16 ' ROYAL SOCIETY OF CANADA 



is inconceivable exists only in an inconceivable space, and thus takes rank 

 with the mathematical imaginaries. 



But the Euclidean geometry is thinkable and consistent, and is in 

 fact the only one which for us is so. Therefore the Euclidean geometry 

 must be a geometry of the universe, and it must be the only one, unless 

 we are prepared to make the inconceivable assumption that the universe 

 may admit of different and contradictory geometries at the same time. 



As for four-dimensional space, even if it were a possibility, it does 

 not necessarily follow from analogy, or from any other reason, that it 

 must falsify our Euclidean geometry of three-dimensions, any more than 

 our geometry of three dimensions falsifies our geometr}^ of two. 



But from the point of view which I have laid down, four-dimensional 

 space need not give us any difficulty. Eor when any intelligent being 

 can imagine or conceive a four-dimensional figure, he has in this very 

 conception a four-dimensional space. 



To say that there may be superior intelligences which possess this 

 power, is to state the existence of a possibility for which we have not the 

 shadow of a proof. 



