angular motion of the prism, or cylinder, round D, because the parts AD, BD, being exactly similar and equal, no reason can be assigned for an angular motion in one direction, which would not be equally valid for an angular motion in the opposite direction, and therefore there will be no angular motion at all.

Since, then, the whole effect of the prism, or cylinder, to produce motion by its weight is exactly counteracted by a single force P, equal to W, acting in the line ECD, that effect is exactly equivalent to a single force W acting at any point in the same line, and therefore at C the middle point (Art. 14).

COR. 1. Hence it follows, that any uniform rod, in a horizontal position, produces the same effect by its weight to turn the rod round any fulcrum in it, as if its whole weight were concentrated in its middle point, the rigidity of the rod being supposed to be still maintained.

COR. 2. Hence also it follows, that any horizontal prism, or cylinder, of uniform density, will balance on its middle point; and the pressure on a fulcrum placed there will be the weight of the prism, or cylinder. And conversely, if such a prism, or cylinder, balance on a point in itself, that point is either its middle point, or in the vertical line which passes through its middle point.

21. PROP. II. 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; and the pressure on the fulcrum is equal to their sum.

Let the two weights P, Q, acting perpendicularly at M A and N on the straight Lever

MCN, whose fulcrum is C,

M

E C N

-B

A

P

Q

balance each other, the Lever being in a horizontal posi

tion. It is required to shew, that P: Q :: CN; CM.

In MN take a point E such, that

ME: MN :: P: P+Q, Euclid, vI. 12,

then ME

MN-ME:: P: P+Q-P,

or ME: NE :: P : Q.

Produce MN both ways to A and B, making MA equal to ME, and NB equal to NE. Then, since

Now suppose AB to be a uniform rod, whose weight is P+Q. Then the weights of AE, and BE, will be P and Q, respectively. And since P and Q, acting at the middle points of the lines AE and BE, balance on C, therefore the portions of the rod, AE and BE, which may be substituted for P and Q by PROP. I., will also balance round C, that is, the whole rod AB balances itself on C, and therefore is its middle point.

But P Q

AE: BE,

:: AB-BE: AB-AE,

:: 2BC-2BN: 2AC-2AM,

:: BC-BN : AC-AM,

:: CN CM.

Also the Pressure on the fulcrum is not altered by substituting the rod for the weights, and therefore

22. Conversely, if two weights, or forces, 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.

To prove this, construct as in the foregoing Proposition, and assume that P: Q :: CN : CM; then from this it will be required to shew, that C is the middle point of the rod AB, and therefore that the rod will balance on C, by COR. 2, PROP. I.

The proof will stand thus,

ME: NE:: P : Q,

:: CN: CM,

:. ME+NE : ME :: CN+CM : CN,
or MN: ME :: MN: CN, :. ME=CN.
Also MN: NE :: MN : CM, :. NE=CM.

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

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

and BC=NB+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.

M

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, balance each other. Then it is to be shewn, that P: Q:: CN: CM; M, and (being the Force which is the nearer to the fulcrum) that the pressure on the fulcrum Q-P.

PV

AR

Ас

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.),

PR: CN: NM;

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

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

Also since Q=P+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 PQ 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 Care 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 Ful- N crum at which the Power acts is less than that at which the Weight acts, as thus

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”, using them.

by

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