It is a peculiar property of equilateral triangles that they are equiangular, and vice versa, it is a peculiar property of equiangular triangles that they are equilateral. It is a property of crystals of the regular system that they are devoid of the power of double refraction, but this is not a property peculiar to them, because liquids and gases are devoid of the same property.

An Accident, the fifth and last of the Predicables, is any quality which may or may not belong to certain objects, and which has no connexion with the classification adopted. The particular size of a crystal does not in the slightest degree affect the form of the crystal, nor does the manner in which it is grouped with other crystals; these, then, are accidents as regards a crystallographic classification. With respect to the chemical composition of a substance, again, it is an accident whether the substance be crystallised or not, or whether it be organised or not. As regards botanical classification the absolute size of a plant is an accident. Thus we see that a logical accident is any quality or circumstance which is not known to be correlated with those qualities or circumstances forming the definition of the species.

The meanings of the Predicables can be clearly explained by our symbols. Let A be any definite group of qualities and B another quality or group of qualities; then A will constitute a genus, and AB, Ab will be species of it, B being the difference. Let C, D and E be other qualities or groups of qualities, and on examining the combinations in which A, B, C, D, E occur let them be as follows:

Here we see that wherever A is we also find C, so that C is a generic property; D occurs always with B, so that it constitutes a specific property, while E is indifferently present and absent, so as not to be related to any other letter; it represents, therefore, an accident. It will now be seen that the Logical Alphabet represents an interminable series of subordinate genera and species; it is but a concise symbolic statement of what was involved in the ancient doctrine of the Predicables.

Summum Genus and Infima Species.

As a genus means any class whatever which is regarded as composed of minor classes or species, it follows that the same class will be a genus in one point of view and a species in another. Metal is a genus as regards alkaline metal, a species as regards element, and any extensive system of classes consists of a series of subordinate, or as they are technically called, subaltern genera and species. The question, however, arises, whether such a chain of classes has a definite termination at either end. The doctrine of the old logicians was to the effect that it terminated upwards in a genus generalissimum or summum genus, which was not a species of any wider class. Some very general notion, such as substance, object, or thing, was supposed to be so comprehensive as to include all thinkable objects, and for all practical purposes this might be so. But as I have already explained (p. 74), we cannot really think of any object or class without thereby separating it from what is not that object or class. All thinking is relative, and implies discrimination, so that every class. and every logical notion must have its negative. If so, there is no such thing as a summum genus; for we cannot frame the requisite notion of a class forming it without. implying the existence of another class discriminated from it; add this new negative class to the supposed summum genus, and we form a still higher genus, which is absurd.

Although there is no absolute summum genus, nevertheless relatively to any branch of knowledge or any particular argument, there is always some class or notion which bounds our horizon as it were. The chemist restricts his view to material substances and the forces manifested in them; the mathematician extends his view so as to comprehend all notions capable of numerical discrimination. The biologist, on the other hand, has a narrower sphere containing only organised bodies, and of these the botanist and the zoologist take parts. In other subjects there may be a still narrower summum genus, as when the lawyer regards only reasoning beings of his own country together with their property.

In the description of the Logical Alphabet it was pointed out (p. 93) that every series of combinations is really the

development of a single class, denoted by X, which letter was accordingly placed in the first column of the table on p. 94. This is the formal acknowledgment of the principle clearly stated by De Morgan, that all reasoning proceeds within an assumed summum genus. But at the same time the fact that X as a logical term must have its negative x, shows that it cannot be an absolute summum genus.

There arises, again, the question whether there be any such thing as an infima species, which cannot be divided into minor species. The ancient logicians were of opinion that there always was some assignable class which could only be divided into individuals, but this doctrine appears to be theoretically incorrect, as Mr. George Bentham long ago stated.1 We may put an arbitrary limit to the subdivision of our classes at any point convenient to our purpose. The crystallographer would not generally treat as different species crystalline forms which differ only in the degree of development of the faces. The naturalist overlooks innumerable slight differences between animals which he refers to the same species. But in a strictly logical point of view classification might be carried on as long as there is a difference, however minute, between two objects, and we might thus go on until we arrive at individual objects which are numerically distinct in the logical sense attributed to that expression in the chapter upon Number. Either, then, we must call the individual the infima species or allow that there is no such thing at all.

The Tree of Porphyry.

Both Aristotle and Plato were acquainted with the value of bifurcate classification, which they occasionally employed in an explicit manner. It is impossible too that Aristotle should state the laws of thought, and employ the predicables without implicitly recognising the logical necessity of that method. It is, however, in Porphyry's remarkable and in many respects excellent Introduction to the Categories of Aristotle that we find the most distinct account of it. Porphyry not only fully and accurately describes the Predicables, but incidentally introduces an example for

1 Outline of a New System of Logic, 1827, p. 117.

illustrating those predicables, which constitutes a good specimen of bifurcate classification. Translating his words1 freely we may say that he takes Substance as the genus to be divided, under which are successively placed as SpeciesBody, Animated Body, Animal, Rational Animal, and Man. Under Man, again, come Socrates, Plato, and other particular men. Now of these notions Substance is the genus generalissimum, and is a genus only, not a species. Man, on the other hand, is the species specialissima (infima species), and is a species only, not a genus. Body is a species of substance, but a genus of animated body, which, again, is a species of body but a genus of animal. Animal is a species of animated body, but a genus of rational animal, which, again, is a species of animal, but a genus of man. Finally, man is a species of rational animal, but is a species merely and not a genus, being divisible only into particular men.

Porphyry proceeds at some length to employ his example in further illustration of the predicables. We do not find in Porphyry's own work any scheme or diagram exhibiting this curious specimen of classification, but some of the earlier commentators and epitome writers drew what, has long been called the Tree of Porphyry. This diagram, which may be found in most elementary works on Logic,2 is also called the Ramean Tree, because Ramus insisted much upon the value of Dichotomy. With the exception of Jeremy Bentham and George Bentham, hardly any modern logicians have shown an appreciation of the value of bifurcate classification. The latter author has treated the subject, both in his Outline of a New System of Logic (pp. 105-118), and in his earlier work entitled Essai sur la Nomenclature et la Classification des Principales Branches d'Art-et-Science (Paris, 1823), which consists of a free translation or improved version of his uncle's Essay on Classification in the Chrestomathia. Some interest attaches to the history of the Tree of Porphyry and Ramus, because it is the prototype of the Logical Alphabet which lies at the basis of logical method. Jeremy Bentham speaks truly

1 Porphyrii Isagoge, Caput ii. 24.

2 Jevons, Elementary Lessons in Logic, p. 104.

3 Chrestomathia; being a Collection of Papers, &c. London, 1816, Appendix V.

of "the matchless beauty of the Ramean Tree." After fully showing its logical value as an exhaustive method of classification, and refuting the objections of Reid and Kames, on a wrong ground, as I think, he proceeds to inquire to what length it may be carried. He correctly points out two objections to the extensive use of bifid arrangements, (1) that they soon become impracticably extensive and unwieldy, and (2) that they are uneconomical. In his day the recorded number of different species of plants was 40,000, and he leaves the reader to estimate the immense number of branches and the enormous area of a bifurcate table which should exhibit all these species in one scheme. He also points out the apparent loss of labour in making any large bifurcate classification; but this he considers to be fully recompensed by the logical value of the result, and the logical training acquired in its execution. Jeremy Bentham, then, fully recognises the value of the Logical Alphabet under another name, though he apprehends also the limit to its use placed by the finiteness of our mental and manual powers.

Does Abstraction imply Generalisation?

Before we can acquire a sound comprehension of the subject of classification we must answer the very difficult question whether logical abstraction does or does not imply generalisation. 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 (p. 27), 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 we class together a great number of objects as dwelling-houses, we overlook the fact that some dwelling-houses are constructed of stone, others of brick, wood, iron, &c. 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 (p. 26). Dwelling-house is a wider term than brick-dwelling-house. House is more general than dwelling-house. But the