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PYTHAGOREAN GEOMETRY 
used the method quite freely before the discovery of the irra 
tional showed them that they were building on an insecure 
and inadequate foundation. 
(5) The irrational. 
To return to the sentence about Pythagoras in the summary 
of Proclus already quoted more than once (pp. 84, 90, 141). 
Even if the reading dXoycov were right and Proclus really 
meant to attribute to Pythagoras the discovery of ‘ the theory, 
or study, of irrationals ’, it would be necessary to consider the 
authority for this statement, and how far it is supported by 
other evidence. We note that it occurs in a relative sentence 
os Srj..., which has the appearance of being inserted in paren 
thesis by the compiler of the summary rather than copied from 
his original source; and the shortened form of the first part 
of the same summary published in the Variae collectiones of 
Hultsch’s Heron, and now included by Heiberg in Heron’s 
Definitions, 1 contains no such parenthesis. Other authorities 
attribute the discovery of the theory of the irrational not to 
Pythagoras but to the Pythagoreans. A scholium to Euclid, 
Book X, says that 
‘ the Pythagoreans were the first to address themselves to the 
investigation of commensurability, having discovered it as the 
result of their observation of numbers; for, while the unit is 
a common measure of all numbers, they were unable to find 
a common measure of all magnitudes, . . . because all magni 
tudes are divisible ad infinitum and never leave a magnitude 
which is too small to admit of further division, but that 
remainder is equally divisible ad infinitum,’ 
and so on. The scholiast adds the legend that 
‘ the first of the Pythagoreans who made public the investiga 
tion of these matters perished in a shipwreck ’. 2 
Another commentary on Eucl. X discovered by Woepcke in 
an Arabic translation and believed, with good reason, to be 
part of the commentary of Pappus, says that the theory of 
irrational magnitudes ‘ had its origin in the school of Pytha 
goras ’. Again, it is impossible that Pythagoras himself should 
have discovered a ‘ theory ’ or ‘ study ’ of irrationals in any 
1 Heron, vol. iv, ed. Heib., p. 108. 
2 Euclid, ed. Heib., vol. v, pp. 415, 417.