1 62 SCIENTIFIC METHOD IN PHILOSOPHY 



if it were a sort of separation of consecutives, and as if 

 it were their differentiation ; and that this also is what 

 is first in numbers, for it is the void which differentiates 

 them." 



This seems to imply that they regarded matter as con- 

 sisting of atoms with empty space in between. But if 

 so, they must have thought space could be studied by 

 only paying attention to the atoms, for otherwise it would 

 be hard to account for their arithmetical methods in 

 geometry, or for their statement that " things are 

 numbers." 



The difficulty which beset the Pythagoreans in their 

 attempts to apply numbers arose through their discovery 

 of incommensurables, and this, in turn, arose as follows. 

 Pythagoras, as we all learnt in youth, discovered the 

 proposition that the sum of the squares on the sides of 

 a right-angled triangle is equal to the square on the 

 hypotenuse. It is said that he sacrificed an ox when he 

 discovered this theorem ; if so, the ox was the first 

 martyr to science. But the theorem, though it has 

 remained his chief claim to immortality, was soon found 

 to have a consequence fatal to his whole philosophy. 

 Consider the case of a right-angled triangle whose two 

 sides are equal, such a triangle as is formed by two sides 

 of a square and a diagonal. Here, in virtue of the 

 theorem, the square on the diagonal is double of the 

 square on either of the sides. But Pythagoras or his 

 early followers easily proved that the square of one whole 

 number cannot be double of the square of another. 1 



1 The Pythagorean proof is roughly as follows. If possible, let the 

 ratio of the diagonal to the side of a square be m/n, where m and n are 

 whole numbers having no common factor. Then we must have m 2 = in 2 . 

 Now the square of an odd number is odd, but m 2 , being equal to 2 2 , is 

 even. Hence m must be even. But the square of an even number divides 

 by 4, therefore 2 , which is half of m% must be even. Therefore n must 



