MATHEMATICS IN GREEK EDUCATION 23 
ding the in- 
le in general 
3S time from 
Pythagoreans, 
: ‘ One of the 
is misfortune 
by teaching 
ippocrates of 
nade himself 
iction of the 
ng two mean 
circles are in 
so taught for 
3 story is that 
Iirough being 
to Athens to 
itay, attended 
geometry that 
the different 
ed of a large 
3reby proving, 
meter, he was 
ary life. 3 
ro glimpses of 
100I- or class- 
.1 as going into 
ter 5 ) and find- 
of astronomy; 
3 theories they 
were drawing 
ler with their 
y of Theodoras 
bhe square root 
at it is incom- 
letetus and the 
randis. 
L. iii. 5. 
younger Socrates thinking whether it was not possible to 
comprehend all such surds under one definition. In these two 
cases we have advanced or selected pupils discussing among 
themselves the subject of lectures they had heard and, in the 
second case, trying to develop a theory of a more general 
character. 
But mathematics was not only taught by regular masters 
in schools; the Sophists, who travelled from place to place 
giving lectures, included mathematics (arithmetic, geometry, 
and astronomy) in their very wide list of subjects. Theo 
doras, who was Plato’s teacher in mathematics and is 
described by Plato as a master of geometry, astronomy, 
logistic and music (among other subjects), was a pupil of 
Protagoras, the Sophist, of Abdera. 1 Protagoras himself, if we 
may trust Plato, did not approve of mathematics as part of 
secondary education ; for he is made to say that 
‘ the other Sophists maltreat the young, for, at an age when 
the young have escaped the arts, they take them against their 
will and plunge them once more into the arts, teaching them 
the art of calculation, astronomy, geometry, and music—and 
here he cast a glance at Hippias—whereas, if any one comes 
to me, he will not be obliged to learn anything except what 
he comes for.’ 2 
The Hippias referred to is of course Hippias of Elis, a really 
distinguished mathematician, the inventor of a curve known 
as the quadratrix which, originally intended for the solution 
of the problem of trisecting any angle, also served (as the 
name implies) for squaring the circle. In the Hippias Minor 3, 
there is a description of Hippias’s varied accomplishments. 
He claimed, according to this passage, to have gone once to 
the Olympian festival with everything that he wore made by 
himself, ring "and seal (engraved), oil-bottle, scraper, shoes, 
clothes, and a Persian girdle of expensive type; he also took 
poems, epics, tragedies, dithyrambs, and all sorts of prose 
works. He was a master of the science of calculation 
{Logistic), geometry, astronomy, ‘ rhythms and harmonies 
and correct writing’. He also had a wonderful system of 
mnemonics enabling him, if he once heard a string of fifty 
1 Theaetetus, 164 E, 168 E. 2 Prctacjoras, 318 n, E. 
3 Hippias Minor, pp. 866 C-368 e.