Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 57
reduced to equality. For let the first be equal to the first, i.e.
to 8. And from 16 subtract 8, and 8 will remain. But from
the third term 32, take away the first 8, and twice the remaining
second, i.e. twice 8, and 8 will remain. So that the three
terms will become by reduction, 8. 8. 8.
The other species of inequality also, will, by the same process
be brought to equality. Hence, it may be indubitably asserted,
that as unity is the principle and element of quantity
that subsists by itself, so equality is the mother of relative
quantity.
CHAPTER 11.
On discovering in cach number, how many numbers of the
same ratio may precede--description of them and an exposition
of the description.
In this affair however, there is a certain profound and admirable
speculation, and as Nicomachus says, ~vvotocea.sov flaopqyz,
or, a theorem replete with intellectual conception, useful
for the purpose of understanding the Platonic generation of
the soul in the Timaeus, and the intervals of the harmonic
discipline. For in the Timaeus we are ordered to produce and
extend three or four sesquialter, or a certain number of sesquitertian
ratios, and sesquiquartan comparisons. But lest this,
which is always indeed attended with the greatest labour,
should frequently be done without advantage, it is now requisite
to investigate how many superparticulars every multiple
has for its attendants. For all multiples will be the leaders
of as many proportions similar to themselves, as they themselves
are distant from unity. But similar proportions are those
which are of the same ratio and denomination. The meaning
however, of multiples, being the leaders of proportions