the artificial clock-maker a treatise of watch, and clock-work, wherein the art of calculating numbers for most sorts of movements is explained to the capacity of the unlearned : also, the history of clock-work, both ancient and modern, with other useful matters, never before published / by w.d. derham, w. (william), - . approx. kb of xml-encoded text transcribed from -bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : - (eebo-tcp phase ). a wing d estc r ocm 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 . 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 , no. a ) transcribed from: (early english books online ; image set ) images scanned from microfilm: (early english books, - ; : ) the artificial clock-maker a treatise of watch, and clock-work, wherein the art of calculating numbers for most sorts of movements is explained to the capacity of the unlearned : also, the history of clock-work, both ancient and modern, with other useful matters, never before published / by w.d. derham, w. (william), - . 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users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf- unicode or text strings within braces); or lossless xml (tei p , characters represented either as utf- unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng clock and watch making. - tcp assigned for keying and markup - apex covantage keyed and coded from proquest page images - emma (leeson) huber sampled and proofread - emma (leeson) huber text and markup reviewed and edited - pfs batch review (qc) and xml conversion the artificial clock-maker . a treatise of watch , and clock-work : wherein the art of calculating numbers for most sorts of movements is explained to the capacity of the unlearned . also the history of clock-work , both ancient and modern . with other useful matters never before published . by w. d. m. a. london , printed for james knapton , at the crown in st. pauls church-yard , . the preface . the following book was at first drawn up in a rude manner , only to please my self , and divert the vacant hours of a solitary country life . but it is now published , purely in hopes of its doing some good in the world , among such , whose genius and leisure lead them to mechanical studies , or those whose business and livelihood it is . many there are , whose fault , or calamity it is , to have time lying upon their hands ; and for want of innocent , do betake themselves to hurtful pleasures . this is the too common misfortune of persons of quality . among some of the courser sort of these , if this book shall find some acceptance , it may be a means to compose their loose spirits ; and by an innocent guile , initiate them in other studies , of greater use to themselves , their family , and country . however it may hinder their commission of many sins , which are the effects of idleness . if there be any one person , in whom these good effects are produced , i shall think my idle hours well bestowed , and bless god for it . however upon the account of the innocence of my end in publishing this book , and that it was written only as the ha●●less ( i may add also the vertuous ) sport of leisure hours ; i think my self excusable to god and the world , for the expence of so much time , in a subject different from my profession . but besides , i think my self under some little obligations of justice and charity , to publish the ensuing papers for the sake of those , whose business the mechanick part is . i take it to be a charity to the trade ; because there are many ( altho excellent in the working part ) who are utterly unskilled in the artificial part of it . and then , it is a debt i pay : because i owe somewhat of health , as well as diversion to the study , and practice of these sort of mechanicks . and the best requital i can make for my trespass , is to publish what i have had better opportunities perhaps of learning , than many workmen have . and further yet , there is another rea& ; son , which much prevailed with me to publish this book , viz. because no body , that i know of , hath prevented me , by treating so plainly and intelligibly of this subject , as to be understood by a vulgar workman . i have often wondered at it , that so useful and delightful a part of mechanical mathematicks should lie in any obscurity , in an age wherein such vast improvements have been made therein , and when many books are daily published upon every subject . i speak here of this art remaining in obscurity ; not as if nothing was ever written of it , and i the in venter of automatical computation . but altho i cannot assume the glory of being the first writer upon this subject , yet very few have as yet done it ; of which i shall next give some account . cardan , kircher , and scottus promised it ; but i do not find they ever published any thing to the purpose of it . our great mr. oughtred i take to be the first that ever wrote to any purpose about the calculation of automata : and i believe he was the first that brought that art under rules , in his little treatise called automtaa . this book was first surreptitiously published in english in a little book , called horolog . dialogues , in the year ; and afterwards far more compleatly in latin , at the theatre in oxon , among mr. oughtred s opusc . mathem . in the year . this last edition it was my misfortune not to meet with , until it was too late , and therefore i have been forced to quote the first , and worst in my book . what mr. oughtred had wrapt up in ●his algebraick obscure characters , was afterwards put into plainer language ▪ by that excellent mathematician sir jon. moor , with some additions of his own ; which you have in his math. compend . and since him , by mr. leyborne , in his pleasure with profit . i hope i shall not be judged to have transgressed the rules of modesty , in coming af●r three such famous men ; neither should i venture that censure , but for two reasons . one is , i find by experience , that what they have written , is understood by very few workmen . and therefore i have endeavoured , with all industry , to make the matter as plain as i could for such . for which reason , i hope the more learned reader will excuse my using many words , when fewer would have served his turn ; and that i have condescended to low things , ( and to him needless ) as teaching the golden-rule , &c. the other reason is , that what those three have written , relates only , or chiefly to the watch-part . to which i have added several other things of my own : particularly the calculation of the clock-part , &c. i have been forced to reduce to rules my self , and to name no more , the historical part hath not been so much as attempted before , that i know of . these reasons will , i hope , excuse me with the most censorious reader , not only for presuming to write after so accurate a ▪ piece , as mr. oughtred s is , but also the novelty of the subject , will i hope procure for me a candid interpretation of the faults and blunders , that i may have unwittingly committed . to the preceeding account of what others have written ( which shews what help i have had from printed books ) i shall subjoyn my acknowledgments , and thanks to the principal of my friends , who have given me thei● assistance in compiling this book . but thei● names i shall not make more publick than mine own , being unwilling to be discovered my self . in the chap. of the terms of art , i owe much to the assistance of l. br .... a judicious workman in white-chappel , who drew me up a scheme of the clock-maker ▪ s language . in the history of the modern inventions , i have had ( among some others ) the assistance chiefly of the ingenious dr. h .... and mr. t .... ▪ the former being the author of some , and well acquainted with others , of the mechanical inventions of that fertile reign of king charles the ii. and the latter actually concerned in all , or most of the late inventions in clock-work , by means of his famed skill in that , and other mechanick operations . there are some other contrivances of this last age ( besides those i have mentioned ) which i have passed over in silence ; because either they are only branches , or improvements of the inventions i have taken notice of , ( such as several ways of repeating work , &c. ) or else , they only collaterally relate to watch-work ( as the inventions of cutting-engines , fusy-engines , &c. ) to treat of all these , would swell my book far beyond its intended bounds ; which i have already somewhat exceeded . i shall therefore commit this task to some better pen , hoping that no person will take it amiss , that i have not mentioned what i have been beholding to him for the relation of . for the resons last mentioned , i have also left out of my book , a chapter of the art of making , and using many sorts of sodders , the way of colouring metals , &c. useful in the practice of clock-work . this i had prepared for the sake of mercurial gentlemen , but omitted printing it , and some other things , out of charity to poor apprentices and other workmen , whose purses i am unwilling my volume should too much exceed . if i have at any time invaded the workman's province , it was not because i pretend to teach him his trade ; but either for gentlemen's sakes , or when the matter led me necessarily to it . i have nothing more to add , but that i would have this little treatise looked upon only as an essay , which i hope will prompt some abler pen to perform the task better , especially in the historical part . for since watch-work oweth so much to our age , and country , t is pity that it should not be remembred : especially when we cannot but lament the great defect of history , about the beginning and improvements of this ingenious and useful art. the contents . chap. i. of the terms of art. the more general terms . p. . names belonging properly to the watch-part . p. . names of the clock-part . p. . chap. ii. the art of calculation . sect. . preliminary rules . to find the turns of a wheel or pinion , . the way of writing down the numbers , . to find the turns of any , or all the wheels in the movement , . to find the beats of the ballance in all the watches going , or in one turn of any wheel , . two strokes to every tooth of the crown ▪ wheel , . sect. . calculation of the watch-part . several ways of performing one and the same motion , . a rule to vary numbers , . the way of working the golden rule , . a very useful rule to vary inconvenient numbers , . rules of perpetual use in proportioning the parts of a watch , . examples of contriving a piece of ordinary watch-work , . examples thereof for minutes , and seconds , . sect. . calculation of the striking-part . general observations and rules relating to the wheel-work of a clock , p. . rules of perpetual use in proportioning the parts of a clock , . examples of calculating the numbers of a small clock , . examples of clocks of longer continuance , . an useful rule to find the number of strokes in one turn of the fusy , . examples of fixing the pinion of report , . sect. . of quarters and chimes . notes concerning the quarters , . of making the chime-barrel , . of dividing it , and setting on the chime-pins , . chimes of psal . , and of a song-tune , . another way of setting chimes on the barrel , . sect. . to calculate numbers to represent the celestial motions . contrivance of movements only to shew these motions , . to add it to a watch that shews the hour of the day , . a motion to shew the day of the month , . to shew the age of the moon , . to shew the day of the year , and sun's place in the ecliptick , his rising or setting , &c. . to shew the tydes , ib. to represent the motion of the planets , fixed stars , &c. . chap. iii. to alter clock-work , p. . example of converting a hour ballance-clock into a pendulum , . to make it go hours , . to change the clock-part , . chap. iv. to size wheels and pinions . to do it arithmetically , . mechanically , . chap. v. of pendulums . irregularities of pendular motions remedied , . cause of the difference of the motion of the same pendulum , . true length of a pendulum that vibrateth seconds , . to find the center of oscillation , . to calculate the lengths , or vibrations of pendulums , . a table of lengths and swings , . to correct the motion of a pendulum , . chap. vi. the antiquity , and general history of watch-work . the ancientest time-engine , . the grecian and roman ways of measuring time , . some horological instruments mentioned by ancient authors , . watch , or clock-work , no new german invention , . the sphere of archimedes , . of po●idonius , . the beginning of our present clock-work , . clocks that perform strange feats , . chap. vii . the invention of pendulum watches . mr. hugens the inventer , p. . others claiming it , . their beginning in england , . the contriver of their carrying a heavy ball , &c. . their use , ibid. the circular pendulum , . chap. viii . of the invention of pocket pendulum watches . inventer , p. . several ways of them , ib. the time when invented , . mr. hugens's watch , . chap. ix . the invention of repeating clocks . the inventer , p. . when and by whom first used in pocket clocks , . chap. xi . numbers for various movements . the way of watch-makers writing down their numbers , . numbers of an day piece , . a month piece , . a two month piece , . a quarter of year piece , . an half year piece , ib. a year piece , . a lesser hours piece , ib. a small week piece , ib. a small month piece , . a small year piece , ib. an day piece pend. inches , . numbers representing the motion of the planet saturn , . of jupiter , ib. monsieur romer's instrument for jupiter's satellites , . numbers for mars , venus , and mercury , . for the dragons head and tail , . numbers for pocket watches of days , ib. of hours , , . the way to amend the numbers , . chap. xi . tables of time. a table for ready casting up the parts of time , . a table to set a watch by the fixed stars , . a table of the variations of the hour by the sun's refraction , . observations concerning refractions , and the variations of the hour , ▪ the artificial clock-maker . chap. i. of the terms of art , or names by which the parts of an automaton are called . it is necessary that i should shew the meaning of those terms which clock-makers use , that gentlemen and others , unskilful in the art , may know how to express themselves properly , in speaking ; and also understand what i shall say in the following book . i shall not trouble the reader with a recital of every name that doth occur , but only such as i shall have occasion to use in the following discourse , and some few others that offer themselves , upon a transient view of a piece of work . i begin with the more general terms : as , the frame ; which is that which contains the wheels , and the rest of the work . the pillars , and plates , are what it chiefly consists of . next for the spring , and its appurtenances . that which the spring lies in , is the spring-box ; that which the spring laps about , in the middle of the spring-box , is the spring-arbor ; to which the spring is hooked at one end . at the top of the spring-arbor , is the endless-screw , and its wheel . that which the spring draweth , and about which the chain or string is wrapped , and which is commonly taper , is the fusy . in larger work , going with weights , where it is cylindrical , it is called the barrel . the small teeth at the bottom of the fusy , or barrel , that stop it in winding up , is the ratchet . that which stops it when wound up , and is for that end driven up by the string , is the garde-caut , or guard-cock , as others ; and garde-du-cord , and gard-du-gut , as others call it . the parts of a wheel are , the hoop , or rim : the teeth : the cross : and the collet , or piece of brass , soddered on the arbor , or spindle , on which the wheel is rivetted . a pinion is that little wheel , which plays in the teeth of the wheel . its teeth ( which are commonly , , , , &c. ) are called leves , not teeth . the ends of the spindle , are called pevetts : the holes in which they run , pevet-holes . the guttered wheel , with iron spikes at the bottom , in which the line of ordinary house-clocks doth run , is called the pully . i need not speak of the dial-plate , the hand , screws , wedges , stops , &c. thus much for general names , which are common to all parts of a movement . the parts of a movement , which i shall consider , are the watch , and clock . the watch-part of a movement is that which serveth to the measuring the hours . in which the first thing i shall consider is the ballance : whose parts are , the rim , which is the circular part of it : the verge , is its spindle : to which belong the two pallets , or nuts , which play in the fangs of the crown ▪ wheel : in pocket-watches , that strong stud in which the lower pevet of the verge plays , and in the middle of which one pevet of the crown-wheel runs , is called the pottans : the wrought piece which covers the ballance , and in which the upper pevet of the ballance plays , is the cock. the small spring in the new pocket-watches is the regulator . the parts of a pendulum are , the verge ▪ pallets and cocks , as before . the ball in long pendulums , the bob in short ones , is the weight at the bottom . the rod , or wire is plain . the terms peculiar to the royal swing , are the pads , which are the pallets in others , and are fixed on the spindle . the fork is also fixed on the spindle , and about inches below , catcheth hold on the rod , at a flat piece of brass , called the flatt , in which the lower end of the spring is fastened . the names of the wheels next follow . the crown-wheel in small pieces , and swing-wheel in royal pendulums , is that wheel which drives the ▪ ballance , or pendulum . the contrate-wheel , is that wheel in pocket-watches , which is next to the crown-wheel , whose teeth and hoop lye contrary to those of other wheels . the great-wheel , or first-wheel , is that which the fusy , &c. immediately driveth . next it , are the second-wheel , third-wheel , &c. next followeth the work between the frame and dial-plate . and first , is the pinion of report ; which is that pinion which is commonly fixed on the arbor of the great-wheel , and in old watches used to have commonly but four leaves ; which driveth the dial-wheel , and this carrieth about the hand . the last part which i shall speak of , is the clock , which is that part which serveth to strike the hours : in which i shall first speak of the great ▪ or first-wheel ; which is that which the weight or spring first drives . in or hour clocks , this is commonly the pin-wheel ; in day pieces ▪ the second-wheel is commonly the ▪ 〈◊〉 this wheel with pins is sometimes called the striking-wheel , or pin-wheel . next to this striking-wheel , followeth the detent-wheel , or hoop-wheel , having a hoop almost round it , in whic● is a vacancy , at which the cloc● locks . the next is the third , or fourth ▪ wheel ( according as it is distant fro● the first-wheel ) called also the warning ▪ wheel . and lastly is the flying-pinion , with a fly or fan to gather air , and so bridle the rapidity of the clock's motion . besides these , there are the pinion o● report , of which before ; which driveth round the locking-wheel , called also the count-wheel , with notches in it commonly , unequally distant from one another , to make the clock strike the hour● of , , , &c. thus much for the wheels of the clock part . besides which there are the rash , or ratch ; which is that sort of wheel , of twelve large fangs , that runneth concentrical to the dial-wheel , and serveth to lift up the detents every hour , and make the clock strike . the detents are those stops , which by being lifted up , or let ●all down , do lock and unlock the clock in striking . the hammers s●rike the bell : the hammer-tails are what the striking-pins draw back the hammers by . latches are what li●t up , and unlock the work. catches are what hold by hooking , or catching hold of . the lifting-pieces do lift up , and unlock the detents , in the clock part . chap. ii. the art of calculation . sect . i. general preliminary rules and directions for calculation . § . for the more clear understanding this chapter it must be observed , that those automata ( whose calculation i chiefly intend ) do by little interstices , or strokes , measure out longer portions of time. thus the strokes o● the balance of a watch , do measure ou● minutes , hours , days , &c. now to scatter those strokes among wheels and pinions , and to proportionat● them , ●so as to measure time regularly is the design of calculation . for th● clearer discovery of which , it will be necessary to proceed leisurely , and gradually . § . and in the first place , you are to know , that any wheel being divided by its pinion , shews how many turns that pinion hath to one turn of that wheel . thus a wheel of teeth driving a pinion of , will turn round the pinion times in going round once . from the fusy to the ballance the wheels drive the pinions ; and consequently the pinions run faster , or go more turns , than the wheels they run in . but it is contrary from the great-wheel to the dial wheel . thus in the last example , the wheel drives round the pinion times : but if the pinion drove the wheel , it must turn times to drive the wheel round once . § . before i proceed further , i must shew how to write down the wheels and pinions . which may be done , either as vulgar fractions , or in the way of division in vulgar arithmetick . e. c. a wheel of moving a pinion of , may be set down thus , / : or rather thus , ) : where the first figure is the pinion , the next without the hook , is the wheel . the number of turns , which the pinion hath in one turn of the wheel , is set without a hook on the right hand : as ) ( , i. e. a pinion playing in a wheel of , moveth round times , in one turn of the wheel . a whole movement ma● be noted thus , / ●● / / / notches in the crown ▪ wheel . or rather as you see here in the margin : where the uppermost number , above the line , is the pinion of report , the dial-wheel , and turns of the pin. of report . the second number ( under the line ) is the pinion , is the great-wheel , and turns of the pinion it driveth . the third numbers , are the second-wheel , &c. the fourth the contrate-wheel , &c. an● the single number under all , is th● crown-wheel . § by the § . before , knowing th● number of turns , which any pinion hath in one turn of the wheel it worketh in ▪ you may also find out how many turns a● wheel or pinion hath , at a greater distance ; as the contrate-wheel , crown ▪ wheel , or &c. for it is but multiplying together the quotients , and the number produced , is the number of turns . an example will make what i say plain : let us chuse these numbers here set down ; the first of which hath turns , the next ▪ and the last . if you multiply and it produceth , for times is , that is , in one turn of the whee● , there are turns of the second pinion , or of the wheel . if you multiply by the last quotient ( that is , times is ) it shews the number of turns , which the third and last pinion hath . so that this third , and last pinion turns times in one turn of the first wheel . another example will make it still more plain the example is in the margin . the turns are , and . these multiplied as before run thus , viz. times is , that is , the pinion ( which is the pin. of the third wheel ) turns times in one turn of the first ▪ wheel . this last product being multiplied by , produces ▪ that is , the pinion ( which is the pin. of the crown-wheel ) turns times in one turn of the first-wheel , of teeth . § . we may now proceed to that , which is the very groundwork of all ; which is , not only to find out the turns , but the beats also of the ballance in those turns of the wheels . by the last paragraph , having found out the number of turns , which the crown-wheel hath in one turn of the wheel you seek for , you must then multiply those turns of the crown-wheel by its number of notches , and this will give you half the number of beats , in that one turn of the wheel . half the number , i say , for the reasons in the following § . for the explication of what hath been said , we will take the example in the last § : the crown-wheel there , has turns in one turn of the first wheel 〈◊〉 this number multiplied by , the notches in the crown-wheel , produceth , which are half the number o● strokes of the ballance , in one turn of the first wheel . the like may be done for any of the other wheels ; as the wheel , or : but i shall not insist upon these , having said enough . i shall give but one example more which will fully , and very plainly illustrate the whole matter . the example is in the margin , and 't is of a hour watch , wherein the pinion of report is , the dial-wheel , the great-wheel i● , the pinion of the secon●-wheel is , &c. the numbe● of notches in the crown ▪ wheel are : the quotients or number of turns in each , are , , . all which being multiplied as be ▪ fore , make : this number multiplied by , produceth ; which la●summ is half the number of beats in on● turn of the dial-wheel . the half number of beats in one turn of the great-wheel , you will find to be : for times is , which is the half number of beats in one turn of the contrate-wheel : and times , is , the half beats in one turn of the second-wheel : and times , is , the half beats in one turn of the great-wheel . and times this last , is before named . if you multiply this by the two pallets , that is , double it , it is , which is the number of beats in one turn of the dial-wheel , or hours . if you would know how many beats this watch hath in an hour , 't is but dividing the beats in hours , into parts , and it gives , the train of the watch , or beats in an hour . if you divide this into parts , it gives and a little more , for the beats in a minute . and so you may go on to seconds and thirds , if you please . thus i have delivered my thoughts as plainly as i can , that i may be well understood ; this being the very foundation of all the artificial part of clock-work . and therefore let the young practiser exercise himself thorowly in it , in more than one example . if i have offended the more learned , quick-sighted reader , by using m●●y words ; my desire to instruct the most ignorant artist ▪ must plead my excuse . § . the ballance or swing hath two strokes to every tooth of the crown-wheel . for each of the two pallets hath its blow against each tooth of the crown-wheel ▪ wherefore a pendulum that swings seconds , hath its crown-wheel but . sect . ii. the way to calculate , or contrive the numbers of a piece of watch work . having in the last section led on the reader to a general knowledge of calculation ; i may now venture him further into the more obscure , and useful parts of that art : which i shall explain with all possible plainness , tho less brevity , than i could wish . § . the same motion may be performed either with one wheel and one or by many wheels and many pinions : provided that the number of turns of all those wheels bear the same proportion to all those pinions , which that one wheel bears to its pinion . or ( which is the same thing ) that the number produced by multiplying all the wheels together , be to the number produced ●y multiplying all the pinions together ; as that one wheel is to that one pinion . thus suppose you had use for a wheel of teeth , with a pin. of leaves , you may make it into wheels and pinions , viz. ● , ) , and ) . for if you multiply the three wheels together , viz. , and ; and the three pinions together by themselves , viz , and , you will find to arise for the wheels , and for the pinions . or if you try the example by the number of turns , it will be the same . for ) ( / . and the quotients and turns of the wheels and pinions multiplied together , are / also , as in the last example . it matters not in what order the wheels and pinions are set , or which pinion runs in which wheel : only for convenien● sake , they commonly set the biggest numbers to drive the rest . § . two wheels and pinions of diff●rent numbers may perform the same m●tion . as , a wheel of drives a pinio● of , all one as a wheel of drives pin. of ; or as a wheel of drives pin. of . the turns of each are . § . if in breaking your train int● parcels ( of which by and by ) any of you● quotients should not please you ; or you would alter any other two number which are to be multiplied together , yo● may vary them by this rule : divid● your two numbers by any two oth● numbers which will measure them ; th● multiply the quotients by the alternat● divisors , the product of these two la● numbers found , shall be equal to the product of the two numbers first give● thus if you would vary times , d●vide these by any two numbers that wi● evenly measure them , as by , and by . the fourth part of is , and divided by gives . multiply by 〈◊〉 the product is ; and multiplied by 〈◊〉 produceth . so that for times ● you shall have found times . the operation is in the margin , that you may see , and apprehend it the better . these numbers are equal , viz. times is equal to times ; both producing . if you divide by , and by , and multiply as before is said , you will have for times , times , equal to also . if this rule seem to the unskilful reader hard to be understood , let him not be discouraged , because he may do without it , altho it may be of good use to him that would be a more compleat artist . § . because in the following paragraphs , i shall have frequent occasion to use the rule of three , or rule of proportion , it will be necessary to shew the unskilful reader , how to work this noble rule . if you find or numbers thus set , with four spots after the second of them , 't is the rule of proportion ; as in this example , . : : . . i. e. as is to : : so is to . the way to work this rule , viz. by the first number to find a fourth , is , to multiply the second number and the third together , and divide their product by the first . thus times is , which divided by , gives ; which is the number sought for , and stands in the fourth place . you will find the great use of this rule hereafter ; only take care to bear it in mind all along . § . to proceed . if in seeking for your pinion of report , or by any other means , you happen to have a wheel and pinion fall out with cross numbers , too big to be cut in wheels , and yet not to be altered by the former rules , you may find out two numbers of the same , or a near proportion , by this following rule , viz. as either of the two numbers given , is to the other : : so is to a fourth : divide that fourth number , as also by . . . . . . . . ( each of which numbers doth exactly measure ) o● by any one of those numbers that bringeth a quotient nearest to an integer ( or whole number . ) thus if you had these two numbers , the wheel , and the pinion , which are too great to be cut in small wheels , and yet can't be reduced into le●s , because they have no other common measure , but unity : say therefore according to the last paragraph , as is to ; or as is to : : so is to a fourth number sought . in numbers thus , : : . . or . : : . . divide the fourth number and by one of the foregoing numbers ; as and by , it gives and . in numbers 't is thus , ) ( ( divide by 't is thus , ) ( ( if you divide and by , it will fall out exactly to be and ) ( ( wherefore for the two numbers and , you may take and ; or and ; or and , o● &c. § . i shall add but one rule more , before i come to the practice of what hath been laid down ; which rule will be of perpetual use , and consists of these five particulars . . to find what number of turns the fusy will have , thus , as the beats of the ballance in one turn of the great-wheel or fusy ( suppose ) to the beats of the ballance in one hour ( suppose ) : : so is the continuance of the watches going in hours ( suppose ) to the number of the turns of the fusy . in numbers 't will stand thus , . : ▪ . . by § . you may remember tha● you are to multiply by , the product is . divide this by , and there will arise in the quotient , which must be placed in the fourth place ▪ and is the number of turns which the fusy hath . . by the beats and turns of the fusy ▪ to find how many hours the watch will go , thus , as the beats of the ballance in one hour , are to the beats in one turn of the fusy : : so is the number of the turns o● the fusy , to the continuance of the watches going . in num●ers thus , . : : ▪ . . to find the strokes of the ballance in one turn of the fusy , say , as the number of turns of the fusy , to the continuance of the watch's going in hours : : s● are the beats in one hour , to the beats o● one turn of the fusy . in numbers it 〈◊〉 thus , . : : ▪ . . to find the beats of the ballance in an hour , say thus , as the hours of the watch's going , to the number of turns of the fusy : : so are the beats in one turn of the fusy , to the beats in an hour . in numbers thus , . : : . . . to find what quotient is to be laid upon the pinion of report , say thus , as the beats in one turn of the great-wheel , to the beats in an hour : : so are the hours of the face of the clock ( viz. or ) to the quotient of the hour-wheel divided by the pinion of report , i. e. the number of turns , which the pinion of report hath in one turn of the dial-wheel . in numbers thus , . : : . . or rather ( to avoid trouble ) say thus , as the hours of the watch's going , are to the numbers of the turns of the fusy : : so are the hours of the face , to the quotient of the pinion of report ▪ in numbers thus , . : : . . if the hours of the face be , the quotient will be ; thus , . : : . . § . having given a full account of all things necessary to the understanding the art of calculation , i shall now reduc● what hath been said into practice , by shewing how to proceed , in calculating a piece of watch-work . the first thing you are to do , is to pitch upon your train , or beats of the ballance in an hour : as , whither a swi●● train , about beats ( which is the usual train of a common hour pocket ▪ watch ) or a slower train of about ( the train of the new pendulum pocket ▪ watches ; ) or any other train . having thus pitched upon your train , you must next resolve upon the number of turns you intend your fusy shall have ; and also upon the number of hours , you would have your piece to go : as suppose turns ; and to go hours , or hours ( which is days ) or &c. these things being all soon determined ; you next proceed to find out the beats of the ballance , or pendulum , in one turn of the fusy , by the last § . part . viz. as the turns of the fusy , to the hours of the watch's going : : so is the train , to the number of beats in one turn of the fusy . in numbers thus , . : : . . which last number are the beats in one turn of the fusy , or great-wheel ; and ( by sect. i. § . of this chap. ) are equal to the quotients of all the wheels unto the ballance , multiplied together . this number therefore is to be broken into a convenient parcel of quotients : which you are to do after this manner . first , half your number of beats , viz. , for the reasons in sect. i. § . of this chap. the half whereof is . next you are to pitch upon the number of your crown-wheel , as suppose . divide by , the quotient will be ( or to speak in the language of one that understands not arithmetick , divide into parts , and will be one of them . ) this is the number left for the quotients ( or turns ) of the rest of the wheels and pinions : which being too big for one or two quotients , may be best broken into three . chuse therefore numbers , which when multiplied all together continually will come nearest . as suppose you take , , and . now times is ; and times is , which is somewhat too much . you may therefore try again other numbers ▪ as suppose , , and . these multiplied as the last , produce ▪ which is as near as can be , and convenient quotients . thus you have contrived your piece ▪ from the great-wheel to the ballance ▪ but the numbers not falling out exactly according as you at first proposed ; you must correct your work thus . first to find out the true number of beats , in one turn of the fusy , you must multiply ▪ aforesaid , which is the true product of al● the quotients , by , the notches of the crown-wheel ; the product of this i● , which is half the number of true beats in one turn of the fusy , by sect. ● § . of this chap. then to find the tru● number of beats in an hour , say by § ▪ part . of this section , as the hours o● the watch's going , viz. , to the ● turns of the fusy : : so is the ha●● beats in one turn of the fusy , to ● the half beats in an hour : the numbe● will stand thus . : : . ● then to know what quotient is to b● laid upon the pinion of report , say by 〈◊〉 . part . of this sect : as the hours 〈◊〉 the watch's going , viz. , to the tur● of the fusy , viz. : : so are the hou●● of the dial-plate , viz. , to the quotient of the pinion of report . in numbers thus , . : : . . having thus found out all your quotients , 't is easie to determine what numbers your wheels shall have : for chuse what numbers your pinions shall have , and multiply the pinions by their quotients , and that produceth the numbers for your wheels , as you see in the margin . thus is the number of your pinion of report , and its quotient ; therefore times , which makes , is the number for the dial-wheel . so the next pinion being , and its quotient , this multiplied produces for the great-wheel . and the like of the rest of the following numbers . thus , as plain as words can express it , i have shewed how to calculate the numbers of a hour watch. § . this watch may be made to go a longer time , by lessening the train , and altering the pinion of report . suppose you could conveniently slacken the train to , the half of which is . then say ( by § part . of this sect. ) as the halfed train , or beats in an hour , viz. , to the halfed beats in one turn of the fusy , viz. : : so are the turns of the fusy , viz. , to the hours of the watch's going : in numbers thus , . : : . so that this watch will go hours . then for the pinion of report , say , by the same § part , as the continuance ; to the turns of the fusy : : so are the hours of the face , to the quotient of the pinion of report . in numbers thus , . : : . . the work is the same as before , as to the numbers ; only the dial-wheel is but , because its quotient is altered to ; as appears in the margin , by the scheme of the work . § . i shall give the reader one example more , for the sake of shewing him the use of some of the foregoing rules , not yet taken notice of in the former operations . suppose you would give numbers to a watch of about beats in an hour , to have turns of the fusy , to go hours , and notches in the crown-wheel . this work is the same as in the last example § . in short therefore thus , as the turns : are to the continuance : : so is the train , to , which are the beats in one turn of the fusy . the numbers will stand thus , . : : ● . half this last is ● . divide this half into parts , and will be for the quotients . and because this number is too big for quotients , therefore chuse : as suppose , , , and ⅗ ( i. e. and fifths ) these multiplied together as before , and with , maketh , which are half the true beats in one turn of the fusy . by this you are to find out your true train first ▪ saying as in the former example , as to : : so , to ; which last is the half of the true train of your watch. then for the pinion of report , say , as , to : : so , to / ● / / ● . which fraction ari●eth thus : if you multiply by it makes ; and divide by , you cannot ; but setting the ( the dividend ) over ( the divisor ) and there ari●eth this fraction ● / / ● / ● , which is a wheel and pinion ; the lower is the pinion of report , and the upper is the dial-wheel , according to sect. i. § . of this chapter . or ( which perhaps will be more plain to the unlearned reader ) you may leave those two numbers , in their divisional posture thus , ) , which does express the pinion and wheel , in the way i have hitherto made use of . but to proceed . these numbers being too big to be cut in small wheels , may be varied , as you see a like example is § . of this section : viz. say , as ▪ is to : : so is , to . or , as , to : : so is , to . in number thus , . : : . . or . : : . . divide , and either of these two fourth and last numbers by , , , , &c. ( as is directed in the rule last cited . ) if you divide by , you will have for your numbers / / / ● / ● / ● or / / . if you divide by ( which will not bring it so near an integer ) you will have / / or / ● / : which last are the numbers set down in the margin ▪ where the numbers of the whole movement are set down . § . having said enough , i think , concerning the calculation of ordinary watches , to shew the hour of the day : i shall next proceed to such as shew minutes and seconds . the process whereof is thus : first , having resolved upon your beats in an hour , you are next to find how many beats there will be in a minute , by dividing your designed train into parts . and accordingly you are to find out such proper numbers for your crown-wheel , and quotients , as that the minute-wheel shall go round once in an hour , and the second-wheel once in a minute . an example will make all plain . let us chuse a pendulum of inches to go days , with turns of the fusy . by mr smith's tables , a pendulum of inches vibrates in an hour . this divided by gives beats for a minute . half these summs are and . now the first work is to break this into good proportions ; which will fall into one quotient , and the crown-wheel . first , for the crown-wheel ; let it have notches . divide afore● by this , the quotient will be . a● so this first work is done : for a crow● wheel of , and a wheel a● pinion , whose quotient is ( ● in the margin ) will go rou● in a minute , to carry a ha● to shew seconds . next for a hand to go round in ● hour , to shew minutes . now becau● there are minutes in an hour , 't is b● breaking into two goo● quotients ( which may be ● and , or and ½ , or &c. and the work is done . thus your number , broken , as near as can be , int● proper numbers . but because it does not fall out exact into the above-mentioned numbers , yo● must correct ( as you were directed before ) and find out the true number ● beats in an hour , by multiplying by , which makes ; and this by ● makes , which is the half of the tru● train . then to find out the beats in on turn of thy fusy , operate as before , vi● as the number of turns , , to the co●tinuan● : : so is to , which are half the beats in one turn of ●he fusy . in numbers thus , . : : . . this must be di●ided by , which are the true ●umbers already pi●ched upon , or beats ▪ ●n an hour . the quotient of this division ●s , which being not too big for one single quotient , needs not be divided into more . the work will stand , as you see in the margin . as to the hour-hand , the great-wheel , which performs only one revolution in turns of the minute-wheel , will shew the hour . or rather you may order it to be done by the minute-wheel , ●s shall be shew'd hereafter . § . i shall add but one example more , and so conclude this section ; and ●hat is , to calculate the numbers of a ●iece whose pendulum swings ▪ seconds , ●o shew the hour , minutes , and seconds , ●nd to go days ; which is the usual per●ormance of those movements called ●oyal pendulums at this day . first , cast ●p the number of seconds in hours ( which are the beats in one turn of 〈◊〉 great-wheel ) these are times 〈◊〉 minutes , and times that , gives ● which are the seconds in hours . h● this number ( for the reasons before ) . the swing-wheel must ne● be , to swing seconds in one of 〈◊〉 revolutions . divide by it , a● is the quotient , or number left to 〈◊〉 broken into quotients . of these quo●ents , the first must needs be for 〈◊〉 great-wheel , which moves round on● in hours . divide by , 〈◊〉 quotient is ; which may be conve● ▪ ently broken into two quotients , as 〈◊〉 and , or and , or and ½ , whi● last is most convenient . a● if you take all the pinions the work will stand as in 〈◊〉 margin . according to this compu● ▪ tion , the great-wheel will 〈◊〉 about once in hours , shew the hour , if you please : the seco● wheel once in an hour , to shew the 〈◊〉 nutes ; and the swing-wheel once in a 〈◊〉 nute , to shew the seconds . thus i have endeavour'd with all possible plainness , to unravel this most mysterious , as well as useful part of watch-work . in which , if i have offended the more learned reader , by unartificial terms , or multitude of words , i desire the fault may be laid upon my earnest intent to condescend to the meanest capacity . sect . iii. to calculate the striking part of a clock . § . altho this part consists of many wheels and pinions , yet respect needs to be had only to the count-wheel , striking-wheel , and detent-wheel : which move round in this proportion ; the count-wheel moveth round commonly ●nce in , or hours . the detent-wheel moves round every stroke the clock striketh , sometimes but once in two strokes . from whence it follows , . that as many pins as are in the pin-wheel , so many turns hath the detent-wheel , in one turn of the pin-wheel . or ( which is the same ) the pins of th● pin-wheel are the quotient of that wheel , divided by the pinion of the deten●-wheel . but if the detent-wheel moveth but once round in two strokes o● the clock , then the said quotient is bu● half the number of pins . . as many turns of the pin-wheel a● are required to perform the strokes of ● hours ( which are ) so many tur●● must the pinion of report have , to turn round the count-wheel once . or thus ▪ divide by the number of striking pins , and the quotient thereof shall b● the quotient of the pinion of report . al● this is , in case the pinion of report b● fixed to the arbor of the pin-wheel , as i● very commonly done . all this i take to be very plain : or 〈◊〉 it be not , the example in the margin wil● clear all difficulties . her● the locking-wheel is ● the pinion of report is ● the pin-wheel is , th● striking-pins are . an● so of the rest . i need onl● to remark hero , that ● being divided by the pins , gives ● which is the quotient of● ▪ the pinion● of report : as was before hinted . as for the warning-wheel , and flying-pinion , it matters little what numbers they have , their use being only to bridle the rapidity of the motion of the other wheels . besides the last observation , there are other ways to find out the pinion of report , which will fall under the next § . § . these following rules will be of great use in this part of calculation , viz. rule . as the number of turns of the great-wheel , or fusy ; . to the days of the clock's continuauc● : : : so is the number of strokes in hours , viz. ▪ . to the strokes in one turn of the fusy , or great-wheel . rule . as the number of strokes in hours , which are , . to the strokes in one turn of the fusy , or great-wheel , : : so are the turns of the fusy , or great-wheel , . to the days of the clock's continuance , or going . rule . as the strokes in one turn of the fu●y , . to the strokes of hours , viz. . : : so is the clock's continuance , . to the number of turns of the fusy , or great-wheel . these two last rules are of no great use ( as the first is ) but may serve to correct your work , if need be , when in breaking your strokes into quotients ( of which presently ) you cannot come near the true number , but a good many strokes are left remaining . in this case , by rule . you may find whether the continuance of your clock be to your mind ▪ and by rule , you may enlarge or diminish the number of turns for this purpose . the praxis hereof will follow by and by . the following rules are to find fit numbers for the pinion of report , and the locking-wheel , besides what is said before § . inference . rule . as the number of strokes in the clock's continuance , or in all it● turns of the fusy , . to the turns of the fusy , : : so are the strokes in hours , which are , . to the quotient of the pinion of report , fixed upon the arbor of the great-wheel . but if you would fix it to any other wheel , you may do it thus , as is before hinted , viz. rule . first , find out the number of strokes , in one turn of the wheel you intend to fix your pinion of report upon ( which i shall shew you how to do in the following § . ) divide by this number , and the number arising in the quotient , is the quotient of the pinion of report . or thus . take the number of strokes in●●e turn of the wheel , for the number of the pinion of report , and for the count ( or locking ) wheel , and vary them to lesser numbers , by sect. . § . of this chapter . rule . the foregoing rules are of greatest use , in clocks of a larger continuance ; altho , where they can be applied , they will indifferently serve all . but this rule ( which will serve larger clocks too ) i add chiefly for the use of lesser pieces , whose continuance is accounted by hours . the rule is to find the strokes in the clock's continuance , viz. as , is to : : so are the hours of the clocks continuance , to the number of strokes in that time . this rule ( i said ) may be made use of for the largest clock ; but then you must be at the trouble of reducing the days into hours whereas the shortest way is to multiply the strokes in one turn of the great-wheel , by the number of turns . thus in an day piece the strokes in one turn are . these multiplied by , the turns , produce ; which are the strokes in the clock's continuance . if you work by the foregoing ruled the hours of days are . then say , ▪ : : . . § . in this paragraph , i shall shew the use of the preceding rules , and by examples make all plain that might seem obscure in them . i begin with small pieces : of which but briefly . and first , having pitched upon the number of turns , and the continuance , you must find , by the last rule , how many strokes are in its continuance . then divide these strokes by the number of turns , and you have the number of striking-pins . or divide by the number of pins , and you have the number of turns . thus a clock of hours , with turns of the great-wheel , hath strokes . for by the last rule , . : : . . divide by , it gives for the striking-pins . or if you chuse for your number of pins , and divide by it , it gives , for the number of ●urns , as you see in the margin . as for the pinion of report , and the rest of the wheels , enough is said in the § . but suppose you would calculate the numbers of a clock of much longer continuance , which will necessitate you to make your pin-wheel further distant from the great-wheel , you are to proceed thus : having re●olved upon your turns , you must find out the number of strokes in one turn of the great-wheel , or fusy , by § ▪ rule . thus in an day piece , of turns , . : : . . so in a piece of days , and turns , ▪ : : . . these strokes so found out , are the number which is to be broken into a convenient parcel of quotients , thus ; first resolve upon your number of striking-pins : divide the last named number by it : the quotient arising hence , is to be one , or more quotients , for the wheels and pinions . as in the last examples ▪ divide by ( the usual pins in an day piece ) and the quotient is ● ; which is a quotient little enough . so in the month-piece : if you take your pins ▪ divide by it , the quotient is . which being too big for one , must b● broken into two quotients , for wheels and pinions , or as near 〈◊〉 can be : which may b● and , or and ½ . th● latter is exactly , and may there●o● stand : as you see is done in the margin . the quotients being thus determined and accordingly the wheels and pinio●● as you see ; the next work is to find 〈◊〉 quotient for the pinion of report , to ●●ry round the count ( or locking ) wh●● once in hours , or as you please . ● you fix your pinion of report on th● great-wheel arbor , you must operate 〈◊〉 the rule . of the last paragraph . as 〈◊〉 the last example in the month-piece : 〈◊〉 rule . before , the strokes in the conti●●ance are . then by rule say , . : : . ● / ● or thus , ) . the first of which two numbers is the pinion , the next is the wheel . which being too large , may be varied to ●● / or ) ; or to ● / or ) , by sect. § . before . these numbers being not the usual numbers of a month-piece , but only made use of by me , as better illustrating the foregoing rules ▪ i shall therefore , for the fuller explication of what has been said , briefly touch upon the calculation of the more usual numbers . they commonly encrease the number of striking-pins , and so make the second-wheel the striking-wheel ▪ suppose you take pins ; divide by it , and the quotient is . which is little enough for one quotient ; and may therefore stand as you see is done in the margin : where the quotient of the first wheel is . in the second wheel of teeth , are the pins , altho its quotient is but , because the hoop-wheel is double , and goes round but once in two strokes of the pin-wheel . the pinion of report here , is the same with the last , if fixed upon the arbor of the great ▪ wheel . but if you fix it on the arbor of the second , or pin-wheel , its quotient then is found by § . infer . . or by § . rule . viz. divide by , and the number arising in the quotient , is the quotient of the pinion of report , which is ¼ . the pinion of report then being , the count-wheel will be , as in the margin . to perfect the reader in this part of calculation , i will finish this section with the calculation of a year-piece of clock-work . the process whereof is the same with the last , and therefore i may be more brief with this , except where i have not touched upon the foregoing rules . we will chuse a piece to go days with turns , and striking-pins . by § . rule . there are strokes in one turn of the great-wheel . for . : : . . this last number divided by the pins , leaves in the quotient , to be broken into two or more quotients , for wheels and pinions . these quotients may be and ; which multiplied , makes , which is as near as can well be , to . the work thus far contrived , will stand as you see in the margin . before you go any further , you may correct your work , and see how near your numbers come to what you proposed at first , because they did not fall out exact . and first , for the true continuance of your clock ▪ if you multiply , , and ( i. e. the quotients un ▪ o the stri●ing-pins , and those pins ) you have the true number of strokes , in one turn of the great-wheel : which , in this example , make . for times , is ; and times that , is . ( this direction i would have noted , and remembered , as a rule useful at any time to discover the nature of any piece of clock-work . ) having thus the true number of strokes desired , by § . rule . you may find the true continuance to be only days . for . : : . . if this continuance doth not please you , you may come nearer to your first proposed number , of days , by a small encrease of the number o● turns ; according to § . rule . viz. by making your turns almost ½ . for . : : . ½ almost . lastiy , for the pinion of report , if you fix it upon the great-wheel , it will require an excessive number : if you fix it upon the pin-wheel ( which is usual ) then by § . rule , the quotient is ; and the pinion of report being , the count-wheel will be ; as you see in the margin . but for the better exercising the reader , let us fix it upon the spindle of the second-wheel . it s quotient is ; which multiplied by ( the pins ) produceth ; which are the strokes in one turn of that second-wheel . then by § ▪ rule , divide by , i. e. set them as a wheel and pinion thus , ) , and vary them to lesser numbers ( by sect. ● § . ) viz. ▪ , or to ) , or th● like . i think it needless to say any thing o● pocket-clocks , whose calculation is the very same , with what goes before . that the unlearned reader may not think any thing going before difficult , i need only to advise him , to look over the working of the rule of proportion , in sect. . § . for i think all will be plain , if that be well understood . sect . . of quarters and chimes . the reader will expect that i should say somewhat concerning quarters and chimes : but because there is little , but what is purely mechanical in it , i shall say the less , and leave the reader to his own invention . § . the quarters are generally a distinct part from the clock-part , which striketh the hour . the striking-wheel may be the first , second , or &c. wheel , according to your clock's continuance . unto which wheel you may fix the pinion of report . the locking-wheel must be divided ( as other locking-wheels ) into , , or more unequal parts , ●o as to strike the quarter , and lock at the first notch ; the half-hour , and lock at the second notch , &c. and in doing this , you may make it to chime the quarters , or strike them upon two bells , or more . 't is usual for the pin-wheel ▪ or the locking-wheel , to unlock the hour-part in these clocks ; which is easily done by some jogg or latch , at the end of the last quarter , to lift up the detents of the hour-part . if you would have your clock strike the hour , at the half-hour , as well as whole hour , you must make the locking-wheel of the hour-part double : i. e. it must have two notches of a sort , to strike , , , , &c. twice apiece . § . as for chimes , i need say nothing of the lifting-pieces and detents , to lock and unlock ; nor of the wheels to bridle the motion of the barrel . only you are to observe , that the barrel must be as long in turning round , as you are in singing the tune it is to play . as for the chime-barrel , it may be made up of certain barrs , that run athwart it , with a convenient number of holes punched in them , to put in the pins , that are to draw each hammer . by this means , you may change the tune , without changing the barrel . this is the way of the royal exchange clock in london , and of others . in this case , the pins or nuts , which draw the hammers , must hang down from the barr , some more , some less , and some stand upright in the barr : the reason whereof is , to play the time of the tune rightly . for the distance of each of these barrs , may be a semi-brief , or &c. of which hereafter . but the most usual way is , to have the pins that draw the hammers , fixed on the barrel . for the placing of which pins , you may make use of the musical notes , or proceed by the way of changes on bells , viz. , , , , &c. the first being far the better way , i shall speak of that chiefly , especially because the latter will fall in to be explained with it . and first , you are to observe what is the compass of your tune , or how many notes or bells there are from the highest to the lowest : and accordingly you must divide your barrel from end to end . thus in the examples following , each of those tunes are notes in compass ; and accordingly the barrel is divided into parts . these divisions are struck round the barrel , opposite to which are the hammer-tails . i speak here , as if there was only one hammer to each bell , that the reader may more clearly apprehend what i am explaining . but when two notes of the same sound come together in a tune , there must be two hammers to that bell , to strike it . so that if in all the tunes you intend to chime , of notes compass , there should happen to be such double notes on every bell , instead of , you must have hammers : and accordingly you must divide your barrel , and strike strokes round it opposite to each hammer-tail . thus much for dividing your barrel from end to end . in the next place , you are to divide i● ( round about ) into as many divisions , as there are musical barrs , semibriefs , minums , &c. in your tune . thus the th psalm-tune hath semibriefs ; the song-tune following , hath barrs of triple time : and accordingly their barrels are divided . each division therefore of the th psalm barrel is a semibrief , ●nd of the song-tune 't is three crotchets ▪ and therefore the intermediate spaces serve for the shorter notes : as , one third of a division , is a crotchet , in the song-tune . one half a division , is a minum ; and one quarter a crotchet , in the psalm-tune . thus the first note in the th psalm , is a semibrief , and accordingly on the barrel , 't is a whole division from to . the second is a minum , and therefore is but half a division from ; and so of the rest . and so also for the song-tune , which is shorter time : the two first notes being quavers , are distant from one another , and from the third pin , but half a third part of one of the divisions . but the two next pins ( of the bell , ) being crotchets , are distant so many third parts of a division . and the next pin ( of the bell ) being a minum , is distant from the following pin ( ) two thirds of a division . a table of chimes to the psalm . the musical notes of psalm . the musical notes of , such command o're my fate , &c a song the chimes of the song , such command o're my fate , &c. pins , to be set on the barrel . you may observe in the tables , that from the end of each table to the beginning , is the distance of two , or near two divisions : which is for a pause , between the end of the tune , and its beginning to chime again . i need not say , that the dotts running about the tables , are the places of the pins that play the tune . if you would have your chimes compleat indeed , you ought to have a set of bells , to the gamut notes ; so as that each bell having the true sound of sol , la , mi , fa , you may play any tune , with its flats and sharps . nay , you may by these means , play both the bass and treble , with one barrel . if any thing going before appears gibberish , i can't help it , unless i should here teach the skill of musick too . as to setting a tune upon the chime-barrel from the number of bells , viz. , , , , i shall here give you a specimen thereof . such command o're my fate , in numbers . , , , . , , , . , . , , , , , , , , , , . ‖ , , , , . , ; , , . , , , , ; , , , , . , , , , . , , , . , ; , ; , , , , . note , in these numbers , a comma [ , ] signifies the note before it , to be a crotchet . a prick'd comma , or semi-colon [ ; ] denoteth a prick'd crotchet . and a period [ . ] is a minum . where no punctation is , those notes are quavers . i shall only add further , that by setting the names of your bells at the head of any tune ( as is done in the tables before ) you may easily transfer that tune , to your chime-barrel , without any great skill in musick . but observe , that each line in the musick , is three notes distant ; i. e. there is a note between each line , as well as upon it : as is manifest by inspecting the tables . sect . . to calculate any of the celestial motions . the motions i here chiefly intend , are the day of the month , the moons age , the day of the year , the tides , and ( if you please ) the slow motion of the suns apogaeum , of the fixed stars , the motion of the planets , &c. § . for the effecting these motions , you may make them to depend upon the work already in the movement ; or else measure them by the beats of a ballance , or pendulum . if the latter way , you must however contrive a piece ( as before in watch-work ) to go a certain time , with a certain number of turns . but then to specificate , or determine the motion intended , you must proceed one of these two ways : either , . find how many beats are in the revolution . divide these beats by the beats in one turn of the wheel , or pinion , which you intend shall drive the intended revolution ; and the quotient shall be the number to perform the same . which , if too big for one , may be broken into more quotients . thus , if you would represent the synodical revolution of the moon , which is days , ¾ hours ) with a pendulum that swings seconds , the movement to go days , with turns of the fusy , and the great-wheel to drive the revolution . divide ( the beats in days ¾ hours ) by ( the beats in one turn of the great-wheel ) and you will have in the quotient : which being too big for one , may be put into two quotients . or . you may proceed as is directed before , in the section of calculating watch-work , viz. chuse your train , turns of the fusy , continuance , &c. and then instead of finding a quotient for the pinion of report , find a number ( which is all one as a pin. of report ) to specificate your revolution , by this following rule . rule . as the beats in one turn of the great-wheel . to the train : : so are the hours of the revolution , to the quotient of the revolution . thus to perform the revolution of saturn ( which is years , days ) with a hour watch , of beats in one turn of the fusy , and , the train : the quotient of the revolution , will be . for , as , to : : so ( the hours in y. and d. ) to . note here , that the great-wheel pinion is to drive the revolution work . but if you would have the revolution to be driven by the dial-wheel , and the work already in the movement ( which in great revolutions , is for the most part , as nice as the last way , and in which i intend to treat of the particular motions ) in this case , i say , you must first know the days of the revolution . and because the dial-wheel goeth round twice in a day , therefore double the number of the days in the revolution , and you have the number of turns of the dial-wheel in that time . this number of turns is what you are to break into a convenient number of quotients , for the wheels and pinions ▪ as shall be shewed in the following examples . § . a motion to shew the day of the month. the days in the largest month are . these doubled are , which are the turns of the dial-wheel , which may be broken into these two quotients ½ and ; which multiplied together make . therefore chusing your wheels and pinions , as hath been directed in the former sections , your work is done . the wheels and pinions may be , as you see done in ▪ the margin . or if a larger pinion than one of be necessary , by reason it is concentrick to a wheel , you may take for the pinion , and for the wheel , as in the margin . the work will lye thus in the movement , viz. fix your pinion , concentrical to the dial-wheel ( or to turn round with it upon the same spindle . ) this pinion drives the wheel : which wheel has the pinion in its center , which carrieth about a ring of teeth , divided on the upper side into days . or , you may , without the trouble of many wheels , effect this motion ; vi● . by a ring divided into or days , and as many fangs or teeth , like a crown ▪ wheel teeth , which are caught and pushed forward once in hours , by a pin in a wheel , that goeth round in that time . this is the usual way in the royal pendulums , and many other clocks ; and therefore being common , i shall say no more of it . § . a motion to shew the age of the moon . the moon finisheth her course ▪ so as to overtake the sun , in days , and a little above an half . this ½ days ( not regarding the small excess ) makes twelve hours , or turns of the dial-wheel , which is to be broken into convenient quotients : which may be , and as in the first example ; or ¾ and , as in the second example in the margin . so that if you fix a pinion of concentrical with your dial-wheel , to drive a wheel of ( according to the last example ) which wheel drives a pinion , which carries about a ring , or wheel of teeth , divided on the upper side into ½ 't will shew the moons age . § . a motion to shew the day of the year , the sun's place in the ecliptick , sun's rising or setting , or any other annual motion of days . the double of is , the turns of the dial-wheel in an year : which may be broken into these quotients , viz. ¼ , and , and , according to the first example ; or ¼ , , and , according to the second . so that a pinion of is to lead a wheel of ; which again by a pinion of , leadeth a wheel of ; which thirdly , by a pinion of , carrieth about a wheel , or ring of , divided into the months , and their days ; or into the signs , and their degrees ; or into the sun's rising and setting , &c. for the setting on of which last , you have a table in mr. oughtred's opuscula . § . to shew the tides at any port. this is done without any other trouble , than the moon 's ring ( before mentioned § . ) to move round a fixed circle , divided into twice hours , and numbered the contrary way to the age of the moon . to set this to go right , you must find out at what point of the compass the moon makes full sea , at the place you would have your watch serve to . convert that point into hours , allowing for every point north or s. lost ′ of an hour . thus at london-bridge 't is vulgarly thought to be high tide , the moon at n. e. and s. w , which are points from the n. and s. or you may do thus : by tide-tables learn how many hours from the moon 's southing , 't is high-water . or thus ; find at what hour it is high-water , at the full or change of the m●on : as at london-bridge , the full tide is commonly reckoned to be hours from the moon 's southing ; or at of clock at the full and change. the day of conjunction , or new-moon , with a little stud to point , being set to the hour so found , will afterwards point to the hour of full tide . this is the usual way ; but it being always in motion , as the tides are not , a better way may be found out , viz. by causing a wheel , or ring to be moved forward , only twice a day , and to keep time ( as near as can be ) with mr. flamsteed's most correct tables . but this i shall commit to the readers contrivance , it being easie , and more of curiosity than use . § . to calculate numbers , to shew the motion of the planets , the slow motion of the fixed stars , and of the sun's apogeum , &c. having said enough before that may be applied here , and they being only curiosities , seldom put in practice , i shall not therefore trouble the reader , or swell my book with so many words , as would be required to treat of these motions distinctly , and compleatly . only thus much in general . knowing the years of any of these revolutions , you may break this number into quotients ; if you will make the revolution to depend upon the year's motion ; which is already in the movement , and described § . before . or if you would have it depend upon the dial-wheel , or upon the beats of a pendulu● , enough is said before to direct in mis matter . in all these slow motions , you may somewhat ●●●●ten your labour , by endless screws to serve for pinions , which are but as a pinion of one tooth . sir jonas moor's account of his large ●phere going by clock-work , will suffi●ently illustrate this paragraph . in this ●phere , is a motion of years , for ●he sun's apogeum , performed by six ●heels , thus , as sir jonas relates it ; for the great-wheel fixed is , a spindle-wheel of bars turns round it times in hours , that is , in hours ; after these , there are four wheels , , , , and , wrought by endless screws that are in value but one : therefore , , , , and multiplied together continually , produceth hours , which divided , by gives days , equal to years . now on the last wheel is a pinion of , turning a great wheel , that carrieth the apogeum number : and divided by , gives the quotient : and times is years . thus i have , with all the perspicuity i ●ould , led my reader through the whole ●rt of calculation , so much of it at least , ●at i hope he will be master of it all ; not ●ly of those motions , which i have par●cularly treated about , but of any other ●t mentioned : such as the revolution of the dragons head and tail , whereby the eclipses of the sun and moon are found , the revolution of the several orbs , according to the ptolemaick system , or of the celestial bodies themselves , according to better systems , with many other such curious performances , which have made the sphere of archimedes of old famous : and since him , that of william of zeland , and another of janellus turrianus of cremona , mentioned by cardan : and of late , that elaborate piece of mr. watson , late of coventry , now of london , in her late majesties closet . chap. iii. to alter clock-work , or convert one movement into another . this chapter i design for the use of such , as would convert old ballance clocks into pendulums , or would make any old work serve for the tryal of new motions , or would apply it to any other such like use . § . to do this , you may draw a scheme of your old work : and so you will see what quotients you have , and what you will want . to do all which , there are sufficient instructions in the preceding chapter . a few instances will make all plain . § . let us chuse for instance an old ballance clock to be turned into a pendulum of inches . the old work is , the great-wheel , the pinion ; the next wheel , the pinion ; the crown-wheel , &c. the scheme of this work is in the margin . the quotients and crown-wheel and pallets multiplied together continually , produce , which are the strokes of the ballance , in one turn of the great-wheel , by sect. l. § , . of the last chapter . and by the quotient of the dial-wheel ( which is ) it appears , that the great-wheel goeth round once in an hour . or you may find the beats in an hour , by § . last cited . having thus found the beats in an hour , of the old work , you must next find the beats in an hour , of a inches pendulum ; which you may do by chap. . § . following ; or by mr. smith's tables , according to whom the number is ▪ . divide this by , and you have the quotient , which is to be added to the scheme of the old work . this quotient is and near ½ ▪ as you see in the margin . the work thus altered , will stand as you see in the margin , viz. a pinion , and a contrate-wheel , must be added . according to this way , the old work will stand as before , only the crown-wheel must be inverted . § . but because the crown-wheel is too big for the contrate-wheel ( which is unseemly ) therefore it will be best , ▪ to make both the contrate , and crown-wheels new ; and encrease the number of the contrate-wheel , but diminish that of the crown-wheel . to do which , pitch upon some convenient number for the crown-wheel . multiply all the quotients , and this new crown-wheel number , as before ; and divide by it . as , suppose you pitch upon for the crown-wheel : if you multiply , and ▪ the product is ; which multiplied by the pallets , makes , which are the beats in one turn of the great-wheel , or in an hour . divide by it , and you have near for the quotient of your contrate-wheel . the work thus ordered , will stand as in the margin . if you would correct your work , to find the true number of beats in an hour , &c. you must proceed , as is shewn sect. . § , and latter end of § . of the last chapter . § . but suppose you have a mind to change the former old watch , into a hour piece , and to retain the old ballance-wheel ( which may be often done : ) in this case , you must add a contrate-wheel , and alter the pinion of report . for the contrate-wheel , chuse such a quotient , as will best suit with the rest of your work ; and then multiply all your quotients , crown-wheel and ● pallets together , and so find the number of turns in the great-wheel , as before . then say by sect. . § . part . before , as the beats in one turn of the great-wheel , to the beats in an hour : : so are the hours of the dial , to the quotient of the pinion of report . thus in the old work before ; to the old quotients and , you may add another of , for the contrate-wheel . those multiplied , as was now directed , make , for the beats in one turn of the great-wheel . and then for the quotient of the pinion of report , say in numbers thus , . : : . . the quotient for the pinion of report is somewhat more than , which overplus may be neglected , as you see by the scheme of the whole work in the margin . if you desire to know what number of turns , the fusy must have in this work ; say by the last quoted § ▪ part , in numbers thus , . : : . almost . so that near turns will do . if you would correct your work , to know the exact beats , &c. you are referred to directions in the end of the last paragraph . § . i shall add but one thing more , to what hath been said in this chapter , and that is , to change the striking part of this old movement , into a hour piece . a scheme of the old work is in the margin . and to alter it , the best way is , to double the number of striking pins , making the , sixteen pins : and the hoop of the detent-wheel double , that the pin-wheel may strike two strokes , in its going round once . the greatest inconvenience here , will be to bridle the rapidity of the strokes ; because a quotient of ( only , added to the old work , will be sufficient for this purpose : which being an inconvenient number , 't will be necessary to be content with the old numbers , or make more wheels and pinions new , than may be thought worth the while . if you would find what number of turns , the fusy will require ; you must find how many strokes are in hours , by sect. . § . r. . before . these are ; which divided by the pins , gives somewhat more than turns of the fusy . lastly , for the pinion of report , you must pursue the directions in the last quoted place , r. . the work thus altered , will stand as in the margin . chap. iv. to size the wheels and pinions , or proportion them to each other , both arithmetically and mechanically . § . for the exact and easie moving of the wheels and pinions together , it is necessary that they should fit each other , by having their teeth and leaves of the same wideness , or near of the same wideness . for many do make the leaves of the pinion narrower than the teeth of its wheel , by reason of their running deep in each other ; which is as if the diameters of the wheel and pinion were less . but this i leave to those , whose practice and observations are greater than mine in these matters . § . to make the teeth of a wheel and pinion alike , the way arithmetically is thus , first you must find the circumference of your wheel and pinion ; which you may best do by the rule of three ( so often made use of before ) the rule is thus , as is to : : so is the diameter to the circumference . or more exactly thus , as , is to , : : so diam . to circum . suppose you have a wheel of inches diameter , and teeth , and would fit to it a pinion of leaves . first : : . , . the circumference of the wheel , is then inches , and tenths of an inch . then say , as the teeth of the wheel , to the circumference of it : : so are the leaves of the pinion , to the circumference thereof . in numbers thus ▪ , : : ▪ , . the pinion then is hundredth parts of an inch round . now to find the diameter , 't is but the reverse of the former rule , viz. as . to : : so the circumference to the diameter . in numbers thus , for the foregoing pinion , . : : , ▪ , the diameter then of the pinion must be two tenths of an inch , to fit the aforesaid wheel of inches diameter . sect . but because this way may be difficult to persons unacquainted with decimal arithmetick , which is very necessary here ; therefore i shall set down a way to do it mechanically . having drawn a circle , divide it into as many parts , as you intend leaves in the pinion you would size . from two of these points in the circle , draw two lines to the center : to which apply two of the teeth of your wheel , guiding them up and down until they touch at the same width on these radii . mark where this agreement is , and a small circle drawn there , will represent the circumference of the pinion sought after . chap. v. of pendulums . sect . among all known motions , none measureth time so regularly , as that of a pendulum . but yet watches governed hereby are not so persect , but that they are subject to the variations of weather , foulness , &c. and the shorter , and lesser the pendulum is , so much the more subject such watches are to these annoyances . there are two ways to obviate these inconveniences in some measure . one way is , to make the pendulum long , the bob heavy , and to vibrate but a little way from its settlement . which is now the most usual way in england . the other is the contrivance of the ingenious mr. hu●ens , which is , to make the upper part of the rod , play between two cheek parts of ● cycloid . sir jonas moor says , that af●er some time , and charge of experiments , he believes this latter to be the better way . and mr hugens calls it admirable . if any desire to know how to make those cycloidal cheeks , fit to all pendulums , i refer him to the aforesaid mr. zulichem's book , because i can't shew how to do it , without the trouble of figures ; and this way is much ceased , since the crown-wheel method ( to which it is chiefly proper ) is swallowed up by the royal pendulums . sect . another thing to be remark'd in pendulums is , that the longer the vibration is , the ●lower it is . for if two isochrone pendulums do move , one the quadrant of a circle , the other not above or degrees , this latter shall move some-what quicker than the former . which is the true reason , why small crown-wheel pendulums go faster in cold weather , or when soul , than at other times . yea , in the best royal pendulum , if you put a divided plate behind the ball , and observe its swings , you may perceive the vibrations to be sometimes shorter ; and that then the watch doth gain too much ▪ somewhat also may perhaps be attributed to the rarity or density of the air ; which i have not yet had an opportunity of observing , by comparing with a good baroscope , the various vibrations of a good royal pendulum . but mr. boyl says , that a pendulum moveth as long , and as fast in a thick medium , as a thin one ; contrary to the opinion of some naturalists , who think the contrary . his opinion is grounded upon the experiment of a pendulum vibrating in his air-pump , the air sucked out , and in the open air ; wherein was no alteration . sect . for the calculation of all pendulums , 't is necessary to fix upon some one , to be as a standard to the rest . i pitch upon a pend ▪ to vibrate seconds each stroke . mr. hugens lays down the length of a pend. to swing seconds to be feet , inches , and tenths of an inch ( according to sir j. moor's reduction of it to english measure . ) the honourable lord bru●cker ( saith sir jonas ) and mr. rook , found the length to be , inches , which a little exceeds the other : and may be , was justened by mr. hugens's rule for the center of oscillation . for mounton's pendulum , that ▪ vibrate times in a minute , it will be found likewise , inches , agreeing to , inches english . therefore for certain , inches may be called the vniversal measure , and relied on , to be the near length of a pend. that shall swing seconds each vibration . but forasmuch as the different size of the ball , will make some difference in the length of this standard pend. , therefore to make this pend. an vniversal measure , to fit all places and ages , you must measure from the point of suspension , to the center of oscillation . which center is found by this rule , as the length of the string from the point of ▪ suspension to the center of a round ball : is to the semidiameter of a round ball : is to the semidiameter , to a fourth number . add two fifths of that fourth number , to the former length , and you have the center of oscillation ; and thereby the true length of this standard pendulum . if it be desired to fit a ball of a triangular , quadrangular , or any other form to this pend , the center of oscillation in any of these bodies , may be found in the last cited book of mr zulichem . if it be asked , what is the meaning of the center of oscillation ? the most intelligible answer ( altho not perfectly true ) is , that it is that point of the ball , at which if you imagine it divided into two parts , by a circle , whose center is in the point of suspension , the lower part of the ball shall be of the same weight ( or near so ) with the upper . § . having thus fixed a standard , i shall next shew how from thence to find the vibrations , or lengths of all other pendulums . which is done by this rule , the squares of the vibrations , bear the same proportion to each other , as their length● do . and so contrary wise . wherefore to find the length of a pend ▪ say ▪ as the square of the vibrations given : to the square of ( the standard ) : : so is the length of the standard ( viz. , ) to the length of the pend. sought . if by the length , you would find the vibrations , 't is the reverse of the last rule , viz. as the length proposed : to the standard ( , ) : : so is the square of ( the vibrations of the standard ) : to the vibrations sought . suppose for example , you would know what length a pend. is , that vibrates strokes in a minute . the square of ( i. e. times ) is . say , . : : , . . a pend. then that vibrates in a minute , is about inches long . on the other hand , if you would know how many strokes a pend. of inches hath in a minute ; say , . , : : . . the square root whereof is , and somewhat more . note , because is always the product of the two middle terms multiplied together , therefore you need only to divide this number by the square of the vibrations , it gives the length sought : by the length , it gives the square of the vibrations . if you operate by the logarithms , you will much contract your labour . for if you seek the length , 't is but substracting the logarithm of the square of the vibrations , out of the logarithm of , which is . , and the remainder is the logarithm of the length sought . if you seek the vibrations , it is but substracting out of the aforesaid logarithm . , the logarithm of the length given , and half the residue is the logarithm of the vibrations required . the following examples will illustrate each particular . to find the length .   logarithms . . squared is . length is more than . . to find the vibrations .   logarithms . . inches long . square of the vibr. . square-toor , or numb . of vibr. is , and somewhat more . . according to the foregoing directions ▪ i have calculated the following table ▪ to pendulums of various lengths : and have therein shewed the vibrations in a minute , and an hour , from to inches . if any desire a more minute account , i refer him to mr smith's tables in his late book . the reason why his calculation and mine differ , is because he measureth the length of the pend. from the point of suspension , to the lower part of the bob ; and i only to the center of the bob. his standards are ½ inches , and inches ; and mine is , , for the reasons aforegoing . a table of swings in a minute , and in an hour , to pendulums of several lengths . pend. length in inches vibrat . in a minute . vibrat . in an hour . , , , , , , , , , , , , , . , , , , , , , the use of this table is manifest , and needs no explication . as to the decimals in the column of minute-swings , i have ●dded them for the sake of calculating the column of hour-swings ; which would have been judged false without them , and would not have been exactly true without them . § ▪ i have but one thing more to add to this chap. of pendulums , and that is , ●o correct their motion . the usual way is , to screw up , or let down the ball. in doing of which , a small alteration will make a considerable ●ariation of time : as you will find by calculation , according to the last paragraph . to prevent the inconvenience of ●crewing the ball too high , or low , mr smith hath contriv'd a very pretty table ●or dividing the nut of a pendulum screw , ●o as to alter your clock but a second in ●day . but by reason no screw and nut can be so made , as to be most exactly strait ●nd true , therefore it may happen , that instead of altering your watch to your mind , you may do quite contrary ; as ●nstead of letting the ball down , you may raise it higher , by the false running of the nut upon the screw . considering this irremediable inconvenience , i am of opinion , that mr hugens's way would do very well , added to this . his way is , to have a small weight , or bob , to slide up and down the pend. rod , above the ball ( which is immoveable . ) but i would rather advise , that the ball be made to screw up and down , to bring the pend. pretty neer its gauge : and that this little bob should serve only for more nice corrections ; as the alteration of a second , or &c. which it will do , better than the great ball. for a whole turn of this little bob , will not affect the motion of the pend. near so much as a small alteration of the great ball. the directions mr hugens gives , about this little corrector , is , that it should be equal to the weight of the wire , or rod of the pend. , or about a th part of the weight of the great ball , which he appoints to be three pounds . perhaps this bob may do its office , if it be made to screw only up and down the lower part of the rod , below the ball. if not , you must make it slide above the ball , or be screwed up and down there . seeing this little bob is not the only corrector ( as in mr zulichem's way ) therefore it is not necessary to insert here , that ingenious person's table , shewing what alterations of time will be made by sliding the bob up and down the rod. only thus much may be observed in that table of his , viz. that a small alteration of the corrector towards the lower end of the pend ▪ , doth make as great an alteration of time , as a greater raising or falling of it , doth make higher . thus the little bob raised divisions of the rod , from the center of oscillation , will alter the watch seconds ; raised , 't will alter it ″ . but whereas , if it be raised to ▪ parts of the rod , it will make the watch go faster minutes , seconds , the watch shall be but ′ . ″ faster , if the bob be raised to ▪ . so that here you have but ″ variation , by raising the bob above parts ; whereas lower , you had the same variation , when raised not above or parts . from what hath been said , it appears , that about half a turn of this little justening bob , will at no time alter the watch , above a second in hours ▪ and that above a whole turn , will not alter it so much , higher on the rod ; supposing that the bob at every turn ascended or descended a whole degree of the rod ; which perhaps it will not do in turns : and consequently , it will require many turns , to alter the watch but one second . chap. vi. the antiquity , and general history of watch , or clock-work . § . it is probable , that in all ages , some instruments or other have been used , for the measuring of time. but the earliest we read of , is the dial of ahaz . concerning which , little of certainty can be said . the hebrew word ma●aloth doth properly signifie degrees , steps , or stairs , by which we ascend to any place . and so this word ma●aloth is rendered ezek. . . and accordingly the lxxii translate the ma●aloth of ahaz , by the words 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and ' 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , i. e. steps or ascents . the like doth the syriack , arabick , and other versions . some pretend to give a description of this dial of ahaz : but it being meer guessing , and little to my purpose , i shall not trouble the reader with the various opinions about it . among the greeks and romans , there were two ways chiefly used to measure their hours . one was by clepsydrae , or hour ▪ glasses . the other by the solaria , or sun-dials . the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , says suidas and phavorinus , was 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 i. e. an astronomical instrument , by which the hours were measured . also , that it was a vessel , having a little hole in the bottom , which was set in the courts of judicature , full of water ; by which the lawyers pleaded . this was , says phavorinus , to prevent babbling , that such as spake , ought to brief in their speeches . as to the invention of those waterwatches ( which were , no doubt , of more common use , than only in the law-courts ) the invention , i say , of them , is attributed , by censorinus , to p. cornelius nasica , the censor ( scipio nasica , pliny calls him . ) the other way of measuring the hours , with sun-dials , seems , from pliny and censorinus , to have been an earlier invention than the last . pliny says , that anaximenes milesius , the scholar of anaximander , invented dialling , and was the first that shewed a sun-dial at lacedoemon . vitruvius calls him milesius anaximander . this anaximander or anaximenes was cotemporary with pythagoras , says laertius ; and flourished about the time of the prophet daniel . but enough of these ancient timeengines , which are not very much to my purpose , being not pieces of watch-work . § i shall in the next place take notice of a few horological machines , that i have met with ; which , whether pieces of clock-work , or not , i leave to the readers judgment . the first is that of dionysius , which plutarch commends for a very magnificent , and illustrious piece . but this might be only a well delineated sun-dial . another piece , is that of sapor king of persia . whether that sapor , who was cotemporary with constantine the great , i ●ow not . ●ardan saith it was made of glass ; that the king could sit in the middle of it , and see its stars rise and set ▪ but not finding whether this sphere was moved by clock-work , or whether it ●ad any regular motion , i shall say no ●ore concerning it . the last machine i shall mention in ●his paragraph , is one i find des●ribed by ●itruvius . which to me seems to be a piece of watch-work , moved by an equal ●nflux of water . if the reader will consult the french ●dition of vitruvius , he will find there a ●ir cut of it . among divers seats which this ma●hine performed ( as sounding trumpets , ●hrowing stones , &c. ) one use of it was , ●o shew the hours ( which were unequal ●n that age ) through every month of the ●ear . the words of vitruvius are , aequa●iter influens aqua sublevat scaphum inversum ( quod ●b artificibus phellos sive tympanam ●icitur ) in quo collocata regula ▪ versatilia ●●mpana denticuli● a ▪ qualibus s●nt perfecta . ●ui denticuli alius alium impelientes , ver●●●iones modicas faciant , ac motiones . item 〈◊〉 reg●loe , aliaque tympana ad eundem modum dentata , quoe una motione coacta , versando faciunt effectus , varietatesque motionum : in quibus moventur sigilla , vertuntur metoe , calculi aut tona projiciuntur , buccinoe canunt , &c. in his etiam , aut in colu●na , an t parastatica horoe describuntur ; quas sigillum egrediens ab imo virguloe , significat , in diem totum : quarum brevitates ant crescentias , cuneorum adjectus aut exemptus , in singulis diebus & mensibus , perficere cogit . the inventer of this famous machine , vitruvius says , was one ctesibius , a barbers son of alexandria . which ctesibius flourished under ptolomy euergetes , says athenoeus , l. . and if so , he lived about years before our saviours days ; and might be cotemporary with archimedes . § . thus having given a small account of the ancient ways of measuring time , it is time to come closer to our business , and say something more particularly of clock-work . which is thought to be a much younger invention , than the fore-mentioned pieces ; and to have had its beginning in germany , within less than these years . it is very probable , that our ballance clocks , and some other automata , might have their beginning there ; or that clock-work ( which had long been buried in oblivion ) might be revived there . but that clock work was the invention of that age purely , i utterly deny ; having ( besides what goes before ) two instances to the contrary , of much earlier date . § . the first example is the sphere of archimedes ; who lived about years before our saviours days . there is no mention of this sphere in archimedes his extant works : but we have an account of it in others . cicero speaks of it more than once . in his d book de nature deorum , are these words ; archimedem arbitrantur plus valuisse in imitandis sphoeroe conversionibus , quam naturam in efficiendis , &c. and in his tusculane questions , the collocutor , proving the soul to be of a divine nature , argues from this contrivance of archimedes , and says , nam cum archimedes lunoe , solis , quinque errantium motus in sphoeram illigavit , effecit , &c. the sense is , that archimedes contrived a sphere , which shewed the motion of the moon , sun , and five planets . but the most accurate description is that of claudian , in these words . jupiter in parvo cum cerneret oethera vitro , risit , & ad superos talia dicta dedit : huccine mortalis progressa potentia curoe ? jam meus in fragili luditur or be labor . jura poli , rerumque fidem , legesque deorum , ecce syracusius transtulit arte senex . inclusus variis famulatur spiritus astris , et vivum certis motibus urget opus . percurrit proprium mentitus signifer annum . et simulata novo cynthia mense redit . jamque suum volvens audax industria mundum gaudet , & humana sidera mente regit . quid falso insontem tonitru salmonea miror ? aemula naturoe parva reperta manus . in english thus : when jove espy'd in glass his heavens made , he smil'd , and to the other gods thus said : strange feats ! when human art so far proceeds , to ape in brittle orbs my greatest deeds . the heav'nly motions , natures constant course , lo ! here old archimede to art transfers . th' inclosed spirit here each star doth drive ; and to the living work sure motions give . the sun in counterfeit his year doth run , and cynthia too her monthly circle turn . since now bold man has worlds of 's own descry'd he joys , and th' stars by human art can guide . why should we so admire proud salmons cheats when one poor hand natures chief work repeats . from this description it appeareth , that in this sphere , the sun , moon , and other heavenly bodies , had their proper motion : and that this motion was effected by some enclosed spirit . what this enclosed spirit was , i cannot tell , but suppose it to be springs , wheels or pullies , or some such means of clock-work : which be●ng hidden from vulgar eyes , might be ●●ken for some angel , spirit , or divine power ; unless by spirit here , you un●erstand some aerious , subtiliz'd liquor , 〈◊〉 vapours . but how this , or indeed any ●hing , but clock-work , could give such ●ue , and regular motions , i am not able ●o guess . § . the next instance i have met with 〈◊〉 ancient clock-work , is that famous ●ne in cicero , which , among other irre●agable arguments ▪ is brought in to ●ove , that there is some intelligent , divine , and wise being , that inhabiteth , ruleth in , and is as an architect of so great a work , as the world is , as the collocutor expresseth himself ▪ his words ( so far as they relate to my present purpose ) are these : cum solarium vel descriptum , aut ex aqua contemplere , intelligere declarari horas arte , non casu , &c. and a little after , quod si in scythiam , aut in britanniam , sphaeram aliquis tulerit hanc , quam nuper familiaris noster effecit posidonius , cujus singulae conversiones idem efficiunt in sole , & in luna , & in quinque stellis errantibus , quod efficitur in coelo singulis diebus , & noctibus ; quis in illa barbarie dubitet , quin ea sphaera sit perfecta ratione ? the sum of the authors meaning is , that there were sun-dials described , or drawn [ with lines , after the manner as our sun-dials are : ] and some made with water ( which were the clepsydrae , or hour-glasses , before-mentioned . ) that posidonius had lately contrived a sphere , whose motions were the same in the sun , moon , and planets , as were performed in the heavens each day and night . the age wherein this sphere was in●ented , was cicero's time , which was a●out years before our saviours birth . and that it was a piece of clock-work , ●s not ( i think ) to be doubted , if it be ●onsidered , that it kept time with those ●elestial bodies , imitating both their an●ual , and diurnal motions , as from the ●escription we may gather it did . it may be questioned , whether those machines were common or not : i believe ●hey were rarities then , as well as mr wat●n's and others are accounted now . but ●nethinks it is hard to imagine , that so ●seful an invention should not be reduced ●to common use ; it being natural , and ●sie to apply it to the measuring of ●ours ( tho unequal ) especially in two ●●ch ages , as those of archimedes and ●ully were , in which the liberal arts so greatly flourished . § . after the times last mentioned , i ●nd little worth remark , till the last age ; 〈◊〉 which clock-work was revived , or ●holly invented anew in germany ▪ as is ●enerally thought , because the ancient ●ieces are german work . but who was ●e inventer , or in what time , i cannot discover . some think sever. boethius invented it about the year . perhaps it was in regiomontanus his time ( if not so early as boethius ) which was above years ago . it is very manifest , it was before cardan's time , because he speaketh of it , as a thing common then . and he lived about years since . § . as to those curious contrivances in clock-work , which perform strange , surprizing feats , i shall say little . dr. heylin tells us of a famous clock and dial in the cathedral church of lunden in denmark . in the dial ( saith he ) are to be seen distinctly the year , month , week-day , and hour of every day throughout the year ; with the feasts , both moveable and fixed ; together with the motion of the sun and moon , and their passage thro each degree of the zodiack . then for the clock , it is so framed by artifical engines , that whensoever it is to strike , two horse-men encounter one another , giving as many blows apiece , as the bell sounds hours : and on the opening of a door , there appeareth a theatre , the virgin mary on a throne , with christ in her arms , and the three kings or magi ( with their several trains ) marching in order , doing humble reverence , and presenting severally their gifts ; two trumpeters sounding all the while , to adorn the pomp of that procession . to this i might add many more such curious performances ; but i rather chuse to refer the reader to schottus , where he may find a great variety , to please him . chap. vii . of the invention of pendulum watches . § . the first that invented the way of applying pendulums to watch-work , was mr christian hugens of zulichem ; as he affirmeth of himself , with very cogent reasons . this excellent invention , he says , he put first in practice in the year ▪ and in the following year , he printed a delineation and description of it . others have claimed the honour of this invention ; among which , the great galileo hath the most to be said on his side . dr. john joachim becher ( who printed a book when he was in england , entituled ; de nova temporis dimetiendi ratione theoria , &c. which he dedicated to the english royal society , anno . ) he , i say , tells us , that the count magalotti ( the great duke of tuscany's resident at the emperors court ) told him the whole history of these pendulum clocks , and denied mr zulichem to be the author of them . also , that one treffler ( clock-maker to the father of the then g. duke of tuscany ) related to him the like history : and said moreover , that he had made the first pend. clock , at florence , by the command of the great duke , and by the direction of his mathematician galilaeus a galilaeo ; a pattern of which was brought into holland . and further he saith , that one caspar doms , a fleming , and mathematician to john philip a schonborn , the late elector of mentz ) told him , that he had seen at prague , in the time of rudolphus the emperor , a pend. clock , made by the famous justus borgen , mechanick and clock-maker to the emperor : which clock the great tycho-brahe used in his astronomical observations . thus far becher . to which i may add , what is said by the academie del cimento , viz. it was thought good to apply the pendulum to the movement of the clock ▪ a thing which galilaeo first invented , and his son vincenzio galilei put in practice in the year . as to these matters thus related by hear-say by becher , and so expressly affirmed by the academy , i have little to reply , but that mr hugens does expressly say , he was the inventer , and that if galilaeo ever thought of any such thing , he never brought it to any perfection . it is certain , that this invention never flourished till mr hugens set it abroad . § . after mr hugens had thus invented these pendulum watches , and caused several to be made in holland , mr fromantil , a dutch clock-maker , came over into england , and made the first that ever were made here : which was about the year . one of the first pieces that was made in england , is now in gresham-colledg , given to that honorable society by the late eminent seth , lord bishop of salisbury : which is made exactly according to mr zulichem's directions . § . for several years this way of mr zulichem was the only method , viz. crown-wheel pendulums , to play between two cycloidal cheeks , &c. but afterwards mr w. clement , a london clock-maker , contrived them ( as mr smith saith ) to go with less weight , an heavier ball ( if you please ) and to vibrate but a small compass . which is now the universal method of the royal pendulums . but dr hook denies mr clement to have invented this ; and says that it was his invention , and that he caused a piece of this nature to be made , which he shewed before the r. society , soon after the fire of london . § . the use of these pendulum watches mr. hugens setteth forth in several instances . particnlarly , he giveth two examples of their great use at sea , in discovering the difference of meridians , more exactly than any other way : which he deduceth from the observations of an english , and french ship. on land , they were found very serviceable , among other uses , particularly to these two . . to measure the time more exactly , and equally than the sun. ● . to be ( as sir christoph . wren first proposed ) a perpetual , and universal measure , or standard , to which all lengths may be reduced , and by which they may be udged , in all ages , and countries . for as our royal society , mr hugens , and mountonus have proposed after sir christopher wren ) this horary foot , or tripedal length , which vib●teth seconds , will fit ill ages and places ▪ but then respect must be had to the center of oscillation , which you have an account of in mr hugens his ●soresaid book , de horologio oscillatorio , ●s hath before been said . § . there is one contrivance more of pendulums , still behind , viz. the circular pendulum : which is mentioned by mr hugens as his own , but is claimed by the ngenious dr. hook as really his . this ●end ▪ doth not vibrate backward and forward , as ●hose we have been speaking of ●o ; but always round , round ; the string being suspended above , at the tripedal length , and the ball fi●ed below , as suppose at the end of the fly of a common ●ack . the motion of this circular pend ▪ is as regular , and much the same with what goeth before : and was thus far made very useful in astronomical observations , by the said dr hook , viz. to give warning at any moment of its circumgyration , either when it had turned but a quarter , half , or any lesser , or greater part of its circle . so that here you had notice not only of a second , but of the most minute part of a second of time. you may find a description of this pendulum , and other matters belonging to it , in dr hook's lectiones cutlerianoe : animad . in hevelius mach ▪ caelest . p. . chap. viii . of the invention of those pocket-watches , commonly called pendulum watches . § . the reason they are called pendulum-watches , is from the regularity of their strokes , and motion . which exactness is effected by the government of a small spiral spring , running round the upper part of the verge of the ballance : which spring is called the regulator . § . the first inventer hereof , was ●hat ingenious and learned member of our royal-society , dr hook : who contrived va●ious ways of regulation . one way was ●ith a load-stone : another was with a ●nder strait spring ▪ one end whereof ●layed backward , and forward , with the ●allance : so that the ballance was to this ●pring as the bob of a pendulum , and the ●ttle spring , as the rod thereof . and se●eral other contrivances he had besides of ●is nature . § . but the invention which best an●ered expectation , was at first , with two ●llances : of which i have seen two sorts , ●ho there were several others . one ●ay was without spiral springs , the ●her with . they both agreed in this , ●hat the outward rims of both the bal●nces , had alike number of teeth ; which ●nning in each other , caused each bal●nce to vibrate alike . but as to the former of these , which had no spiral spring ▪ the verges of its ballances , had each but one pallet apiece , about the middle of the verge . the crown-wheel lay ( contrary to others ) reversed , in the middle of the watch , in the place , and after the manner of the contrate-wheel . the teeth of this crown-wheel , were cut after the manner of contrate-wheel teeth , viz. lying upwards , but very wide apart , so as that the pallets ( which were about one tenth of an inch long , and n●rrow ) might play in and out between each tooth . the verges of the two ballances , were set one on one side , the other on the other side of the crown-wheel , so that the pallets might play freely in its teeth . and when the crown-wheel in moving round , had delivered its self of one pallet , the other pallet on the opposite side , was drawn on to make its beat , by means of the motion which the other ballance had given its ballance , ( the two ballances moving one another , as hath been said in the beginning of this paragraph . ) and so the same back again . it may be here noted , that for the more clear understanding of the last contrivance , i have described the two ballances , as having teeth on the edges of their rims , running in one another . but the contrivance was really thus ▪ there was a small wheel under each ballance , proportioned to the width of the crown-wheel . but the ballances were much larger . and so the teeth of these two little foresaid wheels or ballances , running in one another , moved the larger ballances above them , all one , as if these two great ballances had been toothed and played in each other . § . the other way , with two ballances also , moving each other ( as was said in the beginning of the last § ) had a spiral spring to each ballance , for its regulator . in this invention , only one ballance had the pallets , as the common ballances have : and the crown-wheel operated upon it , according to the usual way . but then when this ballance vibrateth , it giveth the same motion backward and forward , to the other ballance ; as hath been said . the first of these two ways , was never prosecuted so far , as perhaps it deserved . and the excellency of the latter is , that no jirk , or the most confused shake , can in the least alter its vibrations . which it will do in the best pendulum watch with one ballance now commonly used . for if you lay one of these watches upon a table , and by the pendant jirk it backward and forward , you will put it into the greatest hurry ; whereas the last mentioned watch , with two ballances , will be nothing affected with it . but notwithstanding this inconvenience , yet the watch with one ballance and one spring ( which was also dr. hooks invention ) prevailed , and grew common , being now the universal mode : but of the other very few were ever made . the reason hereof , i judge , was the great trouble and vast niceness required in it , and perhaps a little foulness in the ballance-teeth may retard the motion of the ballances . but the other is easier made , and performeth well enough , and in a pocket is scarce subject to the aforesaid disorder , which is caused rather by a turn , than a shake . § . the time of these inventions was about the year , as appears ( among other evidence ) from this inscription , upon one of the aforesaid double ballance-watches , presented to k. charles ii , viz. robert hook inven . . t. tompion fecit . this watch was wonderfully approved of by the king ; and so the invention grew into reputation , and was much talked of at home , and abroad . particularly its same flew into france , from whence the dauphine sent for two ; which that eminent artist mr. tompion made for him . § . dr. hook had long before this , caused several pieces of this nature , to be made , altho they did not take till after . however he had before so far proceeded herein , as to have a patent ( drawn , tho not sealed ) for these , and some other contrivances , about watches , in the year . but the reason why that patent did no further proceed , was some disagreement about some articles in it , with some noble persons who were concerned for the procuring it . the same ingenious dr. had also a grant for a patent for this last way of spring watches in the year : but he omitted the taking it out , as thinking it not worth the while . § . after these inventions of dr. hook , and ( no doubt ) after the publication of mr. hugens's book de horolog . oscil . at paris ( for there is not a word of this , tho of several other contrivances ) after this , i say , mr. hugen's watch with a spiral spring came abroad , and made a great noise in england , as if the longitude could be now found . one of these the lord bruncker sent for out of france , ( where mr hugen ● had a patent for them ) which i have seen . this watch of mr. zulichem's agreed with dr. hook's , in the application of the spring to the ballance : only mr. zulichem's had a longer spiral spring , and the pulses and beats were much slower . that wherein it differs , is . the verge hath a pinion instead of pallets ; and a contrate-wheel runs therein , and drives it round , more than one turn . . the pallets are on the arbor of this contrate-wheel . . then followeth the crown wheel , &c. . the ballance , instead of turning scarce quite round ( as dr. hook's ) doth turn several rounds every vibration . § . as to the great abilities of mr. hugens , no man can doubt , that is acquainted with his books , and his share in the philosophical transactions , &c. but i have some reason to doubt , whether his fancy was not first set on work , by some intelligence he might have of dr hook's invention , from mr oldenburgh , or others his correspondents here in england . but whether or no that ingenious person doth owe any thing herein to our ingenious dr hook , it is however a very pretty , and ingenious contrivance ; but subject to some defects : viz. when it standeth still , it will not vibrate , until it is set on vibrating : which , tho it be no defect in a pendulum clock , may be one in a pocket-watch , which is exposed to continual jogs . also , it doth somewhat vary in its vibrations , making sometimes longer , sometimes shorter turns , and so some slower some quicker vibrations . i have seen some other contrivances of this sort , which i mention not , because they are of younger standing . but these two ( of dr hook and mr hugens ) i have taken notice of , because they were the first that ever appeared in the world . chap. ix . the invention of repeating clocks . § . the clocks i now shall speak of , are such as by pulling of a string , &c. do strike the hour , quarter , or minute , at any time of the day and night . § . these clocks are a late invention of one mr barlow , of no longer standing than the latter end of k. charles ii. about the year . this ingenious contrivance ( scarce so much as thought of before ) soon took air , and being talked of among the london artists , set their heads to work ; who presently contrived several ways to effect such a performance . and hence arose the divers ways of repeating work , which so early might be observed to be about the town , every man almost practising , according to his own invention . § . this invention was practised chief● , if not only , in larger movements , 〈◊〉 k. james ii.'s reign : at which time it as transferred into pocket-clocks . but ●ere being some little contest concern●g the author hereof , i shall relate the ●●e matter of fact , leaving the reader to ●own judgment . about the latter end of k. james ii.'s ●gn , mr barlow ( the ingenious inventer ●ore-mentioned ) contrived to put his ●ention into pocket ▪ watches ; and en●voured ( with the lord chief justice ●bone , and some others ) to get a patent ●it . and in order to it , he set mr tom● the famous artist , to work upon it : ●o accordingly made a piece according ●is directions . ●r quare ( a very ingenious watch ▪ ●er in london ) had some years before 〈◊〉 thinking of the like invention : but bringing it to perfection , he laid by thoughts of it , until the talk of mr bar● patent revived his former thoughts ; ●ch he then brought to effect . this ●g known among the watch-makers , 〈◊〉 all pressed him to endeavour to hin● mr barlow's patent . and accordingly applications were made at court , and a watch of each invention , produced before the king and council . the king , upon tryal of each of them , was pleased to give the preference to mr quare's : of which , notice was given soon after in the gazette . the difference between these two inventions was , mr barlow's was made to repeat by pushing in two pieces on each side the watch-box : one of which repeated the hour , the other the quarter ▪ mr quare's was made to repeat , by a● pin that stuck out near the pendant which being thrust in ( as now 't is done by thrusting in the pendant ) did repeat both the hour , and quarter , with the sam● thrust . it would ( i think ) be very frivolous , to● speak of the various contrivances , and methods of repeating work , and the inventers of them ; and therefore i shall sa● nothing of them . chap. x. numbers for several sorts of movements . i think it may be very convenient to set down some numbers , fit for several movements ; partly , to be as examples to exercise the young reader , in the foregoing art of calculation : and partly , to serve such , who want leisure or understanding to attain to this art. § . but first it may be requisite , to shew the usual way of watch-makers writing down their numbers , which is somewhat different from that in the preceding book . their way representeth the wheel and pinion , on the same spindle ; not as they play in one another . thus the numbers of an old house-watch , of hours , is written thus ; my way : the watch-makers way . ) ▪ ) — ) — — according to my way , the pin. of report [ ] drives the dial-wheel [ : ] the pinion [ ] plays in the great-wheel [ ] &c. but according to the other way , the dial-wheel stands alone ; the great-wheel hath the pinion of report on the same arbor : the wheel [ ] hath the pin : [ ] and the crown-wheel [ ] the pin : [ ] on the same spindles . this latter way ( tho very inconvenient in calculation ) representeth a piece of work handsomely enough , and somewhat naturally . § . numbers of an day piece , with turns the barrel , the pend. vibrates seconds , the shews minutes , seconds , &c. the watch-part . the clock part . ) ) ) — ) — ) ) pins . ) ) ) in the watch-part , the wheel is the minute-wheel , which is set in the middle of the clock , that its spindle may go thro the middle of the dial-plate to carry the minute-hand . also on this spindle is a wheel , which driveth another wheel of , which last hath a pinion , which driveth round the wheel in hours . note here two things : . that the two wheels , are of no other use , but to set the pinion at a convenient distance from the minute-wheel , to drive the wheel , which is concentrical with the minute-wheel . for a pinion driving a wheel , would be sufficient , if the minute-hand and hour-hand had two different centers . ▪ . these numbers , ▪ ) - ) , set thus , ought ( according to the last § ) be thus read , viz. the wheel , hath another wheel on the same spindle ; which wheel divideth , playeth in , or turns round another wheel ; which hath a pinion concentrical with it : which pinion driveth , or divideth a wheel of . for a line parting two numbers ( as - ) denoteth those two numbers to be concentrical , or to be placed upon the same spindle . and when two numbers have a hook between them ( as ) ) it signifies one to run in the other , as hath before been hinted . in the striking-part , there are pins on the second wheel . the count-wheel may be fixed unto the great-wheel , which goeth round once in hours . § . a piece of days , with , or turns both parts : the watch sheweth hours , minutes , and seconds ; and the pend. vibrateth seconds ▪ the watch-part , with turns . with turns . ) ) ) ) ) — ) — ) ) — ) — ) ) ) the striking part . with turns . with turns . ) ) ) pins ) pins ) ) ) double hoop . ) double hoop . ) ) the pinion of report is fixed on the ●nd of the arbor of the pin ▪ wheel . this pinion in the first is , the count-wheel ; thus , ) . or it may be ) . ●n the latter ( with turns ) it may be ) , or ) . § . a two month piece , of days ; with turns ; pend. vibrateth seconds , and sheweth minutes , seconds , &c. watch-part . clock-part . ) ) ) ) ) — ) — ) ) pins ) — )   ) -double hoop ) here the third wheel is the pin-wheel ; which also carrieth the pinion of report , driving the count-wheel . or thus . watch-part . clock-part . ) ) ) ) pins ) — ) — ) — ) ) ) -double hoop ) § . a piece of weeks , with pendulum , turns , and motions , as before . the watch part . ) or thus . ) ) ) ) — ) — ) ) — ) — ) ) ) the clock part . ) or thus . ) ) — ) ) — pins ) — pins — ) ) double hoop ) ) ) § . a seven month piece , with turns , pendulum , and motions , as before . the watch. the clock . ) ) ) ) — ) ) ) — pins ) — ) — ) ) double hoop ) )   § . a year piece , of days , with turns , pendulum , and motions , as before . the watch. the clock . ) ) ) ) — ) ) ) pins ) — ) ▪ ) ) double hoop ) )   if you had rather have the pinion of report , on the spindle of the pin-wheel , it must be ) . § . a piece of hours , pendulum about inches . the watch. the clock . ) ) ) ) pins ) ) ) )   § . a piece of days , with turns , pendulum about inches , to shew minutes , seconds , &c. the watch. the clock may be the same with the day piece before , § . ) ) — ) — ) ) ) the seconds wheel . § . a month piece of days , with pendulum , turns , and motions , as the last . the watch. the clock may have the same numbers , as the clock § . ) ) ) — ) — ) ) ) seconds wheel . § . a year piece of days , with pendulum , turns , &c. as the last . the watch part . ) or thus ▪ with a wheel less , and not to shew minutes and seconds . )   ) ) ) ) — ) ) — ) — ) ) ) ) seconds wheel . ) in the latter of these two numbers , the pinion of report is , on the second wheel . the dial wheel is . the clock-part may have the same numbers , as the year-piece before § . § . an day piece , to shew the hour and minute , pend. about inches long . ) the clock may have the same numbers , as the day piece before , § . ) — ) ) ) automata shewing the motion of the celestial bodies . § . numbers for the motion of the sun and moon . see before in chap. . sect. . § , . § . numbers to shew the revolution of the planet saturn , which consists of days . on the dial-wheel . if you would make it depend upon a wheel going round in a year , thus . ) ) ) ) or thus , ) ) ) note , the lowermost pinion in these , and the following numbers , is to be fixed concentrical to the wheel , which is to drive the motion , viz. the dial-wheel , year-wheel , or &c. § . numbers for the planet jupiter , whose revolution is ½ days . on the dial-wheel . ) or thus , on the year-wheel . ) ) )   )   note here , that the two last numbers of saturn , may be the two first of jupiter also . by the permission of my ingenious friend mr flamsteed , i here insert a description of mr olaus romer , the french king 's mathematician's instrument , to represent the motion of jupiter's satellites ; a copy of which he sent to mr flamsteed in . upon an axis ( which turns round once in days ) are four wheels fixed : one of teeth ; a second of ; the third ; and the last of teeth . on another axis run other wheels ( or pinions you may call them ) which are driven by the asoresaid wheels . the first is a wheel , or pinion of leaves , driven by the wheel , which carrieth round the first satellite . the second is , driven by the wheel , which carrieth round the second satellite . the third hath leaves , driven by the wheel , which carrieth the third satellite . and lastly , is the pinion , driven by the wheel , which carrieth round the fourth satellite . on the first axis is an index , that pointeth to a circle divided into parts , which are the hours in days . on the other axis all the pinions run concentrically , by means of their being hollow in the middle . in the midst of them all , the axis of jupiter himself is fixed , with a little ball at the top , representing jupiter's body . on the ends of small wires , fixed in the four several sockets of the aforesaid pinions , may lesser globules be placed ( at their due distance from jupiter's globule ) to represent the satellites going round that planet . § . numbers for mars , whose revolution is year days . on the dial-wheel ▪   ) the two last numbers of saturn may be the two first of mars also . ) ) § . numbers for venus , whose revolution is in days . on the dial-wheel .   ) note , the last number of jupiter may be the first of venus . ) ) § . numbers for mercury , whose revolution is near days . on the dial-wheel . ) ) § . numbers to represent the motion of the dragon's head and tail , ( near years ) to shew the eclipses of the sun and moon . on the dial-wheel . on the year-wheel . ) ) ) note , the two last numbers of saturn may be the two first of this on the dial-wheel . ) ) as to the placing these several motions on the dial-plate , i shall leave it wholly to the work-mans contrivance . he may perhaps make them to represent the copernican , or some other sys●em . numbers for pocket ▪ watches . § . a watch to go days , with ● turns , to shew minutes and seconds , the train . )   ) — ) — ) . ) on the wheel [ ] is the second's hand placed , and on the wheel [ ] the minute hand . ) § . another of the same , without minutes and seconds , to go with only turns . ) ) ) ) ) § . a pocket-watch of hours , with turns , to shew minutes and seconds , train as the last . ) ) — ) — ) ) ) — seconds hand . § . the usual numbers of hours pendulum watches , with turns , to shew the hour and minute . ) ) — ) — ) ) ) § . the usual numbers of the old hours pocket-watches . with wheels . with wheels . ) ) ) ) ) ) ) ) ) if any of the numbers of the preceding wheels and pinions should not please the reader , he may easily correct them to his mind , by the instructions in the foregoing book . the way in short is this : divide the wheel by the pinion , and so find the number of turns , according to chap. . sect. . § . multiply the pinion you like better , by this number of turns , and the product is the wheel . thus in the day pocket-watch § , if you think the great-wheel too large , you make it instead of ) ( thus , viz. ) ( : i. e. chusing the pinion only , and multiplying it by ( the turns ) the wheel will be . chap. xi tables of time relating to watch-work . a table of time. seconds .             minutes .           hours .         day .       week .     month.   year . the foregoing table will be of good use in calculation , for the ready finding out the parts of time : which is thus . find the parts of time you seek for , the number in the concurrence of squares , is the answer to your question . thus , suppose you seek for the number of seconds ●n a year : in the square under seconds , and in the same line with year ( which is the ●owermost square on the left hand ) is the number sought , viz. , &c. so minutes in a month , are . if you would know any number , where there is the addition of an odd number to it , as the seconds in a month and one day ; add the seconds in a month ( which are — ) and the seconds in a day ( which are — ) and you have the number sought , viz. . a table to set a watch by the fixed stars ▪ night hour . min. sec. night hour . min. sec. explanation of the table . this table shews how much the sidereal , goeth faster than the solar day , in any number of nights for a month . so that observing by your watch , the nice time when any fixed star cometh to the meridian , or any other point of the heavens : if after one revolution of that same star to the same point , your watch goeth ′ . ″ ●lower than the star ; or after two nights ′ . ″ ; or nights , h. ′ . ″ , &c. then doth your watch keep time rightly with the mean motion of the sun. if it vary from the table , you must alter the length of your pend ▪ to make it so keep time . to observe the time nicely , when the star cometh again to the same point of the heavens , 't is necessary to make the observation with a telescope , that hath cross threads in the focus of the object-glass ; and so leaving the telescope fixed in the same posture , till a second observation . you may do this with the telescopular sights of a quadrant , or sextans , and so leaving it standing until anoher night of observation . or for want of this more nice way , you may do it by looking along by the edge of two strings , suspended with plumbets , in a room , at some distance from one another . or by looking at the edge of a chimney , &c. as mr watson hath directed , at the end of mr smith's horo● . disquis . but to make a tolerable observation any of these last ways , 't is necessary to have a candle shine upon the edge of the furthermost string , or chimney ; without which you cannot see exactly when the star cometh thereto . a table shewing the variations made in the true hour of the day , by the refraction of ▪ the sun in the equator , and both the solstices . sun's altitude . deg. sun's refraction . variation at the n. solstice . variation at the equator . variation at the s. solstice . ′ ″ ′ ″ ′ ″ ′ ″ remarks upon the table . the column of the sun's refractions , i owe to that accurate observer of the celestial motions , mr ▪ flamsteed . which refractions , altho in the table the same , yet do differ at different seasons of the year , nay perhaps , according to the different temperature of the air sometimes , in the same day . thus mr flamsteed found the refractions in february , very different from those in april : and it is observed , that the refractions are commonly greater , when the mercury is higher in the barometer . the table therefore doth not shew what the refractions always are , but only about the middle quantity of them , at every degree , of the first of the sun's altitude . and accordingly i have calculated the variations thereby made in the hour of the day . these variations of the hour are greater or lesser , according as the angle of the sun 's diurnal motion is acuter with the horizon . the reason is plain ; because as the sun appears by refraction higher than really he is ; so this false height doth affect the hours in winter , more than the summer half year . there is no ray indeed of the sun , but what cometh refracted to a sun ▪ dial ▪ and consequently , there is no dial but what goeth more or less false ( except at noon in dials that cast a shade , where the refraction makes no variation ▪ ) but the refraction decreaseth apace , as the sun gets higher , and causeth a variation of not above half a minute , at degrees of the sun's altitude ; except when the sun is in , or near the southern tropick . nearer than half a minute , few common sun-dials shew the time . and therefore , partly for this reason , and partly , because mr. flamsteed's observations reach not much farther , i have calculated my table to only degrees . the table needs little explication . for having the sun's height , you have against it , in the next column , the refraction ▪ and in the next the alterations of the hour , at times of the year . taking therefore by a quadrant the sun's altitude , and observe at the same time , the hour of the day by a sun-dial , by the table , you see how many minutes , and seconds , the dial is too fast . as at the sun-rising a sun-dial is too fast ′ . ″ , about june , and ′ . ″ , about mar. . and sept. , and ′ ″ about dec. . addenda . to the fifth part of the rule in § . p. . if you have occasion to lay the pinion of report upon any other wheel , and not the great-wheel , you may do it by this rule , as the beats in one turn of any wheel ; to the beats in an hour : : so are the hours of the dial ; to the quotient of the hour-wheel divided by the pinion of report . to page . suppose in altering an old watch , you would have it shew minutes , as well as hours ; you may do it thus : divide the beats in one turn of the great-wheel , by the beats in an hour ; the quotient will shew in how many hours the great-wheel goeth round once . if the beats in the great wheel exceed the train , you must chuse your minute-wheel first , and multiply it by the quotient ; this will give the pin. of report . but if the train exceeds the beats of the great-wheel , you must chuse the pin. of rep. and multiply the quotient by it : the product is the minute-wheel . but it often falls out , that the train and beats of the great-wheel will not exactly measure one another : if so , the best way is to half the two numbers , as far as they will equally admit of halfing ; or divide them by some common divisor , and so having brought them to as small numbers as you can , you may suppose them to be a wheel and pinion , and reduce them to lesser numbers ▪ by chap. . sect. . § . thus suppose you would make the old dull movement there mentioned , a minute-watch ; you may reduce the numbers of the great-wheel ● ▪ and the train , to a pinion and wheel ) . which pin. being set upon the spingle of the gr. wh. will drive a wheel round once in an hour , to shew minutes . if you make this ▪ wh. drive another of ; ( concentrical to which , is a pin. driving a wheel ( which wheel is concentrical with the minute-wheel ) this will carry a hand round in hours . but in this case , you must place the pin. on the spindle of the gr. wh. so as to slide round stiffly , when you turn the minute-hand to rectifie the watch. finis . notes, typically marginal, from the original text notes for div a -e oughtred of autom . sect . . ) ( ) ( ) ( ) (     by the quotients i commonly mean the ●umber of turns ; which number is set on the right hand , without a hook , as is shewn in the last paragraph . which i no●e ●ere now once for all . ) ● ( ) ( ) ( ) ( ) ( ) (     ) ( ) ( ) ( ) (     sir j. moor ▪ mat. com. p. . ibid. p. . oughtred autom . sect. . ) ought . ib. id. ib.   x   x id. ib. oughtred sect. . sir j. moor ibid. p. . ) ( ) ( ) ( ) (     oughtred sect. . § . ) ( ) ( ) ( ) (     id. ib. § . ) / ) ( ) ( ) ( ) ( ⅗     horol . disq . sect. . § . ) (     ) ( ) ( ½ ) (     § . par. . and § . ● ) ( ● ) ( ● ) ( ½ ● ) (     sir j. moo● ibid. p. . v. sect. . § ▪ . ) ( ) ( ) ( ½     ) ( ) ( pins ) ( ) ( § . infer ● ) ( ) ( ) ( ● ) ( ) ( pins v. sect. ▪ §. . ) ( ) ( . pins ) ( ¼ ) ( ) (   ( pins ) ( ch. . sect. . § . oughtred . § . ) ( ½ ) ( ) ( ½ ( ( id. ib. ) , ( ● ) ( ) ( ¾ ) ( id. ib. ) ( ¼ ) ( ) ( ) ( ¼ ) ( ) ( autom . § . id. ib. mat. com. p. . v. sect. § , . de subtil . l. . notes for div a -e ) ( ) ( ) (     horol . dis . p. . ) ( ½ ) ( ) ( ) ( ( ● ( ½     v. sect. ▪ § . ) ( ) ( ) ( ) (     ) ( ) ( ) ( ) (     ) ●¾ ) ( pins ) ( ) ( ) ( / / ) . pins ) ( ) ( notes for div a -e sir j. moor mat. com. r. . notes for div a -e id. ib. rule . de horol . oscil . p. , , . machina pneumat . exp. . ibid. ibid. fiugenius ●●●i supra , p. . sir j. moor ibid. hugen ▪ moor , ib● ▪ horolog . disquis . ibid. ibid. de centro oscil . prop. . notes for div a -e kings ● ▪ . isai . . . lexic . in verbo 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 de die natali c. . ibid. nat. hist . l. . c. . de archit . l. . c. . in the life of dions euseb . vit. const ▪ l. . de subtil ▪ l. . de architect . l. . c. ● v●d . phi●●nd . not . in vitruv. lib. . § . edit . el●ivir . epigr. in sphoer . archimed . vid. card. de subtil . l. . de nat. deor. lib. . § . m●lyneaux , scioth . telescop . ep. dedic . cosmog . l. . magia univers . p. . proleg : & magia thaumaturg . notes for div a -e hor. oscil . p. ▪ edit . paris . p. . exper ▪ made in the acad ▪ del cimento by mr. wal●er , p. . hugen . ib. horolog . disquis . p. . a supplement to the treatise of watch & clock-work called the artificial clock-maker ... by w.d., m.a. derham, w. (william), - . approx. kb of xml-encoded text transcribed from -bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : - (eebo-tcp phase ). a wing d estc r ocm 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 . 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 , no. a ) transcribed from: (early english books online ; image set ) images scanned from microfilm: (early english books, - ; : ) a supplement to the treatise of watch & clock-work called the artificial clock-maker ... by w.d., m.a. derham, w. (william), - . , [ ] p., [ ] folded leaves of plates : ill. printed for james knapton, london : . "wherein is contain'd, . the anatomy of a watch and clock, . monsieur romer's satellite-instrument : with observation concerning the calculation of the eclipses of jupiter's satellites, and to find the longitude by them, . a nice way to correct pendulum watches, . mr. flamsteed's equation tables, . to find a meridian-line for the governing of watches, and other uses, . to make a telescope to keep a watch by the fixed stars." reproduction of original in the huntington library. created by converting tcp files to tei p using tcp tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between and available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the , texts created during phase of the project have been released into the public domain as of january . anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. % (or pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf- unicode or text strings within braces); or lossless xml (tei p , characters represented either as utf- unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng clock and watch making. - tcp assigned for keying and markup - apex covantage keyed and coded from proquest page images - robyn anspach sampled and proofread - robyn anspach text and markup reviewed and edited - pfs batch review (qc) and xml conversion a supplement to the treatise of watch & clock-work , called the artificial clock-maker . wherein is contain'd , . the anatomy of a watch and clock . . monsieur romer's satellite-instrument : with observations concerning the calculation of the eclipses of jupiter's satellites , and to find the longitude by them . . a nice way to correct pendulum watches . . mr flamsteed's equation tables . . to find a meridian-line for the governing of watches , and other uses . . to make a telescope to keep a watch by the fixed stars . by w. d. m. a. london , printed for james knapton , at the crown in st paul's church-yard . . to the reader , upon a review of my book , in order to another edition , i have thought it necessary to add some things , and to make some small amendments in the body of the book itself . and because i think it a piece of justice owing to the buyers of the first edition , that i should endeavour , as much as i can , to make their edition as compleat as this ; therefore , instead of inserting what this supplement contains into proper places of the book , i have rather chosen to put it rhapsodically together ; and taken care that it be printed so , as to be bought by itself at a small price . also i think my self obliged , to be at the pains to collect the most material alterations , and amendments which i have made in my book , and here to insert them in this supplement ; whereby the reader may supply with his pen ( if he pleaseth ) what is wanting in the first edition . the purchasers both of the first and of this edition will ( we hope ) excuse both the bookseller and me , for reducing this edition into a lesser volume , that it may be more portable for the pocket , and ( we hope ) both book and supplement too , cheaper ; at least , not dearer than the first edition , for the benefit of poor workmen . passages wanting in the first edition . page . line . after pocket-watches , add [ and others ] l. . after wheels , add [ whence it hath its name ] l. penult . dele [ somtimes . ] p. . after l. . add [ the train is the number of beats which the watch maketh in an hour , or any other certain time . ] p. . l. . after wheel , add [ which runs concentrical , or on the same arbor with the second pinion . ] p. . l. . after has , add [ as hath been said . ] p. . l. . add in the margin [ see sect. . § . . ] l. . in the margin add [ see sect. ▪ § . ] p. . l. . add in the margin [ see § . . ] p. . l. . for . r. . p. . l. . after report add [ fixed on the great-wheel . ] p. . l. . add in the margin [ sect. . § . . ] p. . l. . in the margin add [ § . ] p. . l. . after report add [ and the count-wheel . ] . p. . l. . after rules , add [ to find how many strokes the clock striketh in one turn of the fusy , or barrel . l. . after rule . add [ to find how many days the clock will go . ] l. . after rule . add [ to find the number of turns of the fusy or barrel . ] p. . l. . after rule . add [ to fix the pin. of report on the spindle of the great-wheel . ] p. . l. . after turns , add [ of the fusy . ] l. . after then , add [ ( if you make the great-wheel the pin-wheel . ) ] p. . l. . after motions , add [ in watch work ] p. . l. . amend thus [ ) ( , ] p. . l. . r. [ round by a ] p . l. penult . after to the , add [ square ▪ of the. ] p. . l. . add [ or thus with turns . ] ) ) ) ) fig. . fig. . page . th in the appendix . p. , to § . add [ if this crown-wheel be too large you may use these numbers , viz. ) ) ( ) seconds-hand p. . l. after sextans , add [ or any other telescope . ] p. . l. . after . add [ by the directions in chap. . sect. . § . . ] l. . after if add [ as in the movements in ch . . ] i. an explication of the figures . fig. . representeth the parts of a watch and clock described in the book , ch. . the wheels , &c. on the right hand , is the watch-part . they on the left , the clock-part . a. a. a. a. the upper plate of the frame : which you may imagine to be transparent ( as of glass ) to admit of a prospect of the wheel-work underneath it . b. b. b. the lower plate of the frame . c. c. c. c. the pillars . d. d. the spring-boxes of the watch , and clock-part . e. e. the great-wheel of each part . f. f. the fusy of each part , about which the chain , or string is wrapped . g. g. g. g. g. g. g. the ratchet of each part . a. a. a. the hoop , or rim of the second wheel . b. b. the cross thereof . c. the pinion . h. the contrate-wheel . i. the crown ▪ wheel . d. d. the upper and lower pevet thereof . k. a piece of brass , in which the pevet-hole is , in which the pevet d. playeth . l. the pin-wheel , with the striking-pins e. e. e. e. e. m. the detent-wheel . n. the warning wheel , or fourth wheel . o the detent . p. the lifting-piece . q. q. the fan , and flying-pinion . r. the bell. s. the hammer . t. the hammertail . v. v. the chain , or string of the watch , and clock . x. the verge or spindle of the ballance , or pendulum . y. y. y. the rod of the pendulum . z. the fork . . the flatt . . the great-ball . . the corrector or regulator . . . the pallets . ii. fig. . rerpresenteth the satellite instrument of mons . olaus romer , described in the book , page . a. b. the upper plate of the instrument . c. d. the lower plate . k. l. an axis , or spindle , on which four wheels are fixed , and turn round with it , and with the hand l. once in days . e. f. g. h. are the sockets , or hollow arbors of wheels running concentrically . the hollow arbor h. carrieth round the first-satellite p. and belongeth to the wheel , or pinion , which is driven by the fixed wheel . the hollow arbor g. carrieth round the second-satellite s. and belongeth to the wheel which is driven by the wheel . and the like of the arbors f. and e. within all these hollow arbors is another fixed one included , on the top of which is a ball ( i ) representing the planet jupiter : round which the satellites move , represented by the little balls p. s . t. q. this satellite-instrument may be added to a watch , by causing the great-wheel or dial-wheel to drive round the arbor k. l. once in days . to do which there are sufficient directions given in the preceding book . the use of the foregoing instrument . this satellite-instrument may be of good use both at sea and land to assist in finding the longitude by jupiter's satellites : partly , by giving notice when an approaching eclipse is , that we may be ready with a telescope to observe it ; and partly , when any eclipse happeneth , to shew which satellite it is that is eclipsed , which is difficult to be seen in the heavens : and partly , to supply the place of tables , or calculation of the satellite-eclipses , which it may do for a little while , but it must not long be trusted unto . it may seem foreign to my subject , to shew how the longitude may be found by jupiter's satellites : but because i would with all my power advance this way ( which far transcends all others yet known , especially that of the log-line ) therefore i hope the reader will excuse this digression . the way to find the longitude , by an eclipse of any of jupiter's four satellites is briefly this : knowing by tables of the satellite-eclipses ( suppose such as mr flamsteed published in the philos . transact . no , and afterwards gave parker leave to publish in his almanacks , knowing i say ) the time when an eclipse happeneth in any one part of the world ; observe by a telescope , at what time the same eclipse happeneth in any other part of the world , the difference of time giveth the difference of meridians . thus a total immersion of the first satellite was observed at rome , at h. ′ ″ p. m. which mr flamsteed notes at h. ′ ″ . the difference is ′ ″ , or deg . ′ distant from the meridian of the english observatory , where mr flamsteed observ'd it . i once had thoughts of shewing the way to calculate the eclipses of jupiter's satellites , and to make tables thereof , by the help of my very good friend mr flamsteed's , and some other observations : but considering that it would be too great a digression , and especially that monsieur cass●ni hath very ingeniously , and well done it for the first , i shall therefore refer the reader to his tables , reduced to the meridian and style of london , by that very judicious mathematician mr halley ; in philos . trans . no . the reader , i hope will pardon me , if ( before i leave this digression ) i observe a few things which may be of use , not only in the calculation of the eclipses of the outermost satellites , but also may contract the labour of calculation in the first . the first thing to be observed is jupiter's place . for if he be on his aphelion , he moves slowest , and consequently the satellites make their returns to him somewhat sooner , than when he is on his mean distance and perihelion . by mr flamsteed ▪ s first tables the first satellite makes revolutions to jupiter , when he is on his ▪   days h ▪ ′ ″ aphelion in — mean distance — perihelion — the second satellite makes revolutions when jupiter is on his   days h. ′ ″ aphelion in — mean-distance — perehelion — the third satellite makes revolutions when jupiter is on his   days h. ′ ″ aphelion in — mean-distance — perihelion — the fourth , or furthermost satellite makes revolutions to jupiter in his   days h. ′ ″ aphelion in — mean-distance — perihelion — from this account it is easy to compute in what time one revolution of any satellite is at any time performed : which is the next thing to be observed . thus in jupiter's mean-distance the revolution of the   days h. ′ ″ first satellite is — second — third — fourth — the reader may himself , from what hath been said , compute the periods of the satellites in jupiter's other places . from these things laid down , it is easy from an eclipse known , to find the next that will follow . for if you add one , or more revolutions , you have the eclipses following . thus for example , july this year , according to mr flamsteed's computation , the first satellite comes out of jupiter's shadow at h. ′ p. m. ( according to mr cassini's at h. ′ ″ p m. ) consequently the next emersion is on jul. th past of clock in the evening ▪ if you add revolutions , another emersion is on jul. at h. ′ nearly p. m. as here is exemplified in the margin . the last thing i shall take notice of concerning the satellite eclipses is their durations . this varies according as jupiter is nearer unto , or remoter from the th degree of ♒ or ♌ ( as mr flamsteed says . ) about which points are the nodes , or intersections of the plane of the satellite orbit and jupiter's , or the jovial ecliptick . mr cassini makes it in of ♒ or ♌ , and varies in the length of the duration of the eclipses . but according to mr flamsteed ( the accuracy of whose observations is not to be distrusted ) the greatest semiduration of the   h ′ ″ first satellite is second — third — fourth — but as jupiter removeth from his nodes , the semidurations diminish . and when ▪ he is gotte n degrees from either of his nodes , the fourth satellite passeth clear of the shadow , and falleth not into it again , until he comes within degrees of the opposite node . when jupiter is on the limit , or degrees from his nodes , the least semiduration of the eclipse of the   h ′ ″ first satellite is second — third — from this account of the duration of the satellite eclipses , we may , having the immersion into jupiter's shadow , compute the emersion of any satellite out of his shadow : or contrariwise , which will be of use to see both the beginning and end of any eclipse , when visible ; i mean , when not hindred by clouds , day light , or jupiter's body . or if by some of these means we are hindred from seeing the one , we may perhaps hereby see the other . thus ( for instance ) this august . . the first satellite immerges at h. ′ p. m. which cannot be seen , not only by reason of day light , but also because jupiter's shadow lieth a little to the left of his body ; but if you add one whole obscuration ( viz. twice h . ′ ″ the emersion you will find visible at h ′ according to mr flamsteed ; at h. ′ according to monsieur cassini's tables . another instance will make all yet more plain , oct. . at h. ′ p ▪ m. the d satellite will emerge ; from which substracting one obscuration ( viz. twice h. ) you will find the immersion fall at h ▪ ▪ p. m. which may be seen , by reason that jupiter ▪ is at a good distance from his opposition to the sun , so that the shade lies so far on the left hand , as to admit of seeing the d and th satellite between jupiter's body and his shade . i might to these have added divers other remarks , particularly concerning the equation of light , or the time in which light passeth from the sun to jupiter , which is at last settled by that sagacious observer , so often before mentioned , mr flamsteed . but i must forbear , fearing that i have already wearied the readers patience , and shall need ▪ his pardon for detaining him so long on this subject , from so small an occasion , as only a satellite instrument of watch-work . but i was willing from a small occasion , rather than not at all , to say something to excite the observations and enquiries of others concerning this matter , which may be of vast use in navigation , making and correcting maps of countries , &c. many of those , to whom this matter would be of greatest use , scarce ever heard of it ▪ and others ( except monsieur cassini ) have been backward in favouring the world with their observations necessary to calculation . it is indeed a novel subject , and full of difficulties , on which little hath been written , and concerning which the first material observations , to be relied on , were hodierna's and mr rook's . those of the former were published , but not very accurate : those of the latter were more accurate , but not published , and neither of them are yet years old . but neither novelty nor difficulty ought to discourage the curious and the diligent ; to excite whom is partly the design of this digression . iii. to correct the motion of royal pendulums . in chap. . of the preceding book , i judged it to be a good expedient , to bring a pendulum to vibrate nicely , to add a bob underneath the pendulum ball. this i have since found to succeed so much according to expectation , that i think it frivolous to attempt by any of the usual ways to bring a large single ball to vibrate to one single beat , in any considerable quantity of time. but when the great ball is brought pretty near its due length , the little regulating bob will nicely perform the rest . the great ball being of the usual weight and form , to swing seconds , i would have the corrector , or regulating bob , to be about ounces troy , to scrw up and down beneath the ball ; as is directed in chap. before . but after all endeavours of this kind , it must be expected , that the movement will still be exposed to the influences of the weather , and the alterations caused by foulness . for the more easy and quick bringing of a pendulum , that should swing seconds to its true length , i have composed the following table , which sheweth the alterations which will be made in hours by screwing up , or letting down the great ball. if therefore the ball runs upon a rule divided into inches , and tenths of an inch , 't is easy to see how much , or how little the ball needeth to be altered . pendul . length variation of vibrat .   in ten min. sec.   faster .   slower . this table will need little explication . if your ball should be at inches tenths , it would swing seconds . if you alter it to inches , tenth , it would go ′ ″ faster : if to inches tenths , it would go ′ ″ slower . and so of the rest of the table . iv. of the equation of natural days . by reason that the sun's motion in his orbit is not equal , and that although he moved equal arches of the ecliptick in equal times , yet he would come to the meridian with unequal arches of the equator , by whose equal revolutions the equal time is measured ; hence ( i say ) it will happen , that altho a clock should go so exactly , as at the years end to agree with the sun , yet it will vary from the times shewed by the exactest sun-dials . the quantity of which variations may be seen in the following tables for every day in the year . for which tables i am greatly obliged to that most accurate astronomer mr flamsteed so often mentioned . these tables need but little explication . if you would keep your watch to the middle or equal motion of the sun , it must go so many minutes and seconds faster or slower than the sun-dial , as the tables shew . but if you would keep your watch to go by the sun-dial , you may conclude it goes well , if it loseth or gaineth every day , so many seconds as you will find in the table . thus ( for example ) jan. . in leap year , the watch ought to be min. sec. faster than mr flamsteed's tables of aequation of natural-days . the bissextile , or leap-year .   jan. febr. marc. april . may. june . july . aug. sept ▪ octo. nov. dec.   m. s.   m. s   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s         * l*   *           too           *   watch       slow . watch watch watch watch watch watch watch too watch watch watch * i* too too too too too too too fast . ** too too too watch. fast . fast . fast . slow . slow . fast . fast . too slow . slow . slow . too ● ●               slow .       fast . ●                                                 the first after leap-year .   jan. febr. marc. april . may june . july aug. sept. octob. nov. dec.   m. s   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s         * i*   *   watch       too ● ● ●       ●     *   too       slow . ● ● ● ● ● ● ● ● ● watch. watch watch ● watch watch watch watch fast . watch watch watch * i* ● ● ● ● too too too too too too too * i* too too too watch ● ● ● fast . fast . fast . slow slow . fast . fast . too slow . slow . slow . too . ●                 slow .       fast . ●             ●                   ●           ● ●           place these tables in the appendix between page and . the second after leap-year .   jan. feb. marc april ▪ may ▪ june . july . aug. sept. octob. nov. dec.   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s         * l*               too ● ● ●● watch watch watch watch watch * i* watch watch watch watch watch slow . too too too too too watch too too too too too * i* fast . fast . fast . slow . slow too fast . fast . slow . slow . slow . watch ●           fast .   too       too               slow .       fast .                                                     the third after leap-year . m jan. feb. marc. april . may june . july . aug. sep. octob. nov. dec. d m s   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s   m s         * l*               too ● watch watch watch watch watch * l* watch watch watch watch watch slow . too too too too too watch too too too too too * l* fast fast fast . slow . slow . too fast . fast . slow . slow slow watch ●           fast   * too       too ●               slow       fast .     ●     ●                                             the sun dial : on jan. . it ought to be ′ ″ , &c. if you would know on the same days , whether your watch goes well , when kept to go by the sun-dyal if set on jan. . it hath gained on jan. . as much as ′ ″ wanteth of ′- ″ . viz. ″ you may conclude your watch goes well . otherwise you must screw up , or let down the ball or corrector , until it loseth , or gaineth according to the equation tables . the tables will serve for many years , being mede for bissextile , and the years following . by an almanack therefore , or any other way , knowing the year , you may find what table you are to use all that year . by reason of the refractions , or some error in the sun-dial , it may be convenient to compare , or set your watch at some certain hour of the day . noon is a good time for it , if you have a nice meridian-line , or any way to see when the sun is exactly south , because the time of the day is not at all then varied by the refractions , in dials that cast a shade . v. to find a meridian-line . it may happen that we may be at a place , where there is no sun dial , or not one to be relied upon ; or indeed where we have a good one , it may be of great use to us to have a meridian-line . for the finding of which there are divers ways , but i shall shew only two . the first is , draw one or more circles on some plain , as on the bottom of a southern window . ( or you may make the center on the southern edge of the window , and draw only half circles . ) hang up a thread and plumbet exactly over , or in the center of the circles . by a bead or two sliding up and down the thread , mark out exactly the points of the circles , touched by the shade of the beads in some of the morning hours ( the longer before noon the better . ) in the afternoon when the same shade of the beads toucheth the circles , mark that point , or points also . a line drawn thro the center , and in the middle , between these two points in the circle , is the meridian-line , or near so . if you can't hang up a plumbet , a pin set exactly upright will do the matter . another , and better way , is by the pole star , when it is exactly upon the meridian . or if but near so , the error will not be great . you may find the time when the pole-star comes to the meridian , by substracting the suns right ascention from right ascention of the pole-star , and turning the remainder into hours , minutes and seconds , allowing to every degree four minutes of time , whereby you will have the apparent time , when the pole-star comes on the meridian above the pole. i scarce need to observe , that the time when it comes under the pole is hours distant . you may shorten your labour by using tables of the sun 's right ascention in time , which you may find in sir j. moor's mathem . compendium . note , if the sun's r. ascention exceeds the pole-stars r. a. you must add hours to the pole-star's r. a. & then substract . the right ascention of the pole-star is determined by mr flamsteed h ′ . ▪ ″ of time in the year , and the increase of its r. ascention ′ . ″ of time in years . therefore this present year its true r. ascention is h ′ . ″ of time . if the unlearned reader should think this way difficult , he may see when the pole-star comes near the meridian , by hanging up a line and plumbet , and observing when the first star in the great-bear's tail , next her rump , comes under the line on one side of the pole , or when the plumb-line intersects the star in cassiopeia's knee on the other side of the pole. when the pole-star is found to be on the meridian , if you hang up two strings with plumbets , between the pole-star and your eye , this will be a meridian-line , to see when the sun comes to the meridian . or you may do it with a crevis between two boards , or plates of metal , almost touching one another . or ( which is a better way ) with a pair of sights , such as surveyors use ( but much longer ) with a crevis in one sight next the eye ; and a large aperture in the other with a fine ca●s ▪ gut string down the middle . these should be counter-changed , so as to look either at the pole-star by night ; or the other way at the sun by day . page d . appendix ▪ fig. . fig. . your instrument being thus prepar d , plant it in some convenient place , where you may see the pole star , by night , and the sun by day . when the pole-star is on the meridian , look thro the sight with the bigger hole , and turn the whole instrument about until you see the opposite plumb line intersect the pole-star . take care at the same time , that the plumb-lines hang so as to intersect the sights . your instrument , thus plac'd , standeth nicely in the meridian , so as to see when either sun , moon , or stars come on the meridian . when you look by night , 't is necessary that a candle should shine on the plumb-line , that you may see it . if you look at the sun , you must guard your eye against the sun-beams with a coloured glass , or one blackened with the smoak of a candle . i had almost forgotten , to say that it matters not much what length the bottom piece , a. b. is of ( but the longer the better ) provided that the plumb-lines are high enough to see the pole-star , and the sun in the summer solstice , or any time of the year . if the bottom piece be feet long ▪ the plumb-lines had need to be near feet . this instrument is very serviceable to several purposes : particularly . to see the southing of the sun , or moon : which you may do with great exactness . you may see nicely when the very edge of the sun or moon toucheth the meridian , and whilst all their body is passing it . . you may see what stars are , at any time , on the meridian , either northward or southward , and so find the hour of the night . . you may with all exactness continue your meridian-line for many miles , if you please , by looking thro either sight , and seeing what objects the plumb-lines intersect . . if you would be still more nice , you may apply a telescope to this meridian instrument , by placing , for the eye-glass a convex glass , of a convenient focus at a due distance between the plumb-line and either sight , so as thro the sight to see the plumb-line thro the convex glass ( or eye-glass . ) and at a convenient distance from the instrument place another convex-glass for the object-glass . . if i am not much mistaken this meridian-instrument may as well ( and being made telescopulous ) much better serve the design of trying whether the meridian differeth or not ; which some have experimented with more trouble and expence than this instrument comes to . . this instrument is very easily brought to the meridian . for whether it stands upright , aside , or any other way , still the plumb-lines may be brought easily to their due place . . this instrument is prepared with little cost or trouble ; it may be carried from place to place ; or imitated where-ever there is occasion to correct either sun dial or watch. a table , shewing the time when the pole-star is on the meridian . m. january . february . march. april . august . septemb. october . novemb. decemb. d. hour . min.   hour . min.   hour . min.   hour . min.   hour . min.   hour . min.   hour . min.   hour . min   hour . min.   morning . morning . morn . evening . morning . morn . evening . evening . evening . even . ● even     this table is intended for the unskilful reader , to whom it may be of use for some years . but it will in time run out , by reason of the increase of the pole-star's r. ascention , leap year , &c. the hour and minute when the pole-star comes on the meridian is shewn every fifth day . but in may , june , and july it cannot be seen , when it is on the meridian , by reason of daylight . the table will be sufficiently explained by an example or two . jan. . the pole-star comes to the meridian at minutes after of clock in the morning ; at which time you may set your meridian-instrument . so you may do the same , on mar. th at , after of clock at night , at which time also the pole-star is on the meridian . vi. to make a telescope for the government of watches . in chap. . i mentioned a telescope for the governing a watch by the fixed stars . and because it is the nicest way i have mentioned ( by reason you may see a star pass at one beat of a pendulum ) therefore i shall here describe the way to make such a telescope , as is needful for this purpose . prepare your self with two convex glasses : the one ( for the object-glass ) to have its focus , or cons about feet , or according to the length you intend your telescope : the other glass ( for the eye ▪ glass ) about or inches . lodge these glasses in a tube of thin boards , pastboard , or what you think fit . between the object and eye-glass , at the focal distance of the eye-glass ( viz. about inches ) place two fine hairs or threads across , so as to be seen clearly when you look thro the eye-glass . let there be an aperture near these cross hairs , that the light of a candle may shine on them , in the night , when you look at a star. it is convenient that the eye-glass and cross-hairs or threads , should be lodged in a short lesser tube by themselves , so as to go into , and slide backward and forward , in the end of the larger tube ; whereby you may set the eye-glass and cross-strings nearer unto , or farther off from the object-glass . also there must be a conical socket of wood before the eye-glass , such as is usual in all telescopes , to look thro : but its perforation must be very small , so as only to give you leave to see the star through it . your telescope being thus prepared , you must plant it for observation , as is directed in the foregoing book . books printed for james knapton at the crown in st paul's church-yard . a new voyage round the world. describing particularly the isthmus of america , several coasts and islands in the west-indies , the isles of cape verd , the passage by terra del fuego , the south sea coasts of chili , peru , and mexico ; the isle of guam one of the ladrones , mindanao , and other philippine and east-india islands near cambodia , china , formosa , luconia , celebes , &c. new holland , sumatra , nicobar isles ; the cape of good hope , and santa hellena . their soil , rivers , harbours , plants , fruits , animals , and inhabitants . their customs , religion , government , trade , &c. by william dampier . vol. the first , illustrated with particular maps and draughts . the fourth edition , corrected . voyages and descriptions . vol. ii. in three parts , viz. . a supplement of the voyage round the world , describing the countries of tonquin , achin , malacca , &c. their product , inhabitants , manner , trade , policy , &c. . two voyages to campeachy ; with a description of the coast , product , inhabitants , logwood cutting , trade , &c. of jucatan , campeachy , new spain , &c. . a discourse of trade-winds , breezes , storms , seasons of the year , tides and currents of the torrid zone throughout the world ; with an account of natal in africk , its product , negro's , &c. by captain william dampier , illustrated with particular maps and draughts . to which is added , a general index to both volumes . the second edition . a short view of the principal duties of the christian religion . with plain arguments to perswade to the sincere and speedy practice of them . to which is added , a prayer suited to the whole , to be used morning and evening . by a divine of the church of england , for the use of his parishioners , price d. or s. per hundred . the god-fathers advice to his son. shewing ▪ the necessity of performing the baptismal vow , and the danger of neglecting it . with general instructions to young persons to lead a religious life , and prepare them for their confirmation , and worthy receiving the blessed sacrament . very necessary for parents , &c. to give their children , or others committed to their care . by john birket , vicar of milford and hordle in hampshire . the second edition , with a preface . price d. or s per hundred . mr. wingate's arithmetick : containing a plain and familiar method for attaining the knowledge and practice a common arithmetick . the tenth edition , very much enlarged . by john kersey , late teacher of the mathematicks . octavo . finis . notes, typically marginal, from the original text notes for div a -e philos . trans . dec. . no. .   days h. ′ ″ july revol . july revol . july of the unequality of natural time, with its reason and cavses. together with a table of the true æquation of natvral dayes : drawn up chiefly for the use of the gentry, in order to their more true adjusting, and right managing of pendulum clocks, and watches / by john smith ... smith, john, fl. - . approx. kb of xml-encoded text transcribed from -bit group-iv tiff page images. text creation partnership, ann arbor, mi ; 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(eebo-tcp ; phase , no. a ) transcribed from: (early english books online ; image set ) images scanned from microfilm: (early english books, - ; : ) of the unequality of natural time, with its reason and cavses. together with a table of the true æquation of natvral dayes : drawn up chiefly for the use of the gentry, in order to their more true adjusting, and right managing of pendulum clocks, and watches / by john smith ... smith, john, fl. - . [ ], p. printed for joseph watts ..., london : . includes table. reproduction of original in cambridge university library. created by converting tcp files to tei p using tcp tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between and available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the , texts created during phase of the project have been released into the public domain as of january . anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. % (or pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf- unicode or text strings within braces); or lossless xml (tei p , characters represented either as utf- unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng horology -- early works to . clocks and watches. - tcp assigned for keying and markup - aptara keyed and coded from proquest page images - emma (leeson) huber sampled and proofread - emma (leeson) huber text and markup reviewed and edited - pfs batch review (qc) and xml conversion of the unequality of natural time , with its reason and cavses . together with a table of the true aequation of natvral dayes . drawn up chiefly for the use of the gentry , in order to their more true adjusting , and right managing of pendulum clocks , and watches . by john smith , c. m. london , printed for joseph watts , at the half-moon in st. paul's church-yard . . a table of equations , for reducing the unequality of natural days to a mean and equal time . designed chiefly in order to the more true adjusting , and right managing of pendulum clocks and watches . by john smith , c. m. day . janua . sec febru . sec. march sec. april sec. may. sec. june . sec. july . sec. aug. sec. sept. sec. octob. sec. nov. sec. dec. sec. natural dayes longer than the mean day , and clocks gain . ☉ natural dayes shorter than the mean day , and clocks lose . natural dayes shorter than the mean day , and clocks lose .   natural dayes longer than the mean day , and clocks gain . natural dayes longer . nat. dayes shorter than the mean , and clocks lose . nat. dayes shorter than the mean , and clocks lose . nat. days shorter , clocks lose . natural dayes longer than the mean , and clocks gain . natural dayes longer than the mean , and clocks gain . natural dayes shorter than the mean day , and clocks lose . ☉ ☉ nat. dayes longer than the mean day , and clocks gain : ☉ ☉ nat. dayes shorter . ☉ ☉ ☉ longer .   ☉           clocks gain this month clocks lose this month clocks lose this month clocks lose this month clocks gain this month clocks gain this month clocks lose this month clocks lose this month clocks lose this month clocks lose this month clocks gain this month clocks gain this month sum min. sec. min. sec. min. sec. min. sec. min. sec. min. sec. min. sec. min. sec. min. sec. min. sec. min. sec. min. sec. this table contains those seconds of time that each natural day is either longer or shorter than the mean or equal day , or such a one as contains in length the just time of hours ; from which mean day the natural ones differing almost continually in length , this table shall still give you the difference between them , and ( by inspection only ) inform you what quantity of time each natural day is either more than hours long , or less . note , that upon each particular day , a clock that is well adjusted to a mean or hour day , shall then either gain or lose just so much time , as that natural day is either longer or shorter than the mean day : therefore , if a clock , having been first set right to the sun the first day of any particular month , shall either gain or lose in that whole month so much in time as the whole sum of equations for that month amounts to , ( which you shall find noted down at the bottom of every particular column ) then is it well adjusted to the mean , equal or hour day ; but if it have not either got or lost so much as the whole sum of equations for the whole time it has gone in do's amount to , then must its motion be regulated as occasion requires , by skrewing up the bob to make it go faster , in case it has not got enough ; or else letting the bob down lower to make it go slower , in case it hath not lost so much as the whole sum of equations do amount to for the time it has gone in . of the unequality of natural time , &c. the vibrations of a long and weighty pendulum , although it be justly esteemed to be the most exact and steady of all natural motion ; yet is it not capable of regulating the index of a clock to such a pitch of perfection , as continually to point out the same time that shall be given by the sun on the lines of an exact and true dial. the truth of which is sufficiently made evident by the most exact and critical experiments : for , let all the moving parts , of a pendulum clock , be contriv'd with the greatest skill and judgment , and then made up by the most cunning and curious hand , and after all this be adjusted by the utmost care and diligence of man ; yet shall not the motion of it correspond so continually with the hours given by the sun , but that in some considerable quantity of time you shall be sensible of gain or loss in the motion of it . the true reason of which variation proceeds not from any defect that may be attributed to the motion of the pendulum , ( of whose exactness we are by many curious experiments sufficiently sensible ; ) but rather from an unequality legible and easie enough to be discover'd in the diurnal motions of the sun. vulgarly , for the most part , the sun is indeed accounted to be the standard and measure of all equal time , and men generally esteem natural dayes to be all of one length , as containing the just time of hours ; but upon a more exact and curious scrutiny , these vulgar suppositions are found to be false : for , neither is the sun's motion found to be exact , being in appearance to us , sometimes swift , and at other times more slow ; nor are the dayes themselves , accounted from noon to noon , of equal lengths , some dayes containing more time than some , and others less ; there being a natural necessity , that the unequal motions of the sun should produce such inequalities in the lengths of those natural days that are bounded by it . for a natural day being composed of that space of time , in which any one place or point of the earth is moved in its diurnal motion east-ward , from the meridian sun of any one day , to that of the next ; it will follow , that these dayes can never be equal , unless the sun in that space of time be so mov'd in her annual orb , as to cut out equal divisions on the aequinoctial upon the meridian of every day ; which divisions so intersected , are by the learned termed , the right ascensions : for whenever the right ascension either of sun or star is mentioned , we are to understand by it , those degrees of the aequinoctial that are intersected by that meridian , on which either sun or star have then their place . but that the sun between each meridian does not move such just and certain spaces in her own orb , as thus to intersect equal divisions on the aequi-nox , upon every meridian , needs no other evidence than what either calculation it self affords , or globes by an occular inspection demonstrate to us . by calculation , if an exact table of right ascensions be composed , for the meridian , or noon-time of each particular day , there will be found almost a continual difference in the length of those intersections that are made by the sun on the aequi-nox , upon every meridian ; so that there will by this means be found nothing but an almost continual unequalness in the right ascensions ▪ which will be the more apparent , if you make an estimation of the right ascensions of about dayes together , and compare that with those of the same number . the like will appear plainly , if tryed on the globe ; for if you mark out on the ecliptick any , or more day 's motion of the sun , according to his true place found out in an exact ephimeris , and passing these , or more days motion under the meridian , noting what degrees on the aequmoctial are then traced out , which compared with the degrees traced out , by making the same number of day 's motion in some other part of the ecliptick , to pass the meridian , and the difference of right ascensions included between those two equal number of dayes will plainly appear . all which irregularities , or difference in right ascension , proceed from two principal causes : . from the different positions : and , . from the different centres of those orbs in which , and according to which the sun and earth do move ; from whence arises a natural necessity , that between two such regular and equal motions , whose position is thus oblique , those appearing differences should still arise ; for though both sun and earth , the one in his annual the other in it's diurnal revolutions , be rationally supposed to be regular and equal in their own motions ; yet in regard of the different positions of their spheres , the right ascensions that are made by them cannot be equal ; it being impossible that the sun , when near aries and libra , where he moves cross the equinoctial , should then in any particular number of days make so great an alteration in right ascension , as he must do near the two tropicks , where both equinoctial and ecliptick run paralel one to the other ; and accordingly by the best tables of right ascension , 't is found that the right ascensions of daies motion of the sun near the tropick of capricorn , shall arise to above degrees minutes , whereas that of the same number of daies near the equinoctial point of aries shall scarcely amount to degrees . moreover , from the eccentricity of these orbs , another irregularity does happen in the right ascensions , for the centre of the earth , upon which it turns round in it's daily revolutions , being not the same with the centre of the suns orb , it follows that the apparent equinoxes pointed out by an imaginary line drawn through the centre of the earth , and intersecting the ecliptick , shall divide that circle into two unequal parts ; from whence it arises , that the sun must spend more daies in passing through one half of the ecliptick , than he does in passing through the other ; and accordingly by experience he is found to move through that part between libra and aries in daies , but in passing between aries and libra he takes up , which is dayes more ; so that in that part of the year between september and march , he seems to us to be swift in motion , but in the other part between march and september his apparent course is more slow ; which seeming swiftness and slowness of the suns motion , is the cause that the right ascensions near both the tropicks are not alike , but differ much , as do also those that are nigh the equinoxes : for the right ascension of daies motion near the winter tropick , is more by minutes , than that of the same number of daies near that of the summer one ; so also the right ascension of daies time near libra , amounts to above minutes more than those of the same number of daies near aries does . having thus plainly demonstrated that natural daies must needs be unequal , and laid down the causes from whence those unequalities do still arise ; i suppose it may now be concluded to be extreamly unreasonable , for those that are so nice and curious as some are , to expect an exact correspondence between the times given by the motions of a clock , and those divisions of it that are made by these unsteady motions of the sun on the lines of a dial ; for if from the reasons before laid down , there be in nature a necessity for those differences of right ascension before asserted , and that the daies which they bound , must differ also in length , correspondent to what those differences in right ascension do amount to ; how then is it possible that those exact and regular motions of a pendulum , to what pitch soever it be set , should agree with these motions of the sun , and truely divide those daies that are not so regular as it self is : for , suppose a clock should be adjusted to the hour at a time when natural daies are shortest , as about the middle of march , this clock with the same pitch of motion , shall in june or december finish it's diurnal revolutions sooner than the day shall do , by reason the natural daies are now longer than those of march , to which the clock had been formerly adjusted ; and by consequence it shall now gain upon every day , just so much time as these daies in december are longer than those of march. so on the contrary , if a clock be adjusted to go true with the sun in the month of december , at which time the natural day is alwayes longest , this clock when natural daies are shorter , as in march or september , shall not finish it 's daily revolutions so soon as the day it self shall be accomplished , and by consequence go each day so much too slow , as those daies of march or september are more short than them to which the clock before had been exactly adjusted . since therefore there is no tolerable exactness in thus adjusting clocks to the sun it self , because being thus adjusted at times when daies are either shortest or longest , their gaining or losing will be the more extream in the contrary parts of the year : for example ; clocks adjusted to the sun in march shall upon most daies in december gain almost seconds , which in the months time shall amount to near half an hour ; and on the contrary , if adjusted to go true with the sun in december , it shall in march lose the same time , and so for any other , according as daies do differ in length . that clocks therefore may be reduced to a more exact pitch of motion , that their gain or loss may never be so extream , it will be necessary to adjust them , so as that their motion may be agreeable to that of a middle day , or such a one as is a mean between natural daies that are most long , and those others that are most short ; to which pitch if a clock be once adjusted , it 's gain or loss shall then be the less sensible ( for gain and lose it will still ) amounting in december but to about minutes in the whole month , and in march to but about , which is vastly more exact than when it shall happen to be adjusted to the longest or shortest of natural daies , or to any other that is not equal to a mean or middle day , of which there are but few , which in the table are exprest by the character ☉ sol. but to this exact pitch of motion , that may thus correspond to a mean day ( the greatest exactness that a pendulum is capable of being brought to ) there is no way certainly to adjust a clock without the help and assistance of a table of equations , that give the daily differences between a mean day and those which are either longer or shorter than the mean day is ; which equations having formerly been computed by the worthy and ingenious mr. christian hugens de zulechim , ( who is reported to be the first that ever applyed the pendulum to regulate the motion of a clock ) and not long since printed in number . philos . trans . i have made bold , in regard of its exactness , to transcribe in it's more natural form of an equation , by only expressing the equations themselves , without adding them together , and substracting , as mr. hugens has done for a particular use , to shew the nature of a pendulums going , when set right the first of february , and let go the whole year round , without setting afterwards . now for their sakes that desire to know the manner of composing such a table themselves , that thereby they may the better understand the nature of it ; they may note , that the equations are to be found out , and a table composed in the manner following : first find out a mean right ascension , by dividing the degrees of the equinox into parts , and a quarter , equal to the daies of a year , and the product shall be the mean right ascension desired , which will be found to be about three minutes seconds ( according to sir jonas moors account of it in his mathematical compendium ) then by the help of an exact ephemerides ( here lyes the difficulty ) let the natural right ascensions of the sun be computed by calculation , for the meridian position of the sun for every day , to minutes and seconds ; which having done , compare the daily differences of these natural right ascensions with the mean one , by still substracting the lesser from the greater ; and what remains shall be the equations desired ; still noting down either the excess or defect , that is , whether the natural be more than the mean or less : as for example ; suppose the right ascension between the meridians of the st . and d. of january be found to amount to minutes , seconds , this compared with the mean right ascensions , minutes , seconds , and by substracting the lesser from the greater , the remainder will be found to be seconds , and so much the natural right ascension does then exceed the mean one ; this seconds is the equation for that day , it being from noon to noon seconds longer than a mean day is ; and shews you , that a clock when well adjusted to a mean day , shall then gain seconds , because it finishes it's diurnal revolutions sooner by seconds than the day it self does : on the contrary , when the right ascensions of natural daies are less than the mean ones , as they are about the middle of march , by almost seconds , this seconds being the equations belonging to such a day , shall shew you , that upon such a like day a well adjusted clock shall then lose seconds ; for the mean day to which it is adjusted being longer than the natural one by seconds , the natural day shall be finished sooner by seconds than the clock at that time shall accomplish it 's diurnal or daily revolutions , and by consequence it shall then lose seconds . the equations thus found for every particular day , and a table composed of them , shall resemble that which is here inserted , whose use we now come to shew more particular in some cases . for explanation , take notice , that the first column contains the daies common to every month , the other columns that belong to the several months themselves , contain those seconds of time that all natural daies are either longer or shorter than the mean day . note , that in four parts of the table are placed this character ☉ , which denotes the times wherein natural daies having before been longer than the mean day , do then begin to be shorter ; or having before been shorter , do then begin to grow more long : note also , that those daies upon which this character ☉ is affixed , have no equation , they being equal in their length to the mean day ; as for the words inserted among the columns , they are at sight to inform you , that the equations in those parts of the table are either more or else less than the mean day , as the words themselves do fully express ; they also note , that where the equations are more , there clocks shall gain each day so much as the equation belonging to it does then express ; but if the equations are less , they then shall lose ; and how much this gain or loss for every particular months time shall amount to , is by continual addition of the equations belonging to each day summed up , and the quantity of time it amounts to , set down apart at the bottom of every column . note also , that since clocks do either gain or lose , during the whole number of daies included between those daies on whom this character ☉ is affixed , the whole quantity of time either got or lost does amount to the summs that follow , viz. between the st of february and the th of may , the time that a well adjusted clock shall lose , amounts by continual adding the equations together , to about minutes , seconds ; between this th of may and the th of july it shall gain about minutes seconds ; from the th of july to the d of october , it shall lose minutes seconds ; from the d of october to the last of january , it shall gain minutes seconds : all this is to be understood of a well adjusted clock , set right to the sun at the beginning of each time of either gaining or losing . by this table , if you would adjust a clock to a mean time , which is the greatest exactness to which it's possible to be brought , do thus : first set it true to the sun , and note the day , then let it's motion be continued without setting a new , for about or more daies : observe then the time that it has got or lost by the sun , then summ up the whole number of seconds included in the table , between those two daies of first setting and last observation ( allowing seconds to a minute ) and if the gain or loss of your clock be equal to the summ of time that it should have gained or lost by the table , then is it well adjusted ; but if it have not , then must its motion be reduced to a more near agreement , by shortning the pendulum in case the clock have gone too slow , or letting the bob down longer in case it have gone too fast : then set it anew , and try it for about daies more , and then comparing its loss or gain with the summ of those equations contained in the table , as before you did , let the bob be again rectified as the nature of it's motion requires ; and continue to do thus , till you find its gain or loss exactly to correspond with the summ of time given by the equations contained in the table , for the time that the clock has gone . when it is thus well adjusted to a mean time , it will be so exact , as that , being set right at any time of the year , and so let go the whole year about , it shall come right with the same dial by which it was set the same day twelve-month ; but ●n all other parts of the year it shall still differ from the same dial. for example : if set right the first of february , and so continued in motion the whole year about , it shall continually be too slow the whole year , either more or less , till the same day on which it was set : the reason of this is plain enough ; for from the first of february to the th of may , it shall continually lose to the quantity of minutes , seconds ; then from the th of may to the th of july , it shall gain ; but this gaining amounting to but about nine minutes , seconds , it shall still be too slow by minutes , seconds ; because its gaining now shall not be so much as it lost before , by minutes , seconds : then again , from the th of july it shall lose afresh till the st of october ; which second loss amounting to about minutes , seconds , this added to the time that it was too slow on the last account , shall amount to minutes , seconds , and so much it shall be too slow on the st of october ; from whence it shall gain afresh till the last of january , to the quantity of minutes , seconds ; which being equal to what it was before too slow , shall cause it to come right to the same dial with which it was set twelve months before , altho' it went too slow the whole year beside . again , let a clock be set right the d of october , it shall from thence gain time till the last of january ; and this gain shall amount to minutes , seconds ; then from the first of february to the th of may , it shall lose minutes , seconds , which being less than the minutes , seconds , which before it had got , by about minutes , seconds , it shall still be too fast by minutes seconds : then from the th of may , to the th of july , it shall gain anew to the quantity of about minutes seconds ▪ which added to the time it was too fast before , shall amount to minutes , seconds , and so much it shall be too fast on the th of july ; from which time till the d of october , it shall lose this minutes , seconds , and by consequence come right to the same dial with which it was set twelve months before . thus shall one and the same clock , with the same pitch of motion , go alwayes too slow if set at one time of the year , and always too fast if set at another time , if it be let go the whole year about . moreover , if set at some other times , and then continued in its motion for a year , without setting anew , it shall both gain and lose , be sometimes too fast ▪ and sometimes too slow : for if a well adjusted pendulum be set right to the sun the th of may , by the th of july it shall be minutes , seconds too fast : from this th of july to the d of october , having lost minutes , seconds ; from which substracting the minutes , seconds that it was before too fast , there remains minutes , seconds ; and so much it shall be too slow on the d of october ; from which day it shall begin to gain , and continue so to do till the first of january , by which time the clock having got minutes , seconds , which amounting to about minutes , seconds above what it was too slow on the d of october , it shall by consequence be now minutes , seconds too fast ; from whence to the th of may , it shall lose what now it is too fast , and so come right to the same dial with . which it was before set . again , set a clock to the sun the th of july , and if it be well adjusted , it shall by the d of october be minutes , seconds too slow ; from whence to the last of january , it being to gain minutes , seconds , it shall be then minutes , second ▪ too fast ; from which time to the th of may , it losing minutes , seconds , it shall then be minutes , seconds too slow ; which time by the th of july shall again be got , and so the clock shall come right to the same dial. thus , by this table , are these great varieties discoverable in the motion of the best adjusted pendulum , according to the different times of the year that it is set in ; that the same pendulum set right upon the first of february , shall go always too slow till the same day twelve-month ; but if set right the d of october , it shall the whole year round be still too fast , till the same day on which it was set : the same clock being also set to the hour on the th of may , or the th of july , shall on the following year be sometimes too fast , and at other times too slow . these are all strange and unaccountable things to such as understand not the nature of the unequality of time , from whence all these varieties do still arise , and are scarcely to be demonstrated to the understanding by any other way , than by this or some other table of the aequation of time. since therefore there is a necessity for setting clocks a-new to the sun , at some times , that they may be kept as near as possible to the apparent time given by a dial ; i advise that this setting may be , if possible , the first day of every month ; so may you the better discern by the time set down at the bottom of every column , whether your clock have gone right to the mean time , and be as exact in his motion as 't is possible to bring him to : for if it be exactly adjusted , it will then either gain or lose near that time that is set down at the bottom of every month , those sums being nothing else but the aequations of the whole month added together into one summ , as before i directed , when i gave you the method of adjusting a pendulum to the true or mean time : but in case you cannot set it right the first day , then must you be at a little trouble to add together the aequations your self , at such time as you come to set it a-new ; but when the clock is well adjusted , there needs little of this trouble , being assured that it 's brought to the nearest pitch of motion it 's capable of ; and that when it is at any time found to differ from the sun , it must be lookt upon as the natural consequence of the unequality of time , and not any deficiency in the motion of the clock . i speak this of those long and curious pendulums that vibrate within the compass of or inches ; for the less compass a pendulum takes , the more steady is it's motion , not being so subject to rise and fall , as others are , that vibrate in a larger compass . as for those shorter pendulums of a foot long , or under , although they may go very steady for the most part , if frost or foulness hinder not , yet are they not at all intended in this discourse , their motion being apt to an alteration in some cases ; for a short pendulum that goes well when clean , shall go faster than the mean time when foul , because the pendulum is by the foulness hindred from taking its wonted compass : the same effect does frost produce ; for by congealing the oyl in the pevets , its freedom of motion is interrupted , so that the pendulum not fetching it's wonted compass , shall go too fast ; but those long and curious pendulums of inches , that fetch not above three inches compass , they are so exact , that being once adjusted they shall alwayes keep the same time , if their motion continue ; for if the pendulum should fetch a smaller compass , their motion would cease , and themselves stand still . when such a pendulum as this is well adjusted , you may trust to it , as to it's correspondence in motion with the mean time , and only give your self the trouble sometimes to set it a little forward or backward , according to what the unequality of time has made it to differ from the same dial with which you did use to set it ; which times of setting may be , as i said before , once in a month ; yet if the table be well noted , you shall find , there be some times in the year in which a good clock may go a longer time without any material difference from the time given by a dial : for example , suppose you set a clock right to the sun the first of january , this clock if let go till the first of march , shall then be but two minutes four seconds too fast ; for though it gain in january six minutes , no great matter , yet losing in the next month about four minutes , it shall at the end of that month be but two minutes to fast ; nay , if let go another month , as to the last of march is , it shall then differ but about seven minutes , which is no great matter . moreover , if you set a clock the first of may , it shall , if let go till june , lose but two minutes thirty five seconds in the whole , though it shall at the beginning of may be about five minutes to slow , which is no great matter ; so also if it be set right the first of june , it may well go without setting till the first of september ; for though it gain five minutes in june , yet losing eight minutes in august , it shall then be but about three minutes too slow in this three months time ; but at some other times , as from february to may , if a clock were let go 't will be extreamly out , losing above eighteen minutes ; so also in november , december and january , in which time it will gain above thirty minutes . by the table you may also readily find out , what difference there is between the lengths of any two daies in this manner : first , if the daies are both shorter , or both longer than the mean day , then substract the equation in the table belonging to one day from the equation of the other , and the remainder shall shew their difference in length . but if the daies be one longer than the mean , and the other shorter , then add the two equations together , and the summ shall be the time that they differ in length : thus the tenth of january will be found to be thirteen seconds longer than the tenth of may ; also the fifteenth of september will be found to be fifty one seconds shorter than the fifteenth of december . note , that if men be very nice in keeping a clock true to the sun , they should then make use , if possible , of but one time on the dial that they set it with , and that pretty near noon ; for few dials being drawn exactly true , great mistakes may arise , when a clock is set to one hour and then compared with another ; and by reason of refractions great errours may also arise ; for the sun by refractions being made to appear higher than really she is , there can be no certain account taken of the time till near noon , where refractions cease : and when all this care is taken , in regard it is so very hard to distinguish minutes by the shadow of a dial , you will be much more exact if you do thus : let two plain and flat plates or boards , about ten inches square , be joyned so close together that a six-pence may but just go between ; let them be fixed so , that this cranney between them may respect the true south ; this will give you the time to less than half a minute , by observing the first moment that the suns beam is darted through it , and cast upon some dark body that is plac●d on the north-part to receive the light ; so that having thus the exact time , when the sun comes to every meridian , it will be found a much better way to adjust clocks by such a device as this , than by the truest dial. one use more , and that a principal one , i shall add , that is this : when a clock is once well adjusted to the mean or equal day , you may then by this table keep it right to the time given by the sun , although you never set it right to a dial , nor see the sun above once a year ; to perform this do thus : set your clock right to the sun the first day of any one month , and then the table still giving you the time it should lose or gain in that whole months ●ime , 't is then but setting it forward or backward the first day of the next month what it either hath got or lost in the month immediately preceding , and it will then be right with the sun as if it had been set by a dial , and so from month to month you may ( by still setting it either forward or backward according to what the table tells you it will gain or lose ) keep it true to the time given by the sun , though it should never shine so as to give you an opportunity to set it by a dial above once in the whole year ; but be sure your clock be first well adjusted , or else there may be some errour : note , that for this purpose it will be very convenient to paste the table it self on a board , and then putting it into a handsom frame , let it be hung up near the clock , that you may have ready recourse to it on all occasions , especially for this in particular ; for without the assistance of such a table as this , i know no way in the world to keep a clock right to the apparent time , in case clouds should intercept the beams of the sun for any long time together . finis . books sold by joseph watts , at the half-moon in st. paul's church-yard . josephus's wars of the jews . fol. the tryals of thomas walcot , william hone , william lord russel , john rouse , and william blagg . the confession , prayers and meditations of john stern , published by doctor burnet and doctor horneck . the earl of danby's case , and answer to sir robert howard , and his answer to the examination of his case , and reply . the bishops of england may , and ought to vote in cases of blood. baxter against baxter . a serious expostulation with the whiggs in scotland . dr. jane's sermon before the commons , april . . dr. perinchief against toleration . fowler 's catechism of the church of england poetically paraphrased . cave's sermon on the th of january . james's visitation sermon , and on the th of december , . a disquisition upon our saviours sanction of tythes , mat. . . luk. . . stubbs's justification of the dutch war , in two parts , with figures . a table of the equation of days, shewing how much a good pendulum watch ought to be faster or slower than a true sun-dial, every day of the year. tompion, thomas, - . approx. kb of xml-encoded text transcribed from -bit group-iv tiff page image. text creation partnership, ann arbor, mi ; oxford (uk) : - (eebo-tcp phase ). b wing t a estc r ocm 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 . 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 , no. b ) transcribed from: (early english books online ; image set ) images scanned from microfilm: (early english books, - ; : ) a table of the equation of days, shewing how much a good pendulum watch ought to be faster or slower than a true sun-dial, every day of the year. tompion, thomas, - . sheet ([ ] p.) printed for tho. tompion, clockmaker ..., london : . caption title. reproduction of original in: national library of scotland. created by converting tcp files to tei p using tcp tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between and available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the , texts created during phase of the project have been released into the public domain as of january . anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. % (or pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf- unicode or text strings within braces); or lossless xml (tei p , characters represented either as utf- unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng clocks and watches -- calibration -- early works to . time, equation of -- early works to . sundials -- early works to . broadsides -- england -- th century. - tcp assigned for keying and markup - spi global keyed and coded from proquest page images - megan marion sampled and proofread - megan marion text and markup reviewed and edited - pfs batch review (qc) and xml conversion a table of the equation of days , shewing how much a good pendulum watch ought to be faster or slower than a true sun-dial , every day in the year . days . januar. februa . march april . may. june . july . aug. sept. octob. nov. dec. mi.   sec. mi.   sec. m.   sec. m.   sec. m.   sec. m.   sec. m.   sec. m.   sec. mi.   sec. mi.   sec. mi.   sec. m.   sec. watch too fast . watch too fast . watch too fast . w. too f. * watch too slow . w. too slow . * watch too fast . watch too fast . * watch too slow . watch too slow . watch too slow . watch too slow . * * w. too slow . * watch too fast . * watch too fast . * watch too slow .                             set the watch so much faster or slower than the time by the sun , according to the table for the day of the month , when you set it ; and if the watch go true , the difference of it from the sun any day afterward will be the same with the table . london , printed for tho. tompion , clockmaker , at the three crowns in fleet-street , at water-lane end . .