Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

as will be hereafter shown. But of these middles, viz. the
arithmetical and the harmonic, the geometric proportionality
is the medium; for this preserves both in greater and less terms
an equality of ratio. And equality is the medium between the
greater and the less.
CHAPTER XXV.
On the geometrical middle and its properties.
IT is now requisite to discuss the geometrical which follows
the arithmetical middle, and which alone or principally, may
be called proportionality or analogy; because both in the greater
and the less terms of which it consists, there is found to be
the same ratio. For here the same ratio is always preserved,
and contrary to the arithmetical middle, the quantity of the
number is neglected. Thus, in the duple series, 1. 2. 4. 8. 16.
32. etc. or in the triple series, 1.3.9.27. 81. etc. or in any other,
whether quadruple or quintuple, etc. the prior to the following
number, will always be in the same ratio as the following to
another number that is immediately consequent to it. Thus, if
there are three terms 2. 4. 8. it will be 2 : 4 : : 4 : 8, and conversely
8 : 4 : : 4 : 2. And in four terms 2. 4. 8. 16, it will be
2 : 4 : : 8 : 16, and conversely 16 : 8 : : 4 : 2. And in a similar
manner in all the rest.
The peculiarity of this middle is, that in every arrangement
of the terms, the differences are in the same ratio to each other,
as the terms of which they are the differences. For if the
terms are double of each other, the differences also will be
double; if the former are triple, the latter will be likewise
triple; if the one is quadruple, the other also will be quadruple.
And whatever the multiplicity of the terms may be, there