PROFESSOR AT HEIDELBERG 255 



monodromously, and freely, as do bodies in actual space? Answer, 

 expressed according to our analytical geometry : " Let x, y, z, t 

 be the rectangular co-ordinates of a space of four dimensions, 

 then for every point of our tri-dimensional space it follows that 

 x 2 +y 2 + z 2 + t 2 = R 2 , where R is an undetermined constant, 

 which is infinite in Euclidean space." I venture to ask you to let 

 me know if Riemann's essay is already in print, or if there is 

 any prospect of its being published shortly, as seems to me 

 most desirable; in the event of Riemann having taken the 

 same point of departure, my own work would become useless, 

 and I need not go on expending as much time and headache 

 as it has already cost me.' 



Schering replied that ' the most important point in Riemann's 

 treatment of the proposition stated that the magnitude defined 

 by Gauss as the measure of curvature is a differential invariant 

 for homogeneous differential expressions of the second degree 

 and first order, in two variables', and Helmholtz resumes on 

 May 18: 



' I am much obliged for the copy of Riemann's Habilitations- 

 schrift. Herewith I send you a short account of the part of my 

 own studies of this subject which is not covered by Riemann's 

 work, begging you to lay it before the Royal Society to be 

 published in the Gottinger Anzeigen (Proceedings of the Society). 

 I believe a detailed discussion of the whole, consecutively, to be 

 very desirable, and for choice I would have it published in the 

 Proceedings of your Society, along with Riemann's. I there- 

 fore beg to ask if communications are accepted from 

 corresponding members, of which I am one, and when you are 

 bringing out the next volume ? . . . Forgive me for Riemann's 

 sake, for troubling you with these matters.' The paper was 

 published in 1868. 



Helmholtz in the first place endeavoured to distinguish the 

 development of concepts in geometry from the facts of experi- 

 ence, which appear to be necessities of thought, while it was 

 only in the lecture delivered ten years later on the Facts of 

 Perception that he gathered up the results of his researches 

 towards a unified system of philosophy that differed essentially 

 from that of Kant. If this divergence from Kant had been 

 partly apparent in his earlier physiological optics, he only 

 >roclaimed it definitely in the 1868 paper on the axioms of 



