4 ROYAL SOCIETY OF CANADA 



may call the old geometr3^ The old geometer, by submitting his experi- 

 ence to reason and judgment, forms the idea which he formulates in the 

 straight line, and upon this idea he builds up his geometry. 



The system of geometry thus constructed is now called Euclidean, 

 to distinguish it from varieties belonging to the new geometry, and the 

 space which admits of Euclidean geometry is called Kuclidean space. As 

 this geometry assumes the existence of the defined straight line, and as 

 such a line cannot return into itself or have a necessary and absolute 

 termination, Euclidean space may be, and possibly is, infinite. 



Down to about the year 1830 it was generally held by all geometei-s 

 that Euclidean geometry represented the actual of the universe, and that 

 it was the only possible system of geometry. About this time, however, 

 Lobatchewsky questioned the generally received dictum of previous 

 geometers, and expressed the opinion that the geometry of the universe 

 need not perforce be Euclidean, but that some other geometry not 

 founded upon Euclid's definitions and axioms, and to which these were 

 not applicable, might be the real geometry of the universe. 



About 1850 Eiemann and Gauss gave a great impetus to the new 

 geometry, and in more recent years Beltramie, and Dedekind, and Helm- 

 holtz, and Klein, and Clifford, and Hinton, and Ball, and many others 

 have each added their quota to help it along and perfect it into a system. 

 And just about as generally as it is supported by mathematicians it is 

 condemned and repudiated by philosophers. 



It might be here explained that the principal arguments in su])portof 

 the new geometry are drawn from apparent analogies between relations 

 connected with geometry of one and of two dimensions, and that it is by 

 analogy alone that extensions of these new ideas can be made to geometry 

 of higher dimensions. Also, that to postulate the possibility of the new 

 geometry is to postulate the existence of four-dimensional space, and the 

 possible finitude of the universe. So that all these things are, of neces- 

 sity, connected with the subject of transcendental geometry. 



Considering the Euclidean definition of a plane, it is well-known that 

 a triangle in this plane has the sum of its internal angles constant and 

 equal to two right angles. Or to put it in another way, the sum of the 

 internal angles of a triangle formed by three straight lines which inter- 

 sect two and two, but are not concurrent, is invariably two right angles. 



Now the new geometry tells us that since we get our idea of a plane 

 from experience, as also our idea of a line, we cannot be sure that our 

 experimental plane is the Euclidean plane, or that our experimental 

 straight line is the straight line of Euclidean geometry ; so that we have 

 no right to assume that because EucHdean relations seem to hold experi- 

 mentally for any extent of the apparent plane that comes within our 

 observation, therefore they must hold through any extent whatever. 



It tells us that, for all we can know to the contrary, if a" triangle 

 could be taken sufficiently large, the sum of its internal angles might be 



