336 Review of the Principia of Newtons 
the areas, may be divested of incongruity. The 40th pro- 
position of the Principia is the connecting link between the 
preceding investigations of rectilinear, and those subsequent, 
of curvilinear motions. The principle to be proved is very 
important in the theory of central forces, viz. that bodies mo- 
ving in a curve, or in a straight line towards the centre of 
force, if their velocities at ary one point be equal, at equal 
distances from the center, their velocities at all other equal 
distances from the center of force will be equal. The 
The corollary from this proposition, that YP» A” will be 
an expression of the velocity, P being the utmost height to 
which a body will rise under any force whose law is n—1 of 
the distance, and A any other altitude, is one of those con- | 
cise and profound inferences, which more than the propositions 
themselves, render our author’s work very difficult of compre- 
hensionto most readers. This result forms a considerable pro- 
blem of his own fluxional, or differential calculus. By this s¢i- 
ence proceeding according to the precepta, or prescribed rules 
of the Principia, the work, it is certain, may be in many cases 
abridged, and 3 Wi 
constant force, or one which produces constant accel- 
ns or retardations of the motion of the body, be sup- 
i act on it, it is evident that these will be in propor- 
