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Āryabhaṭa's sine table

The astronomical treatise Āryabhaṭīya was composed during the fifth century by the Indian mathematician and astronomer Āryabhaṭa (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns.

Arc and chord of a circle

Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as Āryabhaṭa's table of sine-differences.

Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. The now lost tables of Hipparchus (c.190 BC – c.120 BC) and Menelaus (c.70–140 CE) and those of Ptolemy (c.AD 90 – c.168) were all tables of chords and not of half-chords. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.

Some historians of mathematics have argued that the sine table given in Āryabhaṭiya was an adaptation of earlier such tables constructed by mathematicians and astronomers of ancient Greece. David Pingree, one of America's foremost historians of the exact sciences in antiquity, was an exponent of such a view. Assuming this hypothesis, G. J. Toomer writes, "Hardly any documentation exists for the earliest arrival of Greek astronomical models in India, or for that matter what those models would have looked like. So it is very difficult to ascertain the extent to which what has come down to us represents transmitted knowledge, and what is original with Indian scientists. ... The truth is probably a tangled mixture of both."

Contents

In modern notations

The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in Āryabhaṭīya, and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in Devanagari script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the Hindu-Arabic numerals. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of jya) can be obtained by summing up the differences up to that difference. Thus the value of jya corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of jyas are given in the last column of the table.

In the Indian mathematical tradition, the sine ( or jya) of an angle is not a ratio of numbers. It is the length of a certain line segment, a certain half-chord. The radius of the base circle is basic parameter for the construction of such tables. Historically, several tables have been constructed using different values for this parameter. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value π = 3.1416 known to Aryabhata one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. It has not yet been established who is the first ever to use this value for the base radius. But Aryabhatiya is the earliest surviving text containing a reference to this basic constant.

Sl. No Angle ( A )
(in degrees,
arcminutes)
Value in Āryabhaṭa's
numerical notation

(in Devanagari)
Value in Āryabhaṭa's
numerical notation

(in ISO 15919 transliteration)
Value in
Hindu-Arabic numerals
Āryabhaṭa's
value of
jya (A)
Modern value
of jya (A)
(3438 × sin (A))
1
03° 45′
मखि
makhi
225
225′
224.8560
2
07° 30′
भखि
bhakhi
224
449′
448.7490
3
11° 15′
फखि
phakhi
222
671′
670.7205
4
15° 00′
धखि
dhakhi
219
890′
889.8199
5
18° 45′
णखि
ṇakhi
215
1105′
1105.1089
6
22° 30′
ञखि
ñakhi
210
1315′
1315.6656
7
26° 15′
ङखि
ṅakhi
205
1520′
1520.5885
8
30° 00′
हस्झ
hasjha
199
1719′
1719.0000
9
33° 45′
स्ककि
skaki
191
1910′
1910.0505
10
37° 30′
किष्ग
kiṣga
183
2093′
2092.9218
11
41° 15′
श्घकि
śghaki
174
2267′
2266.8309
12
45° 00′
किघ्व
kighva
164
2431′
2431.0331
13
48° 45′
घ्लकि
ghlaki
154
2585′
2584.8253
14
52° 30′
किग्र
kigra
143
2728′
2727.5488
15
56° 15′
हक्य
hakya
131
2859′
2858.5925
16
60° 00′
धकि
dhaki
119
2978′
2977.3953
17
63° 45′
किच
kica
106
3084′
3083.4485
18
67° 30′
स्ग
sga
93
3177′
3176.2978
19
71° 15′
झश
jhaśa
79
3256′
3255.5458
20
75° 00′
ङ्व
ṅva
65
3321′
3320.8530
21
78° 45′
क्ल
kla
51
3372′
3371.9398
22
82° 30′
प्त
pta
37
3409′
3408.5874
23
86° 15′
pha
22
3431′
3430.6390
24
90° 00′
cha
7
3438′
3438.0000

The second section of Āryabhaṭiya titled Ganitapādd a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.

  • "When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord."

This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function.

  1. Selin, Helaine, ed. (2008).Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2 ed.). Springer. pp. 986–988. ISBN 978-1-4020-4425-0.
  2. Eugene Clark (1930). Theastronomy. Chicago: The University of Chicago Press.
  3. Takao Hayashi, T (November 1997). "Āryabhaṭa's rule and table for sine-differences". Historia Mathematica. 24 (4): 396–406. doi:10.1006/hmat.1997.2160.
  4. B. L. van der Waerden, B. L. (March 1988). "Reconstruction of a Greek table of chords". Archive for History of Exact Sciences. 38 (1): 23–38. Bibcode:1988AHES...38...23V. doi:10.1007/BF00329978. S2CID 189793547.
  5. J J O'Connor and E F Robertson (June 1996). "The trigonometric functions". Retrieved4 March 2010.
  6. "Hipparchus and Trigonometry". Retrieved6 March 2010.
  7. G. J. Toomer, G. J. (July 2007). "The Chord Table of Hipparchus and the Early History of Greek Trigonometry". Centaurus. 18 (1): 6–28. doi:10.1111/j.1600-0498.1974.tb00205.x.
  8. B.N. Narahari Achar (2002). "Āryabhata and the table of Rsines"(PDF). Indian Journal of History of Science. 37 (2): 95–99. Retrieved6 March 2010.
  9. Glen Van Brummelen (March 2000). "[HM] Radian Measure". Historia Mathematica mailing List Archive. Retrieved6 March 2010.
  10. Glen Van Brummelen (25 January 2009). The mathematics of the heavens and the earth: the early 0. ISBN 9780691129730.
  11. Victor J Katz (Editor) (2007). The mathematics of Egypt, Mesopotamia, China, India, and Islam: a sourcebook. Princeton: Princeton University Press. pp. 405–408. ISBN 978-0-691-11485-9.CS1 maint: extra text: authors list (link)

Āryabhaṭa's sine table
Aryabhaṭa s sine table Language Watch Edit The astronomical treatise Aryabhaṭiya was composed during the fifth century by the Indian mathematician and astronomer Aryabhaṭa 476 550 CE for the computation of the half chords of certain set of arcs of a circle It is not a table in the modern sense of a mathematical table that is it is not a set of numbers arranged into rows and columns 1 2 Arc and chord of a circle Aryabhaṭa s table is also not a set of values of the trigonometric sine function in a conventional sense it is a table of the first differences of the values of trigonometric sines expressed in arcminutes and because of this the table is also referred to as Aryabhaṭa s table of sine differences 3 4 Aryabhaṭa s table was the first sine table ever constructed in the history of mathematics 5 The now lost tables of Hipparchus c 190 BC c 120 BC and Menelaus c 70 140 CE and those of Ptolemy c AD 90 c 168 were all tables of chords and not of half chords 5 Aryabhaṭa s table remained as the standard sine table of ancient India There were continuous attempts to improve the accuracy of this table These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama c 1350 c 1425 the founder of the Kerala school of astronomy and mathematics and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places Some historians of mathematics have argued that the sine table given in Aryabhaṭiya was an adaptation of earlier such tables constructed by mathematicians and astronomers of ancient Greece 6 David Pingree one of America s foremost historians of the exact sciences in antiquity was an exponent of such a view Assuming this hypothesis G J Toomer 7 8 9 writes Hardly any documentation exists for the earliest arrival of Greek astronomical models in India or for that matter what those models would have looked like So it is very difficult to ascertain the extent to which what has come down to us represents transmitted knowledge and what is original with Indian scientists The truth is probably a tangled mixture of both 10 Contents 1 The table 1 1 In modern notations 2 Aryabhaṭa s computational method 3 See also 4 ReferencesThe table EditIn modern notations Edit The values encoded in Aryabhaṭa s Sanskrit verse can be decoded using the numerical scheme explained in Aryabhaṭiya and the decoded numbers are listed in the table below In the table the angle measures relevant to Aryabhaṭa s sine table are listed in the second column The third column contains the list the numbers contained in the Sanskrit verse given above in Devanagari script For the convenience of users unable to read Devanagari these word numerals are reproduced in the fourth column in ISO 15919 transliteration The next column contains these numbers in the Hindu Arabic numerals Aryabhaṭa s numbers are the first differences in the values of sines The corresponding value of sine or more precisely of jya can be obtained by summing up the differences up to that difference Thus the value of jya corresponding to 18 45 is the sum 225 224 222 219 215 1105 For assessing the accuracy of Aryabhaṭa s computations the modern values of jyas are given in the last column of the table In the Indian mathematical tradition the sine or jya of an angle is not a ratio of numbers It is the length of a certain line segment a certain half chord The radius of the base circle is basic parameter for the construction of such tables Historically several tables have been constructed using different values for this parameter Aryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures In astronomical computations distances are measured in degrees minutes seconds etc In this measure the circumference of a circle is 360 60 360 minutes 21600 minutes The radius of the circle the measure of whose circumference is 21600 minutes is 21600 2p minutes Computing this using the value p 3 1416 known to Aryabhata one gets the radius of the circle as 3438 minutes approximately Aryabhaṭa s sine table is based on this value for the radius of the base circle It has not yet been established who is the first ever to use this value for the base radius But Aryabhatiya is the earliest surviving text containing a reference to this basic constant 11 Sl No Angle A in degrees arcminutes Value in Aryabhaṭa s numerical notation in Devanagari Value in Aryabhaṭa s numerical notation in ISO 15919 transliteration Value in Hindu Arabic numerals Aryabhaṭa s value of jya A Modern value of jya A 3438 sin A 1 03 45 मख makhi 225 225 224 8560 2 07 30 भख bhakhi 224 449 448 7490 3 11 15 फख phakhi 222 671 670 7205 4 15 00 धख dhakhi 219 890 889 8199 5 18 45 णख ṇakhi 215 1105 1105 1089 6 22 30 ञख nakhi 210 1315 1315 6656 7 26 15 ङख ṅakhi 205 1520 1520 5885 8 30 00 हस झ hasjha 199 1719 1719 0000 9 33 45 स कक skaki 191 1910 1910 0505 10 37 30 क ष ग kiṣga 183 2093 2092 9218 11 41 15 श घक sghaki 174 2267 2266 8309 12 45 00 क घ व kighva 164 2431 2431 0331 13 48 45 घ लक ghlaki 154 2585 2584 8253 14 52 30 क ग र kigra 143 2728 2727 5488 15 56 15 हक य hakya 131 2859 2858 5925 16 60 00 धक dhaki 119 2978 2977 3953 17 63 45 क च kica 106 3084 3083 4485 18 67 30 स ग sga 93 3177 3176 2978 19 71 15 झश jhasa 79 3256 3255 5458 20 75 00 ङ व ṅva 65 3321 3320 8530 21 78 45 क ल kla 51 3372 3371 9398 22 82 30 प त pta 37 3409 3408 5874 23 86 15 फ pha 22 3431 3430 6390 24 90 00 छ cha 7 3438 3438 0000Aryabhaṭa s computational method EditThe second section of Aryabhaṭiya titled Ganitapadd a contains a stanza indicating a method for the computation of the sine table There are several ambiguities in correctly interpreting the meaning of this verse For example the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse 11 When the second half chord partitioned is less than the first half chord which is approximately equal to the corresponding arc by a certain amount the remaining sine differences are less than the previous ones each by that amount of that divided by the first half chord This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function See also EditMadhava s sine table Bhaskara I s sine approximation formulaReferences Edit Selin Helaine ed 2008 Encyclopaedia of the History of Science Technology and Medicine in Non Western Cultures 2 ed Springer pp 986 988 ISBN 978 1 4020 4425 0 Eugene Clark 1930 Theastronomy Chicago The University of Chicago Press Takao Hayashi T November 1997 Aryabhaṭa s rule and table for sine differences Historia Mathematica 24 4 396 406 doi 10 1006 hmat 1997 2160 B L van der Waerden B L March 1988 Reconstruction of a Greek table of chords Archive for History of Exact Sciences 38 1 23 38 Bibcode 1988AHES 38 23V doi 10 1007 BF00329978 S2CID 189793547 a b J J O Connor and E F Robertson June 1996 The trigonometric functions Retrieved 4 March 2010 Hipparchus and Trigonometry Retrieved 6 March 2010 G J Toomer G J July 2007 The Chord Table of Hipparchus and the Early History of Greek Trigonometry Centaurus 18 1 6 28 doi 10 1111 j 1600 0498 1974 tb00205 x B N Narahari Achar 2002 Aryabhata and the table of Rsines PDF Indian Journal of History of Science 37 2 95 99 Retrieved 6 March 2010 Glen Van Brummelen March 2000 HM Radian Measure Historia Mathematica mailing List Archive Retrieved 6 March 2010 Glen Van Brummelen 25 January 2009 The mathematics of the heavens and the earth the early 0 ISBN 9780691129730 a b Victor J Katz Editor 2007 The mathematics of Egypt Mesopotamia China India and Islam a sourcebook Princeton Princeton University Press pp 405 408 ISBN 978 0 691 11485 9 CS1 maint extra text authors list link 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