Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 113
to 10 is a quadruple ratio. If this, therefore, is divided by 2,
it will become duple, which is 20; for 20 is the double of 10.
This, therefore, will be the geometrical mean, between the two
given terms. But to find the harmonic middle, the difference
of the terms must be multiplied into the less term; then the
product must be divided by the sum of the extremes; and in
the last place, the quotient must be added to the less term, and
the sum will be the mean required. Thus the difference between
40 and 10 is 30; but 30)(10=300; and 300 divided by
40+10=50, gives for the quotient 6; and 6+10=16, the
harmonic mean between 10 and 40.
CHAPTER XXX
On the three middles which are contrary $0 the harmonic and
geometric middles.
THE middles which we have now discussed, were invented
and approved of by the more ancient mathematicians; and we
have more largely unfolded them, because these are especially
found in the writings of the ancients, and are most useful to a
genuine knowledge of them. We shall therefore mention the
other middles with brevity, because they are scarcely of any
other use than that of giving completion to the duad. But
these middles appear to be contrary to the former, from which
they nevertheless originate. The fourth middle, however, is
that which is opposite to the harmonic. For in the harmonic,
as the greatest is to the least term, so is the difference between
the greatest and the middle, to the difference between the middle
and the least term, as in the terms 3. 4. 6; but in this
fourth proportionality, three terms being given, as is the greatest
to the least, so is the difference between the middle and
least, to the difference between the greatest and middle terms;