The Complete Aristotle (eng.)

if it is supposed to be commensurate. One infers
syllogistically that odd numbers come out equal to evens, and one
proves hypothetically the incommensurability of the diagonal, since
a falsehood results through contradicting this. For this we found
to be reasoning per impossibile, viz. proving something impossible
by means of an hypothesis conceded at the beginning. Consequently,
since the falsehood is established in reductions ad impossibile by
an ostensive syllogism, and the original conclusion is proved
hypothetically, and we have already stated that ostensive
syllogisms are effected by means of these figures, it is evident
that syllogisms per impossibile also will be made through these
figures. Likewise all the other hypothetical syllogisms: for in
every case the syllogism leads up to the proposition that is
substituted for the original thesis; but the original thesis is
reached by means of a concession or some other hypothesis. But if
this is true, every demonstration and every syllogism must be
formed by means of the three figures mentioned above. But when this
has been shown it is clear that every syllogism is perfected by
means of the first figure and is reducible to the universal
syllogisms in this figure.
24
    Further in every syllogism one of the premisses must be
affirmative, and universality must be present: unless one of the
premisses is universal either a syllogism will not be possible, or
it will not refer to the subject proposed, or the original position
will be begged. Suppose we have to prove that pleasure in music is
good. If one should claim as a premiss that pleasure is good
without adding ‘all’, no syllogism will be possible; if one should
claim that some pleasure is good, then if it is different from
pleasure in music, it is not relevant to the subject proposed; if
it is this very pleasure, one is assuming that which was proposed
at the outset to be proved. This is more obvious in geometrical
proofs, e.g. that the angles at the base of an isosceles triangle
are equal. Suppose the lines A and B have been drawn to the centre.
If then one should assume that the angle AC is equal to the angle
BD, without claiming generally that angles of semicircles are
equal; and again if one should assume that the angle C is equal to
the angle D, without the additional assumption that every angle of
a segment is equal to every other angle of the same segment; and
further if one should assume that when equal angles are taken from
the whole angles, which are themselves equal, the remainders E and
F are equal, he will beg the thing to be proved, unless he also
states that when equals are taken from equals the remainders are
equal.
    It is clear then that in every syllogism there must be a
universal premiss, and that a universal statement is proved only
when all the premisses are universal, while a particular statement
is proved both from two universal premisses and from one only:
consequently if the conclusion is universal, the premisses also
must be universal, but if the premisses are universal it is
possible that the conclusion may not be universal. And it is clear
also that in every syllogism either both or one of the premisses
must be like the conclusion. I mean not only in being affirmative
or negative, but also in being necessary, pure, problematic. We
must consider also the other forms of predication.
    It is clear also when a syllogism in general can be made and
when it cannot; and when a valid, when a perfect syllogism can be
formed; and that if a syllogism is formed the terms must be
arranged in one of the ways that have been mentioned.
25
    It is clear too that every demonstration will proceed through
three terms and no more, unless the same conclusion is established
by different pairs of propositions; e.g. the conclusion E may be
established through the propositions A and B, and through the
propositions C and D, or through the propositions A and B, or A and
C, or B and C. For nothing prevents there being several middles for
the same terms. But in that case there is not one but several
syllogisms. Or again when each of the propositions A and B is
obtained by syllogistic inference, e.g. by means of D and E, and
again B by means of F and G. Or one may be obtained by syllogistic,
the other by inductive inference. But thus also the syllogisms are
many; for the conclusions are many, e.g. A and B and C. But if this
can be called one syllogism, not many, the same conclusion may be
reached by more than three