Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

pared with the first number longer in the other part, the third
with the second, the fourth with the third, and the fifth with
the fourth, etc. the same ratios will be produced as those above
mentioned. Here, however, the differences do not begin from
unity, but they proceed from the binary number through the
same terms ad infinittrm. And the second term of the one
series will be double the first term of the other. The third of
the one will be sesquialter of the second of the other. The
fourth will be sesquitertian of the third; and so on according
to the concord above demonstrated.
Again, squares differ from each other by odd numbers; but
the heteromekeis by even numbers.
Odd differences.
3. 5. 7. 9. 11. 13.
Squares.
Even differences.
4. 6. 8. 10. 12. 14.
Heteromekeis.
But if the first number longer in the other part, is placed
between the first and second square, it is conjoined to both of
them by one ratio; for in each of the ratios, the multiplicity of
the duple is found. If also the second number longer in the
other part, is placed between the second and third square, the
form of the sesquialter ratio to both terms will be produced.
And if the third of these numbers is placed between the third
and fourth square, the sesquitertian ratio will be generated.
And by proceeding in this way in the rest, all the superparticular
species will be found to be produced.
Again; if two of the above squares are added together, and
the number longer in the other part which is a medium be-