MULTIPLICATION 
57 
1, 3, 2, 3, 1, 3, 2, 6, 4, 1, 
4, 1, 7, 2, 3, 1, 2, 7, 6, 7, 7, 5, 5, 5, 2, 7, 1, 
and 1, 5, 4, 5, 5, 1, 5, 5, 5, 1, 1. 
The product is at once given in the text as 19 ‘ quadruple 
myriads 6036 ‘ triple myriads and 8480 ‘ double myriads or 
19.10000 4 + 6036.10000 3 + 8480.10000 2 . 
(The detailed multiplication line by line, which is of course 
perfectly easy, is bracketed by Hultsch as interpolated.) 
Lastly, says Pappus, this product multiplied by the other 
(the product of the tens and hundreds without the bases), 
namely 10.10000 9 , as above, gives 
196.10000 13 +368.10000 12 + 4800 . T.0000 11 . 
(iv) Examples of ordinary multiplications. 
I shall now illustrate, by examples taken from Eutocius, the 
Greek method of performing long multiplications. It will be 
seen that, as in the case of addition and subtraction, the 
working is essentially the same as ours. The multiplicand is 
written first, and below it is placed the multiplier preceded by 
km (= ‘ by ’ or ‘ into ’). Then the term containing the highest 
power of 10 in the multiplier is taken and multiplied into all 
the terms in the multiplicand, one after the other, first into that 
containing the highest power of 10, then into that containing 
the next highest power of 10, and so on in descending order ; 
after which the term containing the next highest power of 10 
in the multiplier is multiplied into all the terms of the multi 
plicand in the same order; and.so on. The same procedure 
is followed where either or both of the numbers to be multi 
plied contain fractions. Two examples from Eutocius will 
make the whole operation clear. 
(1) / arra 1351 
knl t a.Tva X 1351 
p A e 
MMM y a 
1000000 
300000 
50000 
1000 
MMM / er 
300000 
90000 
15000 
300 
NN,€,/30*. 
50000 
15000 
2500 
50 
t aTV a 
1000 
300 
50 1 
( 
0/j.ov M ¿era 
together 1825201.