Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

atios. Let them, therefore, be alternately arranged from unity
as follows: 1. 2. 4. 6. 9. 12. 16. 20. 25. 30. In this arrangement
we shall find, that between 1 which is the first square in capacity,
and 2, there is a duple ratio; and that between 2 and 4,
the ratio is also duple. Here, therefore, a square is joined to
a number longer in the other part, and this to the following
square, in the same ratio, but not with the same differences.
For the difference between 2 and 1 is unity alone, but between
4 and 2 the difference is 2. Again, if 4 is compared to 2 the
ratio is duple, but if 6 is compared to 4 the ratio is sesquialter.
Here then, the terms are discordant in ratios, but equal in
differences. For the difference between 4 and 2 is 2, and there
is the same difference between 6 and 4. In the following also,
after the same manner as in the first numbers, the ratio is
the same, but the differences are not the same. Thus 4 is
joined to 6, and 6 to 9 by a sesquialter ratio; but 6 surpasses
4 by 2, and 9 surpasses 6 by 3. In the terms that follow likewise,
at one time the ratios are the same, but the differences
not the same; and at another, vice versa, the differences are
the same, but the ratios are different. The squares too, and
numbers longer in the other part, surpass each other in their
differences according to the series of natural numbers, but
with a duplication of the terms, as is evident from the following
scheme.
Differences.
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That the agreement and difference however, of these two
species of numbers may become more apparent, let there be
two series of them, in the former of which numbers longer in
the other part come between the squares, and in the latter
squares, between numbers longer in the other part as below: