§ 393. The term required to be divided is known as the Totum Divisum or Divided Whole. It might also be called the Dividend.
§ 394. The classes into which the dividend is split up are called the Membra Dividentia, or Dividing Members.
§ 395. Only two rules need be given for division—
(1) The division must be conducted on a single basis.
(2) The dividing members must be coextensive with the divided whole.
§ 396. More briefly, we may put the same thing thus—There must be no cross-division (1) and the division must be exhaustive (2).
§ 397. The rule, which is commonly given, that each dividing member must be a common term, is already provided for under our definition of the process.
§ 398. The rule that the dividend must be predicable of each of the dividing members is contained in our second rule; since, if there were any term of which the dividend were not predicable, it would be impossible for the dividing members to be exactly coextensive with it. It would not do, for instance, to introduce mules and donkeys into a division of the term horse.
§ 399. Another rule, which is sometimes given, namely, that the constituent species must exclude one another, is a consequence of our first; for, if the division be conducted on a single principle, the constituent species must exclude one another. The converse, however, does not hold true. We may have a division consisting of mutually exclusive members, which yet involves a mixture of different bases, e.g. if we were to divide triangle into scalene, isosceles and equiangular. This happens because two distinct attributes may be found in invariable conjunction.
§ 400. There is no better test, however, of the soundness of a division than to try whether the species overlap, that is to say, whether there are any individuals that would fall into two or more of the classes. When this is found to be the case, we may be sure that we have mixed two or more different fundamenta divisionis. If man, for instance, were to be divided into European, American, Aryan, and Semitic, the species would overlap; for both Europe and America contain inhabitants of Aryan and Semitic origin. We have here members of a division based on locality mixed up with members of another division, which is based on race as indicated by language.