206 THE ELEMENTS DOWN TO PLATO’S .TIME
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(3) The third hypothesis is that of Zeuthen. 1 He starts
with the assumptions (a) that the method of proof used by
Theodorus must have been original enough to call for special
notice from Plato, and (6) that it must have been of such
a kind that the application of it to each surd required to be
set out separately in consequence of the variations in the
numbers entering into the proofs. Neither of these con
ditions is satisfied by the hypothesis of a mere adaptation to
V3, V5 ... of the traditional proof with regard to V 2.
Zeuthen therefore suggests another hypothesis as satisfying
both conditions, namely that Theodorus used the criterion
furnished by the process of finding the greatest common
measure as stated in the theorem of Eucl. X. 2. ‘If, when
the lesser of two unequal magnitudes is continually subtracted
in turn from the greater [this includes the subtraction
from any term of the highest multiple of another that it
contains], that which is left never measures the one before
it, the magnitudes will be incommensurable ’ ; that is, if two
magnitudes are such that the process of finding their G. C. M.
never comes to an end, the two magnitudes are incommensur
able. True, the proposition Eucl. X. 2 depends on the famous
X. 1 (Given two unequal magnitudes, if from the greater
there be subtracted more than the half (or the half), from the -
remainder more than the half (or the half), and so on, there
will be left, ultimately, some magnitude less than the lesser
of the original magnitudes), which is based on the famous
postulate of Eudoxus (= Eucl. Y, Def. 4), and therefore belongs
to a later date. Zeuthen gets over this objection by pointing
out that the necessity of X. 1 for a rigorous demonstration
of X. 2 may not have been noticed at the time; Theodorus
may have proceeded by intuition, or he may even have
postulated the truth proved in X. 1.
The most obvious case in which incommensurability can be
proved by using the process of finding the greatest common
measure is that of the two segments of a straight line divided
in extreme and mean ratio. For, if A B is divided in this way
at G, we have only to mark off' along CA (the greater segment)
1 Zeuthen, ‘ Sur la constitution des livres arithmétiques des Éléments
d’Euclide et leur rapport à la question de l’irrationalité ’ in Oversigt over
det kgl. Danske videnskabernes Selskahs Forhandlinger, 1915, pp. 422 sq.