Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 69
But in these numbers also the sides increase according to the
natural series of numbers. For the first square in power and
capacity, i.e. unity, has 1 for its side. But the second square,
which is the first in energy, ie. 4, has 2 for its side. And the
third, which is 9, and is the second square in energy, has 3 for
its side. And after the same manner all the rest proceed.
Such numbers however, are generated from the series of
natural numbers, not indeed after the same manner as triangles,
but by the continual omission of one number, in the
addition of the terms to each other. For in the natural series
of numbers from unity, 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11, etc. 1 is
the first square in power. Then 1 +3=4, the second square.
1 +3+5=9, the third square. 1 +3+5+7=16, the fourth
square. And thus by continually omitting one term, and adding
the rest, the whole series of squares will be produced.
In these numbers likewise, there is this subtilty of nature, and
immutable order, that each square retains as many units in
its side as there are numbers by the aggregation of which it is
formed. Thus in the first square, because it is produced from
1, the side is 1. In the second square, i.e. 4, because it is generated
from 1 and 3, which are two terms, the side is 2. In
9 because it is generated from three numbers, the side is 3.
And so of the rest.
CHAPTER VII.
On pentagons, their sides, and generation.
A PENTAGON is a number which being expressed by its units,
exhibits the form of a quinquangular figure; and which when
thus described, has five angles, and five equal sides. But pen-