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§ 1. FOR the more clear understand∣ing this Chapter it must be ob∣served, that those Automata (whose Cal∣culation I chiefly intend) do by little In∣terstices, or Strokes, measure out longer
portions of Time. Thus the strokes o•• the Balance of a Watch, do measure ou•• Minutes, Hours, Days, &c.
Now to scatter those strokes among Wheels and Pinions, and to proportionat•• them, ••so as to measure Time regularly is the design of Calculation. For th•• clearer discovery of which, it will be ne∣cessary to proceed leisurely, and gradu∣ally.
* 1.1 § 2. And in the first place, you are to know, that any Wheel being divided by its Pinion, shews how many turns that Pinion hath to one turn of that Wheel. Thus a Wheel of 60 teeth driving a Pi∣nion of 6, will turn round the Pinion 10 times in going round once.
From the Fusy to the Ballance the Wheels drive the Pinions; and conse∣quently the Pinions run faster, or go more turns, than the Wheels they run in. But it is contrary from the Great-Wheel to the Dial Wheel. Thus in the last Exam∣ple, The Wheel drives round the Pinion 10 times: but if the Pinion drove the Wheel, it must turn 10 times to drive the Wheel round once.
§ 3. Before I proceed further, I must
shew how to write down the Wheels and Pinions. Which may be done, either as Vulgar Fractions, or in the way of Divi∣sion in Vulgar Arithmetick. E. C. A Wheel of 60 moving a Pinion of 5, may be set down thus, 60/3: or rather thus, 5) 60: where the first figure is the Pini∣on, the next without the hook, is the Wheel.
The number of Turns, which the Pi∣nion hath in one turn of the Wheel, is set without a hook on the right hand: as 5) 60 (12, i. e. a Pinion 5 playing in a Wheel of 60, moveth round 12 times, in one turn of the Wheel.
* 1.2A whole Movement ma•• be noted thus, 4/•••• 55/5 45/5 40/5 17 Notches in the Crown▪Wheel. Or rather as you see here in the Margin: where the uppermost number, above the line, is the Pinion of Re∣port 4, the Dial-wheel 36, and 9 turns of the Pin. of Report. The second number (under the line) is 5 the Pinion, 55 is the Great-wheel, and 11 turns of the Pinion it driveth. The third numbers, are the Second-wheel, &c.
The fourth the Contrate-wheel, &c. An•• the single number 17 under all, is th•• Crown-wheel.
§ 4 By the § 2. before, knowing th•• number of turns, which any Pinion hath in one turn of the Wheel it worketh in▪ you may also find out how many turns a•• Wheel or Pinion hath, at a greater di∣stance; as the Contrate-wheel, Crown▪wheel, or &c. For it is but multiplying * 1.3together the Quotients, and the number produced, is the number of Turns. An Ex∣ample will make what I say plain: * 1.4let us chuse these 3 numbers here set down; the first of which hath 11 turns, the next 9▪ and the last 8. If you multiply 11 and 9 it pro∣duceth 99, for 9 times 11 is 99, that is, in one turn of the Whee•• 55, there are 99 turns of the second Pinion 5, or of the Wheel 40. If you multiply 99 by the last Quotient 8 (that is, 8 times 99 is 792) it shews the number of turns, which the third and last Pinion 5 hath. So that this third, and last Pinion turns 792 times in
one turn of the first Wheel 55. * 1.5Another Example will make it still more plain The Example is in the Margin. The turns are 10, 9 and 8. These multiplied as before run thus, viz. 10 times 9 is 90, that is, the Pi∣nion 6 (which is the Pin. of the third Wheel 40) turns 90 times in one turn of the First▪wheel 80. This last product 90 being multiplied by 8, pro∣duces 720▪ that is, the Pinion 5 (which is the Pin. of the Crown-wheel 15) turns 720 times in one turn of the First-wheel, of 80 teeth.
§ 5. We may now proceed to that, which is the very groundwork of all; which is, not only to find out the turns, but the Beats also of the Ballance in those turns of the Wheels. By the last Para∣graph, having found out the number of turns, which the Crown-wheel hath in one turn of the Wheel you seek for, you must then multiply those turns of the Crown-wheel by its number of Notches, and this will give you half the number of Beats, in that one turn of the Wheel. Half the number, I say, for the reasons in the fol∣lowing
6 §. For the Explication of what hath been said, we will take the example in the last §: the Crown-wheel there, has 720 turns in one turn of the first Wheel 〈◊〉〈◊〉 This number multiplied by 15, the Notches in the Crown-wheel, produceth 10800, which are half the number o•• strokes of the Ballance, in one turn of the first wheel 80. The like may be done for any of the other Wheels; as the Wheel 54, or 40: but I shall not insist upon these, having said enough.
I shall give but one Example more which will fully, and very plainly illu∣strate the whole matter. * 1.6The example is in the margin, and 'tis of a 16 hour Watch, wherein the Pi∣nion of Report is 4, the Dial-wheel 32, the Great-wheel i•• 55, the Pinion of the secon••-Wheel is 5, &c. the numbe•• of Notches in the Crown▪wheel are 17: the quotients or number of turns in each, are 8, 11 9, 8. All which being multiplied as be▪fore, make 6336: this number multipli∣ed by 17, produceth 107712; which la••∣summ is half the number of Beats in on••
turn of the Dial-wheel. The half num∣ber of Beats in one turn of the Great-wheel, you will find to be 13464: For 8 times 17 is 136, which is the half num∣ber of Beats in one turn of the Contrate-wheel 40: and 9 times 136, is 1224, the half beats in one turn of the Second-wheel: and 11 times 1224, is 13464, the half beats in one turn of the Great-wheel 55. And 8 times this last, is 107712 before named. If you multiply this by the two Pallets, that is, double it, it is 215424, which is the number of Beats in one turn of the Dial-wheel, or 12 hours. If you would know how many beats this Watch hath in an hour, 'tis but dividing the beats in 12 hours, into 12 parts, and it gives 17952, the Train of the * 1.7Watch, or beats in an hour. If you di∣vide this into 60 parts, it gives 299 and a little more, for the beats in a minute. And so you may go on to seconds and thirds, if you please.
Thus I have delivered my thoughts as plainly as I can, that I may be well un∣derstood; this being the very foundation of all the artificial part of Clock-work. And therefore let the young practiser ex∣ercise
himself thorowly in it, in more than one example.
If I have offended the more learned, quick-sighted Reader, by using m••••y words; my desire to instruct the most ignorant Artist▪ must plead my excuse.
* 1.8 § 6. The Ballance or Swing hath two strokes to every tooth of the Crown-wheel. For each of the two Pallets hath its blow against each tooth of the Crown-wheel▪ Wherefore a Pendulum that swings Seconds, hath its Crown-wheel but 30.
HAving in the last Section led on the Reader to a general knowledge of Calculation; I may now venture him further into the more obscure, and useful parts of that Art: Which I shall explain with all possible plainness, tho less brevi∣ty, than I could wish.
* 1.9 § 1. The same motion may be per∣formed either with one Wheel and one
or by many Wheels and many Pinions: provided that the number of turns of all those Wheels bear the same proportion to all those Pinions, which that one Wheel bears to its Pinion. Or (which is the same thing) that the number produced by multiplying all the Wheels together, be to the number produced ••y multiplying all the Pini∣ons together; as that one Wheel is to that one Pinion. * 1.10Thus sup∣pose you had use for a Wheel of 1440 teeth, with a Pin. of 28 leaves, you may make it into 3 Wheels and Pinions, viz. 4 •• 36, 7) 8, and 1) 5. For if you multiply the three Wheels together, viz. 36, 8 and 5; and the three Pinions together by themselves, viz 4, 7 and 1, you will find 1440 to arise for the Wheels, and 28 for the Pinions. Or if you try the example by the number of turns, it will be the same. For 28) 1440 (51 3/7. And the quotients and turns of the 3 Wheels and Pinions multiplied together, are 51 3/7 also, as in the last example.
It matters not in what order the Wheels and Pinions are set, or which Pinion runs
in which Wheel: Only for convenien•• sake, they commonly set the biggest num∣bers to drive the rest.
§ 2. Two Wheels and Pinions of diff••∣rent * 1.11numbers may perform the same m••∣tion. As, a Wheel of 36 drives a Pinio•• of 4, all one as a Wheel of 45 drives Pin. of 5; or as a Wheel of 90 drives Pin. of 10. The turns of each are 9.
§ 3. If in breaking your Train int•• parcels (of which by and by) any of you•• Quotients should not please you; or * 1.12you would alter any other two number which are to be multiplied together, yo•• may vary them by this Rule: Divid•• your two numbers by any two oth•• numbers which will measure them; th•• multiply the Quotients by the alternat•• divisors, the product of these two la•• numbers found, shall be equal to the pro∣duct of the two numbers first give•• Thus if you would vary 36 times 8, d••∣vide these by any two numbers that wi•• evenly measure them, as 36 by 4, and by 1. The fourth part of 36 is 9, and divided by 1 gives 8. Multiply 9 by 〈◊〉〈◊〉 the product is 9; and 8 multiplied by 〈◊〉〈◊〉 produceth 32. So that for 36 times 8••
you shall have found 32 times 9. * 1.13The operation is in the Mar∣gin, that you may see, and ap∣prehend it the better. These numbers are equal, viz. 36 times 8 is equal to 32 times 9; both producing 288. If you divide 36 by 6, and 8 by 2, and multiply as before is said, you will have for 36 times 8, 24 times 12, equal to 288 also.
If this Rule seem to the unskilful Rea∣der hard to be understood, let him not be discouraged, because he may do with∣out it, altho it may be of good use to him that would be a more compleat Artist.
§ 4. Because in the following Para∣graphs, I shall have frequent occasion to use the Rule of Three, or Rule of Propor∣tion, it will be necessary to shew the un∣skilful Reader, how to work this noble Rule.
If you find 3 or 4 numbers thus set, with four spots after the second of them, 'tis the Rule of Proportion; as in this ex∣ample, 2. 4:: 3. 6. i. e. As 2 is to 4:: So is 3 to 6.
The way to work this Rule, viz. by the 3 first number to find a fourth, is,
To multiply the second number and the third together, and divide their product by the first. Thus 4 times 3 is 12, which 12 divided by 2, gives 6; which is the number sought for, and stands in the fourth place.
You will find the great use of this Rule hereafter; only take care to bear it in mind all along.
§ 5. To proceed. If in seeking for your Pinion of Report, or by any other means, you happen to have a Wheel and Pinion fall out with cross numbers, too big to be cut in Wheels, and yet not to be alter∣ed by the former Rules, you may find out two numbers of the same, or a near pro∣portion, by this following Rule, viz. As either of the two numbers given, is to * 1.14the other:: So is 360 to a fourth: Di∣vide that fourth number, as also 360 by 4. 5. 6. 8. 9. 10. 12. 15. (each of which numbers doth exactly measure 360) o•• by any one of those numbers that bring∣eth a quotient nearest to an integer (or whole number.) Thus if you had these two numbers, 147 the Wheel, and 170 the Pinion, which are too great to be cut in small Wheels, and yet can't be reduced
into le••s, because they have no other com∣mon measure, but unity: say therefore according to the last paragraph, As 170 is to 147; or as 147 is to 170:: So is 360 to a fourth number sought. In numbers thus, 170 147:: 360. 311. or 147. 170:: 360. 416. Divide the fourth number and 360 by one of the foregoing numbers; as 311 and 360 by 6, it gives 52 and 60. In numbers 'tis thus,
| 6) | 311 | (52 |
| 360 | (60 |
| 8) | 311 | (39 |
| 360 | (45 |
| 8) | 360 | (45 |
| 416 | (52 |
§ 6. I shall add but one Rule more, be∣fore I come to the practice of what hath been laid down; which Rule will be of perpetual use, and consists of these five particulars.
* 1.15 1. To find what number of turns the Fusy will have, thus, As the Beats of the Ballance in one turn of the Great-Wheel or Fusy (suppose 26928) To the Beats of the Ballance in one hour (suppose 20196)
:: So is the continuance of the Watches going in hours (suppose 16) To the num∣ber of the turns of the Fusy 12. In num∣bers 'twill stand thus, 26928. 20196:▪ 16. 12. By § 4. you may remember tha•• you are to multiply 20196 by 16, the product is 323136. Divide this by 26928, and there will arise 12 in the Quotient, which must be placed in the fourth place▪ and is the number of turns which the Fusy hath.
2. By the Beats and turns of the Fusy▪ to find how many hours the Watch will go, thus,
As the Beats of the Ballance in one hour, are to the Beats in one turn of the Fusy:: So is the number of the turns o•• the Fusy, to the continuance of the Watches going. In num••ers thus,
2196. 26928:: 12▪ 16.
3. To find the strokes of the Ballance in one turn of the Fusy, say, As the num∣ber of turns of the Fusy, to the continu∣ance of the Watch's going in hours:: S•• are the Beats in one hour, to the Beats o•• one turn of the Fusy. In numbers it 〈◊〉〈◊〉 thus,
12. 16:: 20196▪ 26928.
4. To find the Beats of the Ballance in an hour, say thus, As the hours of the Watch's going, To the number of turns of the Fusy:: So are the Beats in one turn of the Fusy, To the Beats in an hour. In numbers thus,
16. 12:: 26928. 20196.
5. To find what Quotient is to be laid upon the Pinion of Report, say thus, As the beats in one turn of the Great-wheel, To the beats in an hour:: So are the hours of the Face of the Clock (viz. 12 or 24) To the Quotient of the Hour-Wheel divided by the Pinion of Report, i. e. the number of turns, which the Pi∣nion of Report hath in one turn of the Dial-Wheel. In numbers thus,
26928. 20196:: 12. 9.
Or rather (to avoid trouble) say thus, As the hours of the Watch's going, Are to the numbers of the turns of the Fusy:: So are the hours of the Face, To the Quo∣tient of the Pinion of Report▪ In num∣bers thus, 16. 12:: 12. 9. If the hours of the Face be 24, the Quotient will be 18; thus, 16. 12:: 24. 18.
§ 7. Having given a full account of all things necessary to the understanding the
Art of Calculation, I shall now reduc•• what hath been said into practice, by shewing how to proceed, in Calculating a piece of Watch-work.
The first thing you are to do, is to pitch upon your Train, or beats of the Ballance in an hour: as, whither a swi•••• Train, about 20000 beats (which is the usual Train of a common 30 hour Pocket▪Watch) or a slower Train of about 16000 (the Train of the new Pendulum Pocket▪Watches;) or any other Train.
Having thus pitched upon your Train, you must next resolve upon the number of turns you intend your Fusy shall have; and also upon the number of Hours, you would have your Piece to go: As suppose 12 turns; and to go 30 hours, or 192 hours (which is 8 days) or &c.
These things being all soon determi∣ned; you next proceed to find out the beats of the Ballance, or Pendulum, in one turn of the Fusy, by the last § 6. part 3. viz. As the turns of the Fusy, To the hours of the Watch's going:: So is the Train, To the number of beats in one turn of the Fusy. In numbers thus, 12. 16:: 20000. 26666. Which last
number are the beats in one turn of the Fusy, or Great-Wheel; and (by Sect. I. § 5. of this Chap.) are equal to the Quo∣tients of all the Wheels unto the ballance, multiplied together. This number there∣fore is to be broken into a convenient parcel of Quotients: which you are to do after this manner. First, half your num∣ber of beats, viz. 26666, for the reasons in Sect. I. § 6. of this Chap. the half whereof is 13333. Next you are to pitch upon the number of your Crown-wheel, as suppose 17. Divide 13333 by 17, the Quotient will be 784 (or to speak in the language of one that understands not Arithmetick, divide 13333 into 17 parts, and 784 will be one of them.) This 784 is the number left for the Quotients (or turns) of the rest of the Wheels and Pini∣ons: which being too big for one or two Quotients, may be best broken into three. Chuse therefore 3 numbers, which when multiplied all together continually will come nearest 784. As suppose you take 10, 9, and 9. Now 10 times 9 is 90; and 9 times 90 is 810, which is somewhat too much. You may therefore try again other numbers▪ as suppose 11, 9, and 8.
These multiplied as the last, produce 792▪ which is as near as can be, and conveni∣ent Quotients.
Thus you have contrived your Piece▪ from the Great-Wheel to the Ballance▪ But the numbers not falling out exactly according as you at first proposed; you must correct your work thus. First to find out the true number of beats, in one turn of the Fusy, you must multiply 792▪ aforesaid, which is the true product of al•• the Quotients, by 17, the notches of the Crown-wheel; the product of this i•• 13464, which is half the number of true beats in one turn of the Fusy, by Sect. •• § 5. of this Chap. Then to find the tru•• number of beats in an hour, say by § 6▪ part 4. of this Section, as the hours o•• the Watch's going, viz. 16, to the 1•• turns of the Fusy:: So is 13464 the ha•••• beats in one turn of the Fusy, to 1009•• the half beats in an hour: the numbe•• will stand thus 16. 12:: 13464. 1009••
Then to know what Quotient is to b•• laid upon the Pinion of Report, say by 〈◊〉〈◊〉 6. part 5. of this Sect: As the hours 〈◊〉〈◊〉 the Watch's going, viz. 16, to the tur•• of the Fusy, viz. 12:: So are the hou••••
of the Dial-plate, viz. 12, To the Quo∣tient of the Pinion of Report. In num∣bers thus, 16. 12:: 12. 9.
* 1.16 Having thus found out all your Quotients, 'tis easie to determine what numbers your Wheels shall have: for chuse what numbers your Pinions shall have, and multiply the Pinions by their Quotients, and that produceth the num∣bers for your Wheels, as you see in the Margin. Thus 4 is the num∣ber of your Pinion of Report, and 9 its quotient; therefore 4 times 9, which makes 36, is the number for the Dial-wheel. So the next Pinion being 5, and its quotient 11, this multiplied produces 55 for the Great-wheel. And the like of the rest of the following numbers.
Thus, as plain as words can express it, I have shewed how to Calculate the num∣bers of a 16 hour Watch.
* 1.17 § 8. This Watch may be made to go a longer time, by lessening the Train, and altering the Pinion of Report. Suppose you could conveniently slacken the Train to 16000, the half of which is 8000.
* 1.18 Then say (by § 6 part 2. of this Sect.) As the halfed Train, or Beats in an hour, viz. 8000, To the halfed beats in one turn of the Fusy, viz. 13464:: So are the turns of the Fusy, viz. 12, To the hours of the Watch's going: in numbers thus, 8000. 13464:: 12. 20 So that this Watch will go 20 hours.
Then for the Pinion of Report, say, by the same § part 5, As 20 the Continu∣ance; To 12 the turns of the Fusy:: So are 12 the hours of the Face, To 7 the quotient of the Pinion of Report. In numbers thus, 20. 12:: 12. 7.
* 1.19 The work is the same as be∣fore, as to the numbers; only the Dial-wheel is but 28, be∣cause its quotient is altered to 7; as appears in the Mar∣gin, by the Scheme of the work.
§ 9. I shall give the Reader one example more, for the sake of shewing him the use of some of the foregoing Rules, not yet taken notice of in the former operations. Suppose you * 1.20would give numbers to a Watch of about 10000 beats in an hour, to have 12 turns
of the Fusy, to go 170 hours, and 17 notches in the Crown-wheel.
This work is the same as in the last Ex∣ample § 7. In short therefore thus, As the turns 12: are To the Continuance 170:: So is the Train 10000, To 141666, which are the beats in one turn of the Fusy. The numbers will stand thus, 12. 170:: 10000 14••666. Half this last is 70••33. Divide this half into 17 parts, and 4167 will be for the quotients. And because this number is too big for 3 quotients, therefore chuse 4: as suppose 10, 8, 8, and 6 ⅗ (i. e. 6 and 3 fifths) These multiplied together as before, and with 17, maketh 71808, which are half the true beats in one turn of the Fusy. By this you are to find out your true Train first▪ saying as in the former example, As 170 to 12:: So 71808, to 5069; which last is the half of the true Train of your Watch. Then for the Pinion of Report, say, as 170, to 12:: So 12, to 1/••4/74/••. Which Fraction ari••eth thus: If you mul∣tiply 12 by 12 it makes 144; and divide 144 by 170, you cannot; but setting the 144 (the dividend) over 170 (the Divi∣sor) and there ari••eth this fraction ••/24/••4/••,
which is a Wheel and Pinion; the lower is the Pinion of Report, and the upper is the Dial-wheel, according to Sect. I. § 3. of this Chapter. Or (which perhaps will be more plain to the unlearned Reader) you may leave those two numbers, in their Divisional posture thus, 170) 144, which does express the Pinion and Wheel, in the way I have hitherto made use of. But to proceed. These numbers being too big to be cut in small * 1.21 Wheels, may be varied, as you see a like Example is § 5. of this Section: viz. Say, as 144▪ is To 170:: So is 360, To 425. Or, as 170, to 144:: So is 360, To 305. In number thus, 144. 170:: 360. 425. Or 170. 144:: 360. 305. Di∣vide 360, and either of these two fourth and last numbers by 4, 5, 6, 8, &c. (as is directed in the Rule last cited.) If you divide by 8, you will have for your numbers 1/14/74/•• 4/••5/•• or 3/48/5. If you divide by 15 (which will not bring it so near an integer) you will have 2/24/8 or 2/••0/4: which last are the numbers set down in the Mar∣gin▪
where the numbers of the whole Movement are set down.
§ 10. Having said enough, I think, concerning the Calculation of ordinary Watches, to shew the hour of the day: I shall next proceed to such as shew minutes and seconds. The process whereof is thus: First, having resolved upon your beats in an hour, you are next to find how many beats there will be in a minute, by dividing your designed Train into 60 parts. And accordingly you are to find out such proper numbers for your Crown-wheel, and quotients, as that the Minute-wheel shall go round once in an hour, and the Second-wheel once in a minute.
An Example will make all plain. Let us chuse a Pendulum of 6 inches to go 8 days, with 16 turns of the Fusy. By * 1.22Mr Smith's Tables, a Pendulum of 6 inches vibrates 9368 in an hour. This divided by 60 gives 156 beats for a mi∣nute. Half these summs are 4684 and * 1.2378. Now the first work is to break this 78 into good proportions; which will fall into one quotient, and the Crown-wheel. First, for the Crown-wheel; let
it have 15 notches. Divide 78 afore•• by this 15, the quotient will be 5. A•• so this first work is done: for a Crow•• wheel of 15, and a Wheel a•• Pinion, whose quotient is 5 (•• in the Margin)* 1.24 will go rou•• in a minute, to carry a Ha•• to shew Seconds.
Next for a Hand to go round in •• hour, to shew Minutes. Now becau•• there are 60 minutes in an hour, 'tis b•• breaking 60 into two goo•• * 1.25 quotients (which may be •• and 6, or 8 and 7½, or &c. and the work is done.
Thus your number 4684, broken, as near as can be, int•• proper numbers.
But because it does not fall out exact into the above-mentioned numbers, yo•• must Correct (as you were directed be∣fore) and find out the true number •• beats in an hour, by multiplying 15 by 5, which makes 75; and this by 6•• makes 4500, which is the half of the tru•• Train. Then to find out the beats in on * 1.26turn of thy Fusy, operate as before, vi•• As the number of turns, 16, To the co••∣tinuan••
192:: So is 4500 to 54000, which are half the beats in one turn of ••he Fusy. In numbers thus, 16. 192:: 4500. 54000. This 54000 must be di∣••ided by 4500, which are the true ••umbers already pi••ched upon, or beats▪ ••n an hour. The quotient of this division ••s 12, which being not too big for one single quotient, needs not be divided into more. The work will stand, as you see in the Margin.* 1.27
As to the Hour-hand, the Great-Wheel, which performs only one revolution in 12 turns of the Minute-wheel, will shew the hour. Or rather you may order it to be done by the Minute-wheel, ••s shall be shew'd hereafter.
§ 11. I shall add but one Example more, and so conclude this Section; and ••hat is, To calculate the numbers of a ••iece whose Pendulum swings▪ Seconds, ••o shew the hour, minutes, and seconds, ••nd to go 8 days; which is the usual per∣••ormance of those Movements called * 1.28••oyal Pendulums at this day. First, cast ••p the number of seconds in 12 hours
(which are the beats in one turn of 〈◊〉〈◊〉 Great-wheel) These are 12 times 〈◊〉〈◊〉 minutes, and 60 times that, gives 432•• which are the seconds in 12 hours. H•• * 1.29this number (for the reasons before) 21600. The Swing-wheel must ne•• be 30, to swing 60 seconds in one of 〈◊〉〈◊〉 revolutions. Divide 21600 by it, a•• 720 is the quotient, or number left to 〈◊〉〈◊〉 broken into quotients. Of these quo••∣ents, the first must needs be 12 for 〈◊〉〈◊〉 Great-wheel, which moves round on•• in 12 hours. Divide 720 by 12, 〈◊〉〈◊〉 quotient is 60; which may be conve••▪ently broken into two quotients, as 〈◊〉〈◊〉 and 6, or 5 and 12, or 8 and 7 ½, whi•• last is most convenient. A•• if you take all the Pinions the work will stand as in 〈◊〉〈◊〉 Margin.* 1.30
According to this compu••▪tion, the Great-wheel will 〈◊〉〈◊〉 about once in 12 hours, shew the hour, if you please: the Seco•• wheel once in an hour, to shew the 〈◊〉〈◊〉 nutes; and the Swing-wheel once in a 〈◊〉〈◊〉 nute, to shew the seconds.
Thus I have endeavour'd with all pos∣sible plainness, to unravel this most my∣sterious, as well as useful part of Watch∣work. In which, if I have offended the more learned Reader, by unartificial terms, or multitude of words, I desire the fault may be laid upon my earnest intent to condescend to the meanest ca∣pacity.
§ 1. ALtho this part consists of many Wheels and Pinions, yet respect needs to be had only to the Count-wheel, Striking-wheel, and Detent-wheel: which move round in this proportion; The Count-wheel moveth round commonly ••nce in 12, or 24 hours. The Detent-wheel moves round every stroke the Clock striketh, sometimes but once in two strokes. From whence it follows,
1. That as many Pins as are in the Pin-wheel, so many turns hath the De∣tent-wheel, in one turn of the Pin-wheel. Or (which is the same) the Pins of th••
Pin-wheel are the Quotient of that Wheel, divided by the Pinion of the De∣ten••-wheel. But if the Detent-wheel moveth but once round in two strokes o•• the Clock, then the said Quotient is bu•• half the number of Pins.
2. As many turns of the Pin-wheel a•• are required to perform the strokes of 1•• hours (which are 78) So many tur•••• must the Pinion of Report have, to turn round the Count-wheel once. Or thus▪ Divide 78 by the number of Striking pins, and the Quotient thereof shall b•• the Quotient of the Pinion of Report. Al•• this is, in case the Pinion of Report b•• fixed to the arbor of the Pin-wheel, as i•• very commonly done.
All this I take to be very plain: or 〈◊〉〈◊〉 it be not, the example in the Margin wil•• clear all difficulties.* 1.31 Her•• the Locking-wheel is 48•• the Pinion of Report is 8•• the Pin-wheel is 78, th•• Striking-pins are 13. An•• so of the rest. I need onl•• to remark hero, that 7•• being divided by the 13 pins, gives 6•• which is the Quotient of••▪ the Pinion••
of Report: as was before hinted.
As for the Warning-wheel, and Flying-Pinion, it matters little what numbers they have, their use being only to bridle the rapidity of the motion of the other Wheels.
Besides the last observation, there are other ways to find out the Pinion of Re∣port, which will fall under the next §.
§ 2. These following Rules will be of great use in this part of Calculation, viz.
Rule 1. As the number of turns of the Great-wheel, or Fusy;
.To the days of the Clock's continuauc••:
::So is the number of strokes in 24 hours, viz. 156▪
.To the strokes in one turn of the Fusy, or Great-wheel.
Rule 2. As the number of strokes in 24 hours, which are 156,
.To the strokes in one turn of the Fusy, or Great-wheel,
::So are the turns of the Fusy, or Great-wheel,
.To the days of the Clock's continuance, or going.
Rule 3. As the strokes in one turn of the Fu••y,
.To the strokes of 24 hours, viz. 156.
::So is the Clock's continuance,
.To the number of turns of the Fusy, or Great-wheel.
These two last Rules are of no great use (as the first is) but may serve to correct your work, if need be, when in breaking your Strokes into Quotients (of which presently) you cannot come near the true number, but a good many strokes are left remaining. In this case, by Rule 2. you may find whether the continuance of your Clock be to your mind▪ And by Rule 3, you may enlarge or diminish the number of turns for this purpose. The praxis hereof will follow by and by.
The 2 following Rules are to find fit numbers for the Pinion of Report, and the Locking-wheel, besides what is said before § 1. Inference 2.
Rule 4. As the number of Strokes in the Clock's continuance, or in all it•• turns of the Fusy,
.To the turns of the Fusy,
::So are the Strokes in 12 hours, which are 78,
.To the Quotient of the Pinion of Re∣port, fixed upon the arbor of the Great-wheel.
But if you would fix it to any other Wheel, you may do it thus, as is before * 1.32hinted, viz.
Rule 5. First, find out the number of Strokes, in one turn of the Wheel you in∣tend to fix your Pinion of Report upon (which I shall shew you how to do in the following §.) Divide 78 by this number, and the number arising in the Quotient, is the Quotient of the Pinion of Report.
Or thus. Take the number of Strokes in•• ••e turn of the Wheel, for the num∣ber of the Pinion of Report, and 78 for the Count (or Locking) wheel, and vary them to lesser numbers, by Sect. 2. § 5. of this Chapter.
Rule 6. The foregoing Rules are of greatest use, in Clocks of a larger conti∣nuance; altho, where they can be ap∣plied, they will indifferently serve all. But this Rule (which will serve larger Clocks too) I add chiefly for the use of lesser Pieces, whose continuance is ac∣counted by hours.
The Rule is to find the Strokes in the Clock's continuance, viz. As 12, is to 78:: So are the hours of the Clocks continu∣ance,
To the number of Strokes in that time.
This Rule (I said) may be made use of for the largest Clock; but then you must be at the trouble of reducing the Days into Hours Whereas the shortest way is to Multiply the strokes in one turn of the Great-wheel, by the number of Turns. Thus in an 8 day piece the Strokes in one turn are 78. These multiplied by 16, the turns, produce 1248; which are the Strokes in the Clock's continuance. If you work by the foregoing Ruled the hours of 8 days are 192. Then say, 12▪ 78:: 192. 1248.
§ 3. In this Paragraph, I shall shew the use of the preceding Rules, and by exam∣ples make all plain that might seem ob∣scure in them.
I begin with small Pieces: of which but briefly. And first, having pitched upon the number of turns, and the conti∣nuance, you must find, by the last Rule, how many Strokes are in its continu∣ance. Then divide these Strokes by the number of turns, and you have the num∣ber of Striking-pins. Or divide by the number of Pins, and you have the num∣ber of Turns.
Thus a Clock of 30 hours, with 15 turns of the Great-wheel, hath 195 strokes. For by the last Rule, 12. 78:: 30. 195. Divide 195 by 15, it gives 13 for the Striking-pins. Or if you chuse 13 for your number of Pins, and divide 195 by it, it gives 15, for the number of ••urns, as you see in the Margin.* 1.33
As for the Pinion of Report, and the rest of the Wheels, enough is said in the § 1.
But suppose you would calculate the numbers of a Clock of much longer con∣tinuance, which will necessitate you to make your Pin-wheel further distant from the Great-wheel, you are to proceed thus: Having re••olved upon your turns, you must find out the number of strokes in one turn of the Great-wheel, or Fusy, by § 2▪ Rule 1. Thus in an 8 day piece, of 16 turns, 16. 8:: 156. 78. So in a piece of 32 days, and 16 turns, 16▪ 32:: 156. 312. These strokes so found out, are the number which is to be broken into a convenient parcel of Quotients, thus;
First resolve upon your number of Stri∣king-pins:
divide the last named number by it: The quotient arising hence, is to be one, or more quotients, for the Wheels and Pinions. As in the last examples▪ Divide 78 by 8 (the usual pins in an 8 day piece) and the quotient is 9••; which is a quotient little enough. So in the Month-piece: if you take your Pins 8▪ divide 312 by it, the quotient is 39. Which being too big for one, must b•• broken into two quoti∣ents, * 1.34 for Wheels and Pinions, or as near 〈◊〉〈◊〉 can be: which may b•• 7 and 5, or 6 and 6½. Th•• latter is exactly 39, and may there••o•• stand: as you see is done in the Margin.
The quotients being thus determined and accordingly the Wheels and Pinio•••• as you see; the next work is to find 〈◊〉〈◊〉 quotient for the Pinion of Report, to ••••∣ry round the Count (or Locking) wh•••• once in 12 hours, or as you please. •• you fix your Pinion of Report on th•• Great-wheel arbor, you must operate 〈◊〉〈◊〉 the Rule 4. of the last paragraph. As 〈◊〉〈◊〉 the last example in the Month-piece: 〈◊〉〈◊〉 Rule 6. before, the strokes in the conti••••∣ance
are 4992. Then by Rule 4 say, 4992. 16:: 78. 499••/124•• or thus, 4992) 1248. The first of which two numbers is the Pinion, the next is the Wheel. Which being too large, may be varied to ••••/9 or * 1.3536) 9; or to ••4/6 or 24) 6, by Sect. 2 § 5. before.
These numbers being not the usual numbers of a Month-piece, but only made use of by me, as better illustrating the foregoing Rules▪ I shall therefore, for the fuller explication of what has been said, briefly touch upon the calculation of the more usual numbers. They commonly encrease the number of Striking-pins, and so make the Second-wheel the Striking-wheel▪ Suppose you take 24 Pins; Di∣vide 312 by it, and the Quotient is 13. Which is little enough * 1.36 for one Quotient; and may therefore stand as you see is done in the Margin: where the Quotient of the first Wheel is 13. In the second Wheel of 72 teeth, are the 24 pins, altho its quotient is but 12, because the Hoop-wheel is double, and goes round but once in two strokes of the Pin-wheel.
The Pinion of Report here, is the same with the last, if fixed upon the arbor of the Great▪wheel. But if you fix it on the arbor of the Second, or Pin-wheel, its quotient then is found by § 1. Infer. 2. or by § 2. Rule 5. viz. Divide 78 by 24, and the number arising in the quotient, is the quotient of the Pinion of * 1.37 Report, which is 3 ¼. The Pinion of Report then being 12, the Count-wheel will be 39, as in the Margin.
To perfect the Reader in this part of Calculation, I will finish this Section with the calculation of a Year-piece of Clock∣work. The Process whereof is the same with the last, and therefore I may be more brief with this, except where I have not touched upon the foregoing Rules.
We will chuse a piece to go 395 days with 16 turns, and 26 Striking-pins. By § 2. Rule 1. there are 3851 strokes in one turn of the Great-wheel. For 16. 395:: 156. 3851. This last number divided by the 26 Pins, leaves 148 in the quotient, to be broken into two or more quotients, for Wheels and Pinions. These quotients may be 12 and 12; which multiplied,
makes 144, which is * 1.38 as near as can well be, to 148. The work thus far contrived, will stand as you see in the Margin.
Before you go any further, you may correct your work, and see how near your numbers come to what you propo∣sed at first, because they did not fall out exact. And first, for the true continuance of your Clock▪ If you multiply 12, 12, and 26 (i. e. the Quotients un▪o the Stri∣••ing-pins, and those Pins) you have the true number of Strokes, in one turn of the Great-wheel: Which, in this example, make 3744. For 12 times 12, is 144; and 26 times that, is 3744. (This Direction I would have noted, and remembered, as a Rule useful at any time to discover the nature of any piece of Clock-work.) Ha∣ving thus the true number of Strokes de∣sired, by § 2. Rule 2. you may find the true Continuance to be only 384 days. For 156. 3744:: 16. 384. If this Conti∣nuance doth not please you, you may come nearer to your first proposed num∣ber, of 395 days, by a small encrease of
the number o•• Turns; according to § 2. Rule 3. viz. by making your turns al∣most 16½. For 3744. 156:: 395. 16½ almost.
Lastiy, For the Pinion of Report, if you fix it upon the Great-wheel, it will require an excessive number: if you fix it upon the Pin-wheel (which is usual) then by § 2. Rule 5, the quotient * 1.39 is 3; and the Pinion of Re∣port being 13, the Count-wheel will be 39; as you see in the Mar∣gin.
But for the better exercising the Reader, let us fix it upon the Spindle of the Se∣cond-wheel 96. Its quotient is 12; which multiplied by 26 (the pins) pro∣duceth 312; which are the Strokes in one turn of that Second-wheel. Then by § 2▪ Rule 5, Divide 78 by 312, i. e. Set them as a Wheel and Pinion thus, 312) 78, and vary them to lesser numbers (by Sect. •• § 5.) viz. 36▪ 9, or to 24) 6, or th•• like.
I think it needless to say any thing o•• Pocket-clocks, whose calculation is the very same, with what goes before.
That the unlearned Reader may not think any thing going before difficult, I need only to advise him, to look over the working of the Rule of Proportion, in Sect. 2. § 4. For I think all will be plain, if that be well understood.
THe Reader will expect that I should say somewhat concerning Quarters and Chimes: but because there is little, but what is purely mechanical in it, I shall say the less, and leave the Reader to his own invention.
§ 1. The Quarters are generally a di∣stinct part from the Clock-part, which striketh the Hour.
The Striking-wheel may be the First, Second, or &c. Wheel, according to your Clock's continuance. Unto which Wheel you may fix the Pinion of Report.
The Locking-wheel must be divided (as o∣ther Locking-wheels) into 4, 8, or more un∣equal parts, ••o as to strike the Quarter, and lock at the first Notch; the half-hour, and
lock at the second Notch, &c. And in doing this, you may make it to chime the Quarters, or strike them upon two Bells, or more.
'Tis usual for the Pin-wheel▪ or the Locking-wheel, to unlock the Hour-part in these Clocks; which is easily done by some jogg or Latch, at the end of the last Quarter, to lift up the Detents of the Hour-part.
If you would have your Clock strike the Hour, at the Half-hour, as well as whole Hour, you must make the Locking-wheel of the Hour-part double: i. e. it must have two Notches of a sort, to strike 1, 2, 3, 4, &c. twice apiece.
§ 2. As for Chimes, I need say nothing of the Lifting-pieces and Detents, to lock and unlock; nor of the Wheels to bridle the motion of the Barrel. Only you are to observe, that the Barrel must be as long in turning round, as you are in Singing the Tune it is to play. As for the Chime-Barrel, it may be made up of certain Barrs, that run athwart it, with a convenient number of holes punched in them, to put in the Pins, that are to draw each Ham∣mer. By this means, you may change
the Tune, without changing the Barrel. This is the way of the Royal Exchange Clock in London, and of others. In this case, the Pins or Nuts, which draw the Ham∣mers, must hang down from the Barr, some more, some less, and some stand up∣right in the Barr: the reason whereof is, to play the Time of the Tune rightly. For the distance of each of these Barrs, may be a Semi-brief, or &c. of which here∣after.
But the most usual way is, to have the Pins that draw the Hammers, fixed on the Barrel. For the placing of which Pins, you may make use of the Musical Notes, or proceed by the way of Changes on Bells, viz. 1, 2, 3, 4, &c. The first be∣ing far the better way, I shall speak of that chiefly, especially because the latter will fall in to be explained with it.
And first, you are to observe what is the Compass of your Tune, or how many Notes or Bells there are from the highest to the lowest: and accordingly you must divide your Barrel from end to end. Thus in the examples following, each of those Tunes are 8 notes in compass; and ac∣cordingly the Barrel is divided into 8
parts. These Divisions are struck round the Barrel, opposite to which are the Hammer-tails.
I speak here, as if there was only one Hammer to each Bell, that the Reader may more clearly apprehend what I am explaining. But when two Notes of the same sound come together in a Tune, there must be two Hammers to that Bell, to strike it. So that if in all the Tunes you intend to Chime, of 8 notes compass, there should happen to be such double Notes on every Bell, instead of 8, you must have 16 Hammers: and according∣ly you must divide your Barrel, and strike 16 strokes round it opposite to each Ham∣mer-tail. Thus much for dividing your Barrel from end to end.
In the next place, you are to divide i•• (round about) into as many divisions, as there are Musical Barrs, Semibriefs, Minums, &c. in your Tune. Thus the 100th Psalm-tune hath 20 Semibriefs; the Song-tune following, hath 24 Barrs of triple time: and accordingly their Bar∣rels are divided. Each division therefore of the 100th Psalm Barrel is a Semibrief, ••nd of the Song-tune 'tis three crotchets▪
And therefore the intermediate Spaces serve for the shorter notes: as, one third of a division, is a Crotchet, in the Song-tune. One half a division, is a Minum; and one quarter a Crotchet, in the Psalm-tune. Thus the first note in the 100th Psalm, is a Semibrief, and accordingly on the Barrel, 'tis a whole division from 5 to 5. The second is a Minum, and there∣fore 6 is but half a division from 5; and so of the rest. And so also for the Song-tune, which is shorter time: The two first notes being Quavers, are distant from one another, and from the third pin, but half a third part of one of the divisions. But the two next pins (of the bell 3, 3) being Crotchets, are distant so many third parts of a division. And the next pin (of the bell 1) being a Minum, is distant from the following pin (4) two thirds of a division.
From what hath been said, you may conceive the surface of a Chime-barrel, to be represented in the Tables following, as stretched out at length: or (to speak plainer) that if you wrap either of these Tables round a Barrel, the Dotts in the Table, will shew the places of the
Pins, to be set on the Barrel.
You may observe in the Tables, that from the end of each Table to the begin∣ning, is the distance of two, or near two divisions: which is for a Pause, between the end of the Tune, and its beginning to Chime again.
I need not say, that the Dotts running about the Tables, are the places of the Pins that play the Tune.
If you would have your Chimes com∣pleat indeed, you ought to have a set of Bells, to the Gamut notes; so as that each Bell having the true sound of Sol, La, Mi, Fa, you may play any Tune, with its Flats and Sharps. Nay, you may by these means, play both the Bass and Treble, with one Barrel.
If any thing going before appears gib∣berish, I can't help it, unless I should here teach the skill of Musick too.
As to setting a Tune upon the Chime∣barrel from the number of Bells, viz. 1, 2, 3, 4, I shall here give you a speci∣men thereof.
Such Command o're my Fate, in num∣bers.
775, 3, 3, 1. 4, 5, 6, 4. 4, 2.
4, 3, 2, 3, 4, 6, 3, 5, 7, 7, 7. ‖
5, 6, 8, 8, 4. 4, 4; 3, 5, 4.
6, 5, 7, 5, 3; 41, 3, 5, 5, 5.
3, 3, 1, 3, 5. 554, 2, 4, 6.
4, 3; 23, 3; 53, 5, 7, 7, 7.
Note, In these numbers, a Comma [,] signifies the note before it, to be a Crotchet. A prick'd Comma, or Semi-colon [;] de∣noteth a prick'd Crotchet. And a Period [.] is a Minum. Where no punctation is, those Notes are Quavers.
I shall only add further, that by set∣ting the Names of your Bells at the head of any Tune (as is done in the Tables be∣fore) you may easily transfer that Tune, to your Chime-barrel, without any great skill in Musick. But observe, that each line in the Musick, is three notes distant; i. e. there is a Note between each line, as well as upon it: as is manifest by inspect∣ing the Tables.
The Motions I here chiefly intend, are the Day of the Month, the Moons age, the Day of the Year, the Tides, and (if you please) the slow motion of the Suns Apogaeum, of the Fixed Stars, the motion of the Planets, &c.
§ 1. For the effecting these Motions, you may make them to depend upon the Work already in the Movement; or else measure them by the beats of a Ballance, or Pendulum.
If the latter way, you must however contrive a Piece (as before in Watch-work) to go a certain time, with a cer∣tain number of turns.
But then to Specificate, or determine the Motion intended, you must proceed one of these two ways: either,
1. Find how many beats are in the Re∣volution. Divide these beats by the beats in one turn of the Wheel, or Pinion, which you intend shall drive the intended Revolution; and the Quotient shall be
the number to perform the same. Which, if too big for one, may be broken into more Quotients. Thus, if you would represent the Synodical Revolution of the Moon, which is 29 days, 12 ¾ hours) with a Pendulum that swings Seconds, the Movement to go 8 days, with 16 turns of the Fusy, and the Great-wheel to drive the Revolution. Divide 2551500 (the Beats in 29 days 12 ¾ hours) by 43200 (the Beats in one turn of the Great-wheel) and you will have 59 in the Quo∣tient: which being too big for one, may be put into two Quotients. Or
2. You may proceed as is directed be∣fore, * 1.40in the Section of Calculating Watch∣work, viz. Chuse your Train, turns of the Fusy, Continuance, &c. And then instead of finding a Quotient for the Pini∣on of Report, find a number (which is all one as a Pin. of Report) to Specificate your Revolution, by this following Rule.
Rule. As the Beats in one turn of the Great-wheel. To the Train:: So are the Hours of the Revolution, To the Quoti∣ent of the Revolution.
Thus to perform the Revolution of Sa∣turn (which is 29 years, 183 days) with a
16 hour Watch, of 26928 Beats in one turn of the Fusy, and 20196, the Train: the quotient of the Revolution, will be 193824. For, As 26928, To 20196:: So 258432 (the Hours in 29 y. and 183 d.) To 193824. Note here, That the Great-wheel Pinion is to drive the Revolution work.
But if you would have the Revolution to be driven by the Dial-wheel, and the Work already in the Movement (which in great Revolutions, is for the most part, as nice as the last way, and in which I in∣tend to treat of the particular Motions) in this case, I say, you must first know the Days of the Revolution. And because the Dial-wheel goeth round twice in a day, therefore double the number of the days in the Revolution, and you have the number of turns of the Dial-wheel in that time. This number of turns is what you are to break into a convenient number of quotients, for the Wheels and Pinions▪ as shall be shewed in the following exam∣ples.
§ 2. A Motion to shew the Day of the Month.
* 1.41 The days in the largest Month are 31. These doubled are 62, which are the turns of the Dial-wheel, which may be broken into these two quotients 15 ½ and 4; which multiplied together make 62. Therefore chusing your Wheels and Pinions, as hath been directed in the former Sections, your work is done. The Wheels * 1.42 and Pinions may be, as you see done in▪ the Margin. Or if a larger Pinion than one of 5 be necessary, by reason it is con∣centrick to a Wheel, you * 1.43 may take 10 for the Pinion, and 40 for the Wheel, as in the Margin.
The work will lye thus in the Move∣ment, viz. Fix your Pinion 10, concen∣trical to the Dial-wheel (or to turn round with it upon the same Spindle.) This Pi∣nion 10 drives the Wheel 40: which Wheel has the Pinion 4 in its center, which carrieth about a Ring of 62 teeth, divided on the upper side into 31 days.
Or, you may, without the trouble of many Wheels, effect this motion; vi••. By a Ring divided into 30 or 31 days, and as many Fangs or Teeth, like a Crown▪
wheel teeth, which are caught and push∣ed forward once in 24 hours, by a pin in a Wheel, that goeth round in that time. This is the usual way in the Royal Pendu∣lums, and many other Clocks; and there∣fore being common, I shall say no more of it.
§ 3. A Motion to shew the age of the * 1.44Moon.
The Moon finisheth her course▪ so as to overtake the Sun, in 29 days, and a little above an half. This 29 ½ days (not regarding the small excess) makes 59 twelve hours, or turns of the Dial-wheel, which is to be broken into convenient quotients: which * 1.45 may be 5, 9 and 10 as in the first example; or 14¾ and 4, as in the second example in the Margin. So that if you fix a Pinion of 10 concentrical with your Dial-wheel, to drive a Wheel of 40 (according to the last example) which Wheel 40 drives a Pinion 4, which carries about a Ring, or Wheel of 59 teeth, divided on the upper side into 29 ½ 'twill shew the Moons age.
* 1.46 § 4. A Motion to shew the day of the Year, the Sun's place in the Ecliptick, Sun's Rising or Setting, or any other annual motion of 365 days.
The double of 365 is 730, the turns of the Dial-wheel in an year: which may be broken into * 1.47 these quotients, viz. 18 ¼, and 10, and 4, according to the first exam∣ple; or 18 ¼, 8, and 5, according to the second. So that a Pinion of 5 is to lead a Wheel of 20; which again by a Pinion of 4, leadeth a Wheel of 40; which third∣ly, by a Pinion of 4, carrieth about a Wheel, or Ring of 73, divided into the 12 months, and their days; or into the 12 signs, and their degrees; or into the Sun's Rising and Setting, &c. For the setting on of which last, you have a Ta∣ble in Mr. Oughtred's Opuscula.
* 1.48 § 5. To shew the Tides at any Port.
This is done without any other trou∣ble, than the Moon's Ring (before menti∣oned § 3.) to move round a fixed circle, divided into twice 12 hours, and num∣bered the contrary way to the age of the Moon.
To set this to go right, you must find out at what Point of the Compass the Moon makes full Sea, at the place you would have your Watch serve to. Convert that point into hours, al∣lowing for every point North or S. lost 45′ of an hour. Thus at London-bridge 'tis vulgarly thought to be high Tide, the Moon at N. E. and S. W, which are 4 Points from the N. and S. Or you may do thus: by Tide-tables learn how many hours from the Moon's Southing, 'tis High∣water. Or thus; find at what hour it is High∣water, at the Full or Change of the M••on: as at London-bridge, the full Tide is com∣monly reckoned to be 3 hours from the Moon's Southing; or at 3 of clock at the Full and Change. The day of Conjuncti∣on, or New-Moon, with a little stud to point, being set to the hour so found, will afterwards point to the hour of full Tide.
This is the usual way; but it being al∣ways in motion, as the Tides are not, a better way may be found out, viz. By causing a Wheel, or Ring to be moved forward, only twice a day, and to keep time (as near as can be) with Mr. Flam∣steed's most correct Tables. But this I
shall commit to the Readers contrivance, it being easie, and more of curiosity than use.
§ 6. To Calculate Numbers, to shew the Motion of the Planets, the Slow Motion of the Fixed Stars, and of the Sun's Apo∣geum, &c.
Having said enough before that may be applied here, and they being only cu∣riosities, seldom put in practice, I shall not therefore trouble the Reader, or swell my Book with so many words, as would be required to treat of these Motions di∣stinctly, and compleatly.
Only thus much in general. Knowing the years of any of these Revolutions, you may break this number into quotients; if you will make the Revolution to de∣pend upon the year's Motion; which is already in the Movement, and described § 4. before. Or if you would have it de∣pend upon the Dial-wheel, or upon the Beats of a Pendulu••, enough is said be∣fore to direct in mis matter.
In all these Slow motions, you may somewhat ••••••••ten your labour, by end∣less Screws to serve for Pinions, which are but as a Pinion of one tooth.
* 1.49 Sir Jonas Moor's account of his large ••phere going by Clock-work, will suffi∣••ently illustrate this paragraph. In this ••phere, is a Motion of 17100 years, for ••he Sun's Apogeum, performed by six ••heels, thus, as Sir Jonas relates it; For the Great-wheel fixed is 96, a Spin∣dle-wheel of 12 bars turns round it 8 times in 24 hours, that is, in 3 hours; after these, there are four Wheels, 20, 73, 24, and 75, wrought by endless Screws that are in value but one: there∣fore 3, 20, 73, 24, and 75 multiplied to∣gether continually, produceth 7884000 * 1.50hours, which divided, by 24 gives 3285000 days, equal to 900 years. Now on the last wheel 75 is a pinion of 6, turning a great Wheel, that carrieth the Apogeum number 114: and 114 divided by 6, gives 19 the quotient: and 900 times 19 is 17100 years.
Thus I have, with all the perspicuity I ••ould, led my Reader through the whole ••rt of Calculation, so much of it at least, ••at I hope he will be master of it all; not ••ly of those motions, which I have par∣••cularly treated about, but of any other ••t mentioned: Such as the Revolution
of the Dragons Head and Tail, whereby the Eclipses of the Sun and Moon are found, the Revolution of the several Orbs, according to the Ptolemaick System, or of the celestial bodies themselves, according to better Systems, with many other such curious performances, which have made the Sphere of Archimedes of old famous: and since him, that of William of Zeland, * 1.51and another of Janellus Turrianus of Cre∣mona, mentioned by Cardan: and of late, that elaborate piece of Mr. Watson, late of Coventry, now of London, in her late Majesties Closet.
Oughtred of Autom. sect. 4.
By the Quotients I com∣monly mean the ••umber of Turns; which num∣ber is set on the right hand, without a hook, as is shewn in the last Paragraph. Which I no••e ••ere now once for all.
Sir J. Moor▪ Mat. Com. p. 109.
Ibid. p. 116.
Oughtred Autom. Sect. 14.
Ought. ib.
Id. ib.
Id. ib.
Oughtred Sect. 12. Sir J. Moor Ibid. p. 109.
Oughtred Sect. 21.
§ 8.
Id. ib. § 22.
Horol. Disq.
Sect. 1. § 6.
§ 6. Par. 3. and § 7.
Sir J. Moo•• Ibid. p. 116.
V. Sect. 1. §▪ 6.
§ 1. Infer 2••
V. Sect. 1▪ §. 3.
Ch. 2. Sect. 2. § 7.
Oughtred. § 26.
Id. ib.
Id. ib.
Autom. § 35. Id. ib.
Mat. Com. p. 117.
V. Sect. 1 § 4, 5.
De Subtil. l. 17.