Madhava's sine table
Madhava's sine table is the table of trigonometric sines constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama (c. 1340 – c. 1425). The table lists the trigonometric sines of the twenty-four angles from 3.75° to 90° in steps of 3.75° (1/24 of a right angle, 90°). The table is encoded in the letters of the Devanagari alphabet using the Katapayadi system, giving entries the appearance of the verses of a poem in Sanskrit.
Madhava's original work containing the sine table has not been found. The table is reproduced in the Aryabhatiyabhashya of Nilakantha Somayaji[1](1444–1544) and also in the Yuktidipika/Laghuvivrti commentary of Tantrasamgraha by Sankara Variar (circa. 1500-1560).[2]: 114–123
The table
The image below gives Madhava's sine table in Devanagari as reproduced in Cultural foundations of mathematics by C.K. Raju.[2]: 120 The first twelve lines constitute the entries in the table. The last word in the thirteenth line indicates that these are "as told by Madhava".
![](../I/Madhava_sine_tabe_in_Devanagari.JPG.webp)
Values in Madhava's table
![](../I/Madhavasine.jpeg.webp)
To understand the meaning of the values tabulated by Madhava, consider some angle whose measure is A. Consider a circle of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. Drop the perpendicular QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle A. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles A, Madhava's table gives the measures of the corresponding angles POS in arcminutes, arcseconds and sixtieths of an arcsecond.
As an example, let A be an angle whose measure is 22.50°. In Madhava's table, the entry corresponding to 22.50° is the measure in arcminutes, arcseconds and sixtieths of an arcsecond of the angle whose radian measure is the value of sin 22.50°
, which is 0.3826834;
- multiply 0.3826834 radians by 180/π to convert to 21.92614 degrees, which is
- 1315 arcminutes 34 arcseconds 07 sixtieths of an arcsecond, abbreviated 13153407.
In the Katapayadi system the digits are written in the reverse order, so that the entry corresponding to 22.50° is 70435131.
Derivation of trigonometric sines from Madhava's table
For an angle whose measure is A, let
Then
Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be
Then according to the rules of the Katapayadi system of Kerala mathematicians we have
Madhava's value of pi
To complete the numerical computations the value of pi (π) must be known. It is appropriate to use the value of π computed by Madhava himself (which is accurate to 11 decimal places). Nilakantha Somayaji has given this value of π in his Āryabhaṭīya-Bhashya as follows:[2]: 119
![](../I/Madhava_value_of_pi.JPG.webp)
A transliteration of the last two lines:
vibudha-netra-gaja-ahi-hutāśana
tri-guṇa-veda-bha-vāraṇa-bāhavaḥ
nava-nikharva-mite vr̥tivistare
paridhi-mānam idaṁ jagadur budhāḥ
The various words indicate certain numbers encoded in a scheme known as the bhūtasaṃkhyā system. The meaning of the words and the numbers encoded by them (beginning with the units place) are detailed in the following translation of the verse: "Gods (vibudha : 33), eyes (netra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (veda : 4), nakṣatras (bha : 27), elephants (vāraṇa : 8), and arms (bāhavaḥ : 2) - the wise say that this is the measure of the circumference when the diameter of a circle is nava-nikharva (900,000,000,000)."
So, the translation of the poem using the bhūtasaṃkhyā system will simply read "2827433388233 is, as the wise say, the circumference of a circle whose diameter is nava-nikharva (900,000,000,000)". That is, divide 2827433388233 (the number from the first two lines of the poem in reverse order) by nava-nikharva (900,000,000,000) to get the value of π. This calculation yields the value π = 3.1415926535922. This is the value of π used by Madhava in his further calculations and is accurate to 11 decimal places.
Example
Madhava's table lists the following digits corresponding to the angle 45.00°:
This yields the angle with measure
The value of the trigonometric sine of 45.00° as given in Madhava's table is
Substituting the value of π computed by Madhava in the above expression, sin 45° is found to be 0.70710681. This is accurate to 6 decimal places.
Accuracy of Madhava's sine values
The table below has entries for the twenty-four angles from 3.75° to 90° in steps of 3.75° (first column). The second column contains the values tabulated by Madhava in Devanagari in the form he used. (These are taken from Malayalam Commentary of Karanapaddhati by P.K. Koru[3] and are slightly different from the table given in Cultural foundations of mathematics by C.K. Raju.[2]: 114–123 ) The third column contains ISO 15919 transliterations of the lines in the second column. The numbers in the second column are decoded into arcminutes, arcseconds, and sixtieths of an arcsecond in Arabic numerals in the fourth column. The values of the trigonometric sines derived from the numbers specified in Madhava's table are listed in the fifth column, using Madhava's value of 3.1415926535922 for π. The actual values of the sines to 8 places are listed in the sixth column.
Angle A, degrees | Sin A in Madhava's notation | Madhava's sin A | Accurate sin A | ||
---|---|---|---|---|---|
in Devanagari script using Madhava's Katapayadi system |
ISO 15919 transliteration | Decoded angle in minutesʹ secondsʹʹ sixtieths of secondʹʹʹ | |||
(1) | (2) | (3) | (4) | (5) | (6) |
03.75 | श्रेष्ठं नाम वरिष्ठानां | śreṣṭhaṁ nāma variṣṭhānāṁ | 224ʹ 50ʹʹ 22ʹʹʹ | 0.06540314 | 0.06540313 |
07.50 | हिमाद्रिर्वेदभावनः | himādrirvēdabhāvanaḥ | 448ʹ 42ʹʹ 58ʹʹʹ | 0.13052623 | 0.13052619 |
11.25 | तपनो भानु सूक्तज्ञो | tapanō bhānu sūktajñō | 670ʹ 40ʹʹ 16ʹʹʹ | 0.19509032 | 0.19509032 |
15.00 | मध्यमं विद्धि दोहनं | maddhyamaṁ viddhi dōhanaṁ | 889ʹ 45ʹʹ 15ʹʹʹ | 0.25881900 | 0.25881905 |
18.75 | धिगाज्यो नाशनं कष्टं | dhigājyō nāśanaṁ kaṣṭaṁ | 1105ʹ 01ʹʹ 39ʹʹʹ | 0.32143947 | 0.32143947 |
22.50 | छन्नभोगाशयाम्बिका | channabhōgāśayāmbikā | 1315ʹ 34ʹʹ 07ʹʹʹ | 0.38268340 | 0.38268343 |
26.25 | मृगाहारो नरेशोयं | mr̥gāhārō narēśōyaṁ | 1520ʹ 28ʹʹ 35ʹʹʹ | 0.44228865 | 0.44228869 |
30.00 | वीरो रणजयोत्सुकः | vīrō raṇajayōtsukaḥ | 1718ʹ 52ʹʹ 24ʹʹʹ | 0.49999998 | 0.50000000 |
33.75 | मूलं विशुद्धं नाळस्य | mūlaṁ viśuddhaṁ nāḷasya | 1909ʹ 54ʹʹ 35ʹʹʹ | 0.55557022 | 0.55557023 |
37.50 | गानेषु विरळा नराः | gāneṣu viraḷā narāḥ | 2092ʹ 46ʹʹ 03ʹʹʹ | 0.60876139 | 0.60876143 |
41.25 | अशुद्धिगुप्ता चोरश्रीः | aśuddhiguptā cōraśrīḥ | 2266ʹ 39ʹʹ 50ʹʹʹ | 0.65934580 | 0.65934582 |
45.00 | शङ्कुकर्णो नगेश्वरः | śaṅkukarṇō nageśvaraḥ | 2430ʹ 51ʹʹ 15ʹʹʹ | 0.70710681 | 0.70710678 |
48.75 | तनुजो गर्भजो मित्रं | tanujō garbhajō mitraṃ | 2584ʹ 38ʹʹ 06ʹʹʹ | 0.75183985 | 0.75183981 |
52.50 | श्रीमानत्र सुखी सखे | śrīmānatra sukhī sakhē | 2727ʹ 20ʹʹ 52ʹʹʹ | 0.79335331 | 0.79335334 |
56.25 | शशी रात्रौ हिमाहारौ | śaśī rātrou himāhārou | 2858ʹ 22ʹʹ 55ʹʹʹ | 0.83146960 | 0.83146961 |
60.00 | वेगज्ञः पथि सिन्धुरः | vēgajñaḥ pathi sindhuraḥ | 2977ʹ 10ʹʹ 34ʹʹʹ | 0.86602543 | 0.86602540 |
63.25 | छाया लयो गजो नीलो | chāya layō gajō nīlō | 3083ʹ 13ʹʹ 17ʹʹʹ | 0.89687275 | 0.89687274 |
67.50 | निर्मलो नास्ति सत्कुले | nirmalō nāsti satkulē | 3176ʹ 03ʹʹ 50ʹʹʹ | 0.92387954 | 0.92387953 |
71.25 | रात्रौ दर्पणमभ्राङ्गं | rātrou darpaṇamabhrāṅgaṁ | 3255ʹ 18ʹʹ 22ʹʹʹ | 0.94693016 | 0.94693013 |
75.00 | नागस्तुङ्ग नखो बली | nāgastuṅga nakhō balī | 3320ʹ 36ʹʹ 30ʹʹʹ | 0.96592581 | 0.96592583 |
78.75 | धीरो युवा कथालोलः | dhīrō yuvā kathālōlaḥ | 3371ʹ 41ʹʹ 29ʹʹʹ | 0.98078527 | 0.98078528 |
82.50 | पूज्यो नारीजनैर्भगः | pūjyō nārījanairbhagaḥ | 3408ʹ 20ʹʹ 11ʹʹʹ | 0.99144487 | 0.99144486 |
86.25 | कन्यागारे नागवल्ली | kanyāgārē nāgavallī | 3430ʹ 23ʹʹ 11ʹʹʹ | 0.99785895 | 0.99785892 |
90.00 | देवो विश्वस्थली भृगुः | devō viśvasthalī bhr̥ guḥ | 3437ʹ 44ʹʹ 48ʹʹʹ | 0.99999997 | 1.00000000 |
Madhava's method of computation
No work of Madhava detailing the methods used by him for the computation of the sine table has survived. However from the writings of later Kerala mathematicians including Nilakantha Somayaji (Tantrasangraha) and Jyeshtadeva (Yuktibhāṣā) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using the power series expansion of sin x:
References
- The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part1-Gaṇitapāda, Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55. https://archive.org/details/Trivandrum_Sanskrit_Series_TSS http://www.sanskritebooks.org/2013/02/trivandrum-sanskrit-series-anantasayana-samskrita-granthavali/
- C.K. Raju (2007). Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16th c. CE. History of Philosophy, Science and Culture in Indian Civilization. Vol. X Part 4. New Delhi: Centre for Studies in Civilizations and Pearson Education in South Asia. ISBN 978-81-317-0871-2.
- Puthumana Somayaji. Karanapaddhati (with a commentary in Malayalam by P.K. Koru). Cherpu, Kerala, India: Astro Printing and Publishing Company. (Published in 1953)
Further references
- Bag, A.K. (1976). "Madhava's sine and cosine series" (PDF). Indian Journal of History of Science. Indian National Academy of Science. 11 (1): 54–57. Archived from the original (PDF) on 5 July 2015. Retrieved 21 August 2016.
- For an account of Madhava's computation of the sine table see : Van Brummelen, Glen (2009). The mathematics of the heavens and the earth : the early history of trigonometry. Princeton: Princeton University Press. pp. 113–120. ISBN 978-0-691-12973-0.
- For a thorough discussion of the computation of Madhava's sine table with historical references : C.K. Raju (2007). Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE. History of Philosophy, Science and Culture in Indian Civilization. Vol. X Part 4. Delhi: Centre for Studies in Civilizations. pp. 114–123.