Ldupuis] presidential ADDRESS 11 



has no meaning, in exact geometry ; and the distance between two 

 points is equally meaningless if the points are to be anything else than 

 those laid down by Euclid's definition, and certainly such points are not 

 those of our experience. And thus the whole doctrine of the triangle, 

 which lies at the foundation of all geometrical measurement, is faulty 

 and uncertain ; and what is worse, we have no means of knowing 

 wherein the faultiness lies, or in which direction it tends. And if we 

 cannot trust our mental conceptions in building up a knowledge of 

 geometr}', how can we trust them as a basis of any other knowledge. 



The straight line is the simplest of all geometrical concepts, since it 

 involves the most elementary idea of continuity without variation. And 

 it is doubtful if any being, endowed with human intelligence, could be 

 placed under circumstances of existence in which it could not form the 

 concept of the straight line. 



Even Clittord's worm in its crooked tube, — of which we shall say 

 more hereafter — if it were to turn on its axis — quite a legitimate opera- 

 tion — would very readily draw an inference as to a tube, or condition of 

 existence, which was bent in every direction alike, and this would be the 

 straight line. 



But the new geometer holds that the universe may be such that 

 Euclidean space may not be its space, but that the latter is so constituted 

 as to have a bend in it, so that its simplest geometrical element may be a 

 curved line. 



The space of the new geometer is then more complex than that of 

 Euclid. And when we consider that Euclidean space is absolute in its 

 simplicity, and that it is thinkable and admits, not only of the straight 

 line, but of ever}^ form that can be imagined or conceived, while spherical 

 and pseudo-spherical spaces are hampered by restriction, and are totally 

 and absolutely inconceivable, it seems to me, ajjart from all other con- 

 siderations, that the probability is overwhelmingly in favour of the space 

 of Euclidean geometry being that of the universe. 



Moreover, if space is curved, the character and direction of its curva- 

 ture must be i^urely arbitrary, and the radius of curvature for any sec- 

 tion must be arbitrary. But just as we can give a strong and valid non- 

 experimental reason why a point acted upon by two forces in difterent 

 directions should move along the diagonal of the parallelogram whose 

 sides rej^resent the forces ; so for every reason wh}' space should have 

 curvature in one direction, an equally potent reason can be given why 

 its curvature should lie in the opposite direction. And the only space in 

 which these arguments may be said to fail, or be said to hold equally' 

 well, is the space of Euclidean geometry, which is another strong pre- 

 sumption in favour of Euclidean space as that of the universe. 



Again, the new geometer, in drawing his conclusions in regard to 

 hyperspace, is compelled to found his arguments very largely upon 



