Section III., 1897. [ 3 ] Trans. R. S. C 



I. — Presidential Address : On the Transcendental Geometry. 



By Prof. N. F. Dupuis, A.M. 



(Read .June 23rrl, 1897.) 



Gentlemen, — As you have done me the honour of electing me to the 

 presidency of this section of the Eoyal Society, the duty has devolved 

 upon me, in accordance with custom, to prepare a presidential address. 



I confess that it is with some degree of misgiving that I have done 

 so, for I cannot forget that we have been pretty plainly told by some of 

 the members of this section that, at least, it is not pleasant to be treated 

 to a lot of hieroglyphics in the way of mathematical formula», in which, 

 from not being able to understand them, they could take no real interest. 



Such a sentiment is hard upon the mathematicians of the section, 

 for if they are not to use mathematical formula», who are to use them ? 

 or what are they to use? Every subject has its own symbology and its 

 special technicalities, and although mathematics may have more of them 

 than the majority of subjects, yet chemistry and physics are not without 

 them. 



The real difficulty is that pure mathematics has little or no affinity 

 with experimental subjects, and we, who are thrown together into a sort 

 of uncognate agglomeration in order to form a section, must try and 

 exercise as much forbearance as possible. 



Whether my subject is one in which many of you are interested or 

 not I do not know, but I will promise you one thing, that I will not treat 

 you to hieroglyphics. I purpose to consider and to criticise, as far as 

 time allows, that system of geometrical speculation known under the 

 names of the new geometry, transcendental geometry, hyperspace, 

 spherical space, pseudo-spherical space, etc. 



Prof. Clitford, who, during his life, was an active advocate of the 

 new geometry, says : " Just as in any branch of physical inquiry we 

 start by making experiments, and basing on our experiments a set of 

 axioms which form the foundations of an exact science, so in geometry 

 our axioms are really, although less obviously, the result of experience. 

 On this ground geometry has been properl}^ termed a physical science." 



I do not agree with Prof. Clitford that geometry is a physical science 

 in anything like the same sense as sciences such as chemistry, and experi- 

 mental physics, and others which we usually call physical sciences. But 

 more of this hereafter. He goes on to say that " the danger of asserting 

 dogmatically that an axiom based on the expei-ience of a limited region 

 holds universally will be apparent." Even admitting the truth of all 

 this, it does not atfect the geometry, which for the sake of distinction we 



