Dirk Jan Struik on Pythagoras

“While most sophists emphasized the reality of change — in particular, the Atomists, followers of Leucippus and Democritus — the Pythagoreans stressed the study of the unchangeable elements in nature and society. In their search for the eternal laws of the universe they studied geometry, arithmetic, astronomy, and music (the quadrivium). Their most outstanding leader was Archytas of Tarentum...and to whose school, if we follow... E. [Eva] Frank, much of the Pythagorean brand of mathematics may be ascribed. ...Numbers were divided into classes: odd, even, even-times-even, odd-times-odd, prime and composite, perfect, friendly, triangular, square, pentagonal, etc. ...Of particular importance was the ratio of numbers (logos, Lat. ratio). Equality of ratio formed a proportion. They discriminated between an arithmetical (2b = a + c), geometrical (b^2 = ac), and a harmonical (frac{2}{b} = frac{1}{a} + frac{1}{c}) proportion that they interpreted philosophically and socially.”

“The Pythagoreans knew some properties of regular polygons... how a plane can be filled by... regular triangles, squares, or regular hexagons, and space by cubes... [They] may also have known the regular oktahedron and dodekahedron—the latter figure because pyrite, found in Italy, crystallizes in dodekahedra, and models... date to Etruscan times.”