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 + —^ •