Madhava's sine table
Encyclopedia
Madhava's sine table is the table
of trigonometric sines
of various angle
s constructed by the 14th century Kerala
mathematician
-astronomer
Madhava of Sangamagrama
. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00° (angles that are integral
multiples
of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is encoded
in the letters
of Devanagari
using the Katapayadi system
. This gives the entries in the table an appearance of the verses
of a poem in Sanskrit
.
Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the Aryabhatiyabhashya of Nilakantha Somayaji
(1444–1544) and also in the Yuktidipika/Laghuvivrti commentary of Tantrasamgraha
by Sankara Variar
(circa. 1500-1560).
as reproduced in Cultural foundations of mathematics by C.K. Raju. 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".
, 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 arcseconds of the angle whose radian measure is the modern value of sin 22.50°. The modern numerical value of sin 22.50° is 0.382683432363 and,
and
In the Katapayadi system
the digits are written in the reverse order. Thus in Madhava's table, the entry corresponding to 22.50° is 70435131.
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
. It is appropriate that we use the value of π
computed by Madhava himself. Nilakantha Somayaji
has given this value of π in his Aryabhatiya
Bhashya as follows:
A loose transliteration of the last two lines:
vibudha-nētra-gaja-ahi-hutāśana
tri-guṇa-vēda-bha-vāraṇa-bāhavaḥ
nava-nikharva-mitē vr̥tivistarē
paridhi-mānamidaṁ jagadur̥ budhāḥ
The various words indicate certain numbers encoded in a scheme known as the bhutasamkhya system
. The meaning of the words and the numbers encoded by them are detailed in the following translation of the verse:
"Gods (vibudha : 33), eyes (nētra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (vēda : 4), naksatras (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 bhutasamkhya 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 pi(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.
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, one gets sin 45° as 0.70710681.
This value may be compared with the modern exact value of sin 45.00°, namely, 0.70710678.
in the form in which it was given by Madhava. (These are taken from Malayalam Commentary of Karanapaddhati
by P.K. Koru and are slightly different from the table given in Cultural foundations of mathematics by C.K. Raju.) The third column contains ISO 15919 transliterations
of the lines given in the second column. The digits encoded by the lines in second column are given 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. These values are computed using the approximate value 3.1415926535922 for π obtained by Madhava. For comparison, the exact values of the trigonometric sines of the angles are given in the sixth column.
(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.
Table (information)
A table is a means of arranging data in rows and columns.Production % of goalNorth 4087102%South 4093110% The use of tables is pervasive throughout all communication, research and data analysis. Tables appear in print media, handwritten notes, computer software, architectural...
of trigonometric sines
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
of various angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s constructed by the 14th century Kerala
Kerala
or Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
-astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00° (angles that are integral
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
multiples
Multiple (mathematics)
In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...
of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is encoded
Character encoding
A character encoding system consists of a code that pairs each character from a given repertoire with something else, such as a sequence of natural numbers, octets or electrical pulses, in order to facilitate the transmission of data through telecommunication networks or storage of text in...
in the letters
Letter (alphabet)
A letter is a grapheme in an alphabetic system of writing, such as the Greek alphabet and its descendants. Letters compose phonemes and each phoneme represents a phone in the spoken form of the language....
of Devanagari
Devanagari
Devanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal...
using the Katapayadi system
Katapayadi system
Kaṭapayādi system of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses...
. This gives the entries in the table an appearance of the verses
Verse (poetry)
A verse is formally a single line in a metrical composition, e.g. poetry. However, the word has come to represent any division or grouping of words in such a composition, which traditionally had been referred to as a stanza....
of a poem in Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
.
Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the Aryabhatiyabhashya of Nilakantha Somayaji
Nilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...
(1444–1544) and also in the Yuktidipika/Laghuvivrti commentary of Tantrasamgraha
Tantrasamgraha
Tantrasamgraha is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics....
by Sankara Variar
Sankara Variar
Sankara Variar was an astronomer-mathematician of the Kerala school of astronomy and mathematics who lived during the sixteenth century CE...
(circa. 1500-1560).
The table
The image below gives Madhava's sine table in DevanagariDevanagari
Devanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal...
as reproduced in Cultural foundations of mathematics by C.K. Raju. 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".
Values in Madhava's table
To understand the meaning of the values tabulated by MadhavaMadhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
, consider some angle whose measure is A. Consider a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. Drop the perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
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 arcminutesMinute of arc
A minute of arc, arcminute, or minute of angle , is a unit of angular measurement equal to one sixtieth of one degree. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc....
, 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 arcseconds of the angle whose radian measure is the modern value of sin 22.50°. The modern numerical value of sin 22.50° is 0.382683432363 and,
- 0.382683432363 radians = 180 / π × 0.382683432363 degrees = 21.926145564094 degrees.
and
- 21.926145564094 degrees = 1315 arcminutes 34 arcseconds 07 sixtieths of arcsecond.
In the Katapayadi system
Katapayadi system
Kaṭapayādi system of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses...
the digits are written in the reverse order. Thus in Madhava's table, the entry corresponding to 22.50° is 70435131.
Derivation of trigonometric sines from Madhava's table
For an angle whose measure is A, letThen
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
Katapayadi system
Kaṭapayādi system of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses...
of Kerala mathematicians we have
Madhava's value of pi
To complete the numerical computations one must have a knowledge of the value of πPi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
. It is appropriate that we use the value of π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
computed by Madhava himself. Nilakantha Somayaji
Nilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...
has given this value of π in his Aryabhatiya
Aryabhatiya
Āryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa.- Structure and style:...
Bhashya as follows:
A loose transliteration of the last two lines:
vibudha-nētra-gaja-ahi-hutāśana
tri-guṇa-vēda-bha-vāraṇa-bāhavaḥ
nava-nikharva-mitē vr̥tivistarē
paridhi-mānamidaṁ jagadur̥ budhāḥ
The various words indicate certain numbers encoded in a scheme known as the bhutasamkhya system
Bhutasamkhya system
Bhūtasaṃkhyā system is a method of recording numbers using ordinary words having connotations of numerical values. The method was popular among Indian astronomers and mathematicians since ancient times. Sanskrit was the language from which words were chosen to write numbers in the bhūtasaṃkhyā...
. The meaning of the words and the numbers encoded by them are detailed in the following translation of the verse:
"Gods (vibudha : 33), eyes (nētra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (vēda : 4), naksatras (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 bhutasamkhya system
Bhutasamkhya system
Bhūtasaṃkhyā system is a method of recording numbers using ordinary words having connotations of numerical values. The method was popular among Indian astronomers and mathematicians since ancient times. Sanskrit was the language from which words were chosen to write numbers in the bhūtasaṃkhyā...
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 pi(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, one gets sin 45° as 0.70710681.
This value may be compared with the modern exact value of sin 45.00°, namely, 0.70710678.
Comparison of Madhava's and modern sine values
In table below the first column contains the list of the twenty-four angles beginning with 3.75 and ending with 90.00. The second column contains the values tabulated by Madhava in DevanagariDevanagari
Devanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal...
in the form in which it was given by Madhava. (These are taken from Malayalam Commentary of Karanapaddhati
Karanapaddhati
Karanapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company, brought this work to the...
by P.K. Koru and are slightly different from the table given in Cultural foundations of mathematics by C.K. Raju.) The third column contains ISO 15919 transliterations
ISO 15919
ISO 15919 Transliteration of Devanagari and related Indic scripts into Latin characters is an international standard for the transliteration of Indic scripts to the Latin alphabet formed in 2001...
of the lines given in the second column. The digits encoded by the lines in second column are given 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. These values are computed using the approximate value 3.1415926535922 for π obtained by Madhava. For comparison, the exact values of the trigonometric sines of the angles are given in the sixth column.
| Angle A in degrees |
Madhava's numbers for specifying sin A | Value of sin A derived from Madhava's table | Modern value of sin A |
||
|---|---|---|---|---|---|
| in Devanagari script Devanagari Devanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal... using Katapayadi system Katapayadi system Kaṭapayādi system of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses... (as in Madhava's original table) |
in ISO 15919 transliteration ISO 15919 ISO 15919 Transliteration of Devanagari and related Indic scripts into Latin characters is an international standard for the transliteration of Indic scripts to the Latin alphabet formed in 2001... scheme |
in Arabic numerals Arabic numerals Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral... |
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श्रेष्ठो नाम वरिष्ठानां | śreṣṭhō nāma variṣṭhānāṁ | |
0.06540314 | 0.06540313 |
| |
हिमाद्रिर्वेदभावनः | himādrirvēdabhāvanaḥ | |
0.13052623 | 0.13052619 |
| |
तपनो भानु सूक्तज्ञो | tapanō bhānu sūktajñō | |
0.19509032 | 0.19509032 |
| |
मध्यमं विद्धि दोहनं | maddhyamaṁ viddhi dōhanaṁ | |
0.25881900 | 0.25881905 |
| |
धिगाज्यो नाशनं कष्टं | dhigājyō nāśanaṁ kaṣṭaṁ | |
0.32143947 | 0.32143947 |
| |
छन्नभोगाशयांबिका | channabhōgāśayāṁbikā | |
0.38268340 | 0.38268343 |
| |
मृगाहारो नरेशोयं | mr̥gāhārō narēśōyaṁ | |
0.44228865 | 0.44228869 |
| |
वीरो रणजयोत्सुकः | vīrō raṇajayōtsukaḥ | |
0.49999998 | 0.50000000 |
| |
मूलं विशुद्धं नाळस्य | mūlaṁ viṣuddhaṁ nāḷasya | |
0.55557022 | 0.55557023 |
| |
गानेषु विरळा नराः | gāneṣu viraḷā narāḥ | |
0.60876139 | 0.60876143 |
| |
अशुद्धिगुप्ता चोरश्रीः | aśuddhiguptā cōraśrīḥ | |
0.65934580 | 0.65934582 |
| |
शम्कुकर्णो नगेश्वरः | śaṃkukarṇō nageśvaraḥ | |
0.70710681 | 0.70710678 |
| |
तनूजो गर्भजो मित्रं | tanūjō garbhajō mitraṃ | 0.75183985 | 0.75183981 | |
| |
श्रीमानत्र सुखी सखे | śrīmānatra sukhī sakhē | |
0.79335331 | 0.79335334 |
| |
शशी रात्रौ हिमाहारौ | śaśī rātrou himāhārou | |
0.83146960 | 0.83146961 |
| |
वेगज्ञः पथि सिन्धुरः | vēgajñaḥ pathi sindhuraḥ | |
0.86602543 | 0.86602540 |
| |
छाया लयो गजो नीलो | chāya layō gajō nīlō | |
0.89687275 | 0.89687274 |
| |
निर्मलो नास्ति सल्कुले | nirmalō nāsti salkulē | |
0.92387954 | 0.92387953 |
| |
रात्रौ दर्पणमभ्रांगं | rātrou darpaṇamabhrāṁgaṁ | |
0.94693016 | 0.94693013 |
| |
नागस्तुंग नखो बली | nāgastuṁga nakhō balī | |
0.96592581 | 0.96592583 |
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धीरो युवा कथालोलः | dhīrō yuvā kathālōlaḥ | |
0.98078527 | 0.98078528 |
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पूज्यो नारीजनैर्भगाः | pūjyō nārījanairbhagāḥ | |
0.99144487 | 0.99144486 |
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कन्यागारे नागवल्ली | kanyāgārē nāgavallī | |
0.99785895 | 0.99785892 |
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देवो विश्वस्थली भृगुः | devō viśvasthalī bhr̥ guḥ | |
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 like Nilakantha SomayajiNilakantha Somayaji
Kelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...
(Tantrasangraha) and Jyeshtadeva
Jyeshtadeva
Jyeṣṭhadeva was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Sangamagrama Madhava . He is best known as the author of Yuktibhāṣā,...
(Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...
) 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.
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 :