Since the time Euclid included his parallel postulate as a ``self-evident truth'', it has been the subject of controversy, and for two thousand years geometers attempted to prove it. It was not until the nineteenth century that these attempts were shown to be futile through the simultaneous development of non-Euclidean geometry by Bolyai, Lobachevsky, and Gauss. Their work demonstrated that geometrical axiomatic systems exist independent of the physical world.

Euclid's * Elements* was the first attempt at an axiomatized mathematical
theory, with rigorous proofs based on his definitions, postulates and common
notions [15, Book I; in v. 1, pp. 153--155,].
A good illustration of their use is the proof of the Pythagorean Theorem
[15, Book I, Proposition 47; in v. 1, pp. 349--350,],
which of course requires the parallel postulate.

Lobachevsky published his exploration of a non-Euclidean geometry
in his * Geometrical Researches on
the Theory of Parallels*, translated in [2], and his * Pangeometry*
[20, pp. 360--374,]. The first work presents Lobachevsky's development
of the basic theorems of his non-Euclidean geometry and their proofs. The
second, written near the end of his life, is more expository, giving a
condensed presentation of the final development of his ideas. The
consistency, and thus the acceptability, of this non-Euclidean geometry was
made beautifully clear later in the century when Euclidean models for it
were constructed, such as Poincaré's conformal
model in the disk [19, pp. 241--242,] [24, p. 2.3f,]
[7,12,13,23].

These revolutionary ideas were popularized and developed further by Riemann, evolving into differential geometry and forming the mathematical basis for the physical theory of relativity. The shock waves of this revolution also affected the humanities, demolishing Kant's philosophy of space, and raising many fundamental questions in epistemology.

Sun Feb 7 00:21:52 MST 1999