Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 95
ticipate of an immutable essence, and are the associates of
sameness; for they are generated by the coacervation of odd,
but never by that of even numbers. For if all the odd numbers
are disposed in an orderly series from unity, viz.
the addition of these will form cube numbers. Thus, since 1
is the first cube in power or capacity, if the two following
numbers, i.e. 3 and 5, are added together, their sum will be the
second cube, which is 8. If the three numbers also that follow
these are added together, viz. 7. 9. 11. their sum 27 will be the
third cube. And the four following numbers, 13, 15, 17, 19
will, when added together, produce the fourth cube, viz. 64.
And so of the rest.
CHAPTER XXIII.
On proportionality, or analogy.
AND thus much may suffice concerning these particulars.
It is now requisite that we should discuss what pertains to
ratios, so far as may be necessary to a knowledge of the theoretic
part of music, astronomy, and geometry, and of the philosophy
of Plato and Aristotle. Proportionality, therefore, is the
collection into one, of two, or more than two ratios.* Or according
to a more common definition, it is the similar habitude
of two or more ratios, though they arc not constitutcd in the
same quantities and differences. But difference is the quantity
between numbers; and ratio is a certain habitude, and as it
were, connexion of two terms to each other; the composition
Analogy is most accurately defined by Proclw on the Thus of Plato, to be,
identity of ratio, and the most beautiful of bonds.