Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

first and second of the one, will correspond to the first and
second of the other; but the third and fourth of the one, to the
second and third of the other; and the fifth and sixth of the
one, to the third and fourth of the other; and so on. Thus let
there be a series of numbers in the natural order, viz. 1. 2. 3.
4. 5. 6. 7. 8. 9. And under it a series of triangles, viz. 3. 6. 10.
15.21. 28.36. 45. Then as 1 is to 2, so is 3 to 6. And as 2 is to
3, so is 10 to 15. Likewise as 3 is to 4, so is 21 to 228; and so of
the rest.
Triangular numbers also are obtained after the following
manner. For in the natural series of numbers from unity, by
first omitting one term, then two terms, afterwards three, then
four, and so on, triangles will be formed in a continued series:
The oerico
of natural 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I S 16 17 18 19 20 21 22 23 24 25 26 27 28
nu~nbers 1
The series
of trian- 1 3 6 10 15
glw t
Ilere it is evident, that between the first and second triangle,
in the natural order of numbers, one number intervenes:
between the second and third triangle, two numbers; between
the third and fourth, three; and so on in the rest.
CHAPTER VI.
On square nzrrnbers, their sides, and generation.
A SQUARE number is that which does itself indeed unfold
breadth, yet not in three angles, as the preceding number, but
four. It likewise is extended by an equal dimension of sides
But such numbers are as follows: