Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 105
from 8, and one side from 27; for 2X2X3=12. And 18, the
greater medium, has two sides from 27, and one side from 8;
for 3X3X2=18.
It is also universally true, that if a square multiplies a square,
the product will be a square." But if a number longer in the
other part, multiplies a square, or vice versa, an oblong number,
and not a square, will always be produced.t Again, if
a cube multiplies a cube, the product will be a cube;$ but if
a number longer in the other part, multiplies a cube, or vice
versa, the product will never be a cube.§ And this happens
from a similitude to the even and the odd. For if an even
multiplies an even number, the product will always be an even
number.** And if an odd multiplies an odd number, an odd
number will be immediately produced.tt But if an odd multiplies
an even, or an even an odd number, an even number
will always be produced.# This, however, will be more easily
known, by consulting that part of the eighth book of the Republic
of Plato, in which the Muses are introduced by the
philosopher speaking of the geometric number, which is the
source of better and worse generations, and which will be
hereafter explained.
+ Thus 4>(9=36, which is a square number. Thus also 9X16rl44, which is
also a square number.
t Thus 2x438 an oblong number. Thus also 6 which is a number loage in
the other part, multiplied by 9, is equal to 54, an oblong number.
$ Thus 8x27~216 a cubc number, the mot of which is 6. Thus too, 8)(6=36 an even number, Thus too, 8X8=64, which is an even
number.
tf Thus 3x5315; and 7~9=63, both which arc odd numbers.
$$ Thus 5x4~20;
and 6x9354, both which are even numbers.