1. The problem--2. The tradition--3. The Pythagoreans in the fifth century--4. The opposites--5. Pythagorean cosmology--6. Pythagorean number theory--7. Astronomy and harmonia--8. The harmony of the spheres--9. The Pythagorean symbola--10. Transmigration--11. Conclusion--Appendixes: 1. Biography and chronology ; 2. Irrationals and incommensurability