SECTION THIRD.

THE PROPERTIES OF THE MECHANICAL POWERS; THE BALANCE, THE LEVER, THE WHEEL, THE PULLEY, THE SCREW, AND THE WEDGE.

PROPOSITION XVIII. (Fig. 9. Pl. I.)

IF, AT THE ENDS OF A BALANCE, AB, WHICH ARE EQUALLY DISTANT FROM THE CENTRE OF MOTION C, TWO EQUAL WEIGHTS BE SUSPENDED, THEY WILL BE IN EQUILIBRIO.

Here AB, that represents the balance, is supposed to be a right line, in which are the three points A, B, C; now, the weights A, B, cannot act upon one another any otherwise than by means of the balance AB, whose fixed point is C.

Suppose, then, that any force applied at A puts the body A into motion, and by means of the balance, the body B; then, since the brachia (or arm) of the beam CA and CB are equal, the arches

Aa, Bb, described by these bodies, will be equal; consequently, the velocities and quantities of matter of A, B, being equal, their momenta, or motions, will be equal; and, because ACB is a right line, they move in a contrary direction; and, therefore, by Ax. 9, these bodies cannot, of themselves, raise one the other, but must remain in equilibrio.

Cor. Hence, equal forces, A, B, applied at equal distances from the centre of motion C, will have the same effect in turning the balance.

PROP. XIX. (Figures 10. 19. 20. and 12. Plates I. and II.) IN ANY STRAIGHT LEVER, IF THE POWER P BE TO THE WEIGHT WAS THE DISTANCE OF THE WEIGHT FROM THE FULCRUM C TO THE DISTANCE OF THE POWER FROM THE FULCRUM, THE

POWER AND WEIGHT (ACTING PERPENDICULARLY ON LEVER)

WILL BE IN EQUILIBRIO.

A lever is any inflexible beam, staff, or bar, whether of metal or wood, &c., that can any way be applied to move bodies. There are four kinds of levers :

1. A lever of the first kind is, that where the fulcrum is between the weight and the power, (fig. 10. Pl. I.)

2. A lever of the second kind is, where the weight is between the fulcrum and the power, (fig. 19. Pl. II.)

3. The lever of the third kind is, where the power P is between the weight and the fulcrum, (fig. 20. Pl. II.)

4. The fourth kind is the bended lever, (fig. 12. Pl. I.)

CASE I. (Fig. 10. Pl. I.)

In the lever of the first kind WCP, instead of the power P, apply a weight P to act at the end of CP. And let the lever WCP be moved into the position aCb. Then will the arches Wa, Pb be as the radii CW, CP; that is, as the velocities of the weight and power. Whence, since P: W:: CW; CP, therefore P: W:: velocity of W: velocity of P: therefore P x velocity of P = W X velocity of W. Consequently the momenta or motions of P and W are equal. And since they act in contrary directions, therefore by Ax. 9. neither of them can move the other, but they will remain in equilibrio.

CASE II. (Figures 19 and 20. Pl. II.)

The levers of the second and third kind may be reduced to the first, thus; make Cp CP, and instead of the power P, apply a weight equal to it at p. Then by Case I., the weight W and power p will keep one another in equilibrio; and (by Cor. Prop. XVIII.) the weight p and power P will have the same effect in turning the lever about its centre, therefore the power P and weight W will be in equilibrio.

Cor. 1. (Figures 10, 11, 12, 13, 14, 15, 16.) In any sort of lever, whether straight or bended, and whether moveable about a single point C, or an axis AB; or whether the lever be fixed to the axis and both together moveable about two centres A, B; or whatever form the levers have; if AB be a right line, and from the ends P, W, there be drawn lines to the centre C, or perpendiculars to the axis AB; and if the power and weight act perpendicular to these lines, and be always reciprocally as these distances drawn to the centre C or aris AB; then they will be in equilibrio.

Cor. 2. (Figures 17 and 18.) In any sort of lever WCP, and in whatever directions the power and weight act on it; if their quantities be reciprocally as the perpendiculars to their several lines of direction, let fall from the centre of motion, they will be in equilibrio. Or they will be in equilibrio, when the weight multiplied by its distance, and the S. angle of its direction, is equal to the power multiplied by its distance, and S. of its direction. W× WC×S.DWC=P×PC×S.EPC.

For the power and weight will be in equilibrio if they be supposed to act at E and D; and (by Ax. 14.) it is the same thing whether they act at E and D, or at P and W. Also by trigonometry, WCXS.W DC, and PCXS.P=CE.

Cor. 3. Hence universally, if any force be applied to a lever, its effect, in moving the lever, will be as that force multiplied by the distance of its line of direction from the centre of motion. Or the effect is as the force by its distance from the centre, and by the sine of the angle of its direction, PX PCX S.P.

Cor. 4. If two bodies be in equilibrio on the lever, each weight is reciprocally as its distance from the centre.

Cor. 5. In the straight lever when the weight and power are in equilibrio, and act perpendicularly on the lever, or in parallel directions; then of these three the power, weight, and pressure upon the fulcrum, any one of them is as the distance of the other two.

For if CP represent the weight W, then CW will represent the power P. And in fig. 10. (pl. Ï.) C sustains both the weights, and therefore the pressure is WP; and figures 19, 20. (pl. II.) C ́sustains the difference of the weights, and therefore the pressure will be WP.

PROP. XX. (Fig. 21. Pl. II.)

IF SEVERAL WEIGHTS BE SUSPENDED ON A STRAIGHT LEVER AB; AND IF THE SUM OF THE PRODUCTS OF EACH WEIGHT, MULTIPLIED BY ITS DISTANCE FROM THE CENTRE OF MOTION C, ON ONE SIDE, BE EQUAL TO THE SUM OF THE LIKE PRODUCTS ON THE OTHER SIDE; THEN THEY WILL BE IN EQUILIBRIO, AND THE CONTRARY.

For the force of each weight to move the lever is as the weight multiplied by the distance (by Cor. 3. last Prop.); and the sum of the products is as the whole forces; which if they be equal, the forces on both sides are equal, and the lever remains at rest.

PROP. XXI. (Fig. 22. Pl. II.)

IF A BENDED LEVER WCP BE KEPT IN EQUILIBRIO BY TWO POWERS, ACTING IN THE DIRECTIONS PB, WA PERPENDICULAR TO THE ENDS OF THE LEVER CP, CW; AND IF THE LINES OF DIRECTION BE PRODUCED TILL THEY MEET IN A, AND AC BE DRAWN, AND CB PARALLEL TO WA;-I SAY THE POWER P, THE WEIGHT OR POWER W, AND THE FORCE ACTING AGAINST THE FULCRUM C, WILL BE RESPECTIVELY AS AB, BC, AC; AND IN THESE VERY DIRECTIONS.

Draw CB, CF parallel to WA, PA; then the angle WFC = WAP=CBP, and the right angled triangles WCF and BCP are similar; whence CF: CB:: CW: CP:: (by Cor. 2. Prop. XIX.) power P: power W. Now since (by Ax. 14.) it is the same thing to what points of the lines of direction PB, WF, the forces P, W be applied; let us suppose them both to act at the point of intersection A; then since the point A is acted on by two forces which are as CF and CB, or as AB and AF; and both these are equivalent to the single force AC (by Cor. 2. Prop. 7.) Therefore the fulcrum C is acted on by the force AC, and in that direction, by Ax. 11.

Cor. 1. Hence the power P, the weight W, and the pressure the fulcrum C sustains; are respectively as WC, PC, and PW. That is, any one is as the distance of the other two.

For since the angles at P, W are right; CA is the diameter of a circle passing through the points A, P, C, W; therefore the angle WPC WAC ACB, and the angle CWP-CAP; therefore the triangles ABC, WCP are similar; and AB: BC :: WC: CP, and BC: AC :: CP : PW. Therefore, &c.

Cor. 2. In any lever WCP, the lines of direction of the powers PW, WF, and of the pressure on the fulcrum C, all tend to one point A.

For if not, the lever would not remain in equilibrio.

PROP. XXII. (Fig. 23. Pl. II.)

IF AB, CD, BE TWO LEVERS MOVEABLE ABOUT A AND C, AND SOME FORCE ACTS UPON THE END B OF THE LEVER AB, IN A GIVEN DIRECTION BF; WHILST THE LEVER AB ACTS UPON CD AT F: IF BE BE DRAWN PERPENDICULAR TO CB, AND AE PARALLEL TO BF: AND IF THESE LEVERS KEEP ONE ANOTHER IN EQUILIBRIO:-THEN I SAY, THE FORCE IN DIRECTION BF,

FORCE AGAINST DC IN DIRECTION EB, AND THE PRESSURE AGAINST THE CENTRE A, ARE RESPECTIVELY AS AE, BE, AB. For since (by Prop. IX.) the lever AB acts upon BC at B, in the direction EB perpendicular to BC; and the lever CD re-acts in direction BE; and (by Ax. 19.) the point A is acted on in direction BA. Therefore the point B is acted on with three forces; BF the force applied at B, and BE the re-action of the lever CD, and AB the re-action of the centre A; and AE is parallel to BF; therefore (by Prop. VIII.) these forces are as AE, BE, and AB.

Cor. If two forces BF, BE, acting perpendicular to the levers AB, DC, keep these levers in equilibrio. The force BE, force BF, and pressure at A, are respectively as radius, Cos. ABD, and S.ABD.

For then EAB is a right-angled triangle; and these forces are as BE, AE, and AB; that is, as radius, S.ABE, and S.AEB; that is, as radius, Cos. ABD, and S.ABD.

PROP. XXIII. (Fig. 24. Pl. II.)

IF AB, BC BE TWO LEVERS MOVEABLE ABOUT THE CENTRES A AND C; IF THE CIRCLES KBM AND DBE BE DESCRIBED WITH THE RADII BC, BA; AND UPON THE CIRCLE BM AS A BASE, WITH THE GENERATING CIRCLE DBE, THE EPICYCLOID BNE BE DESCRIBED; AND THE LEVER CB AND EPICYCLOID BNE BE JOINED TOGETHER IN THAT VERY POSITION, SO AS TO MAKE BUT ONE CONTINUED LEVER CBNE. AND IF THESE LEVERS CBNE, AND AB MOVE ABOUT THE CENTRES C, A; SO THAT THE END D OF THE LEVER AD BE ALWAYS IN THE CURVE OF THE EPICYCLOID DK; I SAY, THAT TWO EQUAL AND CONTRARY FORCES AT D AND K, ACTING PERPENDICULAR TO THE RADII DA, KC,

WILL ALWAYS KEEP THESE LEVERS IN EQUILIBRIO.

For let the levers AB, CBNE come into the position AD, CKDF; then since the epicycloid KD is described by the point D, whilst the arch DB rolled upon the equal arch BK, therefore the end D of the radius AD hath moved through an arch BD equal to the arch BK, through which K has moved. Therefore the points D, K, have equal velocities in any correspondent places D, K. Therefore equal weights H, I, applied to the circles DB and BMG will have equal quantities of motion; and will therefore keep each other in equilibrio, by Ax. 9.

Cor. 1. Hence if one lever AD move uniformly about the centre A, the other CKD will also move uniformly about its centre C. And the arch BD described by D will be equal to the arch BK described by K.

Cor. 2. It is the same thing whether the lever AB act against the convex or concave side of the lever CKD, provided the end D be always in the curve KD.