438 On the fundamental Properiy of the Lever. 
proportional to the perpendicular drawn from the fulerum 
to the line of direction in which the original force was ap- 
plied. As the principle thus assumed, however, is totally 
inadmissible as an intuitive truth, we have attempted to 
demonstrate the proposition-without its assistance. 
Prop. [. 
Tf one arm of a straight lever is any multiple of the 
other, a force acling at the extremity of the one will be in 
equilibrio with a force acting at the extremity of the other, 
when these forces are reciprocally proportional to the length 
of the arms to which they are applied. 
Let AB (Plate X. fig. 1.) be a lever supported on the 
two fulcra F, f, so that Af= fF = FB. Then, if two 
equal weights C,D, of 1 pound each, be suspended from 
the extremities, A, B, they will be in equilibrio, since they 
act at the end of cqual arms Af, BF; and each of the 
fulera f, F, will support an equal part of the whole weight, 
or! pound. Let the fulcrum :f be now removed, and let a 
weight E, of 1 pound, act upwards at the point f; the 
equilibrium will still cantinue; but the weight E, of i 
pound, acting upwards at.f, is equivalent to a weight G of 
‘I pound, acting downwards at B.. Remove, therefore, the 
‘weight E, and suspend the weight G from B; then, since 
‘the equilibrium is still preserved after these two substitu- 
tions, we have a weight C, of one pound, acting at the ex- 
tremity of the arm AFP, in equilibrio with the weights D and 
G, which together make two pounds, acting at the extre- 
mity of the arm FB. But FA is to FB as 2 is to 1; there- 
fore arr equilibrium takes place, when the weights are reci- 
procally proportional to the arms, in the particular case 
when the arms are as'@ to. 1. By making Ff successively 
double, triple, &c. of FB, it may in like manner be shown, 
that, in these cases, the proposition holds true, 
Prop. IT, 
If two forces, acting at the extremities of the two arms 
of a lever, and at equal angles to the arms, are in equilibrio, 
they will be reciprocally proportional to the lengths of the 
arms lo which they ure applied. es 
Let AB, CD (fig. 2.) be two levers jn contact at AB, 
and forming one straight line ABCD.. Bisect AB in f, and 
CD in ¢, and from the extremities A,B, suspend equal 
weights m,m, and from the extremities C, D, equal weights 
n,n, so thatm:n= CD: AB. If the two levers are now 
supported on the fulcra f, 9, they will both be in ae 
: an 
