The lt{athernatrcs Educatiorr                                                                               SECTIOI* BVol....
__/44                                 T H E UIIEEM IIIIOI         !D U IIIION                     Cipayoristayor-dorjye mi...
R. C. GUPIII                                                                                           45the next tabular ...
46                                  f HE   M ATITEM ATI OB   ED IOIIIIO N                                           Refere...
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  1. 1. The lt{athernatrcs Educatiorr SECTIOI* BVol. V I I I , N o . 3 , S e p , 1 9 74 GL IM PSES OFA N C !E N T IN D I A N M A T HE M A T I CSNO . 1 1 Adctition and Subtraction Theorerrrs for ttre Sine and Thelr f.Jse ln CornpuflnE Tabular Slnes. D2Radha Charan Gupta, fuI: Sc.,Pk. D. (Mentber,Internatiortal Clomnrission History of i,n Proft:sor of Mathamatics, Mathentatics) Assistant Birla Instiiuteof Technotog2 O. Mesra, Ranchi P ( Re ce ive d 2 2 Jr r ll l!)7 t) I In tro d u c tion The celebrated Indian astronomer and mathematician Bhrskara tI ( son of Ma h e d v ar a ) was bor n i n d a k a 1 0 3 6 ( :l l l 4 A . D. ). H e w rote w rokson al most every branch of Mathematics and mathematical astronomy as were prevalent in India inthe l2th century of our era. Three of his works (all writen in Sanskrit) are very famous: (i) dtqfqdl Lillvati (on arithmetic, geometry, mensuration, and etc.) which is the most popular book of ancient Indian mathematics. (ii) E.|grrrlq1A Bijaganita ("algebra") devoted to algebra including indeterminate analysis. (iii) feAr;af{R}c[q1 Sidclsnta-.(iromatli(on.astronomy) which was written in daka t072 (:1 150 A. D.) and which is accompaniedby the authors own lucid commentary on it. The composition of some other works is also attributed to him.t TheJyotpatti (aqltqft), consisting of 25 stanzas,is consideredeither the last Chapter XIV of, or an Appendix to, the rflqtsqrq Gol:idhaya part (the other part being called q€qf{r6 Grahagafrita) of his above astronomical work (iii). However, theJyotpatti, devoted to ancient Indian trigonometry, may very well be regarded as an independent tract on the subject. It isin thisJyotpatri (A.D. ll5(l) that the Addition and Subtraction Theorems for the ancient Indian function Sine ( which is equal to radius times the modern sine ) make their first appearancein India. Although certain ancient Greek formulas, such as those used by Ptolerny ( secondcentury A. D. ), are not a proper example of an earlier knowledge of the Theorems becausethe Greeks dealt with chords and not Sines, the Arab Ab[tl Wafa ( tenth century A. D.) is reported to have already obtained these Theoremsz. 2. The Theorems 3 TheJyotpatti,2l-22 statestheTheorems in the following wordsg qrqqlRaaqlETa4 fqq:nlftsqtlat I fzaryqt dqliEri eTFsr4{qeT EJcqw rrtlrt sTqt?d((IT dtsr (ItrqqlTrir(rifror q{,r rrt
  2. 2. __/44 T H E UIIEEM IIIIOI !D U IIIION Cipayoristayor-dorjye mithah-kotijyakihate / Trijydbhakte tayoraikyarp sylccipaikyasya d orjy aki I I 2l | | Clpiintarasya jivi sldttayor-antarasarlrmiti ll2I|1,ll The Sines of the two given arcs are crossly multiplied by (their) Cosines and (theproducts are) divided by the radiur. Their (that is, of the quotients obtained) sum is theSine ofthe sum of the arcs; their difference is the Sine of tbe difference of the arcs.that is, R sin (.4+B):(R sin l). (R cosB)lR + (l? cosA). (R sin.B)/R . . . (t ) TheseAddition and SubtractionTheorems are called, respectively,Samf,sa-bhdvani and Antara-bhdvani (gqrqctsct derr q<Iqtq;f) by the author. In thelataAryrbhata I School the result was knowrr asJiveparasparaRule and is attributed to thefamous Madhava of Safigamagrama (about 1340-1425)who has also given the altcrnateform{ R s in ( l+ B ) : { ( n ti " A l r-U + / I R si n B l -rr (2)where the lam ba, L: ( R s i n l ). (R s i n ,B )/R (g) Mddhavas Rule has been quoted by Nilakantha Somayisyaji if,fv+qa elfatfe) is hisTantra-sar.r.rgraha ( l5O0 A.D. ), and in hir Aryabhativa-bh.rsya where a geometricalproof is also givens. 11o* lhaskara II himself arrived at the Theorems is not stated by him, but, asindicated by his terminology and as explained by several subsequent writers on the subject,it is likely that he derived them by using the theory of the indeterminate cquationo Jr{$! I -t2 Several other proofs are found quoted or given in various Indian works belonging tothe l6th and lTth centuriesT. 3. Bblskara IIs Formulee for Computing Tabular Sines Side by side with the statement of the Addition and Subtraction Thcorems for the Sine, the practical minded author gives two more formulas which are based on these Theo- rems and which form agood example of thelr application. By these formulas we can construct two Sines tables in a quadrant. One of these will cansist of a set of 2+ Sines (attabular interval of 225 minutes) and the other will have 90 Sines (at tabular interval of onedcgree or 60 minutes), the raditrs or Sinus totus (or Sine of 90 degrees),R, in both casesbeing 3438 minutes. The rule for constructing the shorter table is given in theJyotpatti, verses lB-20, osfollows; ...Multiply the Cosine (of any tabular arc ) by l()0 and divide by 1529. The Sine( of that arc ) be diminished by its own 467th part. The sum of the (above two) results is
  3. 3. R. C. GUPIII 45the next tabular Sine; anr.ltheir difference is the preceding tabular Sine. 225 minus one-seventh is the first Sine here. Bvthis method the24 tabular Sines are obtained.That is, R s in ( l+ i) : R s i n A -(R s i n A)1 4 6 7+ 1 0 0 . (R cos A )11529 (4)and R sinh:225-ll7 (5)where i stands for the uniform tabular interval of 225 minutes. In practice we take the positive sign in (4) and put.4-h, 2h,3h,...successively, compute the set of 24 tabular Sines. In other words, we use to Sn11:(466/467 S"+ (100/1529).y42-g"r ). with Sr:225-l17 (=to completc the eet Sr, Sz,...,Szl R). The corresponding rule for computing the table of 90 Sines is given in Jyotpatti,versesl6-lB (first half ) and is equivalcnt to the formula R sin (.4+i):R sin l-(R sin l)/6569+ 10. (n cosA)1573 (6)with R sin i:60 minutes (7)where the tabular interval i is norv equal to 60minutes. Thus we ure the rccurrence rela-tion Sn;r:(6568/6s69). yar_5r" .9"+(10/573).with S r : 60 r ninut e sto compute the full set which is now , S r ,S z r . . , S s o:R ). ( Obviously the formulas (4) and (6) are based on (l) and their rationales can be, tosome extent, worked out by using (5) and (7) respectively.sMoreover, the value (7) is basedon the sinrple relation sin 0:0 approximately, as will be evident by noting that thenumerical value of R, namely 3438 minutcs, is such as to make the length of any very nea-rly equal to the angle subtended by it at the centre of the circle of reference. As lar as the practically accurate value (5) is concerned, the author might havearrived at it by some simple methods such as the following ones:(i) The srrbduplication formulas given in theJyotpaui lC, namely --- R sin (.{ 12):Lt/ n, ( R sin I r+ (n-vers l)rOr R sin(Al2):t/-nJn "err.qtTcan enable one to compute, from the known Sine of 30 degrees( successively, the -R12), and f i n a l l y (5 ).Si n e so f l 5o, 7Lo,(ii) From thetable of 90 Sines ( computed by his method ), one can pick up the tabularentries for 3 and 4 degrees and then apply interpolation to get (5). By accepting this to bethe actual method followed by the author, we also get an explanation ( if necessary) as towhy he first gave the rule for constructing the table of 90 Sines before that for the table of24 Sines.
  4. 4. 46 f HE M ATITEM ATI OB ED IOIIIIO N References and NotesI R. C. Gupta, ((BhaskaraIIs Derivation for the Surface of a Sphere" (Glimpses of Anci- ent Indian Mathematics No. 6), The Mathematics Bducation Vol. VII, No. 2 (June 1973) ,S ec . B , p .5 2 .2 J. D. Bond, The Development of Trigonometric Methods dor,vnto the close of the XVth Century", ISIS. Vol. 4 (1921-22), p. 308.3 The Golidhyrya with the authors own commentary and the Marici commentary (:MC) of Munidvara (about I638 A. D.), part I, p. 137. Edited by D. V. Apte, 1943 (Ananda- srama Sanskrit SeriesNo. 122). In this edition theJyotpatti is given in Chapter V itself, instead of at the end. Thisalso justifies to regard the Jyotpatti as an independent tract.4 B ot h t he f or ms (l ) a n d (2 ) w e re g i v e n b y Ab[ l W afa; seeB ond op. ci t.,p.30B .5 A detailed discussion of the subject is given by the present author in his paper, entitled "Add ition and Subtraction Theorems of the Sine and the Cosine in Medieval fndiar" which is pending publication in the IndianJournal of History of Science.6 e. g. s ee M C p p . 1 5 0 -5 1 . T h e d e ta i l s a re gi ven i n the present author s forthcomi ng paper just cited.7 Proofs from the Yuktibhesa (about 1600), NfC, and Kamaldkara (l7th century) are given in the same paper.B D. A. Somayaji, A Critical Study of the Ancient Hindu Astronomy, p. 9... Karnatak IJniversity, Dharwar,,197I.