Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

similar to themselves, is as follows: The multiplicity of the
duple, as we have before observed, always produces sesquialter
ratios; and the triple is the leader of sesquitertian; but the
quadruple of sesquiquartan ratios.
The first duple, therefore, will only have one sesquialter;
the second will have two; the third three; the fourth four;
and according to this order, there will be the same progression
ad infintiurn. Nor can it ever be possible, that the equable
location from unity, should either surpass or fall short of the
number of the proportions. The first duple, therefore, is the
binary number, i.e. 2, which receives one sesquialter alone, i.e.
3. For 2 when compared to 3, produces a sesquialter ratio.
The number 3, however, because it has not a half, cannot be
compared to any other number in a sesquialter ratio. But 4 is
the second double. This therefore is the leader of two sesquialter
numbers. For 6 compared to it is sesquialter; and to 6
because it has a half, 9 is sesquialter. Hence, there are two
sesquialters, to 4 indeed 6, but to 6, 9. But 9, because it
wants a half, is excluded from this comparison. The third
double is 8. This therefore is the leader of three sesquialters.
For the number 12 is compared to it in this ratio; but to 12
18; and again to 18, 27. But 27 wants a half. The same
thing will also necessarily happen in other numbers, as is evident
in the following table:
For this always occurs by a certain divine, and no human
ordination, that the last sesquialter number is among sesqui-