1. I
,' l^t,, -" 1t r ,ia"{',
l*r / t'i(( (',2
i/
The Hindu Method of SolvingQuadraticEquations
R. C. GUPTA*
'Is it that the Greeks had such a marked Let us apply the rule to solve the
coniemptfor applied Science, leaving even following quadraticequation(written in the
the instruction of their children to slaves? ancient way)
But if so, how is it that the nation that gave
ax'*b x : c
us geom.etry and carried this science so far
I,Iultiplyingboth sidesby 4a,
did not create even rudimentary algebra ?
4 a"xt+4 abx:4 ac
Is it not equally strange that algebra, that
Adding b: to both sides,
corner-stone of modern mathematics,also
4 a'x"14 abxf b!:b!f 4 ac
originatedin India, and at about the same
i.e. (? axf b)':b'*.lac (2)
time that positional numerationdid ?'
Taking root on each side
(Quoted bv JawaharLal Nehru in his
2 ax --b: v b._,_aac
"D isco ve ry of In dia " p . 211)
Hence x:! b-+? n;---b (3)
Actually the modern algebra can be
2a
said to begin with the proof by Paolo
Ruffini (1799) that the equation of fifth The solution (3) can be rvritten down
degree can not be solved in terms ol the by employing the Rule of Brahmagupta
radicals of the coefficients*. Betore this (c.628 A.D.) given in his Brahma Sphuta
time algebrawas mostly identical with the Siddhanta(XVIII-14) and is equivalentto :
solution of equations. The method of
"The quadratic: The absolutequantity
solving a quadratic equationby completing
(i.e. c) multiplied by four times the coeffi-
the squareis called the Hindu Method. It
cient (i.e.a)of the square of the unknorvn,
is basedon Sridhara's Rule. The book of
is increased the squareol' the coellicient
by
Sridhara (circa 750 A. D.) in which this
(i.e. b) of the unknorvn; the square root of
rule was extant not extinct. But many
the result beingdiminishedby the coefficient
subsequentauthors quoted the rule and
of the unknown,is the root."
attributed it to Sridhara. One such author
is Gyanraj (cira 1503A.D.) who gives the Now the quantity (2 ax*b) is the diffe-
rule, in his algebra,as rential coeff. of the L.H.S. of (1) and the
qg{l€d{ri sil ei: ca-{4 gq+( | quantity (btr-4 ac) is called the Discri-
minant of the equation(l). The intermediary
aE4m '.tg'o) .{d1 aiil {dq tl
{ii
step (2) can be written down immediately
"Multiply bo th tlre sides by a quant ir y by applying the aphorism
equal to four times the coeftcient of the
EF[€(I4rrl ^ -
Itdd -r:-- -.:l t4qq{':
-a:-
squareol the unknown; add to both sides
a quantity equal to the square of the coeft- 'Differential-coemcient-square(equals)
cient of the unknown; then take the root',. discriminant'.
*Aftet this demonstratiod
of thc insolvability of quintic cquation algcbraically,a transcendcotalsolutioo
involving ellipuc integrals was given by Hermite in Comptes Rendus (tE5E).
11

2. This aphorism was disclosed by greatEuropeanalgebraistshad 'not quite
Jagadguru swami Shri Bharti Krishna Tirtha, risen to the views taught bl Hindus'.
Sankaracharya Govardhan Math, Puri,
of (Ibid p. 233). Hindus recognisedtwo
during the lectures his interpretationof
on answers for quadratic equation. Thus
the Vedas in relation to Mathematics at Bhaskara gave'50 anC-5 as roots of
the Banaras Hindu Universityin 1949. x'-45x-250
'And the rnost important advanceis
Nature and Number of roots : the theory of quadratic equations made in
'The Hindusearly sawin "oppositionof India is unifying under one rule the three
direction" on a line an interpretationof cases Diophantus'
of (Ibid p. 102)
positiveand negative numbers, In Europe 'The first clear recognition of imagina-
full possessionof these ideas was not ries was Mahavira's extremely intelligent
acquiredbeforeGirard and Descartes (l7th remark (9th century) that, in the natureof
century)' (Cajori p. 233). Hindus were things, a negative number has no square
the first to recognisethe existence of root. Cauchl' made same observation in
absolute negativenumbers and of irrational 1847'(Bell p. 175). For Mahaviracf :-
numbers. (Cajori p. l0l) A Babylonian q{ qqol4r erit qo qqt a,it: sql( |
of sufficiently remote time, who gave4 as
rt €qqa'r-s qril 4ir€(qle diqqq ll
thc root ol
x!:x* 12 (Mahavira's Ganita-sara
sangraha
Chap-
had solvedhis equation completell'because ter I, VerseNo.52)
negative numberswere not in his number-
Same ideas are found in otherHindu
system(Bell p. ll). Diophantus Alexan-
of
Mathematics books. We a_eainquote
dria the famousGreek algebraist regarded
Cajori-'An advancefar beyondthe Greeks
4x*20:4
is the statement of Bhaskarathat "the
as "absurd" (Kramerp. 99) And although
square of a positive number, as also of a
he solved equations of higher degreesbut
negative number,is positive;that the square
accepted only positive roots. Becauseof root of a positivenumber is two.fold posi-
his lack of clear conception of negative tive and negative. Thereis no square root
numbersand algebraicsymbolismhe had to of a negative number,for it is not a square"
study the quadratic equation separately 102).
1p.
under the the threetypes : The Greeks sharply discriminated
( i ) ax eabx - s
between numbers and magnitudes, that the
( ii ) ax':bx*c irrational was not recognised by them
(i i i ) ax ' { c : bx as a number. 81' Hindus irrationals were
takingthe coetEcients always positive. subjected the same processas ordinary
to
'Cardan(1501-1576) callednegative roots of numbersand were indeedregarded them b5,
an equation as "fictitious" and positive as numbers. By doing so the1, greatly
roots as "rea1" (Cajori p. 227). Vieta aided the progress Mathematics.
of
(1540-1603) also statcdto have rejected
is From the foregoi ng statements ofthe
all , e xce pt pos it iv er oots o l a n e q u a ti o n . factsof IJistor-vof Mathematics reader the
(Ibid p. 230). In fact, as Cajori writes, will easily agrce with Hankel with whose
'Before lTth century the majoritl' oi the q u o ta ti on cl osethi s smal larri cl e
I :
27

3. "If oneunderstands algebra the applica-
by or irrational numbers or space-magnitudes,
tion o[ arithmetical operationsto complex th e n the l earned rahrni ns Il i ndustan are
B of
magnitudesof all sorts, whether rational the real inventor of Algebra" (Cajorip. 195).
General Bibliography 6 . Vedic Mathematics JagadguruSwami
by
l. The Encyclopedia Americana (1963ed.): B h arti K ri sna" (B .H .U . 1965).
article by D.J. Struik, Prof. of Math., Developrnentol l{athematicsb-v E.T.
M.I.T., U.S.A. Bel l .
2. The Discovery of India by Jawahar Lal
8 . The Main stream ol Mathematicsby
Nehru.
(Iudian ed. 1960). Edna E. Kramer.
3. Historyot'Hindu Math. by Datta and (Oxf. Univ. Pressl95l)
Singh 9 . A history of Elernentary Mathematicsby
4. qiqa +r tieqrs t Ejo qqq){4 | F. Cajori.
deqs ttqr I (M acmi l l an o. 196l )
C
5. Symposium the History of sciencein
on
rriqrd €K dqa t qatsJil sr4(?d I
India (1961). NationalInst. of Sciences. 10.
India. ; editedand Translated L.C. Jain.
by
'This Revcred Swami used to say that be b a d wr ittcn 16 vol umes, onc of cach of thc
sixteen sutras, and thai tbe MSS of thcsc volumcs wcre doposited at the house of one of his
disciples. Unfortunately, rbc said maouscripts were lost (Vcdic Math' P. X).
28