DEVELOPMENT OF MATHEMATICAL THOUGHT. 713 



1830, the author being Lobatchevski ; and in the 

 appendix to an Introduction to Geometry, published 

 by Wolfgang Bolyai at Maros Vasarheli, a town of 

 Transylvania, the appendix being by the author's son, 

 Johann Bolyai. The elder Bolyai having been a 

 friend and correspondent of Gauss, and his speculations 

 evidently of the same nature as those indicated by the 

 latter in the above-mentioned correspondence, conjectures 

 have been made as to which of the two originated the 

 whole train of thought. 1 The independent investiga- 

 tions of Riemann and Helmholtz started from a differ- 



1 See above, p. 652, note. What 

 is important from our point of 

 view in the investigations of both 

 Riemann and Helmholtz lies in the 

 following points : First, Neither 

 Riemann nor Helmholtz refers to 

 the non - Euclidean geometry of 

 Lobatchevski or Bolyai. This is 

 not surprising in the case of 

 Helmholtz, whose interest was 

 originally not purely mathematical ; 

 in fact, we may incidentally re- 

 mark how, in spite of his profound 

 mathematical ability, he on various 

 occasions carne into close contact 

 with mathematical researches of 

 great originality and importance 

 without recognising them e.g., 

 the researches of Grassmann and 

 Pliicker. As regards Riemann, his 

 paper was read before Gauss, who 

 certainly knew all about Bolyai, and 

 latterly also about Lobatchevski, of 

 whom he thought so highly that he 

 proposed him as a foreign member 

 of the Gottingen Society. Gauss 

 could therefore easily have pointed 

 out to Riemann the relations of 

 his speculations with his own and 

 those of the other mathematicians 

 named. Since the publication of 

 the latest volume of Gauss's works, 

 it has become evident that Gauss 



corresponded a good deal, and 

 more than one would have sup- 

 posed from reading Sartorius's 

 obituary memoir, on the subject 

 of non-Euclidean (astral or imag- 

 inary) geometry, notably with 

 Gerling ; and that several con- 

 temporary mathematicians, such as 

 Schweikart, came very near to 

 Gauss's own position. Second, al- 

 though Riemann, and subsequently 

 also Helmholtz, made use of the 

 term " manifold " (Mannigfaltig- 

 keit), it does not appear in the 

 course of their discussion that they 

 considered the space-manifold from 

 any other than a metrical point 

 of view. In fact, the manifold be- 

 comes in their treatment a magni- 

 tude (Grosse). It is true that 

 Riemann does refer to certain 

 geometrical relations not con- 

 nected with magnitude but only 

 with position, as being of great 

 importance. These two points 

 through which the researches of 

 Riemann and Helmholtz stand in 

 relation to other, and at the 

 time isolated, researches, were 

 dwelt on, the first by Beltrami, 

 and the second by Cayley and 

 Prof. Klein. 



