ly of 2 and 3. This series, of which the
intervals are afterwards filled up, probably represents (1) the diatonic
scale according to the Pythagoreans and Plato; (2) the order and
distances of the heavenly bodies; and (3) may possibly contain an
allusion to the music of the spheres, which is referred to in the myth
at the end of the Republic. The meaning of the words that 'solid bodies
are always connected by two middle terms' or mean proportionals has
been much disputed. The most received explanation is that of Martin, who
supposes that Plato is only speaking of surfaces and solids compounded
of prime numbers (i.e. of numbers not made up of two factors, or, in
other words, only measurable by unity). The square of any such number
represents a surface, the cube a solid. The squares of any two such
numbers (e.g. 2 squared, 3 squared = 4, 9), have always a single mean
proportional (e.g. 4 and 9 have the single mean 6), whereas the cubes
of primes (e.g. 3 cubed and 5 cubed) have always two mean proportionals
(e.g. 27:45:75:125). But to this explanation of Martin's it may be
objected, (1) that Plato nowhere says that his proportion is to be
limited to prime numbers; (2) that the limitation of surfaces to squares
is also not to be found in his words; nor (3) is there any evidence to
show that the distinction of prime from other numbers was known to
him. What Plato chiefly intends to express is that a solid requires a
stronger bond than a surface; and that the double bond which is given
by two means is stronger than the single bond given by one. Having
reflected on the singular numerical phenomena of the existence of one
mean proportional between two square numbers are rather perhaps only
between the two lowest squares; and of two mean proportionals between
two cubes, perhaps again confining his attention to the two lowest
cubes, he finds in the latter symbol an expression of the relation
of the elements, as in the former an image of the combination of two
surfaces. Between fire and earth, the two extremes, he remarks that
there are introduced, not one, but two elements, air and water, which
are compared to the two mean proportionals between two cube numbers.
The vagueness of his language does not allow us to determine whether
anything more than this was intended by him.

Leaving the further explanation of details, which the reader will find
discussed at length in Boeckh and Martin, we may now return to the main
argument: Why d