Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 137
Hence it appears that the eighth perfect number is the 16th
6-92 +320
term of the series produced by the expansion of
1-20+ 64.
As however there are only eight perfect numbers in twenty
terms of this series, it is evident that Ruffus in his Commentary
on the Arithmetic of Boetius, was greatly mistaken in
asserting that every other term in the series is a perfect number.
It is remarkable in this series that the terms alternately end
in 6 and 8. This is also true of the four first perfect numbers;
but the other four end indeed in 6 and 8, yet not alternately.
The other terms also of this series, from their correspondence
with perfect numbers, may be called partially perfect. For
both are resolved by 2 and its powers into parts, the aggregates
of which are equal to the wholes; and both are terminated by
6 and 8.
Again, in the series 1 +8+32+512+8192, &c. produce
1-8-96
by the expansion of the half of each of the terms
1-16,
after the first is a square number. Thus the half of 8, of 32, of
512, kc. viz. 4, 16,216, are square numbers. And if each of the
terms of the series 6+28+4%+8128+ 130816, kc. is doubled,
the sum of the parts of each is a square number. Thus the sum
of the parts of 12, the double of 6 is 16; the sum of the parts
of 56, the double of 28, is 64; the sum of the parts of 992, the
double of 496, is 1024, the root of which is 32; of the parts of
16256, the double of 8128, is 16384, the square root of which
is 128; and of the parts of 216632, the double of 130816, is
262144, the square root of which is 512. And so of the rest;
all the roots after the second increasing in a quadruple ratio.
As this property also extends to the terms that are not perfect
numbers as well as to those that are, it shows in a still greater
degree the correspondence of what I call the partially perfect,
with the completely perfect numbers.