History of Science in South Asia A journal for the history of all forms of scientific thought and action, ancient and modern, in all regions of South Asia  An Indian Sine Table of 36 entries Michio Yano Kyoto Sangyo University  MLA style citation form: Michio Yano.   An Indian Sine Table of 36 entries.   History of Science in South Asia, 7 (2019): 42  51. DOI: 10.18732/hssa.v7i0.43.  Online version available at: http://hssa-journal.org  HISTORY OF SCIENCE IN SOUTH ASIA A journal for the history of all forms of scientific thought and action, ancient and modern, in all regions of South Asia, published online at http://hssa-journal.org  ISSN 2369-775X  Editorial Board:              Dominik Wujastyk, University of Alberta, Edmonton, Canada Kim Plofker, Union College, Schenectady, United States Dhruv Raina, Jawaharlal Nehru University, New Delhi, India Clemency Montelle, University of Canterbury, Christchurch, New Zealand Fabrizio Speziale, School of Advanced Studies in the Social Sciences (EHSS), Paris, France Michio Yano, Kyoto Sangyo University, Kyoto, Japan  Publisher: History of Science in South Asia Principal Contact: Dominik Wujastyk, Editor, University of Alberta Email:  wujastyk@ualberta.ca  Mailing Address: History of Science in South Asia, Department of History and Classics, 2  81 HM Tory Building, University of Alberta, Edmonton, AB, T6G 2H4 Canada This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. Copyrights of all the articles rest with the respective authors and published under the provisions of Creative Commons Attribution-ShareAlike 4.0 License. The electronic versions were generated from sources marked up in LATEX in a computer running GNU/LINUX operating system. PDF was typeset using X TEX from TEXLive. The base font used for Latin script and oldstyle numerals was TEX Gyre Pagella developed by GUST, the Polish TEX Users Group.  An Indian Sine Table of 36 entries Michio Yano Kyoto Sangyo University  1  I N T ROD U C TI O N  RIGONOMETRY IS AN INDISPENSABLE TOOL of Indian mathematical astronomy.  The  T concept of trigonometry originated in Greece and it was transmitted to India  together with astronomy. The Greek concept of a chord was skilfully changed into the Indian sine.1 The earliest Sanskrit text that deals with the sine function is the  ryabhaṭīya of  ryabhaṭa (b. 476 CE).2 Twenty-four sine differences (      in Table 1) in minutes are given here by  ryabhaṭa  s own method of expressing numbers.3 The radius (    or sin 90  4 ) of the table is 3438 minutes. The first arc 225 in minutes was regarded as equal to its sine (  1 =   1 ). By adding       successively we get sin       (=       ) Due to  ryabhaṭa  s method of successive approximation,5 five values, i.e.   6 ,   7 ,   16 ,   17 , and   18 (underlined in Table 1), are off by one minute from the geometrically computed correct values, but  ryabhaṭa  s numbers were faithfully preserved in the Sūryasiddh nta, and we find the same values of       and       in the Chinese text Jiuzhi li (   , 718 CE) during the Tang Dynasty.6 This type of sine table with 24 entries was standard. In Table 2 I have summarized different kinds of sine tables before the time of M dhava (fl. 1380). In the last column I have added table numbers from Pingree (1978). 2  GA RGA SA   H I T   T IS NOTEWORTHY, therefore, that a text entitled Gargasa hit  (GS) offers a sine table of  I 36 entries in a quadrant. On this strange text I contributed   The Gargasa hit : One 1 Toomer (1973) claimed the use of a chord table (with 2    = 6875  ) by Hipparchus, but later he cast doubt on his claim (Toomer 1984). Klintberg (2005) throws doubt on the origin of the Indian sine in the Greek chord, criticizing Toomer  s argument. See also van Brummelen (2009: 43, n. 30). But I would like to propose that Sanskrit term jīva is from the Greek word βι   (biós, bow) which, by the shift of accent, becomes βἰο  (bíos, life). 2 Pingree (1967  8) argued that the Pait mahasiddh nta of the Viṣ udharmottarapur  a predates  DOI: 10.18732/HSSA.V7I0.43  the  ryabhaṭīya. But I do not agree with this view. 3  ryabhaṭīya 1.12: मि ­भि ­फि ­धि ­णि   ि ­ ि ­हस  ­   ि ­ि ष­ श  ि ­ि  व।   लि ­ि    ­ह य­धि ­ि  ­स  ­ श­    व­ ल­  त­फ­ ­  लाध र  या  4 I use capital S to express     sin. 5 See Hayashi 1997. 6 See Yabuuti 1979.  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  MICHIO YANO                    43        versin        1  3;45 225  225  7  2  7;30 224  449  29  3 11;15 222  671  66  4 15;00 219  890  117  5 18;45 215 1105  182  6 22;30 210 1315  261  7 26;15 205 1520  354  8 30;00 199 1719  460  9 33;45 191 1910  579  10 37;30 183 2093  710  11 41;15 174 2267  853  12 45;00 164 2431  1007  13 48;45 154 2585  1171  14 52;30 143 2728  1345  15 56;15 131 2859  1528  16 60;00 119 2978  1719  17 63;45 106 3084  1918  18 67;30  93 3177  2123  19 71;15  79 3256  2333  20 75;00  65 3321  2548  21 78;45  51 3372  2767  22 82;30  37 3409  2989  23 86;15  22 3431  3213  24 90;00  7 3438  3438  Table 1:  ryabhaṭa  s        of the Texts of Jyotiḥ  stra Ascribed to Garga   as section 5 of the joint paper   Garga and Early Astral Science in India   published in this journal (Geslani et al. 2017: 173  82). Since my contribution was only a very brief summary of the text, I would like to offer here a more detailed account. The text of the GS is preserved in a single manuscript belonging to VVRI in Hoshiarpur, Panjab, MS Hoshiarpur VVRI 2069, written in the Malay lam script. I first worked on the text with Prof. David Pingree in 1973. The manuscript we used is a modern copy in Devanagari script possessed by Pingree. After a long interruption, I recently I resumed the work. The manuscript is full of mistakes and some verses do not allow reasonable interpretation. The Sanskrit text I am preparing is only provisional. Our present topic is discussed in Chapters 7, 8 and 10 of the GS. Chapter 7 begins with the reason why the sine function is needed. I quote GS 7.4  6 (verse numbering is my own):  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  44  AN INDIAN SINE TABLE OF 36 ENTRIES Date       499 earliest 24  ca. 550 24 628 24 665  6  904 96 966  9  ca. 1050 24 ca. 1092  6  1281  6       Sources  DSB  3438  ryabhaṭīya (ca. 499), Jiuzhi-li (718), Sūrya-siddh nta (9th century?), Siddh nta iroma i (12th cent. with corrections) 120 Pañcasiddh ntik  of Var hamihira 3270 Br hmasphuṭasiddh nta of Brahmagupta  V.6  III.20 V.16  150 Kha  akh dyaka of Brahmagupta 3437;44 Vaṭe varasiddh nta of Vaṭte vara 200 Kara atilaka of Vijayananda 3415 Siddh nta ekhara of  rīpati 120 Kara aprak  a of Brahmadeva 43 V kyakara a of anonymous author  VIII.13 V.33 VI.27 VI.33  fl. 1380 24 3437;44,48 M dhava quoted by Nīlaka ṭha (S mba iva  stri 1930  57: v. 1, 55) Table 2: Different Kinds of Sine Tables  7.4 By their power, the sky-moving (planets) never move uniformly there (in their orbit). Sometimes they move slowly, (sometimes) swiftly, likewise (sometimes) retrograde. 7.5 Thus they have different motions. Without correction, O twice-born, all the planets would not come to (the position) equal to observation. 7.6 Therefore (I will explain), in the beginning, the division of sines which is the means of its accomplishment [and] which was produced in the old time on the sky having an evenly round form.7 Then our text (GS 7.7) says that the circumference of a circle is 21,600 minutes and the diameter is 6877 minutes.8 GS 7.8  10 mentions some other possible divisions of a quandrant besides 24, i.e.   its half   (48) and   its half   (96). The division of a quadrant into 96 equal parts is attested in the Vaṭe varasiddh nta as shown in Table 2. The first half of GS 7.11 says that chord 60   is the radius of a circle.9 But the second half abruptly changes the topic and says: [The minutes] of the Chord of 1/18 th [of a circle] are [those of its] arc diminished by its own 200th part.10 7 त  वशा     रा  तत   न  व साम  यमा ता ।  दाि   म  द ा  श     तय  व    ा  तथा॥(7.4)  ित स िभ  न तय  त    फ     रण  ि  व ।    तर ण    हा  सव    न   द  त   यता   ता ॥(7.5) तद  य साधन भ तामाद    वाप     पनाम।  या प  रा समव   त  वर प  य ि  न स प  यत ॥(7.6)  8 व   तिव   िहण  त  य   ष   वि  वन   ला ।     या िव     भमानािन स  तस  त  तर व ॥ (7.7) 9 समव   त  य  णार ध    षड श यासम  भव त।  (7.11ab)   (7.11cd)     वाया  त  य   ा    श िनत  धन । 10 ध   यश Our manuscript is di icult to read, and I used Pingree  s reading.  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  MICHIO YANO  45  1  This means: chord 20   = 1200(1    ) = 1194  . Therefore sin 10   = 597  . The same value 200 can be obtained by the linear interpolation using Table 1:11 671    222    (11; 15    10) 222   75 = 671    = 671    74 = 597 225 225  It is strange that the author explained the derivation in a different way. Chord 20   is found again in GS 7.46  47 and GS 7.71. The use of chord is not common in India, where usually sine or   half chord   (jīv rdha or jy rdha) is used. It is possible that the author of our text knew some Arabic sources. Anyway, this value of sin 10   = 597  is used as one of the three starting points for the derivation of 36 sines (see below). From the three starting points all the sines are derived, as GS 7.12ab says.12 For the derivation two rules are used. One is to compute the complementary sine of a given sine. GS 17cd-18ab defines the sine of the complementary arc (sa pūr ajy ), which is nothing but cosine (koṭijy ): 7.17cd  18ab O inspired one, whatever is the result of the subtraction of that (i.e., the square of a sine) from the square of the radius, that is the complementary sine of the 36 arcs of the three signs.13 This means   36      =      2         2 (1) Another rule is to compute the sine of half arc. The wording of GS 7.14cd  15ab14 is not very clear, but it tells how to obtain versed sine (s yaka,   arrow  ). Probably versed sine was used in the following formula:      2 + Vers     2       =    2 2  (2)  The rule is not explicitly stated anywhere in our text, but a similar formula is found in Brahmagupta  s Br hmasphuṭasiddh nta 21.23 and it was a common knowledge. Then GS describes the derivations of sines starting from three different points. In the following diagrams the vertical arrow means the use of (1) and the horizontal arrow that of (2). The three starting points are: (A)   36 , (B)   12 , and (C)   4 . (A) GS 7.16  25: Starting from sin 90   =   36 , a total of 4 sines are derived.   36      18      9     27 11 I owe this explanation to an anonymous referee. 12  त नाथ र त  य ण  व सवर  याना  प     पना। (7.12ab)   ल  भव त॥(17cd)   13   यासाध र व  र त  िवप  र िव  ल षा  यत फ  र  ा ित  राश ना  षि    श  ािप द िहनाम।  (18ab)    य स पण   ध र त   णदर  ला  यद पल  यत ॥ (14cd) 14 य  धन यार    ध र साय म।  (15ab) ति  वश    ध   णार ध    त  धन यार   HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  46  AN INDIAN SINE TABLE OF 36 ENTRIES As results, our text gives   36 = 3438,   18 = 2431,   9 = 131615 , and   27 = 3178.  (B) GS 7.29cd-45: Starting from sin 30   (  12 ), a total of 8 sines are derived.   12      24      6              30           3      33   15     21      Our text states that chord 60   =    , therefore sin 30   =   12 = = 1719. The remain2 ing values are given as:   24 = 2978,   6 = 890,   30 = 3321,   15 = 2093,   21 = 2728,   3 = 499,   33 = 337216 , in this order. (C) GS 7.47  71: Our text starts from   4 (= sin 10   ), which is, as mentioned above, 597. A total of 24 sines are derived.   4                  32           34     2    16                    1     35    8     28    10              26    19     17                      20            14     22    5      31   13     23       29      7           11     25  The 24 sine values derived in the third process are shown in Table 3.                   1  2;30  150         versin    150  3  2  5;0  300  150  13  3  7;30  449  149  29  4 10;0  597  148  54  5 12;30  744  147  83  6 15;0  890  146  117  7 17;30  1034  144  170  8 20;0  1175  142  218  9 22;30  1316  140  270  15 In Chapter 8,   9 is given as 1315. 16 This is wrong because   33 (= sin 82; 30   ) should  be 3409 as in  ryabhaṭa  s   22 . The figure 3372 is  ryabhaṭa  s   21 .  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  MICHIO YANO                   47         versin     10 25;0  1452  137  334  11 27;30  1586  134  400  12 30;0  1719  130  471  13 32;30  1846  127  550  14 35;0  1971  124  634  15 37;30  2093  121  721  16 40;0  2209  116  816  17 42;30  2322  113  915  18 45;0  2431  109  1018  19 47;30  2534  103  1127  20 50;0  2633  99  1240  21 52;30  2728  94  1356  22 55;0  2815  89  1478  23 57;30  2899  84  1603  24 60;0  2978  79  1730  25 62;30  3049  71  1863  26 65;0  3115  66  1997  27 67;30  3178  63  2134  323017  55  2274  3278  46  2416  30 75;0  3321  43  2559  31 77;30  3355  34  2705  32 80;0  3385  30  2852  33 82;30  3409  25  3000  34 85;0  3425  16  3149  35 87;30  3435  10  3288  36 90;0  3438  3  3438  28 70;0 29 72;30  Table 3: Sine table of 36 entries    17 GS 7.56 reads:  णवि  नरद (3233), which is also the reading of JM (see below, Sarma 1977: 48), but  GS 8.8 reads  ि  वित       णव  मर  (3320), which I have changed to  ित  ि  व     णव  मर  (3230).  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  48  AN INDIAN SINE TABLE OF 36 ENTRIES            1  7;30  2 15;0        versin    449  29  890  117  3 22;30 1315  260  4 30;0  1719  460  5 37;30 2093  710  2431  1007  7 52;30 2728  6 45;0  1345  2978  1719  9 67;30 3178  8 60;0  2123  10 75;0  3321  2548  11 82;30 3409  2989  12 90;0  3438  3438  Table 4: Concise sine table  In Chapter 8 of GS all the 36 sines are stated again, this time in the order from   1 to   36 (GS 8.2  10). Some values are slightly different from those given in Chapter 7. The difference of sines (kha  ajy ,        ) and versed sines are also given in GS 8.10  15 and GS 8.16  22 respectively. I have shown these values in Table 3. The values of        and their sum (      ) are good and mostly agree with  ryabhaṭa  s values (Table 1). What is very strange is the values of the versed sines. GS 8.15cd gives definition of versed sine: If these (sine differences) are added in reverse order, they in order are the sum of the versed sines.18 and the following verses just give numerical values . I quote only the beginning part (GS 8.16): vahni (3) vis ve (13) randhradasr  (29) sindhupañca (54) trihastinaḥ (83)/ parvate  ḥ (117) khasaptaik  (170) dh tidasr  (218) khabh ni (270) ca// (8.16) Only four values up to   54   for     = 4 are correct, but after that differences from the correct values conspicuously increase. I do not understand the reason. Even stranger is that correct values are found in the concise sine table in Chapter 10 (Table 4). All the 12 values in the concise table are same as those of  ryabhaṭa, except sin 67; 30   = 3178  (also in the detailed table above), which is better than  ryabhaṭa  s 3177.    18  त     म    मा  य ता     म य    मिपिण ड ा ॥(8.15cd).  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  MICHIO YANO 3  49  JYO T IR M Ī M     S   T IS REMARKABLE that recently I found that Nīlaka ṭha (b. 1444), the most celebrated  I astronomer/mathematician in the M dhava school in Kerala, knew the peculiar sine  table of the Gargasa hit . In his edition of the Jyotirmīm  s  (Sarma 1977), there is a section (according to Sarma,   18. Ṣaṭtri  ajjy nayana (?)  ), called   Derivation of the 36 Rsines  .19 The beginning of this section is missing and Sarma notes that,   The ms. has a gap, which has been suitably filled up  .20 The Sanskrit text of this section is very corrupt and Sarma  s restoration is not successful because he did not know that the parallel passage is found in the Gargasa hit . Actually this section begins with a verse that is identical with GS 7.51cd  52ab. I quote here from the beginning of GS 7.51.   ा  श   या   ित   तत  ि वशितम   ण ।/   त    21   त  माि    व        ित ाद   [   ित ाद  ] दश या ण डस भव ॥ (7.51)   मान  तत  षि  वशत  ण ।   र   दस   षिस  ध भ  That is 2209 (minutes). From this is the sine of the 20th (arc). From this which is 2633 (minutes) there arises the (sum of) the ten sine differences, of which the measure is 145222 (minutes). From it (there originates) the sine of the 26th (arc). This is the derivations of   16 = 2209, from which is   20 (2633), from which is   10 =   1 + ... +   10 (=1452), and from which again is   26 (=3115), which is in the next line (7.52cd). In this way the whole section is borrowed from GS 7.51cd  64ab. We can restore the corrupt text of the Jyotirmīm  s  with the help of the GS. And vice versa, sometimes we can check the readings of the GS by those of the Jyotirmīm  s . We need to know how Garga  s sine table was included in the Jyotirmīm  s . In his commentary on the  ryabhaṭīya, Nīlaka ṭha calls himself G rgyakerala, which means that he belonged to the G rgya clan (gotra) of Kerala. He sometimes quotes Garga  s verses that are found in Var hamihira  s B hatsa hit ,23 an encyclopedic book on divination. This Garga can be different from the author of our GS. Nīlaka ṭha had at least two kinds of texts ascribed to Garga as he says: It is evident that there were two Gargas, one is V ddhagarga and the other is Garga24 . In the same passage, Nīlaka ṭha refers to a mathematical book called Gargasa hit 25 . He quotes a half verse from this Gargasa hit  in his commentary on the   Chapter on Mathematics   (Ga itap da) of the  ryabhaṭīya:26 19 Sarma 1977: viii 20 Sarma 1977: 48, note 1. 21 Sarma (1977: 48) reads    ित ाद   and puts 2233 above the line, but this is wrong. 22 Sarma (1977: 48) puts 1752 above the line, but this is wrong. 23 For example, in the edition of S mba iva  stri (1930  57: v. 3, p. 160) he quotes five verses from   Garga,   but actually they are found in the mūla  text of the B hatsa hit , Chapter 2. But it is possible that the B hatsa hit  at his hand was different from that of Dvivedin  s modern edition. 24 Nīlaka ṭa ad.  ryabhaṭīya, K lakriy p da 10 (S mba iva  stri 1930  57: v. 2, p. 16, line 13): व   ध   र   प  न  र   र     ित    र   वय  प  िस  धम.   25 Idem, line 17:   वप  ण त     र स िहता य   िणतशा     . 26 S mba iva  stri 1930  57: v. 1, p. 11.  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  50  AN INDIAN SINE TABLE OF 36 ENTRIES   प थ  द    ि व  र   या   णर व    ऽनष यत  । The square of the hypotenuse (=diameter) is derived from separately the squares of sine (b hu) and cosine (koṭi).  This is exactly the line we find as GS 7.10cd. Thus we can conclude that Nīlaka ṭha had access to at least two texts ascribed to Garga, one astrological and the other astronomical. What he refers to as   mathematical Gargasa hit    was used when he wrote the Jyotirmīm  s  and the commentary on the  ryabhaṭīya. We do not know whether he actually used this unusual sine table in his computation. Lastly, let us remember that Nīlaka ṭha quotes the very accurate sine table of M dhava as mentioned in Table 2 above. 4  ACK NOW L E DG E M E N T S  The original version of this paper was read at the second International Conference on the History of Mathematics and Astronomy, December 2  8, 2018, Xi  an. 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(2009), The Mathematics of the Heavens and the Earth: The Early History of Trigonometry (Princeton: Princeton University Press), ISBN: 9780691129730. Yabuuti, K. (1979),   Researches on the Chiu-chih li    : Indian Astronomy Under the T  ang    Dynasty  , Acta Asiatica, 36.  HISTORY OF SCIENCE IN SOUTH ASIA 7 (2019) 42  51  Please write to  wujastyk@ualberta.ca  to file bugs/problem reports, feature requests and to get involved. The History of Science in South Asia   Department of History and Classics, 2  81 HM Tory Building, University of Alberta, Edmonton, AB, T6G 2H4, Canada.