Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

CHAPTER XXII.
A demonstration that squares and cubes partake of the nature
of sameness.
IT is a most evident sign that all squares are allied to odd
numbers, because in every arrangement of them, whether in a
duple or triple series of terms, from unity, they are never
found but in the place of an odd number according to the
natural series of numbers. And they will also be found to be
in the place of odd numbers in a quadruple, quintuple, etc.
series, though it is not true in these series, that they are found
in these places alone. For let numbers be disposed in an
orderly series, first the duple, then the triple, etc. as follows:
The places of odd
numbers. 1 3 5 7
The duple series. 1 2 4 8 16 32 64 128
The triple series. 13 9 27 81 243 729 2187
The quadruple series. 1 4 16 64 256 1024 4096 16384
The quintuple series. 1525125 6253125 15625 78125
The sextuple series. 1 6 36 216 12% 7776 46656 279936
Here in the duple and triple series, it will be found that all the
squares are in the places of the odd numbers. Thus 4 and 9
are in the place of 3; 16 and 81 are in the place of 5; and
64 and 729 are in the place of 7. And so of the rest. But in
the quadruple series, the numbers which are in the second and
fourth places (and these are the places of even numbers) viz. 4
and 64 are also squares, as well as those which are in the third,
fifth, etc. places.
Cubes also, though they have three intervals, yet on account
of the equal multiplication by which they are produced, par-