2-32- 



SCIENCE. 



[N. R. Vol. VII. No. 164. 



of Helmholtz and Riemann follow ; that is, 

 there are three, and only three, spaces which 

 fulfill these requirements, namely, the tra- 

 ditional, or Euclidean space, and the spaces 

 in which the group of movements possible 

 is the projective group transforming into 

 itself one or the other of the surfaces of the 

 second degree 



x'^ + y' + z'±l = 0. 



In the appreciation of this work of Lie's, 

 prepared for the Society by Felix Klein, for 

 which the Lobachevski gold medal was 

 given him, he says that Lie's work stands 

 out so prominently over all the others to be 

 compared with it that a doubt as to the 

 . award of the prize would scarcely have been 

 possible. Decisive for this judgment as to 

 the height of the scientific achievement is not 

 only the extraordinary depth and keenness 

 with which Lie, in the fifth section of his 

 book, handles what he has called the Eie- 

 mann-Helmholtz space problem, but es- 

 pecially the circumstance that this treat- 

 ment appears, so to say, as logical conse- 

 quence of Lie's long-continued creative 

 work in the province of geometry, especially 

 his theory of continuous transformation 

 groups. 



The extraordinary importance which the 

 works of Lie possess for the general devel- 

 opment of geometry can scarcely be over- 

 estimated. In the coming years they will 

 be still more widely prized than hitherto. 

 Passing, then, to the consideration of the 

 present state of the space question, Klein 

 takes up the origin of axioms. Whence 

 come the axioms ? A mathematician who 

 knows the non-Euclidean theories would 

 scarcely maintain the position of earlier 

 times that the axioms as to their concrete 

 content are necessities of the inner intu- 

 ition. 



What to the uninitiated appears as such 

 necessity shows itself, after long occupa- 

 tion with the non -Euclidian problems, as the 



result of very complex processes, and espe- 

 cially education and habit. 



Do the axioms come from experience? 

 Helmholtz energetically says yes ! as is 

 well known. But his expositions seem in 

 a definite direction incomplete. 



One will, in thinking over these, willingly 

 admit that experience plays an important 

 part in the formation of axioms, but will 

 notice that just the point especially inter- 

 esting to the mathematician remains un- 

 touched by Helmholtz. 



It is a question of a process which we al- 

 ways complete in exactly the same way in 

 the theoretical handling of any empirical 

 data, and which, therefore, may seem quite 

 clear to the scientist. 



Expressed generally : Always the results of 

 any observations hold good only within definite 

 limits of precision and under •particular condi- 

 tions ; when we set up the axioms we put in the 

 place of these results statements of absolute pre- 

 cision and generality. 



In this 'idealizing' of empirical data lies, in 

 my opinion, the peculiar essence of axioms. 

 Therein our addition is limited in its arbi- 

 trariness at first only by this, that it must 

 cling to the results of experience and, on the 

 other hand, introduce no logical contradic- 

 tion. 



Then enters as regulator also that which 

 Mach calls the ' economy of thinking.' No 

 one will rationally hold fast to a more com- 

 plicated system of axioms when he sees 

 that with a simpler system he already com- 

 pletely attains the exactitude requisite to 

 the representation of the empirical data. 



Klein goes on to mention the possibility 

 of a series of topologically distinguishable 

 space-forms built of limited ("simply com- 

 pendent) space-pieces either all Euclidean, 

 all Lobachevskian or all Riemannian. Be- 

 side these three just mentioned family- 

 types, the parabolic, the hyperbolic, the 

 single elliptic, Klein has shown that the 

 spherical, in which two geodetics always' 



