Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 101
will be the same multiplicity in the differences, as will be evident
from an inspection of the following schemes:
Duple differences.
Duple terms.
Triple differences.
2. 6. 18. 54. 162. 486. 1458.
Triple terms.
Quadruple differences.
3. 12. 48. 192. 768. 3072. 12288.
1. 4. 16. 64. 256. 1024. 40%. 16384.
Quadruple terms.
There is also another peculiarity of this middle, that where
the ratios are duple, every greater term compared to the less
has for its difference the less term. Thus 2 compared to 1 differs
from it by 1. Thus 4 compared to 2 differs from it by 2.
And thus 8 compared to 4 differs from it by 4. And so of the
rest. But if the ratios are triple, the greater term differs from
the less by the double of the less term. Thus 3 differs from 1
by 2 the double of 1. Thus 9 differs from 3 by 6 the double
of 3. And thus 27 differs from 9 by 18 the double of 9. And
in all the rest the same property will be found. If also the
ratios are quadruple, the greater term will differ from the less
by the triple of the less term. Thus 4 differs from 1 by 3 the
triple of 1. Thus 16 differs from 4 by 12, the triple of 4.
And thus 64 differs from 16 by 48, the triple of 16. And so
of the rest. Thus likewise, if the ratios are quintuple, sextuple,