] 66 Proceedings of the Royal Irish Academy. 



nature and characteristics from the space which is the framework of our 

 sensations, and which is given to ' us as visual, tactile, or motor. The study 

 of the laws by which sensations succeed each other is what leads us to adopt 

 the convention of geometrical space, whose properties are based on the 

 displacement of solid bodies, without whose existence in nature there would 

 be no Geometry. All consistent Geometries are equally true ; in fact, the 

 question of truth or falsehood has no meaning. Experiment can tell us 

 only what Geometry is most convenient ; and an easily imaginable change in 

 our experience would lead us to adopt a non-Euclidean Geometry, or even 

 a Geometry of space of four dimensions. 



The speculations of so great a mathematician as Poincare must be treated 

 with respect ; yet I cannot but think that he has, with great skill, combined 

 almost all the errors of previous speculators. 



His only argument against the theory of Kant is that, if it were true, 

 geometrical axioms would be imposed upon us with such a force that we 

 could not conceive the contrary, nor build upon it a theoretical edifice. 



As regards the first of these statements, it is, I think, correct, and is one 

 of the strongest reasons for accepting the theory of Kant. I am quite unable 

 to imagine, or picture to myself as a reality, space of four dimensions, an 

 absolute standard of linear magnitude independent of any arbitrary conven- 

 tion or restriction, or two lines having neither concavity nor convexity which 

 have two points in common without coinciding. 



As to the theoretical edifices of space of more than three dimensions, and 

 non-Euclidean Geometries, it is easy to see, on Kantian principles, how they can 

 be reached. No figure in space, according to Kant, can be cognized without 

 a generation, or successive contemplation of its parts, in time. It thus 

 becomes a continuous series in time, and so amenable to Algebra. Tlie out- 

 come is the science of Analytic Geometry. Algebra is not restricted by the 

 special properties of space, but is applicable to any homogeneous form of 

 intuition capable of being generated by a synthesis in time and reorganized 

 as distinctly simultaneous. Tlie properties of space of four or of any number 

 of dimensions can then be investigated from analogy by means of Algebra ; and 

 the results arrived at are consistent with themselves and with Euclidean 

 Geometry. The non-Euclidean Geometries are arrived at by putting special 

 arbitrary restrictions on a space of three dimensions in a space of four. These 

 Geometries are consistent, and, in a sense, mathematically correct ; but I do not 

 think they are true for our intelligence, nor do they condition objects of our 

 possible experience. It is strange that mathematicians should have confined 

 their erratic speculations to space. They might as well have considered the 

 properties of a universe in which time is of two dimensions, or has a curva- 



