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