On the fundamental Property of the Lever. 439 
and will still form one straight line, the fulcrum f being 
- Joaded with «a weight = 2m, and the fulcrum ¢ with a 
weight = 27. Let us now suppose the extremities B,C, _ 
of the levers to adhere, and form one inflexible line AD; 
and let an inverted fulcrum F be placed at the point of 
junction. The equilibrium of the whole will evidently con- 
tinue, and the fulcra f, ¢, will be loaded as before. Re- 
move the fulcra f,¢, and substitute in their place the 
weights 2m, 2m, acting upwards, and equal to the load 
which they respectively support: the equilibrium will still 
continue. Now, instead of the foree m acting downwards 
at B, substitute an equal and opposite force 2’, acting up- 
wards at A, and instead of the force m acting downwards 
at C, substitute-an equal and opposite force n’, acting up- 
wards at D, and the equilibrium will still be preserved. But 
the two equal forces acting in opposite directions at the 
points A and D, destroy each ether; therefore we have a 
force 2m acting at the extremity of the arm fF, in equili- 
brio with a force-2 7, acting at the extremity of the arm 
gF. But since, by the hypothesis, m: 2 as CD ; AB, and 
since f F is one-half of AB, and ¢ F one-half of CD, we 
have 2m: 2n=9F: fF, an analogy which expresses the 
fundamental property of the lever. 
LEMMA. 
Two equal forces acting at the same point of the arm of a 
lever, and in directions which form equal angles with a per- 
pendiculur 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 directions AP, Bw, will be in equilibrio. But 
a force M, equal to P, and acting in the iemeshot 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; consequently the force W will have the same tendency 
to turn the lever round F as the force w. 
Prop. III.. 
Tf 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 paxpendiculars 
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