GM, gm, perpendicular to DB. On LB describe the semicircle LMB, whose centre is o; draw Mp parallel to DB; also draw the chords BE, BH, EH, and the radius oм.

Now the triangles BEH, BHK, are equiangular; therefore BK: BH :: BH: BE, or BH2= BK. BE, or BH = √/BK. BE. And the equiangular triangles мmp, мON, give

MP: Mm :: MN: Mo. cloid, нh is equal to Gg.

Also, by the nature of the cy

BE to

BK, or as

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, Kk, or as нh to кk, or вH to BK by símilar triangles, or √BK. BE to BK, or BL to BN by similar triangles. That is, the time in Gg : time in Kk :: 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 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

BL:

BN.

::(by sim. tri.)

✓LN

I

LB

LN. ·

:: Nn or мp: 2/BL

LB 2

That is, the time in кk: time in EB:: MP: 2/BL LN.
But it was, time in Gg time in Kk :: ✔BL : √BN; theref.
by comp. time in Gg: time in EB:: MP: 2/BN. NL or 2NM.
But, by sim. tri. мm: 20м or BL:: MP: 2NH.
Theref. time in Gg time in EB :: Mm : BL.

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 мm's, is to LB;

that is, the time in Fg

time in DB ::

Lm : 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. 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

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 the length of the pendulum, also g the space fallen by a heavy body in 1" of time.

then g√:: 1": the time of falling through,

2g

2g

2g

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 vibration in the cycloidal arc; consequently the time of vibration in a small circular arc, is equal to p√; where is the radius 2g

of the circle.

ני

151. So that, if one of these, g or 1, 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 written for / in the theorem, it gives p√

I

I

39/

2g

1: hence

is found g = p2 l = p2 × 394 193-07 inches = 16, feet, for the descent of gravity in 1"; which it has also been found to be, very nearly, by many accurate experi

ments.

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

length of the half-seconds pendulum.

Again 1: 39: 24 inches, the length of the quarterseconds pendulum.

Again, if it were required to find how many vibrations a pendulum of 80 inches long will make in a minute. Here

39

√80: √39 :: 60′′ or l′: 60√ =74√/31·3 =

80

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.

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; 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, be tween 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 ac

counted

counted six in number, viz. the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw

158. Mechanics, is the science of forces, and the effects they produce, when applied to machines, in the motion of bodies.

159. Statics, is the science of weights, especially when considered in a state of equilibrium.

160. Centre of Motion, is the fixed point about which a body moves. And the Axis of Motion, is the fixed line. about which it moves.

161. Centre of Gravity, is a certain point, on which a body being freely suspended, it will rest in any position."

OF THE LEVER.

162. A LEVER is any inflexible rod, bar, or beam, which serves to raise weights, while it is supported at a point by fulcrum or prop, which is the centre of motion. The lever is supposed to be void of gravity or weight, to render the demonstrations easier and simpler. There are three kinds of levers.

166. A Fourth Kind is sometimes added, called the Bended Lever. As a hammer drawing a nail.

W

P

167. In all these instruments the power may be represented by a weight, which is its most natural measure, acting downward: but having its direction changed, when necessary, by means of a fixed pulley.

PROPOSITION XXXI.

168. When the Weight and Power keep the Lever in Equilibrio, they are to each other Reciprocally as the Distances of their Lines of Direction from the Prop. That is, P: W:: CD: CE; where CD and CE are perpendicular to wo and AO, the Directions of the two Weights, or the Weight and Power w and A.

FOR, draw CF parallel to AO, and CB parallel to wo: Also, join co, which will be the direction of the pressure on the propc; for there cannot be an equilibrium unless the directions of the three forces all meet in, or tend to, the same point, as o. Then, because these three forces keepeachother in equilibrio, they are proportional to the sides of the triangle CBO or eFo, drawn in the direction of those forces; therefore

But, because of the parallels, the two triangles CDF, CBB are equiangular, therefore

Hence, by equality,

B

P: W:: CF: EO or CB.

CD CE :: CF: CB.

P : W :: CD: CE.

That is, each force is reciprocally proportional to the distance of its direction from the fulcrum.

=

And it will be found that this demonstration will serve for all the other kinds of levers, by drawing the lines as directed. 169. Corol. When the angle A is the angle w, then is CD: CE :: CW: CA :: P: W. Or when the two forces act perpendicularly on the lever, as two weights, &c; then, in case of an equilibrium, D coincides with w, and ɛ with P'; consequently then the above proportion becomes also P: W:: cw: CA, or the distances of the two forces from the fulcrum, taken on the lever, are reciprocally proportional to those forces.

170. Corol.