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 bu =dn. Make gc = od, and let the weight 
BD dd be divided at the points C and F, by the lines Ce, F 9. 
The weights CF gc, D F 9d, being in equilibrio, by the axiom, 
have no tendency to turn the lever round g, confequently the 
remaining weight BC cd, has the fame tendency to turn the le- 
ver round ¢ as the whole weight BD dd. Hence if bm=c m, 
the weight BCcd acting at the point m, will have the fame 
tendency to turn the lever round 9, as the weight BD dd acing 
at m. © Now BDdd:BCch=bd:bc=nd:mc; and fince 
be=bd—cd, we hayveme=ybd—tcod=nd—ycd=n¢, 
and ad=no+icdazmec+icdumg. Confequently, 
BDdb:BCcb=m9o: 79. 
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LEMMA. 
Two equal forces acting 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, Bw, forming equal angles with the lines AD, BE. 
Produce PA to M. Then, equal forces acting in the direGtions 
AP, Bw, will be in equilibrio. But a force M equal to P, and 
acting in the direG@tion AM, will counteract the force P, ating 
an the direGtion AB, or will have the fame tendency to turn 
the lever round F; and the force W, acing in the direction 
BW, will have the fame tendency to turn the lever round F as 
the 
