Posterior Analytics   

nature. An example of the latter is odd as an attribute of
number-though it is number's attribute, yet number itself is an
element in the definition of odd; of the former, multiplicity or the
indivisible, which are elements in the definition of number. In
neither kind of attribution can the terms be infinite. They are not
infinite where each is related to the term below it as odd is to
number, for this would mean the inherence in odd of another
attribute of odd in whose nature odd was an essential element: but
then number will be an ultimate subject of the whole infinite chain of
attributes, and be an element in the definition of each of them.
Hence, since an infinity of attributes such as contain their subject
in their definition cannot inhere in a single thing, the ascending
series is equally finite. Note, moreover, that all such attributes
must so inhere in the ultimate subject-e.g. its attributes in number
and number in them-as to be commensurate with the subject and not of
wider extent. Attributes which are essential elements in the nature of
their subjects are equally finite: otherwise definition would be
impossible. Hence, if all the attributes predicated are essential
and these cannot be infinite, the ascending series will terminate, and
consequently the descending series too.
If this is so, it follows that the intermediates between any two
terms are also always limited in number. An immediately obvious
consequence of this is that demonstrations necessarily involve basic
truths, and that the contention of some-referred to at the outset-that
all truths are demonstrable is mistaken. For if there are basic
truths, (a) not all truths are demonstrable, and (b) an infinite
regress is impossible; since if either (a) or (b) were not a fact,
it would mean that no interval was immediate and indivisible, but that
all intervals were divisible. This is true because a conclusion is
demonstrated by the interposition, not the apposition, of a fresh
term. If such interposition could continue to infinity there might
be an infinite number of terms between any two terms; but this is
impossible if both the ascending and descending series of
predication terminate; and of this fact, which before was shown
dialectically, analytic proof has now been given.

23

It is an evident corollary of these conclusions that if the same
attribute A inheres in two terms C and D predicable either not at all,
or not of all instances, of one another, it does not always belong
to them in virtue of a common middle term. Isosceles and scalene
possess the attribute of having their angles equal to two right angles
in virtue of a common middle; for they possess it in so far as they
are both a certain kind of figure, and not in so far as they differ
from one another. But this is not always the case: for, were it so, if
we take B as the common middle in virtue of which A inheres in C and
D, clearly B would inhere in C and D through a second common middle,
and this in turn would inhere in C and D through a third, so that
between two terms an infinity of intermediates would fall-an
impossibility. Thus it need not always be in virtue of a common middle
term that a single attribute inheres in several subjects, since
there must be immediate intervals. Yet if the attribute to be proved
common to two subjects is to be one of their essential attributes, the
middle terms involved must be within one subject genus and be
derived from the same group of immediate premisses; for we have seen
that processes of proof cannot pass from one genus to another.
It is also clear that when A inheres in B, this can be
demonstrated if there is a middle term. Further, the 'elements' of
such a conclusion are the premisses containing the middle in question,
and they are identical in number with the middle terms, seeing that
the immediate propositions-or at least such immediate propositions
as are universal-are the 'elements'. If, on the other hand, there is
no middle term, demonstration ceases to be possible: we are on the way