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{{short description|Astronomical treatise written by Nilakantha Somayaji}}
{{Use dmy dates|date=August 2013}}
{{Infobox book
| name = Tantrasamgraha
| title_orig =
| translator =
| image = Opening verses in Tantrasamgraha.JPG
| caption = Opening verses of Tantrasamgraha (in [[Devanagari]])
| author = [[Nilakantha Somayaji]]
| illustrator =
| cover_artist =
| country = India
| language = [[Sanskrit]]
| series =
| subject = [[Astronomy]]/[[Mathematics]]
| genre =
| publisher =
| pub_date = 1500-01 CE
| english_pub_date =
| media_type =
| pages =
| isbn =
| oclc =
| dewey =
| congress =
| preceded_by =
| followed_by =
}}

'''Tantrasamgraha''',<ref name="text">{{cite web|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a5d_s1.pdf|title=Tantrasamgraha with English translation|editor=K.V. Sarma|others=Translated by V.S. Narasimhan|publisher=Indian National Academy of Science|pages=48|language=sa, en|access-date=17 January 2010|url-status=dead|archive-url=https://web.archive.org/web/20120309014402/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a5d_s1.pdf|archive-date=9 March 2012|df=dmy-all}}</ref><ref>''Tantrasamgraha'', ed. K.V. Sarma, trans. V. S. Narasimhan in the Indian Journal of History of Science, issue starting Vol. 33, No. 1 of March 1998</ref> or '''Tantrasangraha''',<ref>Open Library Reference: {{cite web|url=https://openlibrary.org/b/OL2697994M/Tantrasan%CC%87grahah%CC%A3|title= Tantrasaṅgrahaḥ gaṇitam : savyākhyaḥ|author=Nīlakaṇṭha Somayājī|work=Anantaśayanasaṃskr̥tagranthāvaliḥ ;, granthāṅkaḥ 188|publisher=Kerala University, Thiruvananthapuram|language=sa|access-date=18 January 2010}}</ref> (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 mathematics]].
The treatise was completed in 1501 CE. It consists of 432 verses in [[Sanskrit]] divided into eight chapters.<ref name="Andrews">{{cite web|url=http://www-history.mcs.st-and.ac.uk/Biographies/Nilakantha.html|title=Nilakantha Somayaji|author=J J O'Connor |author2=E F Robertson |date=November 2000 |publisher=School of Mathematics and Statistics University of St Andrews, Scotland |access-date=17 January 2010}}</ref> 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 [[Madhava of Sangamagrama|Sangamagrama Madhava]].
In his ''Tantrasangraha'', Nilakantha revised [[Aryabhata]]'s model for the planets [[Mercury (planet)|Mercury]] and [[Venus]]. His equation of the [[Center of mass|centre]] for these planets remained the most accurate until the time of [[Johannes Kepler]] in the 17th century.<ref name=Joseph>George G. Joseph (2000). ''The Crest of the Peacock: Non-European Roots of Mathematics'', p. 408. [[Princeton University Press]].</ref>

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.<ref name="Whish">{{cite journal|author=C.M. Whish|date=1835|title=On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, the Tantra Sahgraham, Yucti Bhasha, Carana Padhati and Sadratnamala|journal=Transactions of the Royal Asiatic Society of Great Britain and Ireland|volume=III|issue=iii|pages=509–23}}</ref> The other books mentioned by C.M. Whish in his paper were [[Yuktibhāṣā]] of [[Jyeshtadeva]], [[Karanapaddhati]] of [[Puthumana Somayaji]] and [[Sadratnamala]] of [[Sankara Varman]].

==Author and date of Tantrasamgraha==

{{main|Nilakantha Somayaji}}

[[Nilakantha Somayaji]], the author of Tantrasamgraha, was a [[Nambudiri]] belonging to the 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 yuga|Kali]] days, of the commencement and of the completion of book. These work out to dates in 1500-01.<ref name="text"/>

==Synopsis of the book==
A brief account of the contents of Tantrasamgraha is presented below.<ref name="Andrews"/> A descriptive account of the contents is available in Bharatheeya Vijnana/Sastra Dhara.<ref>{{cite book|author=N. Gopalakrishnan|title=Baharatheeya Vijnana / Sastra Dhaara ( Handbbok of Ancient Indian Scientific Books)|publisher=Indian Institute of Scientific Heritage|location=Thiruvanannthapuram, India|date=2004|series=Heritage Publication Series|volume=78|pages=18–20|url=http://www.books.user.janyug.com/Pradeesh/data/public/Culture/Ancient/Indian/251.pdf|access-date=12 January 2010}} {{dead link|date=August 2013}}</ref> Full details of the contents are available in an edition of Tantrasamgraha published in the ''Indian Journal of History of Science''.<ref name="text"/>
*''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."<ref name="Glen">{{cite book|author=Glen van Brummelen|title=The mathematics of the heavens and the earth: The early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=128–129|url=http://press.princeton.edu/titles/8956.html}}</ref>
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''.<ref>{{cite journal|author=Radaha Charan Gupta|title=Solution of the astronomical triangle as found in Tantrsasamgraha (A.D.1500)|journal=Indian Journal of History of Science|publisher=Indian National Academy of Science|volume=9|issue=1|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b61_86.pdf|access-date=18 January 2010|url-status=dead|archive-url=https://web.archive.org/web/20120309013400/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b61_86.pdf|archive-date=9 March 2012|df=dmy-all}}</ref> "The spherical triangle is handled as systematically here as in any modern textbook."<ref name="Glen"/>

The terrestrial [[latitude]] of an observer's position is equal to the [[zenith distance]] of the Sun at noon on the [[equinox|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.<ref>*{{cite journal|author=K. Ramasubramanian|author2=M.S. Sriram|name-list-style=amp|date=2003|title=Corrections to the terrestrial latitude in Tantrasamgraha|journal=Indian Journal of History of Science|volume=38|issue=2|pages=129–144|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20008276_129.pdf|access-date=18 January 2010|url-status=dead|archive-url=https://web.archive.org/web/20120309020242/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20008276_129.pdf|archive-date=9 March 2012|df=dmy-all}}</ref>

Tantrasamgraha contains a major revision of the older Indian planetary model for the interior planets [[Mercury (planet)|Mercury]] and [[Venus]] and, in the history of astronomy, the first accurate formulation of the equation of centre for these planets.<ref>*{{cite journal|author=K. Ramasubramanian|author2=M.D. Srinivas|author3=M.S. Sriram|name-list-style=amp|date=25 May 1994|title=Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion|journal=Current Science|volume=66|issue=10|pages=784–790|url=http://www.physics.iitm.ac.in/~labs/amp/kerala-astronomy.pdf|access-date=18 January 2010}}</ref> His planetary system was a partially [[heliocentrism|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 heliocentrism|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.<ref name="Joseph"/><ref>{{Cite journal|title=Model of planetary motion in the works of Kerala astronomers|last=Ramasubramanian|first=K.|journal=Bulletin of the Astronomical Society of India|volume=26|pages=11–31 [23–4]|bibcode=1998BASI...26...11R|year=1998}}</ref> Most astronomers of the Kerala school who followed him accepted his planetary model.<ref name=Joseph/><ref name=Srinivas>K. Ramasubramanian, M. D. Srinivas, M. S. Sriram (1994). "[http://www.physics.iitm.ac.in/~labs/amp/kerala-astronomy.pdf Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion]", ''[[Current Science]]'' '''66''', p. 784-790.</ref>

==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.<ref>{{cite journal|author=M.S. Sriram|date=25 July 2000|title=Meeting reports : Five hundred years of Tantrasangraha - A landmark in the history of astronomy|journal=Current Science|volume=79|issue=2|pages=150–151|url=http://www.ias.ac.in/currsci/jul252000/MEETING%20REPORTS.pdf|access-date=1 February 2010}}</ref>
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 Kerala 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.
<ref>{{cite book|author=M. S. Sriram|author2=K. Ramasubramanian|author3=M. D. Srinivas|name-list-style=amp |title=500 years of Tantrasangraha — A landmark in the history of astronomy|publisher=Inter-University Center, Indian Institute of Advanced Study|location=Shimla|date=2002|pages=185|isbn=81-7986-009-4}}{{cite web |url=http://www.iias.org/p-500years.html |title=Archived copy |access-date=2010-01-18 |url-status=dead |archive-url=https://web.archive.org/web/20100116190950/http://iias.org/p-500years.html |archive-date=16 January 2010 |df=dmy-all }}</ref>

==Other works of the same author==
The following is a brief description of the other works by Nilakantha Somayaji.<ref name="text"/>

*''[[Jyotirmimamsa]]''
*''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.

==References==
{{reflist}}

==Further reading==
Official Website : [https://chenthala.in/tantrasamgraham-tantra-sangraha-of-somayaji/ Tantrasamgraha of Neelakanta Somayaji]

*{{cite journal|author=Ramasubramanian, K|date=1998|title=Model of planetary motion in the works of Kerala astronomers|journal=Bulletin of the Astronomical Society of India|volume=26|issue=11–31|pages=11|bibcode=1998BASI...26...11R}}
*{{cite journal|doi=10.2307/2690896|last=Ranjan Roy|date=December 1990|first1=R.|title=The discovery of the series formula for π by Leibniz, Gregory and Nilakantha|journal=Mathematics Magazine|publisher=[[Mathematical Association of America]]|volume=63|issue=5|pages=291–306|url=http://mathdl.maa.org/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf|access-date=18 January 2010|jstor=2690896}}

{{Indian mathematics}}
{{Indian astronomy}}
{{Scientific Research in Kerala |state=collapsed}}

[[Category:Kerala school of astronomy and mathematics]]
[[Category:Hindu astronomy]]
[[Category:Astronomy books]]
[[Category:Indian mathematics]]
[[Category:Hindu astrological texts]]
[[Category:1501 books]]
[[Category:Indian astronomy texts]]

Tantrasamgraha - Revision history

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imported>SdkbBot: /* Some noteworthy features of Tantrasamgraha */removed erroneous space