Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 111
ratio, arises from a comparison of the difference of the extremes
with the difference between the middle and the less estreme.
For the difference between 12 and 6 is 6, and the difference
between 8 and 6 is 2; and 6 is triple of 2. And the greater
symphony which is the bis diapason, and consists in a quadruple
ratio, is beheld in the comparison of 8 with the difference
between 8 and 6, viz. in the comparison of 8 with 2. Hence 2
proportionality of this kind, is properly and appropriately called
an harmonic middle.
CHAPTER XXIX
How two terms being constituted on either side, the arithmetic,
geometric, and harmonic middle is alternately changed between
them; and on the generation of them.
We ought, however, to show, that as when in musical pipe,
the extreme holes remaining, it is usual, by changing the middle
hole, and opening one and shutting another with the fingers,
to cause them to emit different sounds; or as when two
chords being extended on each side, the musician causes the
sound of the intermediate chord to be either sharp or flat, by
stretching or relaxing it; thus also, two numbers being given,
we may insert between them, at one time an arithmetical, at
another a geometrical, and at another an harmonic middle.
We may, however, be able to change this middle, when placed
between two terms that are either odd or even, so that when
the middle is arithmetical, the ratio and equability of the differences
only is preserved; but when it is geometrical, the conjunction
of the ratios will remain firm; and if there is an harmonic
comparison of the differences, it will not be discordant
with the ratio of the terms. In the first place, therefore, let the
extremities be the even numbers 10 and 40, between which it a