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.[27] Thus the length of the side and the length of the diagonal are incommensurable; that is to say, however small a unit of length you take, if it is contained an exact number of times in the side, it is not contained any exact number of times in the diagonal, and _vice versa_. [27] 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 = 2_n_2. Now the square of an odd number is odd, but _m_2, being equal to 2_n_2, is even. Hence _m_ must be even. But the square of an even number divides by 4, therefore _n_2, which is half of _m_2, must be even. Therefore _n_ must be even. But, since _m_ is even, and _m_ and _n_ have no common factor, _n_ must be odd. Thus _n_ must be both odd and even, which is impossible; and therefore the diagonal and the side cannot have a rational ratio.