J 
>tlier hand 
t + v = \Js. 
the other, 
tance, 849, 
.Pllll, but 
more self- 
important 
ir and says 
istake and 
ulty as the 
d, Tannery, 
r could only 
r use until, 
rchimedes’s 
abetic nota- 
y suspected 
with Greek 
to views are, 
iurely we do 
ut with the 
for instance, 
the figure 3 
7 ; what we 
ly the Greek 
kou reacrapes 
, this would 
red and four 
rerpaKocrLOL 
or of 1000 or 
ible, we say 
ed by four = 
or rpefy 67n 
e that £ thirty 
wo hundred 
housand or a 
(XKOVTCL yi’XiOi 
K0LL Sl(T\l\iOl). 
id Zeuthen), i, 
COMPARISON OF THE TWO SYSTEMS 39 
The truth is that in mental calculation (whether the opera 
tion be addition, subtraction, multiplication, or division), we 
reckon with the corresponding ivords, not with the symbols, 
and it does not matter a jot to the calculation how we choose 
to write the figures down. While therefore the alphabetical 
numerals had the advantage over the ‘ Herodianic ’ of being 
so concise, their only disadvantage,was that there were more 
signs (twenty-seven) the meaning of which had to be com 
mitted to memory : truly a very slight disadvantage. The 
one real drawback to the alphabetic system was the absence 
of a sign for 0 (zero) ; for the 0 for ovSegia or ovSer which 
we find in Ptolemy was only used in the notation of sexa 
gesimal fractions, and not as part of the numeral system. If 
there had been a sign or signs to indicate the absence in 
a number of a particular denomination, e. g. units or tens or 
hundreds, the Greek symbols could have been made to serve 
as a position-value system scarcely less effective than ours. 
For, while the position-values are clear in such a number 
as 7921 ko), it would only be necessary in the case of 
such a number as 7021 to show a blank in the proper place 
by writing, say, X~ koc. Then, following Diophantus's plan 
of separating any number of myriads by a dot from the 
thousands, &c., we could write £ ~^Ka . ^tttS for 79216384 or 
X . - t - S for 70000304, while we could continually add 
sets of four figures to the left, separating each set from the 
next following by means of a dot, 
(e) Notation for large numbers. 
Here too the orthodox way of writing tens of thousands 
was by means of the letter M with the number of myriads 
/3 ¿pot- 
above it, e.g. M = 20000, M ¿cooe = 71755875 (Aristarchus 
Y 
of Samos) ; another method was to write M or M for the 
myriad and to put the number of myriads after it, separated 
by a dot from the remaining thousands, &c., e. g." 
Y 
M pv-XNnS = 1507984 
(Diophantus, IY. 28). ■ Yet another way of expressing myriads 
was to use the symbol representing the number of myriads 
with two dots over it; thus ¿¿^09/3 = 18592 (Heron, Geo 
metrica, 17. 33). The word gvpidSe9 could, of course, be