1. Hermann’s invariant may be found in “Extrait d’une lettre de M. Herman à M. Bernoulli, datée de Padoue le 12. Juillet 1710” published in Histoires de L’Academie Royale des Sciences avec les Mémoires de Mathematique et Physique, Paris, 519–523 (1712). [For a brief survey of Hermann’s life and works, see F. Nagel, Historia Mathematica, 18, 36–54 (1991).] For an historical account of the eccentricity invariant see H. Goldstein, Am. J. Phys. 43, 8 (1975) and 44, 11 (1976) which includes P. S. Laplace, Traité de méchanique céleste, Paris, Tome I, Premiere Partie, Livre II, 165ff (1798–1799); W. R. Hamilton, Proc. R. Irish Acad. 3, Appendix No. III, xxxviff (1847).
2. See Ian Stewart, The Problems of Mathematics, Oxford: Oxford University Press, Chaps. 9 and 17, 1987.
3. See Jean Sivardiére, “Laplace vectors for the harmonic oscillator,” Am. J. Phys. 57(6), (1989).
4. A discussion of Newton’s proof may be found in V. I. Arnol’d, Huygens and Barrow, Newton and Hooke, Basel: Birkhauser, 1990.
5. See Florin N. Diacu, “Painlevé’s Conjecture,” The Mathematical Intelligencer, 15, No. 2 (1993).