54 ORIGIN AND SIGNIFICANCE OF 



space. This investigation has been carried out by 

 Professor Lipschitz of Bonn. 1 It is found that the 

 comprehensive expression for all the laws of dynamics, 

 Hamilton's principle, may be directly transferred to 

 spaces of which the measure of curvature is other than 

 zero. Accordingly, in this respect also, the disparate 

 systems of geometry lead to no contradiction. 



We have now to seek an explanation of the special 

 characteristics of our own flat space, since it appears 

 that they are not implied in the general notion of an 

 extended quantity of three dimensions and of the free 

 mobility of bounded figures therein. Necessities of 

 thought, such as are involved in the conception of such 

 a variety, and its measurability, or from the most 

 general of all ideas of a solid figure contained in it, 

 and of its free mobility, they undoubtedly are not. 

 Let us then examine the opposite assumption as to 

 their origin being empirical, and see if they can be 

 inferred from facts of experience and so established, or 

 if, when tested by experience, they are perhaps to be 

 rejected. If they are of empirical origin, we must be 

 able to represent to ourselves connected series of facts, 

 indicating a different value for the measure of curva- 

 ture from that of Euclid's flat space. But if we can 



'Untersuclmngen iiber die ganzen homogenen Functionen von n 

 Differentialen' (Borchardt's Journal filr MatJiematik, Bd. Ixx. 3, 71 ; 

 Ixxiii. 3, 1) ; Untersuchung eines Problems der Variationsrechnung' 

 ( Ibid. Bd. Ixxiv.). 



