Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

lar, or superpartient ratios, or in those which are generated
from these, i.e. in multiple superparticulars, or multiple superpartients
;-three terms therefore being given with these conditions,
one and the same indubitable method must be observed.
For the first term must be placed as the first. The
second term must be that which remains from the subtraction
of the first from the middle term. But the third term must
be obtained by subtracting the first term from the third, and
also twice the second term that remained from the former subtraction;
for then the remainder will be the third term. By
this process the proportion will be reduced; so that if the proportion
is quadruple, it will first be reduced to a triple, then
to a duple proportion, and last of all to equality. We shall
only demonstrate that this will be the case in multiple pre
portion, as it will be easy to apply the same process to the
other species of inequality also.
Let there be three terms therefore quadruple of each other,
viz. 8. 32. 128. From the middle term, therefore, 32, let the
first term 8 be taken, and 24 will remain. Let 8 then be placed
as the first, and 24 as the second term. But from the third
term 128, take away the first term 8, and twice the remaining
second, i.e. twice 24, and 72 will remain. Hence, the three
terms will be 8. 24. 72; and the quadruple will be reduced to
a triple proportion.
Again, by the same process with these three terms, the triple
will be reduced to a duple proportion. For, let the first be
equal to the first, i.e. to 8. And from the second take away
the first, and 16 will remain. But from the third 72, take
away the first, i.e. 8, and twice the remaining second, i.e. 16,
and 32 will remain. So that the three terms will be 8. 16. 32;
and the habitude will be reduced to a duple proportion.
By the same process however with these terms, they will be