[dupuis] presidential ADDRESS S 



either greater or less than two right angles, and that if either possibility 

 is true for any triangle, however large, it is presumably true to some 

 kind of a proportionate extent, for any triangle however small. 



This hypothesis, in its application to plane geometry, has given rise 

 to the following notation, which may be convenient for reference : The 

 two-dimensional space in which the triangle is supposed to be constructed 

 is called elUptic, parabolic, or hyperbolic, according as the sum of the 

 internal angles of the triangle is greater than, equal to, or less than two 

 right angles. I say apparently plane geometry, because if the word 

 plane be employed in the Euclidean sense, the space is necessarily para- 

 bolic. 



We see then that both the elliptic and the hyperbolic space are 

 founded upon the assumption or hypothesis, that the Euclidean plane is an 

 impossibility in the constitution of the universe, and that the Euclidean 

 straight line is either an impossibility, or is restricted to certain relational 

 directions. 



If you ask for the reason you are told that the possibilities of geometry 

 are fixed and determined by the nature and projîerties of space ; and 

 that space-properties may be such as not to admit of the Euclidean axioms 

 and definitions, and therefore not to admit of such a universe as will 

 answer to every extent, and in all respects to the conclusions of Euclidean 

 geometry. 



Of course, there are no means of proving this. But the new geometer 

 holds that neither are there any means of proving the principles of 

 Euclidean geometry. That all the different species of geometry, for I 

 suppose that that is what we must call them, are equally possible and 

 probable, and are founded upon assumptions which are equally reasonable 

 and plausible, and that although Euclidean geometry may be that of the 

 universe, so also may any of the others, and that it is a mistake to place 

 geometry upon so limited a basis as is involved in merely one species of 

 geometry. 



The following illustrations will possibly make this clearer : 



If we suppose some very minute intellectual lilliputian confined to live 

 and carry on his experiments and observations upon the smooth surface 

 of some frozen lake like Superior, and to be Hmited in his migrations to 

 a few yards or rods from a given point, he would naturally infer that he 

 lived upon a plane, inasmuch as no experiments which he could institute, 

 and no observations that he could make would detect any deviation fro m 

 the absolute plane. He might draw lines on this plane such as he would 

 call parallel and which, he would be full}' convinced, would never meet 

 however far they might be produced. His triangles lying in his habit- 

 able plane would, as far as he could see, have the sum of their internal 

 angles equal to two right angles, and the whole geometry of hi s ii.-?sumed 

 plane would be Euclidean. 



