September 12, 1913] 



SCIENCE 



351 



stated, it proves just nothing. Thus a new 

 investigation has been necessary into the 

 foundations of the principle. There is 

 another problem, closely connected with 

 this subject, to which I would allude: the 

 stability of the solar system. For those who 

 can make pronouncements in regard to this 

 I have a feeling of envy ; for their methods, 

 as yet, I have a quite other feeling. The 

 interest of this problem alone is sufficient 

 to justify the craving of the pure mathema- 

 tician for powerful methods and unexcep- 

 tionable rigor. 



NON-EUCLIDIAN GEOMETRY 



But I turn to another matter. It is an 

 old view, I suppose, that geometry deals 

 with facts about which there can be no two 

 opinions. You are familiar with the axiom 

 that, given a straight line and a point, one 

 and only one straight line can be drawn 

 through the point parallel to the given 

 straight line. According to the old view 

 the natural man would say that this is 

 either true or false. And, indeed, many 

 and long were the attempts made to justify 

 it. At length there came a step which to 

 many probably will still seem unintelligible. 

 A system of geometry was built up in which 

 it is assumed that, given a straight line and 

 a point, an infinite number of straight 

 lines can be drawn through the point, in 

 the plane of the given line, no one of which 

 meets the given line. Can there then, one 

 asks at first, be two systems of geometry, 

 both of which are true, though they differ 

 in such an important paxticular? Almost 

 as soon believe that there can be two sys- 

 tems of laws of nature, essentially differ- 

 ing in character, both reducing the phe- 

 nomena we observe to order and system — 

 a monstrous heresy, of course ! I will only 

 say that, after a century of discussion we 

 are quite sure that many systems of geom- 

 etry are possible, and true ; though not all 



may be expedient. And if you reply that 

 a geometry is useful for life only in pro- 

 portion as it fits the properties of concrete 

 things, I will answer, first, are the heavens 

 not then concrete? And have we as yet 

 any geometry that enables us to form a 

 consistent logical idea of furthermost 

 space? And, second, that the justification 

 of such speculations is the interest they 

 evoke, and that the investigations already 

 undertaken have yielded results of the most 

 surprising interest. 



THE THEORY OP GROUPS 



To-day we characterize a geometry by 

 the help of another general notion, also, for 

 the most part, elaborated in the last hun- 

 dred years, by means of its group. A 

 group is a set of operations which is closed, 

 in the same sense that the performance of 

 any two of these operations in succession is 

 equivalent to another operation of the set, 

 just as the result of two successive move- 

 ments of a rigid body can be achieved by a 

 single movement. One of the earliest con- 

 scious applications of the notion was in the 

 problem of solving algebraic equations by 

 means of equations of lower order. An 

 equation of the fourth order can be solved 

 by means of a cubic equation, because there 

 exists a rational function of the four roots 

 which takes only three values when the 

 roots are exchanged in all possible ways. 

 Following out this suggestion for an equa- 

 tion of any order, we are led to consider, 

 taking any particular rational function of 

 its roots, what is the group of interchanges 

 among them which leaves this function un- 

 altered in value. This group characterizes 

 the function, all other rational functions 

 unaltered by the same group of inter- 

 changes being expressible rationally in 

 terms of this function. On these lines a 

 complete theory of equations which are 

 soluble algebraically can be given. Any 



