*„» Hli 
52 
GREEK NUMERICAL NOTATION 
include addition as the last step in each case, are found in 
Eutocius’s commentary on Archimedes’s Measurement of 
a Circle. We will take the four arithmetical operations 
separately. 
(/3) Addition and Subtraction. 
There is no doubt that, in writing down numbers for the 
purpose of these operations, the Greeks would keep the several 
powers of 10 separate in a manner practically corresponding 
to our system of numerals, the hundreds, thousands, &c., being 
written in separate vertical rows. The following would be 
a typical example of a sum in addition : 
y a v k 8 = 
P y 
1424 
103 
M ft ana 
M A 
1 
M / yoo A rj 
12281 
30030 
43838 
and the mental part of the work would be the same for the, 
Greek as for us. 
Similarly a subtraction would be represented as follows : 
M^xA-r 
M t y v 6 
aK( 
93636 
23409 
70227 
(y) Multiplication. 
(i) The Egyptian method. 
For carrying out multiplications two things were required. 
The first was a multiplication table. This the Greeks are 
certain to have had from very early times. The Egyptians, 
indeed, seem never to have had such a table. We know from 
the Papyrus Rhind that in order to multiply by any number 
the Egyptians began by successive doubling, thus obtaining 
twice, four times, eight times, sixteen times the multiplicand, 
and so on; they then added such sums of this series of multi 
ples (including once the multiplicand) as were required. Thus, 
to multi 
the mull 
13 times 
times it, 
process; 
adding o 
was perf 
tary fash 
ning wit) 
(p- H), t: 
of Xoyia 
methods 
The Ec 
seems ch 
depended 
ment of 
leaved w 
1 I have 
Russia, but 
and looks o 
to an elega; 
Write out i 
(1) the mu 
nearest inti 
plicand, (3 
in the first 
second cok 
the first co 
which are ( 
numbers le 
product. S 
and column 
The explana 
number in tl 
hand columi 
the end; ai 
omit nothin; 
equivalent oj