## 25.10.08

“all things are numbers.” Pythagoras himself spoke of square numbers and cubic numbers, and we still use these terms, but he also spoke of oblong, triangular, and spherical numbers. He associated numbers with form, relating arithmetic to geometry. His greatest contribution, the proposition about right-angled triangles, sprang from this line of thought: “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 this 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 is 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 4p²=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.” (Bertrand Russell, History of Western Philosophy) This shows how Pythagoras’ formulation immediately led to a new mathematical problem, namely that of incommensurables. At his time the concept of irrational numbers was not known and it is uncertain how Pythagoras dealt with the problem. We may surmise that he was not too concerned about it. His religion, in absence of theological explanations, had found a way to blend the “mystery of the divine” with common-sense rational thought.