aryabhataa

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Biography

Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other nameshaving the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical textspells his name thus,[6]including rahmagupta!s references to him "in more than a hundred

places by name"[#]$urthermore, in most instances "Aryabhatta" does not fit the metre either[6]

Time and Place of birth

Aryabhata mentions in theAryabhatiyathat it was composed %,6%& years into the 'ali (uga,when he was )% years old *his corresponds to + -., and implies that he was born in +#6[+]

Aryabhata was born in *aregna /literally, song of the stars0, which is a small town in ihar,1ndia, about %& 2m /3 mi0 from 4atna /then 2nown as 4ataliputra0, the capital city of ihar5tate .vidences ustify his birth there 1n *aregna Aryabhata set up an Astronomical7bservatory in the 5un *emple 6th century

*here is no evidence that he was born outside 4atliputra and traveled to 8agadha, the centreof instruction, culture and 2nowledge for his studies where he even set up a coachinginstitute[9]owever, early uddhist texts describe Ashma2a as being further south, indakshinapathor the ;eccan,while other texts describe the Ashma2as as having foughtAlexander

Education

1t is fairly certain that, at some point, he went to 'usumapura for advanced studies and livedthere for some time[]oth indu and uddhist tradition, as well as hr, based on the belief that it was earlier2nown as 'oum?'al?l?>r /"city of hard stones"0@ however, old records show that the citywas actually 'oum?2ol?>r /"city of strict governance"0 5imilarly, the fact that severalcommentaries on the Aryabhatiya have come from 'erala were used to suggest that it wasAryabhata!s main place of life and activity@ however, many commentaries have come from

outside 'erala
http://en.wikipedia.org/wiki/Bhattahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Brahmaguptahttp://en.wikipedia.org/wiki/Brahmaguptahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-6http://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Kali_Yugahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Yadav2010-3http://en.wikipedia.org/wiki/Aryabhata#cite_note-Ansari-7http://en.wikipedia.org/wiki/Deccanhttp://en.wikipedia.org/wiki/Deccanhttp://en.wikipedia.org/wiki/Alexander_the_Greathttp://en.wikipedia.org/wiki/Aryabhata#cite_note-8http://en.wikipedia.org/wiki/Bh%C4%81skara_Ihttp://en.wikipedia.org/wiki/P%C4%81%E1%B9%ADaliputrahttp://en.wikipedia.org/wiki/P%C4%81%E1%B9%ADaliputrahttp://en.wikipedia.org/wiki/P%C4%81%E1%B9%ADaliputrahttp://en.wikipedia.org/wiki/Patnahttp://en.wikipedia.org/wiki/Patnahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Nalandahttp://en.wikipedia.org/wiki/Nalandahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Tareganahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-9http://en.wikipedia.org/wiki/Aryabhata#cite_note-9http://en.wikipedia.org/wiki/Kodungallurhttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Menon-10http://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Brahmaguptahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-6http://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Kali_Yugahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Yadav2010-3http://en.wikipedia.org/wiki/Aryabhata#cite_note-Ansari-7http://en.wikipedia.org/wiki/Deccanhttp://en.wikipedia.org/wiki/Alexander_the_Greathttp://en.wikipedia.org/wiki/Aryabhata#cite_note-8http://en.wikipedia.org/wiki/Bh%C4%81skara_Ihttp://en.wikipedia.org/wiki/P%C4%81%E1%B9%ADaliputrahttp://en.wikipedia.org/wiki/Patnahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Nalandahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-sarma-5http://en.wikipedia.org/wiki/Tareganahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-9http://en.wikipedia.org/wiki/Kodungallurhttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Menon-10http://en.wikipedia.org/wiki/Bhatta


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Aryabhata/1A5*: ryabhaa, 5ans2rit: 0 or Aryabhata I[3][)]/+#6BCC&-.0[%][+]wasthe first in the line of great mathematician?astronomers from the classical age of1ndianmathematicsand 1ndian astronomyis most famous wor2s are theryabhaya /+ -.,when he was )% years old0[C]and theArya-siddhanta

*he wor2s of Aryabhata dealt with mainly mathematics and astronomy e also wor2ed onthe approximation for pi

Aryabhata mentions "Dan2a" on several occasions in theAryabhatiya, but his "Dan2a" is anabstraction, standing for a point on the eEuator at the same longitude as his Fayini[3)]

Works

Aryabhata is the author of several treatises onmathematicsandastronomy,some of whichare lost is maor wor2,Aryabhatiya, a compendium of mathematics and astronomy, was

extensively referred to in the 1ndian mathematical literature and has survived to moderntimes *he mathematical part of theAryabhatiyacovers arithmetic, algebra,planetrigonometry, and spherical trigonometry 1t also containscontinued fractions, EuadraticeEuations, sums?of?power series, and a table of sines

*heArya-siddhanta, a lot wor2 on astronomical computations, is 2nown through the writingsof Aryabhata!s contemporary,Garahamihira, and later mathematicians and commentators,including rahmaguptaand has2ara 1 *his wor2 appears to be based on the older 5urya5iddhantaand uses the midnight?day rec2oning, as opposed to sunrise inAryabhatiya 1t alsocontained a description of several astronomical instruments: the gnomon/shanku-yantra0, ashadow instrument /chhAyA-yantra0, possibly angle?measuring devices, semicircular and

circular /dhanur-yantraH chakra-yantra0, a cylindrical stic2yasti-yantra, an umbrella?shapeddevice called the chhatra-yantra, andwater cloc2sof at least two types, bow?shaped andcylindrical[9]

A third text, which may have survived in the Arabictranslation, isAl ntforAl-nanf 1t claimsthat it is a translation by Aryabhata, but the 5ans2rit name of this wor2 is not 2nown4robably dating from the th century, it is mentioned by the4ersianscholar and chronicler of1ndia, Ab> IayhnJ[9]

Aryabhatiya

8ain article: Aryabhatiya

;irect details of Aryabhata!s wor2 are 2nown only from theAryabhatiya *he name"Aryabhatiya" is due to later commentators Aryabhata himself may not have given it a nameis disciple has2ara 1calls itAshmakatantra/or the treatise from the Ashma2a0 1t is alsooccasionally referred to asArya-shatas-aShTa/literally, Aryabhata!s 3&90, because there are3&9 verses in the text 1t is written in the very terse style typical ofsutraliterature, in whicheach line is an aid to memory for a complex system *hus, the explication of meaning is dueto commentators *he text consists of the 3&9 verses and 3% introductory verses, and isdivided into fourpdas or chapters:
http://en.wikipedia.org/wiki/IASThttp://en.wikipedia.org/wiki/Sanskrit_languagehttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Aryabhata_the_Elder-0http://en.wikipedia.org/wiki/Aryabhata#cite_note-Aryabhata_the_Elder-0http://en.wikipedia.org/wiki/Aryabhata#cite_note-Publishing2010-1http://en.wikipedia.org/wiki/Common_Erahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Ray2009-2http://en.wikipedia.org/wiki/Aryabhata#cite_note-Yadav2010-3http://en.wikipedia.org/wiki/Aryabhata#cite_note-Yadav2010-3http://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Astronomerhttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_astronomyhttp://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%AD%C4%AByahttp://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%AD%C4%AByahttp://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%AD%C4%AByahttp://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%AD%C4%AByahttp://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%AD%C4%AByahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Roupp1997-4http://en.wikipedia.org/wiki/Siddhantahttp://en.wikipedia.org/wiki/Ujjayinihttp://en.wikipedia.org/wiki/Aryabhata#cite_note-11http://en.wikipedia.org/wiki/Aryabhata#cite_note-11http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Spherical_trigonometryhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Quadratic_equationhttp://en.wikipedia.org/wiki/Quadratic_equationhttp://en.wikipedia.org/wiki/Aryabhata's_sine_tablehttp://en.wikipedia.org/wiki/Varahamihirahttp://en.wikipedia.org/wiki/Varahamihirahttp://en.wikipedia.org/wiki/Brahmaguptahttp://en.wikipedia.org/wiki/Bhaskara_Ihttp://en.wikipedia.org/wiki/Surya_Siddhantahttp://en.wikipedia.org/wiki/Surya_Siddhantahttp://en.wikipedia.org/wiki/Gnomonhttp://en.wikipedia.org/wiki/Water_clockhttp://en.wikipedia.org/wiki/Water_clockhttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Ansari-7http://en.wikipedia.org/wiki/Arabic_languagehttp://en.wikipedia.org/wiki/Persian_peoplehttp://en.wikipedia.org/wiki/Persian_peoplehttp://en.wikipedia.org/wiki/Ab%C5%AB_Rayh%C4%81n_al-B%C4%ABr%C5%ABn%C4%ABhttp://en.wikipedia.org/wiki/Ab%C5%AB_Rayh%C4%81n_al-B%C4%ABr%C5%ABn%C4%ABhttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Ansari-7http://en.wikipedia.org/wiki/Aryabhatiyahttp://en.wikipedia.org/wiki/Bhaskara_Ihttp://en.wikipedia.org/wiki/Sutrahttp://en.wikipedia.org/wiki/Sutrahttp://en.wikipedia.org/wiki/IASThttp://en.wikipedia.org/wiki/Sanskrit_languagehttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Aryabhata_the_Elder-0http://en.wikipedia.org/wiki/Aryabhata#cite_note-Publishing2010-1http://en.wikipedia.org/wiki/Common_Erahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Ray2009-2http://en.wikipedia.org/wiki/Aryabhata#cite_note-Yadav2010-3http://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Astronomerhttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_astronomyhttp://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%AD%C4%AByahttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Roupp1997-4http://en.wikipedia.org/wiki/Siddhantahttp://en.wikipedia.org/wiki/Ujjayinihttp://en.wikipedia.org/wiki/Aryabhata#cite_note-11http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Spherical_trigonometryhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Quadratic_equationhttp://en.wikipedia.org/wiki/Quadratic_equationhttp://en.wikipedia.org/wiki/Aryabhata's_sine_tablehttp://en.wikipedia.org/wiki/Varahamihirahttp://en.wikipedia.org/wiki/Brahmaguptahttp://en.wikipedia.org/wiki/Bhaskara_Ihttp://en.wikipedia.org/wiki/Surya_Siddhantahttp://en.wikipedia.org/wiki/Surya_Siddhantahttp://en.wikipedia.org/wiki/Gnomonhttp://en.wikipedia.org/wiki/Water_clockhttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Ansari-7http://en.wikipedia.org/wiki/Arabic_languagehttp://en.wikipedia.org/wiki/Persian_peoplehttp://en.wikipedia.org/wiki/Ab%C5%AB_Rayh%C4%81n_al-B%C4%ABr%C5%ABn%C4%ABhttp://en.wikipedia.org/wiki/Aryabhata#cite_note-Ansari-7http://en.wikipedia.org/wiki/Aryabhatiyahttp://en.wikipedia.org/wiki/Bhaskara_Ihttp://en.wikipedia.org/wiki/Sutra


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3 Gitikapada: /3% verses0: large units of timeKkalpa, manvantra, andyugaKwhichpresent a cosmology different from earlier texts such as Dagadha!s Vedanga y!tisha/c 3st century -.0 *here is also a table of sines /"ya0, given in a single verse *heduration of the planetary revolutions during a mahayugais given as +%) millionyears

2. Ganitapada/%% verses0: covering mensuration /ketra vyvahra0, arithmetic andgeometric progressions, gnomonH shadows /shanku?chhAyA0, simple, Euadratic,simultaneous,and indeterminateeEuations

% #alakriyapada/)C verses0: different units of time and a method for determining thepositions of planets for a given day, calculations concerning the intercalary month/adhikamAsa0, kShaya-tithis, and a seven?day wee2 with names for the days of wee2

4. G!lapada/C& verses0: LeometricHtrigonometricaspects of the celestial sphere,features of the ecliptic, celestial eEuator,node, shape of the earth, cause of day andnight, rising of Modiacal signson horiMon, etc 1n addition, some versions cite a few

colophonsadded at the end, extolling the virtues of the wor2, etc

*he Aryabhatiya presented a number of innovations in mathematics and astronomy in verseform, which were influential for many centuries *he extreme brevity of the text waselaborated in commentaries by his disciple has2ara 1 /$hashya, c 6&& -.0 and by

=ila2antha 5omayaiin hisAryabhatiya $hasya%/3+6C -.0 e was not only the first to findthe radius of the earth but was the only one in ancient time including the Lree2s and theIomans to find the volume of the earth

Mathematics

Place alue system and !ero

*heplace?valuesystem, first seen in the %rd century a2hshali 8anuscript, was clearly inplace in his wor2 While he did not use a symbol for Mero, the $rench mathematicianLeorges1frahexplains that 2nowledge of Mero was implicit in Aryabhata!splace?value systemas a

place holder for the powers of ten with nullcoefficients [3%]

owever, Aryabhata did not use the rahmi numerals -ontinuing the 5ans2ritictraditionfrom Gedic times, he used letters of the alphabet to denote numbers, expressing Euantities,such as the table of sines in amnemonicform[3+]

Appro"imation of

Aryabhata wor2ed on the approximation forpi/ 0, and may have come to the conclusionthat is irrational 1n the second part of theAryabhatiyam/gaitap


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*his implies that the ratio of the circumference to the diameter is//+ N 3&&0 O 9 N 6)&&&0H)&&&& P 6)9%)H)&&&& P %3+36, which is accurate to five significantfigures

1t is speculated that Aryabhata used the word sanna/approaching0, to mean that not only is

this an approximation but that the value is incommensurable /or irrational0 1f this is correct,it is Euite a sophisticated insight, because the irrationality of pi was proved in .urope only in3#63 by Dambert[36]

After Aryabhatiya was translated intoArabic/c 9)& -.0 this approximation was mentionedin Al?'hwariMmi!s boo2 on algebra[9]

Trigonometry

1n Lanitapada 6, Aryabhata gives the area of a triangle as

tribhu"asya phalashariram samadalak!ti bhu"ardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half?side is the area"[3#]

Aryabhata discussed the concept ofsinein his wor2 by the name of ardha-"ya,which literallymeans "half?chord" $or simplicity, people started calling it"ya When Arabic writerstranslated his wor2s from 5ans2ritinto Arabic, they referred it as"iba owever, in Arabicwritings, vowels are omitted, and it was abbreviated as"b Dater writers substituted it with

"aib, meaning "poc2et" or "fold /in a garment0" /1n Arabic,"ibais a meaningless word0Dater in the 3)th century, whenLherardo of -remonatranslated these writings from Arabic

into Datin, he replaced the Arabic"aibwith its Datin counterpart,sinus, which means "cove"or "bay"@ thence comes the .nglishsince Alphabetic code has been used by him to define aset of increments 1f we use Aryabhata!s table and calculate the value of sin/%&0/corresponding to hasha0 which is 3#3H%+%9 P &C@ the value is correct is alphabetic codeis commonly 2nown as the Aryabhata cipher[39]

Indeterminate e#uations

A problem of great interest to 1ndian mathematicianssince ancient times has been to findinteger solutions to eEuations that have the form ax N by P c, a topic that has come to be2nown as diophantine eEuations*his is an example from h


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are of interest in cryptology,and the I5A -onference, )&&6, focused on the kuttakamethodand earlier wor2 in the 5ulbasutras

Algebra

1nAryabhatiyaAryabhata provided elegant results for the summation of seriesof sEuares andcubes:[)&]

and

Astronomy

Aryabhata!s system of astronomy was called the audAyaka system, in which days arerec2oned from uday, dawn at lankaor "eEuator" 5ome of his later writings on astronomy,which apparently proposed a second model /or ardha-rAtrikA, midnight0 are lost but can be

partly reconstructed from the discussion in rahmagupta!s khan'akhAdyaka 1n some texts,he seems to ascribe the apparent motions of the heavens to the .arth!s rotation e may have

believed that the planet!s orbits as ellipticalrather than circular[)3][))]

Motions of the solar system

Aryabhata correctly insisted that the earth rotates about its axis daily, and that the apparentmovement of the stars is a relative motion caused by the rotation of the earth, contrary to thethen?prevailing view in other parts of the world, that the s2y rotated *his is indicated in thefirst chapter of theAryabhatiya, where he gives the number of rotations of the earth in a

yuga,[)%]and made more explicit in hisg!lachapter:[)+]

1n the same way that someone in a boat going forward sees an unmoving [obect] goingbac2ward, so [someone] on the eEuator sees the unmoving stars going uniformly westward*he cause of rising and setting [is that] the sphere of the stars together with the planets[apparentlyQ] turns due west at the eEuator, constantly pushed by the cosmic wind

Aryabhata described a geocentricmodel of the solar system, in which the 5un and 8oon areeach carried by epicycles *hey in turn revolve around the .arth 1n this model, which is alsofound in the(aitmahasiddhnta/c -. +)C0, the motions of the planets are each governed

by two epicycles, a smaller manda/slow0 and a largerghra/fast0 [)C]*he order of theplanets in terms of distance from earth is ta2en as: the8oon,8ercury,Genus, the 5un,8ars,Rupiter, 5aturn, and the asterisms"[9]

*he positions and periods of the planets was calculated relative to uniformly moving points1n the case of 8ercury and Genus, they move around the .arth at the same mean speed as the5un 1n the case of 8ars, Rupiter, and 5aturn, they move around the .arth at specific speeds,representing each planet!s motion through the Modiac 8ost historians of astronomy considerthat this two?epicycle model reflects elements of pre?4tolemaicLree2 astronomy[)6]Another
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