From the Beginning to Plato

GREEK ARITHMETIC, GEOMETRY AND HARMONICS 259
Phrased this way, Theaetetus’ problem is equivalent to looking at the geometric
mean y between two commensurable straight lines x and z, and asking whether it
is commensurable with the lines. One correct answer to this problem is the
following:
(ii) If y is the geometric mean between straight lines x and z, y is
commensurable with x if and only if x has to z the ratio of a square number
to a square number.
(ii) is equivalent to Elements X.9, which a scholiast ascribes to Theaetetus:
This theorem is the discovery of Theaetetus, and Plato recalls it in the
Theaetetus, but there it is set out in a more particular way, here
universally. For there squares which are measured by square numbers are
said to also have their sides commensurable. But that assertion is
particularized, since it doesn’t include in its scope all the commensurable
areas of which the sides are commensurable.
([8.30] 5:450.16–21)
If Theaetetus was interested in the commensurability of the geometric mean
between commensurable straight lines with the straight lines, it does not seem
unreasonable to suppose that he would also have considered the arithmetic and
harmonic means between commensurable straight lines and seen right away that
these are commensurable with the original lines. 10
So we can imagine that Theaetetus showed that the arithmetic or harmonic
mean between two commensurable straight lines is commensurable with the
original lines, but that this holds for the geometric mean only in the case where
the original lines are related as a square number to a square number; if they are
not so related, he could only say that the geometric mean is commensurable in
square with the original straight lines. He might now wonder whether, if we
insert a mean x between lines y, z which are commensurable in square only, x is
commensurable (at least in square) with y (and z). In fact this can be shown to
hold for none of the means, and so we might imagine Theaetetus having proved:
(iii) The insertion of any of the three means between incommensurable
‘rational’ lines produces an ‘irrational’ line.
We might imagine him pushing on to further ‘irrational’ lines by inserting
further means (cf. Elements X.II5), but I suspect that, if Theaetetus were looking
to the notion of commensurability in square as a kind of limit on
incommensurability, the recognition that any of the means between lines
commensurable in square only goes beyond that limit might have given him
pause. He would then have had a ‘theory’ summarized by (iii). This theory only
gets us to the medial, not to the binomial and apotome. To explain the