Thus we see that Aristotle was not far from the recognition of the four-dimensional existence, both without and within man, and the process of adequately realising the higher dimensional figures to which we shall come subsequently is a simple reduction to practice of his hypothesis of a soul. The next step in the unfolding of the drama of the recognition of the soul as connected with our scientific conception of the world, and, at the same time, the recognition of that higher of which a three-dimensional world presents the superficial appearance, took place many centuries later. If we pass over the intervening time without a word it is because the soul was occupied with the assertion of itself in other ways than that of knowledge. When it took up the task in earnest of knowing this material world in which it found itself, and of directing the course of inanimate nature, from that most objective aim came, reflected back as from a mirror, its knowledge of itself. CHAPTER V THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE LOBATCHEWSKY, BOLYAI, AND GAUSS Before entering on a description of the work of Lobatchewsky and Bolyai it will not be out of place to give a brief account of them, the materials for which are to be found in an article by Franz Schmidt in the forty-second volume of the _Mathematische Annalen_, and in Engel’s edition of Lobatchewsky. Lobatchewsky was a man of the most complete and wonderful talents. As a youth he was full of vivacity, carrying his exuberance so far as to fall into serious trouble for hazing a professor, and other freaks. Saved by the good offices of the mathematician Bartels, who appreciated his ability, he managed to restrain himself within the bounds of prudence. Appointed professor at his own University, Kasan, he entered on his duties under the regime of a pietistic reactionary, who surrounded himself with sycophants and hypocrites. Esteeming probably the interests of his pupils as higher than any attempt at a vain resistance, he made himself the tyrant’s right-hand man, doing an incredible amount of teaching and performing the most varied official duties. Amidst all his activities he found time to make important contributions to science. His theory of parallels is most closely connected with his name, but a study of his writings shows that he was a man capable of carrying on mathematics in its main lines of advance, and of a judgment equal to discerning what these lines were. Appointed rector of his University, he died at an advanced age, surrounded by friends, honoured, with the results of his beneficent activity all around him. To him no subject came amiss, from the foundations of geometry to the improvement of the stoves by which the peasants warmed their houses. He was born in 1793. His scientific work was unnoticed till, in 1867, Houel, the French mathematician, drew attention to its importance.