chose of mathematics; though I shall include those upon the possibility and objective validity à priori, of principles of the mathematical science, which, consequently, are to be looked upon as the principle of these, and which proceed from conceptions to intuition, and not from intuition to conceptions.

In the application of the pure conceptions of the understanding to possible experience, the employment of their synthesis is either mathematical or dynamical, for it is directed partly on the intuition alone, partly on the existence of a phænomenon. But the à priori conditions of intuition are in relation to a possible experience absolutely necessary, those of the existence of objects of a possible empirical intuition are in themselves contingent. Hence the principles of the mathematical use of the categories will possess a character of absolute necessity, that is, will be apodeictic; those, on the other hand, of the dynamical use, the character of an à priori necessity indeed, but only under the condition of empirical thought in an experience, therefore only mediately and indirectly. Consequently they will not possess that immediate evidence which is peculiar to the former, although their application to experience does not, for that reason, lose its truth and certitude. But of this point we shall be better able to judge at the conclusion of this system of principles.

The table of the categories is naturally our guide to the table of principles, because these are nothing else than rules for the objective employment of the former. Accordingly, all principles of the pure understanding are

These appellations I have chosen advisedly, in order that we might not lose sight of the distinctions in respect of the

evidence and the employment of these principles. It will, however, soon appear that—a fact which concerns both the evidence of these principles, and the à priori determination of phænomena-according to the categories of Quantity and Quality (if we attend merely to the form of these), the principles of these categories are distinguishable from those of the two others, inasmuch as the former are possessed of an intuitive, but the latter of a merely discursive, though in both instances a complete certitude. I shall therefore call the former mathematical,* and the latter dynamical principles.† It must be observed, however, that by these terms I mean, just as little in the one case the principles of mathematics, as those of general (physical) dynamics, in the other. I have here in view merely the principles of the pure understanding, in their application to the internal sense, (without distinction of the representations given therein), by means of which the sciences of mathematics and dynamics become possible. Accordingly, I have named these principles rather with reference to their application, than their content; and I shall now proceed to consider them in the order in which they stand in the table.

I.

AXIOMS OF INTUITION.

The principle of these is, "All Intuitions are Extensive Quantities."

* Mathematically, in the Kantian sense.-Tr.

† All combination (conjunctio) is either composition (compositio) or connection (nexus). The former is the synthesis of a manifold, the parts of which do not necessarily belong to each other. For example, the two triangles into which a square is divided by a diagonal, do not necessarily belong to each other, and of this kind is the synthesis of the homogeneous in every thing that can be mathematically considered. This synthesis can be divided into those of aggregation and coalition, the former of which is applied to extensive, the latter to intensive quantities. The second sort of combination (nexus) is the synthesis of a manifold, in so far as its parts do belong necessarily to each other; for example, the accident to a substance, or the effect to the cause. Consequently it is a synthesis of that which, though heterogeneous, is represented as connected, à priori. This combination-not an arbitrary one-I entitle dynamical, because it concerns the connection of the existence of the manifold. This, again, may be divided into the physical synthesis of the phænomena among each other, and the metaphysical synthesis, or the connection of phænomens à priori in the faculty of cognition.

PROOF.

All phænomena contain, as regards their form, an intuition in space and time, which lies à priori at the foundation of all without exception. Phænomena, therefore, cannot be apprehended, that is, received into empirical consciousness otherwise than through the synthesis of a manifold, through which the representations of a determinate space or time are generated; that is to say, through the composition of the homogeneous, and the consciousness of the synthetical unity of this manifold (homogeneous). Now the consciousness of a homogeneous manifold in intuition, in so far as thereby the representation of an object is rendered possible, is the conception of a quantity (quanti). Consequently, even the perception of an object as phænomenon is possible only through the same synthetical unity of the manifold of the given sensuous intuition, through which the unity of the composition of the homogeneous manifold in the conception of a quantity is cogitated; that is to say, all phænomena are quantities, and extensive quantities, because as intuitions in space or time, they must be represented by means of the same synthesis, through which space and time themselves are determined.

An extensive quantity I call that wherein the representation of the parts renders possible (and therefore necessarily antecedes) the representation of the whole. I cannot represent to myself any line, however small, without drawing it in thought, that is, without generating from a point all its parts one after another, and in this way alone producing this intuition. Precisely the same is the case with every, even the smallest portion of time. I cogitate therein only the successive progress from one moment to another, and hence, by means of the different portions of time and the addition of them, a determinate quantity of time is produced. As the pure intuition in all phænomena is either time or space, so is every phænomenon in its character of intuition an extensive quantity, inasmuch as it can only be cognized in our apprehension by successive synthesis (from part to part). All phænomena are, accordingly, to be considered as aggregates, that is, as a collection of previously given parts; which is not the case with every sort of quantities, but only with those which are represented and apprehended by us as extensive.

On this successive synthesis of the productive imagination. in the generation of figures, is founded the mathematics of extension, or geometry, with its axioms, which express the conditions of sensuous intuition à priori, under which alone the schema of a pure conception of external intuition can exist; for example, "between two points only one straight line is possible,' ," "two straight lines cannot enclose a space,' &c. These are the axioms which properly relate only to quantities (quanta) as such.

But, as regards the quantity of a thing (quantitas), that is to say, the answer to the question, How large is this or that object? although, in respect to this question, we have various propositions synthetical and immediately certain (indemonstrabilia); we have, in the proper sense of the term, no axioms. For example, the propositions, "If equals be added to equals, the wholes are equal;" "If equals be taken from equals, the remainders are equal;" are analytical, because I am immediately conscious of the identity of the production of the one quantity with the production of the other; whereas axioms must be à priori synthetical propositions. On the other hand, the self-evident propositions as to the relation of numbers, are certainly synthetical, but not universal, like those of geometry, and for this reason cannot be called axioms, but numerical formulæ. That 7+ 5 = 12, is not an analytical proposition. For neither in the representation of seven, nor of five, nor of the composition of the two numbers, do I cogitate the number twelve. (Whether I cogitate the number in the addition of both, is not at present the question; for in the case of an analytical proposition, the only point is, whether I really cogitate the predicate in the representation of the subject.) But although the proposition is synthetical, it is nevertheless only a singular proposition. In so far as regard is here had merely to the synthesis of the homogeneous (the units), it cannot take place except in one manner, although our use of these numbers is afterwards general. If I say, "A triangle can be constructed with three lines, any two of which taken together are greater than the third," I exercise merely the pure function of the productive imagination, which may draw the lines longer or shorter, and construct the angles at its pleasure. On the contrary, the number seven is possible only in one manner, and so is like

wise the number twelve, which results from the synthesis of seven and five. Such propositions, then, cannot be termed axioms (for in that case we should have an infinity of these), but numerical formulæ.

This transcendental principle of the mathematics of phenomena greatly enlarges our à priori cognition. For it is by this principle alone that pure mathematics is rendered applicable in all its precision to objects of experience, and without it the validity of this application would not be so self-evident; on the contrary, contradictions and confusions have often arisen on this very point. Phænomena are not things in themselves. Empirical intuition is possible only through pure intuition (of space and time); consequently, what geometry affirms of the latter, is indisputably valid of the former. All evasions, such as the statement that objects of sense do not conform to the rules of construction in space (for example, to the rule of the infinite divisibility of lines or angles), must fall to the ground. For, if these objections hold good, we deny to space, and with it to all mathematics, objective validity, and no longer know wherefore, and how far, mathematics can be applied to phanomena. The synthesis of spaces and times as the essential form of all intuition, is that which renders possible the apprehension of a phænomenon, and therefore every external experience, consequently all cognition of the objects of experience; and whatever mathematics in its pure use proves of the former, must necessarily hold good of the latter. All objections are but the chicaneries of an ill-instructed reason, which erroneously thinks to liberate the objects of sense from the formal conditions of our sensibility, and represents these, although mere phænomena, as things in themselves, presented as such to our understandings. But in this case, no à priori synthetical cognition of them could be possible, consequently rot through pure conceptions of space, and the science which determines these conceptions, that is to say, geometry, would itself be impossible.

II.

ANTICIPATIONS OF PERCEPTION.

The principle of these is, "In all phænomena the Real, that which is an object of sensation, has Intensive Quantity, that is, has a Degree."