Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

as in 3. 5. 6. For as 6 is to 3, so is the difference between 5
and 3, i.e. 2, to the difference between 6 and 5. i.e. 1. But
it appears that this middle is opposite, and after a manner contrary
to the harmonic middle, because in the latter, as is the
greatest to the least term, so is the difference of the greater to
the difference of the less terms; but in the former, as is the
greatest to the least term, so is the difference of the less to
the difference of the greater terms. It is the peculiarity of thiD
middle, that the rectangle under the greatest and middle terms,
is double the rectangle under the middle and least terms. Thus
5 )< 6=30; but 3 )< 5=1S. But the two other proportionalities,
viz. the fifth and sixth, are contrary to the geometric proportionality,
and appear to be opposite to it.
For the fifth is when in three terms, as is the middle to the
less term, so is their difference to the difference between the
middle and greater terms; as in the terms 2. 4. 5. For as 4 is
to 2, so is the difference between 4 and 2, i.e. 2, to the
difference between 5 and 4, i.e. 1. But this is contrary to
geometric proportion, because there, as is the ratio of the
greater to the less, i.e. to the middle, and of the middle to the
least term, so is the difference of the greater to the difference
of the less terms, as in 8. 4. 2. For the ratio of 8 to 4, and
of 4 to 2, is the same as the ratio of 84, to the ratio of
4--2. But in the fifth proportionality, as is the ratio of the
greater to the less term, so is the ratio of the difference of the
less, to the difference of the greater terms. The peculiarity
also of this proportionality is, that the rectangle under the
greater term and the middle, is the double of the rectangle
under the two extremes. Thus in the terms 2. 4.5. 5 )< 4=20,
and 20 is the double of 5x2.
The sixth proportionality is when in three given terms, as is
the greater to the middle term, so is the difference of the less,
to the difference of the greatest terms; as in the terms 1. 4. 6.