The Complete Aristotle (eng.)

with every other.
    But how will our conclusion work out in the case of the circle
and the straight line? It would be absurd to suppose that the
motion of one in a circle and of another in a straight line cannot
be similar, but that the one must inevitably move more quickly or
more slowly than the other, just as if the course of one were
downhill and of the other uphill. Moreover it does not as a matter
of fact make any difference to the argument to say that the one
motion must inevitably be quicker or slower than the other: for
then the circumference can be greater or less than the straight
line; and if so it is possible for the two to be equal. For if in
the time A the quicker (B) passes over the distance B’ and the
slower (G) passes over the distance G’, B’ will be greater than G’:
for this is what we took ‘quicker’ to mean: and so quicker motion
also implies that one thing traverses an equal distance in less
time than another: consequently there will be a part of A in which
B will pass over a part of the circle equal to G’, while G will
occupy the whole of A in passing over G’. None the less, if the two
motions are commensurable, we are confronted with the consequence
stated above, viz. that there may be a straight line equal to a
circle. But these are not commensurable: and so the corresponding
motions are not commensurable either.
    But may we say that things are always commensurable if the same
terms are applied to them without equivocation? e.g. a pen, a wine,
and the highest note in a scale are not commensurable: we cannot
say whether any one of them is sharper than any other: and why is
this? they are incommensurable because it is only equivocally that
the same term ‘sharp’ is applied to them: whereas the highest note
in a scale is commensurable with the leading-note, because the term
‘sharp’ has the same meaning as applied to both. Can it be, then,
that the term ‘quick’ has not the same meaning as applied to
straight motion and to circular motion respectively? If so, far
less will it have the same meaning as applied to alteration and to
locomotion.
    Or shall we in the first place deny that things are always
commensurable if the same terms are applied to them without
equivocation? For the term ‘much’ has the same meaning whether
applied to water or to air, yet water and air are not commensurable
in respect of it: or, if this illustration is not considered
satisfactory, ‘double’ at any rate would seem to have the same
meaning as applied to each (denoting in each case the proportion of
two to one), yet water and air are not commensurable in respect of
it. But here again may we not take up the same position and say
that the term ‘much’ is equivocal? In fact there are some terms of
which even the definitions are equivocal; e.g. if ‘much’ were
defined as ‘so much and more’,’so much’ would mean something
different in different cases: ‘equal’ is similarly equivocal; and
‘one’ again is perhaps inevitably an equivocal term; and if ‘one’
is equivocal, so is ‘two’. Otherwise why is it that some things are
commensurable while others are not, if the nature of the attribute
in the two cases is really one and the same?
    Can it be that the incommensurability of two things in respect
of any attribute is due to a difference in that which is primarily
capable of carrying the attribute? Thus horse and dog are so
commensurable that we may say which is the whiter, since that which
primarily contains the whiteness is the same in both, viz. the
surface: and similarly they are commensurable in respect of size.
But water and speech are not commensurable in respect of clearness,
since that which primarily contains the attribute is different in
the two cases. It would seem, however that we must reject this
solution, since clearly we could thus make all equivocal attributes
univocal and say merely that that contains each of them is
different in different cases: thus ‘equality’, ‘sweetness’, and
‘whiteness’ will severally always be the same, though that which
contains them is different in different cases. Moreover, it is not
any casual thing that is capable of carrying any attribute: each
single attribute can be carried primarily only by one single
thing.
    Must we then say that, if two things are to be commensurable in
respect of any attribute, not only must the attribute in question
be applicable to both without equivocation, but there must also be
no