Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

CHAPTER XIX.
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That from the nature of the numbers which are characterized
by sameness, and from the nature of those which are charmeked
by difference, vk. from squares, and numbers LONGER
IN THE OTHER PART, all the habitudes of proportions consist.
LET there be now therefore disposed in an orderly series, not
even and odd numbers, from which squares and numbers
longer in the other part are produced, but the two latter
species of numbers themselves. For thus we shall see a certain
consent of them, in the production of the other species of number.
Let there be two series, therefore, the one of squares from
unity; but the other of numbers longer in the othe~ part from
the duad, viz.
If, therefore, the first term of the one series is compared with
the first term of the other, the quantity of the duple will be
found, which is the first species of multiplicity. But if the
second term of the one is compared with the second term of
the other, the habitude of the sesquialter quantity is produced.
If the third is compared with the third, sesquitertian ratio is
generated. If the fourth with the fourth, sesquiquartan; and
if the fifth with the fifth sesquiquintan ratio. And thus proceeding
ad infiniturn, superparticular ratio will be invariably
produced. In the first ratio, however, which is that of the duple,
the difference between the two terms is only 1; but in the
sesquialter ratio, the difference is 2. In the sesquitertian ratio,
the difference is 3. In the sesquiquartan it is 4. And thus, according
to the superparticular form of numbers, the difference
continually increases by 1. If again, the second square is com-