or common properties* of the class. This exhausts the line of operations connected with the generalization of an object taken as a total or a unity; of these the first alone grows out of pure Similarity, the others suppose a somewhat more complicated action, to be afterwards described.

Take again the genus of round bodies. As before, these are mustered in a class by the attraction of sameness; their classification has the effects already specified of mutual enlightenment and mutual exchangeability. To clench the operation we seize upon some one instance as a representative or typical instance, and our idea of this we call the abstract, or general, idea. We can here adopt a very refined method; we can draw an outline circle, omitting all the solid substance, and presenting only naked form to the eye; this is an abstraction of a higher order than we could gain by choosing a specimen circular object, as a wheel, for it leaves out all the features wherein circular bodies differ, and gives the point of agreement in a state of isolation. The mathematical diagram is thus a more perfect abstract idea than the idea of a river or a mountain, derived from a fair average specimen, or a composed river or mountain; these last scarcely come up to the meaning of an abstraction, although when properly managed they serve all the ends of such. But we may pass in the present case also from an abstract conception, or a diagram, to a Definition by descriptive words, and we may adopt this as our general conception, and use it in all our operations instead of or along with the other.+ The definition is in fact the highest form of the abstract idea, the form that we constantly fall back upon as the test or standard for trying any new claim of admission into the class, or for revising the list of those already admitted.

35. Induction, Inductive Generalization, Conjoined Properties, Affirmations, Propositions, Laws of Nature.-The contrast between Abstraction and Induction as here understood may be expressed thus: in the one a single isolated

A river may be defined' a natural current of water flowing in an open channel towards the sea,' or to that effect.

A circle is defined to be a line everywhere at an equal distance from a point which is the centre.

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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. When we bring all rivers into one class, and define the property common to all, we exemplify the first process; 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 substantive, is not an affirmation. When we affirm the uniform co-existence of two distinct 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 pretty 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 very apt to be mixed up and determined by one and the same effort of the sense and understanding. Still in order to possess the law that rivers form bars and deltas, we require to have observed the peculiarities of rivers, and to have been struck at some moment with 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. This is the first stage of an inductive discovery; it is the suggestion of a law of nature, which we are next to express and 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. But a general affirmation by language makes in this case a proposition, not a definition; it requires a verb for its expression, and carries 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. So, to take a more abstruse example, we identify the conjunction of transparent 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 more labour, than the operation of abstraction, being, however, much more pregnant with results. 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 wonderment at the distorting effect upon objects seen through water or glass. To take in the more abstruse and dissimilar instances,

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as the refractive influence of the air, there is needed a preparation similar to that already exemplified in the identification of burning and rusting. A powerful reach of the identifying faculty must always be implied in great scientific discoverers.

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 refraction occurs, and it is greater as the course is farther from the right angle, being greatest of all when the ray lays 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 reach of the faculty 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; the third law, which connects the distances of the planets from the sun with their periodic times, is a remarkable case. 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 got out of the existing stores of mathematical knowledge; but in all these discoveries of Kepler, we are perhaps to admire the aims and 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 different aspect, is itself the proof of a superior nature, and often the principal turning point of original discovery. The identifying faculty in Kepler showed itself less prominently in the strokes of detail, 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 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 stroke of the scientific intellect; it was the glory alike of Kepler and of Newton. A previously equipped mathematical mind, an indifference or superiority to poetical and fanciful aspects, and a high reach of identifying force, 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 vigorous 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 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