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{{Short description|Indian mathematician-astronomer (476–550)}}
{{Other uses}}
{{pp-vandalism|small=yes}}
{{Use dmy dates|date=September 2019}}
{{Infobox scholar
| image = [[File:Aryabhata.jpeg|Aryabhata]]
| caption = Illustration of Āryabhaṭa
| name = Āryabhaṭa
| fullname = 
| birth_date = 476 CE
| birth_place = Kusumapura / [[Pataliputra]],<br/>[[Gupta Empire]]<br/>(present-day [[Patna]], [[Bihar]], [[India]])<ref name="Bhau">{{cite book | date=1865 | contribution=Brief Notes on the Age and Authenticity of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya | title=Journal of the Royal Asiatic Society of Great Britain and Ireland | author=Bhau Daji | pages=392–406 | url=https://books.google.com/books?id=fAsFAAAAMAAJ&pg=PA392| author-link=Bhau Daji }}</ref>
| death_date = 550 CE (aged 73–74) <ref>{{cite book | last=Singh | first=J. | title=Sterling Dictionary of Physics | publisher=Sterling Publishers Private Limited | year=1999 | isbn=978-81-7359-124-2 | url=https://books.google.com/books?id=eKnrhryjqn0C&pg=PA12 | access-date=2023-04-15 | page=12}}</ref>
| death_place = 
| era = [[Gupta period|Gupta era]]
| main_interests = [[Mathematics]], [[astronomy]]
| notable_ideas = Explanation of [[lunar eclipse]] and [[solar eclipse]], [[Earth's rotation|rotation of Earth on its axis]], [[Moonlight|reflection of light by the Moon]], [[Āryabhaṭa's sine table|sinusoidal functions]], [[Quadratic equation|solution of single variable quadratic equation]], [[Approximations of π|value of π correct to 4 decimal places]], diameter of [[Earth]], calculation of the length of [[sidereal year]]
| major_works = [[Āryabhaṭīya]], Arya-[[siddhanta]]
| influences = [[Surya Siddhanta]]
| influenced = [[Lalla]], [[Bhaskara I]], [[Brahmagupta]], [[Varahamihira]]
}}

'''Aryabhata''' ( [[ISO 15919|ISO]]: {{transliteration|sa|ISO|Āryabhaṭa}}) or '''Aryabhata I'''<ref name="Aryabhata the Elder">{{cite web |last1=O'Connor |first1=J J |last2=Robertson |first2=E F |title=Aryabhata the Elder |publisher=www-history.mcs.st-andrews.ac.uk |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html |access-date=18 July 2012 |url-status=live |archive-url=https://web.archive.org/web/20150711055702/http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html |archive-date=11 July 2015  }}</ref><ref name="Publishing2010">{{cite book|author=Britannica Educational Publishing|title=The Britannica Guide to Numbers and Measurement|url=https://books.google.com/books?id=cuN7rH6RzikC&pg=PA97|date=15 August 2010|publisher=The Rosen Publishing Group|isbn=978-1-61530-218-5|pages=97–}}</ref> (476–550 [[Common Era|CE]])<ref name="Ray2009">{{cite book|author=Bharati Ray|title=Different Types of History|url=https://books.google.com/books?id=9x5FX2RROZgC&pg=PA95|date=1 September 2009|publisher=Pearson Education India|isbn=978-81-317-1818-6|pages=95–}}</ref><ref name="Yadav2010">{{cite book|author=B. S. Yadav|title=Ancient Indian Leaps into Mathematics|url=https://books.google.com/books?id=nwrw0Lv1vXIC&pg=PA88|date=28 October 2010|publisher=Springer|isbn=978-0-8176-4694-3|page=88}}</ref> was the first of the major [[mathematician]]-[[astronomer]]s from the classical age of [[Indian mathematics]] and [[Indian astronomy]]. His works include the ''[[Āryabhaṭīya]]'' (which mentions that in 3600 ''[[Kali Yuga]]'', 499&nbsp;CE, he was 23 years old)<ref name="Roupp1997">{{cite book|author=Heidi Roupp|title=Teaching World History: A Resource Book|url=https://books.google.com/books?id=-UYag6dzk7YC&pg=PA112|date=1997|publisher=M.E. Sharpe|isbn=978-1-56324-420-9|pages=112–}}</ref> and the ''Arya-[[siddhanta]]''.

For his explicit mention of the relativity of motion, he also qualifies as a major early physicist.<ref name="encyclopedia.com">{{Cite web |title=Aryabhatiya |url=https://www.encyclopedia.com/international/encyclopedias-almanacs-transcripts-and-maps/aryabhatiya |access-date=2024-06-20 |website=[[Encyclopedia.com]]}}</ref>

==Biography==
===Name===
While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "[[bhatta]]" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus,<ref name="sarma">{{Cite journal | author=K. V. Sarma | journal=Indian Journal of History of Science | date=2001 | pages=105–115 | title=Āryabhaṭa: His name, time and provenance | volume=36 | issue=4 | url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b67_105.pdf | archive-url=https://web.archive.org/web/20100331152303/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b67_105.pdf | archive-date=31 March 2010| author-link=K. V. Sarma }}</ref> including [[Brahmagupta]]'s references to him "in more than a hundred places by name".<ref name="Bhau" /> Furthermore, in most instances "Aryabhatta" would not fit the metre either.<ref name=sarma/>

===Time and place of birth===
Aryabhata mentions in the ''Aryabhatiya'' that he was 23 years old 3,600 years into the ''[[Kali Yuga]]'', but this is not to mean that the text was composed at that time. This mentioned year corresponds to 499&nbsp;CE, and implies that he was born in 476.<ref name=Yadav2010 /> Aryabhata called himself a native of Kusumapura or [[Pataliputra]] (present day [[Patna]], [[Bihar]]).<ref name="Bhau"/>

====Other hypothesis====
[[Bhāskara I]] describes Aryabhata as ''āśmakīya'', "one belonging to the ''[[Aśmaka]]'' country."  During the Buddha's time, a branch of the Aśmaka people settled in the region between the [[Narmada River|Narmada]] and [[Godavari]] rivers in central India.<ref name=sarma/><ref name="Ansari"/>

It has been claimed that the ''aśmaka'' (Sanskrit for "stone") where Aryabhata originated may be the present day [[Kodungallur]] which was the historical capital city of ''Thiruvanchikkulam'' of ancient Kerala.<ref name="Menon">{{cite book|author=Menon|title=An Introduction to the History and Philosophy of Science|url=https://books.google.com/books?id=qi5Mcrm613oC&pg=PA52|publisher=Pearson Education India|isbn=978-81-317-2890-1|page=52|year=2009}}</ref> This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala has been used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.<ref name=sarma/> K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence.<ref>{{citation | newspaper = [[The Hindu]] | url = http://www.hindu.com/2007/06/25/stories/2007062558250400.htm | title = Aryabhata lived in Ponnani? | date = 25 June 2007 | author = Radhakrishnan Kuttoor | url-status = dead | archive-url = https://web.archive.org/web/20070701212700/http://www.hindu.com/2007/06/25/stories/2007062558250400.htm | archive-date = 1 July 2007 | df = dmy-all }}</ref>

Aryabhata mentions "Lanka" on several occasions in the ''Aryabhatiya'', but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his [[Ujjayini]].<ref>See:<br/> *{{Harvnb|Clark|1930}}<br/> *{{Cite book | date=2000 | title = Indian Astronomy: An Introduction | author1=S. Balachandra Rao | publisher=Orient Blackswan | isbn=978-81-7371-205-0 | page=82 | url=https://books.google.com/books?id=N3DE3GAyqcEC&q=lanka&pg=PA82}}: "In Indian astronomy, the prime meridian is the great circle of the Earth passing through the north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on the Earth's equator."<br/>*{{Cite book | date=2003 | title = Ancient Indian Astronomy | author1=L. Satpathy | publisher=Alpha Science Int'l Ltd. | isbn=978-81-7319-432-0 | page=200 | url=https://books.google.com/books?id=nh6jgEEqqkkC&q=lanka&pg=PA200}}: "Seven cardinal points are then defined on the equator, one of them called Laṅkā, at the intersection of the equator with the meridional line through Ujjaini. This Laṅkā is, of course, a fanciful name and has nothing to do with the island of Sri Laṅkā."<br/>*{{Cite book | title = Classical Muhurta | author1=Ernst Wilhelm | publisher=Kala Occult Publishers | isbn=978-0-9709636-2-8 | page=44 | url=https://books.google.com/books?id=3zMPFJy6YygC&q=lanka&pg=PA44}}: "The point on the equator that is below the city of Ujjain is known, according to the Siddhantas, as Lanka. (This is not the Lanka that is now known as Sri Lanka; Aryabhata is very clear in stating that Lanka is 23 degrees south of Ujjain.)"<br/>*{{Cite book | date=2006 | title = Pride of India: A Glimpse into India's Scientific Heritage | author1=R.M. Pujari | author2= Pradeep Kolhe | author3= N. R. Kumar | publisher=SAMSKRITA BHARATI | isbn=978-81-87276-27-2 | page=63 | url=https://books.google.com/books?id=sEX11ZyjLpYC&q=lanka&pg=PA63}}<br/>*{{Cite book | date=1989 | title = The Surya Siddhanta: A Textbook of Hindu Astronomy | author1=Ebenezer Burgess | author2= Phanindralal Gangooly | publisher=Motilal Banarsidass Publ. | isbn=978-81-208-0612-2 | page=46 | url=https://books.google.com/books?id=W0Uo_-_iizwC&q=lanka&pg=PA46}}</ref>

===Education===
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time.<ref>{{cite book|last=Cooke|author-link=Roger Cooke (mathematician)|title=History of Mathematics: A Brief Course |url=https://archive.org/details/historyofmathema0000cook|url-access=registration|date=1997|chapter=''The Mathematics of the Hindus''|page=[https://archive.org/details/historyofmathema0000cook/page/204 204]|publisher=Wiley |isbn=9780471180821 |quote=Aryabhata himself (one of at least two mathematicians bearing that name) lived in the late 5th and the early 6th centuries at [[Kusumapura]] ([[Pataliutra]], a village near the city of Patna) and wrote a book called ''Aryabhatiya''.}}</ref> Both Hindu and Buddhist tradition, as well as [[Bhāskara I]] (CE 629), identify Kusumapura as [[Pāṭaliputra]], modern [[Patna]].<ref name=sarma/> A verse mentions that Aryabhata was the head of an institution (''{{IAST|kulapa}}'')<!--NOT "kulapati", see source--> at Kusumapura, and, because the university of [[Nalanda]] was in Pataliputra at the time, it is speculated that Aryabhata might have been the head of the Nalanda university as well.<ref name=sarma/> Aryabhata is also reputed to have set up an observatory at the Sun temple in [[Taregana]], Bihar.<ref>{{cite web|url=http://ncsm.gov.in/docs/Get%20ready%20for%20Solar%20eclipse.pdf |title=Get ready for solar eclipse |publisher=National Council of Science Museums, Ministry of Culture, Government of India |access-date=9 December 2009 |url-status=dead |archive-url=https://web.archive.org/web/20110721162632/http://ncsm.gov.in/docs/Get%20ready%20for%20Solar%20eclipse.pdf |archive-date=21 July 2011 }}</ref>

==Works==
Aryabhata is the author of several treatises on [[mathematics]] and [[astronomy]], though ''[[Aryabhatiya]]'' is the only one which survives.<ref>{{Citation |last=Elgarøy |first=Øystein |title=Aryabhata |date=2024-06-18 |work=[[Store norske leksikon]] |url=https://snl.no/Aryabhata |access-date=2024-06-20 |language=no}}.</ref>

Much of the research included subjects in astronomy, mathematics, physics, biology, medicine,  and other fields.<ref name=":1" /> ''Aryabhatiya'', a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times.<ref name=":0" /> The mathematical part of the ''Aryabhatiya'' covers [[arithmetic]], [[algebra]], [[Trigonometry|plane trigonometry]], and [[spherical trigonometry]]. It also contains [[continued fraction]]s, [[quadratic equation]]s, sums-of-[[power series]], and a [[Aryabhata's sine table|table of sines]].<ref name=":0" />

The ''Arya-siddhanta'', a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, [[Varahamihira]], and later mathematicians and commentators, including [[Brahmagupta]] and [[Bhaskara I]]. This work appears to be based on the older [[Surya Siddhanta]] and uses the midnight-day reckoning, as opposed to sunrise in ''Aryabhatiya''.<ref name="Ansari" /> It also contained a description of several astronomical instruments: the [[gnomon]] (''shanku-yantra''), a shadow instrument (''chhAyA-yantra''), possibly angle-measuring devices, semicircular and circular (''dhanur-yantra'' / ''chakra-yantra''), a cylindrical stick ''yasti-yantra'', an umbrella-shaped device called the ''chhatra-yantra'', and [[water clock]]s of at least two types, bow-shaped and cylindrical.<ref name = Ansari>
{{cite journal
 |last=Ansari
 |first=S.M.R.
 |date=March 1977
 |title=Aryabhata I, His Life and His Contributions
 |journal=Bulletin of the Astronomical Society of India
 |volume=5
 |issue=1
 |pages=10–18
 |bibcode = 1977BASI....5...10A |hdl=2248/502
 }}</ref>

A third text, which may have survived in the [[Arabic language|Arabic]] translation, is ''Al ntf'' or ''Al-nanf''. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the [[Persian people|Persian]] scholar and chronicler of India, [[Abū Rayhān al-Bīrūnī]].<ref name = Ansari/>

===Aryabhatiya===

{{Main|Aryabhatiya}}
Direct details of Aryabhata's work are known only from the ''Aryabhatiya''. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name.<ref name="encyclopedia.com" /> His disciple [[Bhaskara I]] calls it ''Ashmakatantra'' (or the treatise from the Ashmaka). It is also occasionally referred to as ''Arya-shatas-aShTa'' (literally, Aryabhata's 108), because there are 108 verses in the text.<ref name=":0">{{Cite web |title=Aryabhata - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Aryabhata_I/ |access-date=2024-06-20 |website=[[Maths History]] |publisher=[[University of St. Andrews]] |language=en}}</ref><ref name="encyclopedia.com" /> It is written in the very terse style typical of [[sutra]] literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four ''pāda''s or chapters:

# ''Gitikapada'': (13 verses): large units of time—''kalpa'', ''manvantra'', and ''yuga''—which present a cosmology different from earlier texts such as Lagadha's ''[[Vedanga Jyotisha]]'' (c. 1st century BCE). There is also a table of sines (''[[jya]]''), given in a single verse. The duration of the planetary revolutions during a ''mahayuga'' is given as 4.32 million years.
# ''Ganitapada'' (33 verses): covering [[mensuration (mathematics)|mensuration]] (''kṣetra vyāvahāra''), arithmetic and geometric progressions, [[gnomon]] / shadows (''shanku''-''chhAyA''), simple, [[quadratic equations|quadratic]], [[simultaneous equations|simultaneous]], and [[diophantine equations|indeterminate]] equations (''kuṭṭaka'').<ref name=":1">{{Cite web |title=આર્યભટ્ટ |url=https://gujarativishwakosh.org/%E0%AA%86%E0%AA%B0%E0%AB%8D%E0%AA%AF%E0%AA%AD%E0%AA%9F%E0%AB%8D%E0%AA%9F |access-date=2024-06-20 |website=[[Gujarati Vishwakosh]] |language=en-GB}}</ref>
# ''Kalakriyapada'' (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (''adhikamAsa''), ''kShaya-tithi''s, and a seven-day week with names for the days of week.<ref name=":1" />
# ''Golapada'' (50 verses): Geometric/[[trigonometric]] aspects of the [[celestial sphere]], features of the [[ecliptic]], [[celestial equator]], node, shape of the earth, cause of day and night, rising of [[zodiacal sign]]s on horizon, etc.<ref name=":1" /> In addition, some versions cite a few [[colophon (publishing)|colophons]] added at the end, extolling the virtues of the work, etc.<ref name=":1" />

The ''Aryabhatiya'' presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (''Bhashya'', c.&nbsp;600&nbsp;CE) and by [[Nilakantha Somayaji]] in his ''Aryabhatiya Bhasya'' (1465&nbsp;CE).<ref name=":0" /><ref name=":1" />

''Aryabhatiya'' is also well-known for his description of relativity of motion. He expressed this relativity thus: "Just as a man in a boat moving forward sees the stationary objects (on the shore) as moving backward, just so are the stationary stars seen by the people on earth as moving exactly towards the west."<ref name="encyclopedia.com"/>

==Mathematics==

===Place value system and zero===
The [[place-value]] system, first seen in the 3rd-century [[Bakhshali Manuscript]], was clearly in place in his work. While he did not use a symbol for [[zero]], the French mathematician [[Georges Ifrah]] argues that knowledge of zero was implicit in Aryabhata's [[place-value system]] as a place holder for the powers of ten with [[Null (mathematics)|null]] [[coefficients]].<ref>{{cite book
 | author = George. Ifrah
 | title = A Universal History of Numbers: From Prehistory to the Invention of the Computer
 | publisher = John Wiley & Sons
 | location = London
 | date = 1998
}}</ref>

However, Aryabhata did not use the Brahmi numerals. Continuing the [[Sanskrit]]ic tradition from [[Vedic period|Vedic times]], he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a [[mnemonic]] form.<ref>
{{Cite book
 | last1 = Dutta
 | given1 = Bibhutibhushan
 | surname2 = Singh
 | given2 = Avadhesh Narayan
 | date = 1962
 | title = History of Hindu Mathematics
 | publisher = Asia Publishing House, Bombay
 | isbn = 81-86050-86-8
}}</ref>

===Approximation of ''π''===
Aryabhata worked on the approximation for [[pi]] (π), and may have come to the conclusion that π is irrational. In the second part of the ''Aryabhatiyam'' ({{IAST|gaṇitapāda}} 10), he writes:
<blockquote>
''{{IAST|caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām}}'' <br/>
''{{IAST|ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.}}''<br />
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."<ref>
{{cite book
 |title= Geometry: Seeing, Doing, Understanding
 |last= Jacobs
 |first= Harold R.
 |date= 2003
 |publisher= W.H. Freeman and Company
 |location= New York
 |isbn= 0-7167-4361-2
 |page= 70|edition= Third
 }}</ref></blockquote>
This implies that for a circle whose diameter is 20000, the circumference will be 62832

i.e,   <math>\pi</math> = <math>62832 \over 20000 </math> =  <math>3.1416</math>, which is accurate to two parts in one million.<ref name="aryabhattapi"/>

It is speculated that Aryabhata used the word ''āsanna'' (approaching), to mean that not only is this an approximation but that the value is incommensurable (or [[irrational]]). If this is correct, it is quite a sophisticated insight, because the irrationality of pi (π) was proved in Europe only in 1761 by [[Johann Heinrich Lambert|Lambert]].<ref>
{{cite book
 | author = S. Balachandra Rao
 | title = Indian Mathematics and Astronomy: Some Landmarks
 | publisher = Jnana Deep Publications
 | orig-year=First published 1994
 | date = 1998
 | location = Bangalore
 | isbn = 81-7371-205-0
}}</ref>

After Aryabhatiya was translated into [[Arabic language|Arabic]] (c.&nbsp;820&nbsp;CE), this approximation was mentioned in [[Al-Khwarizmi]]'s book on algebra.<ref name = Ansari/>

===Trigonometry===
In Ganitapada 6, Aryabhata gives the area of a triangle as
: ''{{IAST|tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ}}''
that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."<ref>{{Cite book
 | author = Roger Cooke
 | title = History of Mathematics: A Brief Course
 | publisher = Wiley-Interscience
 | date = 1997
 | chapter = The Mathematics of the Hindus
 | isbn = 0-471-18082-3
 | quote = Aryabhata gave the correct rule for the area of a triangle and an incorrect rule for the volume of a pyramid. (He claimed that the volume was half the height times the area of the base.)
 | chapter-url = https://archive.org/details/historyofmathema0000cook
 | url-access = registration
 | url = https://archive.org/details/historyofmathema0000cook
 }}</ref>

Aryabhata discussed the concept of ''[[sine]]'' in his work by the name of ''[[ardha-jya]]'', which literally means "half-chord". For simplicity, people started calling it ''[[jya]]''. When Arabic writers translated his works from [[Sanskrit]] into Arabic, they referred it as ''jiba''. However, in Arabic writings, vowels are omitted, and it was abbreviated as ''jb''. Later writers substituted it with ''jaib'', meaning "pocket" or "fold (in a garment)". (In Arabic, ''jiba'' is a meaningless word.) Later in the 12th century, when [[Gherardo of Cremona]] translated these writings from Arabic into Latin, he replaced the Arabic ''jaib'' with its Latin counterpart, ''sinus'', which means "cove" or "bay"; thence comes the English word ''sine''.<ref>{{Cite book
 | author = Howard Eves
 | title = An Introduction to the History of Mathematics
 | publisher = Saunders College Publishing House, New York
 | date = 1990
 | edition = 6
 | page= 237
}}</ref>

===Indeterminate equations===
A problem of great interest to [[Indian mathematicians]] since ancient times has been to find integer solutions to [[Diophantine equations]] that have the form ax + by = c. (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as the [[Chinese remainder theorem]].) This is an example from [[Bhāskara I|Bhāskara]]'s commentary on Aryabhatiya:
: Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7
That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text [[Sulba Sutras]], whose more ancient parts might date to 800&nbsp;BCE. Aryabhata's method of solving such problems, elaborated by Bhaskara in 621&nbsp;CE, is called the ''{{IAST|kuṭṭaka}}'' (कुट्टक) method. ''[[Kuṭṭaka]]'' means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, and initially the whole subject of algebra was called ''kuṭṭaka-gaṇita'' or simply ''kuṭṭaka''.<ref>
Amartya K Dutta, [http://www.ias.ac.in/resonance/Volumes/07/10/0006-0022.pdf   "Diophantine equations: The Kuttaka"] {{webarchive|url=https://web.archive.org/web/20141102224109/http://www.ias.ac.in/resonance/Volumes/07/10/0006-0022.pdf | date = 2 November 2014}}, ''Resonance'', October 2002. Also see earlier overview: [http://www.ias.ac.in/resonance/Volumes/07/04/0004-0019.pdf ''Mathematics in Ancient India''] {{webarchive|url=https://web.archive.org/web/20141102223752/http://www.ias.ac.in/resonance/Volumes/07/04/0004-0019.pdf |date=2 November 2014 }}.</ref>

===Algebra===
In ''Aryabhatiya'', Aryabhata provided elegant results for the summation of [[series (mathematics)|series]] of squares and cubes:<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |date=1991 |isbn=0-471-54397-7 |page=[https://archive.org/details/historyofmathema00boye/page/207 207] |chapter=The Mathematics of the Hindus |quote=He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes. |chapter-url=https://archive.org/details/historyofmathema00boye |url=https://archive.org/details/historyofmathema00boye/page/207 }}</ref>
:<math>1^2 + 2^2 + \cdots + n^2 = {n(n + 1)(2n + 1) \over 6}</math>
and
:<math>1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2</math> (see [[squared triangular number]])

==Astronomy==
Aryabhata's system of astronomy was called the ''audAyaka system'', in which days are reckoned from ''uday'', dawn at ''lanka'' or "equator". Some of his later writings on astronomy, which apparently proposed a second model (or ''ardha-rAtrikA'', midnight) are lost but can be partly reconstructed from the discussion in [[Brahmagupta]]'s ''[[Khandakhadyaka]]''. In some texts, he seems to ascribe the apparent motions of the heavens to the [[Earth's rotation]]. He may have believed that the planet's orbits are [[Ellipse|elliptical]] rather than circular.<ref>J. J. O'Connor and E. F. Robertson, [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Aryabhata_I.html Aryabhata the Elder] {{webarchive|url=https://web.archive.org/web/20121019181214/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Aryabhata_I.html |date=19 October 2012 }}, [[MacTutor History of Mathematics archive]]:
<br/>{{blockquote|"He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses."}}</ref><ref name=Hayashi08Aryabhata>Hayashi (2008), ''Aryabhata I''</ref>

===Motions of the Solar System===
Aryabhata correctly insisted that the Earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the Earth, contrary to the then-prevailing view, that the sky rotated.<ref name="aryabhattapi">[http://www.livemint.com/Sundayapp/8wRiLexg1N2IOXjeK2BKcL/How-Aryabhata-got-the-earths-circumference-right-millenia-a.html How Aryabhata got the earth's circumference right] {{webarchive|url=https://web.archive.org/web/20170115063654/http://www.livemint.com/Sundayapp/8wRiLexg1N2IOXjeK2BKcL/How-Aryabhata-got-the-earths-circumference-right-millenia-a.html |date=15 January 2017 }}</ref> This is indicated in the first chapter of the ''Aryabhatiya'', where he gives the number of rotations of the Earth in a ''yuga'',<ref>Aryabhatiya 1.3ab, see Plofker 2009, p. 111.</ref> and made more explicit in his ''gola'' chapter:<ref>[''achalAni bhAni samapashchimagAni&nbsp;...'' – golapAda.9–10]. Translation from K. S. Shukla and K.V. Sarma, K. V. ''Āryabhaṭīya of Āryabhaṭa'', New Delhi: Indian National Science Academy, 1976. Quoted in Plofker 2009.</ref>
{{Blockquote|In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at the equator, constantly pushed by the [[cosmic wind]].}}

Aryabhata described a [[geocentric]] model of the Solar System, in which the
Sun and Moon are each carried by [[epicycle]]s. They in turn revolve around the Earth. In this model, which is also found in the ''Paitāmahasiddhānta'' (c.&nbsp;425&nbsp;CE), the motions of the planets are each governed by two epicycles, a smaller ''manda'' (slow) and a larger ''śīghra'' (fast).<ref>
{{Cite book
 | last = Pingree
 | first = David
 | author-link = David Pingree
 | contribution = Astronomy in India
 | editor-last = Walker
 | editor-first = Christopher
 | title = Astronomy before the Telescope
 | pages = 123–142
 | publisher = British Museum Press
 | place = London
 | date = 1996
 | isbn = 0-7141-1746-3
 }} pp. 127–9.</ref> The order of the planets in terms of distance from earth is taken as: the [[Moon]], [[Mercury (planet)|Mercury]], [[Venus]], the [[Sun]], [[Mars]], [[Jupiter]], [[Saturn]], and the [[Asterism (astronomy)|asterisms]].<ref name=Ansari/>

The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac. Most historians of astronomy consider that this two-epicycle model reflects elements of pre-Ptolemaic [[Hellenistic astronomy|Greek astronomy]].<ref>Otto Neugebauer, "The Transmission of Planetary Theories in Ancient and Medieval Astronomy," ''[[Scripta Mathematica]]'', 22 (1956), pp. 165–192; reprinted in Otto Neugebauer, ''Astronomy and History: Selected Essays,'' New York: Springer-Verlag, 1983, pp. 129–156. {{ISBN|0-387-90844-7}}</ref> Another element in Aryabhata's model, the ''śīghrocca'', the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying [[heliocentric]] model.<ref>Hugh Thurston, ''Early Astronomy'', New York: Springer-Verlag, 1996, pp. 178–189. {{ISBN|0-387-94822-8}}</ref>

===Eclipses===
Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the [[Moon]] and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by [[Rahu]] and [[Ketu (mythology)|Ketu]] (identified as the pseudo-planetary [[lunar nodes]]), he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the Moon enters into the Earth's shadow (verse gola.37). He discusses at length the size and extent of the Earth's shadow (verses gola.38–48) and then provides the computation and the size of the eclipsed part during an eclipse. Later Indian astronomers improved on the calculations, but Aryabhata's methods provided the core. His computational paradigm was so accurate that 18th-century scientist [[Guillaume Le Gentil]], during a visit to Pondicherry, India, found the Indian computations of the duration of the [[lunar eclipse]] of 30&nbsp;August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.<ref name=Ansari/>

===Sidereal periods===
Considered in modern English units of time, Aryabhata calculated the [[sidereal rotation]] (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds;<ref name="Selin1997">{{cite book|editor=Helaine Selin|editor-link=Helaine Selin|author=R.C.Gupta|contribution=Āryabhaṭa|title=Encyclopaedia of the history of science, technology, and medicine in non-western cultures|url=https://books.google.com/books?id=raKRY3KQspsC&pg=PA72|date=31 July 1997|publisher=Springer|isbn=978-0-7923-4066-9|page=72}}</ref> the modern value is 23:56:4.091. Similarly, his value for the length of the [[sidereal year]] at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days)<ref>Ansari, p. 13, Table 1</ref> is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).<ref>''Aryabhatiya {{langx|mr|आर्यभटीय}}'', Mohan Apte, Pune, India, Rajhans Publications, 2009, p.25, {{ISBN|978-81-7434-480-9}}</ref>

===Heliocentrism===
As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (the ''śīgra'' anomaly) for the speeds of the planets in the sky in terms of the mean speed of the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an underlying [[heliocentrism|heliocentric]] model, in which the planets orbit the Sun,<ref>The concept of Indian heliocentrism has been advocated by B. L. van der Waerden, ''Das heliozentrische System in der griechischen, persischen und indischen Astronomie.'' Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman AG, 1970.</ref><ref>B.L. van der Waerden, "The Heliocentric System in Greek, Persian and Hindu Astronomy", in David A. King and George Saliba, ed., ''From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E. S. Kennedy'', Annals of the New York Academy of Science, 500 (1987), pp. 529–534.</ref><ref>{{Cite book|title=Early Astronomy|author=Hugh Thurston|publisher=[[Springer Science+Business Media|Springer]]|date=1996|isbn=0-387-94822-8|page=188}}</ref> though this has been rebutted.<ref>Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy," ''Isis'', 64 (1973): 239–243.</ref> It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-Ptolemaic [[Greek astronomy|Greek]], heliocentric model of which Indian astronomers were unaware,<ref>Though [[Aristarchus of Samos]] (3rd century BCE) is credited with holding an heliocentric theory, the version of [[Greek astronomy]] known in ancient India as the ''[[Paulisa Siddhanta]]'' makes no reference to such a theory.</ref> though the evidence is scant.<ref>Dennis Duke, "The Equant in India: The Mathematical Basis of Ancient Indian Planetary Models." [[Archive for History of Exact Sciences]] 59 (2005): 563–576, n. 4 {{cite web |url=http://people.scs.fsu.edu/~dduke/india8.pdf |title=Archived copy |access-date=8 February 2016 |url-status=live |archive-url=https://web.archive.org/web/20090318024632/http://people.scs.fsu.edu/~dduke/india8.pdf |archive-date=18 March 2009  }}.</ref> The general consensus is that a synodic anomaly (depending on the position of the Sun) does not imply a physically heliocentric orbit (such corrections being also present in late [[Babylonian astronomical diaries|Babylonian astronomical texts]]), and that Aryabhata's system was not explicitly heliocentric.<ref>{{cite book|last=Kim Plofker|title=Mathematics in India|title-link=Mathematics in India|publisher=Princeton University Press|location=Princeton, NJ|date=2009|page=[https://archive.org/details/mathematicsindia00plof/page/n125 111]|isbn=978-0-691-12067-6}}</ref>

==Legacy==
[[File:Aryabhata Satellite.jpg|thumb|300px|[[Aryabhata (satellite)|India's first satellite]] named after Aryabhata]]
{{more citations needed section|date=March 2017}}

Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The [[Arabic]] translation during the [[Islamic Golden Age]] (c.&nbsp;820&nbsp;CE), was particularly influential. Some of his results are cited by [[Al-Khwarizmi]] and in the 10th century [[Al-Biruni]] stated that Aryabhata's followers believed that the Earth rotated on its axis.

His definitions of [[sine]] (''[[jya]]''), cosine (''[[kojya]]''), versine (''[[utkrama-jya]]''),
and inverse sine (''otkram jya'') influenced the birth of [[trigonometry]]. He was also the first to specify sine and [[versine]] (1&nbsp;−&nbsp;cos&nbsp;''x'') tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, the modern terms "sine" and "cosine" are mistranscriptions of the words ''jya'' and ''kojya'' as introduced by Aryabhata. As mentioned, they were translated as ''jiba'' and ''kojiba'' in Arabic and then misunderstood by [[Gerard of Cremona]] while translating an Arabic geometry text to [[Latin]]. He assumed that ''jiba'' was the Arabic word ''jaib'', which means "fold in a garment", L. ''sinus'' (c. 1150).<ref>{{cite web |title = Online Etymology Dictionary |url = http://www.etymonline.com/ |author = Douglas Harper|date = 2001|access-date = 14 July 2007| archive-url= https://web.archive.org/web/20070713125946/http://www.etymonline.com/| archive-date= 13 July 2007 | url-status= live}}</ref>

Aryabhata's astronomical calculation methods were also very influential.
Along with the trigonometric tables, they came to be widely used in the Islamic world and used to compute many Arabic astronomical tables ([[zij]]es). In particular, the astronomical tables in the work of the [[Al-Andalus|Arabic Spain]] scientist [[Al-Zarqali]] (11th century) were translated into Latin as the [[Tables of Toledo]] (12th century) and remained the most accurate [[ephemeris]] used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing the [[Panchangam]] (the [[Hindu calendar]]). In the Islamic world, they formed the basis of the [[Jalali calendar]] introduced in 1073&nbsp;CE by a group of astronomers including [[Omar Khayyam]],<ref>{{cite encyclopedia|title=Omar Khayyam |encyclopedia=The Columbia Encyclopedia |date=May 2001 |edition=6 |url=http://www.bartleby.com/65/om/OmarKhay.html |access-date=10 June 2007 |url-status=dead |archive-url=https://web.archive.org/web/20071017002631/http://www.bartleby.com/65/om/OmarKhay.html |archive-date=17 October 2007 }}</ref> versions of which (modified in 1925) are the national calendars in use in [[Iran]] and [[Afghanistan]] today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlier [[Siddhanta]] calendars. This type of calendar requires an ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the [[Gregorian calendar]].{{citation needed|date=January 2020}}

[[Aryabhatta Knowledge University]] (AKU), Patna has been established by Government of Bihar for the development and management of educational infrastructure related to technical, medical, management and allied professional education in his honour.  The university is governed by Bihar State University Act 2008.

India's first satellite [[Aryabhata (satellite)|Aryabhata]] and the [[lunar crater]] [[Aryabhata (crater)|Aryabhata]] are both named in his honour, the Aryabhata satellite also featured on the reverse of the [[Indian 2-rupee note]]. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the [[Aryabhatta Research Institute of Observational Sciences]] (ARIES) near Nainital, India. The inter-school [[Aryabhata Maths Competition]] is also named after him,<ref>{{cite news |title=Maths can be fun |url=http://www.hindu.com/yw/2006/02/03/stories/2006020304520600.htm |work=[[The Hindu]] |date=3 February 2006 |access-date=6 July 2007 |url-status=dead |archive-url=https://web.archive.org/web/20071001091954/http://www.hindu.com/yw/2006/02/03/stories/2006020304520600.htm |archive-date=1 October 2007  }}</ref> as is ''Bacillus aryabhata'', a species of bacteria discovered in the [[stratosphere]] by [[ISRO]] scientists in 2009.<ref>{{cite web|url=https://www.sciencedaily.com/releases/2009/03/090318094642.htm|title=New Microorganisms Discovered in Earth's Stratosphere|publisher=ScienceDaily|date=18 March 2009|url-status=live|archive-url=https://web.archive.org/web/20180401232530/https://www.sciencedaily.com/releases/2009/03/090318094642.htm|archive-date=1 April 2018}}</ref><ref name="ISRO Press Release 16 March 2009">{{cite web|title=ISRO Press Release 16 March 2009 |url=http://www.isro.org/pressrelease/scripts/pressreleasein.aspx?Mar16_2009 |publisher=ISRO |access-date=24 June 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120105065022/http://isro.org/pressrelease/scripts/pressreleasein.aspx?Mar16_2009 |archive-date=5 January 2012 }}</ref>

==See also==
* {{IAST|[[Āryabhaṭa numeration]]}}
* {{IAST|[[Āryabhaṭa's sine table]]}}
* [[Indian mathematics]]
* [[List of Indian mathematicians]]

==References==
{{Reflist|30em}}

===Works cited===
* {{cite book
 | first=Roger
 | last=Cooke
 | title=The History of Mathematics: A Brief Course
 | publisher=Wiley-Interscience
 | date=1997
 | isbn=0-471-18082-3
 | url=https://archive.org/details/historyofmathema0000cook
 }}
* {{Cite book
 | title = The {{IAST|Āryabhaṭīya}} of {{IAST|Āryabhaṭa}}: An Ancient Indian Work on Mathematics and Astronomy
 | last=Clark | first=Walter Eugene |author-link=Walter Eugene Clark
 | date=1930
 | publisher=University of Chicago Press; reprint: Kessinger Publishing (2006)
 | isbn=978-1-4254-8599-3
 | url=https://archive.org/details/The_Aryabhatiya_of_Aryabhata_Clark_1930
 }}
* [[Subhash Kak|Kak, Subhash C.]] (2000). 'Birth and Early Development of Indian Astronomy'. In {{Cite encyclopedia
 | editor-last= Selin
 | editor-first = Helaine
 | editor-link = Helaine Selin
 | date = 2000
 | title = Astronomy Across Cultures: The History of Non-Western Astronomy
 | publisher = Boston: Kluwer
 | isbn = 0-7923-6363-9
}}
* Shukla, Kripa Shankar. ''Aryabhata: Indian Mathematician and Astronomer.'' New Delhi: Indian National Science Academy, 1976.
* {{Cite book
 | last1 = Thurston
 | first1 = H.
 | date = 1994
 | title = Early Astronomy
 | publisher = Springer-Verlag, New York
 | isbn = 0-387-94107-X
}}

==External links==
{{Commons category|Aryabhata}}
{{Wikiquote}}
* [https://archive.org/details/The_Aryabhatiya_of_Aryabhata_Clark_1930 1930 English translation] of ''The Aryabhatiya'' in various formats at the Internet Archive.
* {{MacTutor Biography|id=Aryabhata_I}}
* {{cite encyclopedia | editor = Thomas Hockey | last = Achar | first = Narahari | title=Āryabhaṭa I | encyclopedia = The Biographical Encyclopedia of Astronomers | publisher = Springer | date = 2007 | location = New York | page = 63 | url=http://islamsci.mcgill.ca/RASI/BEA/Aryabhata_I_BEA.htm | isbn=978-0-387-31022-0|display-editors=etal}} ([http://islamsci.mcgill.ca/RASI/BEA/Aryabhata_I_BEA.pdf PDF version])
* [https://www.cse.iitk.ac.in/users/amit/story/19_aryabhata.html/ "Aryabhata and Diophantus' son"], [[Hindustan Times]] Storytelling Science column, November 2004
* [http://www.wilbourhall.org/ Surya Siddhanta translations]

{{Indian mathematics}}
{{Authority control}}

{{DEFAULTSORT:Aryabhata}}
[[Category:476 births]]
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