Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

tion of 2 to itself, as by the multiplication of 2 into itself, and
on both these accounts, as we are informed by Camerarius, and
as we have before observed, it was called by the Pythagoreans
justice; the essence of justice consisting in equality. I have
also found that if 4 multiplies any one of the series of squares
1, 4, 9, 16, 25, kc. and unity is added to the product, the sum
will be an hexagonal gnomon. For,
4x1 and +1= 5 the 2ndl
4x4 and +1=17 the 5th hexagonal gnomon.
4x9 and +1=37 the 10th
&c.
Likewise, that if 4 multiplies any gnomon of a square, and
unity is added to the product the sum will be also an hexagonal
gnomon. For,
4x1 and $1= 5 the 2nd)
4x3 and +1=13 the 4th 1
hexagonal gnomon.
4x5 and +1=21 the 6th
&c.
And again, that if 4 multiplies any pentagon, and unity is
added to the product the sum will likewise be an hexagonal
gnomon. Thus,
4X 1 and +1= 5 the 2ndl
4X 5 and r
1=21 the 6th hexagonal gnomon.
4X 12 and 1=49 the 13th
4 )< 22 and + 1=89 the 23rd J
&c.
Farther still, that if 4 multiplies any gnomon of a pentagon,
and unity is added to the product, the sum will also be an
hexagonal gnomon. For,
4X 1 and +1= 5 the 2nd
4~ 4 and +1=17 the 5th hexagonal gnomon.
4X 7 and +1=29 the 8th
4x10 and +1=41 the 11th
&c.
a