Lever. J3 



158. If from A towards B we take Jll = ^JE r and join IH, 

 AIHG will be a parallelogram. But AI, AG, the sides of this 

 parallelogram, represent the magnitudes and directions of the 

 two forces p, q, consequently the diagonal AH represents their 

 resultant; therefore, since AH. thus represents the force exerted 38. 

 against the fulcrum, it may be inferred as a general conclusion 

 that the force exerted against the fulcrum is precisely the resul- 

 tant of the two forces applied to the lever ; and that consequent- 

 ly these two forces act against the fulcrum as if they were im- 

 mediately applied to it according to directions respectively par- 

 allel to those in which they are actually exerted. 144. 



Indeed, this last truth may be rendered evident, by observing 

 that, instead of the force q, may be substituted the two forces 

 AE, AH, the first of which is destroyed by the force p, and the 

 remaining force AH is the single effect to which the two forces 

 p and q are reduced, and by consequence the resultant of these 

 two forces. 



159. By means of the ratios 



q : p : p : : AG : AE : AH, 



above found, we are able to compare the forces q and p, as well 15 

 with each other, as with the force exerted against the fulcrum. 

 But as this ratio is not the most convenient, we proceed to find 

 two others which may be employed for the same purpose. 



(1). According to a principle already established, we have 48. 



AG : AE i AH n sin HAE : sin HAG : sin GAE, 

 or, 



: : sin HAI : sin HAG : sin GAI, 



since the angles HAE, GAE, have the same sines respectively 

 as their supplements HAI, GAI; that is, the forces q, p, g, are Trig. 13. 

 each represented by the sine of the angle comprehended be- 

 tween the directions of the two others. 



(2). It has been shown that with respect to three forces of 

 which one is the resultant of the two others, either two are al- 

 ways to each other reciprocally as the perpendiculars let fall 

 upon their directions from any point taken in the direction of the 

 third. Accordingly, if from any point in AF, as F, for example, 49. 



