350 PHILOSOPHICAL TRANSACTIONS. [aNNO 1794. 



whether they admit of any relation, we are totally unacquainted with. Besides, 

 they act under very different circumstances ; for in the former case, the bodies 

 acted immediately on each other, and in the latter, they act by means of a lever, 

 the properties of which we are supposed to be ignorant of. When forces act on 

 a body, considered as a point, or directly against the same point of any body, 

 we only estimate the effect of these forces to move the body out of its place, 

 and no rotatory motion is either generated, or any causes to produce it, considered 

 in the investigation. When we therefore apply the same proposition to investi- 

 gate the effect of forces to generate a rotatory motion, we manifestly apply it to 

 a case which is not contained in it, nor to which there is a single principle in 

 the proposition applicable. The demonstration given by Mr. Landen, in his 

 Memoirs, is founded on self-evident principles, nor do I see any objections to 

 his reasoning on them. But as his investigation consists of several cases, and is 

 besides very long and tedious, something more simple is still much to be wished 

 for, proper to be introfluced in an elementary treatise of mechanics, so as not 

 to perplex the young student either by tiie length of the demonstration, or want 

 of evidence in its principles. What I here propose to offer will, I hope, render 

 the whole business not only very simple, but also perfectly satisfactory. 



The demonstration given by Archimedes, would be very satisfactory and ele- 

 gant, 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 effects will 

 be the same, so far as respects any progressive motion being communicated to 

 the lever when af 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 dis- 

 tances from the fulcrum to produce such a motion. Besides, the 2 motions are 

 not only different, but the same forces are known to produce different effects 

 in the 1 cases ; for in the former case the 1 equal powers at the extremities of 

 the arms produce equal efl^ects in generating a progressive motion ; but in the 

 latter case they do not produce equal effects in generating a rotatory motion. 

 We cannot therefore reason from one to the other. The principle however may 

 be thus proved. 



Let AC, be 2 equal bodies placed on a straight lever, ap, moveable about p ; 

 bisect AC in b, produce pa to q, and take Ba := pb, and sup- -^ ,, 



pose the end a to be sustained by a prop. TLhen as a and c - o Q o a - 

 are similarly situated in respect to each end of the lever, that 

 is, AP = ca, and Aa = cp, the prop and fulcrum must bear equal parts of the 

 whole weight, and therefore the prop at a will be pressed with a weight equal to 



