From the Beginning to Plato

GREEK ARITHMETIC, GEOMETRY AND HARMONICS 265
To justify this last step Euclid invokes the equivalent of Elements VIII.8 when he
says, ‘However many means fall proportionally between the least numbers, so
many will also fall proportionally between numbers having the same ratio.’
In Euclid’s actual argument step (i) is replaced by something like the
following argument:
(i′) Let d and d′ be the least numbers such that d′:d :: d 1:d n. Then d′ and d
have only the unit as common measure. Now consider d′−d; by the
definition of ‘epimoric’ d′−d is a part of d and a part of d′; therefore it is
the unit.
In asserting that d′ and d have only the unit as common measure Euclid is
apparently relying on the equivalent of VII.22 (‘The least numbers of those
having the same ratio with one another are relatively prime’) and the definition
of relatively prime numbers as those having only the unit as common measure (VII,
def. 12).
The proof ascribed to Archytas by Boethius is even messier. In place of (i′) it
has:
(i′′) Let d+d* and d be the least numbers such that (d+d*:d) :: d 1:d n, so
that, by the definition of ‘epimoric’, d* is a part of d. I assert that d* is a
unit. For suppose it is greater than 1. Then, since d* is a part of d, d*
divides d and also d+d*, but this is impossible. For numbers which are the
least in the same proportion as other numbers are prime to one another and
only differ by a unit.’ Therefore d* is a unit, and d+d* exceeds d by a unit.
After inferring (ii′′) that no mean proportional falls between d+d* and d,
Boethius concludes, presumably by reference to something like VIII.8:
(iii′′) Consequently, a mean proportional between the two original numbers
d 1 and d n cannot exist, since they are in the same ratio as d+d* and d.
In the quoted lines in (i′′) the equivalent of VII.22 is again cited, but, as Boethius
points out, the words ‘only differ by a unit’ are not correctly applied to arbitrary
ratios in least terms but only to epimoric ones. 20
It seems reasonable to suppose that Archytas was responsible for something
like the proof ascribed to him by Boethius, and that Euclid improved it in the Sectio,
perhaps relying on the Elements as an arithmetical foundation. It even seems
reasonable to suppose that Archytas composed some kind of Ur-Sectio, on which
our Sectio was somehow based. However, it seems to me unlikely that our Sectio
is simply an improved version of a work of Archytas. For Euclid’s diatonic scale
is the standard one used by Plato in the Timaeus (35b–36a) for the division of the
world soul into parts, whereas we know from Ptolemy ([8.77], 30.9–31.18, DK
47 A 16) that Archytas’s diatonic was quite different. 21