# Page:Our knowledge of the external world.djvu/178

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 ${\displaystyle m^{2}=2n^{2}}$. Now the square of an odd number is odd, but ${\displaystyle m^{2}}$, being equal to ${\displaystyle 2n^{2}}$, is even. Hence m must be even. But the square of an even number divides by 4, therefore ${\displaystyle n^{2}}$, which is half of ${\displaystyle m^{2}}$, 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.