THEODORUS OF CYRENE 
207 
a length CD equal to GB (the lesser segment), and GA is then 
divided at D in extreme and mean ratio, CD being the 
greater segment. (Eucl. XIII. 5 is the equivalent of this 
A D E C B 
I 1 1 1 1 
proposition.) Similarly, DC is so divided if we set off DE 
along it equal to DA; and so on. This is precisely the 
process of finding the greatest common measure of AG, CB, 
the quotient being always unity; and the process never comes 
to an end. Therefore AG, GB are incommensurable. What 
is proved in this case is the irrationality of V5 — 1). This 
of course shows incidentally that V 5 is incommensurable 
with 1. It has been suggested, in view of the easiness of the 
above proof, that the irrational may first have been discovered 
with reference to the segments of a straight line cut in extreme 
and mean ratio, rather than with reference to the diagonal 
of a square in relation to its side. But this seems, on the 
whole, improbable. 
Theodoras would, of course, give a geometrical form to the 
process of finding the G. C. M., after he had represented in 
a figure the particular surd which he was investigating. 
Zeuthen illustrates by two cases, G5 and V 3. 
We will take the former, which is the easier. The process 
of finding the G. C. M, (if any) of V 5 and 1 is as follows: 
1) V5(2 
2 
V5-2)l (4 
4 (a/5-2) 
(v/5-2) 2 
[The explanation of the second division is this: 
1 = ( v /5-2)(v'5 + 2) = 4 (a/5-2) + (a/5 -2) 2 .] 
Since, then, the ratio of the last term (\f 5 — 2) 2 to the pre 
ceding one, \/5 —2, is the same as the ratio of V5— 2 to 1, 
the process will never end. 
Zeuthen has a geometrical proof which is not difficult; but 
I think the following proof is neater and easier. 
Let ABC be a triangle right-angled at B, such that AB — 1, 
BG = 2, and therefore AC = 5.