The Life and Works of J´anosBolyai Darius Ansari 09/03/2014

In 300 BC, Greek mathematician Euclid published his collection of funda- mental definitions, axioms, theorems, postulates, and mathematical proofs in a comprehensive work titled Elements. Among his postulates he included the parallel (fifth) postulate, which, when translated into English, states:

If a straight line falling on two straight lines make the inte- rior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Euclid’s parallel postulate has historically been among the most controver- sial topics in mathematics. For thousands of years, mathematicians have worked to either prove or disprove Euclid’s postulate, until the beginning of the 19th century, when an entirely new branch of geometry emerged. Today, this new geometry that rejects the parallel postulate is aptly referred to as non-Euclidean geometry, and J´anosBolyai (1802–1860) is widely regarded as one of its most prominent co-founders.

Born and raised in the mountains of modern-day Romania, J´anosBolyai was largely isolated from the then-booming mathematical communities of Eng- land, Germany, and France. Fortunately, however, his father Farkas Bolyai was a mathematician and former student of mathematician Carl Friedrich Gauss (1777–1855); and Farkas took it upon himself to educate his son J´anos after Gauss himself refused. In 1818, J´anosenrolled in the Royal Engineering College in Vienna, proving himself a successful sportsman, violinist, and stu- dent. It was in 1820 that the young Bolyai became fascinated with Euclid’s notorious fifth postulate. In fact, J´anosbeame so obsessed with the parallel postulate that on April 4th, 1820, Farkas wrote–concerning the postulate–to his son:

Detest it as lewd intercourse, it can deprive you of all your leisure, your health, your rest, and the whole happiness of your life. Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there.

1 However, in 1825, J´anoseagerly wrote back to his father to demonstrate his discovery of early non-Euclidean geometry. He informed his father that, out of nothing, he had created a strange new universe.

J´anos’work concerned the geometry of hyperbolic surfaces, much unlike the flat surfaces of Euclidean geometry. By around 1830, J´anosBolyai and Russian mathematician Nikolai Lobachevsky (1792–1856)–unaware of ea- chother’s work–separately published treatises on hyperbolic geometry, which is now considered a traditional form of non-Euclidean geometry. Conse- quently, it is often referred to as Bolyai-Lobachevskian geometry, after its two founding authors.

Of course, Bolyai-Lobachevskian geometry adheres to all of Euclid’s postu- lates except for his fifth postulate, which instead reads:

For any infinite straight line L and any point P not on it, there are many other infinitely extending straight lines that pass through P and which do not intersect L.

Modifications to this single postulate sparked numerous implications. For example, it can be resolved that, in hyperbolic geometry, the sum of all angles in a triangle is less than 180◦; in fact, all three angles of an ideal hyperbolic triangle measure 0◦. Additionally, there are no similar triangles in hyperbolic geometry.

Bolyai and Lobachevsky marked the beginning of publication in non-Euclidean geometry, as their writings on hyperbolic geometry laid the groundwork for scientists such as Albert Einstein (1879–1955). Interestingly enough, how- ever, when Farkas Bolyai showed his son’s work to Gauss, Gauss claimed to have developed coinciding work many years before–and there is evidence to support his claim.

J´anosdied of pneumonia at the age of 57. He left behind more than 20,000 pages of mathematical work. Today, the International Bolyai J´anosPrize of Mathematics is awarded every five years to select mathematicians. Al- though no authentic portrait of J´anosBolyai has been recovered, a picture of him may be found on a Hungarian postage stamp. Beyond the physical representations, however, J´anosBolyai will be forever remembered for his tremendous contributions as a geometer.

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