Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

If, however, the multiples which are produced from the
equal terms are disposed in a converse order, and the same
rules are employed, sesquialter ratio will be generated from
the double terms ; sesquitertian from the triple ; sesquiquartan
from the quadruple, etc. For let there be three double terms
which are generated from equals, and let that which is the
last be placed as the first, viz.
Let the first therefore be made equal to the first, i.e. to 4. But
let the second be equal to the first and second, viz. to 6. And
let the third be equal to the first, to twice the second, and to
the third, viz. to 9.
4 2 1
4 6 9
And it is evident, that sesquialter ratio will be produced
from the double terms.
Again, let the former triple terms be disposed in a converse
order, viz. 9. 3. 1. Let the first therefore be equal to the first,
i.e. to 9. But let the second be equal to the first and second,
i.e. to 12, And the third to the first, to twice the second,
and the third, viz. to 16.
9 3 1
9 12 16
And thus the second species of the superparticular number,
i.e. the sesquitertian is generated.
By the same process also with the quadruple numbers, the
sesquiquartan ratio will be immediately produced, as is evident
from the subject description:
And if the same thing is done with all the parts multiplied ad