The Complete Aristotle (eng.)

all B. If however it is assumed that A belongs to some B, we
shall have the same result as before.
    It is clear then that in all the syllogisms which proceed per
impossibile the contradictory must be assumed. And it is plain that
in the middle figure an affirmative conclusion, and in the last
figure a universal conclusion, are proved in a way.
14
    Demonstration per impossibile differs from ostensive proof in
that it posits what it wishes to refute by reduction to a statement
admitted to be false; whereas ostensive proof starts from admitted
positions. Both, indeed, take two premisses that are admitted, but
the latter takes the premisses from which the syllogism starts, the
former takes one of these, along with the contradictory of the
original conclusion. Also in the ostensive proof it is not
necessary that the conclusion should be known, nor that one should
suppose beforehand that it is true or not: in the other it is
necessary to suppose beforehand that it is not true. It makes no
difference whether the conclusion is affirmative or negative; the
method is the same in both cases. Everything which is concluded
ostensively can be proved per impossibile, and that which is proved
per impossibile can be proved ostensively, through the same terms.
Whenever the syllogism is formed in the first figure, the truth
will be found in the middle or the last figure, if negative in the
middle, if affirmative in the last. Whenever the syllogism is
formed in the middle figure, the truth will be found in the first,
whatever the problem may be. Whenever the syllogism is formed in
the last figure, the truth will be found in the first and middle
figures, if affirmative in first, if negative in the middle.
Suppose that A has been proved to belong to no B, or not to all B,
through the first figure. Then the hypothesis must have been that A
belongs to some B, and the original premisses that C belongs to all
A and to no B. For thus the syllogism was made and the impossible
conclusion reached. But this is the middle figure, if C belongs to
all A and to no B. And it is clear from these premisses that A
belongs to no B. Similarly if has been proved not to belong to all
B. For the hypothesis is that A belongs to all B; and the original
premisses are that C belongs to all A but not to all B. Similarly
too, if the premiss CA should be negative: for thus also we have
the middle figure. Again suppose it has been proved that A belongs
to some B. The hypothesis here is that is that A belongs to no B;
and the original premisses that B belongs to all C, and A either to
all or to some C: for in this way we shall get what is impossible.
But if A and B belong to all C, we have the last figure. And it is
clear from these premisses that A must belong to some B. Similarly
if B or A should be assumed to belong to some C.
    Again suppose it has been proved in the middle figure that A
belongs to all B. Then the hypothesis must have been that A belongs
not to all B, and the original premisses that A belongs to all C,
and C to all B: for thus we shall get what is impossible. But if A
belongs to all C, and C to all B, we have the first figure.
Similarly if it has been proved that A belongs to some B: for the
hypothesis then must have been that A belongs to no B, and the
original premisses that A belongs to all C, and C to some B. If the
syllogism is negative, the hypothesis must have been that A belongs
to some B, and the original premisses that A belongs to no C, and C
to all B, so that the first figure results. If the syllogism is not
universal, but proof has been given that A does not belong to some
B, we may infer in the same way. The hypothesis is that A belongs
to all B, the original premisses that A belongs to no C, and C
belongs to some B: for thus we get the first figure.
    Again suppose it has been proved in the third figure that A
belongs to all B. Then the hypothesis must have been that A belongs
not to all B, and the original premisses that C belongs to all B,
and A belongs to all C; for thus we shall get what is impossible.
And the original premisses form the first figure. Similarly if the
demonstration establishes a particular proposition: the hypothesis
then must have been that A belongs to no B, and the original
premisses that C belongs to some B, and A to all C. If the
syllogism is negative, the hypothesis must have been that A belongs
to some B, and the original premisses that C belongs to no A and to
all B, and this is the middle figure. Similarly