42 
GREEK NUMERICAL NOTATION 
should have expected them to split it up into + as | was 
split up into \ + The orthodox sign for a submultiple 
was the letter for the corresponding number (the denomi 
nator) but with an accent instead of a horizontal stroke 
above it; thus y' = §, the full expression being y /¿epos = 
rpirov juepos, a third part {y is in fact short for rpiros, so 
that it is also used for the ordinal number ‘ third ’ as well 
as for the fraction and similarly with all other accented 
numeral signs) ; A/3' = pi$' = &c. There were 
special signs for namely U'* or C', 1 and for §, namely (&'. 
When a number of submultiples are written one after the 
other, the sum of them is meant, and similarly when they 
follow a whole number; e.g. I/ S' = \ ^ or (Archimedes); 
k6 or' iy' A6' = 29§ ^ 3V = 29§ + ^ + & or 29f°- ; 
p6 l' i£' A S' vd = 49^ X V #4 5T = 49 li 
(Heron, Geom. 15. 8, 13). But ¿y' ro ty' means X5 th times 
ts or t^9 {ibid. 12. 5), &c. A less orthodox method found 
in later manuscripts was to use two accents and to write, 
e.g., £" instead of for A. In Diophantus we find a different 
mark in place of the accent; Tannery considers the genuine 
form of it to be so that y* = and so on. 
% 
(J3) The ordinary Greek form, variously written. 
An ordinary proper fraction (called by Euclid yeprj, parts, 
in the plural, as meaning a certain number of aliquot parts, 
in contradistinction to ¡¿epos, part, in the singular, which he 
restricts to an aliquot part or submultiple) was expressed in 
various ways. The first was to use the ordinary cardinal 
number for the numerator followed by the accented number 
representing the denominator. Thus we find in Archimedes 
l od = yx and fuoXrj 6 id = 1838 T x: (it should be noted, 
however, that the l od is a correction from oia, and this 
oa 
seems to indicate that the original reading was t, which 
would accord with Hiophantus’s and Heron’s method of 
writing fractions). The method illustrated by these cases is 
open to objection as likely to lead to confusion, since l od 
1 It has been suggested that the forms Q and 3 for ^ found in 
inscriptions may perhaps represent half an O, the sign, at all events 
in Boeotia, for 1 obol. 
wouh 
shows 
rnent 
nume 
(for 
(Arisl 
aliquc 
with 
6 .9 
top 
PKCC 
writte 
IV. 3£ 
by th 
avoide 
(1) 
writte] 
numer 
and ot 
The fr 
the wc 
AeTrra 
contains 
is genera' 
number 
12. 5),_ 
Sometime 
notation 
/? e' e' 
povdSes £ 
he. 7|-f 
£ 8i _i 
2 To 
Hultsch 
note wfi 
aliquot 
accents; 
V' S 1" Kt 
But (2) 
is regular 
Metrica 
fraction 
Mi,» №1