PROFESSOR AT HEIDELBERG 261 



where be movable without change of form, and able to rotate 

 in every direction, then the measure of curvature must be 

 constant. He thereby proves that the fundamental assumptions 

 do not require the infinite extensibilty of tri-dimensional space ; 

 space can have the same relation to a quadruply extended 

 complex, as a surface with a constant measure of curvature has 

 to tri-dimensional space. 



Helmholtz's investigation was to a large extent implicit in 

 that of Riemann, but was distinctly original in one particular, 

 so that it was of great importance for all later work, and for the 

 question of the axioms of geometry. He tries to establish the 

 conditions under which the Pythagorean Law as hypothetically 

 assumed and generalized by Riemann would be valid, and 

 makes the condition, which Riemann only introduced at the 

 close of his paper, the basis of his whole treatment of the 

 subject, i.e. that spatial figures should have, without alteration 

 of form, the degree of mobility which is postulated in geometry. 



' For the rest I must observe, that even if the publication 

 of Riemann's work has cancelled the priority of a whole series 

 of my own results, it is of no little importance to me, in regard 

 to such a recondite and hitherto discredited subject, to find that 

 so distinguished a mathematician should have thought these 

 questions worthy of his attention, and it has been to me 

 a certain guarantee of the validity of the way, when I found 

 him upon it as my companion.' 



For Helmholtz the starting-point of the investigation was 

 the fact that all primitive measurement of space rests on the 

 observation of congruence. But since there can be no veri- 

 fication of congruence unless fixed bodies or systems of points 

 can be moved relatively to each other with unaltered form, and 

 unless the congruence of two spatial magnitudes be a fact 

 independent of all motion, he set himself the task of seeking 

 the most universal analytical form of a complex of manifold 

 extension in which the desired mode of motion shall be 

 possible. He inquires in the next place how much the con- 

 ditions which he postulates for the investigation, viz. (i) con- 

 tinuity and dimensions, (2) the existence of mobile solid bodies, 

 (3) free mobility, (4) independence of form of solid bodies on their 

 rotation, restrict the possibility of different systems of geometry. 

 These assumptions led him to a measure of the linear elements, 



