[14] Hume, _Treatise_, Bk. I, Part 1, § 1.
Each of the three categories of relation, i. e. those of substance and accident, of cause and effect, and of interaction between agent and patient involves, according to Kant, a special principle, and these special principles he calls 'analogies of experience'. They are stated thus:[1] (1) In all changes of phenomena the substance is permanent, and its quantity in nature is neither increased nor diminished. (2) All changes take place according to the law of the connexion of cause and effect. (3) All substances, so far as they can be perceived in space as coexistent, are in complete interaction. The justification of the term _analogy_ of experience is as follows. In mathematics an analogy is a formula which asserts the equality of two _quantitative_ relations, and is such that, if three of the terms are given, we can discover the fourth, e. g. if we know that _a_ : _b_ = _c_ : _d_, and that _a_ = 2, _b_ = 4, _c_ = 6 we can discover that _d_ = 12. But in philosophy an analogy is the assertion of the equality of two _qualitative_ relations and is such that, if three of the terms are given, we can discover, not the fourth, but only the relation of the third to the fourth, though at the same time we are furnished with a clue whereby to search for the fourth in experience. In this philosophical sense, the principles involved in the categories of relation are analogies. For instance, the principles of causality can be stated in the form 'Any known event _X_ is to _some other_ event _Y_, whatever it be, as effect to cause'; so stated, it clearly informs us not of the character of _Y_ but only of the fact that there must be a _Y_, i. e. a necessary antecedent, though at the same time this knowledge enables us to search in experience for the special character of _Y_.
[1] The formulation of them in the first edition is slightly different.
The principles to be established relate to the two kinds of temporal relation apprehended in the world of nature, viz. coexistence and succession. The _method_ of proof, which is to be gathered from the proofs themselves rather than from Kant's general remarks[2] on the subject, is the same in each case. Kant expressly rejects any proof which is 'dogmatical' or 'from conceptions', e. g. any attempt to show that the very conception of change presupposes the thought of an identical subject of change.[3] The proof is transcendental in character, i. e. it argues that the principle to be established is a condition of the possibility of _apprehending_ the temporal relation in question, e. g. that the existence of a permanent subject of change is presupposed in any _apprehension_ of change. It assumes that we become aware of sequences and coexistences in the world of nature by a process which begins with a succession of mere perceptions, i. e. perceptions which are so far not the perceptions of a sequence or of a coexistence or indeed of anything;[4] and it seeks to show that this process involves an appeal to one of the principles in question--the particular principle involved depending on the temporal relation apprehended--and consequently, that since we do apprehend this temporal relation, which, as belonging to the world of nature, must be distinct from any temporal relation of our perceptions, the principle appealed to is valid.