[baker] the foundations OF GEOMETEY 118 



that of Lobachevski, and that of Eieinann. Then the question of 

 four-dimensioned space broadened men's visions. Kiemann, Helmholtz 

 and Lie conceived space as a manifold of numbers. The pseudosphere 

 of Beltrami, mathematically possible, physically inconceivable, helped 

 along distrust in our geometry of experience. It is scarcely necessary 

 lo say that in Euclid's sjistem of geometry there is a continual covert 

 reference to the physical universe of experience; especially is this so 

 in the constant use of diagrams. The result of this mental unrest 

 was that it was strongly felt the whole question of the foundations of 

 geometry was in the melting pot, and that something very different 

 from Euclid's system of axioms had to be devised to constitute an 

 unassailable and perfectly logical set of first principles or assumptions 

 from which to make a beginning. 



The importance of establishing the science of geometry with logical 

 accuracy, on a purely rational basis, suppressing completely the role 

 played by experience, will readily be conceived. Philosophically, of 

 course, it is essential in our search for that ideal perfection with which 

 alone we aje content. We must remember also that geometry, the 

 science of the external relations of things, is at the base of dynamics, 

 optics, and other physical sciences, and in laying its foundations securely 

 we are laying theirs. We are also, as in effect I have already said, 

 making a searching and critical examination of those intuitions which 

 lie at the very foundations of our intellectual life, namely, those relating 

 to space. : 



In connection with the unveiling of the Gauss-Weber monument 

 at Gottingen a memorial volume was published, part of which consisted 

 of Professor Hilbert's " Grundlagen der Géométrie," or " Foundations 

 of Geometry," It is dated 1899, His ideas have been developed by 

 Professor George Bruce Halsted of Kenyon College, Gambler, Ohio, 

 who calls his work " Eational Geometry, a Text-Book for the Science 

 ef Space," ^ It is of this system of geometry that I propose to speak. 

 If Hilbert has shown great subtlety of analysis in discussing the neces- 

 sary and sufficient assumptions that may be made the basis of geometry, 

 Halsted has shown consummate ability in the development of those 

 assumptions, and I feel bound to divide my admiration between the two 

 geometers. 



'Rntlnnal Geometry, a Text-book for the Science of Space, based on Hil- 

 bert's Foundations, by George Bruce Halsted, New York, John Wiley & Sons', 

 1904. 



Sec. Ill,, 1906, 8 



