Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

CHAPTER XXVII
On the harmonic middle and its properties.
THE harmonic middle is neither constituted in the same
differences, nor in equal ratios; but in this, as the greatest
term is to the least, so is the difference of the greatest and the
middle, to the difference of the middle and the least term: as
in the numbers 3. A 6, and 2. 3. 6. For as 6 is to 3, so is
the difference between 6 and 4, i.e. 2, to the difference between
4 and 3, i.e. 1. And as 6 is to 2, so is the difference
between 6 and 3, i.e. 3, to the difference between 3 and 2, i.e.
1. Hence in this middle, there is neither the same ratio of the
terms, nor the same differences.
But it has a peculiarity, as we have before observed, contrary
to the arithmetical middle. For in that, there is a greater
ratio in the less, but a less ratio in the greater terms. On the
contrary, in the harmonic middle, there is a greater ratio in the
greater, but a less in the less terms. Thus in the terms 3. 4. 6.
if 4 is compared to 3, the ratio is sesquitertian, but if 6 is compared
to 4, the ratio is sesquialter. But sesquialter is as much
1
greater than sesquitertian ratio, as 3 is greater than t. With
great propriety therefore, is geometrical proportionality said to
be a medium between that in which there is a less ratio in the
greater, and a greater in the less terms, and that in which there
is a greater ratio in the greater, and a less in the less terms.
For that is truly proportionality or analogy, which obtaining as
it were the place of a medium, has equal ratios both in the
greater and the less terms.
This also is a sign that geometrical proportion is a medium
in a certain respect between two extremes; that in arithmetical
proportion the middle term precedes the less, and is preceded