Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

The heptagons formed from the addition of these are, 1.7. 18.
34. 55.
The forms of an octagon of an enneagon, or figures of eight
and of nine angles, are generated according to the same order,
so that the first numbers, or roots of them, are distant from
each other conformably to an equal progression. For in the
generation of octagons five terms are omitted; but in that of
enneagons, six terms. And in short, the number of omissions
continually increasing by 1, the series of plane numbers will be
formed by the addition of the remaining terms.
CHAPTER IX.
On the figurate numbers that are produced from figurate
numbers; and that the triangular number is the principle of
all the rest.-A speculation pertaining to the dcsm'ption of
figurate numbers.
PLANE
numbers thus subsisting, let us in the next place investigate
what is consequent to their subsistence. All the
squares which are disposed under triangles in a natural arrangement
are generated from triangular numbers. Thus the
square 4 is formed from the addition of 1 and 3, i.e. from two
triangular numbers. Thus too, 9 is formed from 3 and 6; and
both of these are triangles. But 16 is from the triangles 6 and
10; and 25 is from the triangles 10 and 15. And the same thing
in the following order of squares will be found to take place
perpetually without any variation. But pentagons are generated
from the aggregation of squares and triangles. Thus the
pentagon 5 is generated from the addition of the square 4 and
the triangle 1. Thus too the pentagon 12 is formed from the
aggregation of the square 9, and the triangle 3. The pentagon