In the variety of the figures of the sixth-, fifth- and fourth-century Pythagoreans it is possible to perceive partly overlapping categories, but hardly any feature common to all of them. This chapter talks about a "family resemblance". This means that certain Pythagoreans had characteristics in common with some Pythagoreans, but not with others. Thus, Hippasus, Theodorus of Cyrene, Philolaus and Archytas shared an interest in mathematics; Democedes, Alcmaeon and Iccus were engaged in medicine; Alcmaeon, Hippo, Philolaus and Ecphantus wrote on natural philosophy; Milo, Astylus of Croton, Iccus and Dicon of Kaulonia were Olympic victors, whereas Milo, Democedes, Hippasus and Archytas were involved in politics. The Pythagorists of comedy and the real Pythagorizers launched the tradition of the existence (and then the coexistence) within Pythagoreanism of different groups, as a result of which two fictional categories of Pythagoreans appeared, the scientific mathematici and the religious acusmatici.

Archytas, a citizen of Tarentum on the Adriatic coast of southern Italy, and a contemporary of Plato, is a significant figure in the history of ancient Greek science. Carl Huffman offers a restrained account of Archytas so far as his work on music and in geometry is concerned. Diogenes Laertius includes an account of Archytas in the Pythagorean Book 8 of his Lives. The chapter discusses the main substance of Diogenes' biographical account. In commenting on Aristotle's Categories the sixth-century-AD Neoplatonist Simplicius quotes extensively from earlier authorities, included among them Archytas. The "Archytean writings" sounds like an abbreviated reference to the Extracts; and if that work demonstrated similarities between Plato's dialogue and passages from Archytas, the Timaeus' account of the receptacle and the movements to which it is subject would be a natural candidate for treatment.