14 ROYAL SOCIETY OF CANADA 



ates for the sake of homogeneity, and in many similar operations. And 

 that his results, as far as any tests can be applied to them^ are consistent 

 and presumably correct. 



Now, in the operations here referred to, the mathematician is em- 

 ploying the symbolic language of algebra, in which the symbols stand 

 for and denote quantities or magnitudes, and operations, which b}^ a cir- 

 cumlocution can always be expressed in words. And there is no more 

 virtue, beyond that of convenience, in writing x* where x stands for a 

 line segment, than there is in speaking of a four-dimensional figure, 

 every dimension of which is the same. And however consistent with the 

 principles of algebraic operation the result may be, it will be capable of 

 being tested, and therefore will be presumably correct, onl}^ when it 

 admits of a real interpretation. 



To say that because x^ denotes the square on the line-segment a;, 

 and x^ denotes the cube on the same, therefore x* must denote a four- 

 dimensional figure of equal dimensions on the line-segment, is no proof 

 of anything, unless we assume, to begin with, that every homogeneous 

 algebraic expression must have an interpretation in real geometry; 

 which is a glaring example of the petitio prmcipii. 



That such usages and formal interpretations have their advantages 

 no mathematician will deny. But in cases where the interpretation 

 cannot be made in real terms, it is difficult to see how such expressions 

 can be looked upon outside of mere matters of formulization. 



Thus we speak of two circles as intersecting in the two circular 

 points at infinity, and we give this as a reason why the two circles cannot 

 intersect in more than two real points. But surely no person will main- 

 tain that two circles, every part of each of which is visible, can extend 

 to infinity, and the very fact that the circular points at infinity are 

 imaginary, shows that they do not. But the nomenclature and the de- 

 termining formula are decidedly advantageous, serving, as they do, to 

 generalize the properties of the circle and to connect it with that group 

 of curves to which it naturally belongs. 



Finally, we must consider the new geometer's idea of space itself. 

 To him space is something which is capable of exercising restriction, of 

 exerting a compelling influence which may prevent the existence of a 

 straight line and of a plane ; it is something which has at least a poten- 

 tial form, which may be curved or bent, and of any or all degrees of 

 curvature ; which is capable of exercising force and thus of resisting 

 curvature; something which can sutler displacement, and which can 

 propagate waves of displacement ; which can be twisted or have vortex 

 motion, and which may have its law of change a function of elapsed time. 



I do not know that any one of the new geometers would endorse all 

 these statements, but they certainly can be drawn from all their writings. 



Now all these assumed properties of space are physical properties 

 which are the attributes of matter, and all matter must possess some of 



