“I am deeply surprised that this task can be spared me, and I am most of all pleased in this that it is the son of my old friend who has in so remarkable a manner preceded me.” The impression which we receive from Gauss’s inexplicable silence towards his old friend is swept away by this letter. Hence we breathe the clear air of the mountain tops. Gauss would not have failed to perceive the vast significance of his thoughts, sure to be all the greater in their effect on future ages from the want of comprehension of the present. Yet there is not a word or a sign in his writing to claim the thought for himself. He published no single line on the subject. By the measure of what he thus silently relinquishes, by such a measure of a world-transforming thought, we can appreciate his greatness. 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 succession—making 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. No mathematical processes beyond this simple one of counting will be necessary. [Illustration: Fig. 19.] Let us suppose we have before us in fig. 19 a plane covered with points at regular intervals, so placed that every four determine a square. Now it is evident that as four points determine a square, so four squares meet in a point. [Illustration: Fig. 20.] Thus, considering a point inside a square as belonging to it, we may say that a point on the corner of a square belongs to it and to three others equally: belongs a quarter of it to each square. [Illustration: Fig. 21.]