TRANSCENDENTAL GEOMETRY. 511 



Thus we see that mathematics may be defined as the science of the 

 relations of concepts. Its vocabulary, too, must be one of fixed con- 

 notation. That is why symbols are so useful ; their connotation does 

 not vary unconsciously. 



Benjamin Peirce defines mathematics as the science that draws 

 necessary conclusions. Mill says, " The problem is given a function, 

 what function is it of some other function ? " It is obvious that nec- 

 essary conclusions can be drawn only so far as there are relations 

 fixed whence to draw them ; the function must be given before we 

 find its relations with other functions. 



Now, I wish to insist, as strongly as I can, that any set of concepts 

 become fit for mathematical handling as soon as their relations are 

 unfolded, and this is what I have so far proved. If you ask, " Whence 

 these concepts ?" my answer is, " From experience." From it comes 

 the " element of intuition " that Stallo says is an element in every geo- 

 metrical axiom. Space itself is but a product of experience. If a 

 man could only hear or taste, would he have our concept of space ? I 

 trow not. 



Let us now, after this long digression, return to our transcendental- 

 ists. Euclidean geometry and non-Euclidean alike are mathematical. 

 Verbally they come to different conclusions, but neither conclusion 

 affects facts. The difference is here, it seems to me. Transcendental 

 geometry is the offspring of analytic, though some have tried to treat 

 it otherwise. The relations that it handles are at first algebraic rela- 

 tions that may apply to anything. Then applying the geometric no- 

 menclature to algebraic expression, calling expressions of the first 

 degree linear, etc., it interprets these results geometrically. Its defini- 

 tions, thus, are different from those of Euclid ; the ideas connoted by 

 its vocabulary are different ; its concepts are not the same. It is not 

 wonderful, then, that it gets a broader field of relations. 



We decide, then, that from their respective definitions the Euclid- 

 ean and the transcendental geometry are true. And this is, perhaps, 

 the most important point to settle, for the transcendentalists have said 

 that, although the geometrical definitions were true, the axioms need 

 not be. We, however, say that the axioms, or what you will, of par- 

 allelism, etc., are part of the connotation of the words defined, and are 

 simultaneously given. Of course, some experience is necessary to 

 make us form any concepts. 



The question now to be answered is, then, Which are the best defi- 

 nitions ? But it must be remembered that, as long as we are dealing 

 with mathematics, we are never dealing with real things. Thus Helm- 

 holtz is wrong in saying that by adding any mechanical axioms or 

 principles we can obtain an empirical science out of geometry, if the 

 science thus obtained is purely mathematical. 



Mathematical concepts can have two virtues in varying degrees, 

 namely, simplicity and resemblance to, or rather correspondence with, 



