Division

Division (from Lat. dividere, to break up into parts, separate), a general term for the action of breaking up a whole into parts. Thus, in political economy, the phrase “division of labour” implies the assignment to particular workmen of the various portions of a whole piece of work; in mathematics division is the process of finding how many times one number or quantity, the “divisor,” is contained in another, the “dividend” (see Arithmetic and Algebra); in the musical terminology of the 17th and 18th centuries, the term was used for rapid passages consisting of a few slow notes amplified into a florid passage, i.e. into a larger number of quick ones. The word is used also in concrete senses for the parts into which a thing is divided, e.g. a division of an army, an administrative or electoral division; similarly, a “division” is taken in a legislative body when votes are recorded for and against a proposed measure.

In logic, division is a technical term for the process by which a genus is broken up into its species. Thus the genus “animal” may be divided, according to the habitat of the various kinds, into animals which live on land, those which live in water, those which live in the air. Each of these may be subdivided according to whether their constituent members do or do not possess certain other qualities. The basis of each of these divisions is called the fundamentum divisionis. It is clear that there can be no division in respect of those qualities which make the genus what it is. The various species are all alike in the possession of the generic attributes, but differ in other respects; they are “variations on the same theme” (Joseph, Introduction to Logic, 1906); each one has the generic, and also certain peculiar, qualities (differentiae), which latter distinguish them from other species of the same genus. The process of division is thus the obverse of classification (q.v.); it proceeds from genus to species, whereas classification begins with the particulars and rises through species to genus. In the exact sciences, and indeed in all argument both practical and theoretical, accurate division is of great importance. It is governed by the following rules. (1) Division must be exhaustive; all the members of the genus must find a place in one or other of the species; a captain who selects for his team skilful batsmen and bowlers only is guilty of an incomplete division of the whole function of a cricket team by omitting to provide himself with good fielders. Rectilinear figures cannot be divided into triangles and quadrilaterals because there are rectilinear figures which have more than four sides. On the other hand, triangles can be divided into equilateral, isosceles and scalene, since no other kind of triangle can exist. (2) Division must be exclusive, that is, each species must be complete in itself and not contain members of another species. No member of a genus must be included in more than one of the species. (3) In every division there must be but one principle (fundamentum divisionis). The members of a genus may differ from one another in many respects, e.g. books may be divided according to external form into quarto, octavo, &c., or according to binding into calf, cloth, paper-backed and so on. They cannot, however, be divided logically into quarto, paper-backed, novels and remainders. When more than one principle is used in a division it is called “cross division.” (4) Division must proceed gradually (“Divisio non facit saltum”), i.e. the genus must be resolved into the next highest (“proximate”) species. To go straight from a summum genus to very small species is of no scientific value.

It is to be observed that logical division is concerned exclusively with universals or concepts; division is of genus and species, not of particulars. Two other kinds of division are recognized:—metaphysical division, the separation in thought of the various qualities possessed by an individual thing (a piece of lead has weight, colour, &c), and physical division or partition, the breaking up of an object into its parts (a watch is thought of as being composed of case, dial, works, &c.). Logical division is closely allied with logical definition (q.v.).