37 
fundamental Property of the Lever. 
weight equal to A. Now take away the weights A and C, 
and put a weight at B equal to their sum ; and then the weight 
at B being equally distant from Q and P, the prop and fulcrum 
must sustain equal parts of the whole weight, and therefore 
the prop will now also sustain a weight equal to A. Hence if 
the prop Q be taken away, the moving force to turn the lever 
about P in both cases must evidently be the same ; therefore 
the effects of A and C upon the lever to turn it about any 
point are the same as when they are both placed in the middle 
point between them. And the same is manifestly tfue if A 
and C be placed without the fulcrum and prop. If therefore 
A C be a cylindrical lever of uniform density, its effect to turn 
itself about any point will be the same as if the whole were 
collected into the middle point B ; which follows from what 
has been already proved, by conceiving the whole cylinder to 
be divided into an infinite number of laminae perpendicular to 
its axis, of equal thicknesses. 
The principle therefore assumed by Archimedes is thus 
established upon the most self-evident principle, that is, that 
equal bodies at equal distances must produce equal effects ; 
which is manifest from this consideration, that when all the 
circumstances in the cause are equal, the effects must be 
equal. Thus the whole demonstration of Archimedes is ren- 
dered perfectly complete, and at the same time it is very short 
and simple. The other part of the demonstration we shall here 
insert, for the use of those who may not be acquainted with it 
B A Z C 
Let X Y be a cylinder, which bisect x— — —■ A ~ - — ■ — y 
in A, on which point it would manifestly rest. Take any 
point Z, and bisect Z X in B, and Z Y in C ; then, from 
