MULTIPLICATION 
55 
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therefore show how to multiply two large numbers each of 
which contains a number of digits (in our notation), that is, 
a certain number of units, a certain number of tens, a certain 
number of hundreds, &c.; it is confined to the multiplication 
of any number of factors each of which is one or other of the 
following ; (a) a number of units as 1, 2, 3, ... 9, (b) a number 
of even tens as 10, 20, 30,... 90, (c) a number of even hundreds 
as 100, 200, 300, ... 900. It does not deal with factors above 
hundreds, e.g. 1000 or 4000; this is because the Greek 
numeral alphabet only went up to 900, the notation begin 
ning again after that with / a, / /3,... for 1000, 2000, &c. The 
essence of the method is the separate multiplication (1) of the 
bases, 7rv6fxeves, of the several factors, (2) of the powers of ten 
contained in the factors, that is, what we represent by the 
ciphers in each factor. Given a multiple of ten, say 30, 3 is 
the Trud/iTyr or base, being the same number of units as the 
number contains tens ; similarly in a multiple of 100, say 800, 
8 is the base. In multiplying three numbers such as 2, 30, 
800, therefore, Apollonius first multiplies the bases, 2, 3, and 8, 
then finds separately the product of the ten and the hundred, 
and lastly multiplies the two products. The final product has 
to be expressed as a certain number of units less Than a 
myriad, then a certain number of myriads, a certain number 
of ‘ double myriads ’ (myriads squared), ‘ triple myriads ’ 
(myriads cubed), &c., in other words in the form 
A 0 + A 1 M + A 2 M 2 +... , 
• 
where M is a myriad or 10 4 and A 0 , A x ... respectively repre 
sent some number not exceeding 9999. 
No special directions are given for carrying out the multi 
plication ( of the bases (digits), or for the multiplication of 
their product into the product of the tens, hundreds, &c., 
when separately found (directions for the latter multiplica 
tion may have been contained in propositions missing from 
the mutilated fragment in Pappus). But the method of deal 
ing with the tens and hundreds (the ciphers in our notation) 
is made the subject of a considerable number of separate 
propositions. Thus in two propositions the factors are all of 
one sort (tens or hundreds), in another we have factors of two 
sorts (a number of factors containing unitl only multiplied