Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 155
have a contrary nature, because this is also the case with their
generators. But again, the cause of this is inequality, and
prior to this, the indefinite duad. And if a mixture should be
made, and as I may say, a marriage of the even and the odd,
the offspring will partake of the nature of each of these,
whether their generators differ by unity, or by some greater
number. For the numbers produced, are either such as are
longer in the other part, or such as are oblong. And again,
from squares mingled with each other, squares are generated;
from numbers longer in the other part, such as resemble them
are produced; but from such as are mixed, squares are never
generated, but numbers that are entirely heterogeneous. And
this is what the most divine Plato says respecting the male
and female governors of his Republic, when they have not
been nourished in the mathematical disciplines, or though
they have been nurtured in them, yet have engaged in wedlock
in a confused and disorderly manner, from which the progeny
being depraved, will be the principle of sedition and discord
to the whole polity."
Let there be given a series of square numbers from unity,
and under them a series of numbers longer in the other part,
from the duad or two, as below:
Then, if 2 is compared to 1, the ratio is duple, and is the root
of the duple ratio. But 6 to 4 is sesquialter, and 12 to 9 sesquitertian,
and by thus proceeding, the superparticular ratio,
the sesquiquartan, sesquiquintan and others will be found, the
differences exceeding each other by unity. For the difference
between 2 and 1 is 1, between 6 and 4 is 2, between 12 and 9
is 3, and so on, the excess being always unity.
In the next place, the terms of one series being compared