It is a peculiar property of equilateral triangles that they are equiangular, or, 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 vitreous and other amorphous transparent solids, such as glass, together with all liquids and gases, are also 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 nature of the crystal, nor does the manner in which it may be 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 crystallized or not, or whether it be organized or not. As regards botanical classification the absolute size of a plant is an accident, due to external circumstances. 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 use of the Predicables can be very concisely explained by our symbols. Thus, let A be any definite group of qualities and B another quality; 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, 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 C is also found, 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 in any way correlated with any of the other letters; it represents, therefore, an accident. It will now be seen that the Logical Abecedarium really 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 any such 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 (vol. i. p. 88), 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, but which with the supposed summum genus will form the species of a still higher genus, which is absurd.

Although there is no absolute summum genus, nevertheless relatively to any branch of knowledge or any special 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 organized 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 living and reasoning beings of his own country.

In the description of the Logical Abecedarium, it was pointed out (vol. i. p. 108) that every series of combinations was really the development of some one single class, denoted by X, which letter indeed was accordingly placed in the first column of the table on p. 109. This is the formal acknowledgment of the principle clearly stated by De Morgan, that all reasoning proceeds within some 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 any smaller 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 me theoretically incorrect, as Mr. George

Bentham indeed long ago stated P. We may always put an arbitrary limit to the subdivisions of our classification at any point convenient to our purpose. The crystallographer would not generally consider as different species of crystalline form those which differ only in the degree of development of the faces. The naturalist overlooks innumerable slight differences between plants or animals which he refers to the same species. But in a strictly logical point of view classification might be carried on so long as there is a single point of difference, however minute, between two objects, and we might thus go on until we arrived at individual objects which are numerically distinct in the logical sense attributed to that expression in the chapter upon Number. We must either, then, call the individual the infima species or allow that there is no such species at all.

The Tree of Porphyry.

The bifurcate method of classification, arising as it does from the primary laws of thought, is the very foundation of all strict scientific method, and its application in formal logic constitutes the method of Indirect Inference, of which the nature and importance were shown in Chapter VI. So slight, however, has been the attention paid to this all important subject, that I shall in this case break the rule which I have laid down for myself, not to mingle the subject of logic as a science with the history of logic.

Both Plato and Aristotle were fully acquainted with the value of bifurcate division which they occasionally employed in an explicit manner. It is impossible, too,

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

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 incidently introduces an example for illustrating those predicables, which constitutes a good specimen of bifurcate classification. Translating his words freely we may say that he takes Substance as the genus to be divided, under which are successively placed as Species-Body, 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.

Thus in the 'Epitome Logica' of Nicephorus Blemmidas,

a 'Porphyrii Isagoge,' Caput ii. 24.