The Complete Aristotle (eng.)

plain that neither is the result of abstraction
prior nor that which is produced by adding determinants posterior;
for it is by adding a determinant to pale that we speak of the pale
man.
    It has, then, been sufficiently pointed out that the objects of
mathematics are not substances in a higher degree than bodies are,
and that they are not prior to sensibles in being, but only in
definition, and that they cannot exist somewhere apart. But since
it was not possible for them to exist in sensibles either, it is
plain that they either do not exist at all or exist in a special
sense and therefore do not ‘exist’ without qualification. For
‘exist’ has many senses.
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3
    For just as the universal propositions of mathematics deal not
with objects which exist separately, apart from extended magnitudes
and from numbers, but with magnitudes and numbers, not however qua
such as to have magnitude or to be divisible, clearly it is
possible that there should also be both propositions and
demonstrations about sensible magnitudes, not however qua sensible
but qua possessed of certain definite qualities. For as there are
many propositions about things merely considered as in motion,
apart from what each such thing is and from their accidents, and as
it is not therefore necessary that there should be either a mobile
separate from sensibles, or a distinct mobile entity in the
sensibles, so too in the case of mobiles there will be propositions
and sciences, which treat them however not qua mobile but only qua
bodies, or again only qua planes, or only qua lines, or qua
divisibles, or qua indivisibles having position, or only qua
indivisibles. Thus since it is true to say without qualification
that not only things which are separable but also things which are
inseparable exist (for instance, that mobiles exist), it is true
also to say without qualification that the objects of mathematics
exist, and with the character ascribed to them by mathematicians.
And as it is true to say of the other sciences too, without
qualification, that they deal with such and such a subject-not with
what is accidental to it (e.g. not with the pale, if the healthy
thing is pale, and the science has the healthy as its subject), but
with that which is the subject of each science-with the healthy if
it treats its object qua healthy, with man if qua man:-so too is it
with geometry; if its subjects happen to be sensible, though it
does not treat them qua sensible, the mathematical sciences will
not for that reason be sciences of sensibles-nor, on the other
hand, of other things separate from sensibles. Many properties
attach to things in virtue of their own nature as possessed of each
such character; e.g. there are attributes peculiar to the animal
qua female or qua male (yet there is no ‘female’ nor ‘male’
separate from animals); so that there are also attributes which
belong to things merely as lengths or as planes. And in proportion
as we are dealing with things which are prior in definition and
simpler, our knowledge has more accuracy, i.e. simplicity.
Therefore a science which abstracts from spatial magnitude is more
precise than one which takes it into account; and a science is most
precise if it abstracts from movement, but if it takes account of
movement, it is most precise if it deals with the primary movement,
for this is the simplest; and of this again uniform movement is the
simplest form.
    The same account may be given of harmonics and optics; for
neither considers its objects qua sight or qua voice, but qua lines
and numbers; but the latter are attributes proper to the former.
And mechanics too proceeds in the same way. Therefore if we suppose
attributes separated from their fellow attributes and make any
inquiry concerning them as such, we shall not for this reason be in
error, any more than when one draws a line on the ground and calls
it a foot long when it is not; for the error is not included in the
premisses.
    Each question will be best investigated in this way-by setting
up by an act of separation what is not separate, as the
arithmetician and the geometer do. For a man qua man is one
indivisible thing; and the arithmetician supposed one indivisible
thing, and then considered whether any attribute belongs to a man
qua indivisible. But the geometer treats him neither qua man nor
qua indivisible, but as a solid. For evidently the properties which
would have belonged to him even if