Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

$her I ( e~epo~ &YO) which is demonstrated to take place in these
numbers." And he concludes with informing us, "that he
shall discuss in its proper place what is delivered by the Pythagoreans
relative to this most splendid and elegant theory." Unfortunately,
he has not resumed this subject in the above mentioned
treatise, nor in any work of his that is extant. And the
only writer I am acquainted with, who has written more fully
concerning these numbers, is Ozanam, who in his Mathematical
Recreations p. 15. observes of them as follows: "The two
numbers 220 and 284 are called amicable, because the first 220
is equal to the sum of the aliquot parts of the latter, viz. 1+2
+4+71+ 142=220. And reciprocally the latter 248 is equal
to the sum of the aliquot parts of the former, viz. 1 +2+4+5
+lO+ll+22+44+55+ 110=284.
"To find all the amicable numbers in order, make use of
the number 2, which is of such a quality, that if you take 1
from its triple 6, from its sextuple 12, and from the octodecuple
of its square, viz. from 72, the remainders are the three
prime numbers 5, 11 and 71 ; of which 5 and 11 being multiplied
together, and the product 55 being multiplied by 4, the
double of the number 2, this second product 220 will be the
first of the two numbers we look for. And to find the other
284, we need only to multiply the third prime number 71, by
4, the same double of 2 that we used before.
"To find two other amicable numbers, instead of 2 we make
use of one of its powers that possesses the same quality, such as
its cube 8. For if you subtract an unit from its triple 24, from
its sextuple 48, and from 1152 the octodecuple of its square 64,
the remainders are the three prime numbers viz. 23, 47, 1151;
of which the two first 23, 47 ought to be multiplied together,
and their product 1081 ought to be multiplied by 16 the double
of the cube 8, in order to have 17296 for the first of the
two numbers demanded. And for the other amicable number