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Madhava's sine table is the
table Table may refer to: * Table (database), how the table data arrangement is used within the databases * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (information), a data arrangement with rows and column ...
of trigonometric sines constructed by the 14th century
Kerala Kerala ( , ) is a States and union territories of India, state on the Malabar Coast of India. It was formed on 1 November 1956, following the passage of the States Reorganisation Act, by combining Malayalam-speaking regions of the erstwhile ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
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astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
Madhava of Sangamagrama Mādhava of Sangamagrāma (Mādhavan) Availabl/ref> () was an Indian mathematician and astronomer who is considered to be the founder of the Kerala school of astronomy and mathematics in the Late Middle Ages. Madhava made pioneering contributio ...
(c. 1340 – c. 1425). The table lists the jya-s or Rsines of the twenty-four
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s from 3.75 ° to 90° in steps of 3.75° (1/24 of a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
, 90°). Rsine is just the sine multiplied by a selected radius and given as an integer. In this table, as in Aryabhata's earlier table, ''R'' is taken as 21600 ÷ 2 ≈ 3437.75. The table is
encoded In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication ...
in the letters of the
Sanskrit Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural ...
alphabet using the
Katapayadi system ''Kaṭapayādi'' system (Devanagari: कटपयादि, also known as ''Paralppēru'', Malayalam: പരല്‍പ്പേര്) of numerical notation is an ancient Indian alphasyllabic numeral system to depict letters to numerals ...
, 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 the ''Aryabhatiyabhashya'' of
Nilakantha Somayaji Keļallur Nīlakaṇṭha Somayāji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehens ...
''The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part 1 - 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/ (1444–1544) and also in the ''Yuktidipika/Laghuvivrti'' commentary of
Tantrasamgraha Tantrasamgraha, or Tantrasangraha, (literally, ''A Compilation of the System'') is an important astronomy, astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathemat ...
by Sankara Variar (circa. 1500–1560). The verses below are given as in ''Cultural foundations of mathematics'' by C.K. Raju. They are also given in the ''Malayalam Commentary of Karanapaddhati'' by P.K. Koru (Published in 1953) but slightly differently.


The table

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 in ISO 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 ÷ 2, with the modern value of with two decimals in the arcthirds are given in the fifth column. 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 the
Katapayadi system ''Kaṭapayādi'' system (Devanagari: कटपयादि, also known as ''Paralppēru'', Malayalam: പരല്‍പ്പേര്) of numerical notation is an ancient Indian alphasyllabic numeral system to depict letters to numerals ...
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.


A simple way of understanding the table

Without going into the philosophy of why the value of ''R'' = 21600 ÷ 2 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 the ''R'' 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 larger ''R'', 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.


Another (more difficult) way to understand the values

Consider some angle whose measure is ''A''. Consider a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
of unit radius and center O. Let the arc PQ of the circle subtend an angle ''A'' at the center O. Drop the
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
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 \anglePOS in
arcminutes A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
,
arcsecond A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
s and sixtieths of an
arcsecond A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...
. 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/ to convert to 21.92614 degrees, which is :1315 arcminutes 34 arcseconds 07 sixtieths of an arcsecond, abbreviated 13153407. For an angle whose measure is ''A'', let :\angle POS = m \text s \text t \text Then: : \begin \sin (A) & = RQ \\ & = \text PS \\ & = \angle POS \text \\ \end


Derivation of trigonometric sines from the table

Each of the lines in the table specifies eight digits. Let the digits corresponding to angle ''A'' (read from left to right) be: : d_1\quad d_2\quad d_3\quad d_4\quad d_5\quad d_6\quad d_7\quad d_8 Then according to the rules of the
Katapayadi system ''Kaṭapayādi'' system (Devanagari: कटपयादि, also known as ''Paralppēru'', Malayalam: പരല്‍പ്പേര്) of numerical notation is an ancient Indian alphasyllabic numeral system to depict letters to numerals ...
they should be taken from right to left and we have: : \begin m & = d_8\times 1000 + d_7\times 100 + d_6 \times 10 +d_5\\ s & = d_4\times 10 + d_3\\ t & = d_2\times 10 + d_1 \end : B = m^s^t^ = \frac\left( m + \frac+ \frac\right) The value of the above angle ''B'' expressed in radians will correspond to the sine value of ''A''. : \sin A = \frac B As said earlier, this is the same as dividing the encoded value by the taken ''R'' value: : \sin A = \frac


Example

The table lists the following digits corresponding to the angle ''A'' = 45.00°: :5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2 This yields the angle with measure: : \begin m & = 2\times 1000 + 4\times 100 + 3\times 10 + 0 \text\\ & = 2430 \text \\ s & = 5\times 10 + 1 \text\\ & = 51 \text\\ t & = 1\times 10 + 5 \text\\ & = 15 \text \end From which we get: : B = \frac\left( 2430 + \frac + \frac\right) = \frac The value of the sine of ''A'' = 45.00° as given in Madhava's table is then just ''B'' converted to radians: : \sin 45^\circ = \frac B = \frac \times \frac Evaluating the above, one can find that sin 45° is 0.70710681… This is accurate to 6 decimal places.


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 Keļallur Nīlakaṇṭha Somayāji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehens ...
( Tantrasangraha) and Jyeshtadeva (
Yuktibhāṣā ''Yuktibhāṣā'' (), also known as Gaṇita-yukti-bhāṣā and ( English: ''Compendium of Astronomical Rationale''), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mat ...
) 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'': : \sin x = x - \frac + \frac - \frac + \cdots


See also

*
Madhava series In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent function (mathematics), functions discovered in 14th or 15th century in Kerala, India by the mathematician and astronomer Madhava o ...
* Madhava's correction term * Madhava's value of * Āryabhaṭa's sine table *
Ptolemy's table of chords The table of chords, created by the Greece, Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's ''Almagest'', a treatise on mathematical astron ...


References


Further references

* *For an account of Madhava's computation of the sine table see : *For a thorough discussion of the computation of Madhava's sine table with historical references : {{DEFAULTSORT:Madhava's Sine Table Trigonometry Indian mathematics Kerala school of astronomy and mathematics