Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

and third, nothing intervenes; for no intervals of space disjoin
6 and 6. Thus also unity multiplied into itself, generates
nothing but itself. For that which is without interval does
not possess the power of generating interval. But every number
multiplied into itself, produces another number greater
than itself, because intervals when multiplied, distend themselves
by a greater length of space. That, however, which is
without interval, has not the power of generating more than it
is itself. From this principle, therefore, i.e. from unity, the
first extension into length proceeds, and which unfolds itself
into all numbers from the duad; because the first interval is a
line; but two intervals are length and breadth, i.e. a superficies;
and three intervals, are length, breadth, and depth, i.e.
a solid. But besides these, no other intervals can be found; so
that the six species of motion subsist conformably to the natures
and number of the intervals. For one interval contains in
itself two motions. Thus in length there is before and behind;
in breadth, the right and the left; and in depth, upward and
downward. But it is necessary that every solid body should
have length, breadth, and depth; and that whatever contains
these three dimensions in itself, should be a solid. Since therefore,
a line surpasses a point by one dimension, viz. by length,
but a superficies surpasses it by two dimensions, i.e. by length
and breadth, and a solid surpasses it by three dimensions, i.e.
by length, breadth, and depth, it is evident that a point itself is
without any corporeal magnitude, or dimension of interval; is
the principle of all intervals; and is naturally incapable of being
divided. Hence a point is the principle of the first interval,
yet is not itself interval; and is the summit of a line, but is not
yet a line. Thus too a line is the principle of a superficies, but
is not itself a superficies; and is the summit of the second interval,
yet retains no vestige of the second interval. And thus
also a superficies is the principle of a solid, but is itself neither