We think of Vienna as the city of music, the birthplace of psychoanalysis, the place of great literature and art, but hardly as the hotbed of the greatest scientific advances. And yet, it was here that a young Hungarian military officer developed a geometric theory that would expand our horizons of the world.
In the 1820s, Janos Bolyai (1802-1860) was a dashing young man in Vienna. He had moved from Klausenberg (now Cluj in Romania) to study at the Imperial Military Academy in Wiener-Neustadt.
It should be confessed that it doesn’t do wonders for the ego of most of us to realize that a 21-year old man, who was a virtuoso violin player, spoke 14 languages, including Tibetan, participated in 13 duels and won them all, discovered also the first variant of Non-Euclidean geometry, a theory that was to influence Einstein and change the course of science. This is someone we could safely call, without being laughed at, “a very stable genius.”
For two millennia, there had been efforts to prove Euclid’s Fifth or Parallel Postulate, all of which had failed (at the end of the nineteenth century the Italian mathematician Beltrami proved that the postulate was impossible to prove). Bolyai’s discovery didn’t prove or disprove Euclid, but demonstrated that other geometries were possible and mathematically consistent. He wrote to his father at the time: “I have discovered such wonderful things that I was amazed … I’ve created a whole new universe out of nothing.”
Now, discovering Non-Euclidean geometry is not quite the same thing as working out that the Fifth Avenue is prime real estate or that there is money to be made in buying casinos, etc. Let’s say that it requires at least slightly more intelligence – and not just on the part of the inventor, but also on the part of his audience. Understanding how “hyperbolic plane geometry” works on a “pseudosphere” with “constant negative curvature” is of a rather different order than laughing at “you are fired” delivered with supreme self-confidence.
There is a problem, however, in not being comprehensible to the greatest majority. It might take a while waiting for someone like Einstein to come along to appreciate you. A following of the two or three people in the world who understand your discoveries is hardly enough for “these discoveries (to) bring (you) honour,” as Bolyai had hoped – at least not in your lifetime. Bolyai, however, fared somewhat better than the Russian mathematician Lobachevsky, who at the same time and independently of Bolyai discovered Non-Euclidean geometry. All Lobachevsky achieved was to convince the establishment that he had lost his mind. Bolyai at least could safely enjoy his reputation of a great shot without causing too much annoyance with his “pseudosphere,” “constant curvatures,” and all the rest of it.