marks, has been considered under the previous topic. The second, the extensive whole, whose parts are kinds or things unfolded by logical division, is more especially before us.

A mathematical whole is an individual, either objective or subjective, viewed as a quantity, and consisting of parts actually separable. These can be evolved only by the whole being cut asunder, i. e., by partition, which must be clearly distinguished from logical division. Such parts are neither marks nor kinds. This whole is of two species. First, the integral whole is one in which its parts originate. They may be homogeneous, as a polygon severed into similar triangles; or heterogeneous, as the human body, consisting of head, trunk, and limbs. Anatomy is a science of partition, of dissection. A sword, which divides into sabre, rapier, etc., is parted into hilt and blade, etc. Secondly, the collective whole is an aggregation of similar parts, one originated by the parts. Such are the notions of an army, a forest, a town. These are formed by the repetition of the notions of a soldier, a tree, a house. We must not confound the notion army, which is a general or class notion, a logical whole, with the notion an army taken as a collective notion, an individual thing formed by a collection of other individual things.

§3. It has been already seen how by specialization we form subordinate groups, or species. Since pure Logic considers only the form, each genus or universal whole can contain under it only two species, marked with A and non-A. For A being a generic difference, i. e., a mark not found in the genus or divisum, but found in some of its members, we know a priori, without any research into the matter of thought, that the members are exclusive of each other and exhaustive of the divisum. This is division by dichotomy, and the members are contradictories. For illustration: animals are rational and irrational, or vertebrate and invertebrate; angles are right and oblique; the oblique are acute and obtuse; the ancients were Greeks and barbarians, or Jews and Gentiles, or bond and free. The process viewed intensively, as thinking in marks, is called determination; viewed extensively, as establishing species, is called specification. In relation to each other, the two species are co-ordinate, as being of equal rank in respect of the divisum; but we remark that either may be of indefinitely greater breadth than the other.

A non-A

The negative member of the dichotomy is characterized by the absence of the mark A, or, in other words, by the negative mark non-A.

Hence we have a peculiar class of concepts called negative, privative, or infinitated concepts. In some cases their sphere is very wide, denoting almost everything, and connoting very little, almost nothing positive. E. g., Ungodly, unhappy, apathy, blindness, senseless, dark, cold, infinite, freedom, shadow, atheist, idle, sober, dead, etc.

§ 4. When the process is pursued beyond a single division,-that is, when a species is regarded as a subaltern genus and subdivided into lower species, then it is requisite at the outset to select some one mark of the original divisum as a ground or principle on or in reference to which the several divisions shall be made. This generic mark so chosen is called the ground of division, fundamentum divisionis. For example, in dividing Mankind we select his religious character or creed as the ground of division, and, subdividing upon the same principle, we obtain a logical series, thus:

The number of distinct forms in which this mark, the principle of division, appears in the things to be divided will determine the extent of the series. This procedure obviously has respect to the matter of thought, and is not strictly pure Logic. We add that, if it is proposed to establish a real division, i. e., one unfolding the true nature of the things contained under the divisum, or, in short, one rigidly scientific, it is requisite to select as a principle of division an essential and original mark of the divisum, and to adhere to it throughout. So logical perfection requires, but it is, in fact, rarely practicable in an extended series.

And this suggests that the distinction made between nominal and real definition may well be carried out relative to division. A nominal or artificial division would be one made for some transient purpose

or to attain a practical end; or one tentative and precursory to a real division; or one popularly accepted and useful, such as the hundreds that may be observed on every page, and in every few minutes of conversation. A real or scientific division would be one proposing to divide notions and things according to their true and essential nature, in order to attain correct objective knowledge of things as they are. Such division develops natural kinds, and is to be looked for in the more refined sciences. The Linnæan artificial divisions of flora were precursory and tentative; those of Jussieu's natural system are real and more rigidly scientific.

§ 5. In divisions not purely logical, but having respect to the matter, it often happens that we have those more than dichotomous; we may have a trichotomy (rpixa, threefold; réurer, to cut), or a polytomy. E. g., "Doctrines are helpful, harmless, hurtful." This arises from two causes. Either it is an abbreviation by which a series of subordinate species is condensed into one co-ordinate statement, as, "Angles are acute, right, and obtuse;" or, "Mankind are Christians, Jews, Mohammedans, polytheists, and atheists;" or, "Plants are endogens, exogens, acrogens." Or it arises from the lack of a sharp definition of our concepts. There is between very many of our thoughts a wide border-land which it is impossible to assign clearly to either, constituting a tertium quid, a third species which it is necessary to insert in order to exhaust the divisum. Thus we have our twenty-four hours divided, with reference to their light, into day, twilight, and night. So we have "White, gray, black;" "Riches, competence, want;" "Young, middle-aged, old," etc. For many of these mediate species we have no names, as between sick and well; strong and weak; long and short; wise and foolish, etc.

We have remarked that in a strictly logical division the two members, A and non-A, are contradictories; no member of that universe can be both, nor can be neither. In a trichotomy or a polytomy the members are disparate notions. Thus, brook, creek, river, are disparate notions contained under the genus streams of water. The two extremes of such a division, as brook and river, are logical contraries. A thing of this genus cannot be both, but may be neither; it may be the tertium quid.

Let it be also noticed that in many cases a notion which seems to have been originally a mere negative of its co-ordinate notion has had thought into it a positive character, so that either may be now thought

as positive and the other as negative; or perhaps both are really positive, and no mere negative exists. Thus, white and black,—the mere negative is dark. So true and untrue or false; happy and unhappy; honor and dishonor; temperate and intemperate, which last has become inverted. So protestant. So also pleasure and pain. Plato taught that pleasure is merely the absence or negation of pain; the Hedonists taught the reverse; but unquestionably both are positive. Also, it was taught anciently that evil is the mere negation of good; and to-day there are those who hold that good is the absence of evil; but both good and evil are positive, and in this case there is no intermediate ground. Actions are either good or bad; there are no indifferent actions.

Finally, a polytomous division admits of one, and only one, strictly privative or negative notion. Thus, "Some men lend, some borrow, some do both, others do neither;" "Plants are monocotyledonous, dicotyledonous, and acotyledonous or flowerless." The intermediate ground, well named the undefined or indifferent part, often takes this negative character; as "Men are very industrious, positively lazy, and neither the one nor the other."

§ 6. The importance of the correlative processes of definition and division cannot well be overrated. They are the reflex respectively of analysis and synthesis, in the balance of which lies the perfection of knowledge." "Such is the excellency of definition and distribution," says an old logician, "that almost they alone do suffice for the absolute putting-down of any art; therefore, the wise Socrates, in Phædro Platonis, saith that if he find any man who can cunningly divide, he will follow his steps and admire him for a god."

We shall do well, then, to observe the following practical directions. From the account given, we first present for forming divisions this CANON: Assemble representative instances of the objects denoted by the divisum, and, having fixed upon a generic mark as a principle of division, select a mark immediately involving this principle for a specific difference; then divide the denotation by affirming the specific difference of the species which it determines, denying it of all other contained objects. In subsequent divisions pursue a similar course,

* When a notion is adequately defined, and thoroughly divided, we have attained a complete knowledge of its characters and kinds, and this process exhausts its content. See Kant's Logik, § 98.

involving in each new specific difference the one immediately preceding, and, of course, the original principle of division.

To this canon we now append the following RULES, useful as a further guide to correct division:

1st. Each division throughout a series should be governed by the same principle, which should be an essential and important mark of the first divisum.

The intervention of a different ground of division in the series gives rise to the logical fault called "Cross division." Thus: "Men are Europeans, Americans, negroes, and pagans." This is an abbreviated series in which the ground of the first division is geographical; the second, color; the third, religion. The members evidently cross or overlap each other; a man may be all of the last three. This very common vice, when more concealed, is detected and the division tested by dichotomy. That is to say, any trichotomy or polytomy, if correct, may be reduced to a dichotomy by taking any one member as positive and including all the others under its negative. If this can be done with each member, without cutting any one, the division is sound. Thus, "Physical substance is animal, vegetable, mineral." Tested: "P. S. is A and non-A (=V+M);" or is "V and non-V (=A+M);" or is "M and non-M (=A+V)." This test applied to the following will clearly demonstrate that it is logically vicious: "The religious sects of Great Britain are Catholic, Calvinist, Episcopal, and Dissenting."

The principle selected must be essential, if we would attain to real, scientific knowledge. It must be important, determining many other attributes, if we would evolve an extended and valuable series. The purpose with which an artificial division is made determines its ground. In civil affairs it would be useless and absurd to divide men into horsemen and footmen; but in military affairs it is important. Words in a grammar are divided according to syntactical relations; in a dictionary, alphabetically. Medical botany and the florist's manual will divide plants differently, and both deviate from Jussieu. We sort our books by size, to fit our shelves; by subjects, for handy reference; by binding, for show. 2d. Dividing members must, as parts, equal the whole divisum. No one must exhaust the divisum; as, "Mankind are rational beings and politicians." Together they must exhaust it; as, "Governments are monarchies, oligarchies, and democracies." This is insufficient; there are other forms of government. Together they must not more than exhaust it; as, "Vertebrates are quadrumana, bimana, quad