The Complete Aristotle (eng.)

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.
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    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 to the basic truths. Similarly if A does not
inhere in B, this can be demonstrated if there is a middle term or
a term prior to B in which A does not inhere: otherwise there is no
demonstration and a basic truth is reached. There are, moreover, as
many ‘elements’ of the demonstrated conclusion as there are middle
terms, since it is propositions containing these middle terms that
are the basic premisses on which the demonstration rests; and as
there are some indemonstrable basic truths asserting that ‘this is
that’ or that ‘this inheres in that’, so there are others denying
that ‘this is that’ or that ‘this inheres in that’-in fact some
basic truths will affirm and some will deny being.
    When we are to prove a conclusion, we must take a primary
essential predicate-suppose it C-of the subject B, and then suppose
A similarly predicable of C. If we proceed in this manner, no
proposition or attribute which falls beyond A is admitted in the
proof: the interval is constantly condensed until subject