The artificial clock-maker a treatise of watch, and clock-work, wherein the art of calculating numbers for most sorts of movements is explained to the capacity of the unlearned : also, the history of clock-work, both ancient and modern, with other useful matters, never before published / by W.D.

CHAP. V. Of Pendulums.

sect 1. AMong all known Motions, none measureth Time so regu∣larly, as that of a Pendulum. But yet Watches governed hereby are not so per∣sect, but that they are subject to the vari∣ations of weather, foulness, &c. And the shorter, and lesser the Pendulum is, so much the more subject such Watches are to these annoyances.

There are two ways to obviate these in∣conveniences in some measure. One way is, to make the Pendulum long, the Bob heavy, and to vibrate but a little way from its settlement. Which is now the most usual way in England. The other is the contrivance of the ingenious Mr. Hu∣••ens, which is, to make the upper part of the rod, play between two cheek parts of •• Cycloid. Sir Jonas Moor says, that af∣••er some time, and charge of Experiments,

he believes this latter to be the better way. And Mr Hugens calls it admirable.

If any desire to know how to make those Cycloidal Cheeks, fit to all Pendu∣lums, I refer him to the aforesaid Mr. Zu∣lichem's Book, because I can't shew how to do it, without the trouble of Figures; and this way is much ceased, since the Crown-wheel method (to which it is chiefly proper) is swallowed up by the Royal Pendulums.

sect 2. Another thing to be remark'd in Pendulums is, That the longer the Vibra∣tion is, the ••lower it is. For if two iso∣chrone Pendulums do move, one the qua∣drant of a circle, the other not above 3 or 4 degrees, this latter shall move some-what quicker than the former. Which is the true reason, why small Crown-wheel Pendulums go faster in cold weather, or when soul, than at other times. Yea, in the best Royal Pendulum, if you put a divided plate behind the Ball, and ob∣serve its swings, you may perceive the Vibrations to be sometimes shorter; and that then the Watch doth gain too much▪ Somewhat also may perhaps be attributed to the rarity or density of the air; which

I have not yet had an opportunity of ob∣serving, by comparing with a good Baro∣scope, the various vibrations of a good Royal Pendulum. But Mr. Boyl says, that a Pendulum moveth as long, and as fast in a thick medium, as a thin one; contrary to the opinion of some Natura∣lists, who think the contrary. His opini∣on is grounded upon the experiment of a Pendulum vibrating in his air-pump, the air sucked out, and in the open air; wherein was no alteration.

sect 3. For the calculation of all Pendu∣lums, 'tis necessary to fix upon some one, to be as a Standard to the rest. I pitch upon a Pend▪ to vibrate Seconds each stroke.

Mr. Hugens lays down the length of a Pend. to swing Seconds to be 3 feet, 3 inches, and 2 tenths of an inch (according to Sir J. Moor's reduction of it to English measure. )

The Honourable Lord Bru••cker (saith Sir Jonas) and Mr. Rook, found the length to be 39, 25 inches, which a little exceeds the other: and may be, was justened by Mr. Hugens's Rule for the Center of Os∣cillation. For Mounton's Pendulum, that▪

vibrate 132 times in a minute, it will be found likewise 8, 1 inches, agreeing to 39, 2 inches English. Therefore for cer∣tain 39, 2 inches may be called the Ʋni∣versal measure, and relied on, to be the near length of a Pend. that shall swing Seconds each vibration.

But forasmuch as the different size of the Ball, will make some difference in the length of this Standard Pend., therefore to make this Pend. an Ʋniversal measure, to fit all Places and Ages, you must mea∣sure from the Point of Suspension, to the Center of Oscillation. Which Center is found by this Rule, As the length of the String from the point of▪ Suspension to the center of a round Ball: is to the Semi∣diameter of a round Ball: is to the Semi∣diameter, to a fourth number. Add two fifths of that fourth number, to the for∣mer length, and you have the center of Oscillation; and thereby the true length of this Standard Pendulum.

If it be desired to fit a Ball of a triangu∣lar, quadrangular, or any other form to this Pend, the center of Oscillation in any of these bodies, may be found in the last cited book of Mr Zulichem.

If it be asked, What is the meaning of the Center of Oscillation? The most in∣telligible answer (altho not perfectly true) is, That it is that point of the Ball, at which if you imagine it divided into two parts, by a circle, whose center is in the point of Suspension, the lower part of the Ball shall be of the same weight (or near so) with the upper.

§ 4. Having thus fixed a Standard, I shall next shew how from thence to find the Vibrations, or Lengths of all other Pendulums. Which is done by this Rule, The squares of the Vibrations, bear the same Proportion to each other, as their Length•• do. And so contrary wise. Wherefore to find the length of a Pend▪ say▪ As the Square of the Vibrations given: To the Square of 60 (the Standard):: So is the length of the Standard (viz. 39, 2) To the length of the Pend. sought.

If by the length, you would find the Vibrations, 'tis the reverse of the last Rule, viz. As the length proposed: To the Standard (39, 2):: So is the Square of 60 (the vibrations of the Standard): To the Vibrations sought.

Suppose for example, you would know what length a Pend. is, that vibrates 153 strokes in a minute. The Square of 153 (i. e. 153 times 153) is 23409. Say, 23409. 3600:: 39, 2. 6. A Pend. then that vibrates 153 in a minute, is about 6 inches long.

On the other hand, if you would know how many strokes a Pend. of 6 inches hath in a minute; Say, 6. 39, 2:: 3600. 23520. The square root whereof is 153, and somewhat more.

Note, Because 141120 is always the Product of the two middle terms multi∣plied together, therefore you need only to divide this number by the Square of the Vibrations, it gives the length sought: by the length, it gives the square of the Vibrations.

If you operate by the Logarithms, you will much contract your labour. For if you seek the length, 'tis but Substracting the Logarithm of the Square of the Vibra∣tions, out of the Logarithm of 141120, which is 5. 149588, and the Remainder is the Logarithm of the length sought.

If you seek the Vibrations, it is but Substracting out of the aforesaid Loga∣rithm

5. 149588, the Logarithm of the length given, and half the Residue is the Logarithm of the Vibrations required. The following examples will illustrate each particular.

To find the Length.

	Logarithms.
141120	5. 149588
153 Squared is 23409	4. 369382
Length is more than 6.	0. 780206

To find the Vibrations.

	Logarithms.
141120	5. 149588
6 inches long	0. 778151
Square of the Vibr.	4. 371437
Square-toor, or numb. of Vibr. is 153, and somewhat more.	2. 185718

According to the foregoing Directions▪ I have calculated the following Table▪ to Pendulums of various lengths: and have therein shewed the Vibrations in a minute,

and an hour, from 1 to 100 inches. If any desire a more minute account, I refer him to Mr Smith's Tables in his late Book. The reason why his calculation and mine differ, is because he measureth the length of the Pend. from the point of Suspension, to the lower part of the Bob; and I only to the center of the Bob. His Standards are 6½ inches, and 41 inches; and mine is 39, 2, for the reasons aforegoing.

A Table of Swings in a Minute, and in an hour, to Pendulums of several lengths.

Pend. length in inches	Vibrat. in a Minute.	Vibrat. in an Hour.
1	375,7	22542
2	265,6	15936
3	216,9	13014
4	187,8	11268
5	168,0	10080
6	153,3	9204
7	142,0	8520
8	132,8	7968
9	125,2	7512
10	118,8	7128
20	84,0	5040
30	68,6	4116
39,2	60.0	3600
40	59,4	3264
50	53,1	3186
60	48,5	2910
70	44,9	2694
80	42,0	2520
90	39,6	2376
100	37,5	2250

The use of this Table is manifest, and needs no explication. As to the Decimals in the column of Minute-Swings, I have ••dded them for the sake of calculating the column of Hour-Swings; which would have been judged false without them, and would not have been exactly true without them.

§ 5▪ I have but one thing more to add to this Chap. of Pendulums, and that is, ••o Correct their Motion.

The usual way is, to screw up, or let down the Ball. In doing of which, a small alteration will make a considerable ••ariation of Time: as you will find by calculation, according to the last para∣graph. To prevent the inconvenience of ••crewing the Ball too high, or low, Mr Smith hath contriv'd a very pretty Table ••or dividing the Nut of a Pendulum Screw, ••o as to alter your Clock but a Second in ••day. But by reason no Screw and Nut can be so made, as to be most exactly strait ••nd true, therefore it may happen, that instead of altering your Watch to your mind, you may do quite contrary; as ••nstead of letting the Ball down, you may raise it higher, by the false run∣ning

of the Nut upon the Screw.

Considering this irremediable inconve∣nience, I am of opinion, that Mr Hu∣gens's way would do very well, added to this. His way is, To have a small Weight, or Bob, to slide up and down the Pend. rod, above the Ball (which is immovea∣ble.) But I would rather advise, that the Ball be made to screw up and down, to bring the Pend. pretty neer its gauge: and that this little Bob should serve only for more nice corrections; as the altera∣tion of a Second, or &c. Which it will do, better than the Great Ball. For a whole turn of this little Bob, will not af∣fect the motion of the Pend. near so much as a small alteration of the Great Ball.

The Directions Mr Hugens gives, about this little Corrector, is, That it should be equal to the weight of the Wire, or Rod of the Pend., or about a 50th part of the weight of the Great Ball, which he appoints to be three pounds.

Perhaps this Bob may do its office, if it be made to screw only up and down the lower part of the Rod, below the Ball. If not, you must make it slide above the Ball, or be screwed up and down there.

Seeing this little Bob is not the only Corrector (as in Mr Zulichem's way) therefore it is not necessary to insert here, that ingenious person's Table, shewing what alterations of Time will be made by sliding the Bob up and down the rod. Only thus much may be observed in that Table of his, viz. That a small alteration of the Corrector towards the lower end of the Pend▪, doth make as great an alte∣ration of Time, as a greater raising or fal∣ling of it, doth make higher. Thus the little Bob raised 7 divisions of the Rod, from the Center of Oscillation, will alter the Watch 15 seconds; raised 15, 2 'twill alter it 30″. But whereas, if it be raised to 154▪ 3 parts of the Rod, it will make the Watch go faster 3 minutes, 15 seconds, the Watch shall be but 3′. 30″ faster, if the Bob be raised to 192▪ 6. So that here you have but 15″ variation, by raising the Bob above 38 parts; whereas lower, you had the same variation, when raised not above 7 or 8 parts.

From what hath been said, it appears, that about half a turn of this little justen∣ing Bob, will at no time alter the Watch, above a second in 24 hours▪ and that

above a whole turn, will not alter it so much, higher on the Rod; supposing that the Bob at every turn ascended or descend∣ed a whole degree of the Rod; which per∣haps it will not do in 20 turns: and con∣sequently, it will require many turns, to alter the Watch but one second.

Notes

• Id. ib. Rule 3.
• De Horol. Oscil. p. 10, 11, 12.
• Machina Pneumat. Exp. 26.
• Ibid.
• Ibid.
• Fiugenius ••••••i supra, p. 141. Sir J. Moor ibid.
• Hugen▪ Moor, ib••▪
• Horolog. Disquis.
• Ibid.
• Ibid. de Centro Oscil. Prop. 23.