Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

ing this table from the Epanthematcr or Arithmetical Flowers
of Thymaridas :
Polygons are formed from triangles. The first triangle in
power 1, is the difference of the first polygons in energy in
the descending series in the table. Thus for instance, the first
triangle in energy is 3, the first square is 4, the first pentagon
is 5, the first hexagon is 6, &c. and the difference between these
is 1. Likewise the first triangle in energy 3, is the difference of
the second polygons in energy 6,9, 12, 15, &c. in the descending
series. And all the gnomons of heptagons have the samc
final numbers, as the first and second have, viz. 1 and 6.
In the series of hexagons, all the perfect numbers are found,
as 6, 28, kc. Likewise, all hexagons are triangles, one triangle
in the series of triangles being alternately omitted, as 1, 6, 15,
28,45, &c. The perfect numbers also contained in the series of
hexagons are triangular, as 6,28, &c. In the series of pentagons,
two of them are alternately even numbers, and two are odd.
And of the even one number is evenly-odd, but the other is
unevenly-even. Thus 1 and 5 are odd, but 12 and 22 even. And
afterwards alternately 12 is uneven1 yeven, but 22 is even1 y-odd.
In the descending series of these numbers, after the series of
units, the first polygons differ by unity, as 3,4,5,6; the second
by 3, as 6,9,12, kc.; the third by 6, as 10, 16,22, &c.; the fourth
by 10, as 15, 25, 35, &c.; and the fifth by 15, as 21, 36, 51, &c.;
so that all these differences are in the series of triangles.
All the second and third polygons from unity, have superparticular
ratios, as 3, 4, 5, 6, and also 6, 9, 12, 15. But the
fourth polygons from unity have superpartient ratios as 10, 15,
22, 28.
Every square is composed from the triangle above it, and the
next antecedent triangle, as 9 from the triangles 6 and 3. Every
pentagon is composed from the triangle above it and twice the
antecedent triangle, as 12 from 6 and twice 3. Every hexagon