VOL. LXXXIV.] PHILOSOPHICAL TRANSACTIONS. 349 



supposes, that if 2 equal bodies be placed on a lever, their effect to turn it about 

 any point is the same as if they were placed in the middle point between them. 

 This proposition is by no means self-evident, and therefore the investigation 

 which is founded on it has been rejected as imperfect. Huygens observes, that 

 some mathematicians, not satisfied with the principle here taken for granted, 

 have, by altering the form of the demonstration, endeavoured to render its de- 

 fects less sensible, but without success. He then attempts a demonstration of 

 his own, in which he takes for granted, that if the same weight be removed to 

 a greater distance from the fulcrum, the effect to turn about the lever will be 

 greater : this is a principle by no means to be admitted, when we are supposed 

 to be totally ignorant of the effects of weights on a lever at different distances 

 from the fulcrum. Besides, if it were self-evident, his demonstration only holds 

 when the lengths of the arms are commensurable. Sir I. Newton has given a 

 demonstration,. in which it is supposed, that if a given weight act in any direc- 

 tion, and any radii be drawn from the fulcrum to the line of direction, the effect 

 to turn the lever will be the same on whichever of the radii it acts. But 

 some of the most eminent mathematicians since his time have objected to this 

 principle, as being far from self-evident, and in consequence have attempted to 

 demonstrate the proposition on more clear and satisfactory principles. The 

 demonstration by Mac Laurin, as far as it goes, is certainly very satisfactory ; 

 but as he collects the truth of the proposition only from induction, and has not 

 extended it to the case where the arms are incommensurable, his demonstration 

 is imperfect. The demonstration given by Dr. Hamilton, in his Essays, depends 

 on this proposition, that when a body is at rest, and acted on by 3 forces, they 

 will be as the 3 sides of a triangle parallel to the directions of the forces. Now 

 this is true, when the 3 forces act at any point of a body ; whereas, considering 

 the lever as the body, the 3 forces act at different points, and therefore the prin- 

 ciple, as applied by the author, is certainly not applicable. If in this demonstra- 

 tion we suppose a plane body, in which the 3 forces act, instead of simply a 

 lever, then the 3 forces being actually directed to the same point of the body, 

 the body viould be at rest. But in reasoning from this to the case of the lever, 

 the same difficulties would arise, as in the proof of Sir I. Newton. But ad- 

 mitting that all other objections could be removed, the demonstration fails when 

 any 2 of the forces are parallel. Another demonstration is founded on this prin- 

 ciple, that if 2 non-elastic bodies meet, with equal quantities of motion, they 

 will, after impact, continue at rest ; and hence it is concluded, that if a lever 

 which is in equilibrio be put in motion, the motions of the 2 bodies must be 

 equal ; and therefore the pressures of these bodies on the lever at rest, to put it 

 in motion, must be as their motions. Now in the first place, this is comparing 

 the effects of pressure and motion, the relation of the measures of which, or 



