398 DEMONSTRATION of the 



removed from the fulcrum, has a greater tendency to turn the 

 lever round its centre of motion,and is, befides, applicable on- 

 ly to a commenfurable proportion of the arms. The founda- 

 tion of Sir Isaac Newton's demonftration is ft ill more inad- 

 miflible. He afTumes, that if a given power act in any direc- 

 tion upon a lever, and if lines be drawn from the fulcrum t© 

 the line of direction, the mechanical effort of the power will be 

 the fame when it is applied to the extremity of any of thefe 

 lines ; but it is obvious, that this axiom is as difficult to be pro- 

 ved as the property of the lever itfelf. M. De la Hire has 

 given a demonftration which is remarkable for its want of ele- 

 gance. He employs the reduclio ad ahsurdum, and thus deduces 

 the proportion from the cafe where the arms are commenfu- 

 rable. The demonftration given by Maclaurin has been 

 highly praifed ', but if it does not involve a petilio principii, it 

 has at leafl the radical defect, of extending only to a commen- 

 furable proportion of the arms. The folutions of Landen and 

 Hamilton are peculiarly long and complicated, and refemble 

 more the demonftration of fome of the abftrufeft points of 

 mechanics, than of one of its fimpleft and moft elementary 

 truths. 



In attempting to give a new demonftration of the fundamen- 

 tal property of the lever, which fhall be at the fame time fimple 

 and legitimate, we fhall aflume only one principle, which has 

 been univerfally admitted as axiomatic, namely, that equal and 

 opposite forces, acling at the extremities of the equal arms of a lever, 

 and at equal angles to these arms, will be in equilibrio. With the 

 aid of this axiom, the fundamental property of the lever may 

 be eftablifhed bj the three following propofitions. 



In Prop. I. the property is deduced in a very fimple manner, 

 when the arms of the lever are commenfurable. 



In 



