1. fhe Mathematics Educatrou SECTION B
V ol, IX , N o , l , M a r c h 1 9 7 5
GL IM PSE S OFA N C IE N T IN O I A N M A T HE illA T I CSNO . 3
1
Sorne Ancient Values of, Pi and Tlrelr uae In Indla
P.O. Mesra, RANCHI (Bihar)
Birla Institttteof Technology
b2 R.G. Gupta* Dept. of Mathematics,
( Re ce ive d 2 2 January 1975)
The study and investigation of the relation between the circunrlerence C and the
diameter D of a circle has always been a part of arrcient and med jt'val mathematical
a ctiv it y in all parts o f th e rv o rl d . T h e p l o b l e m of sqrral i ng the ci rcl e, that i s, fi ndi ng of a
square whose area is the same as that of a giverr circle, t1,as one of the most famous geome-
trical problems of r,ntiquitv. The corrstarrtratio CID is n(,w universally denoted by the
Greek lerrer rs (Pi) bur the first writer to use the symbol definitely in this sensewas William
.fo nes( 1706) r .
Various ar.cient and medieval Indian works contain rules which are taken to imply
certain values of n . Bclow we present some of those ancient approximations which were
used outside Irrdia also.
(1) The Sinplest Approximation, [I : 3.
T his is an Ol d Ba b y l o n i a n (c . 1 7 0 0 B .C .) val uez. Iti sfound i n theB i bl e and the
Talmuds. The same is used in ancient Chinese works such as tlte Chiu ChangSuanSiu (compl-
eted in the first century A;D. ) and the ChouPei-in which the following is statedr
'At the winter solsticethe sun's orbit has a diameter 47600 (Chinese) miles, the
circumferenceof the orbit being 142800miles'.
Somewhat similar statements are found in Indian Epic and Puranic literature. !'or
XII, 44 statess
Bh',smaparua,
irrstance,a published version of the Maha-Bhdrata lrIQIHTI<T),
qd<aaal q€rtlfqt i stt4 gt?rffI I
fsctgtqr d(i {rqq q{sii f{€dr qrr[ nyyrl
Sitrvastvastau sahasripi dve clnye Kurunarrrdana/
Viskambhet"tt 1a1sldjan mall'Jalali trirlisati samam ll44ll
' 0 Kurunandala ! The diameter ol' the sun is eight-plus-two thousand (yojanas)
whence (its) circular periphery,0 king, is equal to thirty (thousandTojanas).
:
Th is im plies n : 3 0 0 0 0 /1 0 0 0 0 3
*Mcmber, International Commissiort on l{istory ol Mathematics

2. IEE Itr IIIEEM IT IOB E D U C ITION
As another example, we quote the following stanza from the
Eltg11q Vilu purdnas
nsqlqrsrq(i FsTilrr't qR*'((qg I
qf"qrQ'lsq
F<rar<thgttsitrcrl c(eqeT ||
'The diameter of the sun is nine thousandltojanas . Three times the diameter is the
cilcumference of its peripheral circle,.
T he B audht )a n a{ u l b a s i tra ,r,l l 2 -Il 3 [ c.500 B .c. ?] al so i mpl y the same approxi m-
ationT. Even some of the later writers who knew better values, mentioneclthe above appro-
ximation for crude and quick calculatiorrs. For example, Brahmagupta (628 A.D.)
in his
qlg€Safgar;d Brifuna-sphuta-siddhtnta (: BSS), XII,40 says8
6qr€a4rrltd5ft qflrfqqil E{Trstif(+ hgti r
Vyr{sa-vyrsirrdha-kr.ti
parid},i-phale vyivaharike trigu ne /
'The diameter and the square of the semi-diarneter (separately)multiplied by three
givc the practical (or gross) circumference and area (of the circle) (respectively).,
l'h:it is,
C:3D (r)
Area : 3(Dl2)t (2)
(n) TheJaina Value, n
- i;
I n Chinat his v a l u e w a s g i v e n b y C h a rr g H eng (c.130A .D .) ancl by C hhi n C hi u-S hao
(c.1 2 50) 0.I t has be e n c a l l e d th e J a i n a V a l u e because t i s fi equentl y usedi nJai l a vr.o1ks.
i
There are plenty of references but the controversies regarding clateand authorship of some
of these works makes it almost impossibleto decide with certainty the earlicst r,se..,t'the
value in India.
The canonical work {Rqcq(fff, S|.ria-ltanr.-ralr'
(Sa'skrit, qdwfia Shryaproj?t6p1;1
ig
stated to have adoPted the rulelo
c:"/Too' (3)
for getting the circumference of the sun's innermost orbit as 315(lB9 l,rjana.rplus little more
fro m or bit ' s m ent io n e d d i a rn e tt' r' o f 9 9 6 4 0y o j a n as,
and for getti ng di rrrerrsi ons ' the suecessi -
ol
ve orbits, but the work is clated variously in the range from abour 5('(, B.C. to about'500 A.D.
As a sample of an origin:rl text for the nrle (3) we quote the following frorn a c()rnm-
entary on the (t4lSiftf q q qa T attuart hridhigama-sutr aLr
lqorqcrti$gqlqr {d qrqftiq: r
Viskambha-klter-da6agunayii rnularlr vrta-pariksepah .
' T he square-root of ten times tl.resqrrare of the diameter is the circrrmference of
th e c ir c le. '
T he alaovervork and the commentary are both attributed to gqf(1lfld Umtisvali (fi1s1
co :tUr y A D. ) but t: rv i r.rl : rn .rrti r' (i :rc l i r di ug date) i s not free from some seri orrs

3. R, C. GUP|IA
2.
controversiesl
There are several other direct and indirect reference to (3) or its equivalentrs. Among
the early non-Jaina rvorks n'hich mention this value are the Sitrla-Siddhdnta,I,58 (K.S.
shukla,s edition, p.l9; Lucknow, 1957); Panca--siddhantika, lY, l, of Varrhamira (sixth
ce n ru r y A . n. ) ; B r ah ma g u p ta ' sB SS , X It,4 0 (s e condhal f) w here i t i s gi ven to be an accurate
(s[&srna)value; etc.
'fhis value appearedr'in the eleventh centur)'Spain in a work of Az--Zarcpli ot
Arzachel (under Indian influence).
It is rurprising to note that even very late mathematicians continued to stick to the
va l rre ( pos s iblybec a rrs eo f tra d i ti o n a n d th e e legant form of the val ue) r.vhenmuch more
:rcculat{jvalues were u'cll-kn()'rr.For instance, the Sidd}tintatatlba-uiuefta (f€af;Adtrfle*O),
II, l +7 ( B enar esedi ti o n , 1 9 2 4 , p .5 0 ) o f Ka ma l rrkara (1658) gi ves i t. In C hi na i t w as sti l l
b e i n g us ed in t he m id d l e o f th e l Bth c e n tu ry rs .
(III) The Archirnedean Value, II : 2217.
Archimede-s (c.225 B.C.) had showrr that the ratio CID is lessthan 2217 and greater
then 223171, but after his trme the value 2217became recognizedas a satisfactory approxim-
a ti o n l 6 . I t n' as giv enrz b ),H e ro n o f A l e x a n d ri a (f i rst century A .D .) and by R abbi N ehemi ah
(c. 1 50) . I t was gen e ra l l y u s e d i n C h i n a i n th e fi frh century A .D . but Tsu-C hhung - C hi h
(4 3 0 -5 01) c ons ider e d i t i n a c c u ra te a n d g a v e th e famous C hi neseval ue (seebel ow )l 8.
A l- B ir I ni ( ele v e n th c e n tu ry ) h a d c re d i t ed B rahrnagupta for know i ng the above
va l tte r e. Lalla ( eigh th c e n tu rl ,) g i v e s th e d i a n re t er of the earth as 1050uni ts and i ts ci rcum-
fercnce to be as 3300 units which imply the :rbove valuer0. fr,ryabhataII (950) has given
i t e xp l ic it ll' as an ac c rrri rte a l u e i n th e fo l l o .,v i n grvords zr
v
aqrgrsstidqrd]ssa'fq q+q cf(fq: r rg Rr r
€d: {qci
' T he diam et er (w h e n ) n i u l ti p l i e d b y 2 2 and di vi dcd by 7 becomes the accurate
ci rcttm f er en e' ,
c
Subseqtrently, the :rbr.,ve vaiue is found mentioned by several lndian arrd foreign wri-
te rs i n clr r ding B his k a ra II (1 1 5 0 ) w h o c o n s i d e re di t as grosszz.
(IV) The Chinese Value, tI : 355/113.
As alle;idy tnentioned, this was given by the Chinese Isu in the 1fifth century. In the
Dhat,ala(tfaor) commentary by elrlc Virasena (c.800) is found a Sanskrit stanza which
gives the rule23
c:3D + (l6D + t 6 )1 1 r3 (4 )
And this would have yielded the Chinesevalue of r if the illogical absolute constant number
l 6 we re not t her e in i t.
The value is stated to occur in the Tantra-samuccala2{ Nlrziyarta (c. 1450). It is used
of
in the Tantra-saitgraha(1500) of Nilakarltha Somayrji (f,tq6€ eltatfe)tu. It is also found in

4. !HE Itr IIIIIEM A!IOB ED gOAI!IOII
his Golasdra, [II, l2 as a close approximation in the following wordsto
les,ioqc'laqrs: cftT: cqls$fqKrgqqFr: r
Vidvaika-samo-vydsah paridheh prdyo'rtha-bha-guna'bhegah I
'(When) the diameter equals ll3, the circumference has 355 parts nearly'.
Some European writers of the sixteenth century also gave the valuez7.
(V) The Egyptian Value II : (16/.19)r
This value is implied in a rule fourrd in the Ahmes Papyrus (c. 1600 B.C.;za.
In India, it is found in the Md.naaa-Sulba-sitrar2s in the Triloka sara of Nemican-
and
d r a ( lO t h c ent u ry )3 0 .
We have already published a separatearticle on Aryabhata I's (born 476 A.D.) Value
o f P i ( 3. 1416)in th e p re s e n tSE R IE SS r.
References and Notes
l. D.E. Smith, Historlt of MathematicsVolumeII, p, 312 ( Dover, New York, l95B).
2. B.L. Van der Waerden, Science
Awakening, pp. 32-312 and 75 (Wily Science Ed.,
N. Y . 1963 ).
3. Smith, op,cit.,p. 302 where the following passageis c1ur.,tt,d nr tht' Bible (l Kings
fir
vii, 23) :
"And he made a molten sea, ten cubits fi<.'mone brim to the otl.er; it was round all
about . . . an d a l i n e o f th i rty c u b i ts d i d compassi t round about" .
4. Y. Mikami, The Deuelopment Mathemalics ChinaandJapan, p. B (Chelsea,N. Y., 196l).
of in
J. Needham, Science andCiailization in China VolumeIII t Mathematicsand etc., p.99
(Cambridge lJniversty Press, 1959, places two works in the Han Period (202 B.C. to220
A.D.) and gives other references.
5. The Mahdbhdratawith Hindi translation, Part III, p.2572 (Geeta Press,Gorakhpur).
6. S r i Ram Sh a rma ' s e d i ti o n u i th H i n d i transl ati on, P art I, p.357 (B arei l y, 1967).
Almost sirnilarly worded stanzais ft,trr,d in the ffltrma-Purirla (sec Sharma's edition,
P ar t [ , p. 4 5 6 ; Ba re i l y 1 9 7 0 ).
7. B.B. Datta, The Science thc Sulba, 149 (Calcutta, 1932).
of p.
8. T he B S S ed i te d b y R S Sh a rma a n d o thcrs, V ol . III p. 857 (N ew D el hi , 1966).
9. M ik m i, p. 7 0 ; Smi th p . 3 0 9 ;N e e d a m, p . 100.
lC. H.R. Kapadia (editor), Ga' ita-tilaka (Baroda, 1937), Introduction, p. XLV; and B.
Datta, "'IheJaina School of Mathenralics", Bullelin Calcutta Math. Soc., Voh.m, XXI
( 1929) pp. I l 5 -l a 5

5. R. C. GUPTA
edited by Khubachandraji, p.170 (Bombay, 1932).
ll. The Sabhasla-Tattuirlhi.dhigama-sitra
12. For a few controversies, see R.C.Gupta, "Circumference of the Jambudvipa in
Jaina Cosmography". Paper presented at Seminar on Lord Mahavira and His Heri-
ta g e , N ew Delhi, 197 3 .
However, D. Pingree, Censusof the Exact Scicnccin Sanskrit,series A, Volume I, p.5B
(Philadelphia, 1970) agrees with the authorship and date as given by us in the present
paper.
13. Few more references are given by us in the paper just cited. We also propose to
p u b l i s h a s epar at epa p e r o n th e J a i n a V a l u e o f Pi .
14. J. D. Bond, "The Development of Trigonometric lethods etc.", .I,S/,S
Volume.l (1921-
-22), p. 314.
15. Ne e d h am , op. c it . s e es e ri a l N o .4 a b o v e ), p . 1 0 2
(
16 . Smi th, op. c it . , p. 307
17 " Smi th, p. 3O 7;and W a e rd e n , p . 3 3 .
i8. Ne e d h am , p. l0l
India,Yol. I, p. 168 (Two parts in one, Delhi, 1964).
19. E.C. Sachau (translator), Alberuni's
20. Tlre Jisladhlutdhida(fucttflgtZ<), Grahaganita
I,56 (S. Dvivedi's edition, Benares,1886,
p .l 0 ).
21. Th e tr t ah. is iddhant ra tfv a r;d ), XV ,9 2
(a (S . D v i vedi ' s edi ti on, B enares,1910,p. 172).
22. Smi th , pp. 307- 310a n d B o n d , p . 3 1 4 .
23. HiralalJain's edition of the ,q1ftft44Jz1ganra (Uqra{et{S) with Dhauali commentary,
Vo l u m e I V , pp. 42 a n d 2 2 1 (A rn a ra v a ti , 1 9 4 2 ).
24. D.M. Bose and others (editors), AConciseHistor2 of Science India, p. 147 (New Delhi
in
l e Tl ).
25. The Tantra-samgraha,Il,fTrivandrum, 1958, p. lB).
26. The Golas-trar l9 (K.V. Sarma's edition), Hoshiarpur, 1970).
p.
27. Smi th, p. 310 and N e e d l ra m, p . I0 l .
28. Smi th, p. 302
29. Datta. Scicnce Sulba,p. 149.
of
30. g a th i l8 ( M anohar L a l ' s e d i ti o n , Bo mb a y , l 9 l B , p. l 0).
31. Glimpses of Ancient Indian l[athematics No. 5, The Mathematics
Education
Volume VII,
No . I ( M ar c h l973 ),S e c ti o n B, p p . l 7 -2 0 .