Madhava's sine table is thetable oftrigonometric sines constructed by the 14th centuryKeralamathematician-astronomerMadhava of Sangamagrama (c. 1340 – c. 1425). The table lists thejya-s or Rsines of the twenty-fourangles from 3.75° to 90° in steps of 3.75° (1/24 of aright angle, 90°). Rsine is just the sine multiplied by a selected radius and given as an integer. In this table, as inAryabhata's earlier table,R is taken as 21600 ÷ 2π ≈ 3437.75.
The table isencoded in the letters of theSanskrit alphabet using theKatapayadi system, giving entries the appearance of the verses of a poem.
Madhava's original work containing the table has not been found. The table is reproduced in theAryabhatiyabhashya ofNilakantha Somayaji[1] (1444–1544) and also in theYuktidipika/Laghuvivrti commentary ofTantrasamgraha bySankara Variar (circa. 1500–1560).[2]: 114–123
The verses below are given as inCultural foundations of mathematics by C.K. Raju.[2]: 114–123 They are also given in theMalayalam Commentary ofKaranapaddhati by P.K. Koru[3] but slightly differently.
The verses are:
श्रेष्ठं नाम वरिष्ठानां हिमाद्रिर्वेदभावनः ।
तपनो भानु सूक्तज्ञो मध्यमं विद्धि दोहनम् ॥ १ ॥
धिगाज्यो नाशनं कष्टं छन्नभोगाशयाम्बिका ।
मृगाहारो नरेशोयं वीरो रणजयोत्सुकः ॥ २ ॥
मूलं विशुद्धं नाळस्य गानेषु विरळा नराः ।
अशुद्धिगुप्ता चोरश्रीः शङ्कुकर्णो नगेश्वरः ॥ ३ ॥
तनुजो गर्भजो मित्रं श्रीमानत्र सुखी सखे ।
शशी रात्रौ हिमाहारौ वेगज्ञः पथि सिन्धुरः ॥ ४ ॥
छाया लयो गजो नीलो निर्मलो नास्ति सत्कुले ।
रात्रौ दर्पणमभ्राङ्गं नागस्तुङ्गनखो बली ॥ ५ ॥
धीरो युवा कथालोलः पूज्यो नारीजनैर्भगः ।
कन्यागारे नागवल्ली देवो विश्वस्थली भृगुः ॥ ६ ॥
तत्परादिकलान्तास्तु महाज्या माधवोदिताः ।
स्वस्वपूर्वविशुद्धे तु शिष्टास्तत्खण्डमौर्विकाः ॥ ७ ॥
The quarters of the first six verses represent entries for the twenty-four angles from 3.75° to 90° in steps of 3.75° (first column). The second column contains the Rsine values encoded as Sanskrit words (in Devanagari). The third column contains the same inISO 15919 transliterations. The fourth column contains the numbers decoded into arcminutes, arcseconds, and arcthirds in modern numerals. The modern values scaled by the traditional “radius” (21600 ÷ 2π, with the modern value ofπ with two decimals in the arcthirds are given in the fifth column.
| AngleA, degrees | R sinA given by Madhava | Modern sinA × (21600 ÷ 2π) to 2 decimals | ||
|---|---|---|---|---|
| In Devanagari script | ISO 15919 transliteration | Decoded angle in minutes′ seconds″ thirds‴ | ||
| (1) | (2) | (3) | (4) | (5) |
| 03.75 | श्रेष्ठं नाम वरिष्ठानां | śreṣṭhaṁ nāma variṣṭhānāṁ | 0224′50″22‴ | 0224′50″21.83‴ |
| 07.50 | हिमाद्रिर्वेदभावनः | himādrirvēdabhāvanaḥ | 0448′42″58‴ | 0448′42″57.58‴ |
| 11.25 | तपनो भानुसूक्तज्ञो | tapanō bhānusūktajñō | 0670′40″16‴ | 0670′40″16.05‴ |
| 15.00 | मध्यमं विद्धि दोहनम् | madhyamaṁ viddhi dōhanam | 0889′45″15‴ | 0889′45″15.61‴ |
| 18.75 | धिगाज्यो नाशनं कष्टं | dhigājyō nāśanaṁ kaṣṭaṁ | 1105′01″39‴ | 1105′01″38.94‴ |
| 22.50 | छन्नभोगाशयाम्बिका | channabhōgāśayāmbikā | 1315′34″07‴ | 1315′34″07.44‴ |
| 26.25 | मृगाहारो नरेशोयं | mr̥gāhārō narēśōyaṁ | 1520′28″35‴ | 1520′28″35.46‴ |
| 30.00 | वीरो रणजयोत्सुकः | vīrō raṇajayōtsukaḥ | 1718′52″24‴ | 1718′52″24.19‴ |
| 33.75 | मूलं विशुद्धं नाळस्य | mūlaṁ viśuddhaṁ nāḷasya | 1909′54″35‴ | 1909′54″35.19‴ |
| 37.50 | गानेषु विरळा नराः | gāneṣu viraḷā narāḥ | 2092′46″03‴ | 2092′46″03.49‴ |
| 41.25 | अशुद्धिगुप्ता चोरश्रीः | aśuddhiguptā cōraśrīḥ | 2266′39″50‴ | 2266′39″50.21‴ |
| 45.00 | शङ्कुकर्णो नगेश्वरः | śaṅkukarṇō nageśvaraḥ | 2430′51″15‴ | 2430′51″14.59‴ |
| 48.75 | तनुजो गर्भजो मित्रं | tanujō garbhajō mitraṃ | 2584′38″06‴ | 2584′38″05.53‴ |
| 52.50 | श्रीमानत्र सुखी सखे | śrīmānatra sukhī sakhē | 2727′20″52‴ | 2727′20″52.38‴ |
| 56.25 | शशी रात्रौ हिमाहारौ | śaśī rātrou himāhārou | 2858′22″55‴ | 2858′22″55.11‴ |
| 60.00 | वेगज्ञः पथि सिन्धुरः | vēgajñaḥ pathi sindhuraḥ | 2977′10″34‴ | 2977′10″33.73‴ |
| 63.25 | छाया लयो गजो नीलो | chāya layō gajō nīlō | 3083′13″17‴ | 3083′13″16.94‴ |
| 67.50 | निर्मलो नास्ति सत्कुले | nirmalō nāsti satkulē | 3176′03″50‴ | 3176′03″49.97‴ |
| 71.25 | रात्रौ दर्पणमभ्राङ्गं | rātrou darpaṇamabhrāṅgaṁ | 3255′18″22‴ | 3255′18″21.58‴ |
| 75.00 | नागस्तुङ्गनखो बली | nāgastuṅganakhō balī | 3320′36″30‴ | 3320′36″30.20‴ |
| 78.75 | धीरो युवा कथालोलः | dhīrō yuvā kathālōlaḥ | 3371′41″29‴ | 3371′41″29.15‴ |
| 82.50 | पूज्यो नारीजनैर्भगः | pūjyō nārījanairbhagaḥ | 3408′20″11‴ | 3408′20″10.93‴ |
| 86.25 | कन्यागारे नागवल्ली | kanyāgārē nāgavallī | 3430′23″11‴ | 3430′23″10.65‴ |
| 90.00 | देवो विश्वस्थली भृगुः | devō viśvasthalī bhr̥ guḥ | 3437′44″48‴ | 3437′44″48.37‴ |
The last verse means: “These are the great R-sines as said by Madhava, comprising arcminutes, seconds and thirds. Subtracting from each the previous will give the R-sine-differences.”
By comparing, one can note that Madhava's values are accurately given rounded to the declared precision of thirds except for Rsin(15°) where one feels he should have rounded up to 889′45″16‴ instead.
Note that in theKatapayadi system the digits are written in the reverse order, so for example the literal entry corresponding to 15° is 51549880 which is reversed and then read as 0889′45″15‴. Note that the 0 does not carry a value but is used for the metre of the poem alone.
Without going into the philosophy of why the value ofR = 21600 ÷ 2π was chosen etc, the simplest way to relate the jya tables to our modern concept of sine tables is as follows:
Even today sine tables are given as decimals to a certain precision. If sin(15°) is given as 0.1736, it means the rational 1736 ÷ 10000 is a good approximation of the actual infinite precision number. The only difference is that in the earlier days they had not standardized on decimal values (or powers of ten as denominator) for fractions. Hence they used other denominators based on other considerations (which are not discussed here).
Hence the sine values represented in the tables may simply be taken as approximated by the given integer values divided by theR chosen for the table.
Another possible confusion point is the usage of angle measures like arcminute etc in expressing the R-sines. Modern sines are unitless ratios. Jya-s or R-sines are the same multiplied by a measure of length or distance. However, since these tables were mostly used for astronomy, and distance on the celestial sphere is expressed in angle measures, these values are also given likewise. However, the unit is not really important and need not be taken too seriously, as the value will anyhow be used as part of a rational and the unit will cancel out.
However, this also leads to the usage of sexagesimal subdivisions in Madhava's refining the earlier table of Aryabhata. Instead of choosing a largerR, he gave the extra precision determined by him on top of the earlier given minutes by using seconds and thirds. As before, these may simply be taken as a different way of expressing fractions and not necessarily as angle measures.
Consider some angle whose measure is A. Consider acircle of unit radius and center O. Let the arc PQ of the circle subtend an angleA at the center O. Drop theperpendicular 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 anglesA, Madhava's table gives the measures of the corresponding anglesPOS inarcminutes,arcseconds and sixtieths of anarcsecond.
As an example, letA 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 ofsin 22.50°, which is 0.3826834;
For an angle whose measure isA, let
Then:
Each of the lines in the table specifies eight digits. Let the digits corresponding to angleA (read from left to right) be:
Then according to the rules of theKatapayadi system they should be taken from right to left and we have:
The value of the above angleB expressed in radians will correspond to the sine value ofA.
As said earlier, this is the same as dividing the encoded value by the takenR value:
The table lists the following digits corresponding to the angle A = 45.00°:
This yields the angle with measure:
From which we get:
The value of the sine ofA = 45.00° as given in Madhava's table is then justB converted to radians:
Evaluating the above, one can find that sin 45° is 0.70710681… This is accurate to 6 decimal places.
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 includingNilakantha Somayaji (Tantrasangraha) andJyeshtadeva (Yuktibhāṣā) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using thepower series expansion of sinx: