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by Brian Vuyk, student at Redeemer University College

The parallel postulate, the fifth and final postulate given by Euclid in his Elements, is one of the most controversial topics in the history of Mathematics. The complexity and length of Euclid’s fifth postulate stands in high contrast to the simplicity and brevity of the preceding four. This has lead to thousands of years of controversy to the point of obsession for mathematicians across the world. Thousands of man-hours have been spent in the attempt to find a method of expressing the parallel postulate in terms of the previous four postulates. This has resulted, however, in an entirely new branch of geometry, termed ‘neutral geometry’, which in entirety may be completely proved using only Euclid’s first four postulates.

Proclus (c. 410 – 485), the early Greek commentator tells how the parallel postulate came under attack nearly immediately. Proclus himself was of the opinion that Euclid’s fifth postulate did not deserve the treatment and assumptions of a postulate, but was rather an unproven theorem. Proclus himself proposed a proof of the parallel postulate, based on the first four postulates. This proof was rejected, however, because Proclus made assumptions based on pictures and sketches he drew without justifying two of his final statements.

The next major contributions to neutral geometry came from a seventeenth century mathematician named John Wallis (1616 – 1703). Wallis chose to approach the problem using a different method than Proclus. Instead of attempting to directly prove the fifth postulate, Wallis instead chose to write a new postulate we call ‘Wallis’ Postulate’ which he believed to be more plausible than Euclid’s fifth postulate. In simplest form, Wallis’ Postulate states the existence of similar triangles. Wallis then used his postulate in conjunction with the four postulates of neutral geometry in order to prove the parallel postulate. This attempt at an explanation has since been discredited, since Wallis’ Postulate can be proved to be logically equivalent to the parallel postulate.