The Complete Aristotle (eng.)

application of geometrical demonstrations to
theorems in mechanics or optics, or of arithmetical demonstrations
to those of harmonics.
    It is hard to be sure whether one knows or not; for it is hard
to be sure whether one’s knowledge is based on the basic truths
appropriate to each attribute-the differentia of true knowledge. We
think we have scientific knowledge if we have reasoned from true
and primary premisses. But that is not so: the conclusion must be
homogeneous with the basic facts of the science.
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    I call the basic truths of every genus those clements in it the
existence of which cannot be proved. As regards both these primary
truths and the attributes dependent on them the meaning of the name
is assumed. The fact of their existence as regards the primary
truths must be assumed; but it has to be proved of the remainder,
the attributes. Thus we assume the meaning alike of unity,
straight, and triangular; but while as regards unity and magnitude
we assume also the fact of their existence, in the case of the
remainder proof is required.
    Of the basic truths used in the demonstrative sciences some are
peculiar to each science, and some are common, but common only in
the sense of analogous, being of use only in so far as they fall
within the genus constituting the province of the science in
question.
    Peculiar truths are, e.g. the definitions of line and straight;
common truths are such as ‘take equals from equals and equals
remain’. Only so much of these common truths is required as falls
within the genus in question: for a truth of this kind will have
the same force even if not used generally but applied by the
geometer only to magnitudes, or by the arithmetician only to
numbers. Also peculiar to a science are the subjects the existence
as well as the meaning of which it assumes, and the essential
attributes of which it 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