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 formula_196. The area of the fundamental parallelogram (in this case a rectangle) of the lattice is formula_82. The area of the open disk, formula_198, of radius formula_199 centered around the origin is formula_200. Furthermore, formula_198 is convex and symmetrical about the origin. Therefore, by Minkowski's theorem there exists a nonzero vector formula_196 such that formula_203. Both formula_204 and formula_205 so formula_206. Hence formula_82 is the sum of the squares of the components of formula_208. Let formula_209 be prime, let formula_210 denote the natural numbers (with or without zero), and consider the finite set formula_211 of triples of numbers. Then formula_212