Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 107
by the greater, by the same part of itself, yet by one part of
the less, and a different part of the greater term. Thus in the
three terms, 2, 3, 4, the term 3 precedes 2, by a third part of
itself, i.c. 1. and is preceded by 4 by the same part of itself.
But 3 does not surpass 2 by the same part of 2, nor is surpassed
by 4, by the same part of 4. For it surpasses 2 by 1,
which is the half of 2, but it is surpassed by 4, by 1, which is
the fourth part of 4. The ratios, however, in the harmonic
middle, have a contrary mode of subsistence. For the middle
term in this ratio, does not surpass the less, nor is surpassed by
the greater, by the same part of itself; but it surpasses the less
by the same part of the less, as is that part of the greater, by
which it is surpassed by the greater. Thus in the harmonic
arrangement, 2. 3. 6., the term 3 surpasses 2 by the half of 2,
and the same 3 is surpassed by 6, by 3 which is the half of
6. But in the geometric proportionality, neither does the rniddle
surpass the less term, nor is it surpassed by the greater
term, by the same parts of itself, nor does it surpass the less
term by the same part of the less, nor is surpassed by the
greater by the same part of the greater; but by that part
of itself, by which the middle surpasses the less term, by the
same part of itself the greater surpasses the middle term.
Thus in the three terms 4. 6. 9, the middle term 6 surpasses 4
by a third part of itself, and 9 also surpasses 6 by a third part
of itself. The harmonic proportion also has another property,
that if the two extremes are added together, and multiplied by
the middle, the product will be double that of the extremes
multiplied by each other. Thus in the terms 3. 4. 6, 3+6=9,
and 9)