All inductive reasoning is but the inverse application of deductive reasoning. Being in possession of certain particular facts or events expressed in propositions, we imagine some more general proposition expressing the existence of a law or cause; and, deducing the particular results of that supposed general proposition, we observe whether they agree with the facts in question. Hypothesis is thus always employed, consciously or unconsciously. The sole conditions to which we need conform in framing any hypothesis is, that we both have and exercise the power of inferring deductively from the hypothesis to the particular results, which are to be compared with the known facts. Thus there are but three steps in the process of induction:-- (1) Framing some hypothesis as to the character of the general law. (2) Deducing consequences from that law. (3) Observing whether the consequences agree with the particular facts under consideration. In very simple cases of inverse reasoning, hypothesis may seem altogether needless. To take numbers again as a convenient illustration, I have only to look at the series, 1, 2, 4, 8, 16, 32, &c., to know at once that the general law is that of geometrical progression; I need no successive trial of various hypotheses, because I am familiar with the series, and have long since learnt from what general formula it proceeds. In the same way a mathematician becomes acquainted with the integrals of a number of common formulas, so that he need not go through any process of discovery. But it is none the less true that whenever previous reasoning does not furnish the knowledge, hypotheses must be framed and tried (p. 124). There naturally arise two cases, according as the nature of the subject admits of certain or only probable deductive reasoning. Certainty, indeed, is but a singular case of probability, and the general principles of procedure are always the same. Nevertheless, when certainty of inference is possible, the process is simplified. Of several mutually inconsistent hypotheses, the results of which can be certainly compared with fact, but one hypothesis can ultimately be entertained. Thus in the inverse logical problem, two logically distinct conditions could not yield the same series of possible combinations. Accordingly, in the case of two terms we had to choose one of six different kinds of propositions (p. 136), and in the case of three terms, our choice lay among 192 possible distinct hypotheses (p. 140). Natural laws, however, are often quantitative in character, and the possible hypotheses are then infinite in variety.