Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 93
ity. For 1 is the whole of 1 to which it is equal, and is the
half of 2. Thus also the difference 2 is the whole of 2 to
which it is equal, and the half of 4. Again, 2 is the difference
of 4 and 6, and 3 is the difference of 6 and 9. And these are
unequal in quantity, but equal in denomination. For in each
the difference is the half of the less, and the third of the greater
number. For 2 is the half of 4, and the third of 6. And
3 is the half of 6, but the third of 9. After the same manner
3 is the difference between 9 and 12, and 4 is the difference
between 12 and 16; and these are unequal in quantity, but
equal in denomination; since each is the third part of the less,
and the fourth part of the greater. And so of the rest. Hence
it is evident that the differences proceed from the whole
through every part. For the first and second differences are
the whole and the second part. The third and fourth differences
are the second and third parts. The fifth and sixth differences,
are the third and fourth parts. And in a similar
manner in the rest.
On the contrary, in the second series, the differences are
equal in quantity, but unequal in denomination. For of the
two first terms 2 and 4, the difference is 2, and this is also the
difference of 4 and 6. These differences, therefore, are equal
in quantity. The difference 2, however, though it is the whole
of the less of the terms, is only the half of the less of the
second terms. And though it is the half of the greater of the
first terms, it is only the third part of the greater of the second
terms. Thus also the difference between 6 and 9 is 3, and the
same likewise is the difference between 9 and 12. But this
difference is the second part of the less of the first, but the
third part of the less of the second terms. The same also is
the third part of the greater of the first, but is only the fourth
part of the greater of the second terms. And so of the rest.