[1078a]
[1]
with the healthy if
it treats of things qua healthy, and with man
if qua man—so this is also true of
geometry. If the things of which it treats are accidentally sensible
although it does not treat of them qua
sensible, it does not follow that the mathematical sciences treat of
sensible things—nor, on the other hand, that they treat of
other things which exist independently apart from these.
Many attributes are essential properties of things as possessing a
particular characteristic; e.g., there are attributes peculiar to an
animal qua female or qua
male, although there is no such thing as female or male in separation
from animals. Hence there are also attributes which are peculiar to
things merely qua lines or planes.And in proportion as the things
which we are considering are prior in formula and simpler, they admit
of greater exactness; for simplicity implies exactness. Hence we find
greater exactness where there is no magnitude, and the greatest
exactness where there is no motion; or if motion is involved, where it
is primary, because this is the simplest kind; and the simplest kind
of primary motion is uniform motion.1 The same principle applies to
both harmonics and optics, for neither of these sciences studies
objects qua sight or qua
sound, but qua lines and numbers2; yet the latter are affections peculiar to
the former. The same is also true of mechanics. Thus
if we regard objects independently of their attributes and investigate
any aspect of them as so regarded, we shall not be guilty of any error
on this account, any more than when we draw a diagram on the ground
and say that a line is a foot long when it is not;
[20]
because the error is not in the
premisses.3 The best way to
conduct an investigation in every case is to take that which does not
exist in separation and consider it separately; which is just what the
arithmetician or the geometrician does.For man, qua man, is
one indivisible thing; and the arithmetician assumes man to be one
indivisible thing, and then considers whether there is any attribute
of man qua indivisible. And the geometrician
considers man neither qua man nor qua indivisible, but qua something solid. For clearly the attributes which would have
belonged to "man" even if man were somehow not indivisible can belong
to man irrespectively of his humanity or indivisibility.Hence for this reason the
geometricians are right in what they maintain, and treat of what
really exists; i.e., the objects of geometry really exist. For things
can exist in two ways, either in complete reality or as matter.4And since
goodness is distinct from beauty (for it is always in actions that
goodness is present, whereas beauty is also in immovable things),
they5 are in error who
assert that the mathematical sciences tell us nothing about beauty or
goodness;for they
describe and manifest these qualities in the highest degree, since it
does not follow, because they manifest the effects and principles of
beauty and goodness without naming them, that they do not treat of
these qualities. The main species of beauty are orderly arrangement,
proportion, and definiteness;
2 Optics studies lines and harmonics numbers because these sciences are subordinate to geometry and arithmetic (Aristot. An. Post. 75b 15).
3 Cf. Aristot. Met. 14.2.9, 10.
4 i.e., potentially.
5 Cf. Aristot. Met. 3.2.4.
Aristotle. Aristotle in 23 Volumes, Vols.17, 18, translated by Hugh Tredennick. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1933, 1989.
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