Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 71
triangle of a class immediately antecedent to it. Thus for instance,
the second square 4 being added to the first triangle 1,
produces the second pentagon 5. Thus too, the third square 9
added to the second triangle 3, forms the third pentagon 12.
And so of the rest.
CHAPTER VIII.
On hexagons and heptagons, and the generation of them.
A general method of discovering the generation of all figare$,
etc.
THE sides of hexagons and heptagons increase after the following
manner. In the generation of triangular numbers,
those terms were added together which followed each other in
a natural order, and surpassed each other by unity alone. But
the generation of squares was from numbers which were added
together, one term being omitted, and which surpassed
each other by 2. In the generation of pentagons two terms
were omitted, and the number by the addition of which they
were formed, surpassed each other by 3. Thus also in the
generation of the hexagon, three terms being omitted, those
are added together that surpass each other by 4; and these
terms indeed, will be the roots and foundations, from the
aggregation of which all hexagons will be produced. These
roots therefore are the numbers 1.5.9. 13. 17. 21, etc. and from
the addition of these, hexagons will be formed, viz. 1. 6. 15.
28. 45. 66, etc.
But in the generation of heptagons, four terms being omitted,
those are to be added that surpass each other by 5; and
these numbers, as was before observed, will be the roots and
foundations of them: