Alex's Adventures in Numberland - Alex Bellos [164]
The parallel postulate states that for any given line and a point not on that line there is at most one parallel line through that point. Janós’s audacity was to suggest that for any given line and a point not on the line, more than one parallel line passes through that point. Even though it was not at all clear how to visualize a surface for which this statement was true, Janós realized that eometry created by this statement, together with the first four postulates, was still mathematically consistent. It was a revolutionary discovery, and he recognized its momentousness. In 1823 he wrote to his father announcing that ‘Out of nothing I have created a new universe’.
Janós was probably helped by the fact that he was working in isolation from any major mathematical institution, and so was less indoctrinated by traditional views. Even after making his discovery, he opted not to become a mathematician. On graduating, he joined the Austro-Hungarian army, where he was reportedly the best swordsman and dancer among his colleagues. He was also an outstanding musician, and it is said that he once challenged thirteen officers to duels on the condition that upon victory he could play the loser a piece on his violin.
Unbeknownst to Janós, another mathematician in an outpost even more distant from the hubs of European academia than Transylvania was making similar advances independently, but he had his work rejected by the mathematical establishment. In 1826 Nikolai Ivanovich Lobachevsky, a professor at Kazan University in Russia, submitted a paper that disputed the truth of the parallel postulate to the internationally renowned St Petersburg Academy of Sciences. It was turned down, so Lobachevsky then decided to submit it for publication in the local newspaper Kazan Messenger, and consequently no one took any notice.
The greatest irony about the toppling of Euclid’s fifth postulate from the plinth of inviolable truth, however, is that several decades beforehand someone at the very heart of the mathematical establishment had indeed made the same discovery as Janós Bolyai and Nikolai Lobachevsky, yet this man had withheld his results from his peers. Quite why Carl Friedrich Gauss, the greatest mathematician of his day, decided to keep his work on the parallel postulate secret is not understood, although the received view is that he wanted to avoid getting embroiled in a feud about the primacy of Euclid with faculty members.
It was only on reading about Janós’s results, which were published in 1831 as an appendix in a book by his father Farkas, that Gauss revealed to anyone that he had also considered the falsity of the parallel postulate. Gauss wrote a letter to Farkas, an old university classmate, in which he described Janós as a ‘genius of the first order’, yet added that he was unable to praise his breakthrough: ‘For to praise it would be to praise myself. The entire content of the essay…coincide[s] with my own discoveries, some of which date back 30 to 35 years…I had intended to write all this down later so that at least it would not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am especially glad that it is just the son of my old friend who takes precedence to me in this matter.’ Janós was distressed when he learned that Gauss had got there first. And when, years later, Janós learned that Lobachevsky had also preceded him, he became haunted by the ludricrous notion