Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 129
In the second place, it is observable, that the difference between
12 and 20 is 8; between 20 and 36 is 16; between 36
and 68 is 32; between 68 and 132 is 64, and so on, which differences
are in a duple ratio.
In the third place, if the number which exceeds its preceding
number by 8, is added to the number immediately preceding
that which is so exceeded, the sum will be the following number,
immediately preceding that which exceeds by 8. Thus
36+24=60, the number immediately preceding 68. Thus too
68+56=124, which immediately precedes 132. And 132 +
120=252. And so of the rest. From all which it is evident,
that the series of unevenly-even numbers ad infinitum may
be easily obtained.
CHAPTER XXXVIII
On the aggregate of the parts of the terms of different series.
THE sum of the parts of each term of the duple series, 2, 4,
8, 16, 32, 64, &c. is equal to the whole of the term less by
unity. Thus the part of 2 is 1; the parts of 4 are 2 and 1,
the sum of which is 3; the parts of 8 are 4, 2, and 1, the aggregate
of which is 7; and the parts of 16 are 8, 4, 2, 1, the sum
of which is 15. And so of the rest.
But each of the terms of the triple series 3, 9, 27, 81, 243,
~rc. exceeds the double of the sum of its parts by unity. Thus
the only part of 3 is 1; and 3 exceeds 2 by 1. The parts of 9
are 3 and 1; and 9 exceeds the double of the aggregate of
these, i.e. 8 by 1. Thus too 27 exceeds the double of 13,
the aggregate of its parts 9, 3 and 1 by 1. Thus also, 81 exceeds
the double of 40 the sum of its parts, 27, 9, 3, and 1, by
1. And thus 243 exceeds twice 121, the aggregate of its parts,
81, 27, 9, 3 and 1 by 1. And so of the rest.