MATHEMATICAL PHYSICS. 5*7 



metric ideas is to be traced among the prehistoric races who carved 

 rough but thoroughly artistic outlines of animals on their weapons. E 

 do not know whether the matter has attracted serious speculation, but 

 I have myself been led to wonder how men first arrived at the notion 

 of an outline drawing. The primitive sketches referred to immediately 

 convey to the experienced mind the idea of a reindeer or the like; but 

 in reality the representation is purely conventional, and is expressed 

 in a language which has to be learned. For nothing could be more 

 unlike the actual reindeer than the few scratches drawn on the surface 

 of a bone ; and it is of course familiar to ourselves that it is only after 

 a time, and by an insensible process of education, that very young chil- 

 dren come to understand the meaning of an outline. Whoever he was, 

 the man who first projected the world into two dimensions, and pro- 

 ceeded to fence off that part of it which was reindeer from that which 

 was not, was certainly under the influence of a geometrical idea, and 

 had his feet in the path which was to culminate in the refined ideali- 

 zations of the Greeks. As to the manner in which these latter were 

 developed, the only indication of tradition is that some propositions 

 were arrived at first in a more empirical or intuitional, and afterwards 

 in a more intellectual way. So long as points had size, lines had 

 breadth and surfaces thickness, there could be no question of exact rela- 

 tions between the various elements of a figure, any more than is the 

 case with the realities which they represent. But the Greek mind loved 

 definiteness, and discovered that if we agree to speak of lines as if they 

 had no breadth, and so on, exact statements became possible. If any 

 one scientific invention can claim preeminence over all others, I should 

 be inclined myself to erect a monument to the inventor of the mathe- 

 matical point, as the supreme type of that process of abstraction w T hich 

 has been a necessary condition of scientific work from the very begin- 

 ning. 



It is possible, however, to uphold the importance of the part which 

 abstract geometry has played, and must still play, in the evolution of 

 scientific conceptions, without committing ourselves to a defense, on 

 all points, of the traditional presentment. The consistency and com- 

 pleteness of the usual system of definitions, axioms and postulates have 

 often been questioned ; and quite recently a more thoroughgoing analy- 

 sis of the logical elements of the subject than has ever before been at- 

 tempted has been made by Hilbert. The matter is a subtle one, and 

 a general agreement on such points is as yet hardly possible. The basis 

 for such an agreement may perhaps ultimately be found in a more ex- 

 plicit recognition of the empirical source of the fundamental concep- 

 tions. This would tend, at all events, to mitigate the rigor of the 

 demands which are sometimes made for logical perfection. 



Even more important in some respects are the questions which have 



