10 1. Euclid Saccheri is remembered today, not for his failed attempt to prove Euclid’s fifth pos- tulate, but because in attempting to do so he managed to prove a great many results that we now recognize as theorems in a mysterious new system of geometry that we now call non-Euclidean geometry. Because of the preconceptions built into the cultural context within which he worked, he could only see them as steps along the way to his hoped-for contradiction, which never came. The next participant in our story played a minor role, but a memorable one. In 1795, the Scottish mathematician John Playfair (1748–1819) published an edition of the first six books of Euclid’s Elements [Pla95], which he had edited to correct some of what were then perceived as flaws in the original. One of Playfair’s modifications was to replace Euclid’s fifth postulate with the following alternative postulate. Playfair’s Postulate. Two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another. In other words, given a line and a point not on that line, there can be at most one line through the given point and parallel to the given line. Playfair showed that this alternative postulate leads to the same conclusions as Euclid’s fifth postulate. This postulate has a no- table advantage over Euclid’s fifth postulate, because it focuses attention on the uniqueness of parallel lines, which (as later generations were to learn) is the crux of the issue. Most modern treatments of Euclidean geometry incorporate some version of Playfair’s postulate instead of the fifth postulate originally stated by Euclid. The Discovery of Non-Euclidean Geometry The next event in the history of geometry was the most profound development in mathe- matics since the time of Euclid. In the 1820s, a revolutionary idea occurred independently and more or less simultaneously to three different mathematicians: perhaps the reason the fifth postulate had turned out to be so hard to prove was that there is a completely consistent theory of geometry in which Euclid’s first four postulates are true but the fifth postulate is false. If this speculation turned out to be justified, it would mean that proving the fifth postulate from the other four would be a logical impossibility. In 1829, the Russian mathematician Nikolai Lobachevsky (1792–1856) published a paper laying out the foundations of what we now call non-Euclidean geometry, in which the fifth postulate is assumed to be false, and proving a good number of theorems about it. Meanwhile in Hungary, Janos ´ Bolyai (1802–1860), the young son of an eminent Hungarian mathematician, spent the years 1820–1823 writing a manuscript that accomplished much the same thing his paper was eventually published in 1832 as an appendix to a textbook written by his father. When the great German mathematician Carl Friedrich Gauss (1777– 1855)—a friend of Bolyai’s father—read Bolyai’s paper, he remarked that it duplicated investigations of his own that he had carried out privately but never published. Although Bolyai and Lobachevsky deservedly received the credit for having invented non-Euclidean geometry based on their published works, in view of the creativity and depth of Gauss’s other contributions to mathematics, there is no reason to doubt that Gauss had indeed had the same insight as Lobachevsky and Bolyai. In a sense, the principal contribution of these mathematicians was more a change of attitude than anything else: while Omar Khayyam, Giovanni Saccheri, and others had also

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