Fundamental Property of the LEVER. 401 



fity. Then by the lemma, this weight has the fame tendency 

 to turn the lever round, as if it were fufpended from the point 

 n, fo taken that b n — dn. Make <pc — <p d, and let the weight 

 BD d b be divided at the points C and F, by the lines Cc, F <p. 

 The weights CF <p c, D F <p d, being in equilibrio, by the axiom, 

 have no tendency to turn the lever round <p, confequently the 

 remaining weight BC c d, has the fame tendency to turn the le- 

 ver round <p as the whole weight BD d b. Hence if b m — c m, 

 the weight B G c b acting at the point m, will have the fame 

 tendency to turn the lever round <p, as the weight BD db acting 

 at n. Now BD d b : BC c b — b d : b c zzn d : mc\ and fince 

 b c ~b d — c d, we have m c — ^b d — -5- c d~nd — ± c d — n <p f 

 and n d — n <p-\- ~ c d zz m c -{- ± c d— ?n <p. Confequently, 



BD d b : BC c b zzm <p : n <p. 



Lemma. 



'two equal forces ailing at the same point of the arm of a lever, and 

 in directions which form equal angles with a perpendicular drawn 

 through that point of the arm, will have equal tendencies to turn 

 the lever round its centre of motion. 



Let AB (fig. 3.) be a lever with equal arms AF, FB. Through 

 the points A, B, draw AD, BE, perpendicular to AB, and AP, 

 Ap, BW, B w, forming equal angles with the lines AD, BE. 

 Produce PA to M. Then, equal forces acting in the directions 

 AP, B w, will be in equilibrio. But a force M equal to P, and 

 acting in the direction AM, will counteract the force P, acting 

 in the direction AB, or will have the fame tendency to turn 

 the lever round F; and the force W, acting in the direction 

 BW, will have the fame tendency to turn the lever round F as. 



the 



