  In the modern world Pythagoras is most of all famous as a mathematician, because of the theorem named after him, and secondarily as a cosmologist, because of the striking view of a universe ascribed to him in the later tradition, in which the heavenly bodies produce “the music of the spheres” by their movements. It should be clear from the discussion above that, while the early evidence shows that Pythagoras was indeed one of the most famous early Greek thinkers, there is no indication in that evidence that his fame was primarily based on mathematics or cosmology. Neither Plato nor Aristotle treats Pythagoras as having contributed to the development of Presocratic cosmology, although Aristotle in particular discusses the topic in some detail in the first book of the Metaphysics and elsewhere. Aristotle evidently knows of no cosmology of Pythagoras that antedates the cosmological system of the “so-called Pythagoreans,” which he dates to the middle of the fifth century, and which is found in the fragments of Philolaus. There is also no mention of Pythagoras’ work in geometry or of the Pythagorean theorem in the early evidence. Dicaearchus comments that “what he said to his associates no one can say reliably,” but then identifies four doctrines that became well known: 1) that the soul is immortal; 2) that it transmigrates into other kinds of animals; 3) that after certain intervals the things that have happened once happen again, so that nothing is completely new; 4) that all animate beings belong to the same family (Porphyry, VP 19). Thus, for Dicaearchus too, it is not as a mathematician or Presocratic writer on nature that Pythagoras is famous. It might not be too surprising that Plato, Aristotle and Dicaearchus do not mention Pythagoras’ work in mathematics, because they are not primarily dealing with the history of mathematics. On the other hand, Aristotle’s pupil Eudemus did write a history of geometry in the fourth century and what we find in Eudemus is very significant. A substantial part of Eudemus’ overview of the early history of Greek geometry is preserved in the prologue to Proclus’ commentary on Book One of Euclid’s Elements (p. 65, 12 ff.), which was written much later, in the fifth century CE.  At first sight, it appears that Eudemus did assign Pythagoras a significant place in the history of geometry. Eudemus is reported as beginning with Thales and an obscure figure named Mamercus, but the third person mentioned by Proclus in this report is Pythagoras, immediately before Anaxagoras. There is no mention of the Pythagorean theorem, but Pythagoras is said to have transformed the philosophy of geometry into a form of liberal education, to have investigated its theorems in an immaterial and intellectual way and specifically to have discovered the study of irrational magnitudes and the construction of the five regular solids. Unfortunately close examination of the section on Pythagoras in Proclus’ prologue reveals numerous difficulties and shows that it comes not from Eudemus but from Iamblichus with some additions by Proclus himself (Burkert 1972a, 409 ff.). The first clause is taken word for word from Iamblichus’ On Common Mathematical Science (p. 70.1 Festa). Proclus elsewhere quotes long passages from Iamblichus and is doing the same here. As Burkert points out, however, as soon as we recognize that Proclus has inserted a passage from Iamblichus into Eudemus’ history, we must also recognize that Proclus was driven to do so by the lack of any mention of Pythagoras in Eudemus. Even those who want to assign Pythagoras a larger role in early Greek mathematics recognize that most of what Proclus says here cannot go back to Eudemus (Zhmud 2012a, 263–266). Thus, not only is Pythagoras not commonly known as a geometer in the time of Plato and Aristotle, but also the most authoritative history of early Greek geometry assigns him no role in the history of geometry in the overview preserved in Proclus. According to Proclus, Eudemus did report that two propositions, which are later found in Euclid’s Elements, were discoveries of the Pythagoreans (Proclus 379 and 419). Eudemus does not assign the discoveries to any specific Pythagorean, and they are hard to date. The discoveries might be as early as Hippasus in the middle of the fifth century, who is associated with a group of Pythagoreans known as the mathematici, who arose after Pythagoras’ death (see below). The crucial point to note is that Eudemus does not assign these discoveries to Pythagoras himself. The first Pythagorean whom we can confidently identify as an accomplished mathematician is Archytas in the late fifth and the first half of the fourth century.