Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

the impression of itself, defines and bounds that which is of itself
indefinite. Hence in the human soul, there is a certain
vestige of divine goodness, which moderates the inequality of
its motions. This vestige is the rational power, by which we
restrain the inordinate tendencies of desire, and the effervescence
of anger, both which partake of the nature of inequality.
This however will be evident, if we demonstrate that all the
species of inequality originate from equality; so that obtaining
as it were, the power of a mother and a root, she pours
forth with primordial exuberance, all the species and orders of
inequality. Let there be three equal terms, therefore, i.e. three
unities, or three of the number 2, or of 3, or of 4, or of any
other number. For that which takes place in one three terms,
will happen in the rest. From the three rules therefore, that
will be given, it will be seen, that in the first place multiples
will be produced, first duple, then triple, then quadruple, and
so on. Again, if the multiples are converted, from these superparticulars
will arise. And from such as are duple indeed, the
sesquialter; from such as are triple, the sesquitertian; from the
quadruple, the sesquiquartan; and the rest after the same manner.
But from the conversion of superparticulars, it is necessary,
that superpartient ratios should be produced; so that the
superbipartient will originate from the sesquialter; the supertripartient
from the sesquitertian ; and the superquadripartient
from the sesquiquartan. From the former superparticulars
however, remaining in a direct position, and not being converted,
multiple superparticulars will arise; but multiple superpartients
will be produced, from thc position of the former
superpartients remaining unchanged.
The three rules however, are as follows: The first number
must be made equal to the first; but the second, to the first
and second; and the third, to the first, to twice the second,
and the third. When this is effected therefore, in equal terms,