=Conclusion b.=--The next paragraph maintains two theses: (_a_) that space is the form of all outer intuition; (_b_) that this fact explains what is otherwise entirely inexplicable and paradoxical, namely, that we can make _a priori_ judgments which yet apply to the objects experienced. The first thesis, that the pure intuition of space is only conceivable as the form of appearances of outer sense, is propounded in the opening sentence without argument and even without citation of grounds. The statement thus suddenly made is not anticipated save by the opening sentences of the section on space.[471] It is an essentially new doctrine. Hitherto Kant has spoken of space only as an _a priori_ intuition. The further assertion that as such it must necessarily be conceived as the form of outer sense (_i.e._ not only as a formal intuition but also as a form of intuition), calls for the most definite and explicit proof. None, however, is given. It is really a conclusion from points all too briefly cited by Kant in the general _Introduction_, namely, from his distinction between the matter and the form of sense. The assertions there made, in a somewhat casual manner, are here, without notification to the reader, employed as premisses to ground the above assertion. His thesis is not, therefore, as by its face value it would seem to profess to be, an inference from the points established in the preceding expositions. It interprets these conclusions in the light of points considered in the _Introduction_; and thereby arrives at a new and all-important interpretation of the nature of the _a priori_ intuition of space. The second thesis employs the first to explain how prior to all experience we can determine the relations of objects. Since (_a_) space is merely the form of outer sense, and (_b_) accordingly exists in the mind prior to all empirical intuition, all appearances must exist in space, and we can predetermine them from the pure intuition of space that is given to us _a priori_. Space, when thus viewed as the _a priori_ form of outer sense, renders comprehensible the validity of applied mathematics.