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lengths, which, by the supposition, are as AC to /ac, the roots of the whole curves. Therefore, the whole times are in the same ratio of √AC to /ac.

143. Corol. 1. Because the axes DC, DC, of similar curves, are as the lengths of the similar parts AC, ac; therefore the times of descent in the curves AC, ac, are as DC to DC, or the square roots of their axes:

144. Corol. 2. As it is the same thing, whether the bodies run down the smooth concave side of the curves, or be made to describe those curves by vibrating like a pendulum, the lengths being DC, DC; therefore the times of the vibration. of pendulums, in similar arcs of any curves, are as the square roots of the lengths of the pendulums.

SCHOLIUM.

145. Having, in the last corollary, mentioned the pendulum, it may not be improper here to add some remarks concerning it..

If the

A pendulum consists of a ball, or any other heavy body B, hung by a fine string or thread, moveable about a centre A, and describing the arc CBD; by which vibration the same motions happen to this heavy body, as would happen to any body descending by its gravity along the spherical superficies CBD, if that superficies were perfectly hard and smooth. pendulum be carried to the situation AC, and then let fall, the ball in descending will describe the arc CB; and in the point B it will have that velocity which is acquired by descending through CB, or by a body falling freely through EB. This velocity will be sufficient to cause the ball to ascend through an equal arc BD, to the same height » from whence it fell at c; having there lost all its motion, it will again begin to descend by its own gravity; and in the lowest point B it will acquire the same velocity as before; which will cause. it to re-ascend to c: and thus, by ascending and descending, it will perform continual vibrations in the circumference CBD. And if the motions of pendulums met with no resistance from the air, and if there were no friction at the centre of motion A, the vibrations of pendulums would never cease. But from these obstructions, though small, it happens, that the velocity of the ball in the point B is a little diminished in every vibration; and consequently it does not return precisely to the same points c or D, but the arcs described con

tinually

tinually become shorter and shorter, till at length they are insensible; unless the motion be assisted by a mechanical contrivance, as in clocks, called a maintaining power.

146. If the cir cumference of a circle be rolled on a right line, beginning at any point A, and continued till the same point A arrive at the line

'DEFINITION.

A

E

again, making just one revolution, and thereby measuring out a straight line ABA equal to the circumference of the circle, while the point a in the circumference traces out a curve line ACAGA; then this curve is called a cycloid; and some of its properties are contained in the following lemma.

LEMMA.

147. If the generating or revolving circle be placed in the middle of the cycloid, its diameter coinciding with the axis AB, and from any point there be drawn the tangent CF, the ordinate CDE perp. to the axis, and the chord of the circle AD: Then the chief properties are these:

CD

The right line
The cycloidal arc AC

The semi-cycloid ACA

the circular arc AD;

double the chord AD;

double the diameter AB, and

The tangent CF is parallel to the chord AD.

PROPOSITION XXX.

148. When a Pendulum vibrates in a Cycloid; the Time of one Vibration, is to the Time in which a Body falls through Half the Length of the Pendulum, as the Circumference of a Circle is to its Diameter.

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And the equiangular triangles Mmp, MON, give

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Mm :: MN: MO.

Hh is equal to Gg.

Also, by the nature of the cycloid,

If another body descend down the chord EB, it will have the same velocity as the ball in the cycloid has at the same height. So that кk and Gg are passed over with the same velocity, and consequently the time in passing them will be as their lengths Gg, кk, or as нh to кk, or вн to BK by similar triangles, or /(BK. BE) to BK, or BE to BK, or as

BL to BN by similar triangles. That is, the time in Gg: time in Kk :: √BL: √/BN. Again, the time of describing any space with a uniform motion, is directly as the space, and reciprocally as the ve locity; also, the velocity in K or кk, is to the velocity at B, as EK to EB, or as LN to LB; and the uniform velocity for EB is equal to half that at the point B, therefore the

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(by sim. tri.): Nn or мp: 2√(BL. LN.)

:

LB

LNLB

That is, the time in Kk: time in EB :: MP: 2√(BL. LN ) But it was, time in Gg time in Kk :: √BL: WBN; theref. by comp. time in Gg: time in EB:: MP: 2 (BN. NL) or 2NM. But, by sim. tri. Mm: 20м or BL: MP: 2NM.

Theref. time in Gg: time in EB :: Mm: FL.

Consequently the sum of all the times in all the Gg's, is to the time in EB, or the time in DB, which is the same thing, as the sum of all the mm's, is to LB;

that is, the time in Fg: time in DB ::

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time in DB::

FBf

Lm

: LB,

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time in DB:: 2LMB: LB.

That is, the time of one whole vibration,

is to the time of falling through half CB,
as the circumference of any circle,

is to its diameter.

149. Corol.

149. Corol. 1. Hence all the vibrations of a pendulum in a cycloid, whether great or small, are performed in the same time, which time is to the time of falling through the axis, or half the length of the pendulum, as 31416 to 1, the ratio of the circumference to its diameter; and hence that time is easily found thus. Put p = 3.1416, and the length of the pendulum, also g the space fallen by a heavy body in 1" of time.

then g::: 1":

theref. 1:p::√

2g

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of one vibration of the pendulum.

150. And if the pendulum vibrate in a small arc of a circle; because that small arc nearly coincides with the small cycloidal arc at the vertex B; therefore the time of vibration in the small arc of a circle, is nearly equal to the time of vibra tion in the cycloidal arc; consequently the time of vibration in a small circular arc, is equal top ✔ where is the radius of the circle.

2g'

151. So that, if one of these, g or 4, be found by experiment, this theorem will give the other. Thus, if g, or the space fallen through by a heavy body in 1" of time, be found, then this theorem will give the length of the second pendulum. Or, if the length of the second pendulum be observed by experiment, which is the easier way, this theorem will give g the descent of gravity in 1". Now, in the latitude of London, the length of a pendulum which vibrates seconds, has been found to be 39 inches; and this being

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written for / in the theorem, it gives p1": hence is

2g

found gpl = p2 x 391 = 193.07 inches = 16 feet, for the descent of gravity in 1"; which it has also been found to be, very nearly, by many accurate experiments.

SCHOLIUM.

152. Hence is found the length of a pendulum that shall make any number of vibrations in a given time. Or, the number of vibrations that shall be made by a pendulum of. a given length. Thus, suppose it were required to find the length of a half-seconds pendulum, or a quarter-seconds pendulum; that is, a pendulum to vibrate twice in a second, or 4 times in a second. Then, since the time of vibration is as the square root of the length,

therefore

DB. On LB descri micircle LMB, wh tre is o; draw Mi to DB; also draw tì BE, BH, EH, and th

OM.

Now the triang BHK, are equian therefore BK: EH :: or BH2

BK. BE, ("

(BK.BE).
And the equiangul
MP Mm :: MN:
Hh is equal to Gg.

If another body e the same velocity a height. So that K velocity, and conse as their lengths Gg, similar triangles, or BL to BN by That is, the time. Again, the time e

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