The Complete Aristotle (eng.)

differentia stated possesses anything that is
co-ordinate with it in a division; for, if not, clearly the one
stated could not be a differentia of the genus. For a genus is
always divided by differentiae that are co-ordinate members of a
division, as, for instance, by the terms ‘walking’, ‘flying’,
‘aquatic’, and ‘biped’. Or see if, though the contrasted
differentia exists, it yet is not true of the genus, for then,
clearly, neither of them could be a differentia of the genus; for
differentiae that are co-ordinates in a division with the
differentia of a thing are all true of the genus to which the thing
belongs. Likewise, also, see if, though it be true, yet the
addition of it to the genus fails to make a species. For then,
clearly, this could not be a specific differentia of the genus: for
a specific differentia, if added to the genus, always makes a
species. If, however, this be no true differentia, no more is the
one adduced, seeing that it is a co-ordinate member of a division
with this.
    Moreover, see if he divides the genus by a negation, as those do
who define line as ‘length without breadth’: for this means simply
that it has not any breadth. The genus will then be found to
partake of its own species: for, since of everything either an
affirmation or its negation is true, length must always either lack
breadth or possess it, so that ‘length’ as well, i.e. the genus of
‘line’, will be either with or without breadth. But ‘length without
breadth’ is the definition of a species, as also is ‘length with
breadth’: for ‘without breadth’ and ‘with breadth’ are
differentiae, and the genus and differentia constitute the
definition of the species. Hence the genus would admit of the
definition of its species. Likewise, also, it will admit of the
definition of the differentia, seeing that one or the other of the
aforesaid differentiae is of necessity predicated of the genus. The
usefulness of this principle is found in meeting those who assert
the existence of ‘Ideas’: for if absolute length exist, how will it
be predicable of the genus that it has breadth or that it lacks it?
For one assertion or the other will have to be true of ‘length’
universally, if it is to be true of the genus at all: and this is
contrary to the fact: for there exist both lengths which have, and
lengths which have not, breadth. Hence the only people against whom
the rule can be employed are those who assert that a genus is
always numerically one; and this is what is done by those who
assert the real existence of the ‘Ideas’; for they allege that
absolute length and absolute animal are the genus.
    It may be that in some cases the definer is obliged to employ a
negation as well, e.g. in defining privations. For ‘blind’ means a
thing which cannot see when its nature is to see. There is no
difference between dividing the genus by a negation, and dividing
it by such an affirmation as is bound to have a negation as its
co-ordinate in a division, e.g. supposing he had defined something
as ‘length possessed of breadth’; for co-ordinate in the division
with that which is possessed of breadth is that which possesses no
breadth and that only, so that again the genus is divided by a
negation.
    Again, see if he rendered the species as a differentia, as do
those who define ‘contumely’ as ‘insolence accompanied by jeering’;
for jeering is a kind of insolence, i.e. it is a species and not a
differentia.
    Moreover, see if he has stated the genus as the differentia,
e.g. ‘Virtue is a good or noble state: for ‘good’ is the genus of
‘virtue’. Or possibly ‘good’ here is not the genus but the
differentia, on the principle that the same thing cannot be in two
genera of which neither contains the other: for ‘good’ does not
include ‘state’, nor vice versa: for not every state is good nor
every good a ‘state’. Both, then, could not be genera, and
consequently, if ‘state’ is the genus of virtue, clearly ‘good’
cannot be its genus: it must rather be the differentia’. Moreover,
‘a state’ indicates the essence of virtue, whereas ‘good’ indicates
not the essence but a quality: and to indicate a quality is
generally held to be the function of the differentia. See, further,
whether the differentia rendered indicates an individual rather
than a quality: for the general view is that the differentia always
expresses a quality.
    Look and see, further, whether the