The Complete Aristotle (eng.)

conclusion is necessary, the middle through which it
was proved may yet quite easily be non-necessary. You can in fact
infer the necessary even from a non-necessary premiss, just as you
can infer the true from the not true. On the other hand, when the
middle is necessary the conclusion must be necessary; just as true
premisses always give a true conclusion. Thus, if A is necessarily
predicated of B and B of C, then A is necessarily predicated of C.
But when the conclusion is nonnecessary the middle cannot be
necessary either. Thus: let A be predicated non-necessarily of C
but necessarily of B, and let B be a necessary predicate of C; then
A too will be a necessary predicate of C, which by hypothesis it is
not.
    To sum up, then: demonstrative knowledge must be knowledge of a
necessary nexus, and therefore must clearly be obtained through a
necessary middle term; otherwise its possessor will know neither
the cause nor the fact that his conclusion is a necessary
connexion. Either he will mistake the non-necessary for the
necessary and believe the necessity of the conclusion without
knowing it, or else he will not even believe it-in which case he
will be equally ignorant, whether he actually infers the mere fact
through middle terms or the reasoned fact and from immediate
premisses.
    Of accidents that are not essential according to our definition
of essential there is no demonstrative knowledge; for since an
accident, in the sense in which I here speak of it, may also not
inhere, it is impossible to prove its inherence as a necessary
conclusion. A difficulty, however, might be raised as to why in
dialectic, if the conclusion is not a necessary connexion, such and
such determinate premisses should be proposed in order to deal with
such and such determinate problems. Would not the result be the
same if one asked any questions whatever and then merely stated
one’s conclusion? The solution is that determinate questions have
to be put, not because the replies to them affirm facts which
necessitate facts affirmed by the conclusion, but because these
answers are propositions which if the answerer affirm, he must
affirm the conclusion and affirm it with truth if they are
true.
    Since it is just those attributes within every genus which are
essential and possessed by their respective subjects as such that
are necessary it is clear that both the conclusions and the
premisses of demonstrations which produce scientific knowledge are
essential. For accidents are not necessary: and, further, since
accidents are not necessary one does not necessarily have reasoned
knowledge of a conclusion drawn from them (this is so even if the
accidental premisses are invariable but not essential, as in proofs
through signs; for though the conclusion be actually essential, one
will not know it as essential nor know its reason); but to have
reasoned knowledge of a conclusion is to know it through its cause.
We may conclude that the middle must be consequentially connected
with the minor, and the major with the middle.
7
    It follows that we cannot in demonstrating pass from one genus
to another. We cannot, for instance, prove geometrical truths by
arithmetic. For there are three elements in demonstration: (1) what
is proved, the conclusion-an attribute inhering essentially in a
genus; (2) the axioms, i.e. axioms which are premisses of
demonstration; (3) the subject-genus whose attributes, i.e.
essential properties, are revealed by the demonstration. The axioms
which are premisses of demonstration may be identical in two or
more sciences: but in the case of two different genera such as
arithmetic and geometry you cannot apply arithmetical demonstration
to the properties of magnitudes unless the magnitudes in question
are numbers. How in certain cases transference is possible I will
explain later.
    Arithmetical demonstration and the other sciences likewise
possess, each of them, their own genera; so that if the
demonstration is to pass from one sphere to another, the genus must
be either absolutely or to some extent the same. If this is not so,
transference is clearly impossible, because the extreme and the
middle terms must be drawn from the same genus: otherwise, as
predicated, they will not be essential and will thus be accidents.
That is why it cannot be proved by geometry that opposites fall
under one science, nor even that the product of two cubes is a
cube. Nor can the theorem of any one science be demonstrated by
means of another science,