MATHEMATICS IN GREEK EDUCATION 
19 
. look equal 
enclicular to 
.meters with 
d (Prop. 35); 
a locus such 
}ject appears 
r e (Prop. 38). 
y of mirrors, 
ch contains, 
md reflexion 
broken line 
a minimum, 
e might say, 
eminus 1 dis- 
agines of war 
Syracuse and 
rt of making 
in Heron’s 
tanics proper, 
ae mechanical 
of the move- 
said to have 
1. 2 astronomy 
nomon, or the 
ous forms of 
1. 3 (2) /xerecopo- 
l other things, 
ent stars cross 
ioptra for the 
is of the sun, 
)H. 4 
iucation lasted 
,s were letters 
d the study of 
41. 19-42. 6. 
^—7 159. 
literature), music and gymnastics; but there is no reasonable 
doubt that practical arithmetic (in our sense), including 
weights and measures, was taught along with these subjects. 
Thus, at the stage of spelling, a common question asked of 
the pupils was, How many letters are there in such and such 
a word, e.g. Socrates, and in what order do they come? 1 This 
would teach the cardinal and ordinal numbers. In the same 
connexion Xenophon adds, ‘ Or take the case of numbers. 
Some one asks, What is twice five?’ 2 This indicates that 
counting was a part of learning letters, and that the multipli 
cation table was a closely connected subject. Then, again, 
there were certain games, played with cubic dice or knuckle 
bones, to which boys were addicted and which involved some 
degree of arithmetical skill. In the game of knucklebones in 
the Lysis of Plato each boy has a large basket of them, and 
the loser in each game pays so many over to the winner. 3 
Plato connects the art of playing this game with mathe 
matics 4 ; so too he associates Trerrda (games with 7Tecraoi, 
somewhat resembling draughts or chess) with arithmetic in 
general. 5 When in the Laws Plato speaks of three subjects 
fit for freeborn citizens to learn, (1) calculation and the science 
of numbers, (2) mensuration in one, two and three dimen 
sions, and (3) astronomy in the sense of the knowledge of 
the revolutions of the heavenly bodies and their respective 
periods, he admits that profound and accurate knowledge of 
these subjects is not for people in general but only for a few. 6 
But it is evident that practical arithmetic was, after letters 
and the lyre, to be a subject for all, so much of arithmetic, 
that is, as is necessary for purposes of war, household 
management, and the work of government. Similarly, enough 
astronomy should be learnt to enable the pupil to understand 
the calendar. 7 Amusement should be combined with instruc 
tion so as to make the subjects attractive to boys. Plato was 
much attracted by the Egyptian practice in this matter: 8 
‘ Freeborn boys should learn so much of these things as 
vast multitudes of boys in Egypt learn along with their 
1 Xenophon, Econ. viii. 14. 2 Xenophon, Mem. iv. 4. 7. 
3 Plato, Lysis, 206 e ; cf. Apollonius Rhodius, iii. 117. 
4 Phaedrus, 274 c-n. s Politicus, 299 E ; Laivs, 820 C. 
0 Laws, 817 e-818a. 7 lb. 809 c, D. 
8 lb. 819 A-c.