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  1. 1. T he M a t h e m a t i c s E d u c a tio n SECTION BV ol. V I I , N o . 2 , J u n e 1 9 7 3 GL Ifv l P S E SOFA N C IE N T I NDI A N M A T H. NO . 6 EBtrdslcara II,s Derivatiom fon tlre Salrface of a Sphere &2R. C. Gupta, Del)t. dMatltemaiics, Dirla Instituteof Teclmology P.O. Mesra, Ranclti,Indi.a. ( Re ce ive d 1 4 M arch 1973 ) The famous TTF{{TSTdBh6skar;icdrya(born A. D. lll4), son of Mahe{vara, was agreat Indian astronomer and mathematician. He is now usually designatedas Bhaskara IIto distinguish him from his name-sake, Bhdskara I, who was active in the early part of theseventh centur)A. D. The Lil:ivati ldlomilt)o f Bhaskara II is the most popular book onancient Indian matlrematicsand is devoted to the elementary mathematics (Arithmetic,Mensuration, and etc.)1. It was translated into Persian b1 Faizi in 1587. Rti kara II also wrote an important treatise on "Algebra" (rlUrffofo) atong rvithhis oun commentarv on it, I{is voluniinous astronomical rvork fqd;ef{ttiqfq Sicldia,rta--Si ro m ani ( : S S ) wa s c o ml )o s e d n A . D . l l 5 0 a n d rvascommentedby the author hi msel fz. iThis commentary is usually called the:fl€iTr{Ts4 Va-sana-Bhztsya (:VB). T|e conrpositionof some other uorks is also attributed to him3. Trvo centuriesearlier than Bh iskara II, there lived another Indian astrclnomer calledArya bhat a I I ( A . D. 9 5 0 ). In h i s M a h d -Si d d h a nta, X V I, 38, i ryabhata II gi ves the fol l o-w i n g r ulea ql fqe{r aqrg: Fqrq d;E6Err}cq gz6aq,o{ n ic rlPa ri dhighno v 1d; ah s y " t k a n d u k a j -a l o p a m a ri ruprsthaphal am II38II k(Earths) circumferencemultiplied bv (its) diameter becomesthe Earths surface-area like th e ( ar ea of t he) ne t c o v e ri n g a b a l l . T h a t i s, surfaceof a sphere:circumference X diameter or S:CX D:4r,R2... (l)where S, C, D,1? are the surface area, circumference,diameter and radius respectively. An equivalent of the rule (l) has been given later on by Bhaskara II in his Liiavatis.In the third chapter, called Bhuvana-Koda, of the GoladhlEya part of his SS and the VBth e re on, t he aut ho r d i s c u s s e s e to p i c i n m o re detai l s. S S , Gol a., III,52 contai nsa state- thme n t of t he r ule ( l). The VB (p. lB7) under SS, Gola., III, 54-57 quotes the following incorrect rule fromLalla (eiglith century) 1flso sftfseT qqen] qelo qiogsaqsq
  2. 2. 50 TrrE ru.ETxErtr.urrcs EDUcATror Vr ttaphalarir paridhighnadr samanrtato bhavati golapl sthaphalam. This text has been interpreted to mean that: The area of the circle (greatestsection of a sphere) multiplied bv the circumference becomes,the area of the surface of sphere. That is, S::"-Rs X2nR-2v2112. This formrrla is obviously very lrong and !o it has been l.ehemently criticised by Bhiisft21a11. rf Lalla, who knerv the work of .Aryabhata r (born A. D, 476), $,as not aware of the correct rule for the surface of a sphere, we may assumethat f,rvabhata I also did not know thc same Irowever, some attempt has been made to credit Arvabhata I rvith the knowle- clge of thc formula (l)by giving a peculiar interpretation to a rule founcl in his iryahhatiya II (Ganit a) , v er s en o . 7 , s e c o n dh 2 l fo . Bhiskaras VB (pp. lB7-lBB) under SS, Gola, III, 54-57 also co.tains a derivatio of the rule (l) by using a sort of crttde integration. A some rvhat free translation of the rele- vant Sanskrit text may be given as follows: Make a model of the Earth in clay or wood and suppose its circlmference to be equal to the minutes in a circle, that is, 21600 units. Mark a poirr on the top ol it. With that point as the centre and with the (arcual) radius equal to the niuety-sixth part of the circumference, that is, 225 minutes (:i say); describe a circle. Again with the same centre with twice that (arcual) radius describe another circle, with three times that, another circle; and so on till 24times. fhus there nill be 24 (horizoirtal) circles. The radii of these circles will be the (correspor.rding tabular) 24 Sines 225 etc. (thati s, R s ini n hic his c q u a l to 2 2 5 to th e n e a re s t mi nute, R si n2h, R si n3i ,...upto R si n24hw h i c h is eqal t o, R i ts e l f ). F ro m th e m th e l c n gths of the ci rcl es can be determi ecl byproportion There the length of the last circle is equal to the minutes in a circle, that is,2 1 6 0 0,and it s r adiu s i s e q u a l to th e T ri j r 1 (S i n e of three si gns or S i ne of 90 degrees, that i sSi n u s t ot us ) , t hat is ,3 4 3 8 . T h e a b o v e Si n e s (o r radi i ) mul ti pl i ed by the mi nrrtesi n a ci rcl eand divided by the Sinus totus become the lengths of the (corresponding) circles. Between arly two (consecutive)circles tliere is an annrrlar figrrre in the form of a bclt.They are 24 in number. There will be more when more ta.bular Sines are used (that is,when finer interval is taken). rn eachannulus (imgined to be a trapezium), the larger lorver circle mav be sllppo-sed to be the base, the upper smallcr circle as tl;e face (or top) and 225 (tirat is, the commo:-ra rcu ai dis t anc ei) as th e a l ti tu d e . T h u s b y th e rtrl e " ai ri trrde mrrl ti pl i e.lbv S al f the srrrr oftho base and the face (that is, the rule for the area of a trapezium) rve " get the artas ol tlre
  3. 3. R c. 5l annular figures separately. The sum of those areas is the surface area of half the sphere, that is the surface area of the whole sphere. That indeed is equal to the product of the diameter and the circumference. Let the circumferencesof the circles starting from the top be Cr,Ct,...Czr and the areas of the corresponding belts (with above circles ar their respective lower e d g e s )be A r , A 2,.. Au . We have At:(hl2) (o-l-c) A,--(hl2)(G+C) A 3: ( hl2 ) (C z * C i t t z E : ( hl 2 ) e * l C :a ) Therefcre, the surface area of the whole spherewill be given by S:2 ( A r I A z * ...a A z t) _2h ( C * C z J _ ...* C " g * * C :r) : 2h x 2 1 6 0 0 (,s r+ ^ s z * ...* ^ s z r_ + R )l R , w h e r e S r , S z , . . . ar eth e ta b u l a r Si n e s . Now Bh-iskara himself gives (VB, p. lS9) the va-lue of tl e bracketed quantity needed above to be 52514. Using this we get s : 2 I 600 x 2 . 225 x. 525t 4 | 3438 : 21600 x 2 x 3 + 3 7 n e a rl y :circumference X diameter, practically. In connection with this derivation, Senguptashas remarked that ,,although we misshere the highly ingeneousmethod of Archimedes (born 287 B. C.) in summing up a trigono-metrical series,there can be no question that the Indian method is perfectly original". Before concluding it may be mentioned that Bha51a.uII has also given an alternateprocedure to derive the formula for the surface of a sphere by dividing the surface into Irrnes(vaprakas) like the natural divisions of the fruit of myrobalan (tsTiqor) his SS Gola., III, in58-61 and the VB (pp. IBB-89) there upon. References and Notesl. H. T. ColebrookesEnglish transl. (lBl7) of the work has been again reprinted by M/l Kitab Mahal, Allahabad , 1967.2. The astronomical work is in two parts namely, Graha-ganita and Golrdhylru. Here
  4. 4. .IT IIF UIT ICI E D TC A TTO|52 T IIEU we are using Bapu Deva Sastrinsedition of the work along with the commentary, Kashi Sanskrit SeriesNo, 72, Benares, 1929.J. BhZskara II wrote a manual of astronomy called fi1uf5(6-€ Karzlna Kat[hala, or flilEAq Brahmatulya (A, D. 1l83 ?); a Commentary on Lallas astronomical work (see K. S. .lJniversity, Shuklas edition 6f p.rtiganita of drldhardcltya Lucknow Lucknow, 1959, p. XXII). His other possibleminor works may be g{a}q(q;4 and Efs66gsq (see S. Dvivedis Ganaka-Tarangini, Benares, 1933, p. 35). His authorship of the "t*)Stq was doubted by S. R. Das (seeH. R. Kapadias edition of the Ganita-Tilaka, Oriental Institute, Baroda, 1937,p. L XIII) and has been refuted by T. S. Kuppanna Sastri (,,The Bijopanaya i Is it a work of Bhrsk.irdclrya"J. Oriental Institute Vol. B, 1959, pp. 399- 409) .4. S. Dvivedis edition, Braj Bhusan Das & Co., Benares, 1910, fasciculusII, p. 192. S eeColebr oo k e s n s l ., Op . c i t., R u l e 2 0 3, p. l l 7.I tra6. See Kurt Elferings German article in Rechenpfennige(Felicitation Volume presentedto Dr. Vogel), Deutschen Museum, Munich l968, pp. 57-677. The value 52514 given LryBhaskara II is on the basis of the Sine tables found in the rvorks like Iryabhatiya, Surya-Siddhdnta and Lallas iiisyadhivrddhida. Otherwise, on t1e basis of the Sine table found in the Mahd-Siddlinta or that which is given by Bh:s- kara II himself, the value should be 52513. Holl,ever, the differenceis insignificant here.o. P. C. Sengupta : "Infinitesimal Calculus in India-Its Origin and Development". J. Dept. of let t er e ( Ca l c u tta U n i v e rs i ty ), Vo l . XX II (1932),arti cl e no 5, p. 17.