Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

of which produces that which is proportional or analogous. For
proportionality arises from the junction of ratios. The least
proportionality, however, is found in three terms, such as 4, 2,
1. For as 4 is to 2, so is 2 to 1; the proportionality consisting
of two ratios, each of which is a duple ratio. Proportionality,
however, may consist in any number of terms greater
than three. Thus in the four terms 8,4,2,1, as 8 is to 4, so is 4
to 2, so is 2 to 1; and in all these the proportionality consists of
duple ratios. This will also be the case in the five terms 16,
8, 4, 2, 1; and in the six terms 32, 16, 8, 4, 2, 1, and so on
ad infiniturn. As often, therefore, as one and the same term so
communicates with two terms placed about it, as to be the
antecedent to the one, and the consequent to the other, this
proportionality is called continued, as in the instances1 above
adduced. But if one term is referred to one number, and
another to another number, it is necessary that the habitude
should be called disjunct. Thus in the terms 1, 2, 4, 8, as 2
is to 1, so is 8 to 4, and conversely as 1 is to 2, so is 4 to 8.
And alternately as 4 is to 1, so is 8 to 2.
CHAPTER XXIV.
On the proportionality which was known to the ancients, and
what the proportions are which those posterior to them have
added.-And on arithmetical proportionality, and its proper-
THE three middles which were known to the more ancient
mathematicians, and which have been introduced into the philosophy
of Pythagoras, Plato, and Aristotle, are, the arithmetical,
the geometrical, and the harmonic. After which habitudes
of proportions, there are three others without an appropriate
appellation; but they are called the fourth, fifth, and sixth, and