Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

tan, the superquadripartient ratio will immediately be generated,
as is evident from the subject description:
It now remains therefore, to show how from superparticular
and superpartient, multiple superparticular, and multiple
superpartient ratios are produced. Of these, however, we shall
only make two descriptions. For from the sesquialter in a direct,
and not in a converted position, duple superparticular
ratio will be generated. Let the direct sesquialter terms therefore
be:
4 6 9
And then according to the former mode, let the first be equal
to the first, i.e. to 4. But let the second be equal to the first
and second, i.e. to 10. And the third, to the first, to twice
the second, and to the third, i.e. to 25:
And duple sesquialter ratios will be produced.
But from the sesquitertian ratio, the terms not being transposed,
the duple sesquitertian will be produced, as is evident
from the subject description :
If, however, we direct our attention to superparticular ratios,
and dispose the terms of them according to the former rules,
we shall find multiple superpartients produced in an orderly
series. For by proceeding according to the rules in this formula
of the superpartient, viz. in 9. 15. 25. the duple superbipartient
ratios 9. 24. 64. will be generated. But from the super-