ARCHYTAS
215
Of the fragments of Archytas handed down to us the most
interesting from the point of view of this chapter is a proof
of the proposition that there can be no number which is
a (geometric) mean between two numbers in the ratio known
as em/xopLos or super particulars, that is, (n + 1) :n. This
proof is preserved by Boetius 1 2 , and the noteworthy fact about
it is that it is substantially identical with the proof of the
same theorem in Prop. 3 of Euclid’s tract on the Sectio
canonist I will quote Archytas’s proof in full, in order to
show the slight differences from the Euclidean form and
notation.
Let A, B be the given £ superparticularis proportio ’ (e?tl-
popiov Sidarripa in Euclid). [Archytas writes the smaller
number first (instead of second, as Euclid does); we are then
to suppose that A, B are integral numbers in the ratio of
n to {n+ 1). |
Take C, BE the smallest numbers which are in the ratio
of A to B. [Here BE means B + E; in this respect the
notation differs from that of Euclid, who, as usual, takes
a straight line BF divided into two parts at G, the parts
BG, GF corresponding to the B and E respectively in
Archytas’s proof. The step of finding C, BE the smallest
numbers in the same ratio as that of A to B presupposes
Eucl. YII. 33 applied to two numbers.]
Then BE exceeds C by an aliquot part of itself and of G
[cf. the definition of empopios dpiOpos in Nicomachus, i. 19. 1].
Let B be the excess [i.e. we suppose E equal to G\.
I say that B is not a number, but a unit.
For, if B is a number and an aliquot part of BE, it measures
BE; therefore it measures E, that is, 0.
Thus B measures both G and BE: which is impossible,
since the smallest numbers which are in the same ratio as
any numbers are prime to one another. [This presupposes
Eucl. YII. 22.]
Therefore B is a unit; that is, BE exceeds G by a unit.
Hence no number can be found which is a mean between
the two numbers C, BE [for there is no integer intervening].
1 Boetius, De inst. mus. hi. 11, pp. 285-6 Friedlein.
2 Musici scriptores Graeci, ed. Jan, p. 14; Heiberg and Menge’s Euclid,
vol. viii, p. 162.
