The foundation of abstract reasoning must always be an adequate host of particulars. To reason about Justice, we must be able to recall a sufficient variety of just actions to bring to view all the characters connoted by justice, and to exclude those that are not connoted. So with regard to Roundness; we must keep in view several circles differing in material, colour, and size, so as to affirm nothing but what belongs to all circles.

The verbal definition provides a mode of seemingly evading this requirement of a plurality of concrete instances. It cannot dispense with the concrete altogether; but it may make one instance suffice. To understand the definition of matternamely, something inert, or resisting-it would be enough to have one example before us, as a cannon ball, provided we understand that all the properties of the ball are to be excluded from our consideration except its inertness. We may, and do in some subjects, contract the habit of looking at an individual concrete in this exclusive way, which is the greatest stretch of abstraction within the competence of the mind. But this is the act of the mature intelligence.

35. INDUCTION, Inductive Generalization, Conjoined Properties, Affirmations, Propositions, Judgments, Belief, Laws of Nature. The contrast between Abstraction and Induction, as here understood, may be expressed thus: in the one a single isolated property, or a collection of properties treated as a unity, is identified and generalized; in the other a conjunction, union, or concurrence of two distinct properties is identified. We exemplify the first process, when we bring all rivers into one class, and define the property common to all; the second process, Induction, is exemplified when we note the fact that rivers wear away their beds, or the fact that they deposit deltas at their mouths. In this case two different things are conjoined; the flow of water over a country to the sea in an open channel, which makes the idea of a river, is associated with the circumstance of depositing or forming land in a particular situation. This conjunction makes an Affirmation, or a Proposition; the idea of a river by itself, or anything expressed by a noun, is not an affirmation. When we affirm the uniform co-existence of two dis

tinct facts, we have a Law of Nature, an intellectual possession respecting the world, an extension of our knowledge, a shortening of labour. Of the two conjoined things, the presence of one is at any time sufficient to assure us of the presence of the other, without farther examination. As surely as we meet with a river, so surely shall we find the carrying down of mud to be deposited at the mouth, if the two facts be really connected as we suppose. An abstraction or definition gives us a general idea; it assembles a class of things marked by the presence of this common feature,—the class river, the class circle, the class red, the class planet, the class just, but does not convey a proposition, a law of nature, a truth.

In forming these inductive generalizations, we need the identifying impetus very much as in abstractive generalizations. The case is distinguished only by being more complex; it is properly a stage beyond the other in the order of discovery, although the two are often accomplished by one and the same effort of the sense and the understanding. Still, in order to arrive at the knowledge that rivers form bars and deltas, we require to have observed the peculiarities of rivers, and to have been arrested by their identity on this point; standing at the mouth of one, and observing the island which parts its stream, we are reminded, by a stroke of reinstating similarity, of the mouth of some other where a similar formation occurs, with perhaps many points of diversity of circumstances. These two coming together will bring up others, until we have assembled in the mind's eye the whole array that our memory contains. Such is the first stage of an inductive discovery; it is the suggestion of a law of nature, which we are next to verify. The conflux of all the separate examples in one view indicates to the mind the common conjunction, and out of this we make a general affirmation, as in the other process we make a general notion or idea. Now, a general affirmation by language makes a proposition, not a definition; it needs a verb for its expression, and is a law or a truth, something to be believed and acted on.

In like manner, it is by an identification of the separate instances falling under our notice, that we are struck with the conjunction, in an animal, of cloven hoofs with the act of ruminating and with herbaceous food. To take a more abstruse example. We identify the conjunction of transparency in bodies with the bending of the rays of light; these transparent bodies are of very various nature,-air, water, glass, crystalline minerals; but, after a certain length of observation, the identity makes itself felt through them all. By an abstractive process, we gain the general idea of transparency; by looking, not simply at the fact of the luminous transmission, but at the direction of the light, we generalize an induction, a proposition, conjoining two properties instead of isolating one. The operation of induction is thus of the same nature, but more arduous, and implying greater labour, than the operation of abstraction. The same cast of mind favours both; the same obstructions block the way. To make a scientific induction, the mind must have the power of regarding the scientific properties and disregarding the unscientific aspects; in discovering the refraction of light, the attention must fasten on the circumstance of mathematical direction, and must not be carried away with vulgar astonishment at the distorting effect of light upon objects seen through water or glass. To take in the more abstruse and dissimilar instances, as the refractive influence of the air, there is needed a preparation similar to that already exemplified in assimilating rust and combustion.

Sometimes an induction from a few identified particulars can be fitted in to a previously established formula or generalization. The above instance of the refraction of light furnishes a case in point; and I quote it as a further example of the identifying operation. The bending of the light on entering or leaving a surface of glass, water, or other transparent material, varies with the inclination of the ray to the surface; at a right angle there is no bending, at all other angles there is bending, and it is greater as the course is farther from the right angle, being greatest of all when the

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ray lies over so much as almost to run along the surface. Now, an important identification was here discovered by Snell, namely, the identity of the rate of refraction at different angles with the trigonometrical relation of the sines of the angles, expressed thus :-the sines of the angles of incidence and refraction bear a constant proportion within the same medium, or the same kind of material. Here the observed amount of the bending at different angles, was found to accord with a foregone relation of the mathematical lines connected with the circle. This too may be looked upon as a discovery of identification, demanding in the discoverer not only great reach of Similarity, but antecedent acquirements in the geometry of the circle, ready to be started by such a case of parallelism as the above. Inductions falling into numerical and geometrical relations, previously excogitated, occur very frequently in the progress of discovery. All Kepler's laws are identifications of this nature; his third law, which connects the distances of the planets from the sun with their periodic times, is a remarkable instance. He had before him two parallel columns of numbers, six in the column, corresponding to the six known planets; one column. contained the distances, another the times of revolution; and he set himself to ascertain whether the relations of these numbers could come under any one rule of known proportions they were not in a simple proportion, direct or inverse, and they were not as the squares, nor as the cubes; they turned out at last to be a complication of square and cube. The law of areas is perhaps an equally remarkable example of a series of particulars embraced in an all-comprehending formula, from the existing stores of mathematical knowledge. In all these discoveries of Kepler, we perhaps should admire the aims, the determination and perseverance of his mind, still more than the grasp of his intellect. We have before remarked, that for a man to extricate himself from the prevailing modes of viewing natural appearances, and to become attached to a totally original aspect, is itself a proof of mental superiority, and often the principal

turning point of great discoveries. The identifying faculty in Kepler showed itself less prominently in the particular strokes, than in the mode of taking up the entire problem, the detection of a common character in the motions of the planets and the relations of the numbers and curves. To make that a pure mathematical problem, which really is one, but has not hitherto been sufficiently regarded as such, is itself a great example of the scientific intellect; it was the glory alike of Kepler and of Newton. A previously equipped mathematical mind, a wide reach of identifying force, and an indifference or superiority to poetical and fanciful aspects, concur in all the authors of discoveries that bind the conjunctions of nature in mathematical laws. The great revolution in Chemistry made by the introduction of definite combining numbers, has been even more rapidly prolific of great consequences, than the discoveries that gave Mechanics, Astronomy, and Optics the character of mathematical sciences. The introduction of numerical conceptions into the subtle phenomena of Heat, through Black's doctrine of latent heat, exhibits a stroke of high intellect not inferior to any of those now adduced. The difficulty of seizing the phenomena of freezing, melting, boiling, and condensing, in a bald, numerical estimate, is attested by the lateness of the discovery, if not sufficiently apparent to any one that considers how very different from this is the impression that these effects have on the common mind. The engrossing sensations of warmth and cold, the providing of fuel and clothing, the prevention of draughts, or the admission of cool air-are the trains of thought usually suggested by the various facts of congelation, liquefaction, &c.; to enter upon those other trains is a consequence of special training and endowment, the explanation of which, according to general laws of mind, has been one of the aims of our protracted examination of the human intellect.

36. DEDUCTION, Inference, Ratiocination, Syllogism, Application or Extension of Inductions.-I have repeatedly urged the value of the identifying process in extending our know