Āryabhaṭa's sine table
Encyclopedia
Āryabhaṭa's sine table is a set of twenty-four of numbers given in the astronomical treatise Āryabhaṭiya
composed by the fifth century 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.
It is only a collection of numbers appearing as a Sanskrit
stanza in Āryabhaṭiya and is couched in the special numerical notation invented by Āryabhaṭa himself. This stanza is the tenth one (excluding the stanza containing the invocation and a stanza which is an explanation of Āryabhaṭa's numerical notation) in the first section of Āryabhatiya titled Daśagītikasūtra.
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 even tried to reconstruct Hipparchus's table of chords from similar tables found in Indian treatises. The inadequacies and imperfections in these arguments have also been pointed by several scholars. "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."
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||
, 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 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.
Without any additional assumptions, no interpretation of this computational scheme has correctly yielded all the numbers in the table compiled by Āryabhaṭa in Āryabhaṭiya.
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:...
composed by the fifth century Indian mathematician
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
and astronomer Āryabhaṭa
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
(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.
It is only a collection of numbers appearing as a 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...
stanza in Āryabhaṭiya and is couched in the special numerical notation invented by Āryabhaṭa himself. This stanza is the tenth one (excluding the stanza containing the invocation and a stanza which is an explanation of Āryabhaṭa's numerical notation) in the first section of Āryabhatiya titled Daśagītikasūtra.
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
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
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
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
. The now lost tables of Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
(c.190 BC – c.120 BC) and Menelaus
Menelaus of Alexandria
Menelaus of Alexandria was a Greek mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines.-Life and Works:...
(c.70–140 CE) and those of
Ptolemy's table of chords
The table of chords, created by the astronomer and geometer 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 astronomy. It is essentially equivalent to a table of values of the sine function...
Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
(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
Madhava series
In mathematics, a Madhava series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Sangamagrama Madhava the founder of the Kerala school of astronomy and mathematics...
of the sine and cosine functions by 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...
(c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava
Madhava's sine table
Madhava's sine table is the table of trigonometric sines of various angles 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°...
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
David Pingree
David Edwin Pingree was a University Professor and Professor of History of Mathematics and Classics at Brown University, and was one of America's foremost historians of the exact sciences in antiquity.-Life:He graduated from Phillips Academy in Andover, Massachusetts in 1950 and thereafter attended...
, 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 even tried to reconstruct Hipparchus's table of chords from similar tables found in Indian treatises. The inadequacies and imperfections in these arguments have also been pointed by several scholars. "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."
The original table
The stanza in Āryabhaṭiya describing the sine table is reproduced below:मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||
In modern notations
The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in AryabhatiyaAryabhatiya
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:...
, 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
Devanagari
Devanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal...
script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919
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...
transliteration. The next column contains these numbers in the 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...
. 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
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
, 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
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...
= 3.1416 known to Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
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
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:...
is the earliest surviving text containing a reference to this basic constant.
| Sl. No | Angle ( A )(in degrees Degree (angle) A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians... , arcminutes) |
Value in Āryabhaṭa'snumerical notation (in Devanagari Devanagari Devanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal... ) |
Value in Āryabhaṭa'snumerical notation (in ISO 15919 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... transliteration) |
Value 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... |
Āryabhaṭa's value of jya (A) |
Modern value of jya (A)(3438 × sin (A)) |
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Āryabhaṭa's computational method
The second section of Āryabhaṭiya titled Ganitapāda 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."
Without any additional assumptions, no interpretation of this computational scheme has correctly yielded all the numbers in the table compiled by Āryabhaṭa in Āryabhaṭiya.