Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

For the sesquialter has for its leaders all the numbers that are
naturally* triple after 3; but for its attendants, all the numbers
that are naturally even after 2. For let the series of natural,
of triple, and of double numbers be described in three
rows as follows:
The first row therefore contains the series of natural numbers;
the second the triple; and the third, the double of them.
Hence, if 3 is compared to 2, or 6 to 4, or 9 to 6, or if all the
superior triple are opposed to all the inferior double numbers,
sesquialter ratio will be produced. For 3 contains in itself 2,
and 1 the half of two. Six also contains in itself 4, and 2 the
half of 4. And 9 contains in itself 6, and the half of 6 which
is 3. And in a similar manner in the rest.
It is likewise requisite to show the method of discovering
the sesquitertian, or second species of the superparticular number.
And the definition indeed of this comparison is as follows:
the sesquitertian is that which when compared to the
less number, contains it once, and a third part of it besides.
But these numbers are found, if all the terms in a continued
series from 4 being made quadruple, are compared with all
the numbers that are made triple from 3. And the leaders in
this case will be quadruple; but the attendants triple. For let
there be a series of numbers in a natural order, and under
these a quadruple, and under the quadruple a triple series.
Let the first triple therefore, be placed under the first quadruple
number; the second under the second; the third under the
By numbers naturally triple, the triples of the natural series 1. 2. 3. 4. 5, kc.
must be understood. And in a similar manner numbers naturally duple, quadruple,
kc. are such as are duple, quadruple, kc. of that series.