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（3） the subject…genus whose attributes； i。e。 essential properties； are
revealed by the demonstration。 The axioms which are premisses of
demonstration may be identical in two or more sciences： but in the
case of two different genera such as arithmetic and geometry you
cannot apply arithmetical demonstration to the properties of
magnitudes unless the magnitudes in question are numbers。 How in
certain cases transference is possible I will explain later。
Arithmetical demonstration and the other sciences likewise
possess； each of them； their own genera； so that if the
demonstration is to pass from one sphere to another； the genus must be
either absolutely or to some extent the same。 If this is not so；
transference is clearly impossible； because the extreme and the middle
terms must be drawn from the same genus： otherwise； as predicated；
they will not be essential and will thus be accidents。 That is why
it cannot be proved by geometry that opposites fall under one science；
nor even that the product of two cubes is a cube。 Nor can the
theorem of any one science be demonstrated by means of another
science； unless these theorems are related as subordinate to
superior （e。g。 as optical theorems to geometry or harmonic theorems to
arithmetic）。 Geometry again cannot prove of lines any property which
they do not possess qua lines； i。e。 in virtue of the fundamental
truths of their peculiar genus： it cannot show； for example； that
the straight line is the most beautiful of lines or the contrary of
the circle； for these qualities do not belong to lines in virtue of
their peculiar genus； but through some property which it shares with
other genera。
8
It is also clear that if the premisses from which the syllogism
proceeds are commensurately universal； the conclusion of such i。e。
in the unqualified sense…must also be eternal。 Therefore no
attribute can be demonstrated nor known by strictly scientific
knowledge to inhere in perishable things。 The proof can only be
accidental； because the attribute's connexion with its perishable
subject is not commensurately universal but temporary and special。
If such a demonstration is made； one premiss must be perishable and
not commensurately universal （perishable because only if it is
perishable will the conclusion be perishable； not commensurately
universal； because the predicate will be predicable of some
instances of the subject and not of others）； so that the conclusion
can only be that a fact is true at the moment…not commensurately and
universally。 The same is true of definitions； since a definition is
either a primary premiss or a conclusion of a demonstration； or else
only differs from a demonstration in the order of its terms。
Demonstration and science of merely frequent occurrences…e。g。 of
eclipse as happening to the moon…are； as such； clearly eternal：
whereas so far as they are not eternal they are not fully
commensurate。 Other subjects too have properties attaching to them
in the same way as eclipse attaches to the moon。
9
It is clear that if the conclusion is to show an attribute
inhering as such； nothing can be demonstrated except from its
'appropriate' basic truths。 Consequently a proof even from true；
indemonstrable； and immediate premisses does not constitute knowledge。
Such proofs are like Bryson's method of squaring the circle； for
they operate by taking as their middle a common character…a character；
therefore； which the subject may share with another…and consequently
they apply equally to subjects different in kind。 They therefore
afford knowledge of an attribute only as inhering accidentally； not as
belonging to its subject as such： otherwise they would not have been
applicable to another genus。
Our knowledge of any attribute's connexion with a subject is
accidental unless we know that connexion through the middle term in
virtue of which it inheres； and as an inference from basic premisses
essential and 'appropriate' to the subject…unless we know； e。g。 the
property of possessing angles equal to two right angles as belonging
to that subject in which it inheres essentially； and as inferred
from basic premisses essential and 'appropriate' to that subject： so
that if that middle term also belongs essentially to the minor； the
middle must belong to the same kind as the major and minor terms。
The only exceptions to this rule are such cases as theorems in
harmonics which are demonstrable by arithmetic。 Such theorems are
proved by the same middle terms as arithmetical properties； but with a
qualification…the fact falls under a separate science （for the subject
genus is separate）； but the reasoned fact concerns the superior
science； to which the attributes essentially belong。 Thus； even
these apparent exceptions show that no attribute is strictly
demonstrable except from its 'appropriate' basic truths； which；
however； in the case of these sciences have the requisite identity
of character。
It is no less evident that the peculiar basic truths of each
inhering attribute are indemonstrable； for basic truths from which
they might be deduced would be basic truths of all that is； and the
science to which they belonged would possess universal sovereignty。
This is so because he knows better whose knowledge is deduced from
higher causes； for his knowledge is from prior premisses when it
derives from causes themselves uncaused： hence； if he knows better
than others or best of all； his knowledge would be science in a higher
or the highest degree。 But； as things are； demonstration is not
transferable to another genus； with such exceptions as we have
mentioned of the 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。
10
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 as