Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

these, and are called by modern mathematicians surds.
In the fourth place, let us consider what we are to understand
by the sesquitertian progeny when conjoined with the
pentad, rn2d thrice increased, atlording two harmonies. By the
sesquitertian progeny, then, Plato means the number 95. For
this number is composed from the addition of the squares of
the numbers 4 and 3, (i.e. 25) which form the first sesquitertian
ratio, and the number 70 which is composed from 40 and
30, and therefore consists of two numbers in a sesquitertian
ratio. Hence, as 95 is composed from 25 and 70, it may with
great propriety be called a sesquiterian progeny. This number
conjoined with 5, and thrice increased, produces ten thousand
and a million. For 100 X 100=10,000, and 10,000 X 100=
1,000,000. But it must here be observed, that these two numbers,
as will shortly be seen, appear to be considered by Plato
as analogous to two parallelopipedons, the former viz. ten
thousand, being formed from 10 X 10 X 100, and the latter
from 1000 X 10 X 100. These two numbers are called by Plato
two harmonies, for the following reason:-Simplicius, in his
commentary on Aristotle's treatise De Coelo, informs us that
a cube was denominated by the Pythagoreans harmony,* because
it consists of 12 bounding lines, 8 angles, and 6 sides;
and 12, 8, and 6 are in harmonic proportion. As a parallelopipedon,
therefore, has the same number of sides, angles, and
bounding lines as a cube, the reason is obvious why the numbers
10,000, and 1,000,000 are called by Plato harmonies. Hence
also, it is evident why he says, "that the other of these harmonies,
viz. a million is of equal length indeed, but more oblong."
For if we call 100 the breadth, and 10 the depth, both of
ten thousand and a million, it is evident that the latter number,
when considered as produced by lOOOX lOX 100 will be
analogous to a more oblong parallelopipedon than the former.
* Sce chap. XXVIII. of this book.