"The question of the truth of the assumptions usually made in our geometry had been considered by J. Saccheri as long ago as 1773; and in more recent times had been discussed by N. I. Lobatschewsky of Kasan, in 1826 and again in 1840; by Gauss, perhaps as early as 1792, certainly in 1831 and in 1846; and by J. Bolyai in 1832 in the appendix to the first volume of his father's Tentamen; but Riemann's memoir of 1854 attracted general attention to the subject... and the theory has been since extended and simplified by various writers, notably A. Cayley... E. Beltrami... by H. L. F. von Helmholtz... by T. S. Tannery... by F. C. Klein... and by A. N. Whitehead... in his Universal Algebra. The subject is so technical that I confine myself to a bare sketch of the argument from which the idea is derived."