63 
GREEK NUMERICAL NOTATION 
less than 1 * or , which is at the same time greater than 
2.67 67 ’ 
4. On trial it turns out that 4 will satisfy the conditions of 
/ 4 \ 2 
the problem, namely that (h? + — ) must be less than 4500, 
so that a remainder will be left by means of which y can be 
found. 
2.67.4 / 4' 2 
Now this remainder is 11 
60 
(—^ f and this is 
V60/ 
equal to 
11.60 2 —2.67.4.60 
60 2 
60 
16 7424 
or 60- 
Thus we must suppose that 2 ^67 + 6 q^) ^¿approximates to 
7424 
gQ2 ’ or that 8048y is approximately equal to 7424.60. 
We .have then to subtract 2 
442640 3025 
/ 4 > 
. 55 
(55 \ 2 
( 67 + 60. 
* 60 2 + 
^60 2 ) 5 or 
60 3 
7424 
+ - , j from the remainder —above found. 
60 4 60 2 
,, „ 442640 , 7424 . 2800 46 40 
Ihe subtraction or ——from 0 ■ gives or —, rf —r. 
60 3 . 60 2 60 3 60 2 60- 
but Theon does not go further and subtract the remaining 
> he merely remarks that the square of approximates 
3025 
46 40 
+ —v As a matter of fact, if we deduct the 
60 60 ' 60 
2800 
lîb 3 
to be 
from 
—, so as to obtain the correct remainder, it is found 
164975 
60 4 
Theon’s plan does not work conveniently, so far as the 
determination of the first fractional terfn (the first-sixtieths) 
is concerned, unless the integral term in the square root is 
large relatively to ^; if this is not the case, the term is 
not comparatively negligible, and the tentative ascertainment 
of x is more difficult. Take the case of V3, the value of which, 
43 55 23 
in Ptolemy’s Table of Chords, is equal to 1 + — + -~ 2 + —^ •