Pythagoras, like many of the greatest characters in history, perhaps never existed: he is a semi-mythical character, who combined mathematics and priestcraft in uncertain proportions. I shall, however, assume that he existed, and that he discovered the theorem attributed to him. He was roughly a contemporary of Confucius and Buddha; he founded a religious sect, which thought it wicked to eat beans, and a school of mathematicians, who took a particular interest in right-angled triangles. The theorem of Pythagoras (the forty-seventh proposition of Euclid) states that the sum of the squares on the two shorter sides of a right-angled triangle is equal to the square on the side opposite the right angle. No proposition in the whole of mathematics has had such a distinguished history. We all learned to “prove” it in youth. It is true that the “proof” proved nothing, and that the only way to prove it is by experiment. It is also the case that the proposition is not _quite_ true—it is only approximately true. But everything in geometry, and subsequently in physics, has been derived from it by successive generalizations. The latest of these generalizations is the general theory of relativity. The theorem of Pythagoras was itself, in all probability, a generalization of an Egyptian rule of thumb. In Egypt, it had been known for ages that a triangle whose sides are 3, 4, and 5 units of length is a right-angled triangle; the Egyptians used this knowledge practically in measuring their fields. Now if the sides of a triangle are 3, 4, and 5 inches, the squares on these sides will contain respectively 9, 16, and 25 square inches; and 9 and 16 added together make 25. Three times three is written “3²”; four times four, “4²”; five times five, “5².” So that we have 3² + 4² = 5². It is supposed that Pythagoras noticed this fact, after he had learned from the Egyptians that a triangle whose sides are 3, 4 and 5 has a right angle. He found that this could be generalized, and so arrived at his famous theorem: In a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides. [Illustration] Similarly in three dimensions: if you take a right-angled solid block, the square on the diagonal (the dotted line in the figure) is equal to the sum of the squares on the three sides. This is as far as the ancients got in this matter. [Illustration]