5 i2 THE POPULAR SCIENCE MONTHLY. ' 



external reality. First, they must be simple, that is, their relations to 

 one another must be easily handled ; second, their relations must cor- 

 respond more or less closely with the relations of some set of external 

 things. They do not correspond absolutely. There are no external 

 things which have the properties of mathematical straight lines except 

 approximately. Take the annexed figure. - - 



One would not hesitate to call A B, B C, C B, 

 straight lines, and to say that the triangle ABC 

 has the sum of its angles equal to 180. The error 

 he would make (they are drawn with compasses) we 

 always make in kind, though not in degree, in apply- 

 ing mathematics to realities. I wish to make clear 

 that the relation between mathematical truths and external facts, is 

 one of resemblance, not identity. What the essence of resemblance is 

 I shall not discuss. 



No external facts can do more than change the utility of the two 

 geometries. At present, for simplicity and accuracy of resemblance 

 to external facts, the Euclidean geometry need not fear being swal- 

 lowed up. If, however, facts should be discovered which could be 

 most simply correlated to transcendental truths, transcendental geom- 

 etry might become important. 



Let us recapitulate. We have tried to show that mathematics 

 deals only with concepts, and that the two geometries are, therefore, 

 also concejDtual. Their apparent discrepancy we tried to account for 

 by showing that they used different concepts. We showed that, al- 

 though concepts might be originated by sensations, they were not, 

 nor were affected bv, external facts. The relation between mathe- 

 matical truths and external facts is one of more or less resemblance, 

 not of identity. Nor can the resemblance be ever proved to be 

 perfect. 



The Euclidean geometry has as great facility in accommodating 

 itself to all known facts as the transcendental, and greater simplicity. 

 It is therefore of greater practical utility. The mathematical truth 

 of each is not affected by experience. 



Thus transcendental geometry, with its egg-shells turned inside- 

 out without cracking, its knots mysteriously untied, its worlds where 

 the background of everything is a man's own head, is from its con- 

 ceptual basis, as a creation of man's mind, true. It is a pretty mathe- 

 matical diversion ; it is, as yet, nothing more. 



