AD + BD or AB is equal in weight to P+Q; and the prism AB is uniform; therefore, by Cor. to Lemma 6, AD is equal in weight to P: in like manner BD is equal in weight to Q.

Now since AM is equal to CN, and MC to NB, the sum AC is equal to the sum CB; and the prism AB will balance upon its middle point C. (Prop. 2; Cor. 1.)

But by Prop. 2. if the prism AD be collected at its middle point M, and the prism BD at its middle point N, the effect will be the same as before; therefore, in this case also, the weights will balance upon C'; that is, the weight P at M, and Q at N, will balance upon C.

Therefore if the weights P, Q be inversely as their distances CM, CN, they will balance each other. Q. E. D.

Also the pressure of the prism AB upon the fulcrum C is equal to the weight of the prism, that is, to the weight P+Q. (Prop. 2, Cor. 2.) And by Prop.2, when AD is collected at M and BD at N, the pressure on C is not altered; that is, when P is at M and Q at N, the pressure upon C is P+Q, the sum of the weights. Q.E.D.

* PROP. IV. If two weights acting perpendicularly on a straight lever on opposite sides of the fulcrum balance each other, they are inversely as their distances from the fulcrum.

on

Let P, Q, be two weights which balance each other the lever MCN; then

NC CM:: P: Q.

M

G

Р

If not, let NC: CM: P: Y, and first, let Y be less than Q; so that QY+E. By Prop. 3,

P and Y will balance each other. And therefore when E is added to Y, by Axiom 4, Y+ E or Q will preponderate against P; but by hypothesis, P and Q balance, which is absurd: therefore Y is not less than Q.

Nor is it greater; for if so, let NC: CM:: X: Q, and since NC: CM:: P: Y, we have P: Y:: X: Q; and since Y is greater than Q, P is greater than X: let P = X + D. Then, since NC: CM:: X: Q, by Prop. 3, X and Q will balance each other. And therefore when D is added to X, by Axiom 3, X + D or P will preponderate against Q; but by hypothesis, P and Q balance, which is absurd; therefore Y is not greater than Q.

Therefore we cannot have NC: CM:: P: Y, Y being a quantity less or greater than Q. Therefore, if P and Q balance, NC: CM :: P: Q.

Q. E.D.

* PROP. V. If two forces, acting perpendicularly on a straight lever in opposite directions and on the same side of the fulcrum, are inversely as their distances from the fulcrum, they will balance each other, and the pressure on the fulcrum will be equal to the difference of the forces.

R

Let MCN be the lever on which two forces P, Q balance each other, the fulcrum P being at C. Let MNC be supposed to be a lever on which two forces P, R, acting perpendicularly at M, C, on opposite sides of

M

the fulcrum N, balance each other. Then by Prop. 4, R: P: MN: NC; and therefore R + P: P :: MN+ NC NC, that is, R+ P: PMC : NC. But (by Prop. 3) the pressure upon the fulcrum Nis equal to R + P, and is in a direction opposite to the

forces P and R. If therefore a force Q equal to R + P, act perpendicularly to the lever MC at N in the direction opposite to P and R, it will supply the place of the fulcrum, and the forces will still balance each other, by Ax. 12. But if we place an immoveable fulcrum at C, it will supply the place of the force R, and the forces P, Q will still balance each other, by Ax. 11. That is, if Q: P: MC: NC, the forces P, Q will balance each other on the lever MCN, of which the fulcrum is at C.

Also the pressure on the fulcrum C is equal to the force R, which is the difference of P + R and P, that is, of Q and P. Q. E. D.

COR. In nearly the same manner, by means of Prop. 4, we may prove the converse proposition; that if two forces, acting perpendicularly on a straight lever, in opposite directions and on the same side of the fulcrum, balance each other, they are inversely as their distances from the fulcrum.

* PROP. VI. To explain the different kinds of levers.

When material levers are used, the two forces which have been spoken of, as balancing each other upon the lever, are exemplified by the weight to be raised or the resistance to be overcome, as the one force, and the pressure, weight, or force of any kind, employed for the purpose, as the other force. The former of these forces is called the Weight, the latter is called the Power.

The preceding Propositions give the proportion of the Power and Weight in the case of equilibrium, that is, when the weight is not raised, but only supported; or when the resistance is not overcome, but

only neutralized. But knowing the Power which will produce equilibrium with the Weight, we know that any additional force will make the Power preponderate. (Ax. 3.)

Straight levers are divided into three kinds, according to the position of the Power and Weight.

1. The Lever of the First kind is that in which the Power and Weight are on opposite sides of the Fulcrum, as in Propositions 3 and 4.

We have an example of a lever of this kind, when a bar is used to raise a heavy stone by pressing down one end of the bar with the hand, so as to raise the stone with the other end: the Power is the force of the hand, the Fulcrum is the obstacle on which the bar rests, the Weight is the weight of the stone.

We have an example of a double lever of this kind in a pair of pincers used for holding or cutting; the Power is the force of the hand or hands at the handle, the Weight is the resistance overcome by the pinching edges of the instrument, the Fulcrum is the pin on which the two pieces of the instrument move.

2. The Lever of the Second kind is that in which the Power and the Weight are on the same side of the Fulcrum, the Weight being the nearer to the Fulcrum.

We have an example of a lever of this kind, when a bar is used to raise a heavy stone by raising one end of the bar with the hand, while the other end rests on the ground, and the stone is raised by an intermediate part of the bar. The Fulcrum is the ground, the Power is the force exerted by the hand, the Weight is the weight of the stone.

We have an example of a double lever of this kind in a pair of nutcrackers. The Power is the force of the hand exerted at the handles; the Weight

is the force with which the nut resists crushing; the Fulcrum is the pin which connects the two pieces of the instrument.

3. The Lever of the Third kind is that in which the Power and the Weight are on the same side of the fulcrum, and the Weight is the further from the fulcrum.

In this kind of lever, the Power must be greater than the weight in order to produce equilibrium, by Prop. 5. Therefore by the use of such a lever, force is lost. The advantage gained by the lever is, that the force exerted produces its effect at an increased distance from the fulcrum.

We have an example of a lever of this kind in the anatomy of the fore-arm of a man, when he raises a load with it, turning at the elbow. The elbow is the Fulcrum, the Power is the force of the muscle which, coming from the upper arm is inserted into the forearm near the elbow, the Weight is the load raised.

We have an example of a double lever of this kind in a pair of tongs used to hold a coal. The Fulcrum is the pin on which the two parts of the instrument turn, the Power is the force of the fingers, the Weight is the pressure exerted by the coal upon the ends of the tongs.

* PROP. VII. If two forces acting perpendicularly at the extremities of the straight arms of a bent lever are inversely as the arms, they will balance each other.

Let MCN be any lever: let P, Q act perpendicularly on the arms CM, CN, and let P: Q:: CN : CM; P and Q will balance.

Produce NC to O, taking CO equal to CM; and at Olet a force R equal to P act perpendicularly on the