percussion, drawing, pushing, pressing, or weighing; and are of the utmost importance in mechanics and the doctrine of forces.

37. If three forces, whose directions concur in one point, are represented by the three contiguous edges of a parallelopiped, their resultant will be represented, both in magnitude and direction, by the diagonal drawn from the point of concourse, to the opposite angle of the parallelopiped.

The demonstration of this is left for the exercise of the student.

38. PROP. To find the resultant of several forces concurring in one point, and acting in one plane.

1st. Graphically.-Let, for example, four forces, A, B, C, D, act upon the point P, in magnitudes and directions represented by the lines pa, pb, pc, pd.

A

B

From the point A draw Ab parallel and equal to PB; from b draw be parallel and equal to PC; from c draw cd parallel and equal to PD; and so on, till all the forces have thus been brought into the construction. Then join pd, which will represent both the magnitude and the direction of the required resultant.

P

a

Ꭰ Ᏸ

γ

This is, in effect, the same thing as finding the resultant of two of the forces A and B; then blending that resultant with a third force c; their resultant with a fourth force D; and so on.

2d By computation. Drawing the lines Aa, Ab', &c. re. spectively parallel and perpendicular to the last force PD; we have

tan drd=

A sin. APD + B sin. BPD+c sin. CPD. Pora+aß+By+7d=a cos.APD+B COS. BPD+CCOS.CPD+D rd = v(8 + đổ) Po sec. drd. The numerical computation is best effected by means of a table of natural sines, &c.

PO

.....

39. Remark. Connected with this subject is the doctrine of moments; for an elucidation of which, however, the stu dent should consult some of the books written expressly on mechanics, as those by Marrat, Gregory, or Poisson.

THE MECHANICAL POWERS, &c.

40. WEIGHT and Power, when opposed to each other, sig. ; or nify the body to be moved, and the body that moves it the patient and agent. The power is the agent, which moves, or endeavours to move, the patient or weight.

41. Machine, or Engine, is any mechanical instrument contrived to move bodies. And it is composed of the mechanical powers.

42. Mechanical powers, are certain simple instruments, commonly employed for raising greater weights, or overcoming greater resistances, than could be effected by the natural strength without them. These are usually accounted six in number, viz. the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw.

43. Centre of Motion, is the fixed point about which a body moves. And the Axis of Motion, is the fixed line about which it moves.

44. Centre of Gravity, is a certain point, on which a body being freely suspended, it will rest in any position.

OF THE LEVER.

45. A LEVER is any inflexible rod, bar, or beam, which serves to raise weights, while it is supported at a point by a fulcrum or prop, which is the centre of motion. The lever is supposed to be void of gravity or weight, to render the demonstrations easier and simpler. There are three kinds

of levers.

1 2 3

A

W

48. A Lever of the Third kind has the power between the weight and the prop. Such as tongs, the bones and muscles of animals, a man rearing a ladder, &c.

P

49. A Fourth kind is some. times added, called the Bended Lever. As a hammer drawing a nail.

C

4

50. In all these instruments the power may be represented by a weight, which is its most natural measure, acting downward; but having its direction changed, when neces sary, by means of a fixed pulley.

51. PROP. When the weight and power keep the lever in equilibrio, they are to each other reciprocally as the distances of their lines of direction from the prop. That is, P: W:: CD: CE; where CD and CE are perpendicular to wo and AO, the directions of the two weights, or the weight and power w and a.

For, draw CF parallel to Ao, and CB parallel to wo: Also, join co, which will be the direction of the pressure on the prop c; for there cannot be an equilibrium unless the directions of the three forces all meet in, or tend to, the same point, as o. Then, because these three forces keep each other in equilibrio, they are proportional to the sides of the triangle CBO or cro, drawn in the direction of those forces; therefore

But, because of the parallels, the
two triangles CDF, CEB are equian.
gular, therefore
Hence, by equality,

A

G

P

B

P: W: CF: Fo or CB.

CD CE :: CF: CB.

P: W: CD: CE.

That is, each force is reciprocally proportional to the distance of its direction from the fulcrum.

Another proof might easily be made out from art. 25, on parallel forces; but it will be found that this demonstration

will serve for all the other kinds of levers, by drawing the lines as directed.

52. Corol. 1. When the angle A is the angle w, then is CD CE CW: CA:: P: w. Or when the two forces act perpendicularly on the lever, as two weights, &c.; then, in case of an equilibrium, D coincides with w, and E with P; consequently then the above proportion becomes also p: w:: cw: CA, or the distances of the two forces from the fulcrum, taken on the lever, are reciprocally proportional to those forces.

53. Corol. 2. If any force r be applied to a lever at a ; its effect on the lever, to turn it about the centre of motion c, is as the length of the lever ca, and the sine of the angle of direction CAE. For the perp. CE is as ca X sin. ▲ a.

54. Corol. 3. Because the product of the extremes is equal to the product of the means, therefore the product of the power into the distance of its direction, is equal to the product of the weight into the distance of its direction.

That is, P X CE = W X CD.

55. Corol. 4. If the lever, with the weight and power fixed to it, be made to move about the centre c; the mo. mentum of the power will be equal to the momentum of the weight; and their velocities will be in reciprocal proportion to each other. For the weight and power will describe circles whose radii are the distances CD, CE; and since the circumferences or spaces described are as the radii, and also as the velocities, therefore the velocities are as the radii cp, CE; and the momenta, which are as the masses and velocities, are as the masses and radii; that is, as P X CE and w X-CD, which are equal by cor. 3.

56. Corol. 5. In a straight lever, kept in equilibrio by a weight and power acting perpendicularly; then, of these three, the power, weight, and pressure on the prop. any one is as the distance of the other two.

prop. will be equal to the sum on the other side, made by multiplying each weight by its distance; namely, (PX AC) + (a X BC) = (R X DC) + (8 X EC).

For, the effect of each weight to turn the level, is as the weight multiplied into its distance; and in the case of an equilibrium, the sums of the effects, or of the products on both sides, are equal. The same would also follow from art. 26. 58. Corol. 7. Because, when B

two weights a and R are in equilibrio, Q R :: CD: CB;

C

therefore, by composition, a +R: Q: BD: CD,

and, QR R:: BD CB.

R

D

That is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other.

SCHOLIUM.

59. On the foregoing principles depends the nature of scales and beams, for weighing all sorts of goods. For, if the weights be equal, then will the distances be equal also, which gives the construc. tion of the common scales, which ought to have these properties:

1st. That the points of suspension of the scales and the centre of motion of the beam, A, B, C, should be in a straight line: 2d, That the arms AB, BC, be of an equal length : 3d, That the centre of gravity be in the centre of motion B, or a little below it: 4th, That they be in equilibrio when empty 5th, That there be as little friction as possible at the centre B. A defect in any of these properties makes the scales either imperfect or false. But it often happens that the one side of the beam is made shorter than the other, and the defect covered by making that scale the heavier, by which means the scales hang in equilibrio when empty but when they are charged with any weights, so as to be still in equilibrio, those weights are not equal; but the deceit will be detected by changing the weights to the contrary sides, for then the equilibrium will be immediately destroyed.

60. To find the true weight of any body by such a false balance :-First weigh the body in one scale, and afterwards weigh it in the other; then the mean proportional between these two weights, will be the true weight required. For, if any body b weigh w pounds or ounces in the scale D, and only w pounds or ounces in the scale E : then we have these VOL. II.

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