Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

third; and let all the triples be arranged under all the quadruples
after the same manner, as follows:
Hence, if the first number is compared with the first, a sesquitertian
ratio will be formed. For 4 contains the whole of 3 in
itself, and a third part of 3 besides, i.e. 1. In a similar manner
8 contains the whole of 6, and a third part of 2. And the
same consequence will take place in the rest ad infinitum. It
must also be observed, that 3, 6, 9, 12, etc. are attendants, and
4, 8, 12, 16, etc. leaders; and that the ratio of the former to
the latter is subsesquitertian, but of the latter to the former sesquitertian.
This also is admirable and most profound in the orders of
these numbers, that the first leader and the first attendant are
conjoined to each other without the intervention of any other
number. But between the second leader, and the second attendant,
one number intervenes. Between those in the third
rank, two numbers intervene. Between those in the fourth,
three. And the intervening numbers are always less by one
than the rank of the numbers themselves. But it is necessary
that this should take place in sesquialter, sesquitertian, or other
superparticular parts. Thus when 4 is compared to 3, no
number intervenes; for 4 succeeds immediately to 3. But
when 8 is compared to 6, which forms the second sesquitertian
ratio, one number intervenes; for 7 comes between 6 and
8. Again, when 12 is compared to 9, which forms the third
sesquitertian ratio, two numbers intervene, viz. 10 and 11.
After the same manner, between those in the fourth order,
three numbers intervene; between those in the fifth, four numbers;
and so on ad infinitum.