We will survey the early history of Greek mathematics through two generational events: this chapter, “The Generation of Archytas,” followed by the next chapter, “The Generation of Archimedes.” This is a substantive claim: Greek cultural life was generally organized by such isolated, generational events. This is the claim of the first section, “The Hypothesis of Generational Events.” Following this general historical statement, the section “What Little We Know” surveys the evidence for Greek mathematics in the first half of the fourth century.