Perbezaan antara semakan "Aryabhata"

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[[ImageFail: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/>
 
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.
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]]''.
# ''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)|colophoncolophons]]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 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
}}</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}} <BRbr />
''{{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)&times;8×8+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>
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
}}</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:
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>
:<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:
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.
 
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
| isbn=0471180823
}}
* {{citation
| title = The {{IAST|Āryabhaṭīya}} of {{IAST|Āryabhaṭa}}: An Ancient Indian Work on Mathematics and Astronomy
| last=Clark | firtst=Walter Eugene
}}
 
== 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}}
 
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[[id:Aryabhata]]
[[he:אריאבהטה]]
[[kn:ಆರ್ಯಭಟ (ಗಣಿತಜ್ಞ)]]
[[kk:Ариабхата Ⅰ]]
[[ht:Aryabhata]]
[[ml:ആര്യഭടൻ]]
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