any new qualities which it may assume are due either to difference of climate, or to difference of soil, or to both these causes conjointly, though our knowledge may not enable us to assign amongst these alternatives the particular cause or combination of causes to which the effect is due. Now ought such an Inference to be classified as a perfect or an imperfect Induction? If we content ourselves with stating the alternatives, the inference should be regarded, so far as it goes, as a Perfect Induction; for within the limits stated the conclusion may be considered absolutely certain. But if, on any grounds, we suppose one of these alternatives to be more probable than the others, and we state this as our conclusion, the inference is, of course, only a probable one, and should rank as an Imperfect Induction.

The same remarks will apply to those cases in which there is any uncertainty as to the nature of the fact of causation. If the inference be, say, that the two phenomena either are one cause and the other effect, or stand to each other in the relation of cause and effect, though we may be unable to determine which of the two is cause and which is effect, or are both of them effects of the same cause (adding any other alternatives which the particular case may require), the inference is, so far as it goes, a Perfect Induction. But, if one or some only of these alternatives be selected, on any grounds short of absolute or moral certainty, to the exclusion of the others, the inference is only probable, and must be regarded as merely an Imperfect Induction.

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Briefly to sum up the contents of this chapter, Imperfect Inductions are the results either of an Inductio per Enumerationem Simplicem (to which I propose to appropriate the expression Empirical Generalisations'), or of the Argument from Analogy (which I call Analogies), or of an imperfect fulfilment of one or other of the Inductive Methods (to which we might, perhaps, advantageously appropriate the expression 'Incomplete Inductions'). In the two former cases there can be no more than an intimation of a Fact of Causation, while in the last we conceive ourselves to be on the way towards establishing one.

CHAPTER V.

On the relation of Induction to Deduction, and on Verification.

THE results of our inductions are summed up in general propositions, which are not unfrequently stated in the shape of mathematical formulæ. These general propositions, the results of inductive reasoning, become, in turn, the data from which deductive reasoning proceeds. Though the major premiss of any single deductive argument may itself be the result of deduction, it will invariably be found, as pointed out long ago by Aristotle', that the ultimate major premiss of a chain of deductive reasoning is a result of induction. There must be some limit to the generality of the propositions under which our deductive inferences can be subsumed, and, when we have reached this limit, the only evidence on which the ultimate major premiss can repose, if it depend on evidence at all, must be inductive. Thus, most of the deductions in the science of Astronomy, and

1 Ἡ μὲν δὴ ἐπαγωγὴ ἀρχή ἐστι καὶ τοῦ καθόλου, ὁ δὲ συλλογισμὸς ἐκ τῶν καθόλου. Εἰσὶν ἄρα ἀρχαὶ ἐξ ὧν ὁ συλλογισμὸς, ὧν οὐκ ἔστι συλλογισμός· ἐπαγωγὴ ἄρα.—Eth. Nic. vi. 3 (3). Cp. Eth. Nic. vi. 6,8 (9); Metaphysics, i. 1; Posterior Analytics, ii. 19.

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many of those in the science of Mechanics, depend ultimately on the Law of Universal Gravitation; but this Law itself is the result of an induction based upon a variety of facts, including both the fall of bodies to the earth and the motion of the planets in their orbits. Again, a large number of geometrical deductions may be traced up to the ultimate major premiss: 'Things that are equal to the same thing are equal to one another.' But this proposition, if not referred directly to induction, is classed under the head of intuitive conceptions, the most probable, though perhaps not the most commonly received, explanation of which is that which derives them from the accumulated experience of generations, transmitted hereditarily from father to spn.

A Deductive Inference combines the results of previous inductions or deductions, and evolves new propositions as the consequence, or, to put the matter in a slightly different point of view, as expressing the total result, of these combinations. I append a few easy examples of the manner in which the results of induction are employed in a deductive argument.

To begin with a very simple instance, but one which will serve as a good illustration of the stage at which our investigations cease to be inductive and become deductive ;-suppose we have ascertained, by previous inductions, that A produces a, B produces B, C produces, D produces, and E produces, we know, by calculation-that is, by deductive reasoning-that the total effect of A, B, C, D, E is 8+3. 중 In this case

the simple rules of Algebra, governing the addition and subtraction of quantities, combined with the special data here furnished, are the premisses from which our deductive reasoning proceeds.

The proposition proved in Euclid, Book i. Prop. 38, that 'Triangles upon equal bases, and between the same parallels, are equal to one another,' is derived from, or is the total result of, the previous deductions (1) that Parallelograms upon equal bases, and between the same parallels, are equal to one another,' (2) that 'Triangles formed by the diagonal of a parallelogram are each of them equal to half the parallelogram' (i. 34), and (3) the previous induction that the halves of equal things are equal.'

What is called the Hydrostatic Paradox, namely, that a man standing on the upper of two boards, which form the ends of an air-tight leather bag, and blowing through a small tube opening into the space between the board, can easily raise his own weight, is a combination of two propositions, both gained from experience by means of induction, these propositions being (1) that fluids transmit pressure equally in all directions, (2) that, the greater the pressure brought to bear on any surface from below, the greater the weight which it will sustain (otherwise expressed by the Mechanical Law that action and reaction are equal).

To take another very simple instance of a similar kind. One of the earliest and easiest problems in the Science of Optics is the following: 'A conical pencil of rays is