1. I , l^t,, -" 1t r ,ia"{, l*r / ti(( (,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 theconiemptfor applied Science, leaving even following quadraticequation(written in thethe instruction of their children to slaves? ancient way)But if so, how is it that the nation that gave ax*b x : cus geom.etry and carried this science so far I,Iultiplyingboth sidesby 4a,did not create even rudimentary algebra ? 4 a"xt+4 abx:4 acIs it not equally strange that algebra, that Adding b: to both sides,corner-stone of modern mathematics,also 4 ax"14 abxf b!:b!f 4 acoriginatedin 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 absolutequantitysolving 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 Sridharas Rule. The book of is increased the squareol the coellicient bySridhara (circa 750 A. D.) in which this (i.e. b) of the unknorvn; the square root ofrule was extant not extinct. But many the result beingdiminishedby the coefficientsubsequentauthors quoted the rule and of the unknown,is the root."attributed it to Sridhara. One such authoris 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 .tgo) .{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 aphorismequal to four times the coeftcient of the EF[€(I4rrl ^ - Itdd -r:-- -.:l t4qq{: -a:-squareol the unknown; add to both sidesa 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 quiteJagadguru swami Shri Bharti Krishna Tirtha, risen to the views taught bl Hindus.Sankaracharya Govardhan Math, Puri, of (Ibid p. 233). Hindus recognisedtwoduring the lectures his interpretationof on answers for quadratic equation. Thusthe Vedas in relation to Mathematics at Bhaskara gave50 anC-5 as roots ofthe Banaras Hindu Universityin 1949. x-45x-250 And the rnost important advanceisNature and Number of roots : the theory of quadratic equations made in The Hindusearly sawin "oppositionof India is unifying under one rule the threedirection" 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 Mahaviras extremely intelligentacquiredbeforeGirard and Descartes (l7th remark (9th century) that, in the natureofcentury) (Cajori p. 233). Hindus were things, a negative number has no squarethe first to recognisethe existence of root. Cauchl made same observation inabsolute 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 €qqar-s qril 4ir€(qle diqqq llthc root ol x!:x* 12 (Mahaviras Ganita-sara sangraha Chap-had solvedhis equation completellbecause ter I, VerseNo.52)negative numberswere not in his number- Same ideas are found in otherHindusystem(Bell p. ll). Diophantus Alexan- of Mathematics books. We a_eainquotedria the famousGreek algebraist regarded Cajori-An advancefar beyondthe Greeks 4x*20:4 is the statement of Bhaskarathat "theas "absurd" (Kramerp. 99) And although square of a positive number, as also of ahe solved equations of higher degreesbut negative number,is positive;that the squareaccepted 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 rootnumbersand 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 toCardan(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 oftheall , 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 whoseBefore 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 ofmagnitudesof all sorts, whether rational the real inventor of Algebra" (Cajorip. 195).General Bibliography 6 . Vedic Mathematics JagadguruSwami byl. 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. HistoryotHindu Math. by Datta and (Oxf. Univ. Pressl95l) Singh 9 . A history of Elernentary Mathematicsby4. qiqa +r tieqrs t Ejo qqq){4 | F. Cajori. deqs ttqr I (M acmi l l an o. 196l ) C5. 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
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