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60 GREEK NUMERICAL NOTATION 
Theon’s and that of dividing My era by y arva as above is that 
Theon makes three subtractions for one term of the quotient, 
whereas the remainder was arrived at in the other case after 
one subtraction. The result is that, though Theon’s method 
is quite clear, it is longer, and moreover makes it less easy to 
foresee what will be the proper figure to try in the quotient, 
so that more time would probably be lost in making un 
successful trials. 
(e) Extraction of the square root. 
We are now in a position to see how the problem of extract 
ing the square root of a number would be attacked. First, as 
in the case of division, the given whole number would be 
separated into terms containing respectively such and such 
a number of units and of the separate powers of 10. Thus 
there would be so many units, so many tens, so many hun 
dreds, &c., and it would have to be borne in mind that the 
squares of numbers from 1 to 9 lie between 1 and 99, the 
squares of numbers from 10 to 90 between 100 and 9900, and 
so on. Then the first term of the square root would be some 
number of tens or hundreds or thousands, and so on, and 
would have to be found in much the same way as the first 
term of a quotient in a long division, by trial if necessary. 
If A is the number the square root of which is required, while 
a represents the first term or denomination of the square root, 
and x the next term or denomination to be found, it would be 
necessary to use the identity {a + x) 2 — a 2 + 2ax + x 2 and to 
find x so that 2ax + x 2 might be somewhat less than the 
remainder A — a 2 , he. we have to divide A — a 2 by 2a, allowing 
for the fact that not only must 2 ax (where x is the quotient) 
but also (2a + x)x be less than A—a 2 . Thus, by trial, the 
highest possible value of x satisfying the condition would be 
easily found. If that value were h, the further quantity 
2 ah + h 2 would have to be subtracted from the first remainder 
A — a 2 , and from the second remainder thus left a third term 
or denomination of the square root would have to be found in 
like manner; and so on. That this was the actual procedure 
followed is clear from a simple case given by Theon of Alex 
andria in his commentary on the Syntaxis. Here the square 
root of 144 is in question, and it is obtained by means of