Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

is denominated the evenly-odd, and the third is the oddly-odd.
And the species indeed which are contrary, and obtain the
place of extremes are the evenly-even, and the evenly-odd.
But the species which is a certain medium, and participates of
each of the extremes, is the number which is called oddly-odd.
Again, the evenly-even number is that which may be divided
into two equal parts, and each of these parts into two
other equal parts, and each of these may be divided in a similar
manner, and the division of the parts may be continued
till it is naturally terminated by indivisible unity. Thus the
number 64 has for its half 32, but the half of this is 16, the
half of 16 is 8, the half of 8 is 4, of 4, two, and the half of 2
is 1, which naturally does not admit of division.
To this number it happens that whatever may be its part is
found to be evenly-even both in denomination and quantity.
And it seems that this number was called evenly-even, because
all its parts are found to be evenly-even both in name and
quantity. We shall however hereafter show how this number
has even parts both in quantity and appellation.
But the generation of these numbers is as follows: All numbers
in a duple ratio from unity, will always be found to be
evenly-even; and it is not possible that they should be produced
in any other way. Thus for instance, the numbers in a
duple ratio from unity are 1.2.4.8.16.32.64.128.256.512. and
so on ad infiniturn; for they are all evenly-even, and the ratio
of their progression is duple.
It is remarkable in this series, that if the number of terms
is even, the two middle terms correspond to each other, and
this also will be the case with the terms above these, and so on
till each term meets with the extremities. Thus for instance,
let there be given a series of evenly-even numbers from 1 to
128. In this series therefore, because the number of terms is