lO ROYAL SOCIETY OF CANADA 



which is capable of vibrating, and of carrying its vibration onwards in a 

 sort of wave motion after the manner of a material body. By means of 

 this wonderful space the chemist could work out explanations of many 

 of his difficulties, and particularly'^ why there is a certain definite number 

 of kinds of atoms. But by the same means he would lose one of his 

 powerful aids in analysis. Dextrose and Lévulose could no longer be 

 distinguished with any certainty by their twisting effect upon a polarized 

 ray of light, for a left-hand twist would become a right-hand one, merel}^ 

 by being turned over in four-dimensional space. 



Examples of this kind might easily be increased ; but to give more 

 is unnecessary, while to enter into an outline of the reasoning by means 

 of which the results have been obtained would take me too far beyond 

 the limits set for this address. 



I have now given what I consider to be a fair statement of the leading 

 results of the transcendental geometry, of that geometry which is being 

 studied and taught in some of the best institutions, and by some of the 

 best mathematicians in the world, not in the descriptive form in which T 

 have presented it for 3'our consideration, but analytically, and by means 

 of the methods and machinery of co-ordinates. 



From the gravity of the situation introduced by the speculations of 

 the new geometer in regard not only to geometrical knowledge, but in- 

 directly to all knowledge, it seems to me that the foundations of this new 

 geometry should be most critically examined both mathematically and 

 philosophically before a general assent is given to it, or before it is held 

 to be altogether untrustworthy. 



The founders have proceeded upon the principal that geometry is a 

 physical subject, derived quite directly from experience, and that in 

 establishing any system of geometry we are not justified in claiming for 

 it any more than we can draw from experience. 



Because we have never experienced a straight line in the Euclidean 

 sense, we have no right to assume that there is such a line ; and because 

 we have never experienced a Euclidean plane it is quite possible that 

 there is and can be no such thing as a plane ; and that our conceptions 

 of such things are merely idle dreams which can have no existence in the 

 real universe, not because they contain any inconsistencies or self-contra- 

 dictions, but because the universe, from the very nature of its space 

 properties, will not admit of them. 



If this principle is to be followed, it is doubtful if we can have any 

 geometry at all in the accurate sense of the word, if, indeed, we can have 

 any knowledge. 



For if we are to adopt the lines and planes and the figures of our 

 experience, instead of the concepts drawn from these by reason and 

 judgment, our lines must be irregular and variable, and our planes 

 curved and uneven. But the determination of a point by two such lines 



