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 KANT'S PROOF OF THE PROPOSITION, ETC. 507 whole field of pure (synthetic) knowledge a priori, he some- what recklessly excluded a whole province of such knowledge, perhaps indeed the most important province, namely Pure Mathematics." Ask Kant for an instance of this real knowledge, and he is found too cautious to mention the name of Metaphysics : he "points to a Euclid " and says, Here ! " Pure Mathematics, in spite of the attacks of the hardiest sceptic, stands, like a Colossus, for the proof of the reality of a knowledge enlarged and augmented through Pure Reason alone." Now from the latter of the points of view, thus briefly indicated, Kant's object is to prove two leading characteris- tics of Mathematics ; first, that it is synthetic knowledge ; second, that it is knowledge a priori; and for the proper understanding of the opening chapters of the Critique it is absolutely necessary to keep the two parts of this proof dis- tinct. Neither of them depends in the very least degree upon the other ; either might be invalid, while the strength of the other remained unweakened. The terms ' synthetic ' and ' a priori ' are far from being indispensable to one an- other ; the often repeated use of them in conjunction, to describe different kinds of knowledge, implies in each case two distinct investigations of the nature of the knowledge so described. In the present essay we shall examine only the process, by which Kant sought to prove the synthetic nature of Mathematics, and leave the other investigation strictly out of account. Kant's final view on the point, which con- cerns us here, is summed up in his brief statement : " Ma- thematical judgments are one and all synthetical " ; we shall see that Kant reached this result only after a long struggle, in the course of which all varieties of mathematical judgments came directly under his consideration : we shall try to dispose of those objections to Kant's doctrine, which deny the per- fection of his induction, and seek to overthrow it by producing particular instances from mathematics of analytic judgments supposed to have been overlooked by Kant. Questions as to the a priori nature of mathematics, or the validity of the distinction between analytic and synthetic judgments gener- ally, will be treated as irrelevant : our procedure is primarily exegetical, and, if at any point it be necessary to defend Kant's doctrine, our defence will be confined entirely within the limits already assigned to our investigation. To the historian of philosophy nothing is more obvious than that Kant was merely following the example of a long and illustrious line of predecessors when he entered upon the realm of Philosophy from the side of mathematical