Some few principles preposited by geometricians are, indeed, really analytical, and depend on the principle of contradiction. They serve, however, like identical propositions, as links in the chain of method, not as principles—for example, a = a, the whole is equal to itself, or (a+b) —> a, the whole is greater than its part. And yet even these principles themselves, though they derive their validity from pure conceptions, are only admitted in mathematics because they can be presented in intuition. What causes us here commonly to believe that the predicate of such apodeictic judgements is already contained in our conception, and that the judgement is therefore analytical, is merely the equivocal nature of the expression. We must join in thought a certain predicate to a given conception, and this necessity cleaves already to the conception. But the question is, not what we must join in thought to the given conception, but what we really think therein, though only obscurely, and then it becomes manifest that the predicate pertains to these conceptions, necessarily indeed, yet not as thought in the conception itself, but by virtue of an intuition, which must be added to the conception. 2. The science of natural philosophy (physics) contains in itself synthetical judgements à priori, as principles. I shall adduce two propositions. For instance, the proposition, “In all changes of the material world, the quantity of matter remains unchanged”; or, that, “In all communication of motion, action and reaction must always be equal.” In both of these, not only is the necessity, and therefore their origin à priori clear, but also that they are synthetical propositions. For in the conception of matter, I do not cogitate its permanency, but merely its presence in space, which it fills. I therefore really go out of and beyond the conception of matter, in order to think on to it something à priori, which I did not think in it. The proposition is therefore not analytical, but synthetical, and nevertheless conceived à priori; and so it is with regard to the other propositions of the pure part of natural philosophy. 3. As to metaphysics, even if we look upon it merely as an attempted science, yet, from the nature of human reason, an indispensable one, we find that it must contain synthetical propositions à priori. It is not merely the duty of metaphysics to dissect, and thereby analytically to illustrate the conceptions which we form à priori of things; but we seek to widen the range of our à priori knowledge. For this purpose, we must avail ourselves of such principles as add something to the original conception—something not identical with, nor contained in it, and by means of synthetical judgements à priori, leave far behind us the limits of experience; for example, in the proposition, “the world must have a beginning,” and such like. Thus metaphysics, according to the proper aim of the science, consists merely of synthetical propositions à priori.