sense one-in truth each of the two units exists potentially (at
least if the number is a unity and not like a heap, i.e. if
different numbers consist of differentiated units, as they say), but
not in complete reality; and the cause of the error they fell into
is that they were conducting their inquiry at the same time from the
standpoint of mathematics and from that of universal definitions, so
that (1) from the former standpoint they treated unity, their first
principle, as a point; for the unit is a point without position.
They put things together out of the smallest parts, as some others
also have done. Therefore the unit becomes the matter of numbers and
at the same time prior to 2; and again posterior, 2 being treated as a
whole, a unity, and a form. But (2) because they were seeking the
universal they treated the unity which can be predicated of a
number, as in this sense also a part of the number. But these
characteristics cannot belong at the same time to the same thing.
If the 1-itself must be unitary (for it differs in nothing from
other 1's except that it is the starting-point), and the 2 is
divisible but the unit is not, the unit must be liker the 1-itself
than the 2 is. But if the unit is liker it, it must be liker to the
unit than to the 2; therefore each of the units in 2 must be prior
to the 2. But they deny this; at least they generate the 2 first.
Again, if the 2-itself is a unity and the 3-itself is one also, both
form a 2. From what, then, is this 2 produced?
Since there is not contact in numbers, but succession, viz.
between the units between which there is nothing, e.g. between those
in 2 or in 3 one might ask whether these succeed the 1-itself or
not, and whether, of the terms that succeed it, 2 or either of the
units in 2 is prior.
Similar difficulties occur with regard to the classes of things
posterior to number,-the line, the plane, and the solid. For some
construct these out of the species of the 'great and small'; e.g.
lines from the 'long and short', planes from the 'broad and narrow',
masses from the 'deep and shallow'; which are species of the 'great
and small'. And the originative principle of such things which answers
to the 1 different thinkers describe in different ways, And in these
also the impossibilities, the fictions, and the contradictions of
all probability are seen to be innumerable. For (i) geometrical
classes are severed from one another, unless the principles of these
are implied in one another in such a way that the 'broad and narrow'
is also 'long and short' (but if this is so, the plane will be line
and the solid a plane; again, how will angles and figures and such
things be explained?). And (ii) the same happens as in regard to
number; for 'long and short', &c., are attributes of magnitude, but
magnitude does not consist of these, any more than the line consists
of 'straight and curved', or solids of 'smooth and rough'.
(All these views share a difficulty which occurs with regard to
species-of-a-genus, when one posits the universals, viz. whether it is
animal-itself or something other than animal-itself that is in the
particular animal. True, if the universal is not separable from
sensible things, this will present no difficulty; but if the 1 and the
numbers are separable, as those who express these views say, it is not
easy to solve the difficulty, if one may apply the words 'not easy' to
the impossible. For when we apprehend the unity in 2, or in general in
a number, do we apprehend a thing-itself or something else?).
Some, then, generate spatial magnitudes from matter of this
sort, others from the point -and the point is thought by them to be
not 1 but something like 1-and from other matter like plurality, but
not identical with it; about which principles none the less the same
difficulties occur. For if the matter is one, line and plane-and
soli will be the same; for from the same elements will come one and
the same thing. But if the matters are more than one, and there is one
for the line and a second for the plane and another for the solid,
they either are implied in one another or not, so that the same