 Aristotle’s testimony on the Pythagoreans gives rise to one remaining puzzle, which is relevant to the authenticity of the core of fragments identified by Burkert. Aristotle specifically identifies the Pythagoreanism that he discusses as contemporary with or a little earlier than the atomists (Metaph. 985b23 ff.). This dating fits Philolaus exactly. Aristotle discusses the Pythagoreans in many places and he is clearly not always referring to Philolaus. Nonetheless, the central Pythagorean metaphysical and cosmological system to which Aristotle refers repeatedly, with limiters and unlimiteds as first principles, the emphasis on the role of harmony in the cosmos and the peculiar astronomical system, which makes the earth a planet orbiting around the central fire, corresponds in great detail with the fragments and testimonia preserved in the later tradition as coming from Philolaus’ book (Metaph. 986a8–21, 987a13–17,990a8–10 etc.; for an opposing view see Zhmud 1998). The problem is that Aristotle never explicitly describes the Pythagoreanism which he discusses as derived from Philolaus’ book. He only refers to Philolaus once by name, and that is in relation to an apothegm which has no obvious connection to his general presentation of Pythagoreanism (EE 1225a30 = Fr. 16). Everywhere else Aristotle refers to the system he describes as the work of “the so-called Pythagoreans” or the Italian philosophers. The former expression shows that Aristotle has his doubts about the connections between these philosophers and Pythagoras himself. Aristotle never mentions Pythagoras himself in connection with the system of “the so-called Pythagoreans” and clearly dates that system after the time of Pythagoras. If Aristotle is relying heavily on Philolaus’ book, however, why does he never mention Philolaus by name when describing the Pythagorean system? Some might argue that Aristotle did not know of a book by Philolaus and hence that there was no such book. Two considerations suggest another explanation. First, Aristotle discusses the Pythagorean system in considerable detail and in at least one passage strongly implies that what he is reporting is based on a written text (Metaph. 1091a13). As Burkert notes, that written source cannot have been ascribed to “the so-called Pythagoreans” and must have been assigned to some author. The evidence suggests that Philolaus’ book was the only book by a Pythagorean circulating in the fifth century (see 1.3 below), and, as we have seen, the reports of the contents of that book agree with what Aristotle reports quite closely. There simply is no other plausible candidate for Aristotle’s written source. Second, Philolaus’ book seems to have been available to Aristotle’s pupil Meno (DK 44 A27–8) and hence is likely to have been available to Aristotle as well. The most reasonable conclusion is that Aristotle is relying largely on Philolaus’ book for his description of fifth-century Pythagoreanism but chooses not to refer to him explicitly. Aristotle makes clear that there are several groups of people included under the heading “so-called Pythagoreans,” by explicitly distinguishing those Pythagoreans who posited the table of opposites from the main Pythagorean system which he describes (Metaph. 986a22). For whatever reason, it was the fashion to refer to a group of related figures as Pythagoreans, and Aristotle follows this fashion rather than singling Philolaus out from the group, even though he relies heavily on Philolaus’ book. Primavesi has recently suggested that some of the puzzles about Aristotle’s account of Pythagoreanism can be mitigated if we recognize that he presents a developmental acount by reconstructing a gradual transition from early arithmological speculations to Philolaus’ theory of principles (2014). 