Wearenowina position to see howthe operation of extracting the square rootwould belikelyto beattacked. First,as in thecase of division,the given whole numberwhose square root is required wouldbe separated, so to speak, intocompartmentseach containing
ARITHMETIC IN ARCHIMEDES. IxXV suchand such anumberofunitsand of the separate powers of10.
Thustherewouldbesomanyunits,somanytens,somanyhundreds,
etc., and it would have to be borne in mind that the squares of numbers*from 1 to 9 would lie between 1 and 99, the squares of
numbers from10 to 90 between100 and 9900,andso on. Thenthe 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," bytrial if necessary. IfA is the number whose square rootis
required, whilearepresents thefirstterm ordenomination of the square root and x the next term or denomination still to be found,itwouldbe necessarytouse theidentity (a+xf -a24-2ax+x*
and to find x so that 2ax+ x2 might be somewhat less than the remainder A a2. Thus by trial the highest possible value of x satisfyingthe conditionwould be easilyfound. If that value were
6, the further quantity 2ab+62 would have to be subtracted from thefirst remainderA a2, and from thesecond remainder thus left a third term or denomination of the square rootwould haveto be derived, andsoon. That thiswas theactual procedure adopted is clear froma simple case given byTheon in his commentary on the
crvi/Tai5. Here the square root of 144 is in question, and it is
obtained by means of Eucl. n. 4. The highest possible denomina- tion (i.e. power of 10) inthe square rootis 10 ; 102subtracted from 144leaves 44, andthismust contain not only twice the product of 10 andthe nexttermofthe squareroot but alsothe square of that next term itself. Now, since 2 .10 itself produces 20, the division of 44 by 20 suggests 2 as the next term of the square root; and
this turns out to be the exact figure required, since 2.20+22-44.
The same procedure is illustrated by Theon's explanation of Ptolemy's method of extracting square roots according to the sexagesimal system of fractions. The problem is to find approxi- matelythe squareroot of 4500 /xoipcuor degrees, and a geometrical figure is usedwhich makes clear the essentially Euclidean basis of the whole method. Nesselmann gives a complete reproduction of the passageofTheon,butthefollowing purelyarithmetical represen- tation of its purportwillprobablybeJpuodclearer, whenlooked at side by sidewith thefigure.
Ptolemy has first found the irtegral part of \/4500 to be 67.
Ixxvi INTRODUCTION.
Now 672=4489, so that the remainder is 11. Suppose now that the rest of the square root is expressed by means of the usual sexagesimal fractions,and that we may thereforeput
N/4500=
2.67#
wherex, yareyet tobe found. Thus x must be such that ^ -
issomewhatlessthan 11,or x must be somewhat less than ~
an
2i.D/
330
or ^, which is at the same time greater than 4. On trial, it
turns outthat 4 will satisfy the conditions of the problem, namely
/ 4\2
that f67+ ^J must be less than 4500, so that a remainder will
beleft by meansofwhich ymay be found.
Now 11
'
: f
7^. J
isthe remainder, andthis isequal to 11.602-2.67.4.60-16 7424
602 GO2"'
, civ' 4\ y . A 7424
Thuswemust suppose that 21 , -T^J) /./-va approximates to -^QT>
orthat 8048?/isapproximately equalto 7424.60.
ARITHMETIC IN ARCHIMEDES. Ixxvii Therefore y is approximately equal to 55. We have then to subtract
' + *-\L f^\* 4*2640 302^
60/608+
\60V ' r 603 + 604 '
fromthe remainder
-^ abovefound.
T, , , ,. . 442640 , 7424 . 2800 46 40 Thesubtractionof
6Q3
from -^ gives -g^T,or -2+ 3;
but Theon does not go further and subtract the remaining -^y4
-
,
instead of which he merely remarks that the square of 55 approximatesto
g-2+ - . Asamatter of fact, if we deduct the
3025 2800 ^ . . .
. . . .
-^TTT- trom -pjprj-, so as to obtain the correct remainder, it is
ulr oU
. , . . 164975 found to be TT^T
GO4
To showthepowerof this methodof extracting square rootsby means of sexagesimalfractions,it is only necessaryto mention that
103 55 23
Ptolemy gives ---- + -^+ ^ as an approximation to v3, which approximation is equivalent to 1*7320509 in the ordinary decimal notation and is therefore correct to 6 places.
But it is now time to pass to the question how Archimedes obtained the two approximations to the value of \/3 which he assumes in the Measurement of a circle. In dealing with this subject I shallfollowthe historical method of explanation adopted by Hultsch, in preference to any of the mostly a priori theories which the ingenuity of a multitude of writers has devised at different times.