150 OF MOTION, FORCES, &c. lengths, which by the supposition, are as Ac to ae, the ac. 7 ! 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 or the square roots of their axes. pc to √ De, 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, 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 superfi cies 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. 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 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 PENDULUMS. 151 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. DEFINITION. 146. If the circumference 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 A A 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 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 The cycloidal arc AC double the chord AD; The semi-cycloid ACA double the diameter AB, and LET ABA be the cycloid; DB its axis, or the diameter of the generating semicircle DEB; CB 2DB the length of the pendulum, or radius of curvature at B. Let the A ball descend from r, and in vibrating, describe the arc FBf. Divide FB into innumerable small parts, one of which is ag; draw FEL, GM, gm, perpendicular to J F 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 Cir cle is to its Diameter. i B DE. ་ BH2 = BK. BE, or вḤ — B And the equiangular triangles мMP, MON, give MN: MO. Also, by the nature of the cycloid, nh MP: мm: If another body descend down the chord EB, it will have That is, the time in Gg time in кk:: √ BL: √ Bn. Again, the time of describing any space with a uniform mo- kk EB Nn *LB LNLB time in EB:: I Consequently the sum of all the times in all the cg'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 ; : : LR, : LB, that is, the time in Fg time in DB ;; LM 149. Corol. 153 PENDULUMS. 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. 2 then √ g : √1⁄2l : : 1′′ 2g ι √.228 which therefore is the time of 2g 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 ἰ where I is the radius > in a small circular arc is equal to P√ zg theref. 1p 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 2g 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. = 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 1 :: √/391 : /1, 1 VOL, II. 21 391 or 4. 1 : 1 :: 391 : 93 inches nearly, the length of the half-seconds pendulum. Again 1: 39: 24 in16 :: ches, the length of the quarter-seconds pendulum. Again, if it were required to find how many vibrations a pendulum of 80 inches long will make in a minute. Here 80391: 60" or 1: 60/ 7/31·3-41.95987, or almost 42 vibrations in a minute. 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. : 291 80 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 at rest. 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. 158. Mechanics, |