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parallel, the alternate angles DCK, Hкс, are equal, and DCK, Cк, are equal, because sc bisects the angle acq: hence HCK HKC, and K = нC.

But, from what has precedeed, P : Q:: HC: HA; and therefore P p or Q: HK: HA.

From B drawing BD parallel to AC, we shall have

PQ AB AC: CD: AC, whence CD: AC:: HK: HA;

T

D

H

a proportion which indicates that the three points, a, K, D, all fall on the diagonal of a parallelogram ABCD. 2. In magnitude. For, with regard to the forces P, Q, repre. sented in magnitude and direction by AB and AF, let T be opposed to those two forces so as to keep the whole system in equilibrio: then it will, of necessity, be equal and opposite to their resultant, R, whose direction is AG. Now, if we sup. pose that the force a is in equilibrio with the two forces P and T (which is consistent with our first hypothe.

Q

P

R

sis) the resultant of these latter will fall in the prolongation of QA, and will be represented by AH = AF. Also, if HD be drawn parallel to AB, and HB be joined, it will be equal and parallel to AG; and we shall have

PT:: AB AD.

Consequently, since AB represents, or measures, the force P, AD will represent or measure the force T; and as that force is in equilibrio with the two forces P and Q, or with their resultant, R, this latter will be represented or measured by AG AD; that is, by the diagonal of the parallelogram

ABGF.

Q. E. D.

30. Corol. 1. If three forces, as A, B, C, acting simultaneously in the same plane, keep one another in equilibrio, they will be respectively proportional to the three sides, DE, EC, CD, of a triangle which are drawn parallel to the di. rections of the forces AD, Db, cd.

For, producing AD, BD, and drawing CF, CE, parallel to them, then the force in CD is equivalent to the two AD, BD, by the supposition; but the force CD is also equivalent to the two ED and CE or FD; therefore, if co represent the force c, then ED will represent its opposite force A, and CE, or FD,

B

E

its opposite force B. Consequently the three forces, A, B, C, are proportional to DE, CE, CD, the three lines parallel to the directions in which they act.

31. Corol. 2. Because the three sides CD, CE, DE, are proportional to the sines of their opposite angles E, D, C ; there. fore the three forces, when in equilibrio, are proportional to the sines of the angles of the triangle made of their lines of direction; namely, each force proportional to the sine of the angle made by the direction of the other two.

32. Corol. 3. The three forces, acting against, and keep. ing one another in equilibrio, are also proportional to the sides of any other triangle made by drawing lines either perpendicular to the directions of the forces, or forming any given angle with those directions. For such a triangle is always similar to the former, which is made by drawing lines parallel to the directions; and therefore their sides are in the same proportion to one another.

33. Corol. 4. If any number of forces be kept in equilibrio by their actions against one another; they may be all reduced to two equal and opposite ones.-For, any two of the forces may be reduced to one force acting in the same plane; then this last force and another may likewise be reduced to another force acting in their plane: and so on, till at last they be all reduced to the action of only two opposite forces; which will be equal, as well as opposite, because the whole are in equilibrio by the supposition.

34. Corol. 5. If one of the forces, as c, be a weight, which is sustained by two strings drawing in the directions DA, DB: then the force or tension of the string AD, is to the weight c, or tension of the string DC, as DE to DC; and the force or tension of the other string BD, is to the weight c, or tension of CD, as ce

to CD.

A

B

E

35: Corol. 6. Since in any triangle CDE we have, by the principles of trigonometry,

DC2

=

DE+EC 2DE. EC COS. DEC,

it follows, that if r, f, be two forces that act simultaneously in directions, which make an angle A, then we may find the magnitude of the resultant, R, by the equation

R = √(F2+f2rf cos. A).

36. Remark. The properties, in this proposition and its corollaries, hold true of all similar forces whatever, whether they be instantaneous or continual, or whether they act by

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 parallelo. piped, 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 concur. ring 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.

PB;

From the point A draw Ab parallel and equal to 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.

-C

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. respectively parallel and perpendicular to the last force PD; we have do=ra + bổ + cc = A sin. APD + B sin. BPD†c sin. CPD. Pd=ra+aß+By+yd=a cos.APD+B COS. BPD+CCOS.CPD+D Pd = √(Pd+do) Pô sec. drd. The numerical computation is best effected by means of a table of natural sines, &c.

tan dpd=

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+

39. Remark. Connected with this subject is the doctrine of moments; for an elucidation of which, however, the student 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 nify the body to be moved, and the body that moves it; or 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 me chanical 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.

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50. In all these instruments the power may be repres sented 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 CF 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 CFO, 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,

E

B

W

D

P: W : CF FO or CB,

CD CE CF CB.

PW:: 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

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