a May 4 a 4 s 8 a 3 a 3 s 6 a 2 a 2 s 4 a June 1 a 1 s 2 a 0 s 0 a 0 s 1 s 1 a 3 s July 2 s 2 a 5 s 3 s 3 a 7 s 4 s 4 a 9 s August . 5 s 5 a 10 s 5 s 5 a 11 s 6 s 5 a 12 s Septēber 6 s 5 a 13 s 6 a 5 s 13 a 6 a 5 s 12 a October 6 a 5 s 12 a 5 a 5 s 11 a 4 a 5 s 9 a Novem. 3 a 4 s 7 a 2 a 3 s 5 a 1 a 2 s 3 a Decemb. 0 a 1 s 1 a 1 s 0 a 1 s 2 s 1 a 3 s The use of the Rectifying Table . NOte that the minutes under the respective years is to be added or substracted to or from the Suns Declination in the former Table , as is noted with the letter a or s : and also note that the first figure in each moneth stands for the first 10 dayes of the moneth , and the second for the second 10 days , & the third for the last 10 dayes , except in March or September , which in March will be the first 9 dayes only , and in September the first 12 dayes . Example . I would know the Suns Declination the 15 day of May 1668. Now because this day of the moneth falls in the second 10 dayes , I look in the Table under the year 1663 , and right against May you shall finde that in the second place of the moneth stands 6 a , which shews me that I must adde 6 minutes to the Suns Declination in the former Table 21 degrees 5 min. that stands against the 15 day of May , and then I find that the Sun will have 21 deg . 11 min. of North Declination , and so for the rest , which will never differ above two minutes from the truth , but seldome so much , and for the most part true . Note that the former Table of the Suns Declination is fitted exactly for the year 1666. by the Rules Mr. Wright gives in his Correction of Errours , and from his Tables , and may indifferently serve for the years 1658 , 1662. 1670 , 1674 , without any sensible errour , and the Table of Right Ascensions will not vary a minute of time in many years . FINIS . Errours in the Horizontal Quadrant . PAge 5 line 6 in an Italian letter should not have been distinct , nor in another letter from the former line . page 5. line 9. for quarter , read half . p. 5. l. 13. r. of a quadrant . p. 11. l. 7. r. 63 d. 26′ . p. 19. l. 7. r. the same day to . p. 23. l. 17. r. and ends at 32′ past 9. p. 27. l. 7. for N R , r. N Z. p. 28. l. 4. r. in the parallel . p. 30. l. 9 , & l. 10. r. 23 d. 31′ . p. 38. l. 4. r. Is to the sine . p. 50. l. 5. r. whereof the Diameter . AN APPENDIX Touching REFLECTIVE DIALLING . By JOHN LYON : Professor of this , or any other part of the Mathematicks , neer Sommerset House in the Strand . LONDON , Printed Anno Domini , 1658. DIRECT DIALLING By a Hole or Nodus . To draw a Dial under any window that the Sun shines upon by help of a thread fastened in any point of the direct Axis found in the Ceiling , and a hole in any pane of glasse , or a knob or Nodus upon any side of the window or window-post . CONSTRUCTIO . FIrst , draw on pastboard or other material , an Horizontal Dial for the Latitude proposed . Then by help of the Suns Azimuth , which may be found by help of a general Quadrant , at any time , or by knowing the true hour of the day with the help of the said Horizontal Dial : and draw that true Meridian from the hole or Nodus proposed , both above on the Cieling , and below on the walls and floor of the Room ; so that if a right line were extended from the said hole or Nodus by any point in any of those lines , it would be in the meridian Circle of the World. To finde a point in the direct Axis of the world , which will ever fall to be in the said Meridian , in which point the end of a thread is to be fastened . FIrst , fix the end of a thread or small silk in the center of the Hole or Nodus , and move the other end thereof up or down in the said meridian formerly drawn on the Cieling or wall , untill by applying the side of a Quadrant to that thread , it is found to be elevated equal to the Latitude of the place ; so is that thread directly scituated parallel to the Axis of the world , and the point where the end of that thread toucheth the meridian either on the Cieling or wall , is that point in the direct Axis sought for , wherein fix one end of a thread , ( which thread will be of present use in projecting of hour-points in any place proposed , then : To find the Hour-points either under the window , or any other convenient place in the Room . Place the center of the said Horizontal Dial in the Center of the Hole or Nodus ; also scituate the said Dial exactly parallel to the Horizon , and the meridian of the said Dial in the meridian of the world , which ( as before ) may easily be done , if at that instant you know the true hour of the day . ) Then take the thread whose end is fixed in a point in the direct Axis , and move it to and fro , until the said thread doth interpose between your eye , and the hour-line on the said Horizontal Dial which you intend to draw , and then keeping your eye at that scituation , make a point or mark in any place where you please , or under the window , so that the said thread or string may interpose between that point or mark so made , and your eye , as aforesaid ; which said point so sound will shew the true time of the day at that hour all the year long , the Sun shining thereon , so will that point , together with the said thread , serve to shew the hour , instead of an hour-line . In like manner , the said thread fixed in the Axis may be again moved to and fro , until the said thread doth interpose between the eye and any other hour-line desired on the said Horizontal Dial and then ( as before ) make another point or mark in any place at pleasure , or under the said window , by projecting a point from the eye , so that the said thread also interpose between that point to be made and the eye , so will that point so found shew the true time of the day for the same hour that did the hour line on the said Horizontal Dial , which was shadowed by the said thread . In like manner may be proceeded ( by help of that thread , and the several hour-lines on the said Horizontal Dial ) to finde the other hour-points which must have the same numbers set to them as have the hour-lines on the said Horizontal Dial. Otherwise to make a Dial from a hole in any pane of glasse in a window , and to graduate the hour-lines below on the Sell , or Beam , or on the ground , that hole is supposed to be the center of the Horizontal Dial , and being true placed , the stile thereof , if supposed continued , will run into the point in the Meridian of the Cieling before found , where a thread is to be fixed ; then let one extend a thread fastned in the center of the Horizontal Dial parallelly to the Horizon , over each respective hour-line , and holding it steady , let another extend the thread fastened in the Meridian , in the Cieling along by the edges of the former Horizontal thread , and so this latter thread will finde divers points on the ground , through which if hour-lines be drawn , and the Sun shine through the hole in the pane of Glasse before made , the spot of the Sun on the ground shall shew the time of the day . For the points that will be thus found on the Beam or Transome , the thread fixed in the Cieling , or instead of it a piece of tape there fixed must be moved so up and down , that the spot of the Sun may shine upon it , and being extended to the Transome or Beam graduated with the hour-lines , as before directed , it there shews the time of the day . Here note , that it will be convenient to have that pane of Glasse darkened through which that spot is to shine . In like manner may a Dial be made from a nail head , a knot in a string tied any where a crosse , or from any point driven into the bar of a window , and the hour-lines graduated upon the Transome or board underneath . To make a Reflected Dial on the Ceiling of the Room is onely the contrary of this , by supposing the Horizontal Diall with its stile to be turned downwards , and run into the true meridian on the ground , where the thread is to be fixed , and to be extended along by the former Horizontal thread ( held over the respective hours as before ) upward , to find divers points in the Cieling , as shall afterwards be shewed . Of Dials to stand in the Weather . These may be also made by help of an Horizontal Dial. DRive two nails or pins into the wall , on which the edge of a Board of competent breadth may rest , then to hold up the other side of the Board , drive two hooks into the wall above , whereto with cord or line the outside of the Board may be sustained , and this Board being Horizontal , place the Horizontal Dial its Meridian-line in the true Meridian of the world . If a Plain look towards the South , the stile of the Horizontal Dial continued by a thread from the center will run into the Plain , which note to be the center of the new Dial , as also that line is the new stile , which must be supported with stayes , when you fix it up . By a thread from the center laid over every hour-line on the Horizontal Dial , cross the Horizontal line of the Plain , which note with the same hours the Horizontal Dial hath . The hour-lines on the Plain are to be drawn from the center before found through those points , and so cut off by the Dial , or continued at pleasure . If the Center of the Dial be assigned before you begin the work , in such Cases you may remove the Horizontal Dial up and down , keeping it still to the true position or hour , till you finde the Axis or stile run into the Center . But if the Plain look into the East or West , then possibly the Axis of the Horizontal Dial will not meet with the Plain : in such Cases you must fix a board so , that it may receive the Axis , ( the board being perpendicular to the Plain ) this stile or Axis is to be fastened to the Plain by two Rests , the hour-lines may be drawn by the eye , or shadowed out by a Light : Bring the thread that represents the Axis or stile into any hour-point ( on the Horizontal Dial ) by your eye or shadow ; at the same time the thread or shadow making marks on the Plain , shews where the hour-lines are to passe . After the same manner any hour-line is to be drawn over any irregular or crooked Plain . Further observe , that any point in the middle , or neer the end of the stile will as well shew the hour of the Day , as the whole stile . Of Refracted Dials . IF you stick up a pin or stick , or assign any point in any concave Boul or Dish , to shew the hour , and make that the center of the Horizontal Dial , assigning the meridian-line on the edges of the Boul , point out the rest of the hour-lines also on the edges of the Boul , and taking away the Horizontal Dial , elevate a string or thread from the end of the said pin fastned thereto over the Meridian-line equal to the Elevation of the Pole or the Latitude of the place ; then with a candle , or if you bring the thread to shade upon any hour-point formerly marked out on the edges of the Boul , at the same time the shade in the Boul is the hour line . And if the Boul be full of water , or any other liquor , you may draw the hour-lines , which will never shew the true hour , unlesse filled with the said Liquor again . Reflected Dialling . To draw a Reflected Dial on any Plain or Plains , be they never so Gibous , and Concave , or Convex , or any irregularity whatsoever , the Glass being fixed at any Reclination at pleasure , ( provided it may cast its Reflex upon the places proposed . ) Together with all other necessary lines or furniture thereon , viz. the Parallels of Declination , the Azimuth lines , the Parallels of Altitude ( or proportions of shadows ) the Planetary Hour-lines , and the Cuspis of those Houses which are above the Horizon , &c. 1. If the Glasse be placed Horizontal upon the Transome of a window , or other convenient place : How upon the Wall or Cieling whereon that Glasse doth reflect to draw the Hour-lines thereon , although it be never so irregular , or in any form whatsoever . CONSTRUCTIO . FIrst , draw on Pastboard or other Material an Horizontal Dial for the Latitude proposed . Then by help of the Azimuth , or at the time when the Sun is in the Meridian ; or by knowing the true hour of Day , whereby may be drawn several lines on the Cieling , Floor , and Walls of the Room : so as in respect of the center of the Glasse they may be in the true Meridian-circle of the World : For if right lines were extended from the center of the said Glasse by any point , though elevated in any of those lines so drawn , it would be directly in the Meridian Circle of the World. Now all Reflective Dialling is performed from that principle in Opticks , which is , That the angle of Incidence is equal to the angle of Reflection . And as any direct Dial may be made by help of a point found in the direct Axis , so may any Reflected Dial be also made by help of any point found in the Reflected Axis . And in regard the reflected Axis for the most part will fall above the Horizon of the Glasse without the window , so that no point there can be fixed , therefore a point must be found in the said Reflected Axis continued below the Horizontal of the said Glasse , until it touch the ground or floor of the Room in some part of the Meridian formerly drawn , which point will be the point in the reversed Axis desired , and may be found , as followeth . One end of the thread , being fixed at or in the center of the said Glasse , move the other end thereof in the meridian formerly drawn below the said Glasse , until the said reversed Axis be depressed below the Horizon , as the direct Axis was elevated above the Horizon , which may be done by applying the side or edge of a Quadrant to the said thread , and moving the end thereof to and fro in the said meridian , until the thread with a plummet cut the same degree as the Pole is above the Horizontal Glasse , and then that point where the end of the thread toucheth the Meridian either on the floor or wall of the room , is the point in the reflected reversed Axis sought for . Now if the Reversed Axis cannot be drawn from the Glasse by reason of the jetting of the window or other impediment , that point in the reverse Axis may be found by a line parallel thereto , by fixing one end of it on the Glasse , and the other end in the meridian , so as that it may be parallel to the floor or wall in which the reversed Axis-point will fall , and finde the Axis point from that other end of the lath : so if the same Distance be set from that point backward in the Meridian on the floor , as is the Lath , the point will be found in the Reversed Axis desired . Thus having found a point in the reflected reversed Axis ; it is not hard , by help whereof and the Horizontal Dial , to draw the reflected hour-lines on any Cieling or Wall , be it never so concave or convex . To do which : First note , that all straight lines in any projection on any Plain , do always represent great Circles in the Sphere , such are all the hour-lines . Place the center of this Horizontal Dial in the center of the Glasse , the hour-lines of the said Dial being horizontal , and the Meridian of the said Dial in the Meridian of the world , which may be done by plumb lines let fall from the meridian on the Cieling : Then fix the end of a thread or silk in the said center of the Dial or Glasse , and draw it directly over any hour-line on the Dial which you intend to draw , and at the further side of the room , and there let one hold or fasten that thread with a small nail . Then in the point formerly found on the reversed Axis on the ●oor , fix another thread there ( as formerly was done in the center of the Diall ) then take that thread , and make it just touch the thread ( on the hour-line of the Horizontal Dial extended ) in any point thereof , it matters not whereabouts , and mark where the end of that thread toucheth the Wall or Cieling , and there make some mark or point . Then again move the same thread higher or lower at pleasure , till it , as formerly touch the said same hour thread , and mark again whereabouts on the wall or Cieling , the end of the said thread also toucheth . In like manner may be found more points at pleasure , but any two will be sufficient for the projecting or drawing any hour-line on any plain , how irregular soever . For if you move a thread , and also your eye to and fro , until you bring the said thread directly between your eye and the points formerly found , you may project thereby as many points as you please at every angle of the Wall or Cieling , whereby the reflected hour-line may be exactly drawn . Again , in like manner remove the said thread fastned in the center of the Horizontal Dial , ( which also is the center of the Glasse ) on any other hour-line desired to be drawn , and as before fasten the other end of the thread , by a small nail , or otherwise at the further side of the room , but so that the said thread may lie just on the hour-line proposed to be drawn on the Horizontal Dial. Then ( as before ) take the thread fastened in the point on the reflected Axis , and bring it to touch the thread of the hour-line in any part thereof , and mark where the end of that thread toucheth the said Wall or Cieling : Then again ( as before ) move the said thread so , as that it only touch the said thread of the hour-line in any other part thereof , and also mark where the end of that thread toucheth the said Wall or Cieling : So is there found two points on the Wall or Cieling , being in the reflected hour-line desired , by help of which two points the whole hour-line may be drawn ; for if ( as before ) a thread be so scituated , that it may interpose between the eye and the said two points found , you may make many points at pleasure , whereunto the said thread may also interpose , which for more conveniency may be made at every angle or bending of the Wall or Cieling , be they never so many : So that if lines be drawn from point to point , that said reflected hour-line will be also exactly drawn . In like manner may the other hour-lines be drawn so , that the Reflex or spot of the Sun from the said Horizontal Glasse scituated in the said window ( as before ) shining amongst the said reflected hour-lines drawn on the wall or Cieling , will exactly shew the hour of the day desired . Now if lines be drawn round about the said Room , equal to the Horizon of the said Glasse , it will shew when the Sun is in or neer the Horizon . To draw the Aequator and Tropicks on any Wall or Cieling to any Horizontal reflecting Glasse . 1 To draw the Reflected Aequator or Equinoctial-line on the Wall or Cieling , which represents a great Circle . TAke the thread fixed in the Center of the Glasse , and move the end thereof to and fro in the meridian line drawn on the Cieling , untill by help of a Quadrant the said thread be elevated equal to the complement of the Latitude , ( which will be alwayes perpendicular to the reversed Axis ) marking in the Meridian where the end of that thread falls , then on that point and the said meridian line on the Cieling erect a perpendicular line , which line may be continued on any plane whatsoever , and is the reflected Equinoctial line desired . Note that all great Circles are right lines , & are alwayes drawn or projected from a right line . 2. To draw the Tropicks . Note , that all Parallels of Declination are lesser Circles , and are Conick Sections . FIrst , make or take out of some Book a Table of the Suns Altitude for each hour of the day , calculated for the place or Latitude proposed , when the Sun is in either of the Tropicks . Then take the thread fixed in the center of the Glasse , and by applying one side of a quadrant to the said thread , and moving one end of it to and fro in the hour-line proposed , elevate the said thread answerable to the Suns height in that hour , when he is in that Tropick you desire to draw , and mark where the end of that thread so elevated toucheth in that hour-line proposed . So may you in like manner finde a several point in each hour-line for the Suns height in that Tropick , whereby a line may be drawn on the Wall or Cieling from point to point formerly made in the said hour-lines , which the Tropick desired . In like manner may any parallel of Declination be drawn : If there be first calculated a Table of the Suns altitude at all hours of the day , when the Sun hath any Declination proposed , whereby may be drawn either the Parallels of the Suns place , or the parallels of the length of the day . To draw the parallels of Declination to any Reflected Glasse most easily , by help of a Trigon first made on past board or other material . FIx the Trigon to the reflected roversed Axis , so that the center of the Trigon may be in the center of the Glasse , then will the Equinoctial on the Trigon be perpendicular to the said Axis : then take the thread fixed in the center of the Glasse , and lay it along either of the Tropicks or other parallels of Declination required , which is drawn on the said Trigon , which thread must be continued so , that the end thereof may touch any hour-line , and on that hour-line mark the point of touch , the thread being still laid on the same parallel of declination on the Trigon : in the same manner finde a point in each hour-line . Lastly , draw a line by those points so found , which will be the Tropick-line or other parallel of declination , as the thread was laid on , on the Trigon . To draw the Azimuth-lines on any Wall or Cieling to any Horizontal reflecting Glasse . Note that all Azimuths are great Circles . FIrst , find a vertical point , either above to the Zenith , or below to the Nadir of the Glasse ( by some called a perpendicular or plumb line ) and mark in what point it cuts the floor of the room , which point I call the reflected vertical point , wherein the end of a thread is to be fixed : For by a point found in the reflected Axis of the Horizon the Azimuths may be drawn , as by a point found in the reflected Axis of the Equinoctial the hour-lines may be drawn . Then on pastboard or other material draw the points of the Compasse or other degrees , and fix the center thereof in the center of the Glasse , and the meridian thereof in the meridian of the world , as was shewn in drawing the hour-lines , being careful to place it horizontal . Then take the thread fixed in the place of the glasse , and draw it over any Azimuth , which is desired to be drawn , and at the further side of the Room fasten that thread with a small nail as it was in drawing the reflected hour-lines : Then take the thread whose end is fastened in the said reflect vertical point , and bring that thread so as just to touch the said horizontal thread , and augment it , until the end thereof touch the wall or Cieling , and there make a mark or point . In like manner , move the said thread , whose end is fastened in the said vertical point , higher or lower at pleasure , till as formerly it touch the said horizontal thread , and mark again whereabouts the end thereof toucheth the said Wall or Cieling : Now by help of these two points found in the reflected Azimuth line , the whole Azimuth line may be drawn ; for if ( as before in drawing the Hour-lines ) a thread be so scituated , that it may interpose between the eye and the said two points , you may make many points at pleasure , to which the said thread so situated may also interpose , which may be made at every angle or bending of the wall or Cieling ( as before ) whereby the reflected Azimuth-line desired may be drawn . In like manner may the other reflected Azimuth lines be drawn . Also there may be lines drawn parallel to the Horizon round about the room , by help of the thread fixed in the center of the Glasse , and a Quadrant for the elevation thereof , which will shew the Suns altitude at any appearance thereof . Thus have I shewed the drawing of a Reflected Dial from an Horizontal Glasse , with all the usual furniture thereon , though the wall or place on which it is to be drawn be never so gibous or irregular , or in what shape soever . Now the Glasse may be exactly situated Horizontal , if you draw a reflected parallel for the present day , and know also the true hour , and so place the Glasse , that the spot or reflex of the Sun may fall thereon on the Cieling , for there is no way by an Instrument to do it , the Glasse is so small . Of Reclining Reflecting Glasses . Reflected Dialling from any Reclining Glasse . I shall now shew how to draw any Reflected Dial , with all the Furniture ( that possible may be ) the Glass being set at any possible Reclination . In the drawing of which there is principally to be considered , 1 The Reflected Horizon . 2 The Reflected Meridian . Note , the Horizon & Meridian are two great circles . 1 To draw the Reflected Horizon according to the situation of any reclining Glasse whatsoever . FIrst , let two pieces of nealed wire be fastened on the window on each side of the said Glasse , the ends thereof being without the room in the air , at whose ends let there be fastned a thread which may be pulled straight at pleasure , by bending of the wire , then bend those wires upward or downward , until the thread fastened at the end of each wire be exactly horizontal with the center of the Glasse , which may be tried by a quadrant : Then I tie a string or thread cross the room , in such sort that I may from most part of the thread see the reflecting glass , and therein the said horizontal thread without the room : Then on the said thread cross the room , I tie a slipping knot to move to and fro at pleasure , which knot I move to and fro on the said thread , until by looking in the said Glasse I finde from my eye the said knot and part of the horizontal thread without , all as it were in a right line , the one interposing the sight of the other . Then being careful to keep the knot in that position , fasten one end of a thread in the place of the center of the reclining reflecting glasse , and bring that thread so , as just to touch the aforesaid knot , augmenting that thread , until the end thereof touch the wall or Cieling , and there make a mark or point , so is there one point found on the Wall or Cieling in the Reflected Horizon of the World. Then I begin again , and remove the position of that thread ( which went overthwart the Room ) either higher or lower at pleasure , still having regard that I may from the most part of the said thread see the Reflecting Glasse , and therein the same horizontal thread without the room . Then , as before , I move the said knot on the said thread to & fro , until ( as before ) by looking in the said Glasse I find from my eye the said knot , and part of the Horizontal thread both in one right line , the one interposing the sight of the other ; and by the said knot I bring that thread , whose end is fastened in the center of the said glasse , and keeping it just to touch the said knot , I continue it , until the end thereof touch the Wall or Cieling , as before , and there I make another mark or point ; so is there two points found in the said reflected Horizon on the wall or Cieling . By which said two points , if a thread ( as before ) be so scituated , that it may interpose between the eye and the said two points , there may be many points made to be in the same interposition of the thread , which ( as before ) may be made at every bending or angle of the Wall or Cieling , whereby the reflected Horizon desired may be drawn , by drawing a line from point to point round about the Room ; Which wil be the true reflected Horizon according to the situation of the glasse . 2 To draw the Reflected Meridian , according to the situation of any Reclining Glasse whatsoever . FIrst , take a lath or thin piece of wood of any convenient length at pleasure , as some one and an half , or two foot long , and at each end thereof make a hole , the one to hang a thread and plummet , and the other is to put a small nail therin to fasten it in some part of the window over the center of the Glasse , so that the thread and plummet may hang without the room : then by help of the Suns Azimuth you may draw the meridian line , ( as before ) as if the Glasse were horizontal , and move the lath with the thread and plummet at the end of it to and fro , until the thread and plummet be in the direct meridian of the world with the center of the Glasse . Then ( as before ) tie a thread crosse the room , in such sort that from or by some part of the said thread both the Reclining glasse and the thread to which the plummet is fastened may be seen at one time . Then ( as before ) on the said thread , which crosses the room , I tie a slipping knot , which I move to and fro on the said string , until by looking in the said Glasse I find from my eye the said knot and some part of the perpendicular thread without , all as it were in one right line , the one shadowing or interposing the sight of the other , being then very careful to keep that knot in the same position , then take the thread ( whose end whereof being fastened in the said center of the Glasse ) and bringing it just to touch the said knot , I augment that thread , until the end thereof touch the said wall or Cieling , and the said thread also touch the knot , as before : then in that place where the end of the said thread toucheth the wall or Cieling , I make a mark , which mark or point will be directly in the reflected meridian of the world , according to the situation of that Glasse . Then again I remove that thread ( overthwart the room ) on which the said knot is , either higher or lower then it formerly was at pleasure , still having regard that from some part of the said thread within , you may see both the Reclining Glasse , and the perpendicular thread without at one time ; and ( as before ) move the said slipping knot on the said thread , until by looking in the said Reclining Glasse , you see the said knot and some part of the perpendicular thread without in one right line , so as the one shadows or hinders the sight of the other , ( as before ) which knot then must not be removed from its situation , then take that thread ( whose end is fastened in the Glasse ) and bring it to touch that knot , the end of the said thread being continued to touch the wall or Cieling : so is that point of touch on the Cieling another point found in the Reflected Meridian of the world . So is there two points found in the said Reflected Meridian , on the wall or Cieling ; by which , if a thread ( as before ) be so situated , that it may interpose between the eye and the said two points , many points thereby in the said reflected Meridian may be made at every bending or angle of the wall or Cieling , whereby the Reflected meridian desired may be drawn , by drawing a line from point to point obliquely in the Room , which will be the true Reflected Meridian of the world , according to the situation of that Glasse . Now this Reflected Horizon and Meridian being first drawn , they will be of great use in drawing the Hour-lines , together with all the furniture that possibly can be drawn on any Diall . To draw the Reflected Hour-lines to any Reclining Glasse on any plane whatsoever , that the Sun will be reflected on : By help of an ordinary Horizontal Dial for that Latitude . FIrst , extend several threads from the center of the Glasse to the extremity of the Reflected Horizon in the Room ( which for more conveniency and use may be the several hour-lines , and may also serve as a bed to situate the Horizontal Diall on the Reflected Horizon ) having regard to situate the center of the Dial on the center of the Glasse , and the Meridian of that Dial on the Reflected Meridian of the World : Then to finde the point in the Reflected reversed Axis on the floor of the Room ; Take a thread , one end thereof being fastened in the center of the Glasse , and move the other end thereof to and fro in the reflected meridian under the Reflected Horizon , until by help of a Quadrant the said thread is found to be depressed under the reflected Horizon , equal to the latitude of the place , and where the end of the said thread intersects or meets the Reflected Meridian either on the floor or wall , that point is the reflected reversed Axis , as was required . In which point fasten one end of a thread , which thread will be of great use in drawing the reflected hour-lines on any wall or Cieling whatsoever . Now if this thread , whose end is fastened in a point on the reflected reversed Axis , be taken and brought to touch any part of any one of the threads of the hour-lines ( produced to and fastened in the reflected Horizon ) the said thread being continued so , as the end thereof may touch the wall or Cieling , and also any part of the said thread touch the hour-line or thread proposed ; that point on the wall or Cieling is in the reflected hour-line desired to be drawn : Also the other point in the same reflected hour-line may be found ; If the said thread , whose end is fastened in the Reflected Axis , be brought to touch some other part of the same hour-thread proposed ; so that when ( as before ) the end of the said thread toucheth the wall or Cieling , some part of that thread may also touch the hour-line desired , which point of touch on the wall or Cieling , is also another point in the said reflected hour-line desired . By which two points so found ( as before ) the reflected hour-line may be drawn by a thread , projecting by those points from the eye , as it was formerly directed in drawing the reflected hour-lines to an Horizontal Glasse . To draw the Reflected Equinoctial line , and also the Tropicks on any wall or Cieling , to any Reclining Reflecting glasse . 1 To draw the reflected Equinoctial line on the Wall or Cieling . TAke that thread , whose end is fastened in the center of the reclining glasse , and move the other end thereof to and fro in the said Reflected meridian formerly drawn , until ( by help of a quadrant ) the said thread is elevated above the reflected Horizon formerly drawn , equal to the Complement of the Latitude , ( which as before will be alwayes perpendicular to the reversed Axis ) and make a point in the said reflected meridian , where the end of the said thread toucheth ; then on that point and the said reflected meridian on the Cieling , raise a perpendicular line , which is the Reflected Equinoctial line desired . 2. To draw the reflected Tropicks , or other Parallels of Declination . FIrst , ( as before ) make or take out of some Book a Table of the Suns Altitude for each hour of the day , calculated for the place or Latitude proposed , when the Sun is in either of the Tropicks , or other parallel of Declination : then take that thread , whose end is fastened in the center of the Glasse , move the other end thereof to and fro in the hour-line proposed , until by applying one side of a quadrant to the said thread you find the said thread elevated above the reflected Horizon answerable to the Suns height in that hour proposed , when he is in that Tropick or degree of Declination proposed . Which altitude required will be found in the foresaid Table for that end calculated , which said thread being of the elevation above the reflected Horizon , as the said Table directeth : then mark where the end of the thread ( so elevated ) toucheth the Wall or Cieling in that hour-line : so is one point found in the reflected parallel of Declination desired to be drawn . In like manner , find in the said Table in the same parallel or degree of declination what altitude the Sun hath at the next hour , and elevate the said thread , whose end is fastened in the center of the Glasse , equal to the Suns altitude in that hour above the said reflected Horizon , by help of the said Quadrant , and where the other end of the said thread falleth in the hour-line proposed , make another mark or point . And so in like manner make the points ( belonging to that parallel of Declination ) in the remaining hour-lines , according to the several Altitudes found in the said Table of Altitudes : Then drawing by hand a line to passe through those several points so found , as before , which line is the reflected parallel of the Suns declination desired . In like manner may be drawn all or any other parallel of Declination , which may have respect to the Suns place , or the length of the day , as shall be desired . Or , To draw the said reflected Tropicks , or other parallels of Declination , without any Tables calculated , only , by help of a Trigon first made on pastboard or other material . Note that all Parallels are lesser Circles . FIrst ( as formerly is shewd in drawing the parallels of Delination to a Reflecting Horizontal Glasse ) fasten the Trigon on the reflected reversed Axis , so that the center of the Trigon may be in the center of the Glasse , then also will the Equinoctial on the Trigon be perpendicular to the said reflected reversed Axis : then take the thread fixed in the center of the said Glasse , ( which is also in the center of the Trigon ) and lay it upon that parallel of Declination , drawn on the said Trigon , whose reflected parallel is required to be drawn on the plane or Cieling : then move the Trigon , the thread lying on the said parallel , until the end of the said thread touch any hour-line on the said wall or Cieling , in which point of touch on that hour-line make a mark , so will that point be in the reflected parallel of Declination desired . In like manner , move the said Trigon , still keeping the thread on the same parallel , until the end of that thread touch another hour-line on the said plane or Cieling , and there also make another mark . And so in like manner find a point in each hour-line through which that reflected parallel must passe ; then drawing a line to passe through those several points on the said plane or Cieling , which line is the reflected parallel of the Suns Declination desired . In like manner may be drawn any other reflected parallel of Declination required . To draw the reflected Azimuth-lines to any reclining Glasse , on any plane whatsoever that the Sun-beams will be reflected on . Here note that Azimuths are great Circles . FIrst , know that the reflected vertical point in the Axis of the Reflected Horizon , will alwayes be found in the reflected meridian . And look how many degrees the reflected Horizon differs from the direct Horizon , so many must the reflected Axis of the Horizon differ from the direct Axis of the Horizon : Hence the reflected vertical point , whereby the reflected Azimuth-lines are drawn , may be thus found . Take that thread whose end is fixed in the center of the Glasse , and move the other end thereof to & fro in the reflected meridian , until by applying one side of a quadrant thereto , you find the said thread depressed just 90 degrees , or perpendicular under the reflected Horizon ; then make a mark or point where the other end of the said thread toucheth the said reflected Meridian on the Wall , Ground , or Floor of the Room , which point so found is the reflected vertical point desired , in which point fasten one end of a thread : Then on pastboard or other material draw the points of the Compasse or other degrees , placing the center thereof in the center of the Glasse , and the meridian thereof in the reflected meridian of the world , which said pastboard must be also situated in the reflected Horizon just as the Horizontal Dial was formerly directed to be situated for drawing the reflected hour-lines : And as the threads from the center fastened in the reflected Horizon were also the hour-lines on the Horizontal Diall , whereby the reflected hour-lines were drawn . So now the threads from the center fastened in the Reflected Horizon may be the Horizontal Azimuth lines , whereby the reflected Azimuth-lines may be drawn : Or if that thread which fastned in the center of the glass be drawn exactly over any Azimuth-line , the end whereof being fastened by a nail or other means in the reflected Horizon on the other side of the Room , there may several points be found in the wall or Cieling , through which the reflected Azimuth line must passe , as followeth : Take that thread , one end of which is fastened in the said vertical point , and bring it just to touch the Azimuth thread formerly fastened , and continue it until the end thereof touch the wall or Cieling , ( and also the thread it self touch the said Azimuth it self , as before ) in which point of touch on the wall or Cieling make a mark , through which point that reflected Azimuth-line must passe . Then move the said string fastened in the said vertical point , so that it may just touch the said thread again , but in another place : then as before continue that thread , untill the end thereof touch the wall or Cieling again , as before , and there make another mark , through which the said reflected Azimuth line must also passe ; In like manner may more points be found for your further guide , in drawing that Azimuth-line . But two points being found will be sufficient . To draw any reflected line by any two points given over any plane whatsoever , without projecting by the eye . FAsten two threads in the place of the center of the said reclining Glasse , drawing the said threads straight , fastening each of the other ends in the two reflected Azimuth-points formerly found on the wall or Cieling . Then situate a thread cross or thwart the room , so as it may crosse those other threads from the center , neer at right angles , and also just touch both of them in that situation . By which said thread crosse the room may any number of points in the said reflected Azimuth-line to be drawn , be found at pleasure : For if the end of another thread be also fastened in the center of the said Glasse , making the other end thereof to touch the wall or Cieling , but so that it may also just touch the said thread , which is fastened crosse the room , which point of touch on the said wall or Cieling is another point in the said reflected Azimuth line required to be drawn . In like manner may more points be found at every angle or bending of the wall or Cieling for the exacter drawing the reflected Azimuth line required , which doth find points , whereby is drawn the same reflected Azimuth line ( or other lines ) as was formerly done by a thread so situated , that it may interpose between the eye and any two points assigned on the wall or Cieling . In like manner , if the thread fastened on the further side of the room were removed on another Azimuth line on the said pastboard , and then fasten it again on the further side of the room ( as before ) you may by help of the said thread fastened in the said vertical point find several points on the wall or Cieling , through which that Azimuth-line will passe ; So may you either by this or the former way draw what Azimuth lines you please , either in points of the Mariners Compasse or degrees , as you please , by drawing it first on pastboard , as before is directed . And note generally , that such relation the point found on the floor or ground in the reflected reversed Axis , hath to the hour-lines drawn on the Horizontal Dial , in drawing the reflected hour-lines ; The same hath the Reflected vertical point found on the floor or ground , to the Azimuths drawn on the pastboard in drawing the reflected Azimuth-lines . To draw the reflected parallels of the Suns altitude , or proportions of shadows to any reclining Glasse on any Plane whatsoever , that the Sun-beams will be reflected on . Here note , that parallels of Altitude are lesser Circles , therefore are not represented by a right line . FIrst , know generally that what respect the parallels of Declination have to the hour-lines , such have the parallels of Altitude to the Azimuths . For if one end of a thread be fastened in the place of the center of the reclining Glasse , and the other end moved to and fro in any reflected Azimuth line , until the said thread be elevated any number of degrees proposed above the reflected Horizon ( the Elevation of which thread being found , by applying a Quadrant thereto , and making a mark or point where the end of the said thread toucheth the said reflected Azimuth drawn on the wall or Cieling , that point so found is the point through which that Almican●er or reflected parallel of the Suns altitude must passe . In like manner , remove the other end of the said thread fastned in the center of the Glasse to another reflected Azimuth-line , and ( as before ) move it higher or lower , untill by applying the edge of a quadrant to that thread , you find the said thread above the reflected Horizon the same number of degrees first proposed , and at the end of the said thread in that Reflected Azimuth-line drawn on the wall or Cieling I make another mark or point , through which the same Reflected Almicanter or parallel of Altitude must also passe : And so in like manner I find a point on each reflected Azimuth-line , through which the same parallel of Altitude must passe . Then drawing by hand a line to passe through these several points so found , as before , that line is the Reflected parallel of the Suns Altitude proposed . In like manner may be drawn all the other parallels of Altitude desired , which will shew the Suns altitude or the Proportion of any shadow to its altitude , at any appearance of the Suns reflex thereon . To draw the Jewish or old unequal hour-lines to any Reclining Glasse on any plane whatsoever that the Sun-beams will be reflected on . Here note that the Jewish hour-lines are great Circles . FIrst , ( by the Rules formerly given ) draw two reflected parallels of Declination of 16 d. 55′ , the one being neer the Summer , and the other neer the Winter-Tropick : for when the Sun hath that Declination , the day is 15 hours long in the Summer , and 9 in the winter : Then ( as is formerly directed ) situate a thread just between the eye , and those three points in the said Reflected Dial , as is expressed in the insuing Table , so may you thereby draw all or any of those Jewish hour-lines desired , which will at any appearance of the spot by the reflex of the Glasse amongst those hour-lines , shew how many of the equal hours is past since Sun-rising , as was desired . Now in this Latitude of 51 deg . 30′ , If the parallels of the Suns declination be drawn , both when the day is 9 and 15 hours long , that is , when it is 16 d. 55′ , any of those Jewish hour-lines will intersect the common hour-lines , either upon the hours , half hours , or quarters . And such a declination may be found , that it shall so do in any Latitude desired . Unequal Hours . 15 H. M Equ . H. 9 H. M. 0 4 30 6 7 30 1 5 45 7 8 15 2 7 00 8 9 00 3 8 15 9 9 45 4 9 30 10 10 30 5 10 45 11 11 15 6 12 00 12 12 00 Unequ . hours 15 H. M. Equ . H. 9 H. M.             7 1 15 1 0 45 8 2 30 2 1 30 9 3 45 3 2 15 10 5 00 4 3 00 11 6 15 5 3 45 12 7 30 6 4 30 To draw the Circles of Position to any reclining Glasse on any plane whatsoever , that the Sun-beams will be reflected on . NOte that all Circles of Position are great Circles of the Sphere , and do alwayes intersect each other in that point of the Reflected meridian which toucheth the Reflected Horizon , which may be called the common intersection ; which said Circles of Position are reckoned upon the Reflected Equinoctial both wayes from the said meridian down to the said Horizon : The Horizon Eastward being the Cuspis of the first House , and the Horizon Westward being the Cuspis of the seventh House ; and the Reflected meridian the cuspis of the tenth House . So that those Meridian-planes , whose Reclination is 60 degrees Westwards , ( being measured from the meridian in the Equinoctial ) lies in the Cuspis of the eighth House , and 30 degr . Westward lies in the Cuspis of the ninth house , and 30 deg . Eastward in the Cuspis of the eleventh House , and 60 deg . Eastward in the Cuspis of the twelfth House : which are all the Houses above the Horizon . Now to draw any Circle of Position , or the Cuspis of any House on any Cieling or wall to any reclining Glasse is done as followeth : First , fasten a thread , in such sort , within the Room , as that it may interpose between the eye and the said common point of intersection on the wall or Cieling , and also between that point where the reflected hour-line of 4 ( being 60 deg . Westward from the said Meridian ) intersects the reflected Equinoctial also on the Cieling , whereby points may be made at every bending or angle of the wall or Cieling , to which the thread so situated may also interpose , by which points the Reflected Cuspis of the eighth House may be drawn . In like manner may the Cuspis of any other House above the Horizon , as the 9 th . or 10 th . which is the Meridian ( or Medium Coell ) or 11 th . or 12 th . be drawn also . For if ( as before ) the said thread be again so fastened within the Room , as that it may also interpose between the eye and the said common point of intersection , and also those points where the reflected hour-line of 2 ( being 30 deg . Westward from the said meridian ) do cut the reflected Equinoctial , whereby may be drawn the reflected Cuspis of the ninth House . Or where the Reflected hour-line of 10 ( being also 30 deg . Eastward from the meridian ) do also cut the said reflected Equinoctial , whereby may be drawn the Cuspis of the 11 th . House . Or where the reflected hour-line of 8 ( being 60 deg . Eastward from the meridian ) do also cut the said reflected Equinoctial , whereby may be drawn the Cuspis of the 12 th . House . The Horizon alwayes being the Cuspis of the first and seventh Houses , and the meridian the Cuspis of the tenth house or Medium Coeli : wherein generally it is to be noted , That in all planes which cut the common Intersection of the meridian and Horizon , ( as doth the Horizontal , and also all meridian planes both Direct and Reclining ) these Circles of Position are all parallel to the meridian , and therefore parallel each to other . For look what respect the hour-lines in all Direct or Reclining Polar Planes , or Direct meridian Planes have to the Axis of the World : Such respect have the Circles of Position , in all Horizontal , or Direct meridian or Reclining meridian Planes , to the Axis of the Prime vertical : For as the hour-lines in the first are all parallel to the Axis of the Equinoctial , in whose Poles they meet : So the Circles of Position in the second are all parallel to the Axis of the Prime Vertical , in whose Poles they also meet . The reason why Glasses reflect a double Spot , is because they are polisht on both sides , which may be remedied with a Pumex-stone . Those that desire to read more of this Subject may see what is written by Kircher , in primitiis Gnomicae Catoptricae , and since him by Magnan and others , VALE . FINIS .