that Pythagoras knew the geometric, arithmetic and harmonic means. In geometry, where
he continued the line of Thale ̄s, the deductive proof of Pythagoras’ theorem is ascribed
to him (empirical formulas for some of the “Pythagorean triplets” – 3, 4, 5, etc. – were
known already in Babylo ̄n). Pythagoras further applied the technique of deductive proof to
arithmetic; to him must go back one of the earliest samples of the theoretical arithmetic –
the theory of even and odd numbers (E, Elem. 9.21–34) using an indirect proof. The
method of deductive proof was further transferred from mathematics to philosophy by
Parmenide ̄s, who had a Pythagorean teacher.
DK 14; RE 24 (1963) 171–203, K. von Fritz; HGP v. 1; Burkert (1972); L. Navia, Pythagoras: An Annotated
Bibliography (1990); Zhmud (1997); C. H. Kahn, Pythagoras and the Pythagoreans: A Brief History (2001).
Leonid Zhmud
PYTHAGORAS OF SAMOS