fhe Mathematics Educatrou

Vol, IX, No, l , March 1975

SECTION B

GLIMPSES OFANCIENT INOIAN MATHEil lATICS NO. 13Sorne Ancient Values of, Pi and Tlrelr uae In Indla

b2 R.G. Gupta* Dept. of Mathematics, Birla Institttte of Technology P.O. Mesra, RANCHI (Bihar)

(Received 22 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

act iv i ty in al l parts of the rvor ld. The ploblem of sqrral ing the c i rc le, that is , f inding 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 corrstarrt ratio CID is n(,w universally denoted by the

Greek lerrer rs (Pi) bur the first writer to use the symbol definitely in this sense was Will iam

.fones (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.

This is anOld Babylonian (c. 1700 B.C.) valuez. I t is found in theBible and the

Talmuds. The same is used in ancient Chinese works such as tlte Chiu Chang Suan Siu (compl-

eted in the first century A;D. ) and the Chou Pei-in which the following is statedr

'At the winter solstice the sun's orbit has a diameter 47600 (Chinese) miles, the

circumference of the orbit being 142800 miles'.

Somewhat similar statements are found in Indian Epic and Puranic l iterature. ! 'or

irrstance, a published version of the Maha-Bhdrata lrIQIHTI<T), Bh',smaparua, XII, 44 statess

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 1a1s ldjan mall 'Jalali tr ir l isati 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).

This impl ies n : 30000/10000 : 3

*Mcmber, International Commissiort on l{istory ol Mathematics

IEE I t r I I IEEMITIOB EDUCITION

As another example, we quote the following stanza from the Eltg11q Vilu purdnas

nsqlqrsrq(i FsTilrr't qR*'((qg IF<rar<thgttsitrcrl qf"qrQ'lsq c(eqeT | |

'The diameter of the sun is nine thousandltojanas . Three times the diameter is thecilcumference of its peripheral circle,.

The Baudht)ana {ulba si t ra,r , l l2- I l3 [c.500 B.c. ?] a lso imply the same approxim-ationT . Even some of the later writers who knew better values, mentionecl the above appro-ximation for crude and quick calculatiorrs. For example, Brahmagupta (628 A.D.) in hisqlg€Safgar;d Brifuna-sphuta-siddhtnta (: BSS), XII,40 says8

6qr€a4rrltd5ft qflrfqqil E{Trstif(+ hgti r

Vyr{sa-vyrsirrdha-kr.t i parid},i-phale vyivaharike trigu ne /

'The diameter and the square of the semi-diarneter (separately) multiplied by threegivc the practical (or gross) circumference and area (of the circle) (respectively).,l 'h:it is,

C:3DArea : 3( Dl2)t

(n) TheJaina Value, n - i;In Chinathis value was given by Charrg Heng (c.130 A.D.) ancl by Chhin Chiu-Shao

(c.1250)0. I t has been cal led the Jaina Value because i t is f iequent ly used inJai la vr .o1ks.There are plenty of references but the controversies regarding clate and authorship of someof these works makes it almost impossible to decide with certainty the earlicst r,se..,t ' thevalue in India.

The canonical work {Rqcq(fff, S|.ria-ltanr.-ralr' (Sa'skrit, qdwfia Shryaproj?t6p1;1 igstated to have adoPted the rulelo

c:"/To o' (3)for getting the circumference of the sun's innermost orbit as 315(lB9 l,rjana.r plus l itt le morefrom orbi t 's ment ioned diarnett ' r 'of 99640 yojanas, and for gett ing dirrrerrs ions ol ' the suecessi-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 att u ar t hridhigama-sutr aLr

lqorqcrti$gqlqr {d qrqftiq: rViskambha-klter-da6agunayii rnularlr vrta-pariksepah .

square-root of ten times tl.re sqrrare of the diameter is the circrrmference of

alaove rvork and the commentary are both attributed to gqf(1lf ld Umtisvali (f i1s1D.) but t \ : rv i r . r l : rn.rr t i r ' ( i : rc l i rd iug date) is not f ree f rom some ser iorrs

( r )(2)

'Thethe circ le. '

The

co:tUry A

R, C. GUP|IA

controversiesl 2.

There are several other direct and indirect reference to (3) or its equivalentrs. Amongthe 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 (sixthcenrury A.n.) ; Brahmagupta's BSS, XIt ,40 (second hal f ) where i t is g iven to be an accurate(s[&srna) value; etc.

'fhis value appearedr' in the eleventh centur)'Spain in a work of Az--Zarcpli otArzachel (under Indian influence).

It is rurprising to note that even very late mathematicians continued to stick to thevalrre (possibly becarrse of t radi t ion and the elegant form of the value) r .vhen much more:rcculat{j values were u'cll-kn()\\ 'rr. For instance, the Sidd}tinta tatlba-uiuefta (f€af;Adtrfle*O),

I I , l+7 (Benares edi t ion, 1924, p.50) of Kamalrrkara (1658) gives i t . In China i t was st i l lbeing used in the middle of the lBth centuryrs.

(III) The Archirnedean Value, II : 2217.

Archimede-s (c.225 B.C.) had showrr that the ratio CID is less than 2217 and greaterthen 223171, but after his trme the value 2217 became recognized as a satisfactory approxim-at ionl6. I t n 'as givenrz b),Heron of Alexandr ia ( f i rst century A.D.) and by Rabbi Nehemiah(c. 150). I t was general ly used in China in the f i f rh century A.D. but Tsu-Chhung - Chih(430-501) consideredi t inaccurate and gave the famous Chinese value (see below)l8.

Al-Bir In i (e leventh century) had credi ted Brahrnagupta for knowing the aboveval t tere. Lal la (eighth centur l , ) g ives the dianreter of the earth as 1050 uni ts and i ts c i rcum-fercnce to be as 3300 units which imply the :rbove valuer0. fr,ryabhata II (950) has giveni t expl ic i t l l 'as an accrrr i r te value in the fo l lo. ,v ing rvords zr

aqrgr sstidqrd]ssa'fq €d: {qci q+q cf(fq: r r g R r r

'The diameter (when) niul t ip l ied by 22 and div idcd by 7 becomes the accuratecirct tmferen ce' ,

Subseqtrently, the :rbr.,ve vaiue is found mentioned by several lndian arrd foreign wri-

ters inclrrding Bhiskara I I (1150) who considered i 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 rule23c:3D + ( l6D + t6)11r3 (4)

And this would have yielded the Chinese value of r if the i l logical absolute constant numberl6 were not there in i t .

The value is stated to occur in the Tantra-samuccala2{ of Nlrziyarta (c. 1450). It is usedin the Tantra-saitgraha (1500) of Nilakarltha Somayrji (f,tq6€ eltatfe)tu. It is also found in

!HE I t r I I I I IEMA!IOB ED gOAI! IOII

his Golasdra, [II, l2 as a close approximation in the following wordsto

les,ioqc'laqrs: cf\tT: cqls$fqKrgqqFr: r

Vidvaika-samo-vydsah paridheh prdyo'rtha-bha-guna'bhegah I

'(When) the diameter equals l l3, 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 and in the Triloka sara of Nemican-dra ( lOth century)30.

We have already published a separate article on Aryabhata I 's (born 476 A.D.) Value

of Pi (3.1416) in the present SERIESSr.

References and Notes

l. D.E. Smith, Historlt of Mathematics Volume II, 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 passage is c1ur.,tt,d fir nr tht' Bible (l Kingsvii, 23) :

"And he made a molten sea, ten cubits fi<.'m one brim to the otl.er; it was round allabout. . .and a l ine of th i r ty cubi ts did compass i t round about" .

4. Y. Mikami, The Deuelopment of Mathemalics in China and Japan, p. B (Chelsea, N. Y., 196l).

J. Needham, Science andCiail ization in China Volume III t Mathematics and etc., p.99(Cambridge lJniversty Press, 1959, places two works in the Han Period (202 B.C. to220A.D.) and gives other references.

5. The Mahdbhdrata with Hindi translation, Part III, p.2572 (Geeta Press, Gorakhpur).

6. Sr i Ram Sharma's edi t ion ui th Hindi t ranslat ion, Part I , p.357 (Barei ly, 1967).

Almost sirnilarly worded stanza is ft,trr,d in the ff ltrma-Purirla (sec Sharma's edition,Part [ , p. 456; Barei ly 1970).

7. B.B. Datta, The Science of thc Sulba, p. 149 (Calcutta, 1932).

8. The BSS edi ted by R S Sharma and othcrs, Vol . I I I p. 857 (New Delhi , 1966).

9. Mikmi, p. 70; Smith p. 309;Needam, p. 100.

lC. H.R. Kapadia (editor), Ga' ita-ti laka (Baroda, 1937), Introduction, p. XLV; and B.Datta, " 'IheJaina School of Mathenralics", Bullelin Calcutta Math. Soc., Voh.m, XXI(1929) pp. I l5- la5

R. C. GUPTA

l l . The Sabhasla-Tattuirlhi.dhigama-sitra edited by Khubachandraji, p.170 (Bombay, 1932).

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-tage, New Delhi , 1973.

However, D. Pingree, Census of the Exact Scicncc in Sanskrit, series A, Volume I, p.5B(Philadelphia, 1970) agrees with the authorship and date as given by us in the presentpaper.

13. Few more references are given by us in the paper just cited. We also propose topubl ish a separate paper on theJaina Value of Pi .

14. J. D. Bond, "The Development of Trigonometric \ lethods etc.", .I,S/,S Volume.l (1921-

-22), p. 314.

15. Needham, op.ci t . (see ser ia l No.4 above), p. 102

16. Smith, op. c i t . ,p.307

17" Smith, p.3O7; and Waerden, p. 33.

i8. Needham, p. l0 l

19. E.C. Sachau (translator), Alberuni's India,Yol. I, p. 168 (Two parts in one, Delhi, 1964).

20. Tlre Jisladhlutdhida (fucttf lgtZ<), Grahaganita I,56 (S. Dvivedi's edition, Benares, 1886,

p. l0) .

21. The tr tah. is iddhanta (rat fvar;d), XV,92 (S. Dvivedi 's edi t ion, Benares, 1910, p. 172).

22. Smith, pp. 307-310 and Bond, p. 314.

23. HiralalJain's edition of the ,\q1ftft44Jz1ganra (Uqra{et{S) with Dhauali commentary,

Volume IV, pp. 42 and 221 (Arnaravat i , 1942).

24. D.M. Bose and others (editors), AConcise Histor2 of Science in India, p. 147 (New Delhi

leTl) .25. The Tantra-samgraha,Il,fTrivandrum, 1958, p. lB).

26. The Golas-trar p. l9 (K.V. Sarma's edition), Hoshiarpur, 1970).

27. Smith, p. 310 and Needlram, p. I0 l .

28. Smith, p. 302

29. Datta. Scicnce of Sulba, p. 149.

30. gathi l8 (Manohar Lal 's edi t ion, Bombay, l9 lB, p. l0) .

31. Glimpses of Ancient Indian l\[athematics No. 5, The Mathematics Education Volume VII,No. I (March l973),Sect ion B, pp. l7-20.