4 o2 D EMONSTRATION of the 



the force M : Confequently the force W will have the fame ten- 

 dency to turn the lever round F as the force w; and this will 

 hold true, whether the arms AF, FB, are ftraight or curvili- 

 neal, provided that they are both of the fame form. 



Prop. III. — If a force acts in different directions at the same point 

 in the arm of a lever, its tendency to turn the lever round its centre 

 of motion, will be proportional to the perpendiculars let fall from 

 that centre on the lines of direction in which the force is applied* 



Let AB, (fig. 4.) be the lever, and let the two equal forces 

 BMj B m, act upon it at the point B, in the direction of the lines 

 BM, B m. Draw BN, B n, refpe&ively equal to BM, B m, and 

 forming the fame angles with the line PB a perpendicular to AB. 

 To BM, B m, BN, B n, produced, draw the perpendiculars AY, Ay, 

 AX, A x. Now, the fide AX — x\Y, and Ax~ A y, on account 

 of the equality of the triangles ABX, ABY ; and if M /, M x, 

 be drawn perpendicular to B a, the triangles ABY, BM /, will 

 be fimilar, and alfo the triangles ABy, B w x : Hence we ob- 

 tain 



AB : AY = BM : B /, and 

 AB : A y — BM : B X 

 Therefore, ex aquo, AY : Ay == B / : B X. 



Complete the parallelograms BM N, B m a n, and B /, B X will 

 be refpeclively one-half of the diagonals Bo, B u. 



Now let two equal forces BM, BN, a 61 in thefe directions 

 upon the lever at B, their joint force will be reprefented by 

 the diagonal B 0, and confequently one of the forces BM will 



be 



