by Brian Vuyk, student at __Redeemer University College__ [1]
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.
Giralomo Saccheri (1667 – 1733), a Jesuit priest, made the next major attempt to prove that parallel postulate. In his book Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw), Saccheri proposed another method for proving the Euclid’s fifth postulate. He studied certain quadrilaterals whose base angles are right angles and whose base-adjacent sides are congruent. These quadrilaterals have subsequently become known as Saccheri quadrilaterals. He then proposed three cases concerning the two interior angles where the vertical lines met the summit. The first was that the angles would be right, the second was that the angles would be obtuse, and the third was that the angles would be acute. He then attempted to prove by contradiction that cases two and three could not be true, leaving the first case. If he was successful, proof of the first case would lead to a proof of the existence of parallel lines.
Saccheri was able to find a contradiction for the second case. He was not, however, able to find a contradiction for the thirds case, concerning acute angles. Although Saccheri did not realize it, his work on this problem led to the discovery of non-Euclidean Geometry.
Soon after Saccheri, a French mathematician name Alexis Claude Clairut (1713 – 1765) attempted to prove the parallel postulate in a manner similar to that of Proclus. Rather than attempting to prove the parallel postulate using only Euclid’s first four postulates, he instead observed the existence of rectangles in the world around him, and created his own axiom, titled ‘Clairaut’s Axiom’, stating simply, “Rectangles Exists”. Using this axiom, and the four neutral postulates, Clairaut was able to prove the existence of parallel lines. This proof was not accepted by everybody. Legendre, an early nineteenth century mathematician rejected based on the fact that it was not justifiable.
Eventually, by the mid eighteenth century, so many failed attempts had been made to prove Euclid’s fifth postulate that G. S. Klugel was able to submit a doctoral thesis, in which he found the flaws in 28 different attempted proofs of the parallel postulate. The lead to a widespread despair among mathematicians of ever solving the parallel postulate, and many gave up in their attempts.
In the early eighteenth century, two mathematicians countries apart simultaneously began to experiment with a new geometry which did not include the parallel postulate. Janos Bolyai, son of famed mathematician Farkos Bolyai first published his discoveries as an appendix to one of his father’s attempts to prove the fifth postulate, published in the Tentamen. Farkos sent a copy of the Tentamen to his longtime acquaintance, Carl Friedrich Gauss. Gauss, in correspondence with the younger Bolyai, explained that he himself had been making unpublished discoveries in non-Euclidean geometry for many years. Deeply disappointed with Gauss’s reply, J. Bolyai fell into a deep depression, and never again published any results of his work.
Gauss, throughout his lifetime, made many discoveries, although he chose not to publish them until his death in 1855. Before this, in 1829, Russian Nikolai Lobachevsky published a book of results from non-Euclidean geometry, followed by further articles and, in 1840, a treatise, of which he sent a copy to Gauss. Following his book, Lobachevsky experienced much criticism at the hands of the academic community. Many regarded his work as a false invention. Gauss, however, admitted Lobachevsky’s work as “masterful”, and recommended Lobachevsky into the Gottingen Scientific Society.
Gauss’ work was finally published subsequent to his death in 1855. It was not until this work was published did the academic community begin to take non-Euclidean geometry seriously. Many mathematicians extended the work of Gauss, Bolyai and Lobachevsky, leading to non-Euclidean geometry as we know it today, including one variation called ‘Hyperbolic Geometry’ in which there are multiple distinct lines parallel to a given line.
After nearly 2000 years of mathematical frustration and development, the study of the Euclidean parallel postulate has led the study of geometry into the development of a geometry in which the Euclidean parallel postulate is taken to be an unprovable theorem, and even into a variation where there are multiple instances of distinct lines parallel to a singular other.
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