Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

that middle which is alone conversant with the difference of
number.
That is called an arithmetical middle, therefore, when three
or more terms being given, an equal and the same difference is
found between all of them; in which the identity of ratio
being neglected, the difference only of the terms is considered.
Thus in the natural series of numbers 1. 2. 3. 4. 5. 6. 7. 8. 9.
10, the differences are equal, but there is not the same ratio
and habitude. If, therefore, three terms are given, the proportionality
is said to be continued. But if there are four or more
terms, the middle is called disjunct. Whether, however, the
terms are three or four, or any other number, there will always
be the same difference of the terms, the ratios only being
changed. Thus the differences of the terms 1.2.3. are the same,
but the ratios are different. For 2 to 1 is a duple, but 3 to 2 is a
sesquialter ratio. And the same thing will take place in the
other terms.
If, also, an equal number of terms are omitted, the differences
will be equal, but the ratios different. Thus if one term
is omitted, the difference will be 2; for then the terms will be
1. 3. 5. the difference between which is 2. But the ratio of 3
to 1 is very different from that of 5 to 3. If two terms are
omitted, the difference will be 3; if three terms are omitted,
it will be 4; and so of the rest, both in continued and disjunct
proportions. But the quality of the ratio will not be the same,
though the terms are distributed by equal differences. If, however,
conversely, there should be the same quality of ratio, but
not the same differences, such proportionality is called geometrical,
but not arithmetical.
It is the peculiarity of this middle, that in three terms, the
sum of the extremes is double the mean; as in 1. 2. 3, 1+3=
4=2~2. But if there are four terms, the sum of the extremes