The Complete Aristotle (eng.)

genus. If on the other hand
this synthesis is applicable to no subject other than the
individual triads, it will be identical with the being of triad,
because we make the further assumption that the substance of each
subject is the predication of elements in its essential nature down
to the last differentia characterizing the individuals. It follows
that any other synthesis thus exhibited will likewise be identical
with the being of the subject.
    The author of a hand-book on a subject that is a generic whole
should divide the genus into its first infimae species-number e.g.
into triad and dyad-and then endeavour to seize their definitions
by the method we have described-the definition, for example, of
straight line or circle or right angle. After that, having
established what the category is to which the subaltern genus
belongs-quantity or quality, for instance-he should examine the
properties ‘peculiar’ to the species, working through the proximate
common differentiae. He should proceed thus because the attributes
of the genera compounded of the infimae species will be clearly
given by the definitions of the species; since the basic element of
them all is the definition, i.e. the simple infirma species, and
the attributes inhere essentially in the simple infimae species, in
the genera only in virtue of these.
    Divisions according to differentiae are a useful accessory to
this method. What force they have as proofs we did, indeed, explain
above, but that merely towards collecting the essential nature they
may be of use we will proceed to show. They might, indeed, seem to
be of no use at all, but rather to assume everything at the start
and to be no better than an initial assumption made without
division. But, in fact, the order in which the attributes are
predicated does make a difference—it matters whether we say
animal-tame-biped, or biped-animal-tame. For if every definable
thing consists of two elements and ‘animal-tame’ forms a unity, and
again out of this and the further differentia man (or whatever else
is the unity under construction) is constituted, then the elements
we assume have necessarily been reached by division. Again,
division is the only possible method of avoiding the omission of
any element of the essential nature. Thus, if the primary genus is
assumed and we then take one of the lower divisions, the dividendum
will not fall whole into this division: e.g. it is not all animal
which is either whole-winged or split-winged but all winged animal,
for it is winged animal to which this differentiation belongs. The
primary differentiation of animal is that within which all animal
falls. The like is true of every other genus, whether outside
animal or a subaltern genus of animal; e.g. the primary
differentiation of bird is that within which falls every bird, of
fish that within which falls every fish. So, if we proceed in this
way, we can be sure that nothing has been omitted: by any other
method one is bound to omit something without knowing it.
    To define and divide one need not know the whole of existence.
Yet some hold it impossible to know the differentiae distinguishing
each thing from every single other thing without knowing every
single other thing; and one cannot, they say, know each thing
without knowing its differentiae, since everything is identical
with that from which it does not differ, and other than that from
which it differs. Now first of all this is a fallacy: not every
differentia precludes identity, since many differentiae inhere in
things specifically identical, though not in the substance of these
nor essentially. Secondly, when one has taken one’s differing pair
of opposites and assumed that the two sides exhaust the genus, and
that the subject one seeks to define is present in one or other of
them, and one has further verified its presence in one of them;
then it does not matter whether or not one knows all the other
subjects of which the differentiae are also predicated. For it is
obvious that when by this process one reaches subjects incapable of
further differentiation one will possess the formula defining the
substance. Moreover, to postulate that the division exhausts the
genus is not illegitimate if the opposites exclude a middle; since
if it is the differentia of that genus, anything contained in the
genus must lie on one of the two sides.
    In establishing a definition by division one should keep three
objects in view: (1) the admission only of elements in