History of mathematical notation back to any school or period before that of the Ionian Greeks, but the subsequent history may be divided into periods, the distinctions between which are tolerably well marked. Greek mathematics, which originated with the study of geometry, tended from its commencement to be deductive and scientific. Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of
Proofs of Fermat's theorem on sums of two squares has two involutions: an obvious one formula_213 whose fixed points formula_214 correspond to representations of formula_82 as a sum of two squares, and a more complicated one, which has exactly one fixed point formula_217. Two involutions over the same finite set must have sets of fixed points with the same parity, and since the second involution has an odd number of fixed points, so does the first. Zero is even, so the first involution has a nonzero number of fixed points, any one of which gives a representation of formula_82 as a sum of two squares. This proof, due to