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III 
PYTHAGOREAN ARITHMETIC 
There is very little early evidence regarding Pj^thagoras’s 
own achievements, and what there is does not touch his mathe 
matics. The earliest philosophers and historians who refer 
to him would not be interested in this part of his work. 
Heraclitus speaks of his wide knowledge, but with disparage 
ment : ‘ much learning does not teach wisdom; otherwise 
it would have taught Hesiod and Pythagoras, and again 
Xenophanes and Hecataeus ’. l Herodotus alludes to Pytha 
goras and the Pythagoreans several times; he calls Pythagoras 
‘ the most able philosopher among the Greeks ’ ('EWrjvcov ov 
7(3 d(r6eveaTctT(p ao(f)L(rTTj. HvOayopy)} In Empedocles he bad 
an enthusiastic admirer : ‘ But there was among them a man 
of prodigious knowledge who acquired the profoundest wealth 
of understanding and was the greatest master of skilled arts 
of every kind ; for, whenever he willed with his whole heart, 
he could with ease discern each and every truth in his ten— 
nay, twenty—men’s lives.’ 3 
Pythagoras himself left no written exposition of his 
doctrines, nor did any of his immediate successors, not even 
Hippasus, about whom the different stories ran (1) that he 
was expelled from the school because he published doctrines 
of Pythagoras, and (2) that he was drowned at sea for 
revealing the construction of the dodecahedron in the sphere 
and claiming it as his own, or (as others have it) for making 
known the discovery of jbhe irrational or incommensurable. 
Nor is tho absence of any written record of Pythagorean 
1 Diog. L. ix. 1 (Fr. 40 in Vorsohratiker, i 3 , p. 86. 1-8). 
2 Herodotus, iv. 95. 
Diog. L. viii. 54 and Porph. V. Pyth. 30 (Fr. 129 in Vors. i 3 , p. 272. 
15-20). v u \ v 
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