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with astrology, chemistry with alchemy. Mathematics was associated with a more refined type of error. Mathematical knowledge appeared to be certain, exact, and applicable to the real world; moreover it was obtained by mere thinking, without the need of observation. Consequently, it was thought to supply an ideal, from which every-day empirical knowledge fell short. It was supposed, on the basis of mathematics, that thought is superior to sense, intuition to observation. If the world of sense does not fit mathematics, so much the worse for the world of sense. In various ways, methods of approaching nearer to the mathematician's ideal were sought, and the resulting suggestions were the source of much that was mistaken in metaphysics and theory of knowledge. This form of philosophy begins with Pythagoras.

Pythagoras, as everyone knows, said that 'all things are numbers'. This statement, interpreted in a modern way, is logically nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music, and the connection which he established between music and arithmetic survives in the mathematical terms 'harmonic mean' and 'harmonic progression'. He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares and cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or, as we should more naturally say, shot) required to make the shapes in question. He presumably thought of the world as atomic, and of bodies as built up of molecules composed of atoms arranged in various shapes. In this way he hoped to make arithmetic the fundamental study in physics as in aesthetics.

The greatest discovery of Pythagoras, or of his immediate disciples, was the proposition about right-angled triangles, that the sum of the squares on the sides adjoining the right angle is equal to the square on the remaining side, the hypotenuse. The Egyptians had known that a triangle whose sides are 3, 4, 5 has a right angle, but apparently the Greeks were the first to observe that 32 + 42 = 52, and, acting on this suggestion, to discover a proof of the general proposition.

Unfortunately for Pythagoras, his theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy. In a right-angled isosceles triangle, the square on the hypotenuse is double of the square on either side. Let us suppose each side an inch long; then how long is the hypotenuse? Let us suppose its length is m/n inches. Then m2/n2 = 2. If m and n have a common factor, divide it out, then either m or n must be odd. Now m2 = 2n2, therefore m2 is even, therefore m is even, therefore n is odd. Suppose m = 2p. Then 4 p2 = 2n2, therefore, n2 = 2p2 and therefore n is even, contra hyp. Therefore no fraction m/n will measure the hypotenuse. The above proof is substantially that in Euclid, Book X.7

This argument proved that, whatever unit of length we may adopt, there are lengths which bear no exact numerical relation to the unit, in the sense that there are no two integers m, n, such that m times the length in question is n times the unit. This convinced the Greek mathematicians that geometry must be established independently of arithmetic. There are passages in Plato's dialogues which prove that the independent treatment of geometry was well under way in his day; it is perfected in Euclid. Euclid, in Book II, proves geometrically many things which we should naturally prove by algebra, such as (a + b)2 = a2 + 2ab + b2. It was because of the difficulty about incommensurables that he considered this course necessary. The same applies to his treatment of proportion in Books V and VI. The whole system is logically delightful, and anticipates the rigour of nineteenth-century mathematicians. So long as no adequate arithmetical theory of incommensurables existed, the method of Euclid was the best that was possible in geometry. When Descartes introduced co-ordinate geometry, thereby