Posterior Analytics   

right angles is not a commensurately universal attribute of figure.
For though it is possible to show that a figure has its angles equal
to two right angles, this attribute cannot be demonstrated of any
figure selected at haphazard, nor in demonstrating does one take a
figure at random-a square is a figure but its angles are not equal
to two right angles. On the other hand, any isosceles triangle has its
angles equal to two right angles, yet isosceles triangle is not the
primary subject of this attribute but triangle is prior. So whatever
can be shown to have its angles equal to two right angles, or to
possess any other attribute, in any random instance of itself and
primarily-that is the first subject to which the predicate in question
belongs commensurately and universally, and the demonstration, in
the essential sense, of any predicate is the proof of it as
belonging to this first subject commensurately and universally:
while the proof of it as belonging to the other subjects to which it
attaches is demonstration only in a secondary and unessential sense.
Nor again (2) is equality to two right angles a commensurately
universal attribute of isosceles; it is of wider application.

5

We must not fail to observe that we often fall into error because
our conclusion is not in fact primary and commensurately universal
in the sense in which we think we prove it so. We make this mistake
(1) when the subject is an individual or individuals above which there
is no universal to be found: (2) when the subjects belong to different
species and there is a higher universal, but it has no name: (3)
when the subject which the demonstrator takes as a whole is really
only a part of a larger whole; for then the demonstration will be true
of the individual instances within the part and will hold in every
instance of it, yet the demonstration will not be true of this subject
primarily and commensurately and universally. When a demonstration
is true of a subject primarily and commensurately and universally,
that is to be taken to mean that it is true of a given subject
primarily and as such. Case (3) may be thus exemplified. If a proof
were given that perpendiculars to the same line are parallel, it might
be supposed that lines thus perpendicular were the proper subject of
the demonstration because being parallel is true of every instance
of them. But it is not so, for the parallelism depends not on these
angles being equal to one another because each is a right angle, but
simply on their being equal to one another. An example of (1) would be
as follows: if isosceles were the only triangle, it would be thought
to have its angles equal to two right angles qua isosceles. An
instance of (2) would be the law that proportionals alternate.
Alternation used to be demonstrated separately of numbers, lines,
solids, and durations, though it could have been proved of them all by
a single demonstration. Because there was no single name to denote
that in which numbers, lengths, durations, and solids are identical,
and because they differed specifically from one another, this property
was proved of each of them separately. To-day, however, the proof is
commensurately universal, for they do not possess this attribute qua
lines or qua numbers, but qua manifesting this generic character which
they are postulated as possessing universally. Hence, even if one
prove of each kind of triangle that its angles are equal to two
right angles, whether by means of the same or different proofs; still,
as long as one treats separately equilateral, scalene, and
isosceles, one does not yet know, except sophistically, that
triangle has its angles equal to two right angles, nor does one yet
know that triangle has this property commensurately and universally,
even if there is no other species of triangle but these. For one
does not know that triangle as such has this property, nor even that
'all' triangles have it-unless 'all' means 'each taken singly': if
'all' means 'as a whole class', then, though there be none in which
one does not recognize this property, one does not know it of 'all