Huygens and Barrow, Newton and Hooke

Huygens and Barrow, Newton and Hooke.- 1. The law of universal gravitation.- § 1. Newton and Hooke.- § 2. The problem of falling bodies.- § 3. The inverse square law.- § 4. The Principia.- § 5. Attraction of spheres.- § 6. Did Newton prove that orbits are elliptic?.- 2. Mathematical analysis.- § 7. Analysis by means of power series.- § 8. The Newton polygon.- § 9. Barrow.- §10. Taylor series.- §11. Leibniz.- §12. Discussion on the invention of analysis.- 3. From evolvents to quasicrystals.- §13. The evolvents of Huygens.- §14. The wave fronts of Huygens.- §15. Evolvents and the icosahedron.- §16. The icosahedron and quasicrystals.- 4. Celestial mechanics.- §17. Newton after the Principia.- §18. The natural philosophy of Newton.- §19. The triumphs of celestial mechanics.- §20. Laplace’s theorem on stability.- §21. Will the Moon fall to Earth?.- §22. The three body problem.- §23. The Titius-Bode law and the minor planets.- §24. Gaps and resonances.- 5. Kepler’s second law and the topology of Abelian integrals.- §25. Newton’s theorem on the transcendence of integrals.- §26. Local and global algebraicity.- §27. Newton’s theorem on local non-algebraicity.- §28. Analyticity of smooth algebraic curves.- §29. Algebraicity of locally algebraically integrable ovals.- §30. Algebraically non-integrable curves with singularities.- §31. Newton’s proof and modern mathematics.- Appendix 1. Proof that orbits are elliptic.- Appendix 2. Lemma XXVIII of Newton’s Principia.- Notes.