Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK TWO
CHAPTER I.
How all inequality may be reduced to equality.
N the former book we have shown how the whole
essence of inequality proceeds from equality as the
principle of its nature. From those things however
which are elements, all things are principally composed,
and by analysis are resolved into the same.
Thus because letters are the elements of speech, the conjunction
of syllables proceeds from them, and are again terminated
in the same as in bounding extremes. Sound also obtains the
same power in music. And every one knows that four elementary
bodies constitute the world. But an ultimate analysis
is again effected into these four elements. Hence, because
we see that all the species of inequality proceed from the
bounding nature of equality, it is requisite that inequality
should be again resolved by us into equality, as into a certain
clement of its proper genus. This, however, is again effected
by a triple rule; and the art of analyzation is as follows: Three
terms being given, which are unequal indeed, but proportionally
constituted, viz. so that the middle may have the same
ratio to the first term, which the last term has to the middle,
in every ratio of inequality, either in multiple, or superparticu-