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lengths, which by the supposition, are as Ac to ae, the roots of the whole curves. Therefore, the whole times are in the same ratio of ✅ AC to
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, de, are as pc to √ De,
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.
145. Having in the last corollary, mentioned the pendulum, may not be improper here to add some remarks concerning it.
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. If the pendulum be carried to the situation AC, and then let fall, the ball in descending will describe the arc CB; and in the point в it will have that velocity which is acquired by descending through CB, or by a body falling freely through EB. 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 в 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 on 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 в 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 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.
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.
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 cr, the ordinate CDE perp. to the axis, and the chord of the circle ap: Then the chief properties are these:
The right line CD the circular arc AD
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 Cir cle is to its Diameter.
MN: MO. Also, by the nature of the cycloid, nh is equal to Gg.
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 Kk and og are passed over with the same velocity, and consequently the time in passing them will be as their lengths &g, xk, or as нh to кk, or вн, to вк by similar triangles, or✔ (BK. BE) to BK, or ✅✅✅ be to √ bk, or as /BL to EN by similar triangles.
That is, the time in Gg time in кk:: √ BL: √ Bn.
Again, the time of describing any space with a uniform motion, is directly as the space, and reciprocally as the velocity; also, the velocity in K or кk, is to the velocity at в, 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 time in кk:
(by sim. tri): : Nn or MP: 2/ (BL, Ln.)
That is, the time in кk time in EB MP 2 (BL. LN.)
Consequently the sum of all the times in all the cg's, is to
: LR, : LB,
that is, the time in Fg time in DB ;; LM
is to its diameter.
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 3·1416, to 1, the ratio of the circumference to its diameter; and hence that time is easily found thus. Put p 3.1416, and I the length of the pendulum, also g the space fallen by a heavy body in 1" of time.
then √ g : √1⁄2l : : 1′′ : : 1′′ : √ the time of falling through 11,
one vibration of the pendulum.
which therefore is the time of
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 в; therefore the time of vibration in the small arc of a circle, is nearly equal to the time of vibration in the cycloidal arc; consequently the time of vibration in a small circular arc is equal to P√ zg where I is the radius
of the circle.
151. So that, if one of these, g or l, 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 pendu lum. 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 394 inches; and this being written for l 391 in theorem, it gives p =1": ́hence is found g ips l=1p2 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.
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 1 :: √/391 : /1, VOL, II.
of the half-seconds pendulum. Again 1: 39: 24 in-
153. In these propositions, the thread is supposed to be very fine, or of no sensible weight, and the ball very small, or all the matter united in one point; also, the length of the pendulum, is the distance from the point of suspension, or centre of motion, to this point, or centre of the small ball. But if the ball be large, or the string very thick, or the vibrating body be of any other figure; then the length of the pendulum is different, and is measured, from the centre of motion, not to the centre of magnitude of the body, but to such a point, as that if all the matter of the pendulum were collected into it, it would then vibrate in the same time as the compound pendulum; and this point is called the Centre of Oscillation; a point which will be treated of in what follows.
THE MECHANICAL POWERS, &c.
154. WEIGHT and Power, when opposed to each other, signify the body to be moved, and the body that moves it M or the patient and agent. The power is the agent, which moves, or endeavours to move, the patient or weight.
155. Equilibrium, is an equality of action or force, between two or more powers or weights, acting against each other, by which they destroy each other's effects, and remain
156. Machine, or Engine, is any Mechanical instrument contrived to move bodies. And it is composed of the mechanical powers.
157. Mechanical Powers, are certain simple instruments, commonly employed for raising greater weights, or overcoming greater resistances, than could be effected by the natural strength without them. These are usually accounted six in number, viz. the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw.