Notices respecting New Books. 465 



had it not actually been held by some philosophers. Mathe- 

 maticians will also agree with Kant in the stress which he lays 

 upon the construction. But there is one point which it seems to 

 us has not been dwelt upon sufficiently. Let us take Euclid, i. 16. 

 If we examine the proof, we shall find that it turns upon the 

 assertion that the angle ACT is less than the angle ACD. Now 

 this may be easily asserted for a particular triangle; but before we 

 can assert it for any triangle, we have to investigate the matter 

 and glance over all the possible cases. If this be intuition, then 

 it is hard to distinguish intuition from the process of generalization 

 which the mind goes through in forming general judgments which 

 Kant would characterize as empirical, such as that contained in the 

 last clause of the following sentence : — " We know nothing but 

 our manner of perceiving them, that manner being peculiar to us, 

 and not necessarily shared in by every being, though, no doubt, by 

 every human being " (p. 37). 



Since Kant's day the domain of Mathematics has been greatly 

 extended, and subjects formerly treated in a manner which was 

 truly empirical are now treated in a manner which is exact. 

 That domain includes even his own science of Logic. If, then, 

 that vast body of knowledge is to be put down as a priori, by 

 a priori can be meant nothing else than exact. If the Modern 

 Logic, as developed and expounded by De Morgan, Boole, Harley, 

 Jevons, Macfarlane, and Venn in this country, by Halsted and 

 Pierce in America, and by Schroder in G-ermany, be sound science, 

 then Kant's ' Critique ' must contain considerable imperfections ; for 

 it proceeds on the assumption that the scholastic logic was perfect. 

 " I mean only to treat of reason and its pure thinking, a knowledge 

 of which is not very far to seek, considering that it is to be found 

 within myself. Common logic gives an instance how all the simple 

 acts of reason can be enumerated completely and systematically " 

 (p. xxiv). " That Logic, from the earliest times, has followed that 

 secure method may be seen from the fact that since Aristotle it 

 has not had to retrace a single step, iinless we choose to consider 

 as improvements the removal of some unnecessary subtleties, or the 

 clearer definition of its matter, both of which refer to the 

 elegance rather than to the solidity of the science. It is remark- 

 able also, that to the present day it has not been able to 

 make one step in advance, so that, to all appearance, it may be 

 considered as completed and perfect" (Sup., p. 364). Whence 

 are the Pure Concepts of the Understanding (the Categories) 

 obtained ? Prom a classification of judgments " differing in some, 

 though not very essential, points from the usual technicalites of 

 logicians." The classification according to Quantity is Universal, 

 Particular, Singular. It may be asked, What of doubly universal 

 judgments, such as, for example, Kprjres ael \pevarai ? It may be 

 pointed out that we may have judgments universal with respect to 

 both Space and Time, or with respect to Space only, or with 

 respect to Time only, or with respect to neither. But the proper 



