But, by construction, MA = ME, and NB=NE;

.. AC=MA+CM=ME+NE=MN,

and BCNB + CN=NE + ME=MN;

.. AC=BC, and.. the rod AB will balance on C. Then, since the weight of AE=P, and weight of BE-Q, by Prop. I., P and Q will also balance on the Lever MCN.

23. PROP. III. 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; and the pressure on the fulcrum is equal to the difference of the forces.

AR

Let the two Forces P and Q, acting perpendicularly at M and N, on the straight Lever MC, in opposite directions, and on the same side of the fulcrum C, M balance each other. Then it is to be shewn, that P: Q :: CN: CM; M and ( being the Force which is p the nearer to the fulcrum) that the pressure on the fulcrum = Q-P.

ΟΥ ΔΩ

VR

Suppose the fulcrum at C removed, and let its resistance (R) be supplied by a Force equal to R, and acting perpendicularly to the Lever in the same direction as P. The equilibrium will not be disturbed.

Then since P and R are exactly counterbalanced by Q, they must produce a pressure at N equal and opposite to Q. Let be removed, and its place supplied by a fulcrum on the contrary side of the Lever to that on which Q acted, sustaining the pressure (namely Q) produced by P and R. The equilibrium is still maintained; and the case is now that of two Forces acting perpendicularly on opposite sides of the fulcrum, and balancing each other; and therefore (by Prop. II.),

P: R CN: NM;

.. P: P+R: CN: CN+ NM:: CN: CM.

But Q, the pressure on the fulcrum which has been supposed to be placed at N, is equal to P+R, by Prop. II.,

.. P Q CN: CM.

Also since QP+R, .. R= Q-P,

that is, the pressure on the fulcrum = the difference of the forces.

24. From the last two Propositions it appears, that if a straight Lever, which is acted on perpendicularly by two weights, or other Forces, P and Q, respectively applied at the distances CM and CN from the fulcrum C, be at rest, then, whether P and Q act on the same side, or on different sides, of the fulcrum, the proportion P : Q :: CN : CM is always true.

Hence also P×CM=Q×CN (Wood's Algebra, Art. 237) is an equation which expresses the conditions of equilibrium in all such

cases.

There is no impropriety in multiplying a Force by a line, because both are expressed in numbers, when they become subjects of calculation. Thus a force of 3 lbs. acting perpendicularly on a straight lever at a distance of 4 feet from the fulcrum will balance another force of 6 lbs. acting at a distance of 2 feet on the opposite side of the fulcrum and in the same direction, because 3×4=12=6×2.

The product P×CM is sometimes called the moment of P about C; and, similarly, Q-CN is the moment of Q about C. Hence, in the last two Propositions, the moments of P and Q are equal.

Also, since if P : Q :: CN : CM, it is proved that P and Q will balance on C, therefore, conversely, if the moments of P and Q with respect to C are equal, they will balance each other.

When the Lever is used to balance a given Force, Q, by the application of another Force, P, Q is usually called "the Weight", and P "the Power".

If CM, the perpendicular distance from the fulcrum at which the Power acts, be greater than CN, the distance at which the Weight acts, the Power required to balance the Weight is less than the Weight; in this case "force" is said to be "gained" by the application of the Lever. But if CM be less than CN, the Power required to balance the Weight is greater than the Weight, and "force" is then said to be "lost".

25. PROP. IV. To explain the different kinds of LEVERS. LEVERS are divided into three classes, according to the relative position of the points where the Power and the Weight are applied with respect to the Fulcrum.

(1) Where the Power (P) and the M Weight (Q) act on opposite sides of the Fulcrum (C), as thus

(2) Where the Power and the Weight act on the same side of the Fulcrum, but the perpendicular distance from the Fulcrum at which the Power M acts is greater than that at which the Weight acts, as thus

(3) Where the Power and the Weight act on the same side of the Fulcrum, but the perpendicular distance from the Fulcrum at which the Power acts is less than that at which the Weight acts, as thus

N

N

M

N

Of the FIRST class the poker, when used to raise the coals, is an instance; the bar of the grate on which the poker rests being the Fulcrum, the force exerted by the hand the Power, and the resistance of the coals the Weight. In the common Balance, the Power and the Weight are equal Forces perpendicularly applied at the ends of equal arms. In the Steelyard, the Power and the Weight are perpendicularly applied at the ends of unequal arms. Pincers, scissors, and snuffers, are double Levers of this kind, the rivet being the Fulcrum.

Since CM may be either greater or less than CN, the Power in Levers of this class may be either less, or greater, than the Weight, and consequently "Force" may be either "gained", or "lost", by using them.

Of the SECOND class, a cutting blade, such as is used by coopers, moveable round one end, which is fastened by a staple to a block, and worked by means of a handle fixed at the other end, is an example. An oar is also such a Lever; the Fulcrum being the extremity of the blade (which remains fixed, or nearly so, during the stroke), the muscular strength and weight of the rower being the Power, and the Weight being the resistance of the water to the motion of the boat, which is counteracted and overcome at

the rowlock. A pair of nutcrackers also is a double Lever of the second class.

Here, since CM is greater than CN, the Power is always less than the Weight, or Force is "gained" by using Levers of the second class.

An example of the THIRD class is the board which the turner (or knifegrinder) presses with his foot to put the wheel of his lathe in motion; the Fulcrum being the end of the board which rests on the ground, the Power being the pressure of the foot, and the Weight being the pressure produced at the crank put on the axletree of the wheel. Fire-tongs and sugar-tongs are double Levers of this kind; the Weight in either case being the resistance of the substance grasped. The limbs of animals are also such Levers: thus, if a weight be held in the hand and the arm be raised round the elbow as a Fulcrum, the Weight is supported by muscles fastened at one extremity to the upper arm, and again attached to the fore-arm, after passing through a kind of loop at the elbow.

Here, since CM is less than CN, the Power is greater than the Weight, or Force is "lost" by making use of Levers of the third class.

26. PROP. V. If two forces, acting perpendicularly at the extremities of the arms of any Lever, balance each other, they are inversely as the arms.

In this Prop. the arms of the Lever are supposed straight, but joined together at the fulcrum so as not to be in the same straight line.

Let the two forces P and Q, acting perpendicularly at the extremities of the straight arms, CM and CN, of any Lever whose fulcrum is C, balance each other; then

P: Q: CN: CM.

For, suppose the arm NC

M

N

C

produced to M', so that CM CM; and suppose a force P', equal to P, to act perpendicularly at M' on the

L. C. C.

2

=

Lever M'CN; then, since P=P', and CM' CM, by Axiom I., Art. 19, the effort of P to turn the Lever MCN round C is equal to that of P on the Lever M'CN. But, by the supposition, P balances Q on the Lever MCN; therefore also P' balances Q on the straight Lever M'CN. Hence, as before proved in Prop. II,

P: Q: CN: CM'; but P=P', and CM CM',

.. P: Q: CN: CM.

=

COR. Here again, as in the two preceding Propositions,

PxCM=QxCN.

27. PROP. VI. If two forces, acting at any angles on the arms of any Lever, balance each other, they are inversely as the perpendiculars drawn from the fulcrum to the directions in which the forces act.

A

Let P and Q be two forces, which, acting at any angles on the arms CA and CB of any Lever ACB, balance each other about the fulcrum C; and let the perpendiculars CM and CN be drawn from the fulcrum C to the lines in which the forces act; then

M

Př

P: Q CN: CM.

N

For, since a force produces the same effect at whatever point in its line of action it is applied (Art. 14), the force Pmay be supposed to be applied at M; and in order that it may be so applied, let a rod, CMA, supposed without weight, be fastened to CA. In like manner, Q may be supposed to be applied at N perpendicularly to the part CN of the rod CNB which is added to CB.

And, since P acting at M perpendicularly to CM balances acting at N perpendicularly to CN, .., by Prop. V, P: Q CN: CM;

and therefore also, when P and Q, acting at A and B in the lines AP, BQ, balance, P: Q: CN: CM.

• More correctly, "to the lines of action of the forces". See Art. II.