A Jesuit logician named Girolamo Saccheri of 17th- and 18th-century Italy used a more sophisticated approach: reductio ad absurdum (disproving a false proposition by logically deducing an absurd consequence). He tried to establish axiom 5 by denying it and seeking a consequent contradiction. In the course of his elaborate chain of deduction he discovered many of the theorems of what is now called hyperbolic geometry, although he had started with the sole purpose of demolishing them. At a certain stage he imagined he had deduced the absurdity he was seeking, and thus he narrowly missed the opportunity for a great achievement. He failed because of his pious belief that Euclid's was the only true geometry.
In 1763 the German mathematician Georg Simon Klügel listed nearly 30 attempts to prove axiom 5 and rightly concluded that the alleged proofs were all unsound. Fifty years later, a new generation of geometers, still working on the same problem, were becoming more and more frustrated. One of them, a Hungarian named Farkas Bolyai, wrote in a letter to his son János Bolyai: "I entreat you, leave the science of parallels alone . . . . I have travelled past all reefs of this infernal Dead Sea and have always come back with a broken mast and torn sail." The son, refusing to heed this warning, continued to think about parallels until, in 1823, he saw the whole truth and declared, in his youthful enthusiasm, "I have created a new universe from nothing!" He understood that, at a certain stage, absolute geometry branches out in two directions depending on whether axiom 5 is asserted or denied. János Bolyai thus recognized two different but equally consistent geometries and published his discovery as a 24-page appendix to a textbook by his father. George Bruce Halsted called it "the most extraordinary two dozen pages in the whole history of thought." In 1832 Farkas Bolyai proudly presented a copy to his friend Carl Friedrich Gauss, then Germany's greatest mathematician, whose reply to the father had a devastating effect on János and led eventually to much criticism of Gauss. The criticism undoubtedly stemmed from the impression that a man of Gauss's stature could have afforded to acknowledge that both the concept and the confidence of its truth are essential to prior claim in any field of intellectual endeavour. The younger Bolyai had conveyed both concept and confidence by choosing the path of publication, while Gauss in his private papers had recorded only his awareness of the concept. For these reasons it seems worthwhile to quote a considerable portion of Gauss's reply (in H.S. Carslaw's translation):
I am unable to praise this work . . . . To praise it would be to praise myself. Indeed, the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations which occupied my mind partly for the last thirty or thirty-five years. So I remained quite stupefied. So far as my own work is concerned . . . my intention was not to let it be published during my lifetime . . . . On the other hand, it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend who takes the precedence of me in such a remarkable manner.
Gauss had indeed been familiar with hyperbolic geometry, even before János was born. He wrote to the elder Bolyai in 1799: "It might well be possible that, however far apart one took the three vertices of a triangle in space, its area was always under a given limit." In a letter to Gauss his pupil Friedrich Ludwig Wachter (1792-1817) remarked that, if axiom 5 is denied, a sphere the radius of which tends to infinity approaches a limiting surface on which certain curves (the geodesics) behave just like the lines of the Euclidean plane. This surface, now known as the horosphere, was destined to play a vital role in the development of the subject.
Non-Euclidean geometry and the theory of relativity have so profoundly affected today's philosophic outlook that it can hardly be imagined how shocking the denial of one of Euclid's postulates must have seemed at the beginning of the 19th century. Such considerations may help to explain why Gauss deliberately renounced his potential claim in favour of János Bolyai and his Russian contemporary Nikolay Ivanovich Lobachevsky, whose first published paper is remarkably like Bolyai's, though quite independent of it.
Lobachevsky made a deeper investigation and wrote several books. Gauss sent him a letter of genuine praise and arranged a corresponding membership for him in the Göttingen Academy. In marked contrast, the unhappy Bolyai received no recognition during his lifetime. In 1848 he read one of Lobachevsky's books (translated into German) and praised it warmly. He was too timid, however, to introduce himself to the prosperous Russian, and there is no evidence that Lobachevsky was aware of his existence. Gauss, who knew them both, was so fully occupied with his own work on other subjects that he never took the trouble to bring them together.
