The fourth dimension

46 THE FOURTH DIMENSION

It is a long step from Gauss’s serenity to the disturbed and passionate life of . Johann Bolyai—he and Galois, the two most interesting figures in the history of mathematics. For Bolyai, the wild soldier, the duellist, fell at odds with the world. It is related of him that he was challenged by thirteen officers of his garrison, a thing not unlikely to happen considering how differently he thought from every one else. He fought them all in successionmaking it his only condition that he should be allowed to play on his violin for an interval between meeting each opponent. He disarmed or wounded all his antagonists. It can be easily imagined that a temperament such as his was one not congenial to his military superiors. He was retired in 1833.

His epoch-making discovery awoke no attention. He seems to have conceived the idea that his father had betrayed him in some inexplicable way by his communications with Gauss, and he challenged the excellent Wolfgang to a duel. He passed his life in poverty, many a time, says his biographer, seeking to snatch himself from dissipation and apply himself again to mathematics. But his efforts had no result. He died January 27th, 1860, fallen out with the world and with himself.

METAGEOMETRY

The theories which are generally connected with the names of Lobatchewsky and Bolyai bear a singular and curious relation to the subject of higher space.

In order to show what this relation is, I must ask the reader to be at the pains to count carefully the sets of points by which I shall estimate the volumes of certain figures.