Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 115
For here the greatest is to the middle term in a sesquialter ratio;
and the difference between 4 and 1, i.e. 3. is in the same
ratio to the difference between 6 and 4, i.e. 2. For as 6:4::3:2.
After the same manner also as the fifth, this proportionality is
contrary to the geometrical, on account of the converse ratio of
the differences of the less to the greater terms.
CHAPTER XXXI
On the four middles which the ancients posterior to those
before mentioned, invented for the purpose of giving completion
to the decad.
AND these indeed are the six middles, three of which remained
from Pythagoras as far as to Plato and Aristotle. But
those that followed them inserted in their commentaries the
three others, which we discussed in the preceding chapter.
And the following age, as we have said, added four other middles,
in order to the completion of the decad. What these are,
therefore, we shall briefly relate. The first of them, which is
in order the seventh, is when in three terms, as the greatest is
to the last, so is the difference of the greatest and least term, to
the difference of the less terms; as in the terms 6. 8. 9. For
9 to 6 is sesquialter, the difference of which is 3. But the
difference of the less terms, i.e. of 8 and 6, is 2, which compared
to the former 3 produces a sesquialter ratio.
Again, the second proportionality of the four, but the eighth
in order is, when in three terms, as the extremes are to each
other, so is their difference to the difference of the greater
terms; as in 6. 7. 9. For 9 to 6 is sesquialter, and their difference
is 3, which compared to the difference of the greater
terms, i.e. to the difference of 9 and 7 which is 2, produces
also a sesquialter ratio.