II 
GREEK NUMERICAL NOTATION AND ARITH 
METICAL OPERATIONS 
The decimal system. 
The Greeks, from the earliest historical times, followed the 
decimal system of numeration, which had already been 
adopted by civilized peoples all the world over. There are, 
it is true, traces of quinary reckoning (reckoning in terms of 
five) in very early times ; thus in Homer Tre/nra^Lv (to 'five ’) 
is used for ' to count O But the counting by fives was pro 
bably little more than auxiliary to counting by tens ; five was 
a natural halting-place between the unit and ten, and the use 
of five times a particular power of ten as a separate category 
intermediate between that power and the next was found 
convenient in the earliest form of numerical symbolism estab 
lished in Greece, just as it was in the Roman arithmetical 
notation. The reckoning by five does not amount to such a 
variation of the decimal system as that which was in use 
among the Celts and Danes; these peoples had a vigesimal 
system, traces of which are still left in the French quatre- 
vingts, quatre-vingt-treize, &c., and in our score, three-score 
and ten, twenty-one, &c. 
The natural explanation of the origin of the decimal system, 
as well as of the quinary and vigesimal variations, is to 
suppose that they were suggested by the primitive practice of 
reckoning with the fingers, first of one hand, then of both 
together, and after that with the ten toes in addition (making 
up the 20 of the vigesimal system). The subject was mooted 
in the Aristotelian Problems, 2 where it is asked : 
' Why do a)l men, whether barbarians or Greeks, count up 
to ten, and not up to any other number, such as 2, 3, 4, or 5, 
so that, for example, they do not say one-qjlus-ûve (for 6), 
1 Homer, Od. iv. 412. 
2 xv. 8, 910 b 28-911 a 4. 
two -pi 
two -pi 
do not 
start 
is the 
and so 
on ten 
chance 
done 
to chan 
Then 
‘ perfect 
'Or ‘ 
because 
of their 
everyth 
Evide 
the num 
as, or 
\eip ai 
men), 
moreove] 
by the 
Befor 
it is pr 
by thei 
Babylon 
The E 
I for th< 
for 
expressei 
thè num 
arrangin 
The grea 
could be 
thè latte. 
The frac