The Complete Aristotle (eng.)

reasoned fact; for the latter are in
possession of the demonstrations giving the causes, and are often
ignorant of the fact: just as we have often a clear insight into a
universal, but through lack of observation are ignorant of some of
its particular instances. These connexions have a perceptible
existence though they are manifestations of forms. For the
mathematical sciences concern forms: they do not demonstrate
properties of a substratum, since, even though the geometrical
subjects are predicable as properties of a perceptible substratum,
it is not as thus predicable that the mathematician demonstrates
properties of them. As optics is related to geometry, so another
science is related to optics, namely the theory of the rainbow.
Here knowledge of the fact is within the province of the natural
philosopher, knowledge of the reasoned fact within that of the
optician, either qua optician or qua mathematical optician. Many
sciences not standing in this mutual relation enter into it at
points; e.g. medicine and geometry: it is the physician’s business
to know that circular wounds heal more slowly, the geometer’s to
know the reason why.
14
    Of all the figures the most scientific is the first. Thus, it is
the vehicle of the demonstrations of all the mathematical sciences,
such as arithmetic, geometry, and optics, and practically all of
all sciences that investigate causes: for the syllogism of the
reasoned fact is either exclusively or generally speaking and in
most cases in this figure-a second proof that this figure is the
most scientific; for grasp of a reasoned conclusion is the primary
condition of knowledge. Thirdly, the first is the only figure which
enables us to pursue knowledge of the essence of a thing. In the
second figure no affirmative conclusion is possible, and knowledge
of a thing’s essence must be affirmative; while in the third figure
the conclusion can be affirmative, but cannot be universal, and
essence must have a universal character: e.g. man is not two-footed
animal in any qualified sense, but universally. Finally, the first
figure has no need of the others, while it is by means of the first
that the other two figures are developed, and have their intervals
closepacked until immediate premisses are reached.
    Clearly, therefore, the first figure is the primary condition of
knowledge.
15
    Just as an attribute A may (as we saw) be atomically connected
with a subject B, so its disconnexion may be atomic. I call
‘atomic’ connexions or disconnexions which involve no intermediate
term; since in that case the connexion or disconnexion will not be
mediated by something other than the terms themselves. It follows
that if either A or B, or both A and B, have a genus, their
disconnexion cannot be primary. Thus: let C be the genus of A.
Then, if C is not the genus of B-for A may well have a genus which
is not the genus of B-there will be a syllogism proving A’s
disconnexion from B thus:
all A is C,
no B is C,
therefore no B is A.
     
    Or if it is B which has a genus D, we have
all B is D,
no D is A,
therefore no B is A, by syllogism;
     
    and the proof will be similar if both A and B have a genus. That
the genus of A need not be the genus of B and vice versa, is shown
by the existence of mutually exclusive coordinate series of
predication. If no term in the series ACD… is predicable of any
term in the series BEF… ,and if G-a term in the former series-is
the genus of A, clearly G will not be the genus of B; since, if it
were, the series would not be mutually exclusive. So also if B has
a genus, it will not be the genus of A. If, on the other hand,
neither A nor B has a genus and A does not inhere in B, this
disconnexion must be atomic. If there be a middle term, one or
other of them is bound to have a genus, for the syllogism will be
either in the first or the second figure. If it is in the first, B
will have a genus-for the premiss containing it must be
affirmative: if in the second, either A or B indifferently, since
syllogism is possible if either is contained in a negative premiss,
but not if both premisses are negative.
    Hence it is clear that one thing may be atomically disconnected
from another, and we have stated when and how this is possible.
16
    Ignorance-defined not as the negation of knowledge but as a
positive state of mind-is error produced by inference.
    (1) Let us first consider propositions asserting a predicate’s
immediate connexion with or disconnexion from a subject. Here,