Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 59
alters as much distant from its leader, as its leader is among
doubles from unity. Thus 9 is the second sesquialter from 4;
and 4 is the second double from 1. Thus too, 27 is the third
sesquialter from 8; and 8 is the third double from 1. And so
of the rest. This last sesquialter likewise, is always incapable
of being divided into two equal parts.
The same thing also takes place in triple numbers; for from
them sesquitertians are generated. Thus, because 3 is the first
triple number, it has one sesquitertian, i.e. 4; of which a third
part cannot be found, and therefore it is deprived of a sesquitertian
number. But the second triple number which is 9, has
for a sesquitertian number 12. And 12 because it has a third
part, has also a sesquitertian 16, which is excluded from a third
part. The number 27, however, because it is the third triple,
has for a sesquitertian 36; and this again is compared in the
same ratio to 48; and 48 has also for a sesquitertian, 64. But
64 has no sesquitertian, because it has no third part. And thus
it will be found in all triple numbers, that the last number of
the same ratio has, preceding it, as many numbers, as the first
of them is distant from unity; and that it is incapable of being
divided into three equal parts:
The description of quadruple numbers is according to the
following scheme, which will be immediately obvious to those