Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

tagonal numbers are such as the following, viz. 1. 5. 12. 22. 35.
51. 70. Their sides also increase after the same manner. For
in every pentagon, the units that form its side, correspond in
number to the class of the pentagon. Thus 1 which is the first
pentagon in power, has 1 for its side. But 5 which is the second
pentagon has 2 for its side. The third pentagon which is
12, has 3 for its side. The fourth which is 22, has 4. And so
on, according to the progression of the series of natural numbers
:
But these numbers which being extended into breadth unfold
five angles, are also generated from the addition of the
natural series of numbers, so that two terms being continually
omitted, and the posterior added to the prior, the several pentagons
are formed, as below:
I
The series of natural
numbers, two
being continually 1 4 7 10 13 16 19 22 25 28 31 34
omitted.
The series of pen- 1 5 12 22 35 51 70 92 117 145 176 210
tagons. 5
Pentagons also are produced from squares and triangles, after
the following manner. The second square added to the first
triangle, will produce the second pentagon. The third square
added to the second triangle, will produce the third pentagon.
The fourth square added to the third triangle, will produce
the fourth pentagon. And so on; each square being added to a