Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two
If three gnomons of squares are assumed, 1, 3, 5, the sum
of them is 9, and the sum of the gnomons 2 and 4, of two
numbers longer in the other part, is 6, which is subsesquialter
to 9. If four gnomons of squares are assumed, 1, 3, 5, 7,
the sum of them is 16, and the sum of the three gnomons
2, 4, and 6 of numbers longer in the other part is 12; but 16
to 12 is a sesquitertian ratio. And by proceeding in this way,
the other ratios will be found to be sesquiquartan, sesquiquintan,
&c.
If squares are compared, and the numbers longer in the
other part which are media between them, the ratio of the first
analogy is duple, i.e. of 1, 2,4; of the second analogy 4,6,9, is
sesquialter; of the third 9, 12, 16, is sesquitertian, and so on.
But if numbers longer in the other part are compared with
squares as media, the ratios will be found to be allied and connected,
viz. the duple with the sesquialter, as in 2, 4, 6; the
sesquialter with the sesquitertian, as in 6, 9, 12; the sesquitertian
with the sesquiquartan, as in 12, 16, 20, and so on. Moreover,
every square and similar number, with a subject number
longer in the other part, and dissimilar, produces a triangular
number. Thus 1 +2=3, 4+6=10, 9+ 12=21, &c. all which
sums are triangular numbers. Likewise, if the first dissimilar
is added to the second similar number, or the second dissimilar
to the third similar number, the sums will also be triangular
numbers. Thus 2+4=6, 6+9=15, 12+16=28, &c. And
thus much respect to what Jamblichus says concerning the
nuptial number.