## János Bolyai (1802 - 1860)

János Bolyai was born in Transylvania, which is now part of Romania (although at the time it was part of the Austro-Hungarian empire). As a child he studied mechanics with his father, Farkas Bolyai - another famous mathematician, who also found the time to become a talented violinist!

Bolyai became interested in Euclidean geometry when he was first introduced to it as a teenager. This is the geometry of flat surfaces, unlike say, the geometry on the surface of a sphere, which not unnaturally is called Spherical geometry.

Euclidean geometry starts off with a set of assumptions and basic ideas - called axioms and postulates. These are accepted as true. Using these axioms and postulates, Theorems are progressively discovered and proved.

So what Bolyai did was to try and prove the postulate - that way it would be settled once and for all. The way he went about proving it is complicated, but worth pursuing.

He started off by assuming that the postulate was false - so he put in its place something different. Using this, he set off to find and prove a result that was also false. By so doing, he would establish that the postulate must be itself true.

But an astonishing thing happened. He found nothing wrong. If he assumed that there were no such parallels he got a new geometry, a strange, fascinating geometry.

If he assumed there was more than one such parallel he again got a new geometry - this time, in fact spherical geometry. The other geometry is the geometry of the surface of a Hyperboloid.

By this means Bolyai pushed forward the boundaries of geometry into strange new worlds. These geometries, now called non -Euclidean geometries, came not a moment too soon - Einstein would need them in his theory of Relativity. By the time that Bolyai had completed this work he was only 21.

János went on to study engineering at college, and on leaving joined the army engineering corps. Apparently the best swordsman and dancer in the army, Bolyai was did not drink alcohol or coffee, and he did not smoke.

He was extremely intelligent, and spoke nine languages (in addition to his native tongue), including Tibetan and Chinese!

Bolyai spent three years (1820 - 1823) working on non-Euclidean geometry only to discover that he had been pre-empted by Gauss before he managed to publish it. Although Gauss regarded Bolyai as a young genius, this was a real blow to Bolyai’s confidence.

During his lifetime Bolyai frequently suffered from bouts of periodic illness, and was eventually forced to give up his army career as a result of these debilitating attacks.

He continued to work on mathematics, only publishing a tiny proportion of roughly 15,000 pages of work that were discovered after his death! These are now held in the Bolyai-Teleki Library in Romania.

## Bolyai’s mathematics

Each Theorem when proved, may be used in the proof of a new Theorem. Here is an example of this at work.

Theorem already proved: alternate angles of a pair of parallel lines are equal; corresponding angles of a pair of parallel lines are equal;

Postulates - angles on a straight line sum to 180^\circ; lines may be extended indefinitely; through a given point may be drawn a line parallel to any other line;

THEOREM: The angles of a triangle add up to 180^\circ

PROOF: Take any triangle. Extend a side of the triangle, and draw a line parallel to another side as shown. Mark equal alternate and corresponding angles as shown:

Because of the properties of parallel lines (alternate angles, the angles marked \angle B; corresponding angles, the angles marked \angle A) there are two sets of equal angles as marked. The angles A, B and C now make a straight line and so

\angle A+\angle B+\angle C=180^\circ \mbox{as required. The proof is complete.}

Bolyai, like many others, didn’t consider the fifth postulate obvious - at least not as obvious as the other postulates. The fifth postulate is one of the ones we have just made use of: though a given point may be drawn a line paralell to a given line.

Notice that it says ‘a’ line, not lines - there is just one such parallel. On this simple point much would emerge.