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what these demonstrations purport to prove, and what they 
do not. 
12. In order to establish the probability of the theory of 
universal gravitation, it must be perfectly obvious to any 
thinking person, that the first thing to be done was to prove 
that a gravitating body could possibly revolve round a 
centre of attraction. Now, there is no attempt whatever to 
prove this mathematically in the Principia .* The theory 
merely rests upon some vague reasoning in the first section, 
under the definition of a centripetal force, founded upon 
the inapplicable illustration of a ball held mechanically by 
a string and swung round ; to which we shall hereafter 
revert. In the first proposition of the second section it is 
simply assumed that gravitating bodies could revolve; and 
the demonstration purports to prove, by a certain mode of 
measuring the areas of a polygonal figure, described by radii 
drawn to a fixed point at intervals, that such bodies will 
describe equal areas in equal times : in other words, the first 
proposition of the Principia purports to demonstrate that 
revolving bodies gravitating to a centre (for that is meant) 
will move in accordance with Kepler^s second law, and describe 
by their radii vectores equal areas in equal times. The two 
“ forces ” employed to produce this motion are a so-called 
centripetal force, intended to represent the constant force of 
gravity, and the innate force ( ff vis insita 33 ) with which a body 
perseveres in its state of - uniform motion in a right line, 
according to the first law of motion. 
13. But in this proposition the “ revolving body 33 is sup- 
posed to move in free space, “ void of resistance,” and the 
areas are described “ in one immovable plane ; 33 and it is to 
these two points I now especially desire to direct attention. 
In the first four corollaries, also, that follow the " demonstra- 
tion,” the same supposition, that the bodies are moving “ in 
spaces void of resistance,” is logically and expressly repeated ; 
and this is necessarily implied in the two additional corollaries. 
But in the last of these it is said — “ 6. The same things hold 
good when the planes in which the bodies are moved, together 
with the centres of force, which are placed in those planes, are 
not at rest, but move uniformly in a right line.” 
14. This is indeed an astounding corollary; and I need 
scarcely say that it is not supported by any attempt at demon- 
stration. Yet what it thus illogically, and, I venture quite 
plainly to say, falsely and absurdly asserts, is coolly introduced 
into the second proposition, which is simply the converse of 
the first with that addition. There is no fresh demonstration 
* Nor elsewhere. Vide Mech. of the Heavens, § 29. 
