54 
GREEK NUMERICAL NOTATION 
It is believed to date from the second century a. d., and it 
probably came from Alexandria or the vicinity. But the 
form of the characters and the mingling of capitals and small 
letters both allow of an earlier date; e. g. there is in the 
Museum a Greek papyrus assigned to the third century B.c. 
in which the numerals are very similar to those on the tablet. 1 
The second requirement is connected with the fact that the 
Greeks began their multiplications by taking the product of 
the highest constituents first, i.e. they proceeded as we should 
if we were to begin our long multiplications from the left 
instead of the right. The only difficulty would be to settle 
the denomination of the products of two high powers of ten. 
With such numbers as the Greeks usually had to multiply 
there would be no trouble; but if, say, the factors were un 
usually large numbers, e.g. millions multiplied by millions or 
billions, care would be required, and even some rule for 
settling the denomination, or determining the particular 
power or powers of 10 which the product would contain. 
This exceptional necessity was dealt with in the two special 
treatises, by Archimedes and Apollonius respectively, already 
mentioned. The former, the Sand-reckoner, proves that, if 
there be a series of numbers, 1, 10, 10 2 , 10 3 ... 10 m ... 10»,.,, 
then, if 10 m , 10» be any two terms of the series, their product 
10 m . 10» will be a term in the same series and will be as many 
terms distant from 10» as the term 10 m is distant from 1; 
also it will be distant from 1 by a number of terms less by 
one than the sum of the numbers of terms by which 10 m and 
10» respectively are distant from 1. This is easily seen to be 
equivalent to the fact that, 10 m being the (m+l}th term 
beginning with 1, and 10» the (n + l)th term beginning 
with 1, the product of the two terms is the (m + n +1 )th 
term beginning with 1, and is io m+ ». 
(hi) Apollonius’s continued multiplications. 
The system of Apollonius deserves a short description. 2 Its 
object is to give a handy method of finding the continued 
product of any number of factors, each of which is represented 
by a single letter in the Greek numeral notation. It does not 
1 David Eugene Smith in Bibliotheca Mathematica, ix 3 , pp. 193-5. 
2 Our authority here is the Synagoge of Pappus, Book ii, pp. 2-28, Hultsch.