Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

Again, of any two numbers whatever, either one of the two,
or their sum, or their difference is divisible by 3. Thus of the
two numbers 6 and 5, 6 is divisible by 3; of 11 and 5 the difference
6 is divisible by 3; and of 7 and 5 the sum 12 is divisible
by 3. The square of 3 also, viz. 9, has this property, that 4,
the sum of its aliquot parts 1, 3, is the square of 2.
Farther still, if any triangular gnomon is multiplied by 3 and
unity is added to the product, the sum will be the gnomon of
a pentagon, as is evident from what is delivered in the 13th
chapter. And lastly, if any gnomon of a square, viz. if any one
of the numbers 1, 3, 5, 7, 9, &c. is multiplied by 3, and unity is
added to the product, the sum will be a pentagonal gnomon, as
we have before shown in the chapter on polygonous numbers.
Thus for instance,
3x1 and +1= 4 the 2ndl
3~ 3 and +1=10 the 4th pentagonal gnomon.
3x5 and +1=16 the 6th 1
3x7 and +1=22 the 8th J
CHAPTER XV
On the properties of the tetrad, pentad, and hexad.
HAVING already said so much about the tetrad in the chapter
on its appellations, there remains but little more to observe
concerning it from the ancients; Theo in the extract we have
given from him respecting the tetractys having nearly exhausted
the subject. I have therefore only to add farther, that the
square of this number has a space equal to the length of the
sides For the sides are 4 in number, each of which is 4, and
the square of it is 16. It is also produced as well by the addi-
For in every order of things there are povq, r~oo80$, and rrrotpogq, i.e. permanency
in, progression from, and a return to causeo