254 HERMANN VON HELMHOLTZ 



earlier work in hydrodynamics, and his new results were laid 

 before the Berlin Academy (April 23, 1868), in the paper ' On 

 Discontinuous Motions of Fluids '. 



In the same year Helmholtz astonished the scientific and 

 mathematical world by the far more comprehensive and 

 fundamental researches which he published in the essay sent 

 to the Gottingen Scientific Society, ( On the Facts that underlie 

 Geometry/ At a later time he endeavoured to present its 

 most important results in a form intelligible to non-mathemati- 

 cians, in the lecture given to the Docentenverein at Heidelberg 

 in 1870, 'On the Origin and Significance of Geometrical 

 Axioms/ These investigations, along with the famous work, 

 ' On the Hypotheses that underlie Geometry,' which Riemann 

 had published as his Habilitationsschrift, in 1854, were epoch- 

 making for the development of the mathematico-philosophical 

 conceptions of the second half of the last century. 



Helmholtz, indeed, had occupied himself with the philosophical 

 analysis of the fundamental conceptions of mathematics and 

 physics at a very early period, as is proved by an interesting 

 sketch published some years before his essay on the Conserva- 

 tion of Energy, which not only shows how he strove in his 

 youth for clearness of fundamental concepts, but already 

 indicates the direction in which he was to do such pioneer 

 work thirty years later. 



On April 21, 1868, he writes to Schering at Gottingen : ' In 

 thanking you for sending me the two little notes about 

 Riemann, there is one question I should like to ask. In your 

 notice of his life I find it stated that he gave a Habilitations- 

 vorlesung on the Hypotheses of Geometry. I have myself 

 been occupied with this subject for the last two years in 

 connexion with my work in physiological optics, but have not 

 yet completed or published the work, because I hoped to make 

 certain points more general. For instance, I cannot yet make 

 everything as universal for three dimensions as I can for two. 

 Now I see by the few indications you give of the results of 

 the work, that Riemann came to exactly the same conclusion 

 as myself. My starting-point is the question: What must be 

 the nature of a magnitude of several dimensions in order that 

 solid bodies (i.e. bodies with unaltered relative measurements) 

 shall everywhere be able to move in it as continuously, 



