287. In the geometrical construction, that serves for the demonstration of the above theorem, which we have used as an example, may be seen what we have just explained with regard to total conceptions containing other conceptions besides those compared. If we place the rectangle and the rhomboid upon the same base, we at once see that there is something common to both, namely, the triangle formed by the base, a part of one side of the rhomboid, and a part of one side of the rectangle. Neither synthesis nor analysis is here required, because there is perfect coincidence, and this in geometry is equivalent to perfect equality. The difficulty is in the two remaining parts, that is, in the trapezoids to which the parallelograms are reduced by the subtraction of the common triangle. The mere sight of the figures teaches nothing concerning the equivalence of the two surfaces; we see only that the two sides of the rhomboidal surface go on extending, but including a less distance in proportion as the angle becomes more oblique, under these two conditions: length of sides, and diminution of distances between two limits, of which one is infinity, and the other the rectangle. The relation of the equivalence of the surfaces may be demonstrated by prolonging the parallel opposite the base, and thus forming a quadrilateral of which the trapezoids are parts; to discover the equality of these trapezoids, it is only necessary to decompose the quadrilateral, attending to the equality of two triangles, each respectively formed by one of the trapezoids and a common triangle. Is any thing here added to the conception of each trapezoid? No. We only compare them. They could not be compared directly, and therefore we included them in a total conception, the mere analysis of which enabled us to discover the relation sought for. The conception does not give this relation; it only shows it; for if the conception of the two figures compared were more perfect, so that we might intuitively behold the relation existing between the increment of the sides and the decrement of their distance from each other, we should see that there is here a constant law, which supplies on one side what is lost on the other; and consequently we should discover, in the very conception of the rhomboid, the fundamental reason of the equality, that is, the permanent value of the surface, notwithstanding the greater or less obliquity of the angles; thus obtaining what we deduced from the above comparison, and generalize with reference to two constant lineal values, base and altitude. The same would happen with respect to the equivalence of all variable quantities differently expressed, could we reduce their conceptions to such clear and simple formulas as those of apparent functions; for example, nx/mx, from which, whatever the value of the variable, there always results the same value of the expression, which is constant, to wit, n/m.
288. Let not these investigations be imagined useless. In this, as in many other questions, it happens that most important truths are the result of a philosophical problem which, in appearance, is merely speculative. Thus, in the present case, we observe Kant explaining the principle of causality, in an inexact, and, as we understand him, in an altogether false sense; but, perhaps, the origin of his equivocation lies in his considering the principle of causality as synthetic, although a priori, whereas it must be regarded as analytic, as we shall show when treating of the idea of cause.
In consideration of the great importance of clear and distinct ideas on the present subject, we will in a few words, sum up the doctrine we have explained concerning mediate and immediate evidence.
There is immediate evidence when, in the conception of the subject, we see its agreement or disagreement with the predicate, without requiring any other means than mere reflection on the meaning of the terms. Judgments of this class are with propriety called analytic, because we have only to analyze the conception of the subject to find therein its agreement or disagreement with the predicate.
There is mediate evidence when, in the conception of the subject, we do not immediately see its agreement or disagreement with the predicate, and therefore have to call in a middle term to make it manifest.
290. Here arises the question whether judgments of mediate evidence are analytic. It is clear that if we mean by analytic only those in which we have solely to understand the meaning of the terms in order to see the agreement or disagreement of the predicate, the judgments of mediate evidence cannot be called analytic; but if by analytic judgment we mean a judgment in which it is only necessary to decompose the conception of the subject in order to find therein its agreement or disagreement with the predicate, we must say that the judgments of mediate evidence are analytic, and the means employed is only the formation of a total conception containing the partial conceptions, the relation of which we seek to discover. In the union of these partial conceptions there is a synthesis, it is true; but there is none in the discovery of their relation, for this is done by analysis.
A judgment is not the less analytic because formed by the union of different conceptions; for then no judgment would be analytic. When we say, man is rational, the two conceptions of animal and rational enter into the conception of man, but do not take from it its analytical character; for this, as its very name imports, consists in the analysis of a conception, being sufficient to show certain predicates in it, without reference to the manner of this conception's formation, whether two or more conceptions are united in it, or not.
291. This clearly shows in what mediate evidence consists. The predicate is indeed contained in the idea of the subject; but, owing to the limitation of our intellect, either these ideas are incomplete, or we do not see them in all their extension, or else we do not well distinguish what we in a confused manner perceive in them; and hence, to know the meaning of the terms does not enable us immediately to see that the predicate is contained in the idea of the subject. Moreover, the objects, even such as are purely ideal, are presented to us separately; and hence, not knowing the sum of them all, we pass successively from one to another, discovering their mutual relations in proportion as we approach them.
292. It may, from what we have said, be inferred that all judgments in the purely ideal order are analytic, since every cognition of this order is obtained by the intuition of whatever is more or less complicated in the conception, and there is no more synthesis than is necessary to bring the objects together, by uniting their conceptions in one total conception, which serves for the discovery of the relation of the partial conceptions.
293. The x, therefore, of which Kant speaks, and the removal of which is one of the most important problems of philosophy, is nothing more than the faculty possessed by the soul to unite the conceptions of different things in one total conception, and to discover in it their mutual relations. This faculty is no new discovery, for the schools have all recognized it under one name or another. No one ever denied to the intellect the faculty of comparing; and comparison is the act whereby the intellect places two or more objects before its sight so as to perceive their mutual relations. In this act the intellect forms a total conception, of which the conceptions compared are a part. Thus we have seen that in geometry to verify the mutual relation of certain figures, we construct a new figure which includes them all, and is a sort of field whereon the comparison is made.
This exposition of analytic and synthetic judgments will suffice for the present; as we proposed to treat of them here only in general, and as related to certainty; consequently we will not descend to their particular application to various ideas, the analysis of which belongs to other parts of this work.
CHAPTER XXX.
VICO'S CRITERION.