(2) It will be observed that though we make a dichotomy at every step, yet one of the two subdivisions is at once laid aside and subjected to no further analysis. This reminds us how distinct is the process thus indicated from that of modern Classification. What is desired in the latter is a complete enumeration of all the subsidiary classes. We start with no preconceived intention of reaching one such special or ultimate class rather than another. But here the rejection at every

step of one of the divisions, and the retention of the other, shows a design of guiding our path towards a definite point: in the case in question what we want to reach is man. What we have before us, in fact, is a method of subdivision rather than a system of classification.

(3) The process gave rise to certain technical terms. The line along which we thus direct our course in order to come down to the particular class or individual at which we aim, is called the predicamental line. The principle of selection adopted at each step in order to break up the class into two divisions is called the fundamentum divisionis.

(4) The non-formal or material character of the process is obvious. Thus compositum does not contain, or in any direct way suggest, the attribute vivens. This latter is therefore an accident which has to be supplied by the special knowledge of the fact that some compositum is, and some is not, vivens. This attribute in fact must be a separable accident of the subject class.

In the common, or Porphyrian Tree, we do as a matter of fact make our division into two contradictory classes at every stage, because this is the most convenient plan. But several of the old writers, e.g. Sanderson, point out that we need not be over particular in adhering to such careful dichotomy. Those whom they had in view in this protest were doubtless the followers of Ramus, whose innovations however in this respect, as in others, seem to be extremely small compared with the outcry raised by their leader and by their opponents. So far as any definite doctrine on this subject of Division is to be found amongst them, it seems to consist in little more than a recommendation to keep to a stringent dichotomy by always dividing into pairs of contradictories. This is certainly what is intended by those1

1 Jevon's Pr. of Sc. 704.

who have with J. Bentham spoken of "the matchless beauty of the Ramean Tree".

What these writers seem to overlook is the important distinction between dichotomy as a process for reaching some. desired class, and a dichotomous final arrangement of our classes. As regards the latter no good reason can be offered why the subdivision should be thus bifurcate, and it would be the merest pedantry to insist upon making it so. In the arrangement of plants, for instance, most systematists start as the primary step with a three-fold division, viz. that into exogens, endogens, and akrogens. Each of these three great families is then subdivided into a varying number of orders, these again into genera, and so on. In looking through such a scheme we shall find few, if any, traces of dichotomy.

Where this principle really does serve a useful purpose is in testing a scheme of classification, and still more in appealing to such a scheme in order to detect some class for which we are in search. This is a topic which will come before us more fully in the next chapter, so I will merely offer a few words of explanation here. Consider, for example, the Analytical Key prefixed to such a work as Bentham's British Flora. We appeal to this when we have some plant actually in hand whose species and technical name we wish to ascertain. Dichotomy is here employed at almost every step. We are set to determine such questions as, whether the plant has a true flower or not; if it has, whether it has both calyx and corolla or not, and so forth. This bifurcate alternative is purposely made to confront us almost every time, and is continued until at length the one finally admissible alternative comprises the species of which we are in search. But to confound such a method as this with a final scientific classification is to con- V found the scaffolding with the building. The path by which the novice proceeds in determining the plant is a very different thing from that systematic arrangement which the botanist adopts as his final result. Not one of the dichotomies by which we thus guide our way need correspond to any of the various orders or families which form the higher classes: in fact we shall find ourselves continually cutting across natural and fundamental units of arrangement. We employ for con

venience a dichotomous path, but we have no intention of retaining a dichotomous result in the end.

Rules for Division. The technical rules for Division correspond in several respects, allowing for the different subjectmatter, to those given for Definition. They are thus given in most of the manuals:

(1) The dividing members are to exhaust the whole class. (2) The divided class must be wider than each of the divisions, and equal to them all collectively.

(3) The dividing members must be distinct and disparate.

(4) The division must be orderly, and each step must as far as possible be a proximate one: i.e. we should not, at a single step, spring to a subdivision so remote that it could have been reached by several steps.

The reader will easily see that the above rules represent a compromise between the formal and the material, as is indeed the case throughout nearly all the treatment of Logic by the later scholastic writers. They leave moreover a good deal to be desired in the way of clearness and terseness. Thus the first three might surely better be summed up in the statement that the subdivisions of any class must be mutually exclusive and collectively exhaustive.

The substance of the third rule is often given in the form of a caution to avoid cross-division. By cross-division is meant any arrangement by which two or more of the subdivisions run across each other, so that some of the objects fall into more than one of the classes. In a bifurcate division such a fault is not likely to be committed, for the two subdivisions will most probably be obviously contradictory of each other, either formally or materially; but when we are concerned with several subdivisions we must be on our guard here. It is said (Fowler, p. 60) that cross-division arises from the adoption of more than one fundamentum divisionis, but this statement, I think, needs some explanation. The fundamentum divisionis is nothing else than the attribute which furnishes the ground of distinction between the members of the class. Now where we are making only two subdivisions we can effect this by resort to one attribute only, viz. by taking it or leaving it; and so obtaining two contradictory classes. But where we are making more

than two subdivisions we must resort to more than one attribute to effect this; in other words, we must admit more than one fundamentum divisionis. What is really meant by the caution in question is rather a postulate of practical good sense than a formal rule. Three or more subdivisions will generally be found to be produced by the plan of starting with some important attribute, and, instead of simply taking or leaving ✓ it, endeavouring to find a series of slight modifications of it. And in doing so we are recommended to be careful that these modifications result in mutually exclusive classes. Thus if we were to divide 'graduates' into those who are of Cambridge and those who are not, this arrangement must be exhaustive and mutually exclusive. But if we divide them into those who are of Cambridge, of Oxford, and of London, we might find that some were reckoned twice over, and some three

times.

CHAPTER XIII.

DIVISION, CONTINUED: CLASSIFICATION.

THE traditional logical process of Division, as we saw, did did not lead to much result. Indeed, in a treatise on Induction, it was somewhat of a departure from strict consistency to touch upon such a procedure at all. It was however desirable to do this, partly because the continuity of technical language in the subject rendered it necessary to give some explanation of the original meaning of certain terms still in use; and partly also because, by seeing what and where were the main deficiencies of the old treatment, we may be better able to see where improvement is needed.

When we lay aside the old restraints imposed by principle and tradition, we may see our way to two very different developments, both of which may be considered to take their start from the customary logical conception of Division. That conception, as we saw, was hampered in execution owing to the fact that though it professed to be, and really aimed at being, a formal process, it nevertheless attended sufficiently to the dictates of common sense to endeavour to render itself practically useful. This compromise naturally acted as a check. What we now propose to do is to see what comes of the attempt to develope each of these two sides, the formal and the material, separately, so that each shall be as little hampered as possible by the other.

I. The first of these developments opens out to us the field of what may be called Symbolic Logic. This is an extension of Logic to which I have already had occasion to allude more than once. As however it is still one which is extremely unfamiliar to most readers, and as it also demands a peculiar mode of treatment necessitating a constant employment of what are