Aryabhata
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[[Image:2064 aryabhata-crp.jpg|thumb|Statue of Aryabhata on the grounds of [[Inter-University Centre for Astronomy and Astrophysics|IUCAA]], [[Pune]].]]
'''Āryabha{{Unicode|ṭ}}a''' ([[Devanāgarī]]: आर्यभट) (AD [[476]] – [[550]]) is the first in the line of great mathematician-astronomers from the classical age of [[Indian mathematics]] and [[Indian astronomy]].
Aryabhata is the father of the [[Hindu-Arabic numeral system|Hindu-Arabic]] number system which has become universal today. His most famous works are the ''[[Aryabhatiya]]'' (AD [[499]] at age of 23 years) and ''Arya-[[Siddhanta]]''.
== Biography ==
Though Aryabhata's year of birth is clearly mentioned in [[Aryabhatiya]], exact location of his place of birth remains a matter of contention amongst the scholars. Some Scholars argue that Aryabhata was born in [[Patna|Kusumapura]], while others argue that Aryabhata was from [[Kerala]].<ref>[http://www.thehindu.com/2007/06/25/stories/2007062558250400.htm The Hindu]</ref>Some believe he was born in the region lying between Narmada and Godavari, which was known as ''[[Ashmaka]]'' and they identify Ashmaka with central India including [[Maharashtra]] and [[Madhya Pradesh]], though early Buddhist texts describe Ashmaka as being further south, ''dakShiNApath'' or the [[Deccan]], while other texts describe the Ashmakas as having fought [[Alexander]], which would put them further north.<ref name = Ansari>
{{cite journal
|last=Ansari
|first=S. M. R.
|year=1977
|month= March
|title=Aryabhata I, His Life and His Contributions
|journal=Bulletin of the Astronomical Society of India
|volume=5
|issue=1
|pages=pp. 10–18
|url=http://hdl.handle.net/2248/502
|accessdate= 2007-07-21}}</ref>
Recently in one of the scholarly studies based upon the astronomical readings in his works, it has been pointed out that Aryabhata's location may have been in [[Kunnamkulam]], Kerala .<ref>[http://www.ias.ac.in/currsci/dec252007/1870.pdf Current Science]</ref>
However, it is fairly certain that at some point, he went to [[Patna|Kusumapura]] for higher studies, and that he lived here for some time.<ref>{{cite book|last=Cooke|authorlink=Roger Cooke|title=|year=1997|chapter=The Mathematics of the Hindus|pages=204|quote=Aryabhata himself (one of at least two mathematicians bearing that name) lived in the late fifth and the early sixth centuries at Kusumapura (now Pataliutra, a village near the city of Patna) and wrote a book called ''Aryabhatiya''.}}</ref> [[Bhaskara I|Bhāskara I]] (AD [[629]]) identifies Kusumapura as Pataliputra (modern [[Patna, India|Patna]]). He lived there in the dying years of the [[Gupta empire]], the time which is known as the golden age of India, when it was already under [[Hun]] attack in the Northeast, during the reign of [[Buddhagupta]] and some of the smaller kings before [[Vishnugupta]].
==Works==
Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost.
His major work, ''Aryabhatiya'', a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature, and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines.
The ''Arya-siddhanta'', a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary [[Varahamihira]], as well as through 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''. This also contained a description of several astronomical instruments, the [[gnomon]] (''shanku-yantra''), a shadow instrument (''chhAyA-yantra''), possibly angle-measuring devices, semi-circle and circle shaped (''dhanur-yantra'' / ''chakra-yantra''), a cylindrical stick ''yasti-yantra'', an umbrella-shaped device called ''chhatra-yantra'', and [[water clock]]s of at least two types, bow-shaped and cylindrical.<ref name = Ansari/>
A third text that may have survived in [[Arabic language|Arabic]] translation is the ''Al ntf'' or ''Al-nanf'', which claims to be a translation of Aryabhata, but the Sanskrit name of this work is not known.
Probably dating from the ninth c., it is mentioned by the Persian scholar and chronicler of India, [[Abū Rayhān al-Bīrūnī]].<ref name = Ansari/>
===Aryabhatiya===
Direct details of Aryabhata's work are therefore known only from the ''Aryabhatiya''.
The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name; it is referred by his disciple [[Bhaskara I]] as ''Ashmakatantra'' or the treatise from the Ashmaka. It is also occasionally referred to as ''Arya-shatas-aShTa'', lit., Aryabhata's 108, which is the number of verses in the text. It is written in the very terse style typical of the [[sutra]] literature, where each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The entire text consists of 108 verses, plus an introductory 13, the whole being divided into four ''pAda''s or chapters:
# ''Gitikapada'': (13 verses) large units of time - ''kalpa'', ''manvantra'', ''yuga'', which present a cosmology that differs from earlier texts such as Lagadha's ''[[Vedanga Jyotisha]]''(ca. 1st c. BC). Also includes the table of sines (''jya''), given in a single verse. For the planetary revolutions during a ''mahayuga'', the number of 4.32mn years is given.
# ''Ganitapada'' (33 verses), covering mensuration (''kShetra vyAvahAra''), arithmetic and geometric progressions, [[gnomon]] / shadows (''shanku''-''chhAyA''), simple, [[quadratic equations|quadratic]], [[simultaneous equations|simultaneous]], and [[diophantine equations|indeterminate]] equations (''kuTTaka'')
# ''Kalakriyapada'' (25 verses) : different units of time and method of determination of positions of planets for a given day. Calculations concerning the intercalary month (''adhikamAsa''), ''kShaya-tithi''s. Presents a seven-day week, with names for days of week.
# ''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.
In addition, some versions cite a few [[colophon (publishing)|colophon]]s added at the end, extolling the virtues of the work, etc.
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'', ca. 600) and by
[[Nilakantha Somayaji]] in his ''Aryabhatiya Bhasya,'' (1465).
==Mathematics==
===Place Value system and zero===
The number [[place-value]] system, first seen in the 3rd century [[Bakhshali Manuscript]] was clearly in place in his work.<ref>P. Z. Ingerman, 'Panini-Backus form', Communications of the ACM 10 (3)(1967), p.137</ref> ; he certainly did not use the symbol, but 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 coefficients<ref>
{{cite book
| title = G Ifrah
| author = A universal history of numbers: From prehistory to the invention of the computer
| publisher = John Wiley & Sons
| address = 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>
{{Harvard reference
| Surname1 = Dutta
| Given1 = Bibhutibhushan
| Surname2 = Singh
| Given2 = Avadhesh Narayan
| Year = 1962
| Title = History of Hindu Mathematics
| Publisher = Asia Publishing House, Bombay
| isbn = 81-86050-86-8 (reprint)
}}</ref>.
===Pi as Irrational===
Aryabhata worked on the approximation for [[Pi]] (<math>\pi</math>), and may have realized that <math>\pi</math> is irrational. In the second part of the Aryabhatiyam (ga{{IPA|ṇ}}itapāda 10), he writes:
<blockquote>
''chaturadhikam śatamaś{{IPA|ṭ}}agu{{IPA|ṇ}}am dvāśaś{{IPA|ṭ}}istathā sahasrā{{IPA|ṇ}}ām <BR>
''Ayutadvayaviśkambhasyāsanno vrîttapari{{IPA|ṇ}}aha{{IPA|ḥ}}.''<br/>
"Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached."
</blockquote>
Aryabhata interpreted the word ''āsanna'' (approaching), appearing just before the last word, as saying that not only that is this an approximation, but that the value is incommensurable (or [[irrational]]). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by [[Johann Heinrich Lambert|Lambert]])<ref>
{{cite book
| title = S. Balachandra Rao
| author = Indian Mathematics and Astronomy: Some Landmarks,
| publisher = Jnana Deep Publications,
| year = 1994/1998
| address = Bangalore,
| isbn = ISBN 81-7371-205-0
}}</ref>.
After Aryabhatiya was translated into [[Arabic language|Arabic]] (ca. 820 AD)
this approximation was mentioned in [[Al-Khwarizmi]]'s book on algebra<ref name = Ansari/>.
===Mensuration and trigonometry===
In Ganitapada 6, Aryabhata gives the area of triangle as
: ''tribhujasya phalashariram samadalakoti bhujardhasamvargah''
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
| year=1997
| chapter = The Mathematics of the Hindus
| isbn=0471180823
| 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).}}</ref>
His great contribution to mensuration and trigonometry is used in the current international mathematics.
===Indeterminate Equations===
A problem of great interest to [[Indian mathematicians]] since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as [[diophantine equations]]. Here is an
example from [[Bhaskara]]'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.
i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general,
diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text [[Sulba Sutras]], the more ancient parts of which may date back to [[800 BCE]]. Aryabhata's method of solving such problems, called the ''ku{{IPA|ṭ}}{{IPA|ṭ}}aka'' (कूटटक) method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm,
as elaborated by Bhaskara in AD [[621]], is the standard method for solving first order Diophantine equations,
and it is often referred to as the [[Aryabhata algorithm]]<ref>
Amartya K Dutta, [http://www.ias.ac.in/resonance/Oct2002/pdf/Oct2002p6-22.pdf Diophantine equations: The Kuttaka], Resonance, October 2002. Also see earlier overview: [http://www.ias.ac.in/resonance/April2002/pdf/April2002p4-19.pdf ''Mathematics in Ancient India,''].</ref>.
The diophantine equations are of interest in [[cryptology]], and the [[RSA Conference]], 2006, focused on the ''kuttaka'' method and earlier work in the [[Sulvasutras]].
==Astronomy==
Aryabhata's system of astronomy was called the ''audAyaka system'' (days are reckoned from ''uday'', dawn at ''lanka'', equator). Some of his later writings on astronomy, which apparently
proposed a second model (''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.
===Motions of the Solar System===
Aryabhata appears to have believed that the earth rotates about its axis. This is made clear in the statement, referring to ''Lanka '', which describes the movement of the stars as a relative motion caused by the rotation of the earth:
:Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in lankA (i.e. on the equator) as moving exactly towards the West. [''achalAni bhAni samapashchimagAni'' - golapAda.9]
But the next verse describes the motion of the stars and planets as real movements: “The cause of their rising and setting is due to the fact the circle of the asterisms together with the planets driven by the provector wind, constantly moves westwards at Lanka”.
''Lanka'' (lit. [[Sri Lanka]]) is here a reference point on the equator, which was taken as the equivalent to the reference meridian for astronomical calculations.
Aryabhata described a [[geocentric]] model of the solar system, in which the
Sun and Moon are each carried by [[epicycle]]s which in turn revolve around
the Earth. In this model, which is also found in the
''Paitāmahasiddhānta'' (ca. AD 425), the motions of the planets are each
governed by two epicycles, a smaller ''manda'' (slow) epicycle and a larger
''śīghra'' (fast) epicycle.
<ref>
{{Harvard reference
| last = Pingree
| first = David
| authorlink = 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
| year = 1996
| ID = ISBN 0-7141-1746-3
}} pp. 127-9.</ref> The order of the planets in terms of distance from earth are taken as: the [[Moon]], [[Mercury (planet)|Mercury]], [[Venus]], the [[Sun]], [[Mars]],
[[Jupiter]], [[Saturn]], and the [[asterism]]s<ref name=Ansari/>.
The positions and periods of the planets was calculated relative to uniformly moving points, which in the case of Mercury and Venus, move around the Earth at the same speed as the mean Sun and in the case of Mars, Jupiter, and Saturn 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 [[Greek astronomy#Hellenistic astronomy|Greek astronomy]].<ref>Otto Neugebauer, "The Transmission of Planetary Theories in Ancient and Medieval Astronomy," ''[[Scripta Mathematica]]'', 22(1956): 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===
He states that the [[Moon]] and planets shine by reflected sunlight. Instead of the prevailing cosmogyny where eclipses were caused by pseudo-planetary nodes [[Rahu]] and [[Ketu]], 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-shadow (verse gola.37), and discusses at length the size and extent of this earth-shadow (verses gola.38-48), and then the computation, and the size of the eclipsed part during eclipses. Subsequent Indian astronomers improved on these calculations, but his methods provided the core. This computational paradigm was so accurate that the 18th century scientist [[Guillaume le Gentil]], during a visit to Pondicherry, found the Indian computations of the duration of the [[lunar eclipse]] of [[1765-08-30]] to be short by 41 seconds, whereas his charts (by Tobias Mayer, [[1752]]) were long by 68 seconds.<ref name=Ansari/>.
Aryabhata's computation of Earth's [[circumference]] as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation was a significant improvement over the computation by the [[Greek mathematics|Greek mathematician]], [[Eratosthenes]] (c. [[200 BC]]), whose exact computation is not known in modern units but his estimate had an error of around 5-10%.<ref>[http://www.nasa.gov/lb/audience/forstudents/5-8/features/F_JSC_NES_School_Measures_Up.html "JSC NES School Measures Up"], ''NASA'', 11th April, 2006, retrieved 24th January, 2008.</ref><ref>[http://www-istp.gsfc.nasa.gov/stargaze/Scolumb.htm "The Round Earth"], ''NASA'', 12th December, 2004, retrieved 24th January, 2008.</ref>
===Sidereal periods===
Considered in modern English units of time, Aryabhata calculated the [[sidereal rotation]] (the rotation of the earth referenced the fixed stars) as 23 hours 56 minutes and 4.1 seconds; 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 30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate in the period.
===Heliocentrism===
Āryabhata claimed that the Earth turns on its own axis and some elements of his planetary epicyclic models rotate at the same speed as the motion of the planet around the Sun. Thus it has been suggested that Āryabhata'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> A detailed rebuttal to this heliocentric interpretation is in a review which describes [[Bartel Leendert van der Waerden|B. L. van der Waerden]]'s book as "show[ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Āryabhata's description,"<ref>Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy," ''Isis'', 64 (1973): 239-243.</ref> although some concede that Āryabhata's system stems from an earlier heliocentric model of which he was unaware.<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[http://people.scs.fsu.edu/~dduke/india8.pdf].</ref> It has even been claimed that he considered the planet's paths to be [[Ellipse|elliptical]], although no primary evidence for this has been cited.<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], [[MacTutor History of Mathematics archive]]'':
<br>{{quote|"He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses."}}</ref> Though [[Aristarchus of Samos]] (3rd century BC) and sometimes [[Heraclides of Pontus]] (4th century BC) are usually credited with knowing the heliocentric theory, the version of [[Greek astronomy]] known in ancient India, ''[[Paulisa Siddhanta]]'' (possibly by a [[Paulus Alexandrinus|Paul]] of [[Alexandria]]) makes no reference to a Heliocentric theory.
==Legacy==
Aryabhata's work was of great influence in the Indian astronomical tradition, and influenced several neighbouring cultures through translations. The [[Arabic language|Arabic]] translation during the [[Islamic Golden Age]] (ca. [[820]]), was particularly influential. Some of his results are cited by [[Al-Khwarizmi]], and he is referred to by the 10th century Arabic scholar [[Al-Biruni]], who states that Āryabhata's followers believed the Earth to rotate on its axis.
His definitions of [[sine]], as well as cosine (''kojya''), versine (''ukramajya''),
and inverse sine (''otkram jya''), influenced the birth of [[trigonometry]]. He was also the first to specify sine and
[[versine]] (1 - cosx) tables,
in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.
In fact, the modern names "''sine''" and "''cosine''", are a mis-transcription of the words ''jya'' and ''kojya'' as introduced by Aryabhata. They were transcribed as ''jiba'' and ''kojiba'' in [[Arabic language|Arabic]]. They were then misinterpreted by [[Gerard of Cremona]] while translating an Arabic geometry text to [[Latin]]; he took jiba to be 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
|year = 2001
|accessdate = 2007-07-14
}}</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 were 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 c.), were translated into Latin as the [[Tables of Toledo]] (12th c.), and remained the most accurate [[Ephemeris]] used in Europe for centuries.
Calendric calculations worked out by Aryabhata and followers have been in continuous use in India for the practical purposes of fixing the [[Panchangam]], or [[Hindu calendar]], These were also transmitted to the Islamic world, and formed the basis for the [[Jalali calendar]] introduced [[1073]] by a group of astronomers including [[Omar Khayyam]]<ref>
{{cite encyclopedia
|title = Omar Khayyam
|encyclopedia = The Columbia Encyclopedia, Sixth Edition.
|date = 2001-05
|url = http://www.bartleby.com/65/om/OmarKhay.html
|accessdate =2007-06-10
}}</ref>, versions of which (modified in [[1925]]) are the national calendars in use in [[Iran]] and [[Afghanistan]] today. The Jalali calendar determines its dates 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 lower in the [[Jalali calendar]] than in the [[Gregorian calendar]].
India's first satellite [[Aryabhata (satellite)|Aryabhata]], was named after him. The [[lunar crater]] [[Aryabhata (crater)|Aryabhata]] is named in his honour. An Institute for conducting research in Astronomy, Astrophysics and atmospheric sciences has been named as Aryabhatta Research Institute of observational sciences (ARIES) near Nainital, India.
The interschool [[Aryabhatta Maths Competition]] is named after him.<ref>{{cite news |title= Maths can be fun |url=http://www.hindu.com/yw/2006/02/03/stories/2006020304520600.htm |publisher=[[The Hindu]] |date = 2006-02-03|accessdate=2007-07-06 }}</ref>
==See also==
*{{IAST|[[Āryabhaṭa numeration]]}}
*[[Aryabhatiya]]
==References==
{{reflist}}
===Other References===
*{{cite book
| first=Roger
| last=Cooke
| authorlink=Roger Cooke
| title=The History of Mathematics: A Brief Course
| publisher=Wiley-Interscience
| year=1997
| isbn=0471180823
}}
*Walter Eugene Clark, ''{{IAST|The Āryabhaṭīya of Āryabhaṭa}}, An Ancient Indian Work on Mathematics and Astronomy'', University of Chicago Press (1930); reprint: Kessinger Publishing (2006), ISBN 978-1425485993.
* [[Subhash Kak|Kak, Subhash C.]] (2000). 'Birth and Early Development of Indian Astronomy'. In {{Harvard reference
| Surname1 = Selin
| Given1 = Helaine
| Year = 2000
| Title = Astronomy Across Cultures: The History of Non-Western Astronomy
| Publisher = Kluwer, Boston
| ID = ISBN 0-7923-6363-9
}}
* Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.
* {{Harvard reference
| Surname1 = Thurston
| Given1 = H.
| Year = 1994
| Title = Early Astronomy
| Publisher = Springer-Verlag, New York
| ID = ISBN 0-387-94107-X
}}
==External links==
* {{MacTutor Biography|id=Aryabhata_I}}
* [http://www.cse.iitk.ac.in/~amit/story/19_aryabhata.html ''Aryabhata and Diophantus' son'', [[Hindustan Times]] Storytelling Science column, Nov 2004]
* [http://www.wilbourhall.org Aryabhatiya in English and Sanskrit (PDF)]
{{Indian mathematics}}
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