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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 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
that form； in which case the proof lays down as its major premiss that
the major is truly affirmed of the middle but falsely denied。 It makes
no difference； however； if we add to the middle； or again to the minor
term； the corresponding negative。 For grant a minor term of which it
is true to predicate man…even if it be also true to predicate
not…man of itstill grant simply that man is animal and not
not…animal； and the conclusion follows： for it will still be true to
say that Calliaseven if it be also true to say that
not…Calliasis animal and not not…animal。 The reason is that the
major term is predicable not only of the middle； but of something
other than the middle as well； being of wider application； so that the
conclusion is not affected even if the middle is extended to cover the
original middle term and also what is not the original middle term。
The law that every predicate can be either truly affirmed or truly
denied of every subject is posited by such demonstration as uses
reductio ad impossibile； and then not always universally； but so far
as it is requisite； within the limits； that is； of the genus…the
genus； I mean （as I have already explained）； to which the man of
science applies his demonstrations。 In virtue of the common elements
of demonstration…I mean the common axioms which are used as
premisses of demonstration； no