1. T he M a t h e m a t i c s E d u c a tio n SECTION B
V 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 a
great Indian astronomer and mathematician. He is now usually designatedas Bhaskara II
to 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 on
ancient Indian matl'rematicsand 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{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.
i
This commentary is usually called the:fl€iTr{Ts4 Va-sana-Bhztsya
(:VB). T|e con'rposition
of some other u'orks is also attributed to him3.
Trvo centuriesearlier than Bh iskara II, there lived another Indian astrclnomer called
Arya 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;E6'Err}cq gz6aq,o{ n ic rl
Pa 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
'(Earth's) circumferencemultiplied bv (its) diameter becomesthe Earth's 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 Goladhl'Eya part of his SS and the VB
th 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-
th
me n t of t he r ule ( l).
The VB (p. lB7) under SS, Gola., III, 54-57 quotes the following incorrect rule from
Lalla (eiglith century)
1flso sftfseT qqen] qelo q'iogsaqsq

2. 50 TrrE ru.ETxErtr.u'rrcs
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 l'rong
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 .
Bhiskara's 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. (that
i 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 n24h
w 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 by
proportion' 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 s
Si 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 e
and 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 ea'channulus (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:-r
a 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 of
tho base and the face (that is, the rule for the area of a trapezium) rve
" get the artas ol' tlre

3. R c. GUPr.tr
5l
annular figures 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 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
miss
here 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 alternate
procedure 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,
in
58-61 and the VB (pp. IBB-89) there upon.
References and Notes
l. H. T. Colebrooke'sEnglish 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. .IT IIF UIT ICI E D T'C 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 Lalla's astronomical work (see K. S.
.lJniversity,
Shukla's 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.
Dvivedi's Ganaka-Tarangini, Benares, 1933, p. 35). His authorship of the
"t*)Stq
was doubted by S. R. Das (seeH. 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) .
4. S. Dvivedi's 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 tra
6. See Kurt Elfering's German article in Rechenpfennige(Felicitation Volume presentedto
Dr. Vogel), Deutschen Museum, Munich l968, pp. 57-67'
7. The value 52514 given LryBhaskara II is on the basis of the Sine tables found in the
rvorks like Iryabhatiya, Surya-Siddhdnta and Lalla's 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.