Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

agreeably to the form of the evenly-odd number. Thus in the
number 24, the quantity of the part is even, being denominated
from the even number. For the fourth part of it is 6, the
second part is 12, the sixth part is 4, and the twelfth part is 2,
which appellations of parts are not discordant from parity of
quantity. The parts however 8, 3, and 1 do not correspond in
denomination to the quantities; for 8 is the third part, 3 is the
eighth part, and 1 is the twentyfourth part. Hence in this instance,
when the denominations are even, the quantities are
found to be odd, and when the quantities are even, the denominations
are odd.
But these numbers are produced in such a way as to designate
their essence and nature even in their very generation;
for they are the progeny of the evenly-even, and the evenlyodd
numbers. For the evenly-odd are produced, as we have
shown from the series of odd numbers; but the evenly-even
from the duple progression. Let all the numbers therefore,
that are naturally odd, be disposed in order, and under these
all the numbers in a duple progression beginning from 4 as
follows :
3 5 7 9 11 13 15 17 19 etc.
4 8 16 32 64 128 256 512 1024 etc.
If therefore the first number in one series, is multiplied by
the first in the other, viz. if 3 is multiplied by 4, or if the same
first is multiplied by the second number in the second series,
i.e. if 3 is multiplied by 8, or the first by the third, i.e. 3 by 16,
and so on as far as to the last term; or if tht: second tern1 in
the first series is multiplied by the first, or second or third, or
in short, by any term in the second series; or the third term in
the first series, by any term in the second, and so of the fourth,
fifth etc. terms in the first series, all the numbers thus produced
will be unevenly-even. Thus 3 )< 4=12, 5 )