66 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- 



I I II 



^ _- -g _ g _ ^ 



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 ; aud 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 which 

 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 j 



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 therefore equal to an infinite quantity, which is impos- 

 sible. 



From the foregoing remarks it appears a legitimate conclusion, 

 that 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- 



