Mr. Bentham also gives a bifurcate arrangement of animals after the method proposed by Duméril in his 'Zoologie Analytique,' this naturalist being distinguished by his clear perception of the logical importance of the method.

A more recent binary classification of the animal kingdom as regards the larger classes may be found in Professor Reay Greene's Manual of the Coelenterata,' p. 18.

Does Abstraction imply Generalization?

Before we can acquire a sound comprehension of the subject of classification we must answer a very difficult question, namely, whether logical abstraction does or does not always imply generalization. It comes to exactly the same thing if we ask whether a species may be coextensive with its genus, or whether, on the other hand, the genus must contain more than the species. To abstract logically is, as we have seen (vol. i. p. 33), to overlook or withdraw our notice from some point of difference. Whenever we form a class we abstract, for the time being, the differences of the objects so united in respect of some common quality. If, for instance, we class together a great number of objects as dwelling-houses, we overlook or abstract the fact that some dwelling-houses are constructed of stone, others of brick, wood, iron, &c. Very often at least the abstraction of a circumstance increases the number of objects included under a class according to the law of the inverse relation of the quantities of extension and intension (vol. i. p. 32). Dwelling-house is a wider term than brick dwelling-house. House, or building, is more general still than dwelling-house. But the question before us is, whether abstraction always increases the number of objects included in a class, which amounts to asking whether the law of the inverse relation of logical quantities is always true. The interest of the question

partly arises from the fact, that so high a philosophical authority as Mr. Herbert Spencer has denied that generalization is implied in abstraction, making this doctrine the ground for rejecting previous methods of classifying the sciences, and for forming an ingenious but peculiar method of his own. The question is also a fundamental one of the highest logical importance, and involves subtle difficulties which have made me long hesitate in forming a decisive opinion.

Let us attempt to answer the question by examination of a few examples. Compare the two classes gun and iron gun. It is certain that there are many guns which are not made of iron, so that abstraction of the circumstance 'made of iron' increases the extent of the notion. Next compare gun and metallic gun. All guns made at the present day consist, I believe, of metal, so that the two notions seem to be co-extensive; but guns were at first made of pieces of wood bound together like a tub, and as the logical term gun takes no account of time, it must include all guns that have ever existed. Here again extension increases as intension decreases. Compare once more 'steam-locomotive engine' and 'locomotive engine.' In the present day so far as I am aware all locomotives are worked by steam, so that the omission of that qualification might seem not to widen the term; but it is quite possible that in some future age a different motive power may be used in locomotives; and as there is no limitation of time in the use of logical terms, we must certainly assume that there is a class of locomotives not worked by steam, as well as a class that is worked by steam. When the natural class of Euphorbiacea was originally formed, all the plants known to belong to it were devoid of corollas; it would have seemed therefore that the two classes' Euphorbiaceae,' and 'Euphorbiaceæ devoid The Classification of the Sciences,' &c., 3rd edit. p. 7.

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of Corollas,' were of equal extent. Subsequently a number of plants plainly belonging to the same class were found in tropical countries, and they possessed bright coloured corollas. Naturalists believe with the utmost confidence that Ruminants' and 'Ruminants with cleft feet' are identical terms, because no ruminant has yet been discovered without cleft feet. But we can see no impossibility in the conjunction of rumination with uncleft feet, and it would be too great an assumption to say that we are certain that an example of it will never be met with. Instances can be quoted, without end, of objects being ultimately discovered which combined properties or forms which had never before been seen together. In the animal kingdom the Black Swan, the Ornithorhyncus Paradoxus, and more recently the singular fish called Ceratodus Forsteri, all discovered in Australia, have united characters never previously known to co-exist. At the present time deep-sea dredging is bringing to light many animals of a new and unprecedented nature. Singular exceptional discoveries may certainly occur in other branches of science. When Davy first succeeded in eliminating metallic potassium, it was a well established empirical law that all metallic substances possessed a high specific gravity, the least dense of all metals then known being zinc, of which the specific gravity is 71. Yet, to the surprise of chemists, potassium was found to be an undoubted metal of less density than water, its specific gravity being o'865.

It is hardly requisite to prove by further examples that our knowledge of nature is incomplete, so that we cannot safely assume the non-existence of new combinations. Logically speaking, we ought to leave a place open for animals which ruminate but are without cleft feet, and for every other possible intermediate form of animal, plant, or mineral. A purely logical classification must take account not only of what certainly does exist, but of what may in after ages be found to exist.

I will go a step further, and say that we must have places in our scientific classifications for purely imaginary existences. A very large proportion of the mathematical functions which are conceivable have no application to the circumstances of this world. Physicists certainly do investigate the nature and consequences of forces which nowhere exist. Newton's Principia' is full of such investigations. In one chapter of his Mécanique Céleste' Laplace indulges in a remarkable speculation as to what the laws of motion would have been if momentum instead of varying simply as the velocity had been a more complicated function of it. I have already mentioned (vol. i. p. 256) that Sir George Airy contemplated the existence of a world in which the laws of force should be such that a perpetual motion would be possible, and the Law of Conservation of Energy would not hold true.

Thought is not bound down to the limits of what is materially existent, but is circumscribed only by those Fundamental Laws of Identity, Contradiction and Duality, which were laid down at the outset. This is the point at which I should differ from Mr. Herbert Spencer. He appears to suppose that a classification is complete if it has a place for every existing object, and this may perhaps seem to be practically sufficient; but it is subject to two profound objections. Firstly, we do not know all that exists, and therefore in limiting our classes we are erroneously omitting multitudes of objects of unknown form and nature which may exist either on this earth or in other parts of space. Secondly, as I have explained, the powers of thought are not limited by material existences, and we may or, for some purposes, must imagine objects which probably do not exist, and if we imagine them we ought (strictly speaking) to find appropriate places for them in the classifications of science.

The chief difficulty of this subject, however, consists in