56 Infinite Divisibility of Matter. 
Let us now suppose a body A to be projected from a point A 
in any direction with a given velocity, and another body B pro- 
et 
A B D EG Ss 
jected at the same instant from the point B in the same direction 
but some distance in advance of A and having just half its velo- 
city. It is evident to common sense that the body A will over- 
take the body B at the point C equally distant from B that B is. 
from A. But apply the law of infinite divisibility and we have 
a different result ; for while the body A moves to the point B, the 
body B moves to the point D; and while A moves from B to D, 
B moves from D to E; and while A moves from D to E, B moves 
from E to G ; and so on, halving to infinity, in which case it is 
clear the body A could never overtake the body B though moving 
with double its velocity. The fallacy then, consists in attempt- 
ing to number the terms of an imaginary infinite series whieh 
are of course innumerable ; and yet, because it is a decreasing 
series, these terms have a sum and a termination; viz. in the 
point C ; 
As the basis for an argument it will readily be granted, 
1st. That the sum of an infinite number of magnitudes, how- 
ever small, is a magnitude infinitely great ; : 
2d. If a body of matter or any other magnitude be divided 
and subdivided to any extent whatever, each of the parts thus pro- 
duced is itself a quantity ; that is, it is greater than nothing; and — 
- 3d. That all these parts together, however numerous, exactly 
make up the original magnitude ; or in other words “the whole 
is equal to the sum of all its parts.” 
In the case of an infinite division, as in every other, each part 
a of any finite quantity A, possesses magnitude or it could clearly 
be no part. As the whole is equal to the sum of all its parts, A 
must be equal to an infinite number of its parts a. But it has 
been granted that the sum of an infinite number of magnitudes, 
however small, is a magnitude infinitely great. ‘The finite quan- 
tity A is taaretore equal to an infinite quantity, which is impos- 
sible. 
_ From the foregoing remarks it appears a legitimate conclusion, 
an infinite division of a finite quantity can result in nothing 
short of its entire annihilation ; as in the case of the bodies A and 
B where the distance between them becomes nothing. And fur- 
