The Complete Aristotle (eng.)

if the
syllogism not universal, the universal premiss cannot be proved,
for the same reason as we gave above, but the particular premiss
can be proved whenever the universal statement is affirmative. Let
A belong to all B, and not to all C: the conclusion is BC. If then
it is assumed that B belongs to all A, but not to all C, A will not
belong to some C, B being middle. But if the universal premiss is
negative, the premiss AC will not be demonstrated by the conversion
of AB: for it turns out that either both or one of the premisses is
negative; consequently a syllogism will not be possible. But the
proof will proceed as in the universal syllogisms, if it is assumed
that A belongs to some of that to some of which B does not
belong.
7
    In the third figure, when both premisses are taken universally,
it is not possible to prove them reciprocally: for that which is
universal is proved through statements which are universal, but the
conclusion in this figure is always particular, so that it is clear
that it is not possible at all to prove through this figure the
universal premiss. But if one premiss is universal, the other
particular, proof of the latter will sometimes be possible,
sometimes not. When both the premisses assumed are affirmative, and
the universal concerns the minor extreme, proof will be possible,
but when it concerns the other extreme, impossible. Let A belong to
all C and B to some C: the conclusion is the statement AB. If then
it is assumed that C belongs to all A, it has been proved that C
belongs to some B, but that B belongs to some C has not been
proved. And yet it is necessary, if C belongs to some B, that B
should belong to some C. But it is not the same that this should
belong to that, and that to this: but we must assume besides that
if this belongs to some of that, that belongs to some of this. But
if this is assumed the syllogism no longer results from the
conclusion and the other premiss. But if B belongs to all C, and A
to some C, it will be possible to prove the proposition AC, when it
is assumed that C belongs to all B, and A to some B. For if C
belongs to all B and A to some B, it is necessary that A should
belong to some C, B being middle. And whenever one premiss is
affirmative the other negative, and the affirmative is universal,
the other premiss can be proved. Let B belong to all C, and A not
to some C: the conclusion is that A does not belong to some B. If
then it is assumed further that C belongs to all B, it is necessary
that A should not belong to some C, B being middle. But when the
negative premiss is universal, the other premiss is not except as
before, viz. if it is assumed that that belongs to some of that, to
some of which this does not belong, e.g. if A belongs to no C, and
B to some C: the conclusion is that A does not belong to some B. If
then it is assumed that C belongs to some of that to some of which
does not belong, it is necessary that C should belong to some of
the Bs. In no other way is it possible by converting the universal
premiss to prove the other: for in no other way can a syllogism be
formed.
    It is clear then that in the first figure reciprocal proof is
made both through the third and through the first figure-if the
conclusion is affirmative through the first; if the conclusion is
negative through the last. For it is assumed that that belongs to
all of that to none of which this belongs. In the middle figure,
when the syllogism is universal, proof is possible through the
second figure and through the first, but when particular through
the second and the last. In the third figure all proofs are made
through itself. It is clear also that in the third figure and in
the middle figure those syllogisms which are not made through those
figures themselves either are not of the nature of circular proof
or are imperfect.
8
    To convert a syllogism means to alter the conclusion and make
another syllogism to prove that either the extreme cannot belong to
the middle or the middle to the last term. For it is necessary, if
the conclusion has been changed into its opposite and one of the
premisses stands, that the other premiss should be destroyed. For
if it should stand, the conclusion also must stand. It makes a
difference whether the conclusion is converted into its
contradictory or into its contrary. For the same syllogism does not
result whichever form the conversion takes. This will be made clear
by the sequel. By contradictory opposition I mean the opposition