646 POPULAR SCIENCE MONTHLY. 



course of intuitive geometry which does not attempt to be rigorously 

 demonstrative, which emphasizes the sensuous rather than the rational. 

 But in a serious work it is now no longer permissible to have nothing 

 to start from. Wherever rigorous mathematics, there pure logic. 



It may be a relief to many that the non-Euclidean geometry has 

 shown the limitations to the arithmetization of mathematics. The 

 opinion that only the concepts of analysis or arithmetic are susceptible 

 of perfectly rigorous treatment Hilbert considers entirely erroneous. 



On the contrary, he says, I think that wherever, from the side of the 

 theory of knowledge, or in geometry, or from the theories of natural or 

 physical science, mathematical ideas come up, the problem arises for mathe- 

 matical science to investigate the principles underlying these ideas and so to 

 establish them upon a simple and complete system of axioms, that the exact- 

 ness of the new ideas and their applicability to deduction shall be in no 

 respect inferior to those of the old arithmetical concepts. 



The arithmetical symbols are written diagrams and the geometrical figures 

 are graphic formulas. 



The use of geometrical signs as a means of strict proof presupposes the 

 exact knowledge and complete mastery of the axioms which underlie those 

 figures; and in order that these geometrical figures may be incorporated 

 in the general treasure of mathematical signs, there is necessary a rigorous 

 axiomatic investigation of their conceptual content. 



In other words, the world has outgrown Wentworth's geometry. 

 More than this, as Frankland puts it, the possibility of explaining 

 'mass ' (the fundamental property of matter) as a function of ' electric 

 charge ' is on the point of banishing both ordinary gross matter and 

 also ether, since the principle of parsimony forbids needless hypothet- 

 ical entities. Now the relation between the two opposite electricities 

 so closely resembles that between Bolyaian and Biemannean space that, 

 as Clifford adumbrated, we may expect to see matter, ether and elec- 

 tricity banished in favor of space, the various and changing geometries 

 of which will be found adequate to account for all the phenomena of 

 the material world. 



Furthermore, these geometries of physical space will be found not 

 to be ' continuous,' but to be the varied and changing ' tactical ' arrange- 

 ments of a discrete, a discontinuous manifold consisting of indivisible 

 units. The notion of continuous extension, so long considered ultimate, 

 will, by this simplification, be subsumed under the finally ultimate 

 notion of juxtaposition, with which Lobachevski begins his great treatise 

 ' Noviya nachala,' in whose very first article he says of it : " This simple 

 idea derives from no other, and so is subject to no further explanation." 



