1. The Mathematics Education SECTION D
V ol. V I I . No.4, Dec. 1973
G L I M P S E S OF A N C IT N T INDIAN M ATH. NO.8
Nhr-ayarla's Method for Evaluattng Gluadratic Surds
D7R. C. Gupta, Department Mathematics, Birla Instituteof Teclmologl,P, O, Mesra,
of
IIAJVCHI, India.
(Received 16 October 1973)
1. Introduction
We frequently come across the name Ndr;1'anu (rr<fqq) among Hindu rvriters on se-
cular sciences(including Astrology, Astronomy and l'{athematics). The one r,r'ith whom rve
are concerned here is called i{a-rdyartaPaldita. He was the son NarasiEha (or Nrsirpha)
and flourished about the middle of l4th Century of our era.
6r:iyana Pan{ita is an important autlror of ancient Indian Mathematics. Two works
from lris pen are now well known. Ire composed the qfq6 slg{t Ganita Kaumudl (:GK) in
the vear 1356 according to the colophonic verse of the work itself. The Gfi, rvhich is devoted
to elementary mathematics (arithmetic, geometry, series,magic squares, and etc.), has been
editedl.
lr{IrZya a's other work is called S}g.ffqfafqtq Bija-Ganitavatagsa (:BG) and is devo-
ted to algebra. The BG is taken to be written eariier than the G,tf sincethe author's R.ja-Ganita
('Algebra') is mentioned in the commentarv part of the GK (part I, p. lg). Dr. K. s. Suk-
la's edition of a part of the BG, which is based on an incomplete manuscript, has been recen-
tly published2,
According to Dr' Shukla (,8G, Intlodnction, p. iv), Narayana 'introduced new slbje-
cts of treatment and made his own contributions to the existing ones. He is the first mathe-
matician to have dealt with the subjectof N4agicSquares('Bhadra-garlita'). Sorneof his copt-
ributions are indeed rer:rarkableand deservespecial credit.........the so-called Fermat,sfacto-
rizatiorr method was given by N:i1ziy3na in his GK (Part II, pp. 245-47) altout three cen-
turies before it struck the mind of the French mathematicians. There are quite a few prob-
lems which were proposed and solved for the first time by Ndrdyana'.
'Ihe GK is indeed a very significant rvork on ancient and medieval Indian mathema-
tics. M. M. Sudhakara Divevedi during his last days is stated to have said that if he were
alive for few more years he would have enthroned G.6 in place of the L lzivati, the most
popular work of ancient Indian Mathematicsa.

2. THE M AITHEM ATICS ED U C AT ION
In this article we shall descril:eNxrlyana's method of finding the approximate value
of a quadratic surd by using thc theory and solutions of indeterminate equation of the second
o rd e r . A s per S hu k l a th e m e th o d w a s g i v e n b y N i rdyaua for the fi rst ti me (B G, Intr., p.i v).
What is meant lty srrch a statement is tlrat an ear'lier referenceto the method is not knou'n
to us at the plesent mornent.
2. Narryaua's Method for Approximating y'N-.
The indeterminate equation
Jtfyzl6---yz (l)
is called Varga-praklti ('Square-Nature') in Hindu mathematics". .f is called the prakrti
or gunaka ('N{ultiplier'), c the ksepa (ka) or'Interpcilator', and x andy, the 'Lesser Root'
and the Greater'Root' respectivelr'.
For finding an approximate valtre of the square-rootof the non-squ:rre nttrnber"Ai the
BG,I, 86 (p. aa) as well as the GK, X, l7 (Part ll, p.244'; girresthe follorving rr.rle.
{(i TTF' q€q q d(cdc} ca dr I
slea F?€qc4T q ' € qE qt;{oqrq?4q tl
pade tatra I
M[larp srrhyat1ryas)'aca tadritpaksepaje
,ob,ai',he "fr:'-ffi il:H::,*:H,::T, re. is,
roo,s,
be determined
"/::r;l-]il ^ roo,o
(as the Multiplier) and trnity as the Interpolatot', and then divide the Greater
(Root) by the Smaller Root. (The resrrlt is'l an approximation to the sqr-tare-root"
That is, if a, b are a pair of roots of the equation
N)cz+l:12 (2)
so that N42+ l -_ b 2 (3)
then { N:olo approximately (4)
One of the cxamples given both jn the .BG and the G.ft is that of finding the sqlrare-
roote of 10. The three pairs of solution menlioned by the attthor (6, l9), (228,721) and
(8658, 27379). So that 'e have the approximations
r ' t o: t g /e :3 .1 6 6 6 7 n e al l y
+rto:lztpze :3.1622807 nearly
{ t o: z z z l g i B6 5 8 :3 .I G2 2 7 7 GG2 arty
ne
The last value being correct to eight decimal places.
The above example is important because y'iD-was one of the approxirnations of rc wid-
ely used in the ancient world6.
The other example given in the BG as rvell as in the G.K is that of finding the square
ro ot of l/ 5.

3. R. C. GU P TII 95
The rationale of the method is simple and follows from the equation (2) which gives
{fi:{@_ll1*
:1f x nearly,
rvh e n T ( s o als o x ) is l a rg e ,
3. Sorne Rernarks orr the rnethod
It can be easily seenthat if (a, b) and {.a', b') arc any two solutions of the equation
(2 ), then
x : ab, *ba ,
):N a' - a ' l b b '
also form a solution of (2). This additive solution (sam=isa-bh.iv2nd)may be called Brahm-
agupta's Lemma (A. D. 628)?. Thtrs trsing the solutions (6, l9) and (228, 721) we get
r:6 x 7 2 1 + 1 9x 2 2 8 :8 6 5 8
t : lox 6 x 2 2 8 + 1 9x 7 2 1:2 7 3 7 5
gir.ing the thircl solution (8658, 27379)noted above. Taking two indenlical solutions, that is
4 :4 '
b: b'
r'c get the tulya-bhlvanE solution, namely
x : 2 ab
):N az tr.bz
rvlriclr is formed fr.om the single solution (a, b). Tl.us from the approximate solution (4) we
get the better appro-ximation
.Jlr:1"u' I bz)l2ab
:("lr+ u2)12a. (5)
rvhere a' stands for the first approximatton bfa.
T hes olut ion (5 )te l l s u s o f a w o n d e rfu l e q ui val enceofthe above method w i th some
other ancient methods. For example, if a better approximation is assumed to be8.
ctrt:d] € (6)
then the small correction e can be found by squaring both sides of (6) and neglecting et. The
value of c thud found, when put in (6), leads to the same solution (5).
Of course the same result is got (as should be expected) by applying the so.called
Newton-Raphson method by taking
( x ) : N- x "
'f
the better root being given by
u-f (e)l f '(e.)

4. 90 TIII M .tr IIHEMT T IC S ED I'C II!IO N
Lastly, as alreadv shollrr b1 the plesent ,1'r'iter'e, same result is also obtained lry.
the
taking the averageof the approximation a and another approximation.A'/a. This last algo-
rithm r.r'as krrorvtrto the Babvlonians of antiquity and rvas later on rrsed by the Greeks and
o t her s inc ludir ' g In d i a n s l o .
'T'hrrs, thesepr-ocedures
all arrlount to the result (5) u'hich can be obtained dilectly by
u s ing t he s ir npleb i n o mi a l a p p l o x i m a ti o n
!
( ot t *r )' :d i rl 2 u
where
'':42+r
References and Notes
l. 'Tlu Ganita A-attnrudi(rvith the author's outn commentary rvhich may be said to form
an integral part of the work itself) edited by Padmakara Dvivedi. The princessof Wales
Sarasvati Bhavana Texts No. 57, Govt. Sanskrit Liltrary, Benarcs (Varanasi); Part I,
1936 and Part II, 1942. The rn'orkis also called Ganitapati Kaumudi and is written in
the st1'leof otirel ancient pati-ganita n'orks of India.
2. N,irayana p2ntlita's Bija Gaqitavatarpsa,Part I, edited by K. S. Shukla and published
by the Akhila Bharatiya Sanskrit Parished in their journal called $taqr, Lucknow, 1970.
of
The Catalogue Publicalions the Varanaseya Sanskrit University dated l968 mentions
of
an earlier edition, by Chandrabhanu Pandeya, of the BG published from (Varanasi.) In
our article the page referencesare according to Shukla's edition.
3. For details seeK. S. Shukla, 'Hindu Methods for Finding Factors or' I)ivisors of a Num-
ber " . G anit a , V o l . 1 7 , N o . 2 (D e c . 1 9 6 6),pp. 109-117.
4. R. C.Jha : Ilidoaduiluti(in Hindi), Chorvkhamba Sanskrit Series Office, Varanasi,
1959,p. 68.
5. For details see B. Datta and A. N' Singh : History of Hindu lvlathcntatics, Single Voltrme
E dit ion, A s ia P rrb l i s h i n gH o u s e , B o m b a y , P art II, pp. l 4l -181.
6. The author of the present article proposesto publish a separatepaper on this (Jaina)
/alue of Pi.
7. Datta and Singh, Op, cit., part II, pp. 146-47.
8. R. C. Gupta, "Barrdhdyana's Valtte of {i". The Mathematics
Education,
Vol. VI, No.3
( S ept . 1972) ,t). 7 8 .
9. G upt a, ( ) P , c i t., p ,7 8 .
10. c. B. Boyer' : A rlistory of A,fathematics,John wiley, New york, 1968, pp. 30-31; and B.
Datta, NSrZialta'sMethod for Finding Approximate Value of a Surdo, Bull. Calcutta
M at h. S oc . , Vo l . 2 3 , N o . 4 (1 9 3 t), p p . l 8 7 -194.