230.] (ß) The statement of the second element of the notion, _i.e._ of the specific character of the universal as particularising, is given by Division in accordance with some external consideration. * * * * * Division we are told ought to be complete. That requires a principle or ground of division so constituted, that the division based upon it embraces the whole extent of the region designated by the definition in general. But, in division, there is the further requirement that the principle of it must be borrowed from the nature of the object in question. If this condition be satisfied, the division is natural and not merely artificial, that is to say, arbitrary. Thus, in zoology, the ground of division adopted in the classification of the mammalia is mainly afforded by their teeth and claws. That is so far sensible, as the mammals themselves distinguish themselves from one another by these parts of their bodies; back to which therefore the general type of their various classes is to be traced. In every case the genuine division must be controlled by the notion. To that extent a division, in the first instance, has three members: but as particularity exhibits itself as double, the division may go to the extent even of four members. In the sphere of mind trichotomy is predominant, a circumstance which Kant has the credit of bringing into notice. 231.] (γ) In the concrete individuality, where the mere unanalysed quality of the definition is regarded as a correlation of elements, the object is a synthetical nexus of distinct characteristics. It is a Theorem. Being different, these characteristics possess but a mediated identity. To supply the materials, which form the middle terms, is the office of Construction: and the process of mediation itself, from which cognition derives the necessity of that nexus, is the Demonstration. As the difference between the analytical and synthetical methods is commonly stated, it seems entirely optional which of the two we employ. If we assume, to start with, the concrete thing which the synthetic method presents as a result, we can analyse from it as consequences the abstract propositions which formed the pre-suppositions and the material for the proof. Thus, algebraical definitions of curved lines are theorems in the method of geometry. Similarly even the Pythagorean theorem, if made the definition of a right-angled triangle, might yield to analysis those propositions which geometry had already demonstrated on its behoof. The optionalness of either method is due to both alike starting from an external pre-supposition. So far as the nature of the notion is concerned, analysis is prior; since it has to raise the given material with its empirical concreteness into the form of general abstractions, which may then be set in the front of the synthetical method as definitions.