On the fundamental Property of the Lever. 437 
round its centre of motion, and is, besides, applicable only 
to a commensurable proportion of the arms. The founda- 
tion of Sir Isaac Newton’s demonstration is still more in- 
admissible. He assumes, that if a given power aet in any 
direction tipon a lever, and if lines be-drawn from the ful- 
crum to the line of direction, the mechanical effort of the 
power will be the same when it is applied to the extremity 
of any of these lines; but it is obvious, that this axiom is 
as difficult to be proved as the property of the lever itself. — 
M. De la Hire has given a demonstration which is ree 
markable for its want of elegance. He employs the re= 
ductio ad absurdum, and thus deduces the proposition from 
the case where the arms are commensurable. The demon- 
stration given by Maclaurin has been highly praised; but if 
it does nat involvea petilio principii, it has at least the radical 
defect, of extending only to a commensurable proportion 
of the arms. The solutions of Landen and Hamilton are 
peculiarly long and complicated, and resemble more the 
demonstration of some of the abstrusest points of me- 
chanics, than of one of its simplest aad most elementary 
truths. 
In attempting to give a new demonstration of the fundae 
mental property of the Jever, which shall be at the same 
time simple aad legitimate, we shall assume only- one prine 
ciple, which has been universally admitted as axiomatic, 
namely, éhat equal and. opposite forces, acting at the en- 
tremities of the equal arms of a lever, and at equal angles 
fo these arms, will be in equilibrio. With the aid of this 
axiom, the fundamental property of the lever may be esla- 
blished by the three following propositions. 
In Prop. I. the property is deduced) in a very simple 
manner, when the arms of the lever are commensurable. 
In Prop. I., which is totally independent of the first, 
the demonstration is general, and extends to any proportion 
between the arms, 
Ju Prop. JIL. the property is established, when the forces 
actin an obliqne direction, and when the lever is either 
rectilineal, angular, or curvilineal. In the demonstrations 
which have generally been given of this last proposition, 
the oblique force has been resolved into two, one of which 
is directed to the fulcrum, while the other is perpendicular 
to that direction. It is then assumed, that the force directed 
to the fulcrum has na tendency to disturb the equilibrium, 
even though it acts at the extremity of a bent arm; and 
hence is easy to demonstrate, that the remaining force i 
E e3 pre 
