f'he Mathematics Education

Vol. VI I I , No.2, June 197*

SECT'ION B

GLIMPSES OFANCIENT INDIAN tv|ATHEMATICS NO. 10

fBrahrna$upta's Forrnulas for the Area and Dla$o-nals of a Cycltc Gluadrllateral

D7 R. C. Gupta Asristant Professor of Mathematics Birla Institute of Techuology

P.O. Mesra, RANCHI.(Received l5 APril 1974)

I fntroduction:

Let ABCD bea plane (convex) quadrilateral with sides AB, BC,CD and DA equal

to a, b, c and d respectively. Let the figure be drawn in such a manner that we may

corrsider, according to the traditional terminoiogy, the side BC to be the base (bh[)

the side AD to be the face (mukha), and the sides AB and DCto be the flank sides (bhu-

jas or arms) of the quadrilateral.

Since a quadrilareral is not uniquely defined by its four sides, its shape and size

arc not fixed. So that, by merely specifying the four sides, the question of finding its

area does not arise. arise. That is why era'rrq Aryabhat II ( 950 A. D. ) in his cgrfir€ra

Mahi-siddhanta (-lvIS), XV, 70 saysr.

+rrinr*t fqqr sg{d qr.[+ si{ qil I

EEd qrsqfdrrqrs] qlss] ${: fq{rs} st lleoll

Karna-jirZrreua vintr caturasre lambakar.n phalalr yadva /Vaktu4r vErlchati gairako yo'sau mfirkhah pidaco va l lTOll

'The mathematician who desires to tell the area or the altitude of a quadrilateralwithout knowing a diagonal, is either a fool or a devil ' .

Brahmagupta ( 628 A. D. )t in his ilr€EiuFsdrd Brahma-sphuta-siddbanta (- BSS)

has given two rules (see below) for finding the area of a quadrilateral in terms of itsfourgiven sides. One of the rules is forgetting a rough value of the area and the otherfor an accurate (s[ksma) value. Now, Brahmagupta's forrnula for the accurate are of a qua-drilateral gives the cxact value only when the quadrilateral 'is cyclic, although he hasnot sptcified this condition, But the condition may be taken to be understood, especially when

we know (see below) that his expressions for the diagonals of the quadrilateral are alsotrue only whcn the figure is cyclic. otherwise the diagonals have remained undefined. In

fact, Brahmagupta docs speak of the circum circle (koqasplg-vltta) and the circum-radius

3+ ' , t Ht M AIIHEnAII ICB EDUOATToN

( hldaya-raj ju ) ot 'rr ianglc arrd quadri lateral in colule(r i .rn rvit l t some other rules2 which

are givcrr bltrvr-.e'rr l r is rrr le fr lr the alea and that fol the diagonals of thc ( cycl ic )

qtradri latcr i t l .

2 Rules for the Area :

' l 'h" BSS, XtI , 2 l \V,r l . I I , p. Bl6) stat t 's

(Ti|$q ft<gg'warg xfter6a)rl<<era: I

glalirrtf ?Euaq gqlqqTilIicE qSqII ll Rt ll

Stlrrl la-phalarlr tr icatu r blruj a-beltupratibrhu-yo gad alaglfi tah /B h r rjayogard ha-catus tay a bhLrjorra-ghatatpada4r sIkiamam //2 | //

' ' fhe produ< t o l 'hal f (he srrrns ol ' ( the two pairs of ) the opposi te s ides of a t r iangleor a qrradrilateral, gives the gross area. Set down half the sum of the sides irr f<rrrr pla-ces (and) diminish them by the (four) sides (respectively). The square-root of the prod-durt (of the four numlters) is the accurate area' .

gross area : * (a * c) . ! , t ( b- fd)

accurate area : l(-s-4t G4t Gt (r-l)where r : (a+1,+c*d12The above ftrrmulas are stated to be applicable to the quatlri lateral as well as to

the triangle in which case we have to takr: the face d to be zero. Thus, in the case of a tria-nql t : r , f s ic lcs a, b . c, we have

gross arca :( b lz) . ( a- lc ) l laccrrrate irrea -\/ s(s_a) ar_r) tr_r)$ ' l r t : ie s:( af-b+c) l ' )

We see that i t does not matter mrrch rvhether the fo lmrr la ( l ) is uscd for cycl ic orothel c l r radlr l l tc la ls, s ince, ; f rer a l l , i t is stated to be a rough one onl l ' . Fermula (2) iskrrown t'r give ex rct are.r r,nly in the ca:e of cyclic qu.adrilateral. However, the formula (5)is applicable to erery triangle. Rut thc foln.irrla (4) has now an additional defect of notv ie ld in, ;21111iq re ( though rouglr ) value of the area ol 'a t r iangle, because wc may getr l i f l r r rent resr i l s by rcgardir ig each of the s ides a, b, c to be 'base' in turn,

Any way, eqtr ivalent rules; which y ie ld formulas ( l ) and (2), have been given bysevelal sultsequer t Indian rvriters with or without some additional comments. Some ofthese wi l l be note, l nr ,u, .

Sri ' lhara ( 4tq< ) in lr is qrdtqfua Pftt iganita (:PG)a has reproduced, word by word,

BSS rule rvhich gives the formrrla (l). However. he adds the following remarks immedia-tcly aftelv <luoting the rules;

'But t l: is result (l) it true only f..r those figures in r,vhich the difference between thealtitude and the flank sides is small. In the case of orher fisures the above result is far

( r )(2)

(4)(s)(6)

R. ( j . GUPTA :.t5

removed from the truth; as for example, in the case of the triangle having I3 for the tr.,"o(flarrk) sides and 24for the base, the gross area is 156, whereas the correct area is 60 (PG,rules I l2-l l4)'.

An ancient commentator of the PG everr goes further and points out an interestingtheoretical defect ofthe rule (l), or (t) other than its grossness. FIe says (P. 160) that therule may yield a rough anlrwer'for thearea even in the case of irnpossible figures, and givesthe example of a triangle of base 20 and flank sides 13 and 7. Since the sum of the twosides is equal to the third (base), no tri:rngle is possible, but the formula (a) wil l give 100for its gross area'.

The MS, XV, 69 (p.165) gives the BSS rrrle for the accurate area, but it is laid down

there for a t r ianglc orr ly ' and not for a quadr i lateral . Bhiskara I I (A.D. l l50) in his

Lil ivati (eitmadl), rule 169, hasalso given the same rule but with the remark tltat it gives

evact area for a triangle and inexact (asphuta) for a quadrilateralG.

3. Sorne Historical and Other Rernarks:

The approximate fornrula (l) was used outside India much before thedate of Brahm-

agupta. The Babylonians of thc ancient.Mesopotamian valley are stated to have used it in

finding the area of e quadrilateralT. The seme formula can be gathernd from the inscripti-

ons (about 100 B.C.) found on the 'Iemple of Horus at Edfn8. In this type of Egyptian

mensurat ional mathenrat ics, t l re t r iangles were regardede as cases of quadr i laterals inwhich one sido (the face) is mad.: zero, just as what is met with in Brahnragupta.

The Chinese mathematical work Wu-t'sao Suan-ching ( about 5th ot 6rh century )applies the formula (l) for coniputing the area of a quadrangr,,ar f ield whose eastern,

western, southern, and northern sides are given to be 35, 45r 25, and l5 pac6s respectivelylo.

The formula (5) Ibr the area of a triangle is generally ca"lled Heron's Formula, but,

according to same medival Arattic scholars, it was kuown everr to Archimedes (third centuryB.C.; t t .

How Brahmagupta arrived at his formula (2), is diff icult to say with certainty. For

an expostion of the attempted prool, of this formula, as given by GaneSa Daivajfra (qivrrien)

in his commentary (1545 A.D. l on tle Lilavati, a paper by M. G. Inamdar may be consul-tedlr. The Ytrkti-bhnsl 1-y6, sixteenth century) also contains a proof of the sameformulal3.

4. Brahrnagupta's Expressions for the Diagonals:The BSS, XIl,24 (Vol. III, p. 836) states

+<rtFragqqrti+agvaurfr;qqTfqil Tqrtq ralrrn gwfagsratrq'l: nqit ca ftq} lrictl

Karnl t i r ita-bhujaghfi taikyam-rrbhay:rthz'in1'onya-bh'j itam gunayet /logena bhujapratiblrda-vacllravoh k'rrnau pade visame ll2Sl I

'The sums of the products of the sides about the diagonals be both divided by eachother; multiply (the quotients obtained) by the sum of the prodr cts of the opporite sides;

35 r I IT MATHEMATICE EDUOATION

the lquare-root (of the results) are the two diagonalr (visame '11'.

I 'hat isAC:,\/mBD:\/M

(7)

(B)

Brahmagupta's Sanskrit stanza, giving these-diagonals, has been quote-dra.vcrbatin byBhrs126 It in"hii Lil ivati witn the remark that 'although indeternrinate, the diagonals aresought to es determinate by Brahmagupta and others'.

It may be noted that the, from (7) and (B), we immediately get

AC. BD:a.cIb.dwhich is called the Ptolemy's Theorem for cyclic quadrilateral after the famous Greek astro-nomer of the second centuiy A. D, The YB (pp, 232-33), horlever, frrl lcrvs the oppositeprocedure of deriving (7) and (8) from (9) and someother relations.

Brahmagupta'.1 expressions for the diagonals are considered to be the t'most remarka'ble in Hindu gu"o-etry and solitary in its excellence" by a recent historian of mathematicslt'The formnla [e) ir stitea to be rediscoveredro in Europe by W. Snell (about l6l9 A.Dr).

Refercncee

I MS edited by S. Dvivedi, Fasciculur II, p. 165;Benares, 19l0 (Braj Bhusan Das & Co.).

2 For a short decription of Brahmagupta's works, see R.C. Gupta, "Brahmagupt_a's Rulefor the Volume of Frustrrm like Solids", The Matl;enatics Education, Vol. VI., No'4(December 1972(, Sce. B,p. l17.

3 BSS, XIl, 26-27. Edited by R.S. Sharma and his team, Vol. III, pp' B33-34; New Delhi,1966 (Indian Inst. of Astronomical:rndSanskrit Research). All page refercnces to BSSare according to this edition.

4 PG, rule l l2a. Editecl. with an ancient commentary, by K.S. Shukla, p. 156 of the text;Lucknorv, iS5') (Lutknow IJuiversity). The editor has placed the author bsllvssn ti50and 950 A.D., while sev.ral earlier scholars placed Sridhera before 850 A.D. (see l 'Gintroduction, p.xxxvii i).

5 Ibid., translation, pp. 87-88.

6 The Lil ivati, part II, p. 156. Edited bv D.V. Apte, Poona, 1937 (Anandasrama SanskritSer ies No. 107).

7 C.B. Boyer, A Hi;tory of Mathematics, p.42; New York, l968 (John Wiley).

B T.L. Heath, A ManualdCreek Mathematics, p. 77; Ncw York 1963 (Dover reprint).

I Ibid., p. 7B.l0 Y. Mikami, The Deuelopment of Mathcmatics inCl.ina and Japan, p. 38; Nerv York, 196l

(Chelsea reprint).

l l Boyer, op.c i t . , p. 149.12 Nagpur Uaiuersitjt Journal, 1946, No.l I pp. 36-42.13 YB (in Maiayalarn) part I, pp. 247-257. Edited by Rama Varma lVlaru Thampuran and

A.R. Aktri leswar Aiyar, Trichur, lg48 (Mangalodayarn Press).l4 Lilsvati, part II, p. lB0.l5 lroward Eves, An Introduction to thc History of Mathematics, p.l}7; New York, 1969 (Holt,

Rinchart and Winston),l6 D E. Smith, Historyt of Mathemah'cs vol. II, p.287; New York, l95B (Dover reprint).

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