Revision 4233833 of "Aryabhata" on mswiki

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[[Fail:2064 aryabhata-crp.jpg|thumb|300px|Arca Aryabhata di tapak [[Inter-University Centre for Astronomy and Astrophysics|IUCAA]], [[Pune]]. Dengan tiadanya maklumat diketahui mengenai rupawannya, apa-apa imej Aryabhata berasal dari konsepsi artis.]]
'''Aryabhata''' ([[IAST]]: {{IAST|Āryabhaṭa}}; {{lang-sa|आर्यभट}}) (476–550 AM) adalah yang pertama dalam turutan [[ahli matematik]]-[[ahli astronomi]] hebat dari zaman klasik [[matematik India]] dan [[astronomi India]]. Karya termasyhurnya adalah ''[[Aryabhatiya]]'' (499 AM, ketika dia berusia 23 tahun) dan ''Arya-[[siddhanta]]''.

== Biografi ==

=== Nama ===
Sungguhpun terdapat kecenderungan bagi salah mengeja sebagai "Aryabhatta" menurut analogi dengan nama-nama lain yang mempunyai akhiran "[[bhatta]]", namanya secara sesuai dieja Aryabhata: tiap teks astronomi mengeja namanya oleh itu,<ref name="sarma">{{citation | author=[[K. V. Sarma]] | journal=Indian Journal of History of Science | year=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}}</ref> termasuk rujukan [[Brahmagupta]] padanya  "dalam lebih daripada seratus tempat mengikut nama".<ref>{{citation | year=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]] | page=392 | url=http://books.google.com/books?id=fAsFAAAAMAAJ&pg=PA392&dq=aryabhata}}</ref> Lebih lanjutnya, dalam kebanyakan contoh "Aryabhatta" tidak muatkan meter juga.<ref name=sarma/>

=== Kelahiran ===
Aryabhata menyebut dalam [[Aryabhatiya]] bahawa ia dikarang 3,600 tahun ke dalam [[Kali Yuga]], ketika dia berusia 23 tahun. Ini bersamaan dengan 499 AM, dan bermakna bahawa dia dilahirkan pada 476 AM.<ref name=sarma/>

Aryabhata tidak memberikan maklumat tentang tempat kelahirannya. Satu-satunya maklumat datang dari [[Bhāskara I]], yang menggambarkan wAryabhata sebagai ''āśmakīya'', "seseorang yang datangnya dari negeri ''aśmaka''." Sungguhpun ''aśmaka'' pada adalnya terletak di barat laut India, ia dibukti ramai bahawa, ketika zaman [[Buddha]], satu cabang suku Aśmaka menetap di kawasan antara [[sungai Narmada]] dan [[sungai Godavari]], di selatan Gujarat–Utara kawasan Maharashtra tengah India. Aryabhata dipercayai dilahirka di sana.<ref name=sarma/><ref name=Ansari/> Bagaimanapun, teks awal Buddha menggambarkan Ashmaka sebelai lebih jauh keselatan, di ''dakshinapath'' atau [[Deccan]], sementara teks lain menggambarkan Ashmakas sebagai menentang [[Alexander]], yang akan meletakkan mereka lebih jauh ke utara.<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=10–18
 |url=http://hdl.handle.net/2248/502 
 |accessdate= 2007-07-21}}</ref>

=== Karya ===
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he lived there 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]] ([[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}}'') at Kusumapura, and, because the university of [[Nalanda]] was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well.<ref name=sarma/>

=== Hipotesis Kerala ===
It has also been suggested that ''aśmaka'' (Sanskrit for "stone") might be the region in Kerala that is now known as Koṭuṅṅallūr, based on the belief that it 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").<ref name=sarma/> It is also claimed that the fact that several commentaries on the Aryabhatiya have come from Kerala suggest that it was Aryabhata's main place of life and activity. But [[K. V. Sarma]], the authority on Kerala's astronomical tradition,<ref name=kch1/> disagrees and cites many commentaries that have come from outside Kerala and the Aryasiddhanta's being completely unknown in Kerala.<ref name=sarma/> In recent (2007) papers, K. Chandra Hari uses a discrepancy in Aryabhata's astronomical values to deduce that he carried out his calculations from a place in Kerala at the same meridian as Ujjayini, possibly Chamravattam (10°N51, 75°E45) in central [[Kerala]]. He further hypothesizes that Asmaka was the Jain country surrounding [[Shravanabelagola]], taking its name from the stone monoliths there.<ref name=kch1>K. Chandra Hari, [http://www.ias.ac.in/currsci/oct252007/1177.pdf "Critical Evidence to Fix the Native Place of Āryabhat̟a-I"], ''Current Science'', Vol. 93, Issue 8, 25 October 2007</ref><ref>K. Chandra Hari, [http://www.ias.ac.in/currsci/dec252007/1870.pdf "Alleged Mistake of Āryabhat̟a — Light onto His Place of Observation"], ''Current Science'' Vol. 93, Issue 12, 25 December 2007, pp. 1870–73.</ref><ref>K. Chandra Hari, [http://www.ias.ac.in/currsci/jan102008/132.pdf "Āryabhat̟a on the Heliacal Rise and Set of Canopus"], ''Current Science'', Vol. 94, Issue 1, 10 January 2008</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>{{Harvnb|Clark|1930}}, p. 68</ref><ref>{{citation | year=2000 | title = Indian Astronomy: An Introduction | author1=S. Balachandra Rao | publisher=Orient Blackswan | isbn=9788173712050 | page=82 | url=http://books.google.com/books?id=N3DE3GAyqcEC&pg=PA82&dq=lanka}}: "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."</ref><ref>{{citation | year=2003 | title = ''Ancient Indian Astronomy'' | author1=L. Satpathy | publisher=Alpha Science Int'l Ltd. | isbn=9788173194320 | page=200 | url=http://books.google.com/books?id=nh6jgEEqqkkC&pg=PA200&dq=lanka}}: "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."</ref><ref>{{citation | title = ''Classical Muhurta'' | author1=Ernst Wilhelm | publisher=Kala Occult Publishers | isbn=9780970963628 | page=44 | url=http://books.google.com/books?id=3zMPFJy6YygC&pg=PA44&dq=lanka}}: "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.)"</ref><ref>{{citation | year=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=9788187276272 | page=63 | url=http://books.google.com/books?id=sEX11ZyjLpYC&pg=PA63&dq=lanka}}</ref><ref>{{citation | year=1989 | title = ''The Surya Siddhanta: A Textbook of Hindu Astronomy'' | author1=Ebenezer Burgess | author2=  Phanindralal Gangooly | publisher=Motilal Banarsidass Publ. | isbn=9788120806122 | page=46 | url=http://books.google.com/books?id=W0Uo_-_iizwC&pg=PA46&dq=lanka}}</ref>

== Karya ==

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 [[fraction]]s, [[quadratic equation]]s, sums-of-power series, and a table of [[sine]]s.

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''. 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/>

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 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. 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. 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]]''(ca. 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 (''kṣetra vyāvahāra''), 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 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. 
# ''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)|colophons]] 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 CE) and by [[Nilakantha Somayaji]] in his ''Aryabhatiya Bhasya,'' (1465 CE).

== Matematik ==
=== Menempatkan sistem nilai dan kosong ===
The [[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 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
 | author =        G. Ifrah
 | title =       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 sebagai tidak rasional ===

Aryabhata worked on the approximation for [[Pi]] (<math>\pi</math>), and may have come to the conclusion that <math>\pi</math> is irrational. In the second part of the ''Aryabhatiyam'' ({{IAST|gaṇitapāda}} 10), he writes: 
<blockquote>
''{{IAST|chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām}} <br />
''{{IAST|Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.''}}<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."  
</blockquote>
This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/20000 = 3.1416, which is accurate to five [[significant figures]].

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 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
 | year = 1994/1998
 | address = Bangalore
 | isbn = 81-7371-205-0
}}</ref>

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

=== Mensurasi dan trigonometri ===

In Ganitapada 6, Aryabhata gives the area of a 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>

Aryabhata discussed the concept of ''sine'' in his work by the name of ''ardha-jya''. Literally, it 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 ''jiab'', meaning "cove" or "bay." (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 ''jiab'' with its Latin counterpart, ''sinus'', which means "cove" or "bay". And after that, the ''sinus'' became ''sine'' in English.<ref> {{Cite book
 | author = Howard Eves
 | title = An Introduction to the History of Mathematics
 | publisher = Saunders College Publishing House, New York
 | year = 1990
 | edition = 6
 | page= 237
}}</ref>

=== Persamaan tidak tetap ===

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]]. This 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
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 BCE. Aryabhata's method of solving such problems is called the ''{{IAST|kuṭṭaka}}'' (कुट्टक) method. ''Kuttaka'' means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and 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]].

=== 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 |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0471543977 |page = 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."}}</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>

== Astronomi ==
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.

=== Mosi sistem suria ===

Aryabhata appears to have believed that the earth rotates about its axis. This is indicated 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 (or 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 that the circle of the asterisms, together with the planets driven by the provector wind, constantly moves westwards at Lanka."

As mentioned above, ''Lanka'' (lit. [[Sri Lanka]]) is here a reference point on the equator, which was the equivalent of 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. They in turn revolve around
the Earth. In this model, which is also found in the ''Paitāmahasiddhānta'' (ca. CE 425), the motions of the planets are each governed by two epicycles, a smaller ''manda'' (slow) and a larger ''śīghra'' (fast).
<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 is 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. In the case of Mercury and Venus, they move around the Earth at the same speed as the mean 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 [[Greek astronomy#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>

=== Gerhana ===

Aryabhata states that the [[Moon]] and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which 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'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 [[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 the Earth's [[circumference]] as 39,968.0582 kilometres was only 0.2% smaller than the actual value of 40,075.0167 kilometres. This approximation was a significant improvement over the computation by [[Greek mathematics|Greek mathematician]] [[Eratosthenes]] (c. 200 BCE), 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>

=== Tempoh sidereal ===

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; 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 is an error of 3 minutes and 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 of the period.

=== Heliosentrisme ===

As mentioned, Aryabhata 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 Earth around 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> A detailed rebuttal to this heliocentric interpretation is in a review that 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 Aryabhata's description."<ref>Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy," ''Isis'', 64 (1973): 239–243.</ref> However, some concede that Aryabhata'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]], but no primary evidence for this has been found.<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 BCE) and sometimes [[Heraclides of Pontus]] (4th century BCE) are usually credited with knowing the heliocentric theory, the version of [[Greek astronomy]] known in ancient India as the ''[[Paulisa Siddhanta]]'' (possibly by a [[Paulus Alexandrinus|Paul]] of [[Alexandria]]) makes no reference to a heliocentric theory.

== Legasi ==
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 CE), was particularly influential. Some of his results are cited by [[Al-Khwarizmi]], and he is mentioned by the 10th century Arabic scholar [[Al-Biruni]], who states that Aryabhata's followers believed that the Earth rotated on its axis.  

His definitions of [[sine]] (''jya''), 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, modern names "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 language|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
|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 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 c.) 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 CE by a group of astronomers including [[Omar Khayyam]]<ref>
{{cite encyclopedia
|title = Omar Khayyam
|encyclopedia = The Columbia Encyclopedia 
|date = 2001-05
|edition = 6
|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 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]].

India's first satellite [[Aryabhata (satellite)|Aryabhata]] and the [[lunar crater]] [[Aryabhata (crater)|Aryabhata]] are named in his honour. 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 |publisher=[[The Hindu]] |date = 2006-02-03|accessdate=2007-07-06 }}</ref> as is ''Bacillus aryabhata'', a species of bacteria discovered by [[ISRO]] scientists in 2009.<ref>[http://www.isro.org/pressrelease/Mar16_2009.htm Discovery of New Microorganisms in the Stratosphere]. Mar. 16, 2009. ISRO.</ref>

== Lihat pula ==
* {{IAST|[[Pembilangan Āryabhaṭa]]}}
* [[Aryabhatiya]]

== Rujukan ==
{{reflist|2}}

=== Rujukan lain ===
* {{cite book
 | first=Roger
 | last=Cooke
 | title=The History of Mathematics: A Brief Course
 | publisher=Wiley-Interscience
 | year=1997
 | isbn=0471180823
}}
* {{citation 
 | title = The {{IAST|Āryabhaṭīya}} of {{IAST|Āryabhaṭa}}: An Ancient Indian Work on Mathematics and Astronomy 
 | last=Clark | firtst=Walter Eugene
 | year=1930
 | publisher=University of Chicago Press; 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   = Boston: Kluwer
 | 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 ==
* http://www.scribd.com/doc/20912413/The-Aryabhatiya-of-Aryabhata-English-Translation - The Aryabhatiya of Aryabhata English Translation 
* {{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]

{{Indian mathematics}}

{{DEFAULTSORT:Aryabhata}}

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