The Complete Aristotle (eng.)

possible: this
can be proved, if the same terms as before are transposed. Also the
conclusion may be true if one premiss is negative, the other
affirmative. For since it is possible that B should belong to the
whole of C, and A to some C, and, when they are so, that A should
not belong to all B, therefore it is assumed that B belongs to the
whole of C, and A to no C, the negative premiss is partly false,
the other premiss wholly true, and the conclusion is true. Again
since it has been proved that if A belongs to no C and B to some C,
it is possible that A should not belong to some C, it is clear that
if the premiss AC is wholly true, and the premiss BC partly false,
it is possible that the conclusion should be true. For if it is
assumed that A belongs to no C, and B to all C, the premiss AC is
wholly true, and the premiss BC is partly false.
    (5) It is clear also in the case of particular syllogisms that a
true conclusion may come through what is false, in every possible
way. For the same terms must be taken as have been taken when the
premisses are universal, positive terms in positive syllogisms,
negative terms in negative. For it makes no difference to the
setting out of the terms, whether one assumes that what belongs to
none belongs to all or that what belongs to some belongs to all.
The same applies to negative statements.
    It is clear then that if the conclusion is false, the premisses
of the argument must be false, either all or some of them; but when
the conclusion is true, it is not necessary that the premisses
should be true, either one or all, yet it is possible, though no
part of the syllogism is true, that the conclusion may none the
less be true; but it is not necessitated. The reason is that when
two things are so related to one another, that if the one is, the
other necessarily is, then if the latter is not, the former will
not be either, but if the latter is, it is not necessary that the
former should be. But it is impossible that the same thing should
be necessitated by the being and by the not-being of the same
thing. I mean, for example, that it is impossible that B should
necessarily be great since A is white and that B should necessarily
be great since A is not white. For whenever since this, A, is white
it is necessary that that, B, should be great, and since B is great
that C should not be white, then it is necessary if is white that C
should not be white. And whenever it is necessary, since one of two
things is, that the other should be, it is necessary, if the latter
is not, that the former (viz. A) should not be. If then B is not
great A cannot be white. But if, when A is not white, it is
necessary that B should be great, it necessarily results that if B
is not great, B itself is great. (But this is impossible.) For if B
is not great, A will necessarily not be white. If then when this is
not white B must be great, it results that if B is not great, it is
great, just as if it were proved through three terms.
5
    Circular and reciprocal proof means proof by means of the
conclusion, i.e. by converting one of the premisses simply and
inferring the premiss which was assumed in the original syllogism:
e.g. suppose it has been necessary to prove that A belongs to all
C, and it has been proved through B; suppose that A should now be
proved to belong to B by assuming that A belongs to C, and C to
B-so A belongs to B: but in the first syllogism the converse was
assumed, viz. that B belongs to C. Or suppose it is necessary to
prove that B belongs to C, and A is assumed to belong to C, which
was the conclusion of the first syllogism, and B to belong to A but
the converse was assumed in the earlier syllogism, viz. that A
belongs to B. In no other way is reciprocal proof possible. If
another term is taken as middle, the proof is not circular: for
neither of the propositions assumed is the same as before: if one
of the accepted terms is taken as middle, only one of the premisses
of the first syllogism can be assumed in the second: for if both of
them are taken the same conclusion as before will result: but it
must be different. If the terms are not convertible, one of the
premisses from which the syllogism results must be undemonstrated:
for it is not possible to demonstrate through these terms that the
third belongs to the middle or the middle to the first. If the
terms are convertible, it is possible to demonstrate everything
reciprocally, e.g. if A and B and C are convertible with one
another. Suppose