/. J44 KANT. condition of (external) perception. The like is true of arithmetic and time. If space were a mere concept, no proposition could be derived from it which should go beyond the concept and extend our knowledge of its properties. The possibility of such extension or synthesis in mathematics depends on ,' the fact that spatial concepts can always be presented or "constructed " in intuition. The geometrical axiom that in the triangle the sum of two sides is greater than the third is derived from intuition, by describing the triangle in imagination or, actually, on the board. Here the object is given through the cognition and not before it. — If space and time were empirical representations the knowledge obtained from them would lack_ necessity, which, as a matter of fact, it possesses in a marked degree. While experience teaches us only that something is thus or so, and not that it could not be otherwise, the axioms, (space has only three dimensions, time only one ; only one straight line is possible between two points,) nay, all the i propositions of mathematics are strictly universal andj apodictically certain : we are entirely relieved from the necessity of measuring all triangles in the world in order to find out whether the sum of their angles is equal to two right angles, and we do not need, as in the case of judg- ments of experience, to add the limitation, so far as it is yet known there are no exceptions to this rule. The apriority is the ratio essendi of the strict necessity involved in the " it must be so " {des Soseinmiissens), while the latter is the ratio cognoscendi of the former. Now since the necessity of mathematical judgments can only be explained through the ideality of space, this doctrine is perfectly certain, not merely a probable hypothesis. — The ivalidity of mathematical principles for all objects of per- i/ception, finally, is based on the fact that they are rules ^ under which alone experience is possible for us. It should be mentioned, further, that the conceptions of change and motion (change of place) are possible only through and in the representation of time. No concept I could make intelligible the possibility of change, that is, of 4 the connection of contradictory predicates in one and the
