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. 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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 ☞ ☞