^g4' ON THE FUNDAMENTAL PROPERTY OF THE LEVER, 



posite force n', acting upwards at D, and the equilibrium 

 will still be preserved. But the two equal forces acting \u 

 opposite directions at the points A and D destroy each 

 other; therefore we have a force 2 m acting at the extremity 

 of the arm/F, in equilibrio with a force '2 n, acting at the 

 extremity of the arm (p F. But since, by the hypothesis, 

 m:n as CD: AB, and since/'F is one half of AB, and 9 F 

 one half of CD, we liave 2 m : 2 nrrf F : /"F, an analogy 

 which expresses the fundamental property of the lever. 



lemma, Lemma. Two equal forces acting at the same point of the 



arm of the letter., 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. 



Demwistrated, Let A B (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, Bit;, forming equal angles with the 

 lines AD, BE. Produce PA to M. Then, equal forces 

 acting in the directions AP, B tc, will be in eqnilibrio. 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 same tendency to turn the lever round F ; and the 

 force W, acting in the direction BW, will have the same 

 tendency to turn the lever round F as the force M; conse- 

 quently the force W will have the same tendency to turn the 

 lever round F as the force w. 



Prop. III. 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 perpen- 

 diculars let fall from that centre on the lines of direction in 

 which the force is applied. 



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

 forces BM, B 7W, act upon it at the point B, in the direction- 

 of the lines BM, Bw. Draw BN, Bw, respectively equal 

 to BM, Bn?, and forming the same angleis with the line 

 PBw perpendicular to AB. ToBM, Bm, BN, Bw, pro- 

 duced, draw the perpendiculars AY, Ay, AX, Ax. Now, 

 , the side AX — AY, and AxrrAy, 011 account of thoequahty 



of 



