Mr. Vince's Observations on the 
34 
fulcrum. Moreover, 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 sup- 
posed, that if a given weight act in any direction, 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 thereof have attempted 
to demonstrate the proposition upon more clear and satisfac- 
tory 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 ex- 
tended it to the case where the arms are incommensurable, his 
demonstration is imperfect. The demonstration given by Dr. 
Hamilton, in his Essays, depends upon this proposition, that 
when a body is at rest, and acted upon by three forces, they 
will be as the three sides of a triangle parallel to the directions 
of the forces. Now this is true, when the three forces act at 
any point of a body; whereas, considering the lever as the 
body, the three forces act at different points, and therefore the 
principle, as applied by the author, is certainly not applicable. 
If in this demonstration we suppose a plane body, in which 
the three forces act, instead of simply a lever, then the three 
forces being actually directed to the same point of the body, 
the body would 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 admitting that all other objec- 
tions could be removed, the demonstration fails when any two 
of the forces are parallel. Another demonstration is founded 
