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Plato's nephew Speusippus was already over 60 when he assumed the headship of the Academy on his uncle's death, and his reign was short (347–339). Aristotle is our principle source for the Old Academy (i.e. prior to 322), which is problematic, as Aristotle tends to be rather tendentious in the way he treats his contemporaries; but apart from Aristotle's evidence, we have very little to go on regarding Speusippus’ philosophy, as only two of his works, On Pythagorean Numbers and the Letter to Philip survive. The most original and influential aspect of Speusippus’ thought is his discussion of the nature of the first principles, in particular, his development of Plato's notions of the One and the indefinite Dyad (which Speusippus called the ‘One and Multiplicity’), and his attempt to explain how the totality of being came to be from these principles. Speusippus was innovative in his understanding of Platonic cosmology, as presented in the Timaeus, although his innovations, e.g. the notion of World‐Soul, did not make their mark until the Neopythagoreanism of the first two centuries A.D. Speusippus’ ideas on happiness and pleasure were already criticized by Plato in the Philebus, but he may have influenced the stoics on this point; while Aristotle, despite his barbed comments, probably learnt a lot from Speusippus’ logic, which included work on genera and species, and on ambiguity as the source of Sophist fallacies.

Xenocrates, who had accompanied Plato on one of his visits to Sicily, became head of the Academy in 339 B.C. Xenocrates stays close to what he takes to be the cosmological doctrine of Plato's Timaeus; indeed Xenocrates’ doctrine may be seen as something of a retreat from Speusippus’ radical position, perhaps in response to Aristotle's criticisms. Dillon reconstructs Xenocrates's cosmological or metaphysical scheme as comprising a pair of first principles, the Monad, or Nous, and the Dyad, or the ‘Everflowing’, to which the Pythagorean tetraktys corresponds as the active counterpart; and a World‐Soul, which receives the forms from the Supreme God's mind, and projects them upon the physical plane. In Logic, Xenocrates remained faithful to Platonic logic, rejecting the Aristotelian categories, although he did argue that the species was prior to the genus; in Ethics, while keen to formalize Plato's teachings, Xenocrates ends up with a position very similar to Aristotle's, in that he emphasizes the needs of the body as well as those of the soul. Xenocrates had a dominant effect on the development of Platonism, because he systematized what he took to be Plato's philosophical system, thus laying the foundation for the ‘Platonic’ system of philosophy; it is Xenocrates’ definition of Form, for instance, which became the standard definition of a Platonic Form.

In this chapter, Dillon considers four minor figures of the Academy: Philippus of Opus, Hermodorus of Syracuse, Heraclides of Pontus, who were all students of Plato; and Crantor of Soli, a contemporary of Polemo. Philippus is best known for editing Plato's Laws for publication, and he was the real author of the Epinomis: as a philosopher, he is distinguished for elevating the rational World‐Soul to the status of the supreme principle, and for identifying astronomy as the true path to the knowledge of God. Hermodorus composed a book on Plato's life and works; philosophically he is interesting for his interpretation of Plato's first principles, and in particular, his denial that matter, or the Unlimited, is a principle. Heraclides, who supervised the Academy during Plato's third trip to Sicily, was nevertheless remarkably free from academic orthodoxy: his most distinctive philosophical position is that the soul is light, or aether, and therefore a quasi‐material substance. Cantor's major contribution to the development of Platonism is the idea of the commentary; Proclus identifies him as the first commentator, because he wrote an exposition, as distinct from an exegesis, of Plato's Timaeus.

This chapter examines how the One shapes different progenies of the One, and the One unfolds its essence into the plural existence of the many. This is the first argument concerning Ge Hong's two eggs theory in the context of various astronomical writings. This cosmogonical answer to the “one and the many” problem basically rejects Plato's doctrine of creation. In particular, Forms cannot be ontological permanents; they can only have relative permanency in the evolving cosmos. Ge Hong's second major argument involves the critical reading of the World Soul. The composite Soul is designed to be a solution to bridge the gap between Being and Becoming. Compared with Ge Hong's method of “two matching talismans,” Plato's natural studies are set within the limits of his idealism.

This chapter completes the Boethian tripartition with a consideration of cosmic music. While the idea of the “music of the spheres” has garnered the most attention, indeed has become a reductive synecdoche of “Pythagoreanism” itself, for the twelfth-century cosmologists, the music of the spheres was the symptom of a more fundamental theory, that of the world soul. Tracing this harmony through both the world’s body (in the form of elemental theory) and the world’s soul (as articulated in Plato’s Timaeus and elaborated by Calcidius and Macrobius, among others), this chapter argues that the “music of the spheres,” along with the aspirational aurality that it entails, is more an epistemic attitude than a cosmic ontology. Hence the attitude it encourages can, and did, survive the gradual eclipse of its materially grounded reality and its absorption into a more generic concept of nature.

Soul is the self-moving, self-constituting entity linking the transcendent with the immanent. Proclus distinguishes many types of souls. This chapter concentrates on the World Soul and the rational human soul. The World Soul infuses the sensible, temporal, divided, and material cosmos with unity and Forms ultimately deriving from the One and Intelligible Being, in a manner reminiscent of human phantasia (imagination). The mathematical psychology of the Timaeus is explained as referring to the Soul’s activity, rather than its essence. Concerning the human soul, with its eternal essence and temporal activity, the authors concentrate on the function and nature of the many ‘vehicles’, assisting the soul’s descent and, in a different manner, its ascent.