206 THE ELEMENTS DOWN TO PLATO’S .TIME 
> 
(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.