Posterior Analytics   

investigates, e.g. in arithmetic units, in geometry points and
lines. Both the existence and the meaning of the subjects are
assumed by these sciences; but of their essential attributes only
the meaning is assumed. For example arithmetic assumes the meaning
of odd and even, square and cube, geometry that of incommensurable, or
of deflection or verging of lines, whereas the existence of these
attributes is demonstrated by means of the axioms and from previous
conclusions as premisses. Astronomy too proceeds in the same way.
For indeed every demonstrative science has three elements: (1) that
which it posits, the subject genus whose essential attributes it
examines; (2) the so-called axioms, which are primary premisses of its
demonstration; (3) the attributes, the meaning of which it assumes.
Yet some sciences may very well pass over some of these elements; e.g.
we might not expressly posit the existence of the genus if its
existence were obvious (for instance, the existence of hot and cold is
more evident than that of number); or we might omit to assume
expressly the meaning of the attributes if it were well understood. In
the way the meaning of axioms, such as 'Take equals from equals and
equals remain', is well known and so not expressly assumed.
Nevertheless in the nature of the case the essential elements of
demonstration are three: the subject, the attributes, and the basic
premisses.
That which expresses necessary self-grounded fact, and which we must
necessarily believe, is distinct both from the hypotheses of a science
and from illegitimate postulate-I say 'must believe', because all
syllogism, and therefore a fortiori demonstration, is addressed not to
the spoken word, but to the discourse within the soul, and though we
can always raise objections to the spoken word, to the inward
discourse we cannot always object. That which is capable of proof
but assumed by the teacher without proof is, if the pupil believes and
accepts it, hypothesis, though only in a limited sense hypothesis-that
is, relatively to the pupil; if the pupil has no opinion or a contrary
opinion on the matter, the same assumption is an illegitimate
postulate. Therein lies the distinction between hypothesis and
illegitimate postulate: the latter is the contrary of the pupil's
opinion, demonstrable, but assumed and used without demonstration.
The definition-viz. those which are not expressed as statements that
anything is or is not-are not hypotheses: but it is in the premisses
of a science that its hypotheses are contained. Definitions require
only to be understood, and this is not hypothesis-unless it be
contended that the pupil's hearing is also an hypothesis required by
the teacher. Hypotheses, on the contrary, postulate facts on the being
of which depends the being of the fact inferred. Nor are the
geometer's hypotheses false, as some have held, urging that one must
not employ falsehood and that the geometer is uttering falsehood in
stating that the line which he draws is a foot long or straight,
when it is actually neither. The truth is that the geometer does not
draw any conclusion from the being of the particular line of which
he speaks, but from what his diagrams symbolize. A further distinction
is that all hypotheses and illegitimate postulates are either
universal or particular, whereas a definition is neither.

11

So demonstration does not necessarily imply the being of Forms nor a
One beside a Many, but it does necessarily imply the possibility of
truly predicating one of many; since without this possibility we
cannot save the universal, and if the universal goes, the middle
term goes witb. it, and so demonstration becomes impossible. We
conclude, then, that there must be a single identical term
unequivocally predicable of a number of individuals.
The law that it is impossible to affirm and deny simultaneously
the same predicate of the same subject is not expressly posited by any
demonstration except when the conclusion also has to be expressed in