s (a series of seven terms, 1, 2, 3, 4, 9, 8, 27); and if
we take this as the basis of our computation, we shall have two cube
numbers (Greek), viz. 8 and 27; and the mean proportionals between
these, viz. 12 and 18, will furnish three intervals and four terms,
and these terms and intervals stand related to one another in the
sesqui-altera ratio, i.e. each term is to the preceding as 3/2. Now if
we remember that the number 216 = 8 x 27 = 3 cubed + 4 cubed + 5 cubed,
and 3 squared + 4 squared = 5 squared, we must admit that this
number implies the numbers 3, 4, 5, to which musicians attach so much
importance. And if we combine the ratio 4/3 with the number 5, or
multiply the ratios of the sides by the hypotenuse, we shall by first
squaring and then cubing obtain two expressions, which denote the ratio
of the two last pairs of terms in the Platonic Tetractys, the former
multiplied by the square, the latter by the cube of the number 10, the
sum of the first four digits which constitute the Platonic Tetractys.'
The two (Greek) he elsewhere explains as follows: 'The first (Greek) is
(Greek), in other words (4/3 x 5) all squared = 100 x 2 squared over 3
squared. The second (Greek), a cube of the same root, is described
as 100 multiplied (alpha) by the rational diameter of 5 diminished
by unity, i.e., as shown above, 48: (beta) by two incommensurable
diameters, i.e. the two first irrationals, or 2 and 3: and (gamma) by
the cube of 3, or 27. Thus we have (48 + 5 + 27) 100 = 1000 x 2 cubed.
This second harmony is to be the cube of the number of which the former
harmony is the square, and therefore must be divided by the cube of
3. In other words, the whole expression will be: (1), for the first
harmony, 400/9: (2), for the second harmony, 8000/27.'

The reasons which have inclined me to agree with Dr. Donaldson and also
with Schleiermacher in supposing that 216 is the Platonic number of
births are: (1) that it coincides with the description of the number
given in the first part of the passage (Greek...): (2) that the
number 216 with its permutations would have been familiar to a Greek
mathematician, though unfamiliar to us: (3) that 216 is the cube of
6, and also the sum of 3 cubed, 4 cubed, 5 cubed, the numbers 3, 4, 5
representing the Pythagorean triangle, of which the sides when squared
equal the square of the hypotenuse (9 + 16 = 25): (4) that it is also
the period of the Pythagorean Metempsychosis: (5) the three ultimate
te