Tantrasamgraha
Encyclopedia
Tantrasamgraha is an important astronomical
treatise written by Nilakantha Somayaji
, an astronomer
/mathematician
belonging to the Kerala school of astronomy and mathematics.
The treatise was completed in 1501 CE. It consists of 432 verses in Sanskrit
divided into eight chapters. Tantrasamgraha had spawned a few commentaries: Tantrasamgraha-vyakhya of anonymous authorship and Yuktibhāṣā
authored by Jyeshtadeva
in about 1550 CE.
Tantrasangraha, together with its commentaries, bring forth the depths of the mathematical accomplishments the Kerala school of astronomy and mathematics, in particular the achievements of the remarkable mathematician of the school Sangamagrama Madhava
.
In his Tantrasangraha, Nilakantha revised Aryabhata
's model for the planets Mercury
and Venus
. His equation of the centre
for these planets remained the most accurate until the time of Johannes Kepler
in the 17th century.
It was C.M. Whish, a civil servant of East India Company
, who brought to the attention of the western scholarship the existence of Tantrasamgraha through a paper published in 1835. The other books mentioned by C.M. Whish in his paper were Yuktibhāṣā
of Jyeshtadeva
, Karanapaddhati
of Puthumana Somayaji
and Sadratnamala
of Sankara Varman
.
, the author of Tantrasamgraha, was a Nambuthiri belonging to Gargya gotra
and a resident of Trikkantiyur, near Tirur
in central Kerala
. The name of his Illam
was Kelallur. He studied under Damodara
, son of Paramesvara. The first and the last verses in Tantrasamgraha contain chronogram
s specifying the dates, in the form Kali
days, of the commencement and of the completion of book. These work out to dates in 1500-01.
In astronomy, the spherical triangle formed by the zenith
, the celestial north pole and the Sun
is called the astronomical triangle. Its sides and two of its angles are important astronomical quantities. The sides are 90° - φ where φ is the observer's terrestrial latitude
, 90° - δ where δ is the Sun's declination
and 90° - a where a is the Sun's altitude
above the horizon
. The important angles are the angle at the zenith which is the Sun's azimuth and the angle at the north pole which is the Sun's hour angle
. The problem is to compute two of these elements when the other three elements are specified. There are precisely ten different possibilities and Tantrasamgraha contains discussions of all these possibilities with complete solutions one by one in one place. "The spherical triangle is handled as systematically here as in any modern textbook."
The terrestrial latitude
of an observer's position is equal to the zenith distance of the Sun at noon on the equinctial day
. The effect of solar parallax on zenith distance was known to Indian astronomers right from Aryabhata
. But it was Nilakantha Somayaji
who first discussed the effect of solar parallax on the observer's latitude. Tantrasamgraha gives the magnitude of this correction and also a correction due to the finite size of the Sun.
Tantrasamgraha contains a major revision of the older Indian planetary model for the interior planets Mercury
and Venus
and, in the history of astronomy, the first accurate formulation of the equation of centre for these planets. His planetary system was a partially heliocentric
model in which Mercury, Venus, Mars
, Jupiter
and Saturn
orbit the Sun
, which in turn orbits the Earth
, similar to the Tychonic system
later proposed by Tycho Brahe
in the late 16th century. Nilakantha's system was more accurate at predicting the heliocentric motions of the interior than the later Tychonic and Copernican models
, and remained the most accurate until the 17th century when Johannes Kepler
reformed the computation for the interior planets in much the same way Nilakantha did. Most astronomers of the Kerala school who followed him accepted his planetary model.
The Conference turned out to be an important occasion for highlighting and reviewing the recent work on the achievements in Mathematics and Astronomy of the Kerla school and the new perspectives in History of Science, which are emerging from these studies. A compilation of the important papers presented at this Conference has also been published.
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
treatise written by 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...
, an 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...
/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....
belonging to the Kerala school of astronomy and mathematics.
The treatise was completed in 1501 CE. It consists of 432 verses 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...
divided into eight chapters. Tantrasamgraha had spawned a few commentaries: Tantrasamgraha-vyakhya of anonymous authorship and 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...
authored by 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,...
in about 1550 CE.
Tantrasangraha, together with its commentaries, bring forth the depths of the mathematical accomplishments the Kerala school of astronomy and mathematics, in particular the achievements of the remarkable mathematician of the school Sangamagrama Madhava
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...
.
In his Tantrasangraha, Nilakantha revised Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
's model for the planets Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
and Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
. His equation of the centre
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
for these planets remained the most accurate until the time of Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
in the 17th century.
It was C.M. Whish, a civil servant of East India Company
East India Company
The East India Company was an early English joint-stock company that was formed initially for pursuing trade with the East Indies, but that ended up trading mainly with the Indian subcontinent and China...
, who brought to the attention of the western scholarship the existence of Tantrasamgraha through a paper published in 1835. The other books mentioned by C.M. Whish in his paper were 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...
of 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,...
, 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...
of Puthumana Somayaji
Puthumana Somayaji
Puthumana Somayaji was a 15th century mathematician from Kerala, India. He was born into the Puthumana Illam, and is believed to have been a contemporary of Vatasseri Namboodiri....
and Sadratnamala
Sadratnamala
Sadratnamala is an astronomical-mathematical treatise in Sanskrit written by Sankara Varman, an astronomer-mathematician of the Kerala school of mathematics, in 1819. Even though the book has been written at a time when western mathematics and astronomy had been introduced in India, it is composed...
of Sankara Varman
Sankara Varman
Sankara Varman is an astronomer-mathematician belonging to the Kerala school of astronomy and mathematics. He is best known as the author of Sadratnamala, a treatise on astronomy and mathematics, composed in 1819...
.
Author and date of Tantrasamgraha
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...
, the author of Tantrasamgraha, was a Nambuthiri belonging to Gargya gotra
Brahmin gotra system
The word "gotra" means "lineage" in the Sanskrit language. Among those of the Brahmin caste, gotras are reckoned patrilineally. Each gotra takes the name of a famous Rishi or sage who was the patrilineal forebearer of that clan...
and a resident of Trikkantiyur, near Tirur
Tirur
Tirur is a town and a municipality in Malappuram district in the Indian state of Kerala spread over an area of 16.55 km2 . It is birth place of Thunchathu Ramanujan Ezhuthachan, the father of Malayalam literature. Thunchan Parambu is highly venerated and its sand is believed to be sacred. The...
in central 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....
. The name of his Illam
Illam
Illam is a malayalee lineage system used for classification and identification of castes in Kerala, South India. The House of Namboothiries are also called the Brahmaalayam or Mana...
was Kelallur. He studied under Damodara
Damodara
Vatasseri Damodara Nambudiri was an astronomer-mathematician of the Kerala school of astronomy and mathematics who flourished during the fifteenth century CE. He was a son of Vatasseri Paramesvara who developed the drigganita system of astronomical computations...
, son of Paramesvara. The first and the last verses in Tantrasamgraha contain chronogram
Chronogram
A chronogram is a sentence or inscription in which specific letters, interpreted as numerals, stand for a particular date when rearranged. The word, meaning "time writing", derives from the Greek words chronos and gramma . In the pure chronogram each word contains a numeral, the natural chronogram...
s specifying the dates, in the form Kali
Kali Yuga
Kali Yuga is the last of the four stages that the world goes through as part of the cycle of yugas described in the Indian scriptures. The other ages are Satya Yuga, Treta Yuga and Dvapara Yuga...
days, of the commencement and of the completion of book. These work out to dates in 1500-01.
Synopsis of the book
A brief account of the contents of Tantrasamgraha is presented below. A descriptive account of the contents is available in Bharatheeya Vijnana/Sastra Dhara. Full details of the contents are available in an edition of Tantrasamgraha published in the Indian Journal of History of Science.- Chapter 1 (Madhyama-prakaranam): The purpose of the astronomical computation, civil and sidereal day measurements, lunar month, solar month, intercalary month, revolutions of the planets, theory of intercalation, planetary revolution in circular orbits, computation of kali days, mathematical operations like addition, subtraction, multiplication, division, squaring and determining square root, fractions, positive and negative numbers, computation of mean planets, correction for longitude, longitudinal time, positions of the planets at the beginning of Kali era, planetary apogees in degrees. (40 slokas)
- Chapter 2 (Sphuta-prakaranam (On true planets)): Computation of risings, and arcs, construction of a circle of diameter equal to the side of a given square, computation of the circumference without the use of square and roots, sum of series, sum of the series of natural numbers, of squares of numbers, of cubes of numbers, processes relating to Rsines and arcs, computation of the arc of a given Rsine, computation of the circumference of a circle, derivation of Rsines for given Rversed sine and arc, computation of Rsine and arcs, accurate computation of the 24 ordained Rsines, sectional Rsines and Rsine differences, sum of Rsine differences, summation of Rsine differences, computation of the arc of an Rsine according to Madhava, computation of Rsine and Rversed sine at desired point without the aid of the ordained Rsines, rules relating to triangles, rules relating to cyclic quadrilaterals, rules relating to the hypotenuse of a quadrilateral, computation of the diameter from the area of the cyclic quadrilateral, surface area of a sphere, computation of the desired Rsine, the ascensional difference, sun's daily motion in minutes of arc, application of ascensional difference to true planets, measure of day and night on applying ascensional difference, conversion of the arc of Rsine of the ascensional difference, etc. (59 slokas)
- Chapter 3 (Chhaya-prakaranam (Treatise on shadow)): Deals with various problems related with the sun's position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates. (116 slokas)
- Chapter 4 (Chandragrahana-prakaranam (Treatise on the lunar eclipse)): Diameter of the Earth's shadow in minutes, Moon's latitude and Moon's rate of motion, probability of an eclipse, total eclipse and rationale of the explanation given for total eclipse, half duration and first and last contacts, points of contacts and points of release in eclipse, and their method of calculation, visibility of the contact in the eclipse at sunrise and sunset, contingency of the invisibility of an eclipse, possibility of the deflection, deflection due to latitude and that due to declination. (53 slokas)
- Chapter 5 (Ravigrahana-prakaranam (Treatise on the solar eclipse)): Possibility of a solar eclipse, minutes of parallax in latitude of the sun, minutes of parallax in latitude of the moon,. maximum measure of the eclipse, middle of the eclipse, time of first contact and last contact, half duration and times of submergence and emergence, reduction to observation of computed eclipse, mid eclipse, non prediction of an eclipse. (63 slokas)
- Chapter 6 (Vyatipata-prakaranam (On vyatipata)): Deals with the complete deviation of the longitudes of the sun and the moon. (24 slokas)
- Chapter 7 (Drikkarma-prakaranam(On visibility computation)): Discusses the rising and setting of the moon and planets. (15 slokas)
- Chapter 8 (Sringonnati-prakaranam (On elevation of the lunar cusps)): Examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it. (40 slokas)
Some noteworthy features of Tantrasamgraha
"A remarkable synthesis of Indian spherical astronomical knowledge occurs in a passage in Tantrasamgraha."In astronomy, the spherical triangle formed by the zenith
Zenith
The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e...
, the celestial north pole and the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
is called the astronomical triangle. Its sides and two of its angles are important astronomical quantities. The sides are 90° - φ where φ is the observer's terrestrial latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
, 90° - δ where δ is the Sun's declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
and 90° - a where a is the Sun's altitude
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...
above the horizon
Horizon
The horizon is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting...
. The important angles are the angle at the zenith which is the Sun's azimuth and the angle at the north pole which is the Sun's hour angle
Hour angle
In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the position of a point on the celestial sphere....
. The problem is to compute two of these elements when the other three elements are specified. There are precisely ten different possibilities and Tantrasamgraha contains discussions of all these possibilities with complete solutions one by one in one place. "The spherical triangle is handled as systematically here as in any modern textbook."
The terrestrial latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
of an observer's position is equal to the zenith distance of the Sun at noon on the equinctial day
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...
. The effect of solar parallax on zenith distance was known to Indian astronomers right from Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
. But it was 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...
who first discussed the effect of solar parallax on the observer's latitude. Tantrasamgraha gives the magnitude of this correction and also a correction due to the finite size of the Sun.
Tantrasamgraha contains a major revision of the older Indian planetary model for the interior planets Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
and Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
and, in the history of astronomy, the first accurate formulation of the equation of centre for these planets. His planetary system was a partially heliocentric
Heliocentrism
Heliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...
model in which Mercury, Venus, Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
, Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
and Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
orbit the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
, which in turn orbits the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
, similar to the Tychonic system
Tychonic system
The Tychonic system was a model of the solar system published by Tycho Brahe in the late 16th century which combined what he saw as the mathematical benefits of the Copernican system with the philosophical and "physical" benefits of the Ptolemaic system...
later proposed by Tycho Brahe
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...
in the late 16th century. Nilakantha's system was more accurate at predicting the heliocentric motions of the interior than the later Tychonic and Copernican models
Copernican heliocentrism
Copernican heliocentrism is the name given to the astronomical model developed by Nicolaus Copernicus and published in 1543. It positioned the Sun near the center of the Universe, motionless, with Earth and the other planets rotating around it in circular paths modified by epicycles and at uniform...
, and remained the most accurate until the 17th century when Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
reformed the computation for the interior planets in much the same way Nilakantha did. Most astronomers of the Kerala school who followed him accepted his planetary model.
Conference on 500 years of Tantrasamgraha
A Conference to celebrate the 500th Anniversary of Tantrasangraha was organised by the Department of Theoretical Physics, University of Madras, in collaboration with the Inter-University Centre of the Indian Institute of Advanced Study, Shimla, during 11–13 March 2000, at Chennai.The Conference turned out to be an important occasion for highlighting and reviewing the recent work on the achievements in Mathematics and Astronomy of the Kerla school and the new perspectives in History of Science, which are emerging from these studies. A compilation of the important papers presented at this Conference has also been published.
Other works of the same author
The following is a brief description of the other works by Nilakantha Somayaji.- JyotirmimamsaJyotirmimamsaJyotirmimamsa is a treatise on the methodology of astronomical studies authored by Nilakantha Somayaji in around 1504 CE. Nilakantha somayaji was an important astronomer-mathematician of the Kerala school of astronomy and mathematics and was the author of the much celebrated astronomical work...
- Golasara : Description of basic astronomical elements and procedures
- Sidhhantadarpana : A short work in 32 slokas enunciating the astronomical constants with reference to the Kalpa and specifying his views on astronomical concepts and topics.
- Candrachayaganita : A work in 32 verses on the methods for the calculation of time from the measurement of the shadow of the gnomon cast by the moon and vice versa.
- Aryabhatiya-bhashya : Elaborate commentary on Aryabhatiya.
- Sidhhantadarpana-vyakhya : Commentary on his own Siddhantadarapana.
- Chandrachhayaganita-vyakhya : Commentary on his own Chandrachhayaganita.
- Sundaraja-prasnottara : Nilakantha's answers to questions posed by Sundaraja, a Tamil Nadu based astronomer.
- Grahanadi-grantha : Rationale of the necessity of correcting old astronomical constants by observations.
- Grahapariksakrama : Description of the principles and methods for verifying astronomical computations by regular observations.