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