Walter Burkert was Professor Emeritus of Classics, University of Zurich.

• * Introduction I. Platonic and Pythagorean Number Theory * The Platonic Theory of Ideal Numbers * The Philosophy of the Pythagoreans according to Aristotle * The Later Non-Aristotelian Tradition and Its Sources, Speusippus, Xenocrates, and Heraclides Ponticus * Pythagoreanism in Plato and the Origin in Platonism of the Pythagorean Tradition II. Pythagoras in the Earliest Tradition * Source Problems * Historical Background * Metempsychosis and "Shamanism" Acusmata Acusmatici and Mat hematici * Early Evidence for Pythagoras as a Scientist? III. Philolaus * The Special Position of Philolaus' Book in the Pythagorean Tradition * The Spurious and the Genuine in the Philolaus Fragments * Reflections of Pythagorean Philosophy in the Fifth Century B.C.? IV. Astronomy and Pythagoreanism * The Structure of the World and the Planetary System * The Theory of Planetary Movements * The Cosmos of Philolaus * Harmony of the Spheres and Astral Immortality V. Pythagorean Musical Theory * Speculation, Experimentation, and Fiction * Number Symbolism and Calculation of Proportions in Philolaus VI. Pythagorean Number Theory and Greek Mathematics * Did the Pythagoreans Lay the Foundations of Greek Mathematics? * Pythagorean Arithmetic * Pythagorean Geometry and Mathematical Secrets * Number and Cosmos * Abbreviations * Bibliography * Greek Words Discussed * Index of Passages * General Index

Walter Burkert was Professor Emeritus of Classics, University of Zurich.

• * Introduction I. Platonic and Pythagorean Number Theory * The Platonic Theory of Ideal Numbers * The Philosophy of the Pythagoreans according to Aristotle * The Later Non-Aristotelian Tradition and Its Sources, Speusippus, Xenocrates, and Heraclides Ponticus * Pythagoreanism in Plato and the Origin in Platonism of the Pythagorean Tradition II. Pythagoras in the Earliest Tradition * Source Problems * Historical Background * Metempsychosis and "Shamanism" Acusmata Acusmatici and Mat hematici * Early Evidence for Pythagoras as a Scientist? III. Philolaus * The Special Position of Philolaus' Book in the Pythagorean Tradition * The Spurious and the Genuine in the Philolaus Fragments * Reflections of Pythagorean Philosophy in the Fifth Century B.C.? IV. Astronomy and Pythagoreanism * The Structure of the World and the Planetary System * The Theory of Planetary Movements * The Cosmos of Philolaus * Harmony of the Spheres and Astral Immortality V. Pythagorean Musical Theory * Speculation, Experimentation, and Fiction * Number Symbolism and Calculation of Proportions in Philolaus VI. Pythagorean Number Theory and Greek Mathematics * Did the Pythagoreans Lay the Foundations of Greek Mathematics? * Pythagorean Arithmetic * Pythagorean Geometry and Mathematical Secrets * Number and Cosmos * Abbreviations * Bibliography * Greek Words Discussed * Index of Passages * General Index