The Mathematics Educat ionVol. VI I , No. 2, June 1973

SECTION B

GLIfv lPSES OFANCIENT INDIAN MATH. NO. 6

EBtrdslcara II,s Derivatiom fon tlre Salrfaceof a Sphere

&2R. C. Gupta, Del)t. dMatltemaiics, Dirla Institute of Teclmology P.O. Mesra, Ranclti, Indi.a.( Received 14 March 1973 )

The famous TTF{{TSTd Bh6skar;icdrya (born A. D. l l l4), son of Mahe{vara, was agreat Indian astronomer and mathematician. He is now usually designated as Bhaskara IIto distinguish him from his name-sake, Bhdskara I, who was active in the early part of the

seventh centur') ' A. D. The Lil: ivati ldlomilt)o f Bhaskara II is the most popular book onancient Indian matl 'rematics and 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 rvith

his ou''n commentarv on it, I{ is voluniinous astronomical rvork fqd;ef{tt iqfq' Sicldi a,rta--

Siromani ( :SS) was coml)osed in A. D. l l50 and rvas commented by the author himsel fz.

This commentary is usually called the:fl€iTr{Ts4 Va-sana-Bhztsya (:VB). T|e con'rpositionof some other u'orks is also attributed to him3.

Trvo centuries earlier than Bh iskara II, there l ived another Indian astrclnomer calledAryabhata I I (A.D.950). In his Mahd-Siddhanta, XVI, 38, i \ ryabhata I I g ives the fol lo-wing rulea

ql fqe{r aqrg: Fqrq d;E6'Err}cq gz6aq,o{ n ic rl

Par idhighno v1d;ah sy"t kandukaj-alopamarir kuprsthaph alam I I38I I'(Earth's) circumference multiplied bv (its) diameter becomes the Earth's surface-area l ikethe (area of the) net cover ing a bal l . ' That is,

surface of 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 Goladhl'Eya part of his SS and the VBthere on, the author discusses the topic in more detai ls. SS, Gola. , I I I ,52 contains a state-ment of the rule ( l ) .

The VB (p. lB7) under SS, Gola., III, 54-57 quotes the following incorrect rule fromLalla (eiglith century)

1flso sftfseT qqen] qelo q'iogsaqsq

50 TrrE ru.ETxErtr.u'rrcs EDUcATror

Vr ttaphalarir paridhighnadr samanr tato bhavati golapl sthaphalam.This text has been interpreted to mean that:

'The area of the circle (greatest section of a sphere) multiplied bv the circumferencebecomes, the area of the surface of sphere'.That is,

S::"-Rs X2nR-2v2112.

This formrrla is obviously very l\ 'rong and !o it has been l.ehemently crit icised byBhiisft21a 11.

rf Lalla, who knerv the work of .Aryabhata r (born A. D, 476), $,as not aware of thecorrect rule for the surface of a sphere, we may assume that f,rvabhata I also did not knowthc 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 iryahhatiyaI I (Gani ta), verse no. 7, second h2l fo.

Bhiskara's VB (pp. lB7-lBB) under SS, Gola, III, 54-57 also co.tains a derivatio' ofthe 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 beequal to the minutes in a circle, that is, 21600 units. Mark a poirr on the top ol it. Withthat point as the centre and with the (arcual) radius equal to the niuety-sixth part of thecircumference, that is, 225 minutes (:i say); describe a circle. Again with the samecentre' with twice that (arcual) radius describe another circle, with three times that, anothercircle; and so on ti l l 24times. 'fhus there nil l be 24 (horizoirtal) circles.

The radii of these circles wil l be the (correspor.rding 24 tabular) Sines 225 etc. (thatis, R sini n 'h ichis cqual to 225 to the nearest minute, R sin2h, R sin3i , . . .upto R sin24hwhich is eqal to,R i tsel f ) . From them the lcngths of the c i rc les can be determi 'ecl byproportion' There the length of the last circle is equal to the minutes in a circle, that is,21600, and i ts radiusis equal to the Tr i j r '1 (Sine of three signs or Sine of 90 degrees, that isSinus totus), that is ,3438. The above Sines (or radi i ) mult ip l ied by the minrr tes in a c i rc leand divided by the Sinus totus become the lengths of the (corresponding) circles.

Between arly two (consecutive) circles tl iere is an annrrlar f igrrre in the form of a bclt.They are 24 in number. There wil l be more when more ta.bular Sines are used (that is,when finer interval is taken).

rn ea'ch annulus (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:-rarcuai d istance i ) as the al t i tude. Thus by the r t r le "air i t r rde mrr l t ip l ie. l bv Sal f the srrrr oftho base and the face (that is, the rule for the area of a trapezium) " rve get the artas ol' t lre

R c. GUPr.tr 5l

annular f igures separately. The sum of those areas is the surface area of half the sphere,Trr.'ice that is the surface area of the whole sphere. That indeed is equal to the productof the diameter and the circumference.'

Let the circumferences of the circles starting from the top be Cr,Ct,...Czrand the areas of the corresponding belts (with above circles ar their respective loweredges) be Ar, A2,. . Au.

We have

At:(hl2) (o-l-c)

A,--(hl2) (G+C')

A3:(hl2) (Cz*Ci

t tzE:(hl2) e* lC:a)

Therefcre, the surface area of the whole sphere wil l be given by

S:2 (Ar I Az*. . .aAzt)

_2h (C*CzJ_.. .* C"g** C:r)

:2h x 21600 ( ,sr+^sz*. . .*^szr_+ R)lR,

where Sr, Sz, . . .are the tabular Sines.

Now Bh-iskara himself gives (VB, p. lS9) the va-lue of t l e bracketed quantityneeded above to be 52514. Using this we get

s : 2 I 600 x 2 \. 225 x. 525t 4 | 3438

:21600 x2x3+37 near ly

:circumference X diameter, practically.

In connection with this derivation, Senguptas has remarked that ,,although we miss

here the highly ingeneous method 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.u II has also given an alternateprocedure to derive the formula for the surface of a sphere by dividing the surface into Irrnes(vaprakas) l ike the natural divisions of the fruit of myrobalan (tsTiqor) in his SS Gola., III,58-61 and the VB (pp. IBB-89) there upon.

References and Notes

l. H. T. Colebrooke's English transl. (lBl7) of the work has been again reprinted by M/lKitab Mahal, Allahabad , 1967.

2. The astronomical work is in two parts namely, Graha-ganita and Golrdhylru. Here

52

J.

TIIEU . ITI IFUITICI EDT'CATTO|

we are using Bapu Deva Sastrins edition of the work along with the commentary, Kashi

Sanskrit Series No, 72, Benares, 1929.

BhZskara II wrote a manual of astronomy called fi1uf5(6-€ Karzlna Kat[hala, or flilEAq

Brahmatulya (A, D. 1l83 ?); a Commentary on Lalla's astronomical work (see K. S.

Shukla's edition 6f p.rtiganita of drldhardcltya Lucknow .lJniversity,

Lucknow, 1959,

p. XXII). His other possible minor works may be g{'a}q(q;4 and Efs66gsq (see S.

Dvivedi's Ganaka-Tarangini, Benares, 1933, p. 35). His authorship of the "t*)Stq

was doubted by S. R. Das (see H. R. Kapadia's 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).

S. Dvivedi's edition, Braj Bhusan Das & Co., Benares, 1910, fasciculus II, p. 192.

See Colebrooke's t ransl . , Op. c i t . , Rule 203, p. l l7.

See Kurt Elfering's German article in Rechenpfennige (Felicitation Volume presented to

Dr. Vogel), Deutschen Museum, Munich l968, pp. 57-67'

The value 52514 given Lry Bhaskara II is on the basis of the Sine tables found in the

rvorks l ike Iryabhatiya, Surya-Siddhdnta and Lalla's i i isyadhivrddhida. 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 difference is insignificant here.

P. C. Sengupta : "Infinitesimal Calculus in India-Its Origin and Development". J. Dept.

of let tere (Calcut ta Universi ty) , Vol . XXII (1932), ar t ic le no' 5, p. 17.

4.

I6.

7.

o.