Background

- 2750 B.C., Sacred geometry
- 500 B.C., Thales, Pythagoras
- 380 B.C., Plato
- 330 B.C., Aristotle
- 280 B.C., Euclid (packaged sacred geometry in an Aristotlean form)
- 300, Byzantium
- 800, Baghdad
- 1100, Cordova
- 1200, Palermo
- 1570, Dee (first translation of Euclid into English, but also revived the Pythagorean tradition) and Viete (symbolic algebra)
- 1637, Descartes and Fermat (analytic geometry)
- 1638, Galileo (publishes
*Discourses*on laws of motion) - 1687, Newton (publishes the
*Principia*, including the calculus, universal gravitation, and the equations of motion))

- 1733, Saccheri (proposes alternatives to Post. 5, only hypothetical)
- 1763, Klugel (suggests Post. 5 cannot be proved as a theorem)
- 1766, Lambert (discovers non-euclidean geometry)
- 1781, Robert Simson (publishes outstanding edition of Euclid in English)
- 1794, Gauss (develops non-euclidean geometry, tries to prove the world is Euclidean)

- 1826, Lobachevsky (develops hyperbolic geometry)
- 1827, Gauss (differential geometry, curvature)
- 1829, Lobachevsky (publishes hyperbolic geometry)
- 1831, Bolyai (publishes hyperbolic geometry)
- 1854, Riemann (lecture "On the hypotheses that lie at the foundations of geometry, manifolds, curvature)
- 1856, Weierstrass (began teaching in Berlin)
- 1860, Riemann (studied Dirichlet)
- 1868, Beltrami (proves that Post. 5 is not provable)
- 1872, Klein (Erlangen programm, works on hyperbolic geometry)
- 1873, Hilbert (studied with Weierstrass)
- 1878, Cantor (set theory, Cantor sets, fractals, continuum hypothesis)
- 1882, Poincare (works on hyperbolic geometry)
- 1884, Frege (axioms for natural numbers)
- 1885, Klein and Poincare (work on Riemann program of geometric function theory)
- 1888, Peano (axioms for vector spaces)
- 1889, Poincare (discovers chaotic dynamics)
- 1899, Hilbert (publishes
*Foundations of Geometry*, repairs Euclid, ideas of consistency, independence, completeness) - 1899, Poincare (reviews Hilbert's book [Reid, p. 62])

- 1900, Hilbert (23 problems)
- 1931, Husserl (student of Weierstrass, writes on the foundations of geometry)
- 1931, Godel (proves undecidability and incompleteness, refers to Liar's paradox [Wang, p. 62; Van Heijenoort, p. 89])
- 1945, von Neumann and Ulam (study chaos in 1D maps)
- 1960, Derrida (translates and comments upon Husserl)
- 1975, Mandelbrot (establishes fractal geometry)
- 1961, Ueda (discovers chaotic attractor)
- 1963, Lorenz (discovers chaotic attractor)
- 1993, Grim & Mar (establish chaotic logic, resolve Liar's paradox)

Revised 06 February 1996 by Ralph Abraham <abraham@vmi.vismath.org>