gg Mr. Vince's Observations on the 
offer will, I hope, render the whole business not only very 
simple, but also perfectly satisfactory. 
The demonstration given by Archimedes would be very sa- 
tisfactory and elegant, provided the principle on which it is 
founded could be clearly proved ; viz. that two equal powers 
at the extremities , or their sum at the middle of a lever , would 
have equal effects to move it about any point. Now, that the ef- 
fects will be the same, soffar as respects my progressive motion 
being communicated to the lever when at liberty to move 
freely, is sufficiently clear ; but there is no evidence whatever 
that the effects will be the same to give the lever a rotatory 
motion about any point, because a very different motion is 
then produced, and we are supposed to know nothing about 
the efficacy of a force at different distances from the fulcrum 
to produce such a motion. Besides, the two motions are not 
only different, but the same forces are known to produce dif- 
ferent effects in the two cases ; for in the former case the two 
equal powers at the extremities of the arms produce equal ef- 
fects in generating a progressive motion ; but in the latter case 
they do not produce equal effects in generating a rotatory mo- 
tion. We cannot therefore reason from one to the other. The 
principle, however, may be thus proved. 
Let A C, be two equal bodies placed on a straight lever, 
A P moveable about P ; bisect AC in B, ^ab^c p 
produce PA to Q, and take BO = BP, | ®''° 
and suppose the end Q to be sustained by a 
prop. Then as A and C are similarly situated in respect to 
each end of the lever, that is, A P = C Q, and A Q = C P, 
the prop and fulcrum must bear equal parts of the whole 
weight ; and therefore the prop at Q will be pressed with a 
