✓ \ ST ^ 0 s ' ^ Digitized by the Internet Archive in 2015 https://archive.org/details/selectmechanicalOOferg" SELECT Mechanical Exerdfes : Shewing how to contact different CLOCKS, ORRERIES, and SUN-DIALS, ON PLAIN AND EASY PRINCIPLES. WITH SEVERAL MISCELLANEOUS ARTICLES; AND NEW TABLES, I, For expeditioufly computing the Time of any NEW or FULL MOON within the Limits of 6000 Years before and after the 18th Century. II. For graduating and examining the ufual Lines on the SECTOR, PLAIN SCALE, and GUNTER. Illuftrated with COPPER-PLATES. To which Js prefixed, A fhort Account of the Life of the Author. By JAMES FERGUSON, F. R. S. LONDON: Printed for W. Strahan: and T. Cadell, in the Strand, MDCCLXXIIIa - ' •r 0 i ..{ , ' / , .1 ? * • ♦- • - ■ > x.» - .* .... *». ■' X •*■ I "% > f 1 i + •«/ V, . : ■ ’? •A ‘ \#r' . 4 .* V ... b c: ... j 3 T / - 7 f V • J ^ : v . .. o ' r, ; W ' 1 ! ! 1 ;,/i -. ■ ■iZ-iJ. :n A .v r — ."7 a >r V/* . ; -r: : ) ' ; ■ ' V ' • . - f r \ t ; t . , 1 > ':i ' *: •: f. ■ t * ^ .,,i • . a- % ’ ' . y ■ . * - * - . 1. 1 . - .?»V, f I • T _ ? ; ' ^ f r ' T TS ' \ r < ■ : • ^ * X. (* ' x- » ?A . ■ < i 4 y . • ...j % ■ ■ '• . 'vl'. SELECT EXERCISES. A Clock Jhewing the Hours, Minutes , and Seconds, having only three Wheels and two Pinions in the whole Movement . Invented by Dr. Franklin of Phila- delphia. fTlHE dial-plate of this clock is reprefented by Fig. i . of Plate I. The hours are engraven in fpiral fpaces, along two diameters of a circle containing four times 60 mi- nutes. The index A goes round in four hours, and counts the minutes from any hour it has palled by, to the next following hour. The time, as it appears in the figure, is either 324. B minutes 2 SELECT EXERCISES. minutes pail XII, or paft IIII, or paft VIII ; and fo on, in each quarter of the circle, pointing to the number of minutes after the hours the index laffc left in its motion. Now, as one can hardly be four hours miftaken in efti- mating the time, he can always tell the true hour and minute, by look- ing at the clock, from the time he rifes till the time he goes to bed. The l'mall hand B, in the arch at top, goes round once in a minute, and fhews the feconds, as in a common clock. Fig. 2. fhews the wheel-work of this clock. A is the firil or great wheel, it contains 160 teeth, goes round in four hours, and the index A (Fig. i .) is put upon its axis, and moved round in the fame time. The hole in the index is round, it is put tight upon the round end of the axis, fo as to be carried by the motion of the wheel, but may be fet at any time to the proper hour and minute, without afFecfting either the wheel or its axis. This wheel of 160 teeth turns a pinion B of SELECT EXERCISES* 3 B of io leaves ; and as 10 is but a 16th part of 160, the pinion goes round in a quarter of an hour. On the axis of this pinion is the wheel C of 120 teeth; it alfo goes round in a quarter of an hour, and turns a pinion Z), of 8 leaves, round in a minute ; for there are 15 minutes in a quarter of an hour, and 8 times 15 is xs?o. On the axis of this pinion is the fecond-hand B (Fig. 1.) and alfo the common wheel E (Fig. 2.) of 30 teeth, for moving a pendulum (by pallets) that vibrates feconds, as in a common clock. This clock is not defigned to be \vound up by a winch, but to be drawn up like a clock that goes only 30 hours. For this purpofe, the line muft go over a pulley on the axis of the great wheel, as in a common 30 hour clock. Several clocks have been made according to this ingenious plan of the Do&or’s ; and I can affirm, that I have feen one of them, which meafures time exceedingly well. B 2 The 4 SELECT EXERCISES. The fimpler that any machine is, the better it will be allowed to be, by every man of fcience. Another Clock that Jhezvs the Hours, Mi- nutes, and Seconds, by means of only three Wheels and tzvo Pinions in the 'whale Movement, As Dr. Franklin, whom I rejoice to call my friend, is perhaps the lad perfon in the world, who would take any thing amifs that looks like an amendment or improvement of any fcheme he propofes, I have ventured to offer my thoughts concerning his clock, and how one might be made as fimple as his, with fome advan- tages. But I mud confefs, that my alteration is attended with fome in- conveniences, of which his are en- tirely free. — I fhall mention both, to the bed of my knowlege, that they who chufe to have fuch fimple and cheap SELECT EXERCISES. 5 cheap clocks may have them made in either way they pleafe. The DoiStor’s clock cannot well be made to go a week without drawing up the weight ; and if a perfon wakes in the night, and looks at the clock, he may poflibly be miftaken four hours in reckoning the time by it, as the hand cannot be upon any hour, or pafs by any hour, without being upon or palling by four hours at the fame time. To avoid thefe inconve- niences, I have thought of the fol- lowing method. In Fig. i. of Plate II. the dial-plate of fuch a clock is reprefented ; in which there is an opening abed be- low the center. Through this open- ing, part of a flat plate appears, on which the twelve hours are engraved, and divided into quarters. This plate is contiguous to the back of the dial- plate, and turns round in 12 hours; fo that the true hour, or part thereof, appears in the middle of the opening, B 3 at 6 SELECT EXERCISES. at the point of an index A, which is engraved on the face of the dial -plate. — B is the minute-hand, as in a com- mon clock, going round through all the 60 minutes on the dial in an hour; and, in that time, the plate feen thro' the opening abed fhifts one hour un- der the fixt engraven Index A. By thefe, you always know the hour and minute, at whatever time you view the dial-plate. — In this Plate is another opening e fg h , thro' which the feconds are feen on a flat mover able ring, almoft contiguous to the back of the dial-plate ; and, as the ring turns round, the feconds upon it are fhewn by the top-point of a fleur- de-lis C, engraved on the face of the dial-plate. Fig. 2. reprefents the wheels and pinions in this clock. A is the firfl: or great wheel; it contains 120 teeth, and turns round in 12 hours. On its axis is the plate on which the 12 hours above mentioned are engraved. This SELECT EXERCISES. 7 ■ * * plate is not fixed on the axis, but is only put tight upon a round part thereof, fo that any hour, or part of an hour, may be fet to the top of die fixed index A (Plate I.) without af- fecting the motion of the wheel. For this purpofe, twelve fmall holes are drilled through the plate, one at each hour, among the quarter divifions: and, by putting a pin into any hole in view, the plate may be fet, without affefting any part of the wheel-work. This great wheel A, of 120 teeth, turns a pinion B of 10 leaves round in an hour ; and the minute-hand B (Fig. 1.) is on the axis of this pinion, the end of die axis not being fquare, but round; that the minute-hand may be turned occalionally upon it, with- out affecting any part of the move- ment. On the axis of the pinion B is a wheel C of 120 teeth, turning round in an hour, and turning a pinion D of 6 leaves in 3 minutes ; for 3 mi- nutes is a loth part of an hour, and 6 B 4 is 8 SELECT EXERCISES. is a 20th part of 120. On the axis of this pinion is a wheel E of 90 teeth, going round in 3 minutes, and keep- ing a pendulum in motion that vi- brates feconds, by pallets, as in a common clock, where the pendulum wheel has only 30 teeth, and goes round in a minute. — But, as this wheel goes round only in 3 minutes, if we want it to fhew the feconds, a thin plate mull be divided into 3 times 60, or 180 equal parts, and numbered 10, 20, 30, 40, 50, 60; 10, 20, 30, 40, 50, 60; 10, 20, 30, 40, 50, 60 ; and fixed upon the fame axis with the wheel of 90 teeth, fo near the back of the dial-plate, as only to turn round without touching it : and thefe divifions will fhew the feconds, thro’ the opening e f g h in the dial-plate, as they Hide gradually round below the point of the fixed fleur-de-lis C. As the great wheel A, and pulley on its axis over which the cord goes (as in a common 30 hour clock) turns. rouncf SELECT EXERCISES. 9 round only once in 12 hours, this clock will go a week with a cord of common length, and always have the true hour, or part of that hour, in fight at the upper end of the fixed in- dex A on the dial-plate. Thefe are two advantages it has beyond Dr. Franklin ’s clock : but it has two difad- vantages of which his clock is 'free. For, in this, although the 12 hour wheel turns the minute-index B, yet, if that index be turned by hand, to fet it to the proper minute for any time, it will not move the 12 hour plate to fet the correfponding part of the hour even with the £op of the index A: and therefore, after having fet the minute- index B right by hand, the hour-plate mull be fet right by means of a pin put into the fmall hole in the plate juft below the hour. ’Tis true, there is no great matter in this ; but I have feme fufpicion that the pendulum wheel E having go teeth inftead of the common number 30, may be fome difadvantage JO SELECT EXERCISES, difadvantage to the fcapement, on account of the fmallnefs of the teeth ; and ’tis certain, that it will caufe the pendulum-ball to defcribe but fmall arcs in its vibrations. Indeed fome men of fcience think fmall arcs are beft ; but if they really are, I mull confefs myfelf ignorant of the reafon. For, whether the ball defcribes a large or a fmall arc, if the arc be nearly cycloidal, the vibrations will be per- formed in equal times ; the time then depending entirely on the length of the pendulum-rod, not on the length of the arc the ball defcribes. The larger the arc is, the greater is the momentum of the ball ; and the greater the momentum is, the lefs will the times of the vibrations be affedted by any unequal impulfe of the pendulum- wheel upon the pallets. But the word thing about this clock (and what every one will allow to be a difadvantage) is, that the weight of the flat ring on which the feconds are engraved. SELECT EXERCISES. it engraved, will load the pivots of the axis of the pendulum-wheel with a great deal of friction, which ought by all pollible means to be avoided ; and yet I have feen one of thefe clocks (lately made) that goes very well, not- withftanding the weight of this ring. For my own part, I think it might be quite left out ; for feconds are of very little ufe in common clocks not made ' for aftronomical obfervations ; and table-clocks never have them. A Clock fhewing the apparent daily Mo- tions of the Sun and Moon , the Age and Phafes of the Moon , •. with the Time of her coming to the Meridian, and the Times of High and Low Water , by ha- ving only two Wheels and a Pinion added to the common Movement. The dial-plate of this clock is re- pefented by Fig. i. of Plate III. It contains *2 SELECT EXERCISES. contains all the 24 hours of the day and night. S is the Sun, which ferves as an hour-index, by going round the dial-plate in 24 hours ; and M is the Moon, which goes round in 24 hours 507 minutes, from any point in the hour-circle to the fame point again, which is equal to the time of the Moon’s going round in the heavens, from the meridian of any place to the fame meridian again. The Sun is fixed to a circular plate (as Fig. 3.) and carried round by the motion of that plate, on which the 24 hours are en- graven, and within them is a circle divided into 29^- equal parts for the days of the Moon’s age, accounted from the time of any new Moon to the next after; and each day Hands di- rectly under the time (in the 24 hour circle) of the Moon’s coming to the meridian, the XII under the Sun Hand- ing for mid-day, and the oppofite XII for midnight. — Thus, when the Moon is 8 days old, fhe comes to the meri- dian SELECT EXERCISES. 13 dian at half an hour pall VI in the af- ternoon ; and when fhe is 1 6 days old, Ihe comes to the meridian at I o’clock in the morning. The Moon M (Fig. 1.) is fixed to another circular plate, of the fame diameter with that which carries the Sun ; and this Moon-plate turns round in 24 hours 504 minutes. It is cut open* fo as to lhew fome of the hours, and days of the Moon’s age, on the plate below it that carries the Sun, and, acrofs this opening, at a and b are two fhort pieces of fmali wire in the Moon-plate. The wire a fhews the day of the Moon’s age, and time of her coming to the meridian, on the plate below it that carries the Sun ; and the wire b fhews the time of high water for that day, on the fame plate. Thefe wires mull bq placed as far from one another, as the time of the Moon’s coming to the meridian differs from the time of high water at the place where the clock is intended to ferve. — At London-bridge, 14 SELECT EXERCISES. it is high water when the Moon is twd hours and an half paft the meridiam Above this plate, that carries the Moon, there is a fixed plate AT* fup- ported by a wire A, the upper end of which is fixed to that plate, and the lower end is bent to a right angle, and fixed into the dial-plate at the lower- tnoft or midnight XII. This plate may reprefent the Earth, and the dot at L London, or any other place at which the clock is defigned to fhew the times of high and low water. Around this plate is an elliptical fhade upon the plate that carries the Moon M: the higheft points of this fhade are marked High Water , and the loweft points Low Water. As this plate turns round below the fixed plate N, the high and low water points come fucceffively even with L, and Hand juft over it at the times when it is high or low water at the given place ; which times are pointed out by the Sun $ among the 24 hours on the dial-plate i and s SELECT EXERCISES. 15 and, in the arch of this plate, above XII at noon, is a plate H that rifes and falls as the tide does at the given place. Thus, when it is high water (fuppofe at London) one of the higheft points of the elliptical fhade ftandsj uft over L, and the tide-plate H is at its great- eft height : and when it is low water at London, one of the loweft points of the elliptical fhade Hands over L, and the tide-plate H is quite down,, fo as to difappear beyond the dial-plate. As the Sun S goes round die dial- plate in 24 hours, and the Moon M goes round it in 24 hours yoi minutes, the Moon goes- fo much flower than the Sun as to make only 284, revolu- tions in the time the Sun makes 294.; and therefore the Moon’s diftance from the Sun is continually changing ; fo that, at whatever time the Sun and Moon are together, or in conjunction, in 297 days afterward they will be in conjunction again, Confequently, the plate that carries the Moon moves fo 6 much SELECT EXERCISES. much flower than the plate that car* ries the Sun, as always to make the wire a fhift over one day of the Moon’s age on the Sun’s plate in 24 hours. In the plate that carries the Moon, there is a round hole m, thro’ which the phafe or appearance of the Moon is feen on the Sun’s plate, for every day of the Moon’s age from change to change. When the Sun and Moon are in conjunction, the whole fpace feen through the hole m is black : when the Moon is oppoflte to the Sun (or full) all that fpace is white : when ftie is in either of her quarters, the fame fpace is half black half white : and different in all other pofitions, fo as the white part may refemble the vilible or enlightened part of the Moon for every day of her age. To fhew thefe various appearances of the Moon, there is a black fhaded fpace (Fig. 3.) as NfFl, on the plate that carries the Sun. When the Sun and Moon are in conjunction, the whole SELECT EXERCISES. if Whole fpace feen through the round hole is black, as at N: when the Moon is full, oppofite to the Sun, all the fpace feen through the round hole is ■white, as at F: when the Moon is in her firft quarter, as at f, or in her laft quarter, as at /, the hole is only half fliaded ; and more or lefs accordingly for each pofition of the Moon with regard to her agej as is abundantly plain by the Figure. The wheel-work and tide-work of this clock is reprefented by Fig. 2. in which A and B are two wheels of equal diameters. A has 57 teeth, its axis is hollow, it comes through the dial of the clock, and carries the Sun- plate with the Sun (£, in Fig. 1.). B has 59 teeth, its axis is a folid fpindle, turning within the hollow axis of A, and carrying the Moon-plate with the Moon (M, in Fig. 1.). A pinion Cof 19 leaves takes into the teeth of both the wheels, and turns them round. This pinion is turned round, by the common clock-work, in 8 hours ; and, C as i8 SELECT EXERCISES. as 8 is a third part of 24, fo 19 is a third part of 57 : and therefore the wheel A , of 57 teeth, that carries the Sun, will go round in 24. hours ex- actly. But, as the fame pinion C (that turns the wheel A of 57 teeth) turns alfo the wheel B of 59 teeth, this laid wheel will not turn round in lefs than 24 hours 504. minutes of time: for as 57 teeth are to 24 hours, fo are 59 teeth to 24 hours jot minutes, very nearly. On the back of the Moon-wheel of 59 teeth is fixed an elliptical ring Z>, which, as it turns round, raifes and lets down a lever E F, whofe center of motion is on a pin at F ; and this, by means of an upright bar G, raifes and lets down the tide-pnare //, twice in the time of the Moon’s revolving from the meridian to the meridian again. The upper edge of this plate is fhewn at H, in Fig. 1. and it moves between four rollers R, R, R, R , in Fig. 2. I have made one of thefe clocks to go by the movement of an old watch, in the following manner . The ’■ '"U. A SELECT EXERCISES. \9 The firft, or great wheel of a watch 3 goes round in four hours. I put a wheel of 20 teeth on the end of the axis of that wheel, to turn a wheel of 40 teeth on the axis of the pinion C; by which means, that pinion is turn- ed round in 8 hours, the wheel A in 24 hours, and the wheel B in 24 hours 50* minutes. — I never faw nor heard of any other clock of this kind. An ajlrommical Clock , Jhevuing the apparent daily Motions of the Sun , Moon , and Stars , voith the Times of their rifing , fouthing , and fetting , the Places of the Sun and Moon in the ‘ Ecliptic , and the Age and Phafes of the Moon, for every Day of the Tear . The dial-plate of this clock is repre- fented by Fig. 1. of Plate IV. It con- tains all the 24 hours of the day and C 2 night, ao SELECT EXERCISES. night, and each hour is divided into 1 2 equal parts, fo that each part an- fwers to $ minutes of time. Within thefe diviiions of the hour- circle is a flat ring, the face of which is juft even (or in the fame plane) with the face of the hour-circle. This ring is divided into 297 equal parts (num- bered 1, 2, 3, 4, &c. from the right hand toward the left) which are the days of the Moon’s age from change to change : the ring turns round in 24 hours, and has a fleur-de-lis upon it, ferving as an hour-index to point out the time of the day or night in the 24 hour circle. Within this ring, and about four tenths of an inch below its flat fur- face, is a flat circular plate, on which the months and days of tire year are engraved ; and within thefe, on the fame plate, is a circle containing the figns and degrees of the ecliptic, di- vided in fuch a manner, as that each particular SELECT EXERCISES. 21 particular day of the year Hands over the fign and degree of the Sun’s place on that day. Within this circle, on the fame plate, the ecliptic, equinoctial, and tropics, are laid down ; and all the ftars of the firft, fecond, and third magnitude that are ever feen in the latitude of London, according to their refpeCtive right afcenfions and declinations ; thofe of the firft magnitude being diftinguifh- ed by eight points, thofe of the fecond by fix, and thofe of the third by five. This plate turns round in 23 hours 56 minutes 4 feconds 6 thirds of time, which is the length of a fydereal day ; and confequently it makes 3C6 revo- lutions (as the ftars do in the heavens) in the time the Sun makes 365 ; the number of fydereal days in a year exceeding the number of folar days by one. Over the middle of this plate, and about four tenths of an inch from it, is a fixed plate E, to reprefent the C 3 Earth ; 22 SELECT EXERCISES. Earth ; round which, the Sun, Moon, and Stars move in their proper times, viz. the Sun in 24 hours, the Moon in 24 hours 507 minutes, and the Stars in 23 hours 56 minutes 4 feconds 6 thirds. The Sun S is carried round by a wire A, which is fixed into the infide of the Moon’s age ring, even with the fleur-de-lis ; the Moon M is carried round by a wire B, which is fixed to the axis of a wheel below the Earth E, and the Star-plate is turned round by a wheel at the back of the dial-plate. Over the dial-plate is a glafs, as in common clocks. On this glafs is an ellipfis II, drawn with a diamond, to reprefent the horizon of the place for which the clock is to ferve, and acrofs this horizon is a ftraight line e E d (even with the two XII’ s) to reprefent the meridian. All the Stars that are feen at any time within this ellipfis are above the horizon at that time, arid all thofe that are without it are then I SELECT EXERCISES. 23 then below the horizon. When any Star on the plate comes to the left- hand fide of the horizon (the Stars moving from left to right) the like Star in the heavens is rifing ; when it comes under the meridian line e E, the like Star in the heavens is on the meridian of the place ; and when it comes to the right-hand fide of the horizon, the like Star in the heavens is fetting. When the point of the ecliptic, that the Sun’s wire A interfedds, comes to the left-hand fide of the horizon, on any day of the year cut by that wire, the fleur-de-lis will be at the time of the Sun’s rifing, in the 24 hour-circle ; and at the time of his fetting when the interfeddion of the Sun’s wire and ecliptic comes to the weflern fide of the horizon. — The like is to be under- flood with regard to the rifing and fet- ting of the Moon, when the point of the ecliptic which her wore B inter- C 4 feels 24 SELECT EXERCISES. fetfts comes to the left and right hand fides of the horizon. Every 24 hours, the Moon’s wire fhifts over one day of her age in the circle of 297 equal parts on the flat ring above mentioned. Each of thefe day-fpaces is divided into four equal parts, for fhewing the Moon’s age to every fixth hour thereof. Thus, as the Moon’s wire B ftands in the figure, it fhews the Moon to be 8 days 18 hours old. It fhifts quite round the ring, and carries the Moon round from the Sun to the Sun again, in 29 days 1 2 hours and 45 minutes. The Sun, on its wire A, goes 365 times round in 365 days ; and, in that time, the Star-plate, with the months and days of the year upon it, goes 366 times round. So that, for every revo- lution of the Sitn, the Star-plate ad^ vances forward, under the Sun’s wire, through the fpace of one day in the circle of months : and by this means the 3 SELECT EXERCISES. 2 5 the Sun’s wire fhews the day of the month throughout the whole year; and at the fame time, for each parti- cular day of the year, it Ihews the Sun’s place in the ecliptic, in the cir- cle of ligns. — The Moon’s wire B cuts the Moon’s place in the circle of ligns, for e ery day of her age, throughout the year. The whole circle of figns Ihifts round, under the Sun’s wire A, in 365 days 5 hours 48 minutes 38 fe- conds, which is the time the Sun takes in going quite round the ecliptic : and the Moon’s wire Ihifts over all the circle of figns in 27 days 7 hours and 43 minutes, which is the time of the Moon’s going round the ecliptic. And thus, by thefe different motions of the Sun and M’oon, there are always 29 days 12 hours and 45 minutes be- tween any conjunction of the Sun and Moon, and the next fucceeding one. The moon ill is a round ball, half black half white : it turns round its axis, e 0 23 35-4 7 6 2 3 32 28.; ’ 0 27 3 i -3 S 7 2 3 28 32. s ! O 3 1 27.2 c 8 23 24 36.5 ) 0 35 23.1 IC 9 23 20 41.C 0 39 19.0 i ] 10 23 1 6 4 S.i 0 43 14.9 I 2 1 1 23 1 2 49.2 0 47 10.8 13 12 2 3 8 53-3 0 5 1 6.7 < M 1 3 2 3 4 57-4 0 55 2.6 Is H 23 1 I. s 0 58 58.5 16 15 22 57 5.6 I 2 54-4 1 7 16 22 53 9*7 I 6 5°*3 ' 18 17 22 49 13.8 i 10 46.2 *9 18 22 45 17.9 I H 42.1 20 '9 22 4 i 22.C I 18 38.0 , 21 20 22 37 26. 1 I 22 33-9 22 21 22 33 30.2 I 26 29.8 ; 23 22 22 29 34-3 I 3 ° 2 5*7 24 23 22 25 38.4 I 34 21.6 2 5 24 22 2 1 4 2 -5 I 38 * 7-5 26 23 22 17 46.6 I 42 * 3-4 27 26 2 2 13 50.7 I 46 9*3 28 27 22 9 54.8 I 5 ° 5-2 ■ 29 28 22 5 58.9 1 54 1. 1 3 C 2 9 22 2 3 «c I 57 57 .o 4 ° 39 2 1 22 44.0 2 37 16.0 5 ° 49 20 43 2 v0 3 16 35 -° 60 59 20 4 6.0 3 55 54.0 : 7 ° 69 *9 24 47 -o 4 33 1 3 -° 80 79 18 45 28.0 5 M 32.0 90 89 18 6 9.0 5 53 51.0 too 99 *7 26 50.0 6 33 10. 0 2 CO ! '99 10 53 ‘ 40.0 1 '3 6 20.0 3OO 2 '99 4 : 20 30.0 1 '9 39 3 °° 360 3 139 0 : 24 36.0 2 *3 35 24.0 3 164 0 4 56 ; 2 ^3 55 3-5 366 3 6 5 0 I 0.6 2 : 3 58 594 The SELFXT EXERCISES. 35 The firft column denotes the num- ber of revolutions of the Stars, from the meridian to the meridian again, in a common year of 365 days : the next other columns (titled Days. H. M. S ) fhew the times in which thefe re- volutions are made : and thofe in the right-hand part of the Table (titled H. M. S.) fhew how much any Star gains daily upon the time fhewn by a well-regulated clock or watch. Therefore, to know whether the clock goes true or not, obferve the time by the clock when any Star dif- appears behind a chimney (or any other fixed objeft) as feen through a hole in a thin plate of metal fixed in a window-fhutter ; and if the fame Star difappears on every fucceeding night as much fooner by the clock as to agree with the times fhewn in the right-hand part of the Table (as fup- pofe 39 minutes 19 feconds in jo days, or 1 hour 18 minutes 38 feconds in 30 D a days) a 6 SELECT EXERCISES. days) the clock goes true : otherwife it does not, and mhft be regulated ac- cordingly, by fcrewing up or letting down the ball of the pendulum, as it goes too flow or too fall. To find the Length ofi a Pendulum that Jhall make any given Number ofi Vibrations in a Minute , and vice verfa. A pendulum whofe length is 39.2 inches, from the point of fufpenfion to the center of ofcillation, makes 60 vi- brations in a minute ; and this is called the ftandard length. Then, for any other number of vibrations in a mi- nute, fay, As the fquare of the given number of vibrations is to the fquare of 60, fo is the length of the ftandard to the length fought. — Thus, fuppofe the given number of vibrations to be 30 per minute: the fquare of 30 is 900, and the fquare of 60 is- 3600 ; then, as SELECT EXERCISES. 37 as 900 is to 3600, fo is 39.2 to 156.8 ; fo that the length required for 30 vi- brations/*^ minute is 156.8 inches. If the length of the pendulum be given, and the number of vibrations it makes in a minute be required ; fay, As the given length is to die Itandard length (39.2 inches) fo is the fquare of 60 vibrations to the fquare of the number required : the fquare root of which fhall be the number of vibra- tions made by the pendulum in a mi- nute. Thus, fuppofe the given length to be 1 56.8 inches : As 1 56.8 is to 39.2, fo is 3600 (the fquare of 60) to goo j the fquare root of which is 30, the number of vibrations that this pen- dulum will make in a minute. The length of a pendulum that would make only one vibration in a minute ^3920 yards, or 141 120 inches: and the length of a pendulum that would make 240 vibrations in a mi- nute (or 4 in a fecond) is 2.45 inches, D 3 In 38 SELECT EXERCISES. In thefe calculations it is fuppofed that the weight of the pendulum- rod hears little or no fenfible proportion to the weight of the ball. But as this cannot be the cafe in practice, the center of ofcillation will always be further from the point of fufpenfion than the calculation makes it 5 and this muft be found by trial. To divide the Circumference of a Circle into any given Number of equal Parts, vuhe- iher even or odd. As there are very uncommon and odd numbers of teeth in fome of the wheels of aflronomical clocks, and which confequently could not be cut by any common engine ufed by clock- makers for cutting the numbers of teeth in their clock- wheels, I thought proper to fliew how to divide the cir- cumference of a circle into any given odd or even number of equal parts, fa SELECT EXERCISES. 39 fo as that number may be laid down upon the dividing plate of a cutting engine. There is no odd number, but from which, if a certain number be fub- tradted, there will remain an even number, eafy to be fubdivided. Thus, fuppofing the given number of equal divifions of a circle on the dividing plate to be 69 ; fubtradt g, and there will remain 60. Every circle is fuppofed to contain 360 degrees : therefore fay, As the given number of parts in the circle, which is 69, is to 360 degrees, fo is 9 parts to the correfponding arc of the circle that will contain them : which arc, by the Rule of Three, will be found to be 46^^. Therefore, by the line of chords on a common fcale, or rather on a fedtor, fet off 464^ (or 46/^) degrees with your compafles in the periphery of the circle, and divide that arc or portion of the circle into 9 equal parts, and the reft of the circle D 4 into SELECT EXERCISES. 40 into 60 ; and the whole will be divided into 69 equal parts, as was required. Again, fuppofe it were required to divide the circumference of a circle into 83 equal parts ; fubtraft 3, and 80 will remain. — Then, as 83 parts are to 360 degrees, fo (by the Rule of Pro- portion) are 3 parts to 13 degrees and one hundredth part of a degree ; which fmall fraction may be negleft- ed. Therefore, by the line of chords, and compaffes, fet off 13 degrees in the periphery of the circle, and divide that portion or arc into 3 equal parts, and the reft of the circle into 80; and the thing will be done. Once more, fuppofe it were required to divide a given circle into 365 equal parts : fubtrafl 5, and 360 will remain. Then, as 365 parts are to 360 degrees, fo are 5 parts to 4- f V- 0 - degrees. There- fore, fet off 4 -pVV (or 4^) degrees in the circle; divide that fpace into 5 equal parts, and the reft of the circle into 360 ; and the whole will he di- vided SELECT EXERCISES. 4 J vided into 365 equal parts, as was required. I have often found this rule or me- thod very ufeful in dividing circles into an odd number of equal parts, or wheels into odd numbers of equal fiz’d teeth with equal fpaces between them: and now I find it juft as eafy to divide any given circle into any odd number of equal parts, as to divide it into any even number. And, for this purpofe, I prefer the line of chords on a feCtor to that on a plain fcale ; becaufe the feCtor may be opened fo, as to make the radius of the line of chords upon it equal to the radius of the given circle, unlefs the radius of the circle exceeds the whole length of the feCtor when it is opened fo as to refemble a ftraight ruler or fcale ; and this is what very feldom happens. Any perfon who is ufed to handle the compaffes, and the fcale or feCtor, may very eafily, by a little practice, take off degrees, and fractional parts of 4 2 SELECT EXERCISES. of a degree, by the accuracy of his eye, from a line of chords, near enough the truth for the above-mentioned purpofe. Suppofing the Difiance between the Centers of two Wheels , one of which is to turn the other , be given ; that the Number of Teeth in one of thefe Wheels is different from the Number of Teeth in the other , and it is required to make the Diameters of thefe Wheels in fuck Proportion to one another as their Numbers of Teeth are , fo that the Teeth in both Wheels may be iff equal Size , and the Spaces between them equal , that either of them may turn the other eafily and freely : it is required to find their Diameters . Here it is plain, that the diftance between the centers of the wheels is equal to the fum of both their radii in the working parts of the teeth. — Therefore, 5 V. SELECT EXERCISES. 43 Therefore, as the number of teeth in both wheels, taken together, is to the diftance between their centers, taken in any kind of meafure, as feet, inches, or parts of an inch ; fo is the number of teeth in either of the wheels to the radius or femidiameter of that wheel, taken in the like meafure, from its center to the working part of any one of its teeth. Thus, fuppofe the two wheels mull be of fuch lizes, as to have the diftance between their centers 5 inches; that one wheel is to have 75 teeth, and the other to have 33, and that the lizes of the teeth in both the wheels is equal, fo that either of them may turn the other. The fum of the teeth in both wheels is 108 ; therefore fay, As 108 teeth is to 5 inches, fo is 75 teeth to 3-A/-5- inches; and as 108 is to 5, fo is 33 to 1 -AV So that, from the center of the wheel of 75 teeth to the work- ing part of any tooth in it, is 3 inches and 47 hundred parts of an inch ; and, from 44 SELECT EXERCISES. from the center of the wheel of 33 teeth to the working part of either of its teeth, is 1 inch and 53 hundred parts of an inch. The Defcription and Ufe of a New Machine called the MECHANICAL PARADOX. The vulgar and illiterate take almofl every thing, even the moil important, upon the authority of others, without ever examining it themfelves.- — Al- though this implicit confidence is fel- dom attended with any bad confe- quences in the common affairs of life, it has neverthelefs, in other things, been much abufed ; and in political and religious matters, has produced fatal effects. On the other hand, knowing and learned men, to avoid this wcaknefs, have fallen into the contrary extreme : fome of them be- lieve every thing to be unreafonable, or impoilible, that appears fo to their SELECT EXERCISES. 45 fir ft apprehenfion ; not adverting to the narrow limits of the human un- derftanding, and the infinite variety of objedls, with their mutual opera- tions, combinations, and affedtions, that may be prefented to it. It muft be owned, that credulity has done much more mifchief in the world than incredulity has done, or ever will do ; becaufe the influences of the latter extend only to fuch as have fome fhare of education, or afFedt the reputation thereof. — And fince the human mind is not necelFarily im- pelled, without evidence, either to be- lief or unbelief; but may fufpend its aflent to, or diflent from, any propo- fition, till after a thorough examina- tion ; it is to be wifhed, that men of literature, efpecially philofophers, would not haftily, and by firft appear- ances, determine themfelves with re- fpedt to the truth or falfehood, poffi- bilitv or impoliibility of things. A 46 SELECT EXERCISES. A perfon who has made but little progrefs in the mathematics, though in other refpefts learned and judicious* would be apt to pronounce it impof- fible that two lines, which were no where two inches afunder, may con- tinually approach toward one another, and yet never meet, although conti- nued to infinity : and yet the truth of this propolition may be eafily demon- flrated. — And many, who are good mechanics, would be as apt to pro- nounce the fame, if they were told, that although the teeth of one wheel fliould take equally deep into the teeth of three others, it fliould affedt them in fuch a manner, that in turning it any way round its axis, it fliould turn one of them the fame •way, another the contrary •way , and the third no •way at all. On a very particular occafxon, about eighteeen years ago, I contrived a fmall machine of this fort, which has been fhewn SELECT EXERCISES. 4 f fhewn and explained to many ; and which I Ihall here defcribe, and ex- plain fome of the ufes it has been ap- plied to. It is reprefented to view by Fig. 1 4 of Plate V. in which, A is called the immoveable plate, becaufe it lies ftill on a table whilft the machine is at work. B C is a moveable frame, to be turned round an upright axis a (fixed into the center of the immoveable plate) by taking hold of the knob n, which is fixed into the index h. On the faid axis is fixed the im- moveable wheel D, whole teeth take into the teeth of the thick moveable wheel E, and turn it round its own axis, as the frame is turned round the fixed axis of the immoveable wheel D ; and in the fame direction that the frame is moved. The teeth of the thick wheel E take equally deep into the teeth of the three wheels F , G, and H-, but operate on thefe wheels in fuch a manner, that whilft 4 g SELECT EXERCISES. whilft the frame is turned round, the ■wheel H turns the fame i way that the wheel E does ; the wheel G turns the contrary way, and the wheel F turns no way at all. Before we explain the principles on which thefe three different effects de- pend, it will not be improper to fix fome certain criteria for bodies turn- ing or not turning round their own axes or centers ; and to make a dif- tin&ion between abfolute and relative motion. i. If a body fhews all its fides pro- greffively round toward a certain fixed point in the heavens, the body turns round its own axis or center, whether it remains flill in the fame place, or has a progreffive motion in any orbit whatever. — For, unlefs it does turn round its own center, it cannot poffibly have one of its fides toward the weft at one time, toward the fouth at an- other, toward the eaft at a third time, and toward the north at a fourth. — This 4 SELECT EXERCISES. 49 This is the cafe with the Moon, which always keeps one fide toward the earth ; but fhews the fame fide to every fixed point of the ftarry heaven in the plane of her orbit, in the time fhe goes once round her orbit ; becaufe in the time that fhe goes round her orbit, fhe turns once round her own axis or center. — On the contrary, if a body ftill keeps one of its fides toward a fixed point of the heaven, the body does not turn round its own axis or center, whether it keeps in one and the fame place, or has a progreffive motion in any orbit or direct ion what- ever. — This is the cafe with the card of the compafs in a fhip, which ftill keeps one of its points toward the magnetic north, let the fhip be at reft, or fail round a circle of many miles diameter. Both thefe cafes may be exemplified either by a cube or a globe, having a pin fixed into either of its fides to hold it by : we fhall fuppofe a cube, becaufe its fides are flat. — Sit down at a table, E and 5 o SELECT EXERCISES. and hold the cube by the pin, which may be called its axis, and keep one of its fides toward any fide of the room. Whilft you do this, you do not turn the cube round its axis, whe- ther you ftill keep it in the fame place, or carry it round any other fixed body on the table. — But if you -try to keep any fide of the cube toward the fixed body, whilft you are carrying it round the fame, you will find that you can- not do fo, without turning the pin round (which is fixed into the cube) betwixt the finger and thumb whereby you hold it ; unlefs you rife and walk round the table, keeping your face always toward the fixed body on the table ; and then, both yourfelf and the cube will have turned once round ; for the cube will have fhewn the fame fide progreflively round to all fides of the room, and your face will have been turned toward every fide of the room, and every fixed point of the horizon. 8 2. If SELECT EXERCISES. 51 2. If a fliip turns round, and at the fame time a man Hands on the deck without moving his feet, he is turned abfolutely round by the motion of the fliip, though he has no relative mo- tion with refpe< 5 h to the fliip. — But if, whilft the fliip is turning round, lie endeavours to turn himfelf round the contrary way ; he thereby only undoes the effeft that the turning of the fliip would otherwife have had upon him- felf ; and is, in fa< 5 t, fo far from turn- ing abfolutely round, that he keeps himfelf from turning at all ; and the fliip turns round him, as round a fixed axis ; although, with refpedl to the fliip, he has a relative motion. Fig. 2. is a fmall plan, or flat view of the machine, in which, the fame letters of reference are put to the wheels in it, as to thofe in Fig. 1. for the conveniency of looking at both j:he figures, in reading the defcription of them. WSEN is the round im- moveable plate: D the immoveable E 2 wheel 1 SELECT EXERCISES. 5 2 wheel on the fixed axis in the center of that plate : E the thick moveable wheel, whofe teeth take into the teeth of the wheel D ; and F is one of the thin wheels, over which G and H may be put ; and then, F, G, and H will make a thicknefs equal to the thick- nefs of the wheel E, and its teeth will take equally deep into the teeth of them all. The frame that holds thefe wheels is reprefented by the parallelo- gram abed ; and if it be turned round, it can give no motion to the wheel Z), becaufe that wheel is fixed on an axis which is fixed into the great immove- able plate. Take away the thick wheel E, and leave the wheel F where it lies, on the lower plate of the frame. Then turn the frame round the axis of the immoveable plate JVSEN (denoted by A in Fig. i.) and it will carry the wheel F round with it. — In doing this, F will Hill keep one and the fame fide toward the fixed central wheel Z>, as SELECT EXERCISES. 53 as the Moon ftill keeps the fame fide toward the Earth : and although F will then have no relative motion with rcfpeft to the moving frame, it will be abfolutely turned round its own center g (like the man on the fhip whilft he flood without moving his feet on the deck) for the crofs mark on its fide next S will be progreflively turned toward all the fides of the room. But, if we would keep the wheel F from turning round its own center, and fo caufe the crofs mark upon it to keep always toward one fide of the room; or, like the magnetic needle, to keep the fame point ftill toward one fixed point in the horizon ; we mull produce an eflfedl upon F, refembling what the man on the fhip did, by en- deavouring to turn himfelf round the contrary way to that which the fhip turned, fo as he might keep from turning at all ; and by that means keep his face ftill toward one and the fame point of the horizon.— And this E 3 is 54 SELECT EXERCISES. is done, by making the numbers of teeth equal in the wheels D and F (fuppofe 20 in each) and putting the thick wheel E between them, fo as to take into the teeth of them both. For then, as the frame is turned round the axis of the fixed wheel D, by means of the knob n, the tvheel E is turned round its axis by the wheel D ; and, for every fpace of a tooth that the frame would turn the wheel F, in di- rection of the motion of the frame, the wheel E will counteract that motion, by turning the wheel F juft as far backward with refpeCt to the motion of the frame ; and fo will keep F from turning any way round its own cen- ter : and the crofs mark near its edge will be always directed towards one fide of the room. — Whether the wheel E has the fame number of teeth as D and F have, or any different number, its effeCt on F will be ftill the fame. If F had one tooth lefs in number than D has, the effeCt produced on F, by the turning of the frame, would be SELECT EXERCISES. 55 he as much more than counteracted by the intermediate wheel E , as is equal to the fpace of one tooth in F : and therefore, whilft the frame was turned once round, fuppofe in direc- tion of the letters W S EN on the im- moveable plate, the wheel F would be turned the contrary way, as much as is equal to the fpace taken up by one of its teeth. But, if F had one tooth more in number than D has, the ef- fect of the motion of the frame (which is to turn F round in the fame direc- tion with it) would not be fully coun- teracted by means of the intermediate wheel E ; for as much of that effeCt would remain as is equal to the fpace of one tooth in F : and therefore, in the time the frame was turned once round, the wheel F would turn, on its own center, in direction of the mo- tion of the frame, as much as is equal to the fpace taken up by one of its teeth : and here note, that the wheel E (which turns F) always turns in direction of the motion of the frame. E 4 And 5 6 SELECT EXERCISES. And therefore, if an upright pin be fixed into the lower plate of the frame, under the center of the wheel F, and if the wheel F has the fame number of teeth that the fixed wheel D has, the wheel G one tooth lefs, and the wheel H one tooth more ; and if thefe three wheels are put loofely upon this pin, fo as to be at liberty to turn either way ; and the thick wheel E takes in- to the teeth of them all, and alfo into the teeth of the fixed wheel D 5 then, whichever way the frame is turned, the wheel H will turn the fame way, the wheel G the contrary way, and the wheel F no way at all.— The lefs num- ber of teeth G has, with refpeft to thofe of D, the fafter it will turn backward j and the greater number of teeth H has, with refpeft to thofe in D, the fafter it will turn forward 5 reckoning that motion to be backward which is contrary both to the motion of the frame and of the thick wheel E , and that motion to be forward which is in the fame direction with the motion of the SELECT EXERCISES. 57 the frame and of the wheel E. — So that the turning or not turning of the three wheels, F, G, H, or the direction and velocity of the motions of thofe that do turn round, depends entirely on the relation between their numbers of teeth and the number of teeth in the fixed wheel D, without any regard to the number of teeth in the move- able wheel E, Having folved the paradox , and defcribed the caufe of the different effects which are produced upon the three wheels F, G, and H, we Jhall now proceed to Jhew fonie ufes that may be made of the machine. This machine is fo much of an Or- rery, as is fufficient to fhew the dif- ferent lengths of days and nights, the viciflitudes of the feafons, the retro- grade motion of the nodes of the Moon’s orbit, the dired: motion of the apogeal point of her orbit, and the months in which the Sun and Moon sqauft be eclipfed. On 58 SELECT EXERCISES, On the great immoveable plate A (fee Fig. x.) are the months and days of the year, and the ligns and degrees of the zodiac fo placed, that when the annual index h is brough t to any given day of the year, it will point to the de- gree of the fign in which the Sun is on that day. — This index is fixed to the moveable frame B C, and is carried round the immoveable plate with it, by means of the knob n. The carry- ing this frame and index round the immoveable plate, anfwers to the Earth’s annual motion round the Sun, and to the Sun’s apparent motion round the ecliptic in a year. The central wheel D (being fixed on the axis a, which is fixed in the center of the im mo cable plate) turns the thick wheel E round its own axis by the motion of the frame ; and the teeth of the wheel E take into the teeth of the three wheels F , G, H, whofe axes turn within one another, like the axes of the hour, minute, and fecond hands SELECT EXERCISES. 59 hands of a clock or watch, where the feconds are fhewn from the center of the dial-plate. On the upper ends of thefe axes are the round plates I, K, L; the plate / being on the axis of the wheel F, K on the axis of G, and L on the axis of 11. So that, whichever way thefe wheels are affedted, their refpedtive plates, and what they iupport, muft be af- fected. in the fame manner ; each wheel and plate being independent of the others. The two upright wires M and N are fixed into the plate /; and they fupport the fmall ecliptic 0 P, on whi.ch, in the machine, the ligns and degrees of the ecliptic are marked. — This plate alfo fupports the fmall terreflrial globe e on its inclining axis f, which is fixed into the plate near the foot of the wire N. This axis inclines 237 degrees from a right line, fuppofed to be per- pendicular to the furface of the plate. /, and 66‘ to the plane of the fmall ecliptic / 6o SELECT EXERCISES. ecliptic O P which is parallel to that plate. On the Earth e is the crefcent g, which goes more than half way round the Earth, and Hands perpendicular to the plane of the fmall ecliptic 0 P, diredtly facing the Sun Z : its ufe is to divide the enlightened half of the Earth next the Sun from the other half which is then in the dark ; fo that it reprefents the boundary of light and darknefs, and therefore ought to go quite round the Earth ; ‘ but cannot, in a machine, becaufe, in fome pofitions, the Earth’s axis would fall upon it. — The Earth may be freely turned round on its axis by hand, within the crefcent, which is fup- ported by the crooked wire w, fixed to it, and into the upper plate of the moveable frame B C. In the plate K are fixed the two up- right wires P_and R : they fupport the Moon’s inclined orbit ST in its nodes, which are the two oppofite points of the SELECT EXERCISES. 6 r the Moon’s orbit where it interfedls the ecliptic O P. The afcending node is marked a , to which the defcending node is oppofite, below e , but hid from view by the globe e. The half a Te of this orbit is on the north fide of the ecliptic 0 P, and the other half tSa is on the fouth fide of the ecliptic. The Moon is not in this machine : but, when fhe is in either of the nodes of her orbit in the heavens, fhe is then in the plane of the ecliptic : when fhe is at T in her orbit, fhe is in her great- eft north latitude ; and when fhe is at S, fhe is in her greateft fouth latitude. In the plate L is fixed the crooked wire U U, which points downward to the fmall ecliptic 0 P, and fhews the motion of the Moon’s apogee therein, and its place at any given time. The ball Z represents the Sun, which is fupported by the crooked wire XT, fixed into the upper plate of the frame at J. A ftraight wire W proceeds from the Sun Z, and points always toward the 6z SELECT EXERCISES. the center of the Earth e ; but toward different points of its furface at dif- ferent times of the year, on account of the obliquity of its axis, which keeps its parallelifm during the Earth’s an- nual courfe round the Sun Z ; and therefore muff incline fometimes to- ward the Sun, at other times from him, and twice in the year neither to- ward nor from the Sun, but fidewife to him. The wire W is called the folar ray. As the annual index h fhews the Sun’s place in the ecliptic for every day of the year, by turning the frame round the axis of the immoveable plate A , according to the order of the months and figns, the folar ray does the fame in the fmall ecliptic 0 P : for, as this ecliptic has no motion on its axis, its figns and degrees flill keep parallel to thofe on the immoveable plate. At the fame time, the nodes of the Moon’s orbit ST (or points where it i'nterfe&s the ecliptic OP) are moved backward, SELECT EXERCISES. 63 backward, or contrary to the order of figns, at the rate of igt degrees every Julian year ; and the Moon’s apogeal wire U U is moved forward, or ac- cording to the order of the figns of the ecliptic, nearly at the rate of 41 de- grees every Julian year; the year being denoted by a revolution of the Earth e round the Sun Z ; in which time the annual index h goes round the circles of months and figns on the immove- able plate A. Take hold of the knob n, and ttirn the frame round thereby ; and in doing this, you will perceive that the north pole of the Earth e is conftantly before the crefcent^, in the enlightened part of the Earth toward the Sun, from the 2eth of March to the 23d of Septem- ber ; and the fouth pole all that time behind the crefcent in the dark : and, from the 23d of September to the 20th of March, the north pole is conftantly in the dark, behind the crefcent, and the fouth pole in the light before it : which SELECT EXERCISES. which fhews, that there is but on6 day and one night at each pole, in the whole year ; and that, when it is day at either pole, it is night at the other. From the 20th of March to the 23d of September, the days are longer than the nights in all thofe places of the northern hemifphere of the Earth which revolve through the light and dark, and fliorter in thofe of the fouth- ern hemifphere.— From the 23d of September to the 20th of March, the reverfe. There are 24 meridian femicircles drawn on the globe, all meeting in its poles ; and as one rotation or turn of the Earth on its axis is performed in 24 hours, each of thefe meridians is an hour diftant from the other, in every parallel of latitude. — Therefore, if you bring the annual index h to any given day of the year, on the im- moveable plate, you may fee how long the day then is at any place of the Earth, by counting how many of thefe meridians SELECT EXERCISES. 65 meridians are in the light, or before the crefcent, in the parallel of latitude of that place ; and this number being fubtradted from 24 hours, will leave remaining the length of the night. — And if you turn the Earth round its axis, all thofe places will pafs diredtly tinder the point of the folar ray, which the Sun paffes vertically over on that day, becaufe they are juft as many degrees north or fouth of the equator, as the Sun’s declination is then from the equinodtial. At the two equinoxes, viz. on the acth of March and 23d of September, the Sun is in the equinoctial, and con- fequently has no declination. On thefe days, the folar ray points diredtly to- ward the equator, the Earth’s poles lie under the inner edge of the crefcent, or boundary of light and darknefs ; and, in every parallel of latitude, there are twelve of the meridians, or hour- circles, before the crefcent, and twelve behind it ; which fhews that the days F and 66 SELECT EXERCISES. and nights then are each twelve hours long at all places of the Earth. And, if the Earth be turned round its axis, you will fee that all places on it go equally through the light and the dark hemifpheres. On the 2 ill of June, the whole fpace within the north polar circle is en- lightened, which is 23 f degrees from the pole, all around ; becaufe the Earth’s axis then inclines 23J- degrees toward the Sun ; but the whole fpace within the fouth polar circle is in the dark ; and the folar ray points toward the tropic of Cancer on the Earth, which is 231 degrees north from the equator. — On the 20th of December the reverfe happens, and the folar ray points toward the tropic of Capricorn, which is 23?- degrees fouth from the equator. If you bring the annual index h to the beginning of January, and turn the Moon’s orbit S T by its fupporting wires ^_and Ii till the afeending node (marked SELECT EXERCISES, 67 (marked a) comes to its place in the Ecliptic 0 P, as found by an Ephe- meris, or by Aftronomical Tables, for the beginning of any given year ; and then move the annual index by means of the knob n, till the index comes to any given day of the year afterward, the nodes will Hand againft their places in the ecliptic on that day. — And if you move the index onward, till either of the nodes comes directly againft the point of the folar ray, the index will then be at the day of the year on which the Sun is in con- junction with that node. At the times of thofe new Moons which happen within feventeen days of the con- junction of the Sun with either of the nodes, the Sun will be eclipfed : and at the times of thofe full Moons, which happen within twelve days of either of thefe conjunctions, the Moon will be eclipfed. — Without thefe limits there can be no eclipfe either of the Sun or Moon ; becaufe, in nature, the F 2 Moon’s 68 SELECT EXERCISES. Moon’s latitude, or declination from the ecliptic, is too great for the Moon’s fhadow to fall on any part of the Earth, or for the Earth’s fhadow to touch the Moon. Bring the annual index to the begin- ning of January, and fet the Moon’s apogeal wire U U to its place in the ecliptic for that time, as found by Aftronomical Tables: then move the index forward to any given day of the year, and the wire will point on the fmall ecliptic to the place of the Moon’s apogee for that time. The Earth’s axis f inclines always toward the beginning of the lign Can- cer on the fmall ecliptic O P. — And, if you fet either of the Moon’s nodes, and her apogeal wire, to the begin- ning of that fign, and turn the plate A about, until the Earth’s axis inclines toward any lide of the room (fuppofe the north fide) and then move the. an- nual index round and round the im- moveable plate A, according to the order SELECT EXERCISES, 69 order of the months and figns upon it, you will fee that the Earth’s axis and beginning of Cancer will ftill keep toward the fame fide of the room, without the leaf! deviation from it ; but the nodes of the Moon’s orbit S T will turn progreflively towards all the fides of the room, contrary to the order of figns in the fmall ecliptic OP, or from eaft, by fouth, to weft, and fo on : and the apogeal wire U U will move the contrary way to the motion of the nodes, or according to the order of the figns. in the fmall ecliptic, from weft, by fouth, to eaft, and fo on quite round. — A clear proof that the wheel F, which governs the Earth’s axis and the fmall ecliptic, does not turn any way round its own center; that the wheel G, which governs the Moon’s orbit 0 P, turns round its own center backward, or contrary both to the motion of the frame B C and thick wheel E ; and that the wheel H, which governs the Moon’s apogeal wire U U, F 3 turns I 70 SELECT EXERCISES. turns round its own center, forward, or in direftion both of the motion of the frame, and of the thick wheel E, by which the three wheels, F, G, and H, are affected. The wheels D, E, and F, have each 39 teeth in the machine ; the wheel G has 37, and H 44 ; as fhewn in Fig. 3. The parallel ifm of the Earth’s axis is perfect in this machine ; the mo- tion of the apogee very nearly fo ; the motion of the nodes not quite fo near the truth, though they will not vary fenfibly therefrom in one year. — But they cannot be brought nearer, unlefs larger wheels, with higher numbers of teeth, are ufed. In nature, the Moon’s apogee goes quite round the ecliptic in eight years and 312 days, in direftion of the Earth’s annual motion ; and the nodes go round the ecliptic, in a contrary direftion, in eighteen years and 225 days. — In the machine, the apogee goes round the ecliptic 0 P in eight years SELECT EXERCISES. 7 1 years and four-fifths of a year, and the nodes in eighteen years and a half. Notwithftanding the difference of the numbers of teeth in the wheels F, G , and H, and their being all of equal diameters, they take tolerably well into the teeth of the thick wheel E, becaufe they are made of foft wood. —But, if they were made of metal, the wheel E in Fig. i. ought to be made of the fhape of E (feen edge- wife) in Fig. 3. with very deep teeth : and the wheels F, G, and H, in Fig. 1. of diameters proportioned to their re- fpeftive numbers of teeth, as F, G , and H , in Fig. 3. And then the teeth of thefe three wheels would be of equal fizes with thofe of the wheel E wherein they work : and the motions would be free and eafy, without any pinching or {hake in the teeth. F 4 An 72 SELECT EXERCISES. An Orrery, f sewing the Motions of the Sun % Mercury , Venus , Earth, Moon , and Nodes of the Moon's Orbit ; the different Lengths of Days and Nights , the Viciffitudes of Seafons , Phafes of the Moon, and all the Solar and Lunar Eclipfes. The ufe of this Orrery, and the manner of uling it, being already de- scribed in my Aftronomy, I lhall not repeat thofe matters here ; but fhall only defcribe the wheel-work of it, which is not done in that book. It is not copied from any other Orrery whatever, and I can with truth fay, that it fhews the revolutions of the Moon and Planets nearer the truth than any other Orrery does, that has fallen under my examination. I there- fore freely give the following account of it to the Public, in the bell manner that I can ; and do wifh the descrip- tion may be generally underllood. To any SELECT EXERCISES. 73 any Clock-maker I hope it will be plain, and to every Orrery-maker I believe it will be quite fo. The fifth Plate is a plan of the wheel-work, in which the diameter of each wheel is equal to the femidia- meter or radius thereof in the Orrery I have made' : the numeral figures at each wheel fhew the number of teeth in that wheel, and the fhaded parts fhew where the teeth of any one wheel takes into the teeth of another, as the one turns the other. The fixth Plate is a fedtion or fide- view of all the wheel- work that could be brought into fight. But in this, fome few wheels could not be fhewn ; for, in the Orrery itfelf, take a view of the wheels on any fide you pleafe, fome of them will be unavoidably hid from fight by others that are between them and the eye. Thofe in Plate VII. that come in fight have the fame numeral figures fet to them as the like ones have in Plate VI. and 74 SELECT EXERCISES. and alfo the fame letters of reference where there is room to infert them. And therefore, in reading the deferip- tion of Plate VI. it will be requifite to look firfi: at it, and then at Plate VII. ; by which means the Reader will fee the pofition of thefe wheels with re- fpedl to each other, as they are placed higher or lower in the frames which contain them. AAA A is a round immoveable plate fupported by four pillars ; fome of the wheels are below it, but the greateft number of them are above it. It fup- ports and bears the weight of them all. B is the axis of the handle or winch by which all the wheels are turned : on its axis is a wheel C of 74 teeth, which turns a wheel D of 32, and D turns a wheel E of 73 teeth, on whofe axis is a wheel F of 32, turning a wheel G of 160 teeth, which turns awheel H of 32, and H turns a wheel I of the fame number, on the top of whofe axis is k SELECT EXERCISES. 75 is a fmall wheel K of 12 teeth (juft tinder the Earth) which turns a wheel L of the fame number and fize ; and L turns fuch another wheel M of the fame number. The axis of M inclines 237 degrees, and the Earth on the top of it is turned round by it. The wheel H of 32 teeth turns a wheel N of the fame number, on the top of whofe axis is an index which goes round a circle of 24 hours, (on the plate that covers the wheel-work) in the time the Earth turns round its axis. — The wheels D and E could not be fliewn in Plate VII. becaufe the wheel C of 74 teeth hides them from fight. On the axis of the wheel N is a wheel O of 64 teeth, turning a con- trate wheel P of 30, on whofe axis is an endlefs fere w, of a fingle thread 1 , turning a wheel ^_of 63 teeth, which carries the Moon round the Earth in her orbit, from change to change, in 29 days 12 hours 45 minutes. This wheel of 63 teeth turns a wheel R of 24, 76 SELECT EXERCISES. 24, which turns a wheel S of 63 teeth round in 29 days 12 hours 45 minutes, on whofe axis is an index that fhews the days of the Moon’s age on a circle of 297 equal parts, on the plate that covers the wheel-work. On B, the axis of the handle, is a pinion T of 8 leaves, turning a wheel U of 25 teeth, which turns another wheel V of the fame number and fize, on whofe axis is a pinion W of 7 leaves, turning a wheel X of 69 teeth, on whofe axis is a pinion T of 7 leaves, turning a wheel Z of 83 teeth once round in 363 days 5 hours 48 minutes 57 feconds, and carrying the Earth round the Sun in that time. For, in this wheel are four fhort pillars, whofe upper ends are fixed into the lower plate of a moveable frame aaaaaaa (Plate VI.) that turns round on a fixed upright pin in the center of the plate AAA A, and contains the above-men- tioned wheels belonging to the Earth and Moon: fo that the whole frame goes SELECT EXERCISES. 77 goes round the center pin in the fame time with the wheel Z. This laft wheel cannot be feen in Plate VII. becaufe it lies within the wheel G, which is only a thick ring having 160 teeth on its outfide. Its innermoft fide is reprefented by a dot- ted circle in Plate VI. and it is kept in its place by three rollers, marked ***, which turn upon pins fixed in the great immoveable plate A AAA. As the uppermoft edge of the con- trate wheel F (fee Plate VII.) mull come a little way through the plate A AAA, in order to turn the ring- wheel G that lies on the upper fide of this plate, and this wheel turns the wheel H of 32 teeth that belongs to the Earth’s diurnal motion; it is plain, that as the wheel H muft go round G in a year by the annual part of the work, G muft be thick enough to turn H at fuch a diftance from or above the plate AAA A, that H may go over the top of F without touching it : other- 6 wife. 7 S SELECT EXERCISES. wife, when TI came round to F, it could not pafs by, but would flop the annual motion. In the center, juft above the upper furface of the moveable frame-plate a a a a, is a fixed wheel b of 40 teeth taking into the teeth of the wheel c, which is alfo 40 in number ; and thefe take into the teeth of a wheel d, whofe number is 40 alfo. The axis of this laft wheel is hollow, and the top of it is fixed tight at K (fee Plate VII.) in the piece KLM that carries the Earth. — This part of the work keeps the Earth’s inclined axis in a conftant parallelifm in its annual courfe round tite Sun. For, as d is connected with the fixed wheel b, by means of the in- termediate wheel c, and c rolls or goes round b by the annual work, and as b, c , and d have equal numbers of teeth, d muft always preferve its parallelifm throughout its annual motion. The axis of b is fixed into the immoveable plate AAA A ; and it is hollow, to let the 7 SELECT EXERCISES. 7$ the axes of fome wheels below that plate turn within it. The folid fpindle, or axis of the wheel / of 32 teeth, turns within the hollow axis of the wheel d of 40 ; and on the top of this folid fpindle is the fmall wheel K of 12 teeth, which turns the Earth round its axis by the wheels L and M, of equal number and fize with K, as already mentioned. The hollow axis of the parallelifm- wheel d is within an upright focket, whofe lowermoft end is fixed into the top-plate (marked 56 in Plate VII.) of the moveable frame aaaa, and on the top of this focket is fixed a fmall wheel e of 16 teeth, which take into the teeth of another wheel f of the fame num- ber and fize ; on the axis of which is a long pinion g of 16 leaves, which take into the wheel h of 16 teeth, whofe axis is hollow, and has a black cap on the top of it, covering juft one half of the Moon. Now, as the focket, on whofe top the wheel c is placed, so Select exercises. placed, is fixed into the annual moving frame, it is plain, that, whichever fide or tooth of the wheel e is once toward the Sun, will always be fo ; and there- fore, as the wheel f, the pinion g, and the wheel h, go all round the wheel e by the work that carries the Moon round the Earth, and all thefe have equal numbers of teeth, the wheel h will always keep the Moon’s cap facing toward the Sun, and fhew her to be always full as feen from the Sun, but continually changing her phafes as feen from the Earth in her going round it. For, when the Moon is between the Earth and the Sun (as reprefented in Plate VII.) her cap will hide the the whole of her from the Earth: but, when lhe is oppofite to the Sun, all the half or fide of her next the Earth will then appear like a full Moon, be- fore the circular edge of the cap : and when fhe is mid-way between thefe pofitions, or in either of her quadra- tures, lhe will appear juft half en- lightened as feen from the Earth. The Select exercises. 8i The axis of the wheel of 63 teeth, which carries the Moon round the Earth, is hollow, and turns round upon the above-mentioned fixed foc- ket. To the top of this axis (juft un- der the wheel e of 1 6 teeth, Plate VII.) is fixed the bar i f which carries the Moon round the Earth by the motion of the wheel On the top of the axis of the wheel c of 40 teeth, is a wheel k of 59, turn- ing a wheel / of 56, which caufe the nodes of the Moon’s orbit to go once round, with a retrograde motion, thro’ all the figns and degrees of the eclip- tic, in 18E years. The axis of / is hollow, and turns upon the hollow axis of £>j and on the axis of l is a cir- cular plate m (Plate VII.) fixed obliquely on that axis, and parallel to the Moon’s orbit. The work that carries the Moon round the Earth carries alfo the piece g round upon this oblique plate ; and, as the lower end of the Moon’s axis (which turns within the hollow axis G Of 9 * SELECT EXERCISES. of her cap) is fixed into the piece it caufes the Moon to rife and fall in her oblique orbit, according to her north or fouth latitude or declination from, the ecliptic. As the nodes of her or- bit are even with the plane of the ecliptic, one half of her orbit is on the north fide, and the other half on the fouth fide of the ecliptic. On the axis of the wheel X, which has 69 teeth, is a pinion n of 10 leaves, turning a wheel 0 of 73 teeth, which carries Venus round about the Sun in 224 days 17 hours. The axis of the wheel 0 is hollow (becaufe another axis turns within it) and on the top of it is fixed the lower plate of the frame pppp-, which carries Venus round the Sun, and has wheels within it belong- ing to Venus and to Mercury. Under the loweft plate of this frame is a fixed wheel q of 74 teeth, of the fame diameter as the wheel Z of 83, which gives the Earth its annual mo- tion} fo that, in Plate VI. one and the fame SELECT EXERCISES. 83 fame circle reprefents both the wheels. —A pinion r of 8 leaves takes into the teeth of the fixed wheel q, and is car- ried round q by the motion of the frame pppp, that carries Venus round the Sun. C'onfequently, in the time this pinion is carried round the wheel, it will turn 9^ times round its axis, equal to the number of Venus’s days and nights in the time lire goes round the Sun. The wheel q of 74 teeth is fixed on the fame (above-mentioned) focket on which the wheel b of 40 teeth is fixed. The top of this focket goes through the lower plate of the frame pppp t and a wheel s of 28 teeth is fixed upon the top of this focket, juft above the fame plate. Another wheel t of 28 teeth takes into the teeth of s, and is carried round it by the motion of the frame: and a third wheel u of 28 teeth (which is alfo carried round by the frame) takes into the teeth of t : the axis of u is hollow, it turns upon G 2 the fi 4 SELECT EXERCISES. the folid fpindle or axis of the pinion r of 8 leaves, and on its top is fixed the curved piece v (Plate VII.) that carries Venus on her inclined axis, which, by means of the three laft- mentioned wheels of 28 teeth, is kept in a con- ftant parallelifm in going round the Sun. On the top of the axis of the pinion r of 8 leaves, and juft above the curved piece v (Plate VII.) is a fmall wheel w of 12 teeth, which turns another wheel x of the fame number and fize ; and this laft wheel turns a third wheel y of the fame number, which is fixed on the axis of Venus, and turns her times round her axis in the time fhe goes round the Sun; which is juft as often as the pinion r turns round in the time it is carried round the fixed wheel q of 74 teeth. On the top of the axis of the middle wheel t of 28 teeth, is another wheel of the fame number and fize, which turns a wheel / S of 18 teeth; and this ?. wheel SELECT EXERCISES. 85 wheel turns another wheel $ of the fame number, whofe axis is a hollow focket, on which a bar y (Plate VII.) is fixed ; and this bar carries Mercury round the Sun in 87 days 23 hours. On the axis of the wheel X (already mentioned) of 69 teeth is a wheel t of 78, which turns a wheel x of 64 round its axis in 25 days 6 hours. The axis of this wheel turns within the above- mentioned hollow arbors in the cen- ter ; and on its top is the fmall wheel ?r of 3>a teeth, which turns another wheel | of the fame number and fize : this laft wheel is fixed on the Sun’s axis, it turns in the fixed piece 9 (Plate VII.) and turns the Sun round his axis in 25 days 6 hours. The Sun’s axis inclines 77 degrees from a perpendicular to the ecliptic ; Venus’s axis 75 degrees, and the Earth’s axis 231. The Earth turns round within a black cap, that always covers the half of it which at any inflant of time is G 3 turned 86 SELECT EXERCISES. turned quite away from the Sun : the edge of the cap reprefents the folar horizon , or circle bounding light and darknefs : it is fupported by a crooked wire p, whofe lower end is fixed into the plate that covers the wheels, and is carried round by the annual motion- work. An index (called the annual index ) goes round the ecliptic, by the fame work, keeping always oppofite to the Sun, and fhewing the days of the months, and the Sun’s apparent place in the ecliptic 4s feen from the Earth. On looking at Plate VI. it may per- haps appear, even to a very ingenious paechanic, that the wheels C, D, and E are fuperfiuous ; and that the wheel F, which gives motion to the toothed ring G, might have been upon the axis B of the handle. For, as F has 32 teeth, and H , that is turned by the teeth of G, (and turns the Earth round its axis) has alfo 32 teeth, F and H lypiild turn round in equal times; and confe- SELECT EXERCISES. 87 confequently, a turn of the handle would have anfwered to a turn of the Earth on its axis. — This indeed would have been the cafe if the Earth had no annual motion : but as H goes round G in a year, the fame way that G turns round, H lofes five turns in going round G (for 5 times 32 is 160, the number of teeth in GJ , and then the handle would have turned 370 times round in the time the Earth made 365 rotations. — To prevent this, and fo make the turns of the Earth and handle agree together, C has 74 teeth, and E only 73. So that the wheel E will turn 5 times oftener round than the handle does in 365 turns thereof; and confequently make the Earth’s daily rotation equal to a turn of the handle, or to 24 hours of mean folar time. \ O 4 Another SELECT EXERCISES. U Another Orrery. This is the Orrery mentioned in my Tables and Trafis (page 169, 2d edition) which I intended to keep for my fon, who was then ferving an apprentice- fhip to a mathematical inllrument- maker. But, as it has pleafed God to call him from this world to a better, I fhall now freely communicate it to the Public. It fhews the length of day and night at all places of the Earth, every day of the year, with the Sun’s true place, declination, time of riling and fetting, the hour of the day, the Sun’s altitude, azimuth, and the variation of the com- pafs at any place. Alfo the Moon’s periodical and fynodical revolution, her motion on her axis, her latitude, altitude, azimuth, riling and fetting ; her mean anomaly and elliptic equa- tion ; with the days of all the new and full Moons and eclipfes, for 6000 years before SELECT EXERCISES. 89 before and after the Chriftian a:ra. — • The outfide figure of this Orrery is exa&ly fhewn in the eighth plate of my Aftronomy ; but the infide work differs much from what is reprefented in the fecond figure of that plate : and this infide work is what I fhall now defcribe. A large wheel of 235 teeth is fixed in the box that contains the work, the center of the wheel being in the cen- ter of the box, directly under the Sun’s center. On this wheel runs a pinion of 19 leaves, carried round the teeth of the wheel by the annual motion of the Earth ; and by this means, the pi- nion is turned round its own axis for every 19 teeth that it is carried on- ward, in going round the wheel. — Now, fuppofing this pinion to be car- ried round the wheel in 365^ days, the pinion will be turned round its own axis in 29 days 12 hours 44 minutes 25 feconds, and a bar on the axis of the pinion will carry the Moon round the t 90 SELECT EXERCISES. the Earth, from change to change, in that time. This comes fo near to the truth, as to vary but one day in the Moon’s courfe in 335 years ; and thefe are the neareft numbers forfuch Ample wheel-work that can poffibly be found for mean lunations. But, in nature, the Earth moves un- equally round the Sun, fo that there are 8 days more between the vernal and autumnal equinox, than between the autumnal and vernal. — And there- fore, in common Orreries, where this circumflance is taken no notice of, the Earth’s pofition to the Sun cannot be right at both the equinoctial points. In order to avoid this error, I f 5 rft divided the ecliptic into 360 equal parts for degrees ; and then, after having put the names of the figns to it, I laid down the days of the year from an ephemeris againft the degrees of the Sun’s place in the ecliptic, for each day refpedtively throughout the year. By this means, the daily fpaces, anfwer- SELECT EXERCISES. 9 * anfvvering to the Earth’s unequal mo- tion round the Sun, were fo divided, as to be continually and gradually leflening from the 30th of December till the firft of July ; and then as gra- dually lengthening from the firft of July till the 30th of December; as the Earth’s progrellive annual motion is fwifteft of all on the 30th of Decem- ber, and flowed: of all on the firft of July- The days of the months being un- equally divided, fo as to anfwer to the Earth’s unequal motion round the Sun, I made thefe divifions a pattern or fcale for dividing the 235 teeth of the above-mentioned wheel into fuch unequal fpaces as would agree with the fpaces allotted for the days an- fwering to them. But thefe gradual inequalities of the teeth were fo very fmall, and the difference fo little be- tween the wideft and narrowed:, that the pinion (whofe leaves were all equal) run very fmoothly through all the teeth of the wheel ■, as the leaves of 92 SELECT EXERCISES. of the pinion were fized to thefe teeth, which were at a mean rate between the greateft and leaft diftances from one another. By this contrivance, the mean lunation was always 29 days 1 2 hours 44 minutes 25 feconds through- out the whole year ; and the pinion was among the leaft diftant teeth of the wheel when the annual index was at the firft of July, and among the moil diftant teeth when the index was at the 30th of December. For the parallelifm of the Earth’s axis, a wheel of 59 teeth was fixed in the middle of the work, with its cen- ter directly over the center of the wheel of 235 teeth, and the teeth of another wheel of 59 took into the teeth of the former ; and thofe into the teeth of a third wheel of the fame number, on the top of whofe axis the piece that carries the Earth on its oblique axis was fixed. And, as the Earth was moved round the Sun, thefe three wheels kept the parallelifm of the Earth’s axis, as defcribed in the Me- chanical SELECT EXERCISES. 93 chanical Paradox , and the former Or- rery. Above the middlemoft wheel of 59, and on its axis, is fixed a wheel of the fame number, which takes into a wheel of 56 teeth : this lafl wheel is juft below the Earth, and turns the nodes of the Moon’s orbit quite round backward, in 184. years. Above the laft-mentioned wheel of 59 teeth, and on the fame axis with it, is a wheel of 55, turning a wheel of 62 teeth below the Earth; and this wheel of 62 moves the Moon’s apogee plate quite round forward, in 8 years 312 days. And from this plate a wire rifes, and points out the place and motion of the apogee in the Moon’s ecliptic. [By comparing this defcription -with that of the Orrery in the 8 th plate of my /Iflro- nomy (fee § 3 99.), it 'will be very eafily underfood : particularly thofi parts which fhew the parallelifm of the Earth's axis , the motion of the Moan's nodes t and apogee .} As 94 - SELECT EXERCISES. As the Moon goes round the Earth, fhe comes to her mean changes, nodes, and apogee, in the proper times ; and, at all intermediate times, her diflance from her apogee and nodes are fhewn in her ecliptic, orbit, and apogee plate ; on which laft her mean anomaly and elliptic equation are fhewn : by which means her true place in the ecliptic, and her latitude, may be very nearly found for any given time. The days of the months, through- out the year, are laid down in a dia- gonal manner, in a fpiral line of four revolutions, marked o, i, 2, 3, for leap year, and the firft, fecond, and third vears after. The annual index, in thefe fpirals, being at the given day of any month, for either of thefe years, all the other motions and phenomena will be right for that day : and, by means of thefe diagonals, the luna- tion is brought Hill nearer the truth than as above fpecified. Within this fct of fpirals are tables, which fhew the places of the Sun, Moon, SELECT EXERCISES. 95 Moon, Afcending Node, and Apogee, fox' the noon of the firftday of January, in any year within the limits of 6000 years both before or after the Chriftian tera. And, by means of thefe tables, the Orrery may, in lefs than two mi- nutes of tiiue, be rectified for the be- ginning of any of thefe years ; and then, all the motions, not only for that year, will be right, but alfo for 334 years afterward, without needing any rectification. A New Geometrical Method of conftrufting Sun-Dials. Draw at pleafure the hoi’izontal line AC B (Plate VIII. Fig. 1.) and on the point C, as a center, defcribe the circle DGEF. Draw the diameter DCE , fo as to make an angle (D CA) with ACB equal to the co-latitude of the place for which the dial is to ferve ; and draw FCG at right angles (or per- pendi- 96 Select exercises. pendicular) to 1 ) CE : then A CB fhall reprefent the horizon of the place, DCE the equinoctial, ECG the axis of the world and flile of the dial, G the north pole, F the fouth pole, and the arc B G the elevation of the pole above the north point of the horizon ; which elevation is equal to the latitude of the place. From the point E draw the right line EH parallel to the horizon ACB, and from the point D draw D H parallel to C F. So E H fhall be equal to the longefl diameter of an ellipfis (Fig. 2.) and D E equal to the fhorteft diameter thereof. Divide the circle DGEF into 24 equal parts, beginning at D ; and conned the divifion-points which are equidiflant from D by the flraight lines ab, c d, ef, &c. continuing thefe lines down to the points v, w, x , y, z, in that part of the line HE that falls without the circle. From the point / 3 , where GCF in- terfeds HE, draw (2 BCE (Fig. 2.) perpendi- SELECT EXERCISES. 97 perpendicular to H (3 E (Fig. 1.) and draw ACD in Fig. 2. parallel to H 13 E in Fig. r. So, in Fig. 2» BCE and ACD fliall crofs each other at right angles in the point C . — On this point, as a center, with the length (3 H or ( 3 E in Fig. 1. as a radius, defcribe the circle ABD E in Fig. 2. and divide it into 24 equal parts, beginning at A , and conned; the divifion-points, which are equidiflant from A, by the ftraight lines a f, b g, c h, d i, e k, &c. Then, from Fig. 1. take CD in yourcompaf- fes, as a radius ; and, with that extent, on C as a center in Fig. 2. defcribe the circle FGHI, and divide it into 24 equal parts, beginning at G. Through thefe divifion-points />, v, 0, x, n, j, &c. which are equidiftant from G, draw the right lines 5 7, 4 8, 3 9, 210, &c. meeting the lines within the former circle at the points 5, 7, 4, 8, 3, 9, 2, 10, 1, ii, on the fide AGD-, and at 7,5, 8, 4, 9, 3, 10, 2, 1 1, 1, on the fide AID. Then, through thefe points of meet- ing, draw by a fteady hand the eliipfis H A 1 9 8 SELECT EXERCISES, A i 2345678, 8cc. whofe longeft diameter AD is equal to II 13 E in Fig. 1. and its fhorteft diameter GI equal to DCE in the fame figure, as above mentioned. This done (which may be much fooner done than deferibed) lay the edge of a ftraight ruler to the center C in Fig. 2. and to the above-men- tioned divifion- points 5, 7, 4, 8, &c. in the ellipfes, and draw ftraight lines from C through thefe points, as in Fig. 3. ; and they will be the true hour-lines on a horizontal dial. Laftly, from the center C, in Fig. 2. draw the ftraight line C G parallel to CG in Fig. 1. for the axis of the ftile, or edge thereof that calls a fhadow on the time of the day ; and C G fhall be parallel to the axis of the world when the dial is truly fet, as the like edge of the ftile of every dial muft be. [A r . B. Straight lines, parallel to BCE in Fig. 2. being drawn through the ellipfis from the points (Fig. 1.) V, IV, SELECT EXERCISES. 99 345 6 44 87 30976 59 1 17 55696 »5 29 360c 30 59 14400 45 89 32400 60 "9 57600 A I ' SELECT EXERCISES. 121 A TABLE Jhewing how much the Mercury would fink in a Barometer at given Heights above the Earth's plane Surface ; and confequently , how the Heights of Hills may be found thereby • At the height Merc. finks At the height of Merc. finks At the height of Merc. finks At the height of Merc. finks At the height of Merc. finks 0 S £ p- 0 £ CO CO k— i 0 a 0 g. ^ sf * CO ^ • CO 0 3 0 g- *T 3 2 5 * CO = 8 2 Si • c-t o» 0 3 0 g. Sf £? c/> £ (A Feet. Feet. Feet. Feet. Feet. 100 0 .11 3700 3 -83 73 cc 7 .05 10900 9 . g 7 14500 12 *37 200 0 .22 380c 3 * 9 2 74 co 7 -13 1 1000 9 *94 14600 12 .44 300 0 -33 3900 4 .02 7 5 00 7 .22 1 1 100 fO .01 14700 12 .50 400 0 .4, 4000 4 .12 7600 7 * 3 ° 1 1 200 10 .08 14800 12 .57 500 0 .54 4100 4 .21 7700 7 * 3 8 1 1 300 10 .16 14900 12 .63 600 0 .65 4200 4 * 3 ° 780c 7 * 4 6 1 1 400 10 .23 1 5000 12 .70 700 0 .76 4300 4 -39 7900 7 -55 1 1500 10 .30 1 5 100 12 .76 800 0 .87 4400 4 .49 8000 7 * 6 3 1 1600 1° .37 15200 12 .83 900 0 .98 4500 4 * 5 8 8100 7 - 7 i 1 1700 10 .44 15300 12 .89 1 000 1 .09 4600 4 -67 8200 7 -79 1 1800 10 .52 15400 12 .96 1 100 1 .19 4700 4 *77 8300 7 * 8 7 1 1900 10 .59 15500 13 .02 1200 i .30 4800 4 .86 8400 7 *95 1 2000 10 .66 15600 13 .09 1 3 °° 1 .40 4900 4 -95 8500 8 .03 1 2 100 10 .73 1 4700 r 3 -15 1 40c 1 .50 500c 5 -°4 8600 8 .ii 1 2200 10 .80 15800 13 .21 1500 1 .61 5100 5 -13 8700 8 .19 12300 10 .87 1 5900 13 .28 i6co 1 .72 5200 5 .22 8800 8 .27 12400 10 .94 16000 *3 *34 1700 1 .82 53 co 5 * 3 ! 890c 8 -35 12500 11 .01 1 61 00 13 .40 1800 1 *93 54 °° 5 * 4 ° 9000 8 -43 12600 1 1 .08 16200 1 3 *47 1900 2 .03 55 °° 5 -49 9100 8 .51 1 2700 1. .15 16300 13 *53 2000 2 .14 5600 5 *S 8 920c 8 .58 12800 11 .22 16400 13 *59 2100 2 .24 570c 5 * 6 7 9300 8 .66 1 2900 1 1 .29 16500 13 .6$ 2200 2 -34 5800 5 * 7 6 9400 8 .74 13000 1 1 .36 16600 13 .71 2300 2 .44 59 co 5 ,8 S 9500 8 .82 1 3100 i 1 *43 1 6700 ■3 - 7 8 2400 2 .54 6000 5 *94 9600 8 .89 13200 1 1 .50 16800 13 .84 2500 2 .65 6100 6 .02 9700 8 .9 7 13300 1 1 .56 16900 13 .90 2600 2 *75 6 200 6 .1 1 980c 9 *°5 13400 11 .63 17000 ■3 - 9 6 2700 2 .85 6300 6 .20 990c 9 .12 13500 11 .70 17100 14 .02 2800 2 .95 6400 6 .28 10000 9 .20 13600 11 .77 17200 I4 .08 2900 3 .05 6500 6 «37 roioo 9 * 2 7 13700 1 1 .84 17300 H * T 5 3000 3 .15 6600 6 .45 1 0200 9 *34 13800 11 .90 17400 14 .21 3100 3 * 2 5 67CO 6 .54 10300 9 * 4 2 13900 11 .97 17500 H * 2 7 3200 3 -34 68co 6 .63 IO4OO 9 - 5 ° 14000 12 ,04 17600 ! 4 *33 3300 3 -44 690c 6 .71 IO5OO 9 *57 14100 12 .11 17700 H *39 340° 3 *54 7000 6 .80 10600 9 14200 12 .17 17800 H -45 3500 3 .63 : 1 7100 6 .88 ' IO 7 OO 9 * 7 2 1430c 12 .24 I 79 0c *4 * 5 ' 3600 3 *73 'J 7200 | 6 .97 1 1 0800 9 -79 I 44 OO} 12 . 3 C 1 800c *4 *57 Bj 122 SELECT EXERCISES. By this Table, and a common ba- rometer, the height of any hill may be found, if its height, taken in per- pendicular meafure, be not much above half a mile. — Thus, if the mer- cury be 2 inches and 95 hundred parts of an inch lower in the tube at the top of the hill, than what it was ob- ferved to be at the bottom, the per- pendicular height of the hill is 2800 feet, which is 160 feet more than half a mile. But as there are many hills much higher than 2800 feet, and the com- mon barometer-fcale is only 3 inches long, let a fcale 14 inches long, di- vided into inches, and hundred parts of an inch by diagonal lines, be ap- plied to the tube, and have a Aiding index acrofs it in the common way: and this, I apprehend, will do for the higheft mountain on the earth. — For, fuppofing the quickftlver was obferved to be 13 inches and 21 hundred parts of an inch lower in the tube when at the SELECT EXERCISES. 123 the top of the hill than it was when at the foot: againft 13.21 inches in the Table is 15800 feet for the height of the hill, which wants only 40 feet of being 3 miles high. T 0 divide the Area of a given Circle into any required Number of equal Parts, by concentric Circles. In Fig. 4. of Plate IX. let A B D E be a circle, whofe area is required to be divided into 5 equal parts by con- centric circles, as F, G, H, I. Divide the femidiameter A C into 5 equal parts, as A 1, 1 2, 2 3, 3 4, 4 5; and on the middle point e as a center, with the radius e A, defcribe the femi- circle Aab c dC. From the points of equal divifion at 1, 2, 3, and 4, and perpendicular to AC, raife the per- pendiculars 1 a, zb, 3 c, 4 d, till they meet the femicircle in the points a, b, c, and d; through which points, draw the 124 SELECT EXERCISES. the concentric circles F , G , H y /, and the thing will be done. Suppoling that five blackfmiths fhould agree to buy a grinding-ftone among them, each paying an equal fhare of the price, and that each man fhould therefore have the ufe of the ftone, to wear off a fifth part of it, till it came to the laft man, who was to wear it out : the firft man fhould wear the ftone from E to F, the fecond from F to G, the third from G to H, the fourth from H to /, and the fifth from / to the center or axle C. By this eafy method, which I learnt of Mr. Hutton, teacher of the Mathe- matics at Newcaftle, the area of any circle may be divided by concentric circles into any required number of equal parts. For, into whatever num- ber of equal parts the radius be divided, the area of the circle will be divided into the like number of parts, all equal among themfelves. To SELECT EXERCISES. 125 To make tvoo equal Circles , vahofe Areas, taken together , Jhall be equal to the Area of a given Circle : or four equal Crefcents , the Sum of whofe Areas fhall be equal to the Area of a given Square. In Fig. 5. of Plate IX. let c d k i be the given circle. In this circle de- fcribe the fquare ef ml-, and, on the middle points of any two of its fides, as at e and f as centers, defcribe the two circles AaBEA and BbCEB: the areas of thefe two circles, taken together, fhall be equal to the area of the given circle cdki. Draw the diagonal A EC, which will divide the fquare into two tri- angles ABC and ADC, right angled at B and D. Now, as the fides A B and B C of the triangle AB Care equal, and fo are the fides A D and DC of the triangle ADC, and the areas of cir- cles being as the fquares of their dia- meters, 2 126 SELECT EXERCISES. meters, and the hypothenufe A C be- ing fquared is equal to the fquares of the fides AB and BC, 0xAD2.nA.DCy the larger femicircle Ac B dC is equal to the two lefier femicircles A a B e A and B dCfB. Confequently, if you fubtraft the two common portions AcBeA and BdCfB , the two re- maining crefcents AaBcA and BbCdB will be equal to the two triangles AEBA and BCEB, which make one half of the fquare efml : and there- fore the fum of the areas of all the four outward crefcents is equal to the area of the whole fquare. Of fquaring the Circle. Although there has not yet been any method found for doing this to mathematical exaiStnefs, yet, by means of the following numbers, it may be brought fo very near the truth, as to be within a grain of fand in a fquare mile, SELECT EXERCISES. 127 mile, fuppofing 100 grains of fand (placed in a ftraight line, and touch- ing one another) to be equal to the length of an inch ; and confequently 40144896000000 to cover a fquare mile. If the diameter of a circle be given, and the length of the fide of a fquare fo nearly equal to the circle as to be true to 14 places of figures be re- quired ; fay, As 1 is to the diameter of the given circle, fo is 0.88622692545276 to the fide of the fquare required, in fuch meafures as the diameter of the circle was taken. If the length of the fide of a fquare be given, and the diameter of a circle equal (as nearly as above mentioned) to the fquare be required ; fay, As 1 is to the fide of the given fquare, taken in any meafure, as feet, inches, 6’c. fois 1. 12837916709551 to the diameter of a circle (taken in the fame kind of meafures) whofe area is equal to the area of the given fquare. In practice, it is fufficient to take out the decimal parts to four places . of 128 SELECT EXERCISES. of figures ; for, even by fo fmall a number, we come fo near the truth as to be within a ten thoufandth part of the whole area of being perfectly true. And this is nearer than any one can pretend to delineate on paper. Thus, fuppoling the diameter of a circle to be 12 inches, and that it is required to find the length of the fide of a fquare (or to make a fquare) whofe area fhall be equal to the area of the circle; fay, As 1 is to 12, fo is .8862 to 10.6344 inches, the length of the fide of the fquare required. Or, fuppofing the fide of a fquare to be 12 inches, and that it is required to find the diameter of a circle whofe area fhall be equal to the area of a fquare; fay, As 1 is to 12, fo is 1.1284 (inftead of 1.128379) to 13.5408 inches, the diameter of the circle required. Hence, as a fquare veflel, juft one foot wide and one foot deep, would hold a cubic foot of water ; a cylin- drical veffel 13.54 inches wide and one foot deep would a cubic foot of water too; SELECT EXERCISES. 129 too ; at leaft fo near the truth, that no difference could be perceived. The diameter of any circle is in pro- portion to its circumference, as 1 is to 3.1415926535897932384626434; or as 1 is to 3.1416, near enough for practice. Any circle is equal to a parallelo- gram, whofe length is equal to half the circumference of the circle, and breadth equal to half the diameter. Therefore multiply half the circum- ference by half the diameter, and the product fhall be equal to the area of the circle, in fquare meafure. The fquare root of this area is the fide of a fquare equal to the circle. K To I$0 SELECT EXERCISES. To Jhew that an Angle may he continually dimmi/hed , and yet never he reduced to nothing : and confequently r that Matter is infinitely divijible. In Fig. 6. of Plate IX, let AB be a flraight line, produced to an infinite length beyond B, and flraight through- out. On this line let there be an infi- nite number of equilateral triangles placed', as A a b r b c d, d e f, f g h, &c. tvhofe bafes Ab-, bd, df, fh, &e. touch one another upon the right line A B * and let the fide a b of the firft triangle be of any given length, as fnppofe an inch, and each fide of each triangle be of the fame length with a b. Then, from the point A draw the flraight line A c to the top of the fe- cond triangle bed-, and A c fhall cut ab in the middle point at m. From the point A draw the flraight line Ae to the top of the third triangle def\ and Ac fhall cut ab at n, in two 3 thirds SELECT EXERCISES. 131 thirds of its length from a, leaving only one third remaining, from n to b. From the point A draw the right line Ag to the top of the fourth triangle f g h ; and Ag fliall cut a b at 0, in three fourth parts of its length from a ; and confequently leave one fourth of it remaining, from o to b. Here it is plain, that every line drawn from A to the top of the next triangle beyond that to which the laft preceding line was drawn, will make a lefs angle with the line A B than the laft preceding line did. But no right line drawn from the point A to the top of any triangle placed upon A B, even at an infinite diftance from A , could ever coincide with the line A By although every fucceeding line will make a lefs angle with A B than the line laft drawn before it did : and therefore the angle at A will be con- tinually diminifhing, but can never come to nothing. Confequenily, the K a whole i 3 2 SELECT EXERCISES. whole line ab will never be exhauftcd or quite cut off by any line drawn from A to the top of any triangle : and therefore, a part of it will hill remain between a and b; which proves that matter is infinitely divifible. A New Experiment in Electricity, floe wing the Motions of the Sun , Earth , and Moon -, by Edward King, Efq\ of Lin- coln’s Inn. The Sun and Earth go round the common center of gravity between them in a folar year* and the Earth and Moon go round the common cen- ter of gravity between them in a lunar month. — Thefe motions are repre- fented by an eledtrical experiment, as follows : In Fig. y. of Plate IX. the ball S re- prefents the Sun, E the Earth, and M the Moon, connected by bended wires a c and bd : a is the center of gravity between SELECT EXERCISES. 133 between the Sun and Earth, and b is the center of gravity between the "Earth and Moon. Thefe three balls, and their connecting wires, are hung and fupported on the fharp point of a wire A, which is (luck upright in the prime conductor B of the eleCtrical machine •, the Earth and Moon hang- ing upon the fharp point of the wire cae, in which wire is a pointed fhort pin, flicking out horizontally at c ; and there is juft fuch another pin at d , flicking out in the fame manner, in the wire that connects the Earth and Moon. When the globe C of the eleCtrical machine is turned, the above-men- tioned balls and wires are electrified : and the eleCtrical fire, flying off hori- zontally from the points c and d, caufe S and E to move round their common center of gravity a ; and E and M to move round their common center of gravity b. And, as E and M are light when compared with S and E, there K 3 -is i 3 4 SELECT EXERCISES, is much lefs friction on the point b than upon the point a ; fo that E and M will make many more revolutions about the point b than S and E make about the point a— I ha.d this experi- ment from my ingenious friend Mr. King ; and have adjufted the weights of the balls fo, that E and M go twelve times round b in the time that S and E go only once round a . — It makes a, good amuling experiment in electri- city ; but is fo far from proving that the motions of the planets in the Heavens are owing to a like caufe, that it plainly proves they are not. For the real Sun and Planets are not connected by wires or bars of metal ; and confequently there can be no fuch metallic points as a and b between them. And without fuch points, the electric fluid would never caufe them to move : for, take away thefe points in the above-mentioned experiment, and the balls will continue at reft, let them be ever fo ftrongly electrified. TABLES TABLES FOR Calculating the true Time of an y NEW or FULL MOON, from the Creation of the World to A. D. 7800 ; near enough the Truth for any common Almanack. ADVERTISEMENT. In all the Britifh Lunar Tables hitherto publi/Jjed , the time depending upon the Moons annual equation is fubtratted from the mean time of Neva and Full Moon when the Suns anomaly is lefs than fix figns, and added when greater : in the elliptic equa- tion^ the time depending on the Moorfs ano- maly is added to the mean time of New and Full Moon when her anomaly is lefs than fix figns , and fubtrafted therefrom when her anomaly is greater. In the following Tables I have made thefe equations always addi- tive, which renders the calculations much e after . For this purpofe , / have put down all the radical mean times of New and Full Moon 1 3 hours 59 minutes fooner than in the former Tables ; the. great ef annual equation being 4 hours 1 1 minutes , and the great efi elliptic equation 9 hours 48 minutes \ the fum of both thefe ( to the near efi full minute) is 13 hours 59 minutes . The numbers un- der A are the degrees of the Suns mean ano- maly, and thofe under B the degrees of the Moon's . — I was led to this , by obferving that the late eminent M. Clairaut at Paris has made all the equations additive in his Lunar Tables . SELECT EXERCISES, *37 TABLE I. The mean Times of New and Full Moon in January , from A. D. 1 700 to A. D. 1 800, according to the Old Stile . Years. New Moon. Full Moon. D. h. m. A _B D. h. m. A B L . 1700 '8 0 43 202 1 22 J 9 S 217 - 1 94 1701 26 22 16 221 336 12 3 54 206 1 43 1702 16 7 5 210 286 1 12 43 *95 93 1703 5 15 53 199 236 20 10 *5 214 69 L . 1704 2 3 13 26 2l8 21 1 8 *9 4 203 18 1705 12 22 *5 207 161 27 16 37 222 354 1706 2 7 3 I96 1 1 1 17 1 2 5 21 1 304 1707 21 4 36 215 87 6 10 *4 200 254 L. 1708 9 13 24 203 3 6 24 7 46 218 229 1709 28 10 57 22 2 12 *5 16 35 207 1 79 1710 17 l 9 46 21 I 322 3 1 24 196 129 1711 7 4 34 200 272 ' 21 22 5 6 215 lo 5 L. 1712 2 5 2 7 219 247 IO 7 45 204 55 17*3 *4 10 56 208 197 2 9 5 18 223 3 ° 1 7 l 4 3 *9 44 197 H 7 18 4 6 212 340 1715 22 17 *7 2l6 122 7 22 55 201 290 L. 1716 11 2 5 20 5 72 2 5 20 27 220 265 1717 0 10 54 194 22 *5 5 16 209 215 1718 *9 8 2 7 213 358 4 14 5 1 98 165 1719 8 *7 IS 202 308 23 11 37 2*7 140 L. 1720 26 48 220 283 n 20 26 205 90 1721 15 2 3 37 209 233 I 5 J 5 194 ' 4 ° 1722 5 8 2 5 I98 * 8 3 20 2 47 213 16 1723 ?4 5 53 217 158 9 u 3 6 202 325 L* 1724 12 47 206 108 27 9 9 221 30* ’725 1 23 3 ? *95 58 16 J 7 57 210 251 1726 20 21 8 214 34 6 3 46 1 99 201 *727 10 5 5 6 203 344 25 6 18 218 *77 L. 1728 28 3 29 222 3*9 1 3 9 7 207 126 1729 17 12 j8 21 1 269 2 17 56 I96 76 1730 6 21 6 200 219 21 >5 28 21 5 S 2 i 73 i 2 S 18 38 219 194 11 0 16 204 1 L. 1732 H 3 28 208 *44 28 21 49 223 337 43? SELECT EXERCISES, TABLE I. ( Mean T 'imes of New and Full Moon in January , Old Stile) continued . New Moon. Full Moon. Years. D. h. m.; A B D. h. m. A B j >733 3 12 16 197 94 18 6 38 212 287 1734 22 9 49 216 69 7 *5 27 201 236 1735 11 18 37 205 19 26 12 59 220 212 L. > 73 $ 0 3 26 194 3 2 9 *4 21 48 209 162) * 7$7 19 0 59 21 3 3°5 4 6 37 198 1 12 > 73 § 8 9 47 202 2 54 23 4 9 21 7 ! 87 1739 27 7 20 220 230 12 12 58 205 37 L. 174° 16 9 209 180 0 21 47 1 94 347 . I 74 i 5 0 57 ; 1 98 130 l 9 *9 l 9 213 3 2 3 1742 2 3 22 3 °; 217; 105 9 4 8 202* 2 73 : 1743 13 7 i8; 706 55 28 1 40 221 248 L. 1744 i 16 ’7 ! 95 5 16 20 2 9 210 1 9B5 1745 26 *3 401 214 34 » 5 19 18 199 148; 174 6 9 22 28 203 291 24 16 50 218 124; 1747 28 20 1 222 26 6 14 1 39 207' 731 L. 1748 17 4 5 °; 2 1 1 216 2 10 28 196 23: 1749 6 /3 3 * ZOO 166 21 8 0 2 1 5 359 , 1750 2 S it I I 2l8 141 10 16 49 203 308 1751 H 19 59 207 0 1 37 J92 258! L. 1752 3 f 48 196 41 17 23 10 21 I 2 34 1 7 5 3 22 2 f 21 2 1 5 16 7 7 59 200 184 1754 11 ii 9 i 204 326 26 5 3 1 219' 1 59 1755 6 19 *93 276 r 5 .H 20 208 109 L. 1756 18 }7 212 252 3 23 9 197; 59 1757 8 2 19 201 202 22 20 4 i 2l6‘ 34 1758 26 Y> 23 5 2 220 177 12 5 3 ° 205 s 344 1759 16 r 8 4 1 209 127 1 H *9 1 94 294 L. 1760 4 17 29 198 77 19 1 1 5 1 213 270 1 76 1 23 *3 2 217 5 2 8 20 40 202 219 1762 12 23 5 ° 206 '2 2 7 18 12 221 J 95 I 7 6 3 2 8 39 195 312 >7 3 1 2 I C *45 L. 1764 20 6 12 214 287 .5 11 5 ° 199 95 * 7 6 5 9 }5 0 203 2 37 ‘ 24 9 22 218' 70 1766 28 12 33 221 213 *3 18 1 1 206 2C SELECT EXEjRCISES. s 39 TABLE I Fuji Moon in . ( Mean Timej of New an $ January , Old Stilt ) conceded . New Moon. ‘ v Full Moon. x ears. D. h. m. A B D. h. m. A B 1767 >7 ' 2 1 22 210 1.63 3 1 3 . 0 >95 330 L. 1 768 6 6 10 ■ 99 . 1 >3 21 9 3 ? 214 306 > 7^9 2 5 3 43 218 88 10 9 21 203 255 > 77 ° >4 12 3 1 2°7 38 29 p 53 ^ 222- 23 1 L. f 77 1 3 21 20 196; } 4 8 18 >5 42 21 1 ,181 1772 21 ia 53 • 2 * 5 l 323 7 9 3 1 2 CO > 3 1 ? 77 3 1 1 3 4 1 204; 273 25 22 3 219; 106 >774 0 12 30 >93 223 >5 S 2 208 56 ? 775 >9 JO 3 212 198 4 >5 4 1 > 97 ; . 6 L. 1776 7 18 5 1 201 148 22 >3 >3 216' 341 1777 26 l6 24 22c 124 1 1 22 2 205; 291 1778 . 1 6 1 12 209 74 1 6 5 ° > 94 ; 2 4 > 1 1779 5 1 198 24 20 4 23 2! 3 •217 U 178° 2 3 7 34 2 1 7 359 8 >3 12 202 1 6.7 I78I 1 2 16 22 206 3°9 27 10 44 221 142 ' 7 ^, 2 1 1 1 >95 259 1 6 >9 33 210 .92 >783 20 22 44 2*3 234 6 4 22 I98 42 L. (784 9 7 3 2 202 184 24 1 54 217 i* 7 * 1785 28 5 5 221 160 >3 10 43 206 3 2 7 I786 J 7 >3 54 210 I IQ 2 >9 32 >95 2 77 I787- 6 22 42 >99 59 21 >7 4 214 252 L. I788 24 20 >5 218 35 10 1 53 203' 202! 1789 >4 5 3 207 345 28 2 3 2 5 222 178 179° 3 >3 52 196 295 18 8 >4 21 I 128 ? 79 * 22 1 1 2 5 215 270 7 >7 3 200 73 L. 1792 10 20 >3 204 220 25 >4 35 219 53 >793 0 5 2 >93 17 ° >4 2 3 2 4 208 3 >794 >9 2 35 212 >45 4 8 13 >97 3>3 ? 795 8 11 23 201 95 2 3 5 45 216 288 L. > 7 S 6 26 8 219 7 1 1 1 >4 34 204 238 >797 >5 >7 44 208 21 0 23 22 JI 93 188 ? 79 8 5 2 33 >97 3 3 1 >9 20 55 - 212 164 >799 24 0 6 216 306 . 9 5 44 201 > >4 L. 1800 1 2 8 54 205 256 ?7 3 j6 220 89 1 4 o SELECT EXERCISES; TABLE II. The mean Times of New and Fall Moon in January , from A. D. 1752 to A. D. 1800, according to the New Stile . Years. New Moon. Full Moon. D. h. m. A B D. h. m. A B 1 7 5 3 3 *3 37 186 350 18 7 59 200 184 1754 22 1 1 9 204 326 7 16 47 189 *33 1 75 5 1 1 *9 58 *93 276 26 x 4 20 208 109 L. 1756 29 x 7 3 i 212 252 H 2 3 9 •1 97 59 r 757 *9 2 >9 201 201 4 7 57 186 9 •758 8 1 1 8 190 * 5 * 23 5 3 C 205 344 1 759 27 8 41 209 127 12 *4 ! 9 ! Q 4 294 L. 1760 >5 *7 29 198 77 0 2 3 7 '*3 244 1761 5 2 18 187 27 19 20 40 202 219 1762 2 3 23 5 ° 206 2 9 5 28 1 9 1 169 1763 13 8 39 28 *95 3 12 28 3 1 2 10 1 45 L. 1764 1 17 184 262 16 11 5 C 199 95 1765 20. *5 0 203 2 37 5 20 38 188 45 2 766 9 23 49 202. 187 2 4 18 1 1 zo6 20 1767 28 21 22 210 163 x 4 3 0 *95 33 ° L. 176b *7 6 10 199 1 1 3 2 11 48 184 280 1769 6 H 59 188 63 21 9 21 203 2 55 1770 2 5 12 3 i 207 38 10 18 9 192 205 1 77 1 H 2J 20 196 348 0 2 58 1 8 1 1 5 5 L. 1772 3 6 9 185- 298 18 0 3 i 200 131 1 77 3 22 3 4 1 204 273 7 9 *9 189 80 *774 1 1 12 3 o *93 223 26 6 52 208 5 6 •1775 0 21 *9 182 »73 *5 *5 4 X 1 97 6 L. 1776 iS >8 201 148 4 0 29 186 316 1777 8 3 40 190 198 22 22 2 205 191 1778 27 1 12 209 74 12 6 5 ° 194 2 4 l 1779 16 10 1 198 24 t *5 39 >83 * 9 1 L* 178© 4 18 5 ° 187 334 *9 *3 12 202 167 1781 2 3 22 206 3°9 8 22 c 191 1 1 7 1782 *3 2 1 1 195 2 59 2 7 *9 33 210 9 2 1783 2 10 c 184 209 1 84 <7 4 22 198 4 2 L. 17^4 20 7 3 2 203 5 *3 10 187 352 1785 9 16 21 192 '34 24 10 43 206 3 2 7 SELECT EXERCISES. Ht TABLE II. concluded . New Stile. New Moon. Full Moon. x ears. D. h. m A 8 D. h. m. A B 1786 28 *3 54 210 1 10 l 3 x 9 3 2 *95 2 77 >7 8 / 17 22 4 2 199 59 3 4 20 184 227 L. 1788 6 7 3 1 188 9 21 1 53 203 202 1789 2 5 3 3 207 345 10 10 4 1 192 152 1790 H 13 5 2 196 295 2 9 8 H 21 1 128 1 79 1 3 22 +» .85 2 45 18 *7 3 200 78 L. 1792 21 20 *3 204 220 7 1 5 1 1 89 28 1 793 11 5 2 *93 170 2 5 2 3 24 208 3 *794 0 13 5 1 182 1 20 1 5 8 x 3 197 3*3 1795 >9 1 1 2 3 201 95 4 x 7 1 186 263 L. 1796 7 20 12 190 45 22 *4 34 204 238 1 797 26 17 44 208 21 1 1 23 22 *93 188 i 79 8 16 2 33 1 97 3 3i 1 8 1 1 182 138 1799 5 I I 22 186 281 20 5 44 201 1 14 C. 1800 24 8 54 205 2 5 6 9 *4 3 2 190 63 TABLE III. Mem Lunations . Lun. D. h. m. A B Months. Days. 1 2 9 12 44 29 26 For Suit, 2 59 1 2’8 S 8 , 5 2 January - • 0 3 88 H 1 2 87 77 February - - 3 X O 4 118 2 5 6 1 16 103 March - 59 5 H7 x 5 40 146 129 April - - - 90 it O 6 l 77 4 24 *75 *55 May - - - 120 c r 7 206 17 8 204 181 June - - - 141 0 8 236 5 5 2 2 33 207 July - - - 181 c CT 9 265 18 36 262 232 Auguft - - 212 p 10 295 7 20 291 258 September - 243 g> 1 1 3 2 4 20 5 320 284 October - - 2 73 0- 12 354 8 49 349 310 November 3°4 13 383 21 33 18 336 December - 334 In L^ap years, in January and February, add a day to the time found by thefe Tables. *42 SELECT EXERCISES, TABLE IV, The fir ft Equation. (A) A h. m. A h. nr. A h. m. A h. m. A h. m. l 4 7 37 . T 43 73 0 *3 109 0 1 2 *45 1 45 2 4 2 38 I 39 74 0 12 1 10 0 14 146 1 48 3 3 53 39 I 3 6 75 0 1 1 1 1 1 0 15 *47 1 52 4 3 54 40 I 32 7 6 0 1 C 1 12 0 17 148 1 5 6 5 3 5 ° 4 1 I 29 77 0 9 **3 0 18 1 49 1 59 t 3 46 42 I 26 78 0 S 1 14 0 20 15 ° 2 3 7 3 4 2 43 I 2.3 79 ' 0 7 ”5 0 22 I 5 I 2 ■ 7 8 3 37 44 I 19 80 0 6 1 16 0 23 152 2 1 1 9 3 33 45 I 16 81 0 5 117 0 25 *53 2 J 5 10 3 29 46 I 13 82 0 4 118 0 27 x 54 2 J 9 1 1 3 24 47 I 10 8 3 0 3 1/9 0 29 i 55 2 23 12 3 20 48 I 7 84 0 2 120 0 3 1 156 2 27 *3 3 16 49 I 4 85 0 2 1 2 1 0 34 *57 2 3 1 H 3 12 5 ° I 1 , S6 0 1 122 0 36 *58 2 35 r 5 3 7 5 1 O S*> 87 0 1 1 23 0 38 1 59 •2 39 1 6 3 3 52 O 5 6 88 0 1 124 6 40 160 2 43 */ 2 59 53 0 53 89 0 .6 425 0 43 1 6 1 2 48 18 2 55 54 O 5 1 90 0 0 1 26 0 45 162 2 5 2 ] 9 2 5 i 55 0 48 9 1 0 c 127 0 48 16 J 2 56 20 2 47 >6 0 45 92 0 c 1 28 0 5 i 164 3 0 21 2 43 57 0 43 93 0 c 129 0 53 165 3 5 22 2 39 58 0' 40 94 0 0 130 0 55 166 3 9 2 3 2 35 59 0 95 0 I 1 3 1 0 59 167 3 13 2 4 2 31 66 0 3 6 96 0 1 J 3 2 1 2 168 3 18 25 2 27 61 0 34 97 0 I 133 1 5 169 3 22 26' 2 *3 62 o' 3^2 g 8 0 2 134 1 8 170 3 27 27 . 2 *9 6 3 0 3 ° 99 0 2 135 1 11 171 3 3 1 28 2 J 5 64 0 28 100 0 3 136 1 >4 172 3 35 29' 2 12 : 65 0 26 101 0 4 1 37 1 1 7 ' 7 ? i 40 30 2 8 66 0 24 102 0 5 138 1 20 1 74 3 44 3 1 2 4 6 7 0 22 103 0 6 i 39 1 24 l 7 S 3 49 32 2 0 68 0 20 104 0 7 140 1 27 176 3’ 53 33 ' 1 57 - 6 9 0 19 io 5 0 8 H 1 1 30 1 77 .3 5 8 34 1 54 7c 0 l 7 106 0 9 142 1 34 178 4 2 35 I 5 ° 71 0 1 5 107 0. LC i 43 1 37 1 79 4 7 36 1 46 72 0 14 108 0 1 1 144 1 41 1 80 4 1 1 SELECT EXERCISES. *43 TABLE IV. ( Equation A) concluded. A h. m. A ;h. m. A h. m. A h. ib. A h. m. 181 4 15 217I 6 45 2 53 8 1 2 289 8 7 3 2 5 6 3 2 182 4 20 218 ■|6 4 cS 254 8 1 3 290 8 5 326 6 29 183 4 2 4 219 <6 5 2 2 55 8 *4 291 8 3 3 2 7 6 2 5 184 4 2 9 220 6 55 236 8 15 292 8 2 328 6 22 ■85 4 33 221 6 5 S 2 57 8 16 2 93 8 0 329 6 18 186 4 3 8 222 7 2 z 5 8 8 *7 294 7 5 * 330 6 *4 1*87 4 42 223 7 5 259 8 18 2 95 7 56 33 1 6 10 188 4 47 224 7 8 260 8 *9 296 7 54 33 2 6 7 189 4 5 1 225 7 1 1 26l 8 20 297 7 5 2 333 6 3 190 4 55 226 7 H 262 8 20 298 7 50 334 5 59 191 5 0 227 7 1 7 2 63 8 21 299 7 4 S 335 5 55 192 5 4 228 7 20 264 8 21 300 7 46 33 6 5 5 1 *93 5 9 . 229 7 2 3 265 8 21 3 GI •7 44 337 5 47 194 5 *3 2 5 ° 7 26 266 8 22 302 7 4 1 338 5 43 *95 5 *7 23 1 7 29 267 8 22 305 7 39 339 5 39 196 5 22 232 7 3 1 268 8 22 3°4 7 37 34 ° 5 35 : 197 S 26 2 3 3 7 34 269 8 22 3°5 7 34 34 1 5 3 * 198 5 3 ° 234 7 37 270 8 22 306 7 3 1 34 2 5 2 7 ! *99 5 34 2 35 7 39 271 8 22 3°7 7 29 343 5 22 200 5 39 236 7 4 * 272 8 21 308 7 26 344 5 *9 ; 201 5 43 2 37 7 44 2 73 8 21 309 7 2 3 345 5 *5 202 5 47 238 7 46 274 8 20 310 7 2 1 346 - > 10 2°3 5 5 1 2 39 7 48 275 8 20 3 1 1 7 18 347 5 6 2O4 5 55 240 7 50 276 8 *9 3 12 7 *5 34 8 5 2 205 > 59 241 7 53 277 8 *9 3 1 3 7 12 349 4 5 8 206 6 3 242 7 55 278 8 18 3*4 7 9 350 4 54 2O7 6 7 2 43 7 57 279 8 1 7 3*5 7 6 35 * 4 49 208 6 1 1 244 7 59 280 8 16 316 7 3 35 2 4 45 209 6 *5 2 4 S 8 0 281 8 *5 3*7 7 0 353 4 4 * 210 6 *9 246 8 2 282 8 4 3*8 6 5 6 354 4 37 21 I 6 23 247 8 4 2^3 •8 *3 3*9 6 53 355 4 34 212 6 26 248 8 , 6 284 8 12 32c 6 5 ° 35 6 4 28 21 3 6 3 ° 2 49 8 7 285 8 1 1 3 21 6 46 357 4 24 2I4 6 34 250 8 9 286 8 10 322 6 43 3 5 8 4 20 2‘ls 6 37 251 8 10 287 8 9 3 2 3 6 39 359 4 *5 216 6 4 l tv) un K) 8 1 1 288 8 8 3 2 4 6 360 4 11 i 4 4 SELECT EXERCISES. TABLE V. The fecond Equation. (B) B h. m. B h. m. B h. m. B h. m. B h. m. i 9 59 37 t6 2 73 x 9 21 109 18 5 1 H 5 *5 4 2 10 10 3 * 16 1 1 74 *9 23 1 10 18 47 146 H 5 6 3 10 21 39 16 *9 75 *9 25 1 1 1 18 43 H 7 x 4 48 4 10 32 40 16 27 76 x 9 27 I 12 18 3 8 148 *4 39 5 to 43 4 1 16 35 77 l 9 29 M 3 18 34 ! 49 *4 3 1 6 10 54 42 16 43 7 8 l 9 3 ° II 4 18 29 150 x 4 2 3 7 1 1 5 43 16 5 ° 79 '9 32 1 x 5 18 24 ' 5 ' x 4 x 4 8 1 1 16 44 16 58 80 •9 33 1 16 18 19 152 x 4 5 9 11 27 45 17 5 81 ! 9 34 1 17 18 14 1 53 x 3 57 to 11 38 46 *7 12 82 l 9 35 1 18 18 8 x 54 x 3 48 i i 1 1 49 47 17 IQ 83 l 9 35 U 9 18 3 '55 x 3 39 12 1 1 59 43 l 7 26 . 84 l 9 3 6 120 l 7 57 156 l 3 3 1 l 3 12 IC 49 *7 33 85 l 9 3 6 121 l 7 5 1 1 57 x 3 22 *4 12 20 50 17 39 86 l 9 36 I 22 1 7 45 158 x 3 x 3 ! 5 12 3 1 3 i 17 46 8 7 x 9 3 6 1 23 ' l 7 4 c *59 1 3 4 16 12 42 S 2 r 7 5 2 88 l 9 36 I24 l 7 34 160 1 2 55 l 7 12 S 2 53 17 5 8 89 l 9 35 125 l 7 28 161 12 46 1 8 *3 2 » 54 [8 4 9 ° i 9 35 126 1 7 22 162 12 37 x 9 *3 13 55 18 9 9 1 x 9 34 127 x 7 x 5 l6 3 12 28 20 x 3 23 56 18 x 5 92 *9 33 128 l 7 9 164 12 18 21 1 3 33 >7 18 20 93 19 32 129 *7 2 165 12 9 22 x 3 43 58 18 25 94 l 9 3 i 130 16 5 6 166 12 0 2 3 ‘3 53 59 18 3 ° 95 l 9 3 ° X 3 X 16 49 167 1 1 5 1 24 *4 3 60 1 8 35 96 l 9 28 1 3 2 16 42 168 1 1 4 1 z 5 *4 *3 61 18 4 ° 97 >9 2> 1 33 16 35 169 1 1 32 26 ! 4 23 62 18 44 98 >9 24 x 34 16 28 l 7 ° 1 1 23 27 x 4 S 2 6 3 18 48 99 ‘9 22 1 35 16 21 i 7 \ 1 1 x 4 28 *4 42 64 18 52 100 l 9 20 [ 3 6 16 M 172 1 1 4 29 x 4 5 2 *5 18 56 toi x 9 18 1 37 16 6 x 73 10 55 30 1 66 19 0 102 *9 x 5 1 38 *5 59 1 7 4 10 45 31 *5 10 6 7 *9 4 * 03 >9 12 ‘39 1 5 5 1 .«?5 10 36 32 l 5 19 68 69 '9 * 104 l 9 Q ‘ 4 ° x 5 44 176 10 26 ,33 l S 28 *9 IC 105 l 9 6 .41 x 5 36 1 77 10 1 7 34 *5 37 70 •9 1 ■ 106 l 9 2 142 x 5 2S *7 10 7 35 x 5 45 T\ 19 16 107 18 59 x 43 15 20 179 9 5 s ,36, <5 54 .7 2 ‘9 I c 1 08 18 55 4 - l 5 12 1 8c 9 4 8 SELECT EXERCISES. 145 TABLE V. ( Equation B) concluded. B h. rrj . B h. m. B h. m. B ih. m. B h. m. 181 9 38 2I 7 4 16 2 53 0 3 8 289 iO 2 C 3 2 5 3 5 * 182 9 29 218 4 8 2 54 0 35 290 |o 2 3 326 3 59 >83 9 19 219 4 0 2 S 5 0 3 * 291 0 26 3 2 7 4 8 184 9 10 220 3 5 2 256 0 27 292 0 29 328 4 17 185 9 0 221 3 45 2 57 0 24 2 93 0 3 2 329 4 26 186 8 5 1 222 3 37 238 0 21 2 94 0 36 33 ° 4 35 ■87 8 4 1 223 3 3 ° 259 0 *9 295 0 40 33 * 4 44 188 8 3 2 224 3 22 260 0 16 296 0 44 33 2 4 54 189 8 22 225 3 *5 26l 0 14 297 0 48 333 5 2 190 8 13 226 3 8 262 0 12 298 0 5 2 334 5 *3 191 8 4 227 3 1 263 0 10 299 0 56 335 5 23 192 7 55 228 2 54 264 0 8 300 1 1 336 5 33 *93 7 45 229 2 47 265 0 6 301 1 t 337 S 43 194 7 3 6 230 2 40 266 0 5 302 I 1 1 338 5 53 1 9 5 7 27 231 2 34 267 0 4 3°3 1 16 339 6 3 196 7 18 232 2 27 268 0 3 304 1 22 34 ° 6 13 197 7 8 2 33 2 21 269 0 2 305 I 27 34 * 6 23 1 98 6 59 2 34 2 14 270 0 1 306 1 32 34 2 6 34 1 99 6 5 ° 2 35 2 8 271 0 1 3°7 I 38 343 6 44 2 C 0 6 4 1 236 2 2 272 0 1 308 I 44 344 6 54 201 6 3 2 2 37 1 5 6 2 73 0 0 3°9 I 5 C 345 7 5 202 6 23 2 3 « 1 5 1 2 74 0 0 310 I 57 34 6 7 16 203 6 2 39 1 45 2 75 0 0 3 1 1 2 3 347 7 26 204 6 5 240 1 39 276 0 1 312 2 10 348 7 37 2 C '5 5 57 241 1 33 277 0 1 313 2 *7 349 *7 / 48 206 5 4 8, r 242 1 28 278 0 2 3*4 2 24 35 ° 7 58 20 7 5 39 2 43 1 22 279 0 3 3*5 2 3 i 35 * 3 2 208 5 31 244 1 *7 2S0 0 4 316 2 ,38 35 2 8 2C 209 5 22 2 45 1 1 2 28 1 0 6 3*7 2 46 353 8 3 * 210 5 13 246 1 7 282 0 7 518 2 53 354 8 4 2 211 5 5 247 1 2 283, 0 9 3*9 3 1 355 8 53 212 4 57 248 0 58 284 0 11 320 3 9 35 6 9 4 2! 3 4 49 249 0 53 285 0 13 3 21 3 *7 357 9 15 214 4 4 ° 250 0 49 28$ 0 14 322 3 2 5 358 9 26 215 4 3 2 2 5 * 0 46- 287 0 16 3 2 3 3 34 359 9 37 210 4 2 4 252 0 4 2 288 0 17 '424 3 42 60 9 4b L 146 SELECT EXERCISES. TABLE VI. Supplemental to Table I. for finding the mean 'Time of New or Full Moon in January for 6000 Tears before or after any given Tear in the 1 Sth Century , according to the Julian or Old Stile. Years. P- h. m. A B | Years. D. h. m. A 8 100 4 8 1 i 3 2 55 3100 16 10 4 1 347 2 53 200 8 16 22 7 * 5 * 3200 20 18 5 2 35 1 148 3 00 *3 0 33 10 46 3300 2 5 3 3 354 44 400 *7 8 43 *3 301 3400 29 1 1 *4 357 299 500 21 16 54 ' >7 *97 3500 4 6 40 33 2 169 600 26 1 5 20 92 3600 8 14 5 i 335 64 700 O 20 3 2 354 322 3700 1 z 2 3 2 338 3 i 9 800 5 4 43 358 217 3800 17 7 13 34 2 2*5 900 9 1 2 54 1 1 12 3900 21 15 24 345 1 10 1000 *3 2 1 5 4 8 4000 z 5 2 3 35 349 6 1 100 18 5 16 8 263 4100 0 19 1 3 2 3 2 35 1200 22 13 26 1 1 *59 4200 5 3 12 326 * 3 ° 1300 26 21 37 *4 54 43 co 9 1 1 2 3 329 26 I4OO 1 1 7 4 349 284 4400 *3 1 9 34 333 28 1 I5OC 6 1 *5 35 2 1 79 4500 r8 3 45 336 *77 l6oo 10 9 26 355 74 4600 22 1 1 56 340 7 2 I7OO ‘4 17 37 359 330 4700 26 20 7 343 3 2 7 l800 *9 1 48 2 225 4800 1 *5 33 317 *97 1900 2 3 9 5 8 5 1 20 4900 5 2 3 44 3 2 * 9 2 2000 2 7 1.8 9 9 1 6 5000 10 7 55 3 2 4 348 2 100 2 13 36 343 2 45 5100 *4 16 6 3 2 7 2 43 2200 6 21 47 ’ ? 4 6 * 4 * 5200 *9 0 1 7 33 * *38 2300 1 1 5 38 350 -36 53 °° 2 3 8 28 534 34 24OO *5 *4 9 353 291 5400 27 16 39 337 289 25OO *9 22 2 C 356 ,8 7 i 5,-00 2 12 5 5* 2 *59 260O 2 4 6 3 1 0 82! i 5600 6 20 16 3*5 54 2700 28 H 41 3 337 | 5700 1 1 4 27 318 309. 2800 3 10 8 3 37 207 ] j8oo *5 1 2 38 322 205 29OO 7 18 *9 341 102 ? 9 °° l 9 20 49 3 2 5 IOC 3000 12 2 30 344-1 358! S 6000 24 3 0 328 355 SELECT EXERCISES. *47 To calculate the true Time of New, or Full Moon. For any propofed year, within the. limits of Table I. for Old Stile, or Table II. for New Stile, write out the mean time of New or Full Moon in January, with the numbers or argu- ments under A and B. With thefe ar- guments find the equations in Tab. IV. and V. which being added to the mean time of New or Full Moon in January, will give the true time thereof in that month. For the time of New or Full Moon in any month after January, add as many lunations from Table III. to the mean time in January, as the given month is after January, and alfo the numbers A and B for thefe lunations, to the numbers A and B belonging to the mean time in January. Then, with the refpeftive fums of thefe num- bers, if under 360, find the corre- L 2 fponding I 4 S SELECT EXERCISES. fponding equations in Tab. IV. and V. which added to the mean time will give the true time of the required New or Full Moon, in days, hours, and minutes, from the beginning of Ja- nuary. When either of the fums A or B exceeds 360, fubtraefc 360 from it ; and with the remainder enter the correfponding Table, and take out the equation. Then, from the number of days made by the lunations added to the New or Full Moon day in January, fubtradl the number of days againft the given month in Table III. and the remainder will be the day of the re- quired New or Full Moon in that month. If this number of days be equal to the number you fuotract them from, the required New or Full Moon falls not in the given month* but on the laft day of the month pre- ceding it : and, when that is the cafe, you mull add a lunation to it froni- Table III. with the equations A and B for / SELECT EXERCISES. 149 B for that lunation ; and then you will have the true time of New or Full Moon in the given month. If the number of the given month after January be equal to or exceed the number of days accounted from the beginning of January on which the New or Full Moon therein falls, you mull take out one lunation more from Table III. than what anfwers to the number of the given month after January; otherwife you would have the New or Full Moon not in the month you want, but in the month next be* fore it. And this will always be the cafe in fome month or other of the year wherein the New or Full Moon in January falls before the nth day thereof. In leap years, in the months of Ja- nuary and February, the Tables give the New or Full Moon a day fooner than it really falls : and therefore, in thefe years and months, a day mull be added to the time found by thefe Tables, They always begin the day L 3 at SELECT EXERCISES. 150 at noon, and reckon the hours on- ward from that time to the noon of the following day. Example I. For the true Time of New Moon in June A. D. 1772, New Stile. To New Moon in fan. 1772, Tab. If. Add 6 lunations from Tab. III. The fums are - - Firit equation (A) for 360, Tab. IV. Second equation (B) for 93, Tab. V. The whole makes - Table 111 . againfl: June, fubtradt Remains the true time, 'viz. June D. h. m A B 3 6 9 185 298 177 4 H I 7 > Hi PCl 0 O CO 0 0 rr> 453 4 1 ! 360 IQ 7,2 93 181 IO 10 1 a 30 10 16 So the true time of the required New Moon is the 30th of June, at 16 minutes paft X in the evening. Example II. For the true Time of Full Moon in J tine A. D. 1772, New Stile. D. h. m. A B To Full Moon in Jan. 1772, Tab. II. Add 5 lunations from Tab. III. 18 O 3 1 200 *3 1 147 1 5 40 146 1 29 The fums are - 165 16 1 1 34 6 260 Firll equation (A) for 346, Tab. IV. 5 10 Second equation (B) for 260, Tab. V. 0 16 The whole makes 165 21 37 Table 111. againftjune, fubtradl «*» Remains the true time, viz. June 14 21 37 Namely, the 15th of June, at 37 minutes paft IX in the morning. 9 To SELECT EXERCISES. To calculate the true Time of New or Full Moon in any given Tear and Month after the 18 th century , which begins with A. D. 1700, and ends with A. D. 1800. Here we mufi go by the Old Stile , and then reduce the time to the New , by the Table further on, JJjewing the number of days whereby the files do differ . In Tab. I. find a year in the 18th century of the fame number with that in the century propofed : then from that year, in the 18th century, write out the mean time of New or Full Moon in January, with the numbers A and B belonging thereto. This done, find a year in Table VI. which, w hen added to the faid year in the 18th century, lliall make up the number of the given year. Take out the time and numbers A and B for that year in Table VI. and add them to thofe in January in Table I. and the films fhall be the mean time L 4 of 15* SELECT EXERCISES. of New or Full Moon in January for the given year, and the arguments fpr finding the equations in that month. Then, for any other month in the given year, work as already taught. When the fum of days for January exceeds 31 (as in the following ex- ample) fubtracSt a lunation (Tab. III.) from the time, and alfo the numbers A and B for that lunation from the former numbers, and fet down the remainders for the mean time in Ja- nuary, and equation-arguments for that month. If either of the numbers A or B to be fubtraifted be greater than the number you would fubtraft from, add 360 to the lefler number, and then make the fubtraftion. Example SELECT EXERCISES. 1 53 Example III. For the true Time of New Moon in May A. D. 1909, Old Stile. To A. D. 1709 add 200, and the fun* will be 1909. D. h. m. A B To New Moon in Jan. 1709, Tab. T. 28 10 57 222 12 Add, for zoo years, from Tab. VI. 8 16 22 7 111 The fums are - 37 3 *9 229 163 Subtract 1 lunation, Tab. III. 29 1 2 44 2 Q 26 Rem. mean New Moon in Jan. 1909 7 H 35 200 »!? Add 4 lunation^ from Tab. III. 118 2 1 1 6 121 The fums are - - 125 »7 3 i 316 240 Firft equation (A) for 316, Tab. IV. 7 3 Second equation (B) for 240, Ta b. V. 1 39 The whole makps - 120 2 J 3 Table III. again ft May, fubtratt 120 Remains the true time, viz. May b 2 *3 So the true time of the required New Moon is the 6th of May, at 13 minutes pail II in the afternoon. There was no difference between the Old Stile and the New in the year of Chrift 200. — In any century after that year, to find the difference be- tween the Old and New Stile, divide the number of the given century by 4, and (without regarding the re- 3 mainder 154 SELECT EXERCISES. mainder when there is any) add 3 to the quotient ; then fubtradl the fum from the number of the century, and the remainder fhall be the number of days which muft be added to the Old Stile time, to reduce it to the New. Thus, in the year 1909, the difference will be found to be 12 days: and therefore, in the preceding example, the day of New Moon in May will be the 1 8th, according to the New Stile. — At the end of thefe precepts and examples, I fhall fubjoin a Table of thefe differences, which I have copied from a certain Author, who has co- pied much more from my book of Aftronomy into one of his, without doing the common juftice of acknow- ledging, in thefe cafes, from whom he copied. To SELECT EXERCISES. 155 T 0 calculate the true Time of New or Full Moon in any given Tear and Month be- tween the Chrijlian £ra and the 18 th Century, Old Stile . Find a year in the 18th century of the fame number with that in the cen- tury propofed, and take out the mean time of New or Full Moon in January, from Table I. for the faid year in the 1 8th century, with its numbers A and B. Then, from Table VI. take out the time and numbers A and B for as many years as, when fubtracTed from the above-mentioned year in the 18th century, fLall leave the given year remaining. Subtract the times and numbers taken from Table IV. from thofe taken from Table I. and fet ' t ^ down the remainders for the mean time of New or Full Moon in January in the given year, and the arguments A and B for finding the equations in January : and then, for any other month in that year, work as above taught. When j 5 6 select exercises. When fubtradfion of days cannot be made (as in the following example) add a lunation, and its numbers A and B, from Table III. to the firft time and numbers taken out: then fubtradt, and fet down the remainders for the mean time of New Moon in January, and the arguments for finding the equations belonging to it. Example IV. For the true Time of Full Moon in April, Old Stile , in the Year of Chrift 79 6. From A. D. 1796 fubtracl 1000, and 796 will remain. To Full Moon in Jan. A. D. 1796 Add 1 lunation from Table III. The fums are - - From which, fubt. for 1000 years, 7 Table VI f - - j Hem. lor Full Moon, Jan. A.D. 796 Add 3 lunations, for April, Tab. 111 . The fums are Firft equation (A) for 356, Tab. IV. Second equation (B) for 333, Tab. V. The whole makes - Table III. for April, fubtraft Remains the true time, w'«. April D. h. m. A 1 B 1 1 T 4 34 204 z§8 2 9 12 44 29 26 4 * 3 18 2 33 20^ x 3 21 5 4 8 27 6 *3 229 256 88 H 12 87 77 1 1 5 20 2 5 3 'd 333 7 3 * 2 1 ib 8 3c 92 '26 8 30 So SELECT EXERCISES. 157 So that, in April A. D. 796, the true time of Full Moon was the 28th day, at 30 min. paft VIII in the evening. To calculate the true Time of New or Full Moon in any given Tear and Month before the Chriftian ara, according to the Old Stile. Find a year in the 18th century, which being added to the given num- ber of years before Chrift diminifhed by one, Ihall make any number of com pleat centuries. Find this number of centuries in Table VI. and fubtradt the time and numbers A and B belonging to it from thofe in the year of the 18th century; and the remainders will be the mean time of New or Full Moon for January in the given year, and the numbers A and B belonging thereto. Then, for any other month in that year, work as above. When 1 58 SELECT EXERCISES. When the number of davs taken J from Tab. VI. exceeds the number taken from Tab. I. add a lunation. And when fubtradtion cannot be made in the numbers A or B (as is the cafe in the 3d and 4th lines (A) in the fol- lowing example) add 360 to the lefler number, and then make fubtradtion. Example V. For the true Time of New Moon in May , Old Stile, the Tear before Chrif 585. The years 584 added to 1716, make 2300, or 23 centuries. D. h. m. A B To New Moon in Jan. 1716, Tab. I. 1 1 2 5 20c 72 Add 1 lunation from Table III. 29 12 44 2 9 jL The fums are - 40 ‘4 49 2 34 9 s Subtract, fromTAB.VI. for 2300 years 11 5 ??? New Moon in Jan. bef. Chrift 585 2 9 8 5 1 •244 62 Add 4 lunations, for April, Tab. ill. 1 1 8 2 S 6 1 16 103 The fums are, for April Firft equation (A) for 360, Tab. IV. 147 1 1 47 36° 165. 4 1 1 Second equation (B) for 165, Tab.V, 12 9 The whole makes - - - 148 4 Table III. for May, fubtraft 120 Remains the true time, viz. May 28 4 7 So that the true time was the 28th of May, at 7 minutes paft IV in the after- SELECT EXERCISES. 159 afternoon. — At that time, a total eclipl’e of the Sun put an end to the long war between the Medes and Lydians, by frightening both the armies with a fudden darltnefs, which overfpread the field of battle, juft as they were ready to begin the decifive engagement. Some chronologifts believe, that the world was created at the time of the autumnal equinox, in Gcftober, the year 400 7* before the year of Chrift’s birth, and that the Moon was full upon the third day of the creation week, which muft have been Wednefday. — But, by a calculation from thefe Tables, I find that the faid Full Moon was on Tuefday, the 23d of October, at 45 minutes paft VI in the morning; and Dr. Halley’s Tables make it but one minute fooner. In that year, the au- tumnal equinox was on Wednefday, October the 24th. Thefe Tables are adapted to the meridian of London; but they will anfwer 1 6b SELECT EXERCISES. anfwer for any other, by adding 4 minutes to the time found by them for every degree eaftward from the meridian of London, or fubtrafting 4 minutes for every degree weft-ward. They are as near the truth as can be,- by having only two equations ; and I have never found them to differ half an hour from Meyer’s, which have' thirteen^ And thus, by a very ftlort and eafy method, the time of any New or Full Moon, within the limits of 6000 years either before or after the chriftian tera, may be found, fufficiently near the truth for any common purpofe. — I have tried various methods for niak» ing fuch tables as fhould render the calculations of New and Full Moons from them If ill fhorter, but have hitherto found it impoflible, unlefs I fhould have contented myfelf with fuch as Would fometimes vary a whole hour from the truth. A TABLE SELECT EXERCISES. i5i A TABLE Jhezving the Number of Days between the Old and New Stile in different Periods of Tune . Years be- fore Chrift. Ntw Stile. Days Diff. Years be- fore Chrift. New Stile Days Diff + Years af. ter Chrift. New Stile. Days diff. + L. 6 coo 47 L. 2000 17 L. 2000 *3 590c 46 IQOO 16 2100 580c 45 1800 15 2200 l 5 5700 44 17CO H 23OO 16 L. 5600 44 L. 1600 1 4 L. 2400 16 S5°° 43 1500 13 2500 17 5400 42 1400 12 2600 18 5300 4 1 1 300 I I 2700 l 9 L. 5200 4 > L. 1200 1 I L. 2800 19 5 100 40 1 100 IO 2900 20 5000 39 1000 0 3000 21 4900 38 900 8 3 IOC 22 L. 4800 38 r ■00 0 0 8 L, 3200 22 4 7 oc 37 700 7 3300 23 460c 36 600 6 3400 24 45 00 35 500 5 3500 25 L. 4.4.00 35 0 0 ■'t* 5 L. 3600 2 5 4300 34 300 4 37 oo 26 4200 33 2 C 0 3 3800 27 4100 32 Bef 100 2 3900 28 L. 4000 32 L. 0 2 L. 40C0 28 3900 3 1 Aft , 100 _ i 4100 29 3800 30 200 + 0 4200 3 ° 3700 29 3C0 1 4300 31 L. 3600 2 9 L. 4c 0 1 L. 4400 3 1 35 ° C 28 $co 2 4500 32 340c 2 7 600 3 4 6 30 33 33 °° 26 700 4 4700 34 L. 3200 26 L. 8co 4 L. 4800 34 3100 25 900 5 4900 35 3000 24 IOCO 6 5000 3 6 2900 23 I ICO 7 5 IOC 37 L. 2800 23 L. 1 zoo 7 L. .5200 37 2700 22 1 30c 8 530c 38 2600 21 1400 9 54 °c 39 .2500 20 1500 10 5500 40 Li 2400 20 L. 1600 10 L. 5630 40 2300 l 9 1700 1 1 5700 4i 2 200 18 1800 1 2 5800 42 2100 17 1900 n 5900 43 M The days in the columns marked — are to be fubtra&ed from the Old Stile time to reduce it to the New $ tlrofe in the columns marked 4- are to be added to the Old Stile, in order to reduce it to the New. All the years int this Table are Leap years in the Old Stile, but only thofe which are marked L are Leap years in the New. The difFerence between the two Stiles was nothing in the aoodth year after the year of ChriiFs birth : from that time* backward or forward, it varies 3 days in every 400 years. SELECT EXERCISES. 162 The Weight of Gold compared with the Weights of other Materials , of equal Bulk with the Gold . An hundred pound weight of pure fine Gold is equal in bulk to An hundred pound weight of coinage Gold is equal in bulk to Pound weight. 90.3 of Guinea Gold. 69.1 of Quickfilver. 58.1 of Lead. 56.5 of pure Silver. 54.1 of coinage Silver. 49.4 of Bifmuth. 45.8 of Copper. 44.1 of hammered Brafs. 40.3 of cait Brafs. 40.2 of forged Iron. 39.8 of Spring vSteeJ. 37.2 of Block Tin. 35.7 of Silver Ore. 30.5 of Gold Litharge. 25.5 of Lapis Calamir.aris. 24.2 of Loadftone. 21.3 of Copper Ore. 20.8 of Sapphires. 17.9 of Diamonds. 16.2 of Cornelian Stone. 16.0 of Cryftal Glafs, 13.5 of Lapis Lazuli. 1 5.0 of Plate Glafs. *4.5 of red Coral. 1 3 8 of Iceland Crylial. 13.8 of white Marble. 13.4 of Rock Cryftal. 13. 1 of homogeneai Pyrites, 1 1.9 of Sulphur. 9.5 of black Lead. 5.4 of Frankincenfe. 5.1 of common Water. 1 5.0 of Camphire. 4.7 of Proof Spirits. 4.4 of pure Spirits. 3.9 of Oil of Turpentine. 3.7 of -/Ether. 1.2 of Cork. Pound weight. 1 10.8 of pure fine Gold. 76.6 of Quickfilver. 6 4. 4 of Lead. 62.6 of pure Silver. 59.4 of coinage Silver. 54.7 of Bifmuth. 50.8 of Copper. 48.3 of hammered Brafs. 44.6 of call Brafs, 44.5 of forged Iron. 44.0 of Spring Steel. 41.2 of Block Tin. 39.5 of Silver Ore. 33.8 of Gold Litharge. 28.2 ef Lapis Calaminaris. 26.8 of Loadftone. 23.0 of Copper Ore. 22.5 of Sapphires. 19.9 of Diamonds. 18.4 of Cornelian Stone. 17.7 of Cryftal Glafs. 17.2 of Lapis Lazuli. 16.6 of Plate Glafs. 16. 1 of red Coral, 15.3 of Iceland Cryftal. 15.3 of white Marble. 15.0 of Rock Cryftal, 14.6 of homogeneai Pyrites. 13.2 of Sulphur. 10.5 of black Lead. 6.0 of Frankincenfe. 5.6 of common Water. 5.3 of Camphire. 5.2 of Proof Spirits. 4.9 of pure Spirits. 4.3 of Gil of Turpentine. 4.1 of TEther. 1.3 of Cork. SELECT EXERCISES. 163 Number . of Shil- lings and Pence in the Troy F ound of Wdight of Weight of K. §5 Dates of the feveral Standard of the Silver at each zo Shillings in reckon- ing of Stan- fine Silver contained in 20 Shillings Mint Indent ures. Standard Silver coined dard Silver at each of thefe Pe- iri reckon- ing at each particular Period* Fine Copper at each rioas. Silver. ! Alloy. Period. A. L). oz pW. oz. pw. ft. d. oz. pw. g r * 02. pw. gr. 1066 I I 2 O 18 21 4 II 5 O IO 8 3 c> ^ 1037 1 1 2 O 18 20 0 12 0 O 1 1 2 0 1.* I300 1 1 2 1° 18 20 3 I I ! 7 i IO 19 6 1347 1 1 2 ° 18 22 6 IO 13 8 9 17 8 £ ^ 1354 I t-» Vo 1395 l I 2 O 18 2 5 0 8 12 0 8 17 i 4 g I402 ) s A 1412 1 1 2 O 18 32 0 7 10 0 6 18 18 f^ I422 1 1 2 O 18 30 0 8 0 0 7 8 0 $ ^ v» •■* 1422 1 1 2 O 18 37 6 6 8 0 5 18 10 1 p I426 I 1446 j I I «» O 18 3 ° 0 8 6 0 7 8 0 14611 '464 | 1482 }■ I I 2 O 18 37 6 6 8 b 5 18 10 ^ fs 'O ■483 1 » 494 J ! 1 NSg» O .&> •"* i?o; I I 2 O 18 40 0 6 0 0 5 i i 0 Se 1 S°9 i "532 : ! l I 2 O 18 45 0 S 6 16 4 18 6 ~g«* 1 543 IO O 2 0 48 0 5 0 0 4 3 8 *545 6 O 6 0 48 0 5 0 0 2 10 0 ^3 r> *»s. Co K, 1546 ; 1 547 [ 4 O 8 0 48 0 5 0 0 1 13 8 1543 . \ 6 *549 6 O 0 72 0 3 6 1 6 I 13 8 ** S- 9 l 1 5 3 2 1 1 7 6| '• 377 6 5 2 4 Henry VII!. S 1 543 1 3 $ £ 1.1635 4 7 l 4 9 f >545 0 *3 1 O.6984 2 9 i 8 0 1546 I ‘ 547 f 0 9 3 l O.4656 1 10I 1 2 0 Edward VI 1 548 3 ■349 0 9 3 l O.4656 2 9 \ 1 2 0 1 5 5 1 0 4 7 i O.2328 ! 4 5 2 4 0 1 553 _ 1 0 I .0286 3 . 3 1 4. 5 58 Mary I. 1 5 Oo , 1*0333 Elizabeth. i 5 8 3 ! 1 0 8 5 2 5 4 * 1601" 1605 James I. 1627 Charles F. 1661 1 \ V 1 Charles II. 1671 ■ r 1 0 0 I. OOOO 5 2 5 7 1685 James IF. 1720 1764-- George I. George III. SELECT EXERCISES. 165 Prices of Goods at the above-mentioned Times. From A. D. 1000 to A. D. 1066. A horfe x 1 . 1 7 s. 6 d. A cow 6 s. An ox 7 s. 6 d. A fwine 2 s. A fheep 1 s. 3 d. Wheat per quarter is. 6d. From A. D. 10 66 to A. D. 1199. A horfe 12 s. 5 d. An ox 4 s. 8 d. A fow 3 s. A colt 2 s. 4-^d. A calf 2 s. 47 d. A fheep is. 8 d. Wheat per quarter 3s. id. From A. D. 1199 to A. D. 1307. A horfe 1 1 . ns. An ox 4 s. 8 d. A fow 3 s. A cow 17 s. o|d. A lamb 4 s. A heifer 2 s. ifd. A goofe 1 s. o|d. A cock 47 d. A hen 3d. Wheat per quarter 1 1. 3 s. 2 4 d. From A. D. 1307 to 1418. A horfe 18 s. 4d. An ox 2 1 . 6 s. 1 d. A cow 7s. 2d. A calf 4s. 2d. A fheep 2 s. 7 d. A goofe 9 d. A cock 34 d. A hen 22 d. Wheat per quarter jys. Ale per gallon 74 d. Day labourer’s wages 44 d. From l66 SELECT EXERCISES. From A. D. 1418 to A. D. 1524. A horfe 2 1 . 4 s. An ox 1 1 . 15 s. 8i d. A cow 15 s. 6 d. A colt 7 s. 8 d. A Iheep 5 s. A hog 5 s. A calf 4 s. id. A cock 3d. A hen 2 d. Wheat per quarter 115.3d. Ale per gallon 24 d. Pay labourer’s wages 34 d. From A. D. 1524 to A. D. 1604. An ox 1 1 . 16 s. y d. A Iheep 4 s. 34 d. A calf 5 s, 6 d. A lamb 4 s. 44 d. A goofe is. A capon 1 s. Beef per ftone 1 1 d. Coals per chaldron 7 s. 94 d. Wheat per quarter 15 s. From A. D. 1624 to A. D. 1646. A pheafant 5 s. 6 d. A turkey 3 s. 9d. A goofe 2 s. A partridge is. A pul- let 1 s. 6 d. A pigeon 6 d. From A. D. 1 730 to A, D. 1 760. A horfe 10 1 . An ox 81 . A cow 7 1 . 7 s. A hog 1 1 . 15 s. A Iheep 1 1 . 6 s. A turkey 4 s. A cock is. 3d. Seamens wages per day 9 d. Common labour- ers is. 8 d. In the preceding Table, by com- paring the number of Ihillings in the SELECT EXERCISES. 167 pound weight of filver in the former times with the number in the pound weight at prefent, it will be found that the above-mentioned articles were not fo cheap as is now generally be- lieved. Concerning Gold. The flandard for coinage gold is n ounces of pure gold and 1 ounce of copper. A cubic foot of coinage or guinea gold is worth 60117 guineas 5 Ihil- lings 27 d. or 63123 pounds 2 fhillings 2"f d. The weight of a cubic foot of fuch gold is 1107 pounds 14 ounces Avoir- dupoife; which, in Troy weight, is 1346 pounds 4 ounces 10 penny- weight 5 grains. — A lump of this gold, equal in bulk to 2059-^-5- cubic feet, would weigh 2772820.7 Troy pounds ; and if coined into guineas, would pay the national debt. M 4 The x68 SELECT EXERCISES. The Number of different Ways in which all the Letters of the Alphabet might be coni -* bined , or put together, from i Letter to 25. Or, the Number of Changes which might be rung on any Number of Bells not exceed- ing the Number of Letters in the Alphabet . Thus, 2 letters may be put 2 different ways toge- ther; 3 letters, 6 different ways ; 4 letters, 24 ways ; 5 letters, 120 ways; 6 letteis, 720 ways; and foon, as in the following Table. 1 A 2 B 3 C 4 D 5 E 6 F 7 G 8 H 9 I 10 K 1 1 L 12 M ! 3 N 14 O 15 P 16 CL 17 R 18 S <9 T 20 U 21 V 22 w 2 3 X *4 Y z \ t 6 H 1 20 720 5 ° 4 ° 40320 362880 3628800 39916800 479001600 6227020800 87178291200 1 307674368000 20922789888000 335687428096000 6402373705728000 121645100408832000 243 2902008 1 76640000 5 1 090942 1 7 1709440000 1 1 24000727777607680000 258520167388.84976640000 620448401733239439360000 15511210043330985984000000 Now* SELECT EXERCISES. 169 Now, fuppofmg all the 25 letters could be put down in 30 feconds of time, or each combination of them made in. that time (which might be done) it would require 57461442099517020244 Julian years to make all the various combinations which thefe letters would admit of. And consequently, if the world had already lafled 6000 years, it would require 9576907016586170 fuch ages to make all thefe combina- tions, without ever flopping for one fingle Second of time. Suppofng a fquare Cijlern to be a Mile wide and a Mile deep) or to contain a Cubic Mile of Water ; and that a Cubic Yard of Wa- ter ffould run off from it every Minute until it was quite emptied . Qu. How much Time would all the Water take to run put of the Cijlern ? Anf 5451776000 minutes (for fo many cubic yards there are in a cubic mile) or 10365 Julian years 139 days 7 hours 20 minutes. Concern - j 70 SELECT EXERCISES. Concerning the Strength of Steam. From the Reverend Mr. Mitchell’* Treatife on Earthquakes. “ There are many effe&s produced by the vapour of water, when in- tenfely heated, which make it pro- bable that the force of gunpowder is not near equal to it. The effects of an exceeding fmall quantity of water, upon which melted metals are acci- dentally poured, are fuch, I think, as could no ways be expefted from the like quantity of gunpowder. Found- ers, if they are not careful, often ex- perience thefe effedls to their coft. — An accident of this kind happened about forty years ago, at the calling of two brafs cannon at Windmill- Hill, Morefields. The heat, fays Cranmer, of the metal of the firlt gun drove fo much damp into the mould of the fe- con4, which was near it, that, as foon as the metal was let into it, it blew up 4 with SELECT EXERCISES. i 7 r with the greateft violence, tearing up the ground fome feet deep, breaking down the furnace, untiling the houfe, killing many fpedfators on the fpot with the fleams of the melted metal, and fcalding many others in a mofl miferable manner.” Volcanos prove that there are fires within the earth, far below its fur- face ; and over fome of thefe fires there may be caverns of water. When any of the water finds its way through the bottom of fuch a cavern, and falls down into the fire, the water will be immediately rarefied into fleam; and the elaflicity thereof will heave up the ground above it, and make an earthquake. The deeper the fire is, the further will the earthquake be extended. MATHE- .;r . . * - r ' r / v > • * ■ C> “/ * f ■ > « * «<■ »J .», V4 - | . 3 j' MATHEMATICAL TABLES FOR Dividing the LINES on SCALES and SECTORS. I In thefe Tables I have only num- bered the whole degrees, the inter- mediate lines fhewing how many parts each of them is divided into: as, where there are three fuch lines, they denote the degree to be divided into quarters ; where two, into thirds ; and where one, into halves. SELECT EXERCISES. *75 Natural Chords. o a CrQ — t < 7 > CO Chord. Parts. a Crq ~t f6 O Chord . Parts. « CTQ cd 0 Chord. Parts. O n Crq >-t rt) A c n Chord . Parts. I 4 * 4.36 I 4 - 143.86 1 + 282.66 I 4 - 420.09 8.72 148.21 286.98 4 2 4-35 1 3*°9 1 5 z - 5 6 291.30 428.61 i f 7-45 9 156.91 *7 295.61 2 5 432.87 21.81 1 61.26 299.93 437.13 26.17 165.61 304.24 44 I -39 30-53 169.96 3 o8 -55 445-65 2 349 ° 10 1 74 - 3 1 18 3 12.86 26 449 - 9 ° 39.26 178.66 3 1 7' 1 7 454 «i 5 43.62 1 83.01 32 1.48 438.40 47.98 1S7.35 325.79 462.65 3 5 2, 35 1 1 191.69 *9 33°*°9 2 7 466.89 56.71 196.02 334-39 471.13 61.07 200.37 338.69 475-37 65.43 204.71 34 2 -99 479-61 4 69.79 12 209.05 20 347- 2 9 28 483.84 74 .I 5 2 1 3*39 35 u 59 488.07 78.51 217.73 355 - 88 492.3° 82.87 222.07 360.18 496.53 S 87.23 13 226.40 21 364*47 29 500.76 91. 59 230.74 368.76 . 5 ° 4 - 9 8 95-95 235.07 373-04 5O9.2O ICO.3I 239.40 377-33 s^ 2 6 IO4.67 H 2 43-73 22 3S1.61 30 5 l 7*f>3 IO9.O3 248.06 385.89 5 2I - 8 5 II3.38 2 5 2 - 39 390.18 526.06 1 1 7-74 256.72 394.46 530.27 7 122.09 *5 261-05 23 398.73 31 5 34-47 126.45 265.38 403.01 538.68 1 30.80 269.70 407.28 542.88 *35-16 274.02 4U.55 547.08 8 ‘39 5 i 16 278.34 24 415.82 1 32 551.27 tj6 SELECT EXERCISES/ Natural Chords . U a O O n> CTQ Chord . CTQ Chord . n> CTC Chord . n> Crp Chord . *-1 n n > cn * Parts. *1 O O c/> Parts. n> CO Parts. — ! rt n Parts. i. 555-46 I 4 - 688.19 + 817.45 1 "¥ 942.79 559 6 5 692.23 821.43 946 63 563.84 696.37 825.41 950.47 33 568.03 4 1 700.41 49 829.38 57 9 S 4 - 3 * 572.21 7 ° 4*5 5 853-55 958.14 57 6 -39 70S.58 837-3* 961.97 580.56 712.71 841.27 965.79 34 5 8 4-74 4 2 716.73 50 8 45- 2 3 58 969.61 588.92 720.81 849,1 8 973*43 593- c 9 724 87 853**3 977.24 597*25 728.94 857.08 981.04 35 601.41 43 733.00 5 1 861.02 59 984,84 605.57 737.06 864.96 987.64 609.72 741.11 868.89 992.43 613.88 745 ? 6 872.82 996.22 36 618.03 44 749.21 S 2 876.74 60 1000.00 622. 18 75 3* 2 5 880.66 1003.77 626.32 757.29 884.57 1007.54 630.46 761.33 888.48 101 1. 3 1 37 634 60 45 765.36 53 892.39 61 101 5.07 638.74 769 59 8c,6 29 1018.83 642.87 773 * 4 2 900.19 1022.58 647-00 777-44 904.09 1026.33 38 651.13 4 6 78 ! .46 54 907 98 62 1030.07 655.26 785-47 911.85 1033.81 659.38 789 48 91574 1037.54 663.50 793-49 919.62 63 1041.27 39 667.61 • 47 797-49 j 55 923.49 1044.99 671.72 80I.49 927.30 1048.7 1 675- 8 3 805.49 93122 1052.42 679.94 809 48 56 935.08 1056. 1 3 40 684.04 48 813.47 938.94 64 1059 8; SELECT EXERCISES. '77 Natural Chords. o a CTQ CD CD CM Chord, Parts, d n> Ore? O CD C* Chord . Parts. O 0 CTQ 1 n> 0 C ft Chord . Parts. d cd CTQ ft) CD CO Chord , Parts. t 1063.52 1 4 1164.95 I 4 1262.03 I 4 1 354*39 1067.22 1 168.49 1265.41 1 357.60 1070.91 1172.03 1268.78 1 360.80 6 S 1074.59 72 1 1 75 * 57 79 1272.15 86 1363.99 1078.27 1179.09 I2 75 * 5 1 1367.18 1081.94 1 182.61 1278.87 i 37°*36 1085.61 i 186.13 1282.22 1 373-33 66 1089.27 73 1 189.64 80 1285.57 87 1376.70 1092.93 1288.91 1379.86 1096.58 1 196.64 1292.24 1383.02 1 100.23 1200.14 I2 9 S *57 1386.17 67 1103.87 74 1203.63 81 1 298.89 88 1389.31 1107,5 1 1207. 1 1 1 302.20 I 39 2 -45 1 1 1 1.14 1210.58 1393.58 1 1 14.76 1214.05 1308.81 1398.70 68 1 1 18.38 75 1217.52 82 1312. 11 89 I4OI.8I 1 121.99 1220.98 1315.40 I4O4.92 1 125.60 1224.43 13 18.69 1408.02 1 128.21 1 227.88 1321.97 I4I 1.12 69 1 132.81 76 1231.32 83 I 3 2 5 ,2 4 90 I4I4. 21 1 1 36.40 I2 34*75 1328.50 1139.99 1238.18 1331.76 1 * 43*57 £241.60 1 33 5 -° 1 7 ° 1 1 47* 1 5 77 1245.02 84 1338.26 1 150.72 1248.43 1341.50 1154.29 1 25 1 .84 1 344*73 1157.85 1255.24 1 347.96 7 1 1 161.40 78 1258.64 85 1351.18 End of the Table of Natural Chords. N 17* SELECT EXERCISES, Natural Sines. Degrees. Sine . Parts. Degrees. Sine. Parts. Degrees. Sine. Parts. Degrees. Sine . Parts. 4 - 4.36 4 1 43-49 4 279.83 1 4 410.72 8.72 147.81 284.02 414.69 13.09 1 52.12 288.20 418.66 1 > 7*45 9 1 5 6 -43 17 292.37 25 422.62 2I.8l 160.74 296.54 426.57 26.17 165.05 300.71 430.51 30-53 l6 9-35 : 3O4.86 434-45 2 34 - 9 ° 10 173.65 18 309.02 26 43^-37 39.26 ' 77-94 3 1 3 - i 6 442.29 43.62 182.23 3 * 7 - 3 ° 446.20 47.98 I86.52 321.44 450.10 3 52.34 11 I9O.8I 19 325.57 z 7 453-99 56.69 195.09 329.69 457.87 61.O5 199-37 333 - 8 ' 4 6i »75 65.4° 203.64 337-92 465.61 4 69.76 12 207.91 20 342.02 28 469.47 74.11 •21 2.18 346.12 473 - 3 * 78.46 216.44 ■350.2' 477.16 82.81 220.70 354-^9 48O.99 : 5 87.16 13 224.95 21 358.37 29 484.81 91.50 229.20 362.44 488.62 95.85 233-45 366.50 492.42 IOO.I9 237.69 370.56 496.21 6 IO4.53 H 241.92 22 374.6' 3 o 5OO.OO IO8.87 246.1 5 378.65 5 ° 3-77 113.20 250.38 382.68 5 ° 7-54 117.54 254.60 386.71 511.29 7 I 21.87 '5 258.82 23 39°-73 3 i 515.04 126.20 263.03 394-74 518.77 I3O.53 267.24 398-75 522.50 134.85 271.44 402.75 526.21 8 139.17 16 1 * 75- 6 4 24 406.74 32 529.92 SELECT EXERCISES. 179 Natural Sines. Degrees. Sine, Parts. Degrees. Sine, Parts. Degrees. Sine, Parts. Degrees. 0 Sine, Parts. x 4 533 - 6 ' JL 646.12 4 746.06 JL 4 8 3 1 -47 537 - 3 ° 649.49 748.96 833.86 54°-97 652.76 751.84 836.29 33 544.64 41 656.06 49 754 - 7 ' 57 838-67 54 - 8 . 2 Q 659-3; 757-56 84I.O4 55 1 *94 662.62 760.41 8 43-59 5 55-57 665.88 763.23 8 45-73 34 559* 1 9 4 2 669.13 5° 766.O4 58 848.05 562.80 672.37 768.84 850.55 566.41 675-59 771.62 852.64 565.99 678.80 774-39 8 S 4 - 9 i 35 573-58 43 682.00 5* 777-15 59 857-17 577-'4 685.18 779.88 859 - 4 ' 580.70 688.35 782.6l 861.63 584.25 691.51 7 8 5 - 3 2 863.84 36 587.79 44 694.66 5 ? 788.01 60 866.03 59 1 • 3 1 697.79 790.69 868.20 594.82 700.91 793-35 870.36 598.32 704.01 796.00 872.50 37 6oj.8j 45 707.11 53 798.64 61 874.62 605.29 710.19 891.25 8 76.73 608.76 7 1 3.25 803.86 878. 8z 612.22 716.30 806.44 S80.89 38 615,66 46 7 I 9 - 34 * 54 809.02 62 882.95 619.09 722.36 811.57 884.99 622.51 725.37 814.12 887.01 625.92 728.37 816.64 889.02 39 629.32 47 731-35 55 819.15 63 891.01 632.71 734 - 3 ? 821.65 892.98 636.08 737.28 824.13 894.93 639.44 740.22 826.59 896.87 40 642.79 48 743 -H 56 829.04 64 . 898.79 N a i8o SELECT EXERCISES, 6 5 66 6 7 68 69 70 7 1 72 73 74 Natural Sines • Sine. Parts. 900.70 902.59 904.46 906.31 908.14 909.96 91 1.76 9 I 3-55 9 I 5-3i 917.06 918.79 920.50 922.20 923.88 925.54 927.18 928.81 930.42 932.01 933-S 8 935* >4 936.67 938.' 9 939.69 942.64 945.52 948.32 951.06 95 3-7 2 956.30 958.82 46i;26 a n Crq 75 76 77 78 79 80 81 82 33 84 85 90 Sine. Parts. 963.63 96s*93 970.30 974*37 978.15 981.63 984.81 987.69 990.27 992. ;5 994.52 996.19 1000.00 c 3 £ u-, o JU 3 U* o TJ g w Co g * «3 5J O 00 O «« w nj g o £ . <-3 1^2 f* ^ (U ^ o § =3 S ^ ^ s c - « 2 92 ■ rt t I . * j7 l ‘ C 1^< O £ ~ Z cn p, O u S^.S c 2 nD • rfl u * M u W o Ly, ^ 3*f ? O ,2 6 CO G *>^3 ^2o nC 5 3 —• e 22 - o o 8 <£: 8 •5 ^ « >s7 •-G cJ H J2 S2 G ! < rt « i 2 ■? O' 3 w ^ P..I SELECT EXERCISES. 181 Natural 'Tangents. Degrees. 'Tang* Parts. Degrees. Tang * Parts. Degrees. Tang* Parts. Degrees. Tang. Parts. I 4 4 * 3 6 1 4 144.99 I 291.47 4- 450.47 8/73 149.45 296.21 455*73 1 3-°9 153 - 9 1 300.97 461.01 i * 7 - 4 6 9 158.38 17 305.73 2 5 466.31 21.82 162.86 310.51 47»- 6 3 26.I9 *67*34 476.98 30*55 171-83' 320.10 482.34 i 2 34 - 9 2 10 176.33 18 324.92 26 4 8 7*73 59- 2 9 1 80.83 329.75 493 • 1 5 43.66 ,185,34 334.60 498.58 48.03 189.86 339-45 504.04 3 5 2 - 4 ‘ 1 1 194.38 *9 344 33 2 7 5 ° 9*53 56.78 198.91 349.22 5 1 5*°3 6l.l6 203.45 354.12 520.57 65.34 208.00 359*°3 526.13 4 69-93 12 212.56 20 363.97 28 53 i* 7 i 74 - 3 ' 217.12 368.92 537.32 78.70 221.69 373.88 542.96 83.09 226.26 37 8 * 8 7 548.62 5 87.49 13 230.87 21 383.86 29 554 - 3 1 91.89 235-47 388.88 560.03 96.29 240.08 393 - 9 1 565.77 IOO.69 244.70 398.96 571-55 IOJ.IO H 249-33 22 404.03 30 577-35 109.52 253-97 409.1 1 583.18 113*93 258.62 414.21 589.04 118.36 263.28 419*33 594.94 7 122.78 15 267.95 23 424.47 3 1 600.86 127.22 272.63 429.63 606.81 131.65 277.32 434.81 612.80 - 136.09 282.03 440.01 I 618.82 8 140.54 16 286.74 2 4 445-23 1 32 624.87 182 select exercises. Natural Tangents. o a G re OTQ Tang. OkT Tang. re CTQ Tang. re Ore? Tang. "72 Parts. re re Parts* “i re re Parts. Parts. CO Cfl CO CO •j I 4 630.9; z 4 846.56 1 4 1 120.41 I ? 1496.61 637.07 854.08 1 130.29 15 10.84 643.22 861.66 1 140.28 1 5 2 5* 2 5 33 649.41 41 869.29 49 1 * 5°*37 57 1539.86 655.63 876.98 1160.56 1554.67 661.89 884.73 1 170.85 1569.69 ' 668.18 892.53 1 181.25 1584.90 34 674.51 42 900.40 5 o 1 1 9 1 *7 5 58 1600.33 680.88 908.34 1202.37 1615.98 687.28 916.33 1213.10 1631.85 693.72 9 2 4-39 1223.94 1647.95 35 700.21 43 932.52 5 1 1234.90 59 1664.28 706.73 940.71 1 2 45*97 1680.85 713.29 948.96 1 2 57* 1 7 1697.66 719.90 957.29 1 268.49 17 * 4-73 36 726.54 44 965.69 52 1279.94 60 1732.05 733-23 974.16 1291.52 1749.64 739.96 982.7O 1303-23 1767.49 74 6 .74 99 *- 5 i * 3 * 5-°7 1785.63 37 753-55 45 1 000.00 53 1327.04 61 1804.05 760.42 1008.76 ' 339-16 1 822.76 7 6 7-33 1017.61 , 35 , - 4 2 1841.77 774.28 1026.53 '363-83 1 861.09 38 781.29 46 io 35*53 54 1376,38 62 1880.73 788.34 1 1044.61 I389.O9 1900.69 795-44 1053.78 I4OI.95 1920.98 802.58 1063.03 * 4 * 4-97 1941.62 39 809.78 47 1072.37 55 1428.15 63 1962.61 817.03 1081.79 * 44 **49 1983.96 824.34 1091.31 * 455 «P I 2005.69 831.69 1100.91 1468.70 2027.80 40 839.10 48 1 110.61 5 6 1482.56 64 2050.30 2 SELECT EXERCISES. 183 Natural Tangents. 0 O O a 0 7 Q Tang. si Crq Tang. n> Crq Tang. r* Orq Tang. n> Parts. rt Parts. ►i n> n> Parts. n> 0 Parts. CO CO co ; I 4 2073.21 4 2506.52 1 4 - 3124,00 _t_ 4 4086.66 2096.54 2538.63 3 * 71.59 4*65-3° 2120.30 2571.5° 3220.53 4246.85 6 S 2144.5 1 69 2605.09 73 3270.85 77 4331.48 2169.17 26 39-45 3322.64 44*9-36 2194.30 2674.62 3375 - 94 - 4510.71 2219.92 2710.62 3430.84 4605.72 6 6 2246.04 70 2747.48 74 3487.41 78 4704.63 2272.67 2785.23 3545*73 4807.69 2299.84 2823.9I 3605. 88 4915.16 2327.56 2863.56 3667.96 5027.34 67 2 355- 8 5 71 29O4.2I 75 3732.05 79 5 * 44-55 2384.73 2945.9O 3798.27 5267.15 2414.21 2988.68 3866.71 5395.52 2444-33 3032.60 3937 - 5 * 5530.07 68 2475.09 1 72 3O77.68 76 4010.78 80 5671.28 As the Tangents are never laid down further than to 80 degrees on common fcales, it would be need- lefs to carry them further in this Table, The femi-tangents may be laid down on a fcale by taking out the tangents of half the number of degrees in this Table. — Thus, the femi- tangent of a whole degree is the whole tangent of half a de- gree : the femi- tangent of 2 degrees is the whole tangent of 1 degree : the femi-tangent of 3 degrees is the whole tangent of degree : the femi-tan- gent of 4 degrees is the whole tangent of 2 de- grees : and fo on. They are never fubdivided further than to half degrees. i8 4 SELECT EXERCISES, Natural Secants. a Ol 1 'a to C rq Sec. to Cro Sec. 1 to ag Sec. n> crq Sec. ~t c> o o* Farts. •"* a> CO QO Parts. to to C /3 Parts. -t rts n> cn Parts. O 1 000.00 j a 1260.47 3 4 - 1 459*46 3 4 1732.67 IO 1015.43 38 1269.02 47 1466.28 55 1743*45 *5 1033.28 1277.78 1473.19 1754.40 16 ioao.30 39 1286.76 1480.19 1765.52 17 1045,69 1295.97 1487.28 1776.81 18 1051.46 40 I 3 ° 5 * 4 I 48 1494.48 5 6 1788.29 19 1057.62 1310.22 1501.77 1799.95 20 1064.18 1 3 1 5 *°9 1509.16 181 1.80 21 1071. 14 1320.02 1516.65 1823.84 22 io 7 8 -53 4 i 1 325.01 49 1524.25 57 1836.08 2 3 1086.36 1330.07 1 5 3 1 -96 1848.51 24 1094.64 I 335 ,I 9 I 539*79 1861.16 2 5 1103.38 1340.38 1547.69 1874.01 26 1 1 12.60 42 i 345* 6 3 5 ° l 5 S 5*7 2 5 8 1887.08 27 1 I22 *33 1350.95 1563.87 1900.37 28 1132.57 # 1356.34 1572.13 1913.88 29 ! H 3*35 1361.80 1580.51 1927.62 30 1154.70 43 1 3 ^ 7-3 3 5 1 I589.O2 59 1941.60 1 160.59 I 37 2 *93 1597.64: 1955.82 3 i 1 166.63 1 378.56 1606.39 1970.29 1 172.83 i 3 8 4*34 I6I5.26 1985.02 3 2 1179*18 44 1390.16 5 2 I624.27 60 2000.00 1 185.69 1396.06 1633.41 2015.25 33 1 192.36 1402.03 I642.68 2030.77 1 199.20 1408.08 1652.O9 2046.57 34 1206.22 45 1414.21 53 I661.64 61 2062.67 1213.41 1420.42 1671.33 2079.05 35 1220.77 1426.72 l68l.I7 2095.74 1228.33 ! 433*°9 169I.16 21 12.74 36 1236.07 46 1439.56 54 I7OI.3O 62 2130.05 1 244.00 1446.10 171 1.60 2147.70 37 1252.14 1452.74 1722.05 2165.68 SELECT EXERCISES 185 — Natural Secants . a Sec. C 3 Sec . O Sec. a Sec. r& Cp Parts. Op Parts. op Parts. fb Qp Parts. 3 4- 2184.01 4 2585.91 . 3 1 4 3 * 93-22 1 4 4027.23 63 2202.69 2613.13 72 3236.07 4283.66 2221.74 2640.97 3280.15 4362.99 2241.16 68 2669.47 3325.51 77 4445.41 2260.97 2698.64 3372 . 2 1 4531.09 64 2281.17 2728.50 73 3420.30 4620.22 2301.79 2759.09 3469.86 47 * 3-03 2322.82 69 2790.43 3520.94 78 4809.73 2344.29 2822 54 3573*61 4910.58 6 5 2366.20 2855.45 74 3627.96 5015.85 2388.57 2889.20 3684.O5 5125.83 241 1.42 70 2923.80 3 74 1 * 9 ^ 79 5248.84 2434.76 2959.3! 3801.83 5361.23 66 2458.49 2995.74 75 3863.70 5487.40 2482.95 3 ° 33* I 5 39 2 7 ' 7 i 5619.76 '2507.84 7 i 3071 -S 5 3993-93 80 5758-77 j 2 533- 2 9 3111.01 4062.5 1 TVip F.nrl 67 [2559.30 j 3151-55 76 4 * 33-57 Natural Rhumbs . R. Parts. |R. Parts. R. Parts. R.I | Parts. 1 4 49.09 4 438.19 4 810.48 1 4 1 15 1.62 98.14 485 96 855.10 1 191.40 147.12 5 ? 3 - 4 2 899.22 1230.46 1 196.02 3 580.56 s 942.79 7 1268.78 224.82 627.36 985.82 1306.34 293.46 673.78 1028.20 1 34 3- 12 341.92 719.80 1069.98 I 379»°7 2 390.18 4 -65.36 6 1 1 1 1. 14 8 1414. 2 1 This and all the preceding Tables are fitted to one and the fame Radius on the plain fcale. O 186 SELECT EXERCISES, A TABLE fhewing in what Parallel of Latitude any given Number of Geographical Miles make a Degree of Longitude 5 and their Projection on a plain Scale . Miles. Latitude, .. . Nat. Chord. Parts. 1 Miles. Latitude. Nat. Chord. Parts. 6o 0 ' OO OO 00.00 30 0 ' 60 O 1000.00 59 ro 28! ; 182.86 29 61 6 1016.58 58 14 50 258.17 28 62 1 1 1032.82 57 18 nf 3 16.22 27 63 15! 1048.79 56 21 2f 365*43 26 64 191 1064.60 55 2 3 33 l 408.28 z 5 63 22* 1080.I I 54 25 504 447.49 24 66 25 f 1095.45 53 2 7 57 483.OO 23 67 27! IIIO.53 5 2 2 9 55 ! 516.42 22 68 294 1125.49 5 1 3 < 47 ! 547-71 21 69 3of ■ II4O.15 5 ° 33 33 ! 577-37 20 70 3*1 1 1 54 - 79 49 35 15 605.57 19 7 i 3 2 ? 1 168.99 48 36 52 63 2.40 18 72 3 2 ! 1183.21 38 26 658. 28 17 73 3 2 ! 3^1 97. 20 46 39 5 6 ! 683.12 16 74 3 2 1211.05 45 4 1 707.08 *5 75 3 i 1224.66 44 42 50 73 °* 3 ° >4 76 3 °! I238.27 43 44 13 1 752.8+ *3 77 2 9 ! I25I.70 42 45 34 ! 774-s 8 12 78 2 ?| 1264.87 41 4 6 53 ! 795.82 1 1 79 26 I277.98 40 48 Ilf &i 6.52 10 80 24! I 29I.O7 39 49 2 7 ! 836,65 9 8l 22j I303.82 38 5 ° 4 2 i 856.36 8 82 20 j i 3 i6 * 5 8 37 5 1 55 ! 875.52 7 83 18 1329.16 36 53 8 894.48 6 84 15 I 34 I * 5 ° 35 54 19 91 2.90 5 85 13 1 3 5 3*97 34 55 2 9 930,97 4 86 io§ 1 366.26 33 56 38 948.69 3 87 8 1378.40 3 * 57 46 966.06 2 88 5! 1390.47 3 1 58 53 ! 983.19 1 89 2f 1402.36 30 60 c 1000.00 0 90 O 1414.21 SELECT EXERCISES. 187 Latitudes , for the Dialing Scale, Deg. Parts. Deg. Parts. Deg. Parts. I 24.7 3 1 6+7.5 61 931.1 2 49-3 3 2 662.2 62 936.0 3 73-9 33 676.+ P 940.8 4 98.4 34 690.2 64 945-4 5 • 122.8 35 703.6 949.6 6 I 47 *° 36 716.6 .66 953-9 7 • 17.. 1 37 729.2 .67 957.8 8 149.9 38 741.4 6s : 961.5 9 2 18,6 39 753-2 69 9 6 5 -i i° 241.9 40 764.7 70 968.5 1 1 265*0 4 i 775 - 8 71 971 6 12 287.9 42 786.5 72 974-5 13 310.4 43 796.8 73 977-4 *4 33 2 -5 44 806.7 74 980.1 15 3 . 54-3 45 816.5 75 982.5 16 37 - 8 46 825.9 76 984,8 *7 396.0 47 834- 8 77 986 9 18 417.6 48 843.6 78 988.8 19 437 * 8 49 851.9 79 990.6 20 . 457*7 ,50 860.0 80 992.4 21 477*3 5 i 867.8 85 998.2 22 496.1 5 2 875-3 90 ICOO.O 23 514.6 53 882.5 G to - 0 • 24 532.8 54 889.5 75 0 T2 <8.2 g> 2q 550-5 55 89b. 2 to OJ 0 S ^ -T3 26 567.8 56 902.6 faD C QJ O G > C 27 584.6 57 ,908.8 OD- r-t 4) J £5 87.0 I 5 753-4 * C\ 20 113.8 20 769.0 25 139-6 25 784.6 Js 3 ° 164.5 3 ° 800.2 35 188.8 35 816.0 40 212.0 40 831.9 45 234.6 45 847.8 50 256.7 5 ° 863.8 <2 55 278.0 55 8 So. 1 I 0 298.8 mi 0 896.5 5 319.2 5 913.1 45 10 339.0 10 930.0 15 358.4 *5 947.1 b-» 20 377-3 20 964.5 feo 25 395-9 25 982.0 S 3 3 ° 414.2 30 1000.0 Vo 35 432.2 35 1018.3 £ 40 449-7 40 1036.9 ^S 45 467.1 45 1055.8 ■♦o 5 ° 484.2 5 ° 1075.2 Vb __ 55 501. I 55 1095.0 -u* II 0 5 * 7-7 V 0 ii* 5-4 *3 5 534 -* 5 1 136.2 Is 10 5 5°-4 10 1 * 57-5 *5 566.4 15 1179.6 *5 20 582.4 20 1202.2 c >§ 25 598-3 25 1225.4 v$ 30 614.0 3 ° 1249.7 43 35 629.6 35 12746 40 645.2 40 1300.4 45 660.8 45 1327.2 5 ° 676.3 50 > 3 ? 5 -i 55 691-7 55 •383-9 III 0 707.1 VI 0 1414. 2 7 SELECT EXERCISES. i8 9 Inclination of Meridians. Deg. Parts. J Deg. Parts. Deg. Parts. ! 2 4-3 3 1 530.8 61 909.8 • w 2 47.7 32 543.8 62 923-3 -a 3 79.4 33 556.8 6 3 93 6 -9 4 92.4 34 569.7 64 950.6 c3 , 4 5 113.8 35 582.4 65 964.5 6 1 34* 5 3 6 595*i 66 978.5 O 7 154.6 37 607.7 67 992.8 cs 8 174-2 33 620.3 68 1007.3 13 9 193.4 39 632.8 69 1022.0 -0 S3 10 212.0 40 645.2 70 1036.9 P3 1 1 230.1 41 657.7 7* 1052.0 to ta J 2 247.9 42 67O.O 72 1067.4 a 0 13 265.3 43 682.4 73 1083.1 53 *4 282.2 44 694.7 74 1099.1 Uh 0 *5 298.8 45 707 . 1 75 1U5.4 0 a - 16 315 -I 46 7 1 9-5 76 1 132.0 13 17 33 r -J 47 731.8 77 1 148.9 5J3 18 346.8 48 744.2 78 1166.3 .s *9 362.2 49 756.5 79 1 184.1 U . 20 377-3 50 769.0 80 1202.2 U 2 1 392.2 5 1 781.4 81 1220.8 Cl, O 22 406.9 52 793-9 82 1240.0 2 3 421.4 53 806.5 33 1259.6 O 24 435-7 54 819.1 84 1279.7 2 5 449-7 55 831.8 85 1300.4 O 26 463.6 5 6 844.5 86 1321.8 TO 27 477-3 57 857.4 87 * 343 - 8 5 28 490.9 53 870.4 88 1366.5 29 504.4 59 883.4 89 1389.9 30 5 * 7-7 60 896.5 go 1414.2 The preceding Tables are for gra- duating the Lines of Natural Chords, Sines, Tangents, and Secants, &c. on common plain Scales and Se&ors. — The following Tables are for laying down Gunter’s logarithmic Lines of Numbers, Sines, Tangents, Verfed Sines, Meridional Parts, &c. on his Scales. SELECT EXERCISES, 191 Numbers. The Logarithm of 1 ~ 0. N. Parts. N. Parts. N. Parts. 1. 01 4.32 i* 3 i 1 17.27 1.61 206.83 1.02 8.60 1.32 120.57 1.62 209.52 1.03 12.84 1-33 123.85 1.63 212.19 1.04 l 7'°3 1*34 127.10 1.64 214.84 1,03 21.19 i* 3 ? 130*33 i. 65 217.48 1.06 2 S* 3 1 1.36 133*54 1.66 220.1 1 1.07 29.38 i *37 136.72 1.67 222.72 1.08 33 - 4 * 1.38 139.88 1.68 225.3 1 1.09 37.43 1 *39 H3.01 1.69 227.89 1. 10 41.39 1.40 146.13 1.70 230.45 1. 1 1 45 * 3 * 1.41 149.22 1.71 233.00 1.12 49.22 1.42 152.29 1.72 * 35*53 1.13 53.08 *•43 1 55-37 1*73 238.05 j 1. 14 56.90 1.44 158.36 1.74 240.55 1.15 60.70 i «45 161.37 1 *75 243.0+ I. l6 64.46 1.46 1 64-35 1.76 245.51 Ul 7 68.19 1.47 167.32 i *77 247.97 1. 18 71.88 1.48 170.26 1.78 250.42 1.19 7 5-55 1.49 I 73*19 1.79 252.85 1.20 79.18 1.50 I76.O9 1.80 255.27 1. 21 82.79 I *S I I78.98 1.81 257.68 1.22 86.36 1 *5 * l8l.84 1.82 260.07 1.23 89.91 i *53 I84.69 1.83 262.45 1.24 93.42 !*54 I87.52 1.84 264.82 1.23 96.91 *•55 I 90-33 1.85 267.17 1.26 ico.37 1.56 193.12 1.86 269.51 i **7 103.80 i -57 ! 9 ?- 9 ° 1.87 271.84 . 1 1.28 107.21 1.58 198.66 1.88 274.16 | 1-29 110.59 I.59 201.34 1.89 276.46 1 1.30 113.94 1.60 204.. 1 2 1. 90 s 278.75 t 9 2 SELECT EXERCISES. Gunter's Line of Numbers. ! N. Parts. N. Parts. N. Parts. 1.91 281.03 2.42 383.82 3-°5 484.30 1.92 283.30 2.44 3 ^ 7-59 3.10 491.36 i *93 285.56 2.46 390.94 3- 1 5 498.31 *•94 287.80 2.48 394*45 3.20 505-15 1.95 290.03 2.50 397-94 3 - 2 5 51 1.88 1*90 292 26 2.52 401.40 3 - 3 ° 5 1 8.5 1 1.97 294.47 2.54 404.83 3 - 3 S 525.04 1.98 296.66 2.56 408.24 3.40 531.48 1.99 298.85 2.58 41 1.62 3-45 537-S2 2.00 -301.03 2.60 414.97 3 - 5 ° 544.07 2.02 3 ° 5-35 2.62 418.30 3-55 550-23 2.04 309.63 2.64 421.60 : 3-6o 556-30 2.06 313-87 2.66 424.88 3.65 562.29 2.08 318.06 2.68 428.13 3 - 7 ° 568.20 2.10 322.22 2.70 43 *- 3 6 3-75 574-^3 2.12 326.34 2.72 434 - 5 / i 3- So 579-73 2.14 33 °- 4 ' 2.74 437*75 3.85 585.46 2.16 334-43 2.76 440.91' 3 - 9 ° 591.06 2.18 338.46 2.78 444.04 3*95 596.60 2.20 342.42 2.80 : 447 - r6 4.00 602 06 2.22 346.35 2.82 450.25 4.05 607.45 2.24 350.25 2.84 453 - 3 ^ 4.10 61 2.78 2.26 254.11 2.86 456.37 4*5 61 8.05 2.28 357-93 2.88 459-39 4.20 623.25 2.30 361.73 2.90 462.40 4.25 628.40 2.32 365.49 2.92 ; 465.38 430 633-47 ^•34 369.22 2.94 468.35 4-35 638.49 2.36 3 7 2 - 9 1 2.96 471.29 4.40 643.49 2.38 376.58 2.98 474.22 4-45 648.36 2H° 380.21 3.00 477.12 4.50 655.2? SELECT EXERCISES. / 193 Gunter's Line of Numbers . N. Parts. N. Parts. N. Parts. 4*55 658.01 6.05 781.7; 7-55 877-95 4.60 662.76 6. 10 7 s S -33 7.60 880.8 1 4.65 667.4; 6.15 788.75 7.65 883.66 4 * 7 ° 672. 10 6.20 79 2 *39 7.70 886.49 4-75 676.69 6.25 795.88 7-75 889.30 4.80 681.24 6.30 799-34 7.80 892.09 4.85 685.74 6 *35 802.77 7.85 894.87 4.90 690.20 6,40 806.I8 7.90 897.63 4-95 694.61 6.4; 8O9.56 7*95 900.37 5.00 698.97 6.50 812.9I 8.00 903.09 5*°5 703-29 6*55 8l6.24 8.05 905.71 5.10 707-57 6.60 819.54 8.10 908.48 5 -i 5 711.81 6.65 822.82 8 - i 5 911.16 5.20 716.00 6,70 826.O7 8.20 913.81 5.25 720.16 6.75 829.3O 8.25 916.45 5.30 724.28 6.80 832.51 8.30 919.08 5*35 728.35 6.85 835.69 8.3; g21.bg 5 * 4 ° 732.39 6.90 838.84 8.40 924.28 5*45 736.40 6.9; 84I.98 8.45 926.86 5.50 74 °- 3 6 7.00 845.IO 8.50 929.42 5-55 744.29 7.05 848,19 8.55 93 J *97 5.60 74 8 -i 9 7.10 851.26 8.60 934-50 565 752.05 7 -iS 854.3O 8.65 937.02 5.70 755- 8 7 7.20 8 57‘33 8.70 939*52 5*75 759* 6 7 7-25 86C.34 8.75 942.81 5.80 7 6 3-43 7 « 3 o 863.32 8.80 944.48 5.85 767.16 7*35 866.29 8.85 946.94 5.90 77 °. 8 5 7.40 869.23 8.90 949*39 5*95 774 - 5 z 7.45 872.I6 8-9; 951.82 6.00 77 8 * 1 5 . 7.50 875.06 9.CO 954.24 P I 9 4 SELECT EXERCISES. Gunter's Line of Numbers . N. Parts. N. Parts. N. Parts. 9.05 9.10 9.15 9.20 9.25 9.30 9-35 956.6 5 959.04 961.42 963.79 966.14 968.48 970.8? 9.40 9*45 9.50 9-55 9.60 9.65 9.70 973.13 975-43 977.72 980.00 982.27 984.53 986 77 1 9*75 9.80 9.85 9.90 9- 95 10.00 989.00 991.23 993.44 995-63 997.82 1000.00 i The End. All the number of parts into which the double Line pf Numbers (from i to 10) on a fcale two feet long can well be divided, are inferted in this Table. The Line of Numbers is marked with thf numeral figures i, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. — The fpace between 1 and 2 may be di- vided into 100 parts ; from 2 to 3, into 50; but all the others into no more than 20 each, without hurting the eye to look at fuch fmall divifions. But, in common prattice, the fpaces from the fir ft 1 to the fecond 1 are divided only into 10 parts each ; and after that, from 1 to 2, into 50 ; from 2 to 5, into 20 each ; and from 5 to 10, each fpace is divided into no more than 10 parts; leaving the intermediate fub- divifions to be eftimated by the eye, which may be eafijy done. In the following Tables, D. Hands for Degrees, and for Minutes of a Degree. SELECT EXERCISES. 195 Gunter's Line of Sines . D. M. | Parts. D. M. Parts. D. M. Parts. 0 20 2235.25 3 15 1246.47 6 10 968.91 2 5 2138.34 20 1235.49 *5 963.10 3 ° 2059.16 2 5 1224.78 : 20 957*38 35 1992.21 3 ° I214.32 2 5 95 1 -7 2 40 1934.22 35 I204.I2 30 946.14 45 1883.07 40 I I94.I5 35 940.63 5 ° 1837.32 45 1 184.40 40 935-19 55 > 795*93 5 ° 1174.87 45 929.82 1 0 1758.14 55 >165.54 50 924.52 5 I 7 2 3-39 4 0 1 1 56.42 55 919.28 10 169I.2I 5 1147.48 7 0 914.11 15 l66l.25 10 i>3 8 *7 2 5 908.99 20 l633.22 >5 I 130.13 10 903.94 2 5 1606.90 20 1 I2I.7I *5 898.84 30 I582.O8 2 5 1115. 10 20 894.01 35 I558.6l 30 I 105.36 25 889.13 40 >536.34 35 1097.40 3 ° 884.30 45 1515.15 40 1089*60 35 8 79-53 So 1494.96 45 1081.92 40 874 81 55 1475.66 5 ° 1074*39 45 870.15 2 0 1457.18 55 1066.98 50 865.53 5 >439.46 5 0 1059.70 55 860.96 10 1422.43 5 1052.64 8 0 856.44 >5 I406.O5 10 1045.50 5 851.97 20 1390.27 *5 1038.57 10 846.67 2 5 I 375*°3 20 1031.75 *5 843.17 30 1360.32 2 5 1025.04 20 838.84 35 1346.09 3 ° 1018.43 2 S 834*55 40 1 33 2 * 3 1 35 101 1.92 30 830.30 45 1318.96 40 1005*50 35 826.09 5° 1 306,00 45 999.18 40 821.93 55 1293.42 5 ° 992.96 45 817.80 3 0 1281 .20 55 986.82 50 813.72 5 1 269.3 1 6 0 980.77 55 809.67 10 1 2 57.74 5 974.78 9 0 805.67 P 2 1 96 SELECT EXERCISES Gunter's Line of Sines. D. M. Parts. D. M. Parts. } D. M. 9 5 10 1 5 20 2 5 30 35 40 45 5° 55 10 o 10 20 3° 40 50 1 1 o 10 20 3° 40 5° 12 o 10 20 30 40 5° 13 o 10 20 3° 40 5° 14 o 801 .70 797.77 793*87 790.01 786.18 782.39 778.63 774*9* 771.22 767.56 763.93 760.33 753-23 746.24 739-37 732.61 725.95 719.40 712.95 706.60 700.34 694.18 688.1 3 682.12 676.22 670.40 664.66 659.00 653.42 647.9 1 642.48 637.11 631.8 1 626.59 62 1 .42 616.32 14 10 20 30 40 5° 15 o 10 20 3° 40 50 16 o 10 20 30 40 5° 17 o 10 20 30 40 50 18 o 10 20 30 40 50 19 o 10 20 30 40 5° 20 o 61 1.29 606.3 1 60 ! .40 596.54 591*7; 587.OO 582.32 577.68 573.10 568.57 564.09 559.66 555.28 55° 9; 546.66 542.42 538.22 534x6 529 95 525.88 521.86 5‘7.8? 5 ' 3*93 510.02 506.15 502.32 498.52 494*77 491.04 487.36 483.71 480 .0 9 476 . 5 ° 47 2 -95 469.43 465*95 15 3° 45 22 o 15 3° 45 23 o *5 30 45 24 o *5 30 45 25 o *5 30 45 26 o ! 5 30 45 27 o *5 3o 45 28 o *5 3o 45 29 o Parts. 460.78 45 5*67 450.64 445.67 440.77 435-9 2 43i*i4 426.42 421.76 417.16 412.61 408.12 403.68 399*3° 394-97 300.60 386.46 382.27 378*14 374*°5 370.01 366.02 362,06 358.16 354.29 35°*47 346.69 34 2 -95 339-25 335*59 3 8l -97 328.39 32485 321*34 317.87 314*43. SELECT EXERCISES. 197 Gunter's Line of Sines . Deg.| Parts. Deg. Parts. |Deg. Parts. 0 • 0 o« 1 4 31 1.02 1 ? 208.24 X a 89.31 307.66 205.85 55 86.64 So 304-33 203.48 84.01 V§ * 3° 301.03 39 201. 1 3 5 6 8i -43 .* 0 297.76 198.80 78.89 4-1 « 1 , »H 294-53 196.49 57 76.41 291.33 194.20 58 73-97 o 'Z 3 * 288.16 40 191.93 71.58 « « 3 1 1 Q 4 285.02 187.46 69.23 0 4-» C3 flv 281.92 4 i 183.06 59 66.93 .5 +3 ^ ,, i ♦ , r** 278.84 * 78.73 60 64.68 13 si 3 (U 32 275.79 4 2 174.49 62.47 272.77 170.32 61 58.I8 * **** • «< 269.78 43 166.22 62 54.07 O 258.1 1 45 *50.51 66 39- 2 7 O ^ ^ M 4-1 255.26 46 146.76 67 35-97 D 34 252.44 I 43-°7 68 32.83 -Eg 249.64 139.44 69 29.85 2 * - 246.87 47 *35-87 70 27.OI . O g* - ON 244.13 48 132.37 7 * 2 4-33 .9 C _ C 35 241.41 *28.93 72 21.79 ja 2 § 238.71 125.54 73 19.40 *So.S 0 cj 00 236.05 49 122.22 74 17.16 O eq 63.5 327.7 37 478:5 652.3 *4 169.7 332.2 483.5 48 658.3 175.9 2 7 33 6 .7 488.5 664.3 1 5 182.1 34 i. 1 38 493.6 670.3 188.3 345-7 498.7 49 676.4 I SELECT EXERCISES 205 Gunter' $ Line of Meridional Parts , » Deg. Parts. Deg. Parts. Deg. Parts. Deg. Parts. 1 i 682.5 60 905.5 68 1 126.1 V 76 1 442.0 688,7 ¥ 9 1 1 • 5 1 134.2 * 454-5 So 694.9 9 * 7-5 U 4 2 -3 1467.2 701.1 923.7 1 * 5 0 -5 1480.2 707.4 6l 929.8 69 1158.9 77 * 493-4 S 1 7 1 3*7 936.0 1167.3 1 506.9 720.1 942.3 H75.8 1520,6 726.5 948.6 1 1 84.4 * 534*6 5 2 733 -o 62 954-9 70 1 1 93 - 1 78 1548.9 739-5 961.4 1 202.0 * 5 6 3-5 746.1 967.9 1210.9 * 579-3 53 752.7 9744 1 2 19.9 * 593-5 759-4 63 980.9 7 i 1 229.1 79 1609.0 766.1 987.6 1238.4 1625.0 54 772.9 994-3 1247.8 1641.3 779-7 1001. 1 1257.3 1657.9 786.6 64 1007.9 72 1266.9 80 1675.0 5 S 793-5 10 1 4.7 1276.7 1692.5 800.5 1021.6 1286.6 1710.4 807.6 1028.7 1 296.6 1728.8 5 6 814.7 6 5 1 °35 *7 73 1 306.8 81 1747.8 821.9 1042.8 1317.2 1767.2 829.2 I050.1 1327-7 1787.2 57 836.5 1057.5 1 33^*3 1807.8 849.9 66 1065.7 74 I 349 - 1 82 1829.0 85 1 *3 1072. 1 1360 1 1850.9 58 858.8 1079.6 137**2 1873.6 866.4 1087.1 1382.5 1896.9 874.1 67 1094.8 75 1 394.0 83 1921. 1 59 881.8 1102. 5 1405.7 1946.2 889.6 1 1 10.2 1417.6 1972.2 897.5 1 1 18.1 '429.7 1999.2 0.3 20 6 SELECT EXERCISES. Of the Conf ruction of the plain Scale, Sector, and Gunter* s Scale. As it often happens that thefe three moft ufeful inftruments are badly di- vided, either through the fault of pat- terns erroneous in their firft conftruc- tion, or worn out by much ufe, or by the ignorance or negledt of the di- vider, it was thought that rules and Tables for graduating them accu- rately, and examining thofe which have been already made, might be very acceptable both to the workman and young mathematician defirous of projecting them, or to any perfon who is obliged to ufe them. The foundation of thefe, and indeed of molt other fcales, is a line of equal parts, fo fubdivided as that one of the leaft of thefe parts fhall fcarce take up a vilible fpace, as into thoufand parts pf an inch, or even more minute fo that; SELECT EXERCISES. 207 that an error of one unit in the laft place, in taking off any diftance, may not affedt the graduations of your in- tended inftrument. Furnifhed with fuch a fcale, and a beam compafs with a regulating fcrew at one end, you may lay down any number, of four, or even five places of figures, with great exadtnefs. The belt way of dividing a right line into any pofllble number of parts, and that which is pradtifed by our belt di- viders of mathematical inftruments, is by help of diagonal lines ; both be- caufe the fmalleft diagonal fubdivi- fions are as perceptible on fuch a fcale as the largeft, and that this is the moll fimple method of fubdividing, and may be extended further than any other way yet known. In order to have an exadt diagonal fcale, it is abfolutely neceliary that the parallel lines, through which the dia- gonal lines are drawn, be equidiftant among themfelves ; that the diagonals be 208 SELECT EXERCISES. be fo likewife ; and that, in the two outermoft parallels in which the dia- gonals terminate, the ends of each diagonal muft be juft even with the oppofite ends of thofe next to it on each fide ; that is, in a line perpendi- cular to the parallels. In one inch, 25 diagonal lines may be very eafily placed, and confequently 50 in two inches : and if thefe diagon- als run through 20 equidiftant paral- lel lines, they will divide every two inches of length of the diagonal fcale into 1000 equal parts, which will be quite fufficient for graduating the lines of Rhumbs, Chords, Sines, Tan- gents, and Secants, on common plain fcales, where the radius (or 60 degrees of the Chords) takes up only two inches in length of the whole line : and to that line of Chords, the Rhumbs, Sines, Tangents, and Secants, are adapted. There is generally another line of Chords of 3 inches radius on thefe 3 fcales, SELECT EXERCISES. 209 fcales, at the right-hand end; and juft above that line is a line marked M, Lon. for lhewing how many geogra- phical miles are contained in a degree of longitude in any given parallel of latitude. To lay down thefe two lines, there muft be a diagonal fcale con- taining 1000 parts in 3 inches ; and thefe parts or divilions muft be con- tinued on to 1420, which will be fuffi- cient for the purpofe, as the whole length of each of thefe two lines takes up only 1 414.2 1 parts. But, for laying down the above- mentioned lines of Rhumbs, Sines, Chords, Tangents, and Secants ; where the radius 2 inches contains 1000 equal parts, and the length of the Tangent line (in which there are only 80 de- grees laid down) is u r V inches, the diagonal parts or equal divilions muft be carried to 5760; of which, 5671.28 will be equal to the Tangent of 80 degrees, and 5758.77 to the Secant of the fame number of degrees ; as fhewn by 210 SELECT EXERCISES. by the Tables of Natural Tangents and Secants. For Gunter’s lines of Numbers, Rhumbs, Sines, Verfed Sines, Tan- gents, and Meridional Parts, every one of which takes up 22-% inches of his two foot fcale, a diagonal fcale mull be firlt made, in which 22A inches fhall contain 2000 equal parts : and, for this purpofe, there will be no need for more than 10 equidiftant parallel lines along the fcale, and one inch of its length will not include quite 8 equidiftant diagonal lines. For Sectors, of whatever length the radius of the Chords (or of 60 degrees) be, which is generally very near the whole length of the Sector when fhut, a diagonal fcale mull be prepared, in which that length fhall contain 1000 equal parts. — The diftance between thefe long parallels fhould not be lefs than a tenth part of an inch ; other- wife the eye cannot well eftimate the decimal parts exprefled in the Tables, which 211 SELECT EXERCISES. which are hundredth parts of the fpaces between the long parallel lines, in all except the Dialing Tables of Latitudes, Hours, and Inclination of Meridians, and in the Table of Gun- ter’s Meridional Parts; in which, the fpaces between the parallels are fup- pofed to contain only ten parts each. Being thus provided with proper diagonal fcales, patterns for gradu- ating all the lines on Scales and Sec- tors may be made ' in the following manner. ' l , Having drawn the lines on your in- tended brafs pattern, analogous to thofe which mull contain the graduations on fcales, fix your diagonal fcale clofe by the upper edge of the pattern to be graduated : and then applying one fide of a fquare along the upper edge of the diagonal fcale, and the other fide diredtly acrofs both that fcale and the pattern, cut the firll divifion of each line on the pattern direftly even with the very beginning of the diagonals ; and 2i2 SELECT EXERCISES. and each divifion after that, againft the fame number of parts, found among the diagonals, as anfwer to the num- ber of parts in the Tables belonging to each degree in the refpe&ive lines : and, where the lines are long enough to admit of the quarters of degrees, thefe may be put in, according to the numbers in the Tables. But, in the common plain fcales, where the length of the radius of the Chord line is only two inches, the fpaces taken up by the degrees are too fmall to al- low of fubdivifions, except in the Tan- gents and Secants after the 50th de- gree. Thus, in the Tables of Natural Chords, Sines, Tangents, and Secants, againft the firft degree is 17.45 parts both for the Chords and Sines, and 17.46 for the firft degree of the Tan- gents. Therefore, finding thofe parts among the diagonals, and laying the crofs edge of the fquare to them, cut the firft degree of thefe lines on the pattern, SELECT EXERCISES. 213 pattern, clofe by the faid edge of the fquare: and proceed on in the fame manner with all the reft of the degrees in the feveral lines, cutting them right againft the fame number of parts found among the diagonals which the Tables fhew to belong to them refpe6tively. Thus, the 20th degree in the line of Chords muft be cut even with 347.29 parts among the diagonals ; the like degree of the Sines even with 342.02 parts ; and the fame degree of the Tangents even with 363.97 parts in the diagonal fcale. Where the Sines end on the plain fcale, the Secants begin ; namely, even with 1000 parts among the dia- gonals. But the Secant degrees are fo fmall at firft, that there can be no putting in any of them lefs than the icth, which anfwers to 1015.43 parts among the diagonals ; and the next that can be put in is the 15 th, which anfwers to 1035.28 parts: after which, all the degrees as far as 80 may be put into 214 SELECT EXERCISES. into the line of Secants, cutting them even with the fame numbers of parts found among the diagonals as Hand againft them in the Table of Natural Secants. On Seniors, there are two Tangent lines on each leg. The firft of thefe goes only to 45 degrees, the length of the leg admitting of no more : and the others begin at 45 degrees, at a fourth part of the length of the leg from the center of the joint to that part near the end of the leg where the 45th degree of the former line Hands, and goes on generally to 76 degrees. The firft of thefe, which goes from o de- grees to 45, is called the lower Tan- gents ; and the laft, which goes from 45 to 76, is called the upper Tangents. The lower Tangent degrees are laid down by the diagonal fcale according to the numbers found againft them in the Table of Natural Tangents ; 1000 parts of the diagonal fcale taking up juft as much length as the whole 45 degrees SELECT EXERCISES. 215 degrees of thefe Tangents do. But, as the radius of the upper Tangents is only a fourth part of the length of that of the lower ones, in order to lay them down on the Seftor-pattern, all the numbers in the Table of Natural Tangents above 45 degrees muft be di- vided by 4 , and their quotients fought for, among the diagonals, for laying down the refpe&ive degrees of thefe Tangents. To fave the operator this trouble, I have taken it, and made the following Table, which confifts of thefe quotients, and confequently con- tains the number of parts (to be found among the diagonals) for dividing the line of upper Tangents, the 76th de- gree of which is even with 1002.69 parts of the diagonal fcale. 21 6 SELECT EXERCISES. Supplemental Table, for laying down the Line of upper Tangents on Patterns for dividing the Lines on Sectors. Deg. Parts. Deg.j Parts. (Deg. Parts. Deg. Parts. 45 250.00 4 - 328.77 1 X 441.87 1 4 626.33 252.19 53 331-76 446.41 634.46 254.40 334-79 61 451.01 642.87 256.63 337-85 455-69 69 651.2*7 46 258.88 340.36 460.44 659.86 261.15 54 344.09 465.27 668.65 263.44 347.27 62 470.18 677.65 265.51 35°*49 475- 1 7 70 686.87 47 268.09 353*74 480.24 696.31 270.45 55 357-04 485.40 705.98 272.83 360.37 63 490.65 7 I 5-^9 275.23 363-75 495.99 7 1 726.05 48 277.73 367.17 501.42 73 6 -47 280.10 5 6 370.64 506.95 747-17 282.57 374*15 64 512-57 758.15 285.07 377-71 518.30 7 2 769.42 49 287.59 381.31 524-13 781.00 290.14 57 384.96 530.07 792.70 292.71 388.67 65 536-13 805.13 295.31 392.42 542.29 73 817.71 5 ° 297.94 396.22 548-57 830.66 300.59 58 400.08 554.98 843.98 303.27 403.99 66 561.51 857.71 305.98 407.96 568.17 74 871.85 5 1 308.72 4^-99 574.96 886.43 311*49 59 416.07 581.89 901.47 3 H *29 420.21 67 588.96 916.99 317.12 424.41 596.18 75 933 - 01 5 2 319.98 428.68 603.55 949-57 322.88 60 433 -oi 61 1.08 966.68 323.81 437 - 4 1 68 618.77 984.38 The SELECT EXERCISES. 217 The diagonal fcale remaining where it did, for dividing the line of lower Tangents, [See the Remark further ori\ cut the 45 th degree by . the fquare againft 250 parts found among the diagonals, having put the crofs fide of the fquare to it : fo this fecond line of Tangents fhall begin juft at a fourth part of the length of the former line from the center of the joint; and then proceed with the reft of the degrees from 45 to 76, cut- ting them in the pattern even with the fame number of parts found among the diagonals as belong to them in the preceding Table. The Secants on the Sedlor begin alfo at a fourth part of the length of the leg from the center of the joint; and therefore, to lay down the degrees of the Secants on the Sec- tor pattern, all the numbers in the Table of Natural Secants from o to 754 mult be divided by 4, and their quotients taken among the parts on the diagonal fcale. The following Table, for this purpofe, is the quotients of the numbers in the Table of Natural Secants divided by 4. The Secant of 754 degrees would reach to 1015.55, which is 15.55 parts more than the diagonal fcale contains. R Sup- ai8 SELECT EXERCISES. Supplemental Table, for dividing the Line of Secants on Sett or Patterns . Deg. Parts. Ueg.i Parts. |Deg.| Parts. Deg.f Parts. 0 250.00 315.12 364.86 433-*7 IO 253.86 38 3 1 7* 2 5 47 366.57 55 435.86 * 5 258.82 3 * 9*44 368.30 438.60 16 260.07 39 3 2I * 6 9 370.03 44 *- 3 8 *7 261.42 3 2 3*99 37 1 - 82 444*20 1 8 262.86 40 326.2,- 48 373 * 6 2 56 447*07 *9 264.40 3 2 7-55 375*44 449.99 20 266.04 3 ?* 77 37 7 * 2 9 45 2 *95 2 t 267.78 330.00 379 * 16 455.96 22 269.63 4 1 33 1 * 2 5 49 381.06 57 459.02 2 3 2 7**<;9 332 . 5 ^ 382.99 462.13 2 4 273.66 333 * 8 ° 384-95 465.29 2 5 275.84 335- c 9 380.92 468.50 26 278.15 42 336 . 4 * 5 o 3*8.93 5 8 471*77 27 280.58 337*74 390-79 475.09 28 283.14 539 -°* . 393-03 478.47 29 285.84 340.45 395- 1 3 481.90 30 288.67 43 34*. 8 3 5 * 397-23 59 485.40 290.15 343- 2 3 399 - 4 ' 488.95 3 1 291.66 344.64 401.60 492.57 293.21 346.08 403.81 496.25 3 2 294.79 44 347*54 5 2 406.O7 60 ; 00.00 296.42 349.01 408.3; . 503.81 33 298.09 350.51 410.67 507.69 299.80 352.02 4 * 3-02 51 I.64 34 3o» 55 45 333-55 53 415.41 61 515.67 303-35 355.10 4 * 7*83 519.76 35 305 19 356.68 420.29 5 2 3-93 307.08 358.25 422.79 528.18 36 309.02 46 359 8 9 54 425.32 62 532 - 5 * 311.00 j 361.52 427.90 536.92 37 313.03 { 36c?. 1 8 430.51 1 < 4*. 42 SELECT EXERCISES. 2 ! 867.46 565.24 6 7 639.82 7 39 - ^ 3 880.23 64 570.29 646.48 74^-93 893.40 575-45 553.28 758.29 74 906.99 580.70 660.24 7 1 707.89 921.OI 586.07 68 667.35 777-75 933-49 65 5^-53 674.66 787.89 9^0.46 597 -H 682.1 2 798.30 73 965.92 602.85 6 ?g -77 72 809.02 981.93 608.69 69 697.61 ! 820.0! 998.48 On Sedtors, the line of Lines (which is divided into equal parts) and the lines of Chords, Sines, Tangents, and Secants, are laid down on both the, legs. They are all drawn from the center of the joint, and ought to be ftridtly at equal angular diftances from each other, at the other ends of the legs. So that, whether the Sedtor be open or Ihut, the fame opening of the compalfes that reaches crofs-wife from io on the line of Lines on one R 2. leg *20 SELECT EXERCISES. leg to jo on the other (at the ends furtheft from the joint) fhould reach from 60 to 60 degrees of the Chords, from 90 to 90 of the Sines, from 45 to 45 of the lower Tangents, from end to end of the upper Tangents, and likewife from end to end of the lines of Secants. I generally find all thefe very well laid down, except the lines of upper Tangents and Secants ; which, for want of this precaution, are trou- blefome to ufe on molt Sectors. I apprehend that Sedtors would be much more convenient than they now dre, if their lines of Chords contained all the degrees of o to 90. For then, in laying down an angle of any num- ber of degrees lefs than 90, one open- ing of the cbmpafles would do 5 whereas, as they now are, it requires two operations to lay down an angle of any number of degrees above 60. And befides, if the line of Chords con- tained all the 90 degrees, the lower Tangents, inftead of ending at 45 de- 6 grees, SELECT EXERCISES. 22 1 grees, would go on to 55, by carry- ing them out a very fmall fpace beyond the end of the Chords: and alfo, the line prolonged on which the Sines are laid down (they going no further than 60 degrees of the Chords) would receive thereon all the Secants as far as 45 degrees ; fo that, all thefe Tangents and Secants might be taken off’ without a fecond opening of the Sedtor, as is cuftomary in common ones ; which would be a very great convenience to thofe who ufe it. And then, beginning the line of upper Tangents at 55 degrees, and of upper Secants at 55, with a fourth part of the Tabular numbers from the center of the joint, both Tangents and Se- cants might be carried cn to 80 de- 1 grees. — For thefe purpofes, the dia- gonal fcale muff be fo divided, as that 1 414.2 of its equal parts fhall be equal in length to the whole line of Chords ; and then, 1440 of thefe parts would .extend but a very little further. And R 3 the 222 SELECT EXERCISES. the line of Lines (which is a line- of equal parts) muft be fo divided, as that ten of its grand divifions, to which the numeral figures are l'et, fhall be precifely equal in length to 60 de- grees of the line of Chords. — In com- mon Seftors, 6 inches long when Ihut, each grand divifion of the line of Lines is fubdivided into 20 equal parts, every one of which is fuppofed to be fub- divided into 5 ; by which means, the 10 grand divifions of that line are fup- pofed to contain 1000 equal parts, viz. fhe tabular number anfwering to the Radius, or 60 degrees of the Chords. The annexed fmall Table is for laying down the line of Polygons on Sedtors where the line of Chords goes on to 90 degrees. Thus the figure (or num- ber) 4 muft ftand even with 1414.21 parts of the diagonal fcale ; the figure 5 againft 1175.57 parts; the figure 6 againft 1000.00 parts; and fo on N° | t'arts. 4 1414-21 5 1 * 75*57 6 1000*00 7 867.89 8 7 6 5-36 9 684.O4 10 6| 8.O3 1 i 5 6 9 - 5 S 1 2 5 ' 7.63 SELECT EXERCISES. 223 on to 12, as in the Table. — But, by thefe numbers, the line of Polygons could be laid down only from 6 to 12 on common Sectors, where the line of Chords goes no further than 60 de- grees. But, for thofe who chufe to make Sectors in the common way, the here- annexed Table fhews the numbers in the diagonal fcale by which the line of Polygons is to be laid down. Thus, the figure 4 muft an- fwer to 1000.00 parts of the diagonal fcale, the figure 5 to 813.47 parts, the figure 6 to 707.11 parts, the figure 7 to 613.69, and fo on to the laft divifion 12, N° ! Farts. 4 1000.00 5 831.25 6 707.11 7 613.69 8 54 M 9 9 483.69 10 427 -°i 1 1 40273 1 2 366.02 224 SELECT EXERCISES. REMARK. As all thofe lines which are properly called Sectoral Lines *, terminate in an arch whofe center is the center of the joint, the diagonal fcale ought to be fo placed, as that the long parallel lines upon it may be ilridlly parallel to each fe&oral line on the pattern to be divided from the fcale : and alfo, that when one fide of the fquare is laid clofe to the upper edge of the dia- gonal fcale, and the other fide of the fquare (that lies acrofs the fcale and pattern) to the center of the joint, that fide of the fquare may then be at the beginning of the diagonal divifions on the fcale. Then all the divifions cut by that fide of the fquare will be true, and each divifion at right angles to its own refpedlive line. Without this precaution, the innermoft feftoral lines * The lines which are drawn from the center of the joint almoft to the other ends of the legs, would SELECT EXERCISES. 225 would not be divided to their whole proper lengths : and fo they would not all have the fame radius, and confer quently the meafures taken off from them by the compaffes would not agree together. But, when the Sector-pattern is truly divided according to this method, it may be applied clofe to the fide of the Sedtor to be divided from it ; becaufe, as the lines on the intended Se< 5 tor will be parallel to the like lines on the pat- tern, one fide of a fquare may be ap- plied to the upper edge of the pattern, and the other fide will lie acrofs both the pattern and Sedtor at right angles : and then, by applying that fide to each divifion of the pattern, and cutting each fuch divifion clofe by it on the Sedtor, all the divifions on the Sedtor lines will be true, although they be not cut at right angles to thofe lines to which they belong refpedtively. As the dialing lines of Latitudes, flours, and Inclination of Meridians, have %z6 SELECT EXERCISES. have no dependance on the radius of any of the above-mentioned lines, and are indifcriminately put upon. Sectors and plain Scales, they may be made of any convenient length where there is fufficient room. But, as they de- pend upon one another, they muft be all laid down from one fcale of equal parts. The lines of Hours and Inclina- tion of Meridians are of equal length, yvhich ought to be fix inches at leaft ; and the length of the line of Latitudes is equal to that of four hours and an half, in the line of Hours. To lay down thefe lines, you muft have a diagonal fcale of fuch a length, as that 1414.2 of its equal parts lhall contain as great a length as the line of Hours is intended to be of. And then, the fame number of parts, which hand in the Tables againft the degrees of Latitudes, Inclination of Meridians, Hours and parts of an Hour, muft be found among the diagonals; and the refpe&ive divifions in the lines cut by the SELECT EXERCISES. 227 the fide of a fquare applied to thefe parts in the diagonal fcale, after it has been fixed clofe to the edge of that on 'which thefe lines are to be divided. — Thus, 10 degrees in the line of Lati- tudes (reckoned from the beginning thereof, which mull be even with the beginning of the diagonal divifions) mull Hand even with 241.9 parts a- mong the diagonals : 10 degrees of the line of Inclination of Meridians mull be even with 212 parts of the diagonals, the hour of I againll 298.8 parts ; and fo on, as in the Tables. Gunter' s Lines, of Rhumbs, Num- bers, Sines, Verfed Sines, Tangents, and Meridional Parts (on the fcale that goes by his name) are all laid down by one diagonal fcale of equal parts; and 2000 of thefe parts mull include a length equal to the whole length of the line of Numbers, which confifts of 18 grand divifions of dif- ferent lengths, marked with the nu- meral figures i, 2, 3, 4, 5, 6, 7, 8, 9, *28 SELECT EXERCISES. i» 2, 3, 4, $, 6, 7, 8, 9, io ; the firft grand divifion being the fpace between the firft i and the firft 2, and th i I aft between the fecond 9 and the 10. — On this fcale, the grand divifions from the firft 1 to the fecond x are generally fubdivided into 10 parts each, altho’ they might bear four times that num- ber from 1 to 3 ; 20 divifions each •from the firft figure 3 to the figure 7, and after that, only 10 divifions each, to the fecond figure 1, which is at the middle of the line. The grand divi- fions in the other half of the line, are of the fame lengths with thofe in the former half ; but in the latter half of the line, each grand divifion from the figure 1 to 3 is fubdivided into 50 parts ; from 3 to 7, into 20 parts each ; and from 7 to 1 o at the end of the line, each grand divifion is alfo fubdivided into 20 parts only, on account of the flbortnefs of the fpaces. Being provided with a diagonal fcale pf 2000 equal parts, which include a length SELECT EXERCISES? 2^ length equal to the intended length of the line of Numbers, fix the lower edge of it clofe to the upper edge of the fcale intended to be divided, and applying one fide of the fquare to the upper edge of the diagonal fcale, and the other fide to the beginning of the diagonal divifions,- cut the firft crofs line in the line of Numbers (where the firft 1 is to ftand) clofe by that fide of the fquare ; and then, moving the fquare onward till the fame fide of it comes to the number of parts among the diagonals which anfwer in the Table of Gunter’s Numbers to the in- tended fubdivifions between x and 2, cut thefe divifions accordingly, in the line of Numbers, clofe by the fide of the fquare which was fet to the parts in the diagonals anfwering to thefe fubdivifions ; and fo on till the whole line be divided. Thus, the Tables fhew, that the figure 2 , at the end of the firft grand divifion (marked in the Table 2.00) 0 mull 230 SELECT EXERCISES. muft anfwer to 301.03 parts found among the diagonals; the divifion- line for the figure 3 muft anfwer to (or ftand even with) 477.12 parts, the divifion-line for 4 againft 602.60 parts ; and fo on, to the end of the firft half of the line, where the fecond figure 1 ftands againft 1000. The other half of the line is divided the fame way, by the other 1000 diagonal parts of the fcale. — The fubdivifions which the operator chufes to put into this line muft be cut even with the like number of parts found among the diagonals as anfwer to them in the Table. In order to divide the lines of Rhumbs, Sines, Verfed Sines, Tan- gents, and Meridional Parts, on Gun- ters Scale, the diagonal fcale muft be placed the contrary way to what it •was for dividing the line of Numbers, becaufe all thefe lines are divided backward, or from the right hand toward the left. Therefore, turn the diagonal SELECT EXERCISES. 231 diagonal fcale, and placing its con- trary edge to the upper edge of the fcale to be divided, fix it fo, as that, when the fquare is applied, the be- ginning of the diagonal divifions fihail Hand juft even with the end of the line of Numbers : and then, applying the crofs fide of the fquare to the fame number of parts among the diagonals which anfwer to the degrees, half de- grees, &c. in the Tables of Rhumbs, Sines, Verfed Sines, Tangents, and Meridional Parts, cut thefe divifions in the proper lines clofe by that fide of the fquare. As Tables of this kind were never all in print before (at leaft fo far as I ever heard of) and they ferve not only for dividing the lines accurately on Scales and Se<5fors, but alfo for ex- amining and proving whether the Scales and Sectors which are fold at fhops be accurately divided or not, I hope they will be acceptable, not only to the makers of thefe inftruments in 5 1 general, 2 3 2 SELECT EXERCISES. general, but alfo to thofe who ufe them. How to examine the Div i/Ions of the Lines on SeStors and Scales. If the line of Lines (which is a line of equal parts) on the Sector be accu- rately divided, which may be eafily tried with a pair of compaffes, it will ferve for examining all the other lines which are drawn from the center of the joint ; for all their divilions ought to anfwer to the equal parts of that line, as they do to the like equal parts of a diagonal fcale from which they are fuppofed to be laid down accord- ing to the preceding directions. Whatever the length of the Sector be, if the line of Chords upon it goes no further than 60 degrees, the line of Lines contains io grand divilions, marked i, 2, 3, 4, 5, 6, 7, 8, 9, io : but if the Chords go on to 90, the line of Lines SELECT EXERCISES. 233 Lines ought to contain 14T4.2 equal parts, fuppofing each grand divifion to be fubdivided into 100. On Sectors fix inches long, each grand divifion of this line is fubdi- vided into 20 equal parts ; and if each of thefe parts be fuppofed to be fub- divided into 5 (which is left to be eftimated by the eye) each grand di- vifion will contain 100 parts ; and con- fequently the whole ten will contain 1000. On Sedlors 12 inches long, thefe grand divifions are feldom fubdivided into more than 20 parts each, altho’ there might very well be 50, and then each part could be eafily divided into two by eftimation of the eye ; and con- fequently the whole line would con- tain 1000 equal parts. This being underftood, fet one foot of the compaffes in the center of the joint, and extending the other foot to the fame number of parts in the line of Lines as agree with the tabular S number SELECT EXERCISES. 234 number of parts for any degree of the Chords, Sines, Tangents, or Secants, turn that foot to the line of degrees you want to examine ; and if it falls into the degrees anfwering refpectively to the tabular numbers, the line is truly divided ; otherwife not. But if the Sector be too long for the points of the compafles to take in the whole length of the line of Lines between them, open it fo far, as that the compafles may reach conveniently acrofs the Seftor from 1 o in the line of Lines on one leg to 10 in the line of Lines on the other : and then, taking the tabular numbers of parts in the compafles acrofs the Seiftor, in each line of Lines which anfwer to the ta-, bular number of parts for each degree of the line of Lines, Chords, Tan- gents, or Secants, apply that extent acrofs the Sector to the like degrees of thefe lines : and if the points of the compafles fall direcflly into thefe gra- duations, the lines are accurately di- vided ; otherwife not. To SELECT EXERCISES. 235 To examine the lines on plain Scales or Gunter’s Scales, open the Sector fo, as that the length of the radius of the Chords, or whole length of the line of lines, taken in the compares, their points fhall then reach from 10 in one line of Lines on the Senior to 10 on the other: and then, taking the tabular numbers for the degrees of the Chords, Sines, or Tangents, acrofs the SeCtor from one line of Lines to the other, fet one foot of the compalTes in the beginning of the line to be examined, and the other foot forward among the degrees of that line ; and if it falls juft into each given degree, the line is truly divided; hut falfely if it does not. Thus you may examine the whole line of Lines, the line of Chords to 60 degrees, and the Tangents to 45 : but, in order to examine the degrees of the Tangents above 45, and all the de- grees of the Secants, you muft take the following method. S 2 The 2 36 SELECT EXERCISES. The Sector remaining at the fame opening as before, take the fupple- mental tabular numbers in your com- pafles acrofs the Sedtor from one line of Lines to the other, which anfwer to the rcfpedtive degrees of the Tan- gents and Secants, and apply that extent acrofs the Sedtor to the like de- grees in thefe lines : if the compafs- points fall exadtly into them, the lines are truly graduated. For Gunter’s lines of Numbers, Rhumbs, Sines, Verfed Sines, Tan- gents, and Meridional Parts, fet one foot of the compafles in the beginning of the line of Numbers (at the firft figure i) and open* the compafles till the other foot falls into the fecond i at the middle of the line. Then open the Sedtor fo, as that the fame open- ing of the compafles fhall reach from io in the line of Lines on one leg to jo in the like line on the other. Then, taking the tabular numbers in your compafles acrofs the Sedtor in thefe two SELECT EXERCISES. 237 two laft-mentioned lines, which an- fwer to the divifions of the lines you want to examine, apply that extent forward from the left-hand end of the line of Numbers, but backward from the right-hand end of the other lines ; and if thefe divifions or graduations are found by the compafles to agree with the tabular numbers belonging to them, the lines are truly divided ; otherwife not. N. B. In the Sectoral Lines, which proceed from the center of the joint, each has three parallel ftraight lines drawn for limiting the greater and leffer divifions : in each of thefe, it is the innermoft line to which the points of the compafTes muft be applied ; as that is the only line of the three that proceeds directly from the center of the joint. — I have dwelt the longer on this fubjedt, becaufe thefe inftrumencs are in the hands of every Mathemati- cian, and it is of the utmoft import- ance to have them rightly divided. S 3 Having SELECT EXERCISES. 238 Having fhewn how to divide and examine the Lines on Scales and Sec- tors, it is natural to fuppofe that the divider would be willing to know, not only how to examine the Tables them- felves, but alfo how to conftruct them. The Tables of Natural Sines, Tan- gents, and Secants, are copied from thofe in Sherwin’s Tables thereof. The right Sine. of an Arc is half the Chord of double that Arc. T hereforc, double the Natural Sine of half the given Arc (or number of degrees) and that will be the Chord of the whole Arc, or number of degrees required ; which, in the Tables, is carried no further than to 90. The conftrudlion of the Table of Natural Rhumbs is the fame as that pf the Chords. The Table of Numbers is only the logarithms of thefe Numbers. The logarithm of unity (or 1) being no- thing, is not inferred in the Table : the logarithm of 1.01 (the fame as that of 10 J SELECT EXERCISES. 2 39 jox without the charadteriftic) is 43214, which may be put down 4.32 ; the lo- garithm of 1.02 is 86002, which may be put down 8.60 ; and fo on, for laying down the line of Numbers on Gunter’s Scale. For the fame fcale, the Sines are the refpective logarithmic Sines fubtradted from radius, or the Co-fecants lefs radius. The Tangents (for the fame fcale) are the logarithmic Tangents, reject- ing radius. The Verfed Sines are double the logarithmic Secants of half the given number of degrees. For the Meridional Parts, divide the Meridional Parts in any Table thereof by 60. For the Sine Rhumbs, having found the degrees in every point and quar- ter-point of the Mariner’s Compafs, find the logarithmic Sines thereof, and take their complements arithmetical from radius, or from 8 points. S 4 The SELECT EXERCISES. 240 The fame holds good for Tangent Rhumbs, remembering they extend only to 4 points on the fcale, and that both the Rhumb and its complement Rand on the fame point of the line ; a logarithm and its reciprocal being equally diftant from radius on the oppofite fides. By thefe rules I have made the fore- going Tables from the now common Tables of Logarithms, Sines, Tan- gents, Secants, and Verfed Sines, with all the care and accuracy that I pof- fibly could, without regarding the time and pains required to conftruft them. Of the plain Scale , Sehlor> and Gun ter V Scale . The lines on the plain Scale are ufe- ful in mo ft branches of the mathema- tics, and its ufe in each of them is to be found in almoft every treatife of the practical mathematics. The SELECT EXERCISES. 241 The Sector is principally ufeful as a univerfal plain Scale, fitted to any radius within the compafs of its open- ing ; only obferving that the equal Parts, Sines, Chords, and Tangents under 45 degrees, are not taken along one line, as on the plain Scale, but acrofs the Seftor from any degree on one leg to the fame degree of the fame line on the other. The Gunter’s lines are for readily working proportions ; in which, re- gard mull be had to the terms, whe- ther arithmetical or trigonometrical ; that the firft and third term may be of the fame name, and the fecond and fourth of the fame name likewife : then, railing your proportion accord- ing to thefe rules, take the extent on their proper line, from the firft term to the third, in your compalles ; and applying one point of the compalles to the fecond, the other applied to the right or left according as the fourth term is to be more or lefs than the 242 SELECT EXERCISES. the fecond, will reach to the fourth, Three examples will explain this. i. If 4. yards of cloth coPl 18 {hillings, Then 32 yards will coll ? 2 . As radius to the hypothenufe 1 20, So the fine of the angle oppofue the bafe 30°* 17 $0 the bafe. 3 - As the co-fine of the latitude 51° 30' (— the fine of 38 3 30') is to radius, So is the fine of the Sun’s declin. 20° 14/ to the fine of the Sun’s amplitude. Note, The Line of Numbers (on Gun- ter’s Scale or on the Sector) is intended to fupply the Table of Logarithms, Thofe of Sines, Tangents, (and Se- cants, which are found in the fame plane with their Co-fines) and the Verfed Sines, fupply the places where their logarithms are neceffary in cal- culation, Now, SELECT EXERCISES. 243 Now, in the firft queftion, on the Line of Numbers take in your cora- pafies the diftance between 4 and 32 ; apply one foot thereof on the fame line at 18, and the other will reach to 144, the fail lings required. In the fecond, the diftance between radius (or the Sine of go 0 ) and the Sine of 30° 17', taken from the Line of Sines, and one foot applied to the hypothenufe 120 on the Line of Num- bers, and the other applied to the left (as the legs of a right lined and right angled triangle are lefs than the hy- pothenufe) that foot will reach to 604, the length of the bafe required. The third is wrought wholly on the Line of Sines. The diftance between the Sines of 38° 30' and 20° 14' taken in your compaffes, fet one foot on the radius or Sine of go% and the other will reach to 33°!, the Sun’s ampli- tude required. In the fame manner are ufed the Tangents, Secants, and Verfed Sines, in 244 SELECT EXERCISES. in proportions where they are res- quired: though fometimes the Verfed Sine is taken when the other foot ftands on the Line of Sines, as in find- ing the azimuth, &c. which is eafily performed when the art of raifing the proportion is known. A Jhort Account of the Logarithms . The invention of Logarithms, fays the great Dr. Halley, is juflly efteemed one of the mod ufeful difcoveries in the art of numbers. This afiertion is corroborated by the united teftimonies of every learned mathematician who has made mention of them, from the time of their firft publication to this day. Lord Napier, of Merchifton in Scot- land, in the year 1614, publifhed the firft fpecimen of thefe ufeful num- bers, under the title of Mirifici Logarith- moruni Canonis defcriptio ; a book which was SELECT EXERCISES. 245 •vvas received with tranfport in the ma- thematical world. And tho’ the au- thor had referved the method of con- ftrudting his Table, till the fenfe of the learned upon his invention fhould be known ; yet Kepler, Speidell, and others, abroad and at home, labour- ed at the computation of Logarithms, and conftrudted fmall tables thereof, conformable to the plan of Lord Na- pier. The defcription and ufe of the Ca- non being in Latin, induced Mr. Ed- ward Wright, a learned mathemati- cian of thofe times, and to whom we owe the principles of that falfely call- ed Mercator's Sailing, to tranflate it for the benefit of fuch of his pupils and others, who not underftanding the original, were yet defirous of be- ing acquainted with fo valuable a per- formance. This he effected with great care ; but unhappily dying before the publication, that office devolved to his fon, Samuel Wright, who, with the affiftance 7 246 SELECT EXERCISES. a Alliance of Mr. Briggs, then Geo- metry Profeffor at Grefham College* and whofe name fbines with great luftre in a hiftory of Logarithms, pub- lifhed it in the year 1616 or 1618 *. This tranflation was dedicated by the publiflrer to the Eaft India Com- pany, who, it feems, had employed Mr. Edward Wright in mathematical affairs : in which dedication he fays, “ his father’s care to make the tranf- “ lation bear a true refemblance of “ the original was fo great, that he “ procured the author's perufal of it ; “ who, after great pains taken therein , “ gave approbation to it. And it is “ apparent enough that he (Edward “ Wright) efteemed it, and intended “ to have recommended it as a book “ of more than ordinary worth, efpe- “ cially to feamen. But fhortly after “ he had it returned out of Scotland, “ it pleafed God to call him away, * My copy is printed at London for Simon Waterfon, 1 6 1 8 . “ before SELECT EXERCISES. 24 1 “ before he could publifh it.” Thus we fee that the edition of Edward Wright had fomc advantages over the fii ft edition, from the revifal of the inventor, who alfo wrote a preface to the fame; which, as it fets forth the purfuits that led to this difeovery, and as the book is very rare, I fhall infert in the words of its noble author, to- gether with a fpecimen and deferip- tion of the work itfelf. “ The Author’s Preface to the admirable u Table of Logarithms . a Seeing there is nothing, right 9 5527 5198 1 56 5^98121 *5 4 999984.7 4 * 20 5818 5146843 5146326 1 7.0 999983.1 4 ° 21 6109 5098054 5098035 18.7 999981.3 ' 39 22 6399 5 ° 5 1 5 34 5051514 20.5 999979.5 3 s 23 6690 5007083 5007061 22.4 999977.0 37 24 6981 4964524 4964499 24.5 999975 - 6 36 2 5 7272 49 2 3703 4923676 26,5 999973.6 35 26 7 S 6 3 4884483 4884454 28.7 99997* -4 34 2 7 7 8 54 4846743 4846712 30.9 999969.2 33 28 8145 48(0376 4810343 33.2 999966.8 3 2 29 8436 4775286 4775 2 50 35*6 999964.4 3 * 30 8726 4741385 474*347 3S.1 99996!. 9 30 Min. Deg. 89. 252 SELECT EXERCISES. The columns marked Sines are the natural fines anfwering to the degree and minute of the firft and laft co- lumns ; the adjoining ones their loga- rithms, and the middle one the dif- ferences between the logarithms. The Author explains every one of thefe columns ; but I fhall only infert, in his words, what relates to thofe con- taining the logarithms. “ § ii. The 3d column containeth “ the logarithms of the arches and “ fines towards the left hand. “ § 13. And they are alfo the loga- 6 ‘ rithms of the complements of the “ arches and fines towards the right “ hand, which we call Antilogarithms. “ § 14. The fifth column containeth 4 ‘ the logarithms of the arches and “ fines towards the right hand. “ § 16. They are alfo the antiloga- “ rithms of the arches and fines to- “ wards the left hand, or the loga- “ rithm of the complements. “ § 17. Laftly, the 4th or middle “ column containeth the differences ** between SELECT EXERCISES. 253 <* between the logarithms of the 3d “ and 5th columns ; and fo this co- “ lumn is twofold, abounding and “ defective. “ § 18. Thofe differences arc abound- “ ing which arife out of the fubtrac- “ tion of the logarithms of the fifth “ column from the logarithms of the “ third column. “ § 19. But the differences arifing “ by fubtra&ion of the logarithms of “ the third column out of the loga- “ rithms of the fifth column, are de- “ fedlive ; which therefore are lefs “ than nothing. “ § 20. The abounding differences “ are called the differential numbers . - '..I fj I _vl “ of the arches towards the left hand. “ § 22. And are alfo the logarithms “ of the tangents of the left hand “ arches. “ §23. But the defedfive differences “ are called the differential numbers “ of the right hand arches. “ § 25, And are alfo the logarithms T 3 “of 254 SELECT EXERCISES. “ of the tangents of the right hand “ arches.” And as admonitions, amongft others, he gives the following : “ 28. Here it is to be noted, that “ if you make the logarithms of the “ 3d column defective, fetting before “ them this mark — , they fhall be “ made the logarithms of the hypo- “ thenufes or fecants of the right hand “ arches of the feventh column. ic 30. And if you make the loga- * £ rithms of the fifth columns defec- “ tive, they fhall be the logarithms of “ the hypothenufes, or of the fecants “ of the left hand arches of the firfl “ column/’ Thus the nature of the Canon re- quired fome knowlege of algebraic addition and fubtradtion ; and it was befides troublefome thereby to find the intermediate numbers ; the loga- rithms being only given for fuch num- bers SELECT EXERCISES. 255 bcrs as. were equal to tire natural fines of each minute. The latter inconve- nience E. Wright endeavoured to re- move by a fcale, of which a draught and aefcription is given in the tranf- lation of the admirable Canon, but which Mr. Briggs performed much better in the fame work, by means of a fubfidiary table of Ample applica- tion. The firft of thefe inconveniencies was well removed by John Speidell, who was a very ingenious man, and firft completed the Canon for trigono- metrical ufe, by adding three other columns, which were the comple- ments arithmetical of Lord Napier’s, to wit. Secant, and Co-fecant and Co- tangent lefs radius. By this means he faved the application of the plus and minus, and rendered the calculus of triangles more eafy, This he pub- liflied in the year 1619, under the title of New Logarithms ; and to the fixth impreffion thereof in 1624, is T 4 added a 56 SELECT EXERCISES. added a Table of the Logarithms of Natural Numbers to 1000, in which the logarithm of 1 is nothing, and that of one thoufand, 690775. — Kepler too, in the year 1724, publifhed at Marpurg his Chilias Logarithmorum ad totidem Numeros rotundas ; in which his logarithm of 1 was 1611809.59, and of 100000, nothing; which alfo an- fwered to the logarithm of 90°. And as Napier had given the logarithms to every degree and minute of the qua- drant, fo Kepler gives the neareft cor- refponding degree and minute, an- fwering to each of his logarithms. But of all that bellowed their labour on the firlt fort of logarithms, Benja- min Urlinius is moll worthy of men- tion, who calculated a table of fuch logarithms to every 10 feconds, which was printed at Cologne in 1625, and to which Vlaac mull have been much obliged in conltrudting his Canon Mag- nus. Thefe attempts ferved to facilitate and enlarge the ufe of Napier’s firlt fyftem SELECT EXERCISES. 257 fyftem of logarithms, but were all neverthelefs fubjedt, in fome meafure, to its inconveniencies. The perfec- tion of thofe noble numbers was re- served as an additional honour to their firft inventor ; and the prefent fyftem, than which there can fcarcelybe hoped for a better, is likewife the invention of Lord Napier. Mr. Henry Briggs, at the time when the Canon Mirificus appeared, was Geo- metry Profeflor at Grefham College. This man joined to a great genius and wonderful fagacity in mathematical affairs, a moft indefatigable induftry, with lingular purity of manners, can- dour, and generality. Inflamed by this truly admirable work, he writes, on the 10th of March 1615, to Mr. (af- terwards ^rchbilhop) Ulher, “ that “ he was wholly taken up and em- “ ployed about the noble invention of “ logarithms, then lately discovered.” And again, “ Naper Lord of Markin- “ lion hath Set my head and hands at “ work 258 SELECT EXERCISES. “ work with his new and admirable “ logarithms ; I hope to fee him this “ Summer, if it pleafe God ; for I “ never faw a book which pleafed me “ better, and made me more wonder.” And we find he actually computed new logarithms, wherein he made o the fine total, as in the Canon, but made 10000000000 the logarithm of one-tenth part of the whole fine, or of 5 degrees 44 minutes 21 feconds, judg- ing them more commodious than thole of Napier ; which, in the Summer of 1616, he carried with him to Edin- burgh, where he was received by the Baron with great kindnefs, and Raid with him a month. It was in the courfe of this time, by Mr. Briggs’s account, that Lord Napier communicated to him his in- tended alteration of the Canon ; and that- he had caufed it to be printed in its prefent form, till his health and leifure fhould permit him to calculate another, wherein he. intended to make o the SELECT EXERCISES. 2 59 o the logarithm of unity, and io that of the fine total. Briggs, greatly pleafed with that fuggeftion, which he confefled was much more eligible than his own, threw afide thofe he had begun, and, as well from his own inclination as the earned: entreaty of Lord Napier, on his return to London, fet about the calculation of new ones, after that form : of which having computed 1000 to 8 places of figures, befides the index, he carried them, in the Summer of 1617, to the Baron, who was highly pleafed therewith, and prefled his beloved friend Briggs, as he ftiles him, to continue the work. Which injunction he afterwards more ftrongly enforced, by the public tefti- monies he gave of both the capacity and induftry of his friend 5 and the expectations the world might form from his continuing the great work, of which his Lordfhip, in regard to thefe improved logarithms, challen- ged little more than the invention and mode of conftruCtion. Nor 2 6o SELECT EXERCISES. Nor were the fentiments of Briggs in regard to Lord Napier lefs lively than thofe of the latter for him ; he mentions him every where with the utmolt veneration, warmth, and af- fedtiop ; and feems perfectly enrap- tured by the noblenefs and utility of the great invention. And indeed no- thing lefs could have fullained him through the immenfe labour of fuch computations as he underwent in bringing it into pradtice. On his return to London, he printed the Cbilias Prima ; and feven years af- ter, in 1624, h e produced his Arithme- tic Logarithmica , wherein he gives the logarithms of 31000 natural numbers to 14 places of figures, befides the in- dex ; a work which will appear ftu- pendons to any one who, by the fame method, will take the trouble to compute the logarithms of only two or three fuch numbers ; which me- thod he may find in the faid work, in Malcolm’s Arithmetic, Keil’s Euclid, Ward’s Mathematics, and feveral other writers 5 SELECT EXERCISES. 261 writers ; and of which an idea may be formed from the account of Dr. Halley, who fays, that to have his logarithm true to 14 places of figures, Mr. Briggs, by continual extraction, was obliged to find the root of more than the 140 million of millioneth power. Euclid Speidell fays, he was told, that only finding the logarithm of 2 true to 15 places, for the aforefaid Table, was the work of eight perfons a whole year ; and Pardies, in his Geo- metry, praifes God, who, for the pub- lic good, has roufed up perfons to whom he had given fufficient patience to furmount the fatigue of a labour feemingly infupportable : “ for,” fays he, “ we know that more than 20 per- “ fons, employed for that purpofe, “ fpent upwards of 20 years with in- “ defatigable afliduity in the calcula- “ tion.” This laft teftimony, which fome have thought an infinuation that the logarithms were firfl: invented in France, rather proves the genius and 2 ability 262 SELECT EXERCISES. ability of our excellent countrymari Briggs, who, in eight years, with fcarce as many perfons, performed a much greater work than 20 French- men in thrice the time: for the largeft French tables of thofe times were car- ried to only 1 1 places of figures, and thefe are fuppofed to have been taken from Vlaac. Though the Chilias Prima was print- ed in the latter end of the year 1617, it was neverthelefs not publifhed till after the death of Napier, which hap- pened on the 3d of April, 1618; for, in the preface thereto, Briggs fays, “ why thefe logarithms differ from “ thofe fet forth by their mod illuf- “ trious inventor, of ever worfhipful “ memory, in his Canon Mirificus , it is “ to be hoped his pofthumous book “ will fhortly make appear.” In 16 20, two years after the Chilias. Prima of Briggs was publillied, Mr. Edmund Gunter fet forth a Canon of Sines and Tangents, adapted to thefe loga- SELECT EXERCISES. 263 logarithms, which was the firft of that kind, and to which he gave the name of Artificial Sines and Tangents. This muft have coft Gunter fome labour, though he appears to have been much beholden to the Chilias, which conlifts of 8, and thefe only of 7 places, be- fides the index, of which the laft place is anomalous, when compared with thofe which Briggs and Vlaac after- wards computed. Thefe he repub- lifhed in his book De Sedlore et Radio , in 1623, together with the Chilias Pri- maoi his old colleague. But Briggs himfelf lived to complete a Table of logarithmic Sines and Tan- gents, to the one hundredth part of every degree, and to 14 places of fi- gures, befides the index ; to which he intended to have written a full defcrip- tion and ufe, when death put an end to his labours, at the age of 74, on the 26th January, 163^, thirteen years after his beloved Napier. It is remarkable of thefe two great men, that they both undertook thefe mighty 264 SELECT EXERCISES. mighty labours at rather an advanced age. The Compendium of Honour mentions Napier to have died in his 67th year; and as it is generally fup- pofed the logarithms were invented by him about 16 10, he mult have been near 60 years of age, and, as himfelf informs us, weak and infirm of body, when he began the calculation of his Canon ; and in 1614, when Briggs firft applied his thoughts that way, he mult have been upwards of 57 : which may teach us, there is no time of our life too late to acquire honeft fame, when we are obedient to the voice of genius, and keep our eafe in due fubje&ion to induftry. Briggs, when dying, recommended his laft-mentioned work to the care and completion of Henry Gellibrand, then Aftronomy Profeflor at Grefham College, who added thereto the de- fcription and ufe of the Canon, and publifhed it at London*, in 1633, un- * At Gouda, according to Tome writers. der SELECT EXERCISES. 265 der the title of Trigonometria Britan - nica. Thus were thefe numbers com- pletely planned by the wifdom of the ingenious Lord of Merchifton, and the great building rear’d, in nice con- formance thereto, by the molt induf- trious and learned Henry Briggs, in a few years ; and it is difficult to de- termine in thefe two great men, which is the moll admirable ; the fagacity of the inventor, or the indefatigable application of the calculator. In the Arithmetica Logarithmica , Mr. Briggs had calculated the logarithms of all the natural numbers from 1 to 20000, and from 90000 to ioioooj leaving the interval to be filled by the ingenious ; to any of whom, in his preface, he offers paper properly rul- ed, and necefiary inftru£tions, he pur- pofing, in the mean time, to employ himfelf on his Triangular Canon, and having left no labour which an ordi- nary Ikill might not perform. But Adrian Vlaac, of Targou, or Gouda U in 2 66 SELECT EXERCISES. in Holland, completed the 70000 ki- termediate numbers with fuch expe- dition, that, in 1628, he publifhed the fecond edition of the Arithmetica Loga- rithmica , wherein was contained all the logarithms from 1 to 100000 to 10 places of figures, together with a Table of Artificial Sines, Tangents, and Secants (as Gunter had called them) to every minute of the quadrant. Confidering alfo that the ufual me- thod of angular fupputation was by minutes and feconds, Vlaac % in the fame year with the Trigonomctria Bri- tannic a, publifhed his Trigonometria Ar- tficialis ; wherein is contained the lo- garithmic Sines and Tangents to every 10 feconds, and the logarithms from 1 to 20000: which fhews Vlaac to have been a very afiiduous and ingenious man ; though Dr. Newton, Norwood, and alfo Wingate, feem to cenfure both him and others, who rafhly pub- lifhed new editions of Briggs’s Loga - rithmica Arithmetica without his leave ; thereto preventing the additions he intended CONTENTS. A Table /hewing the Standard Weight , Va- lue and comparative View of Englifh Silver Money , from King William the Conqueror , A. D. I1066, to A, D. 1765, Page 163 Prices of Goods between thefe Dates, 1 65 Concerning Standard Gold, - 167 The Number of Ways in which all the Let- ters of the Alphabet might be combined 9 or put together , from 1 Letter to 25. Or, the Number of Changes that might be rung on any Number of Bells not exceed- ing the Number of Letters in the Alphabet., 168 The Time it would take to do this , - 1 69 Concerning the Strength of Steam . From the Reverend Mr . Mitchell's Treatife on Earthquakes , - - - 1 70 Mathematical Tables for dividing the Lines on Scales and Sectors , - 175 — 205 The Conftrudiion of the Plain Scale, SeBor, and Gunter* s -Scale, by the Tables , 206 CONTENTS. How to examine the Divifions of the Lines on Sectors and Scales , - Page 232 Hoiu to conjlru£l thefe Mathematical Ta - hies , - - - - 238 Some Ufes of the Plain Scale y Setlor, and Gunter s Scale , - 240 A fhort Account of the Logarithms , r£ fimple Way of Confiru£lion y 31 How to regulate a Clock by the Motion of the Stars fo as to meafure mean Solar Time exaBly-i 33 7 o find the Length of a Pendulu?n that fhall make any given Number of Vibrations in a Minute , ^zzJ vice verfa, - "*36 The Defcription of a new Machine , called the Mechanical Paradox, - 44 An Orrery , fhewing the Motions of the Sun , Mercury , Venus , Earth , Moon , and Nodes of the Moon's Orbit ; different Lengths of Days and Nights , Vicijfitudes of Seafons , Phafes of the Moon , dwJ <2// ^ *SW<2r Mr. MacLaurin, who commended it in prefence of a great many young gentlemen who attended his leisures. He defired me to read them a lecfture on it, which I did with- out any hefitation, feeing I had no reafon to be afraid of fpeaking before a great and good man who was my friend. — Soon after that, I fent it in a prefent to the Reverend and ingeni- ous Mr. Alexander Irvine, one of the m milters at Elgin in Scotland. I then made a fmaller and neater Orrery, of which all the wheels were of ivory, and I cut the teeth in them with a file. — This was done in the be- ginning of the year 1 743 ; and, in May that year, I brought it with me to Lon- don, where it was foon after bought by Sir Dudley Rider. I have made fix Orreries fince that time, and there are not any two of them in which the wheel- work is alike : for I could never bear LIFE OF THE AUTHOR, xxxvii bear to copy one thing of that kind from another, becaufe I ftill faw there was great room for improvements. I had a letter of recommendation from Mr. Baron Edlin at Edinburgh to the Right Honourable Stephen Poyntz, Efq; at St. James’s, who had been pre- ceptor to his Royal Highnefs the late Duke of Cumberland, and was well known to be poffefTed of all the good qualities that can adorn a human mind. — • To me, his goodnefs was really beyond my power of expref- fion ; and I had not been a month in London till he informed me that he had wrote to an eminent profeflbr of mathematics to take me into his houfe, and give me board and lodging, with all proper inftruftions to qualify me for teaching a mathematical fchool he (Mr. Poyntz) had in view for me, and would get me fettled in it. This I ihould have liked very well, efpecially as I began to be tired of drawing pic- tures, in which, I confefs, I never c 3 ftrovc xxxviii AN ACCOUNT OF THE ftrove to excel, becaufe my mind was fcill purfuing things more agreeable. He foon after told me he had juft re- ceived an anfwer from the mathema- tical matter, defiring I might be fent immediately to him. On hearing this, I told Mr. Poyntz, that I did not know how to maintain my wife during the time I muft be under the matter's tui- tion. What, fays he, are you a mar- ried man? 1 told him I had been fo ever firice May in the year 1739. He laid he was forry for it, becaufe it quite defeated his fcheme; as the matter of the fchool he had in view for me muft be a batchelor. He then afked me what bufinefs I intended to follow? I anfwered, that I knew of none bcfides that of draw- ing pictures. On this he deftred me to draw the pictures of his lady and children, that he might fhew them in order to recommend me to others ; and told me, that, when I was out of bufi- nefs, I lhould come to him, and he would LIFE OF THE AUTHOR, xxxix would find me as much as he could: and I foon found as much as I could execute : but he died in a few years after, to my inexpreflible grief. Soon afterward, it appeared to me, that, although the Moon goes round the Earth, and that the Sun is far on * the outfide of the Moon’s orbit, yet the Moon’s motion muft be in a line that is always concave toward the Sun: j and upon making a delineation repre- fenting her ahfolute path in the Hea- vens, I found it to be really fo. I then made a Ample machine for de- lineating both her path and the Earth’s on a long paper laid on the floor. I carried the machine and delineation to the late Martin Folkes, Efquire, Prefident of the Royal Society, on a Thurfday afternoon. He expreffed great fatisfaclion at feeing it, as it was a new difeovery ; and took me that evening with him to the Royal Society, where I fliewed the delinea- tion, and the method of doing it. When kl AN ACCOUNT OF THE When the bufinefs of the Society was over, one of the members defired me to dine with him next Saturday at Hackney; telling me that his name was Ellicott, and that he was a watch- maker. I accordingly went to Hackney, and was kindly received by Mr. John Elli- cott, who then fhewed me the very fame kind of delineation, and part of the machine by which he had done it ; telling me that he had thought of it twenty years before. I could eafily fee, by the colour of the paper, and of the ink lines upon it, that it mull have been done many years before I faw it. He then told me, what was very certain, that he httd neither ftolen the thought from me, nor had 1 from him. And from that time till his death, Mr. Ellicott was one of mv bed friends. The figure of this machine and delineation is in the 7 th Plate of my book of Aftronomy. Soon after the flile was changed, I bad my Rotula new engraved ; but have LIFE OF THE AUTHOR. xli have negle&ed it too much by not fit- ting it up and advertifing it. After this, I drew out a fcheme, and had it engraved, for fhewing all the pro- blems of the Rotula except the Eclipfes : and, in place of that, it fhews the times of riling and fetting of the Sun, Moon, and Stars and the pofitions of the Stars for any time of the night. In the year 1747, I publifhed a Dif- fertation on the Phenomena of the Harvelt Moon, with the defcription of a new Orrery, in which there are only four wheels. But having never had grammatical education, nor time to ftudy the rules of juft compofition, I acknowledge that I was afraid to put it to the prefs ; and, for the fame caufe, I ought to have the fame fears ftill. But having the pleafure to find that this my firft work was not ill received, I was emboldened to go on, in pub- lilhing my Aftronomy, Mechanical Lecftures, Tables andTrafts relative to feveral Arts and Sciences, The Young Gentleman % xlii AN ACCOUNT OF THE Gentlem&n and Lady’s Aftronomy, a fmall treatife on Electricity, and the following fheets. In the year 1748, I ventured to read Lectures on the Eclipfe of the Sun that fell on the 14th of July in that year. Afterwards I began to read Aftrono- mical Ledtures on an Orrery which I made, and of which the figures of all the wheel-work are contained in the 6th and 7th Plates of this book. I next began to make an apparatus for Ledtures on Mechanics, and gra- dually increafed the apparatus for other parts of Experimental Philofo- phy, buying from others what I could not make for myfelf, till I brought it to its prefent ftate — I then entirely left off drawing pidtures, and employed myfelf in the much pleafanter bufinefs of reading Ledtures on Mechanics, Hydroftatics, Hydraulics, Pneumatics, Eledtricity, and Aftronomy : in all which, my encouragement has been greater than I could have expected. q The LIFE OF THE AUTHOR. xliii The belt machine I ever contriv- ed is the Eclipfareon, of which there is a figure in the 1-3 th Plate of my. A- ftronomy. It fhews the time, quan- tity, duration, and progrefs of Solar Eclipfes, at all parts of the earth. My next bell contrivance is the Univerfal Dialing Cylinder, of which there is a figure in the 8th Plate of the Supple- ment to my Mechanical Ledlures. It is now thirty years fince I came to London ; and during all that time, I have met with the highell inllances of friendfhip from all ranks of people both in town and country, which I do here acknowledge with the utmoll re- fpe£t and gratitude ; and particularly the goodnefs of our prefent gracious Sovereign, who, out of his privy purfe, allows me fifty pounds a-year, which is. regularly paid without any deduc- tion. I ERRATA; Pag. 73. 1 . 4. for fifth read fixth. Pag, 187. /. 8. /« the Table ^ in the fecond column , /or 149.9 read 194.9. 242. /. 4. /r^w the bottom , for plane place. Pag . 261. /. 14. for roufed read raifed. Page 244. afterivard throughout , for Napier Nepair ; that was the way in which the Baron of Merchiftgit himfelf fpelt his Name . &P£2(A.L <65 '6