The Complete Aristotle (eng.)

universally to a subject
when it can be shown to belong to any random instance of that
subject and when the subject is the first thing to which it can be
shown to belong. Thus, e.g. (1) the equality of its angles to two
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
triangles’.
    When, then, does our knowledge fail of commensurate
universality, and when it is unqualified knowledge? If triangle be
identical in