398 DEMONSTATION of the 
removed from the fulcrum, has a greater tendency to turn the 
lever round its centre of motion, and i is, befides, applicable on- 
ly toa commenturable proportion of the arms. The founda- ~ 
tion of Sir Isaac NEwron’s demonftration is ftill more inad- 
miffible. He affumes, that if a given power ac in any direc- 
tion upon a lever, and if lines be drawn from the fulcrum te 
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. Dr 1a Hire has 
given a demonftration which is remarkable for its want of ele- 
gance. He employs the reductio ad absurdum, and thus deduces 
the propofition from the cafe where the arms are commenfu- 
rable. The demonftration given by Macraurin has been 
highly praifed ; but if it does not involve a petitio principit, it 
has at leaft the radical defedt, of extending only to a commen- 
furable proportion of the arms. The folutions of LanpEN and 
HamittTon 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 affume only one principle, which has 
been univerfally admitted as axiomatic, namely, that equal and 
opposite forces, acting 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 eftablithed by the three following propofitions. 
In Prop. FE. the property is deduced in a very fimple manner, 
when the arms of the lever are commenfurable. 
In. 
