710 SCIENTIFIC THOUGHT. 



Bolyai in Hungary, Lobatchevski in Kasan, Grassmann 

 in Stettin. Most of these were unknown to each 

 other. However, near the beginning of the last third 

 of the century three distinct publications created a 

 great stir in the mathematical world, brought many 

 scattered but cognate lines of reasoning together, and 

 made them mutually fertile and suggestive. These 

 three were first, the publication in 1860 of Gauss's 

 correspondence with Schumacher, in which two letters 

 of the former, dated May and July 183 1, 1 became 

 known, * where he referred to his extensive but un- 

 written and unfinished speculations on the foundations 

 of geometry and the theorem which refers to the 

 sum of the angles in a triangle. The second was the 

 publication in 1867 of the first and only part of Her- 

 mann Hankel's " Lectures on the Complex Numbers 

 and their Functions." 2 The third was the posthumous 

 publication in the same year of Kiemann's paper, dated 

 1854, 3 "On the Hypotheses which lie at the Foundation 

 of Geometry." Almost simultaneously there appeared 

 the first of Helmholtz's two important papers 4 on the 



1 See ' Briefwechsel zwischen 

 Gauss und Schumacher,' ed. Peters. 

 1860, vol. ii. pp. 260, 268. 



a The small volume contains so 

 much original and historical matter 

 that I have on several occasions 

 referred to it. See above, pp. 645, 

 653. 



3 Riemann, ' Math. Werke,' 1st 

 ed., p. 254 sqq. 



4 The first publication of Helm- 

 holtz was a lecture on "the actual 

 foundations of geometry," which 

 he delivered on the 22nd May 1868 

 to the Medical Society at Heidel- 

 berg. This communication, which 



referred to investigations carried on 

 for many years, notably in con- 

 nection with the theory of the 

 colour -manifold, was occasioned 

 by the publication of Riemann's 

 paper in the ' Transactions ' of the 

 Gottingen Society. He had heard 

 of this through Schering, to whom 

 he wrote on the 21st April 1868 

 before having seen Riemann's 

 paper : "I have myself been oc- 

 cupied with the same subject dur- 

 ing the last two years, in connection 

 with my researches in physiological 

 optics. ... I now see, from the 

 few hints which you give as to the 



