7. If a force act perpendicularly on the straight arm of a bent lever at its extremity, the effect to turn the lever round the fulcrum will be the same, whatever be the angle which the arm makes with the other arm, so long as the length is the same.

B

If a force Q act perpendicularly on CB at its extremity B, C being the fulcrum, and an equal force R act perpendicularly on an equal arm CD, at its extremity, the effect to turn the lever round C in the two cases is equal.

8. When a force acts upon a rigid body it will produce the same effect to urge the body in the line of its own direction, at whatever point of the direction it acts.

9. If a body which is moveable about an axis be acted upon by two equal forces, in two planes perpendicular to the axis, the forces being perpendicular at the extremities of two straight arms of equal length from the axis; the two forces will produce equal effects to turn the body, at whatever points the arms meet the axis.

10. If a string pass freely round a fixed body, so that the direction of the string is altered, any force exerted at one extremity of the string will produce at the other extremity the same effect as if the force had acted directly.

11.

If in a system which is in equilibrium, there be substituted for the force acting at any point, an immoveable fulcrum at that point, the equilibrium will not be disturbed.

12. If in a system which is in equilibrium there be substituted for an immoveable point or fulcrum the force which the fulcrum exerts, the equilibrium will not be disturbed.

13. A perfectly hard and smooth surface, acted on at any point by any force, exerts a reaction which is perpendicular to the surface at that point; and if the surface be supposed to be immoveable, the force will be supported, whatever be its magnitude.

14. A heavy material straight line, prism, or cylinder, of uniform density, may be supposed to be composed of a row of heavy points of equal weight, uniformly distributed along the line.

15. A heavy material plane of uniform density may be supposed to be composed of a collection of parallel straight lines of equal density, uniformly distributed along the plane.

16. A heavy solid body of uniform density may be supposed to be composed of a collection of particles, the weight of each of which is as the portion of the body which it occupies, and which may be considered as heavy points.

POSTULATES.

1. A prism or cylinder of uniform density, and of given length, may be taken, which is equal to any given weight.

2. A force may be taken equal to the excess of a greater given force over a less.

3. A force may be taken in a given ratio to a given force.

PROPOSITION I. If a weight be supported on a horizontal rod by two forces acting vertically at equal distances from the weight, the forces are equal to each other, and their sum is equal to the weight.

Let the two forces P, Q act perpendicularly at the extremities of the equal arms CA, P

CB of the horizontal lever AB; and let them balance each other. The forces P, Q, will be equal;

R

for if not, let one of them, as P, be the less, and by Post. 2 take X, the force which is the excess of Q above P, so that P+X is equal to Q; therefore, by Ax. 2, P+X will balance Q. But since P balances Q, if we add to P the force X it will preponderate, by Ax. 3; which is absurd. Therefore P is not less than Q; and in the same manner it may be shewn that Q is not less than P. Therefore P and Q are equal.

Hence, since P and Q are equal, by Ax. 5, the pressure on the fulcrum C is equal to the sum of the two forces P, Q. Hence, by Ax. 12, if, instead of a fulcrum, there be a force R, acting at C perpendicularly to the lever, and equal to the sum of P and Q, this force will balance the pressure at C, just as the fulcrum does, and there will be an equilibrium; that is, a vertical force or weight R will be supported by two forces P, Q, acting vertically at equal distances CA, CB; and these forces will be equal; and the weight R is equal to the sum of P and Q.

Q. E. D.

* PROP. II. A horizontal prism or cylinder of uniform density will produce the same effect by its weight as if it were collected at its middle point.

Let AB be the prism or cylinder, and C its middle

The half AC of the prism may (by Ax. 14) be supposed to be made up of small equal weights, distributed along the whole of the line AC, as at P,R; and the half BC may in like manner be conceived to be made up of small equal weights distributed along BC; as at Q, S; of which the weight at Q is equal to the weight at P, that at S to that at R, and so on.

Let F be a fulcrum about which the prism AB tends to turn by its weight. In CB, produced if necessary, take CG equal to CF, and suppose a fulcrum placed at G.

Let the weights at P, Q, R, S be denoted by P, Q, R, S.

The two weights P and Q produce upon the fulcrums F and G pressures which together are equal to the sum of the weights P+Q, (Ax. 5,) or to the double of P, since P and Q are equal. But the pressure upon each of these fulcrums is equal, (Ax. 6,) hence the pressure upon each of them is P; therefore the pressure upon the fulcrum G, arising from the two weights P and Q, is P; in like manner the pressure upon the fulcrum G, arising from R and S, is R; and so of the rest and the whole pressure on G, arising from the whole prism AB, is the sum of all the weights P,R &c. from A to C; that is, it is half the weight of the prism.

But if the whole prism be collected in its middle point C, the pressure upon the two fulcrums F and

G will be the whole weight of the prism, and the pressures on the two fulcrums are equal, by Prop. 1. Therefore, in this case also, the pressure on the fulcrum G is equal to half the weight of the prism. Therefore the prism, when collected at its middle point, produces the same pressure on the fulcrum G as it did before.

Therefore, when a uniform prism is collected at its middle point, it produces the same effect by its weight as it did before.

Q. E.D.

COR. 1. A uniform prism or cylinder will balance itself upon its middle point.

COR. 2.

When a prism or cylinder thus balances upon its middle point, the pressure upon the fulcrum is equal to the weight of the prism.

* PROP. III. If two weights, acting perpendicularly on a straight lever on opposite sides 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 their sum.

Let P, Q be the two weights, MCN the lever. Let NC be the less of the two

AMD C

P

B

NC, CM. Take MD and MA each equal to NC, and NB equal to ND. Let there be a uniform prism of the length AB, equal in weight to P+Q (Post. 1). Since MD is equal to CN, adding CD to both, MC is equal to DN. Therefore AD, which is double of MD, is double of CN; and BD which is double of DN, is double of CM. Therefore AD: BD :: CN: CM; and by supposition, CN CM: P: Q; therefore AD : BD :: P: Q, and componendo AD + BD : AD :: P + Q : P.

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