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Aryabhata (IAST: ryabhaa, Sanskrit: ) (476550 CE) was the first in the lineof great mathematician-astronomersfrom the classical age ofIndian mathematics andIndian astronomy. His most famous works are theryabhaya (499 CE, when he was 23years old) and theArya-siddhanta.

[edit] Biography

[edit] Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with othernames having the "bhatta" suffix, his name is properly spelled Aryabhata: everyastronomical text spells his name thus,[1] including Brahmagupta's references to him "inmore than a hundred places by name".[2] Furthermore, in most instances "Aryabhatta"does not fit the metre either.[1]

[edit] Birth

Aryabhata mentions in theAryabhatiya that it was composed 3,600 years into the KaliYuga, when he was 23 years old. This corresponds to 499 CE, and implies that he wasborn in 476 bc .[1]

Aryabhata provides no information about his place of birth. The only information comesfrom Bhskara I, who describes Aryabhata as makya, "one belonging to the amakacountry." It is widely attested that, during the Buddha's time, a branch of the Amaka

people settled in the region between theNarmada and Godavari rivers in central India,today the South GujaratNorth Maharashtra region. Aryabhata is believed to have beenborn there.[1][3]However, early Buddhist texts describe Ashmaka as being further south, indakshinapath or the Deccan, while other texts describe the Ashmakas as having foughtAlexander, which would put them further north.[3]

[edit] Work

It is fairly certain that, at some point, he went to Kusumapura for advanced studies andthat he lived there for some time.[4] Both Hindu and Buddhist tradition, as well asBhskara I (CE 629), identify Kusumapura asPaliputra, modernPatna.[1] A verse

mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and,because the university ofNalanda was in Pataliputra at the time and had an astronomicalobservatory, it is speculated that Aryabhata might have been the head of the Nalandauniversity as well.[1] Aryabhata is also reputed to have set up an observatory at the Suntemple in Taregana, Bihar.[5]

[edit] Other hypotheses

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It was suggested that Aryabhata may have been from Kerala, but K. V. Sarma, anauthority on Kerala's astronomical tradition, disagreed[1] and pointed out several errors inthis hypothesis.[6]

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.[7]

[edit] Works

Aryabhata is the author of several treatises onmathematicsandastronomy, some ofwhich are lost. His major work,Aryabhatiya, a compendium of mathematics andastronomy, was extensively referred to in the Indian mathematical literature and hassurvived to modern times. The mathematical part of theAryabhatiya coversarithmetic,algebra,plane trigonometry, and spherical trigonometry. It also contains continuedfractions, quadratic equations, sums-of-power series, and a table of sines.

TheArya-siddhanta, a lost work on astronomical computations, is known through thewritings of Aryabhata's contemporary, Varahamihira, and later mathematicians andcommentators, including Brahmagupta and Bhaskara I. This work appears to be based onthe olderSurya Siddhantaand uses the midnight-day reckoning, as opposed to sunrise inAryabhatiya. It also contained a description of several astronomical instruments: thegnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), acylindrical stickyasti-yantra, an umbrella-shaped device called the chhatra-yantra, andwater clocksof at least two types, bow-shaped and cylindrical.[3]

A third text, which may have survived in the Arabic translation, isAl ntforAl-nanf. It

claims that it is a translation by Aryabhata, but the Sanskrit name of this work is notknown. Probably dating from the 9th century, it is mentioned by the Persian scholar andchronicler of India, Ab Rayhn al-Brn.[3]

[edit] Aryabhatiya

Direct details of Aryabhata's work are known only from theAryabhatiya. The name"Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it aname. His disciple Bhaskara I calls itAshmakatantra (or the treatise from the Ashmaka).It is also occasionally referred to asArya-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, theexplication of meaning is due to commentators. The text consists of the 108 verses and 13introductory verses, and is divided into fourpdas or chapters:

1. Gitikapada: (13 verses): large units of timekalpa, manvantra, andyugawhichpresent a cosmology different from earlier texts such as Lagadha's VedangaJyotisha (c. 1st century BCE). There is also a table of sines (jya), given in a single

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verse. The duration of the planetary revolutions during a mahayuga is given as4.32 million years.

2. Ganitapada (33 verses): covering mensuration (ketra vyvahra), arithmetic andgeometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic,simultaneous, and indeterminate equations (kuTTaka)

3. Kalakriyapada (25 verses): different units of time and a method for determiningthe positions of planets for a given day, calculations concerning the intercalarymonth (adhikamAsa), kShaya-tithis, and a seven-day week with names for thedays of week.

4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere,features of the ecliptic, celestial equator, node, shape of the earth, cause of dayand night, rising ofzodiacal signs on horizon, etc. In addition, some versions citea few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy inverse form, which were influential for many centuries. The extreme brevity of the text

was elaborated in commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) and byNilakantha Somayaji in hisAryabhatiya Bhasya, (1465 CE).

[edit] Mathematics

[edit] Place value system and zero

Theplace-value system, first seen in the 3rd century Bakhshali Manuscript, was clearlyin place in his work. While he did not use a symbol for zero, the French mathematicianGeorges Ifrahargues that knowledge of zero was implicit in Aryabhata's place-valuesystem as a place holder for the powers of ten with null coefficients [8]

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic traditionfrom Vedic times, he used letters of the alphabet to denote numbers, expressingquantities, such as the table of sines in a mnemonic form.[9]

[edit] Approximation ofpi

Aryabhata worked on the approximation forpi (), and may have come to the conclusionthat is irrational. In the second part of the Aryabhatiyam (gaitapda 10), he writes:

caturadhikam atamaaguam dvaistath sahasrmayutadvayavikambhasysanno vttapariha."Add four to 100, multiply by eight, and then add 62,000. By this rule the circumferenceof a circle with a diameter of 20,000 can be approached." [10]

This implies that the ratio of the circumference to the diameter is((4+100)8+62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significantfigures.

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It is speculated that Aryabhata used the word sanna (approaching), to mean that notonly is this an approximation but that the value is incommensurable (or irrational). If thisis correct, it is quite a sophisticated insight, because the irrationality of pi was proved inEurope only in 1761 by Lambert.[11]

After Aryabhatiya was translated intoArabic(c. 820 CE) this approximation wasmentioned in Al-Khwarizmi's book on algebra.[3]

[edit] Mensuration and trigonometry

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 thearea."[12]

Aryabhata discussed the concept ofsine in his work by the name ofardha-jya. Literally,it means "half-chord". For simplicity, people started calling it jya. When Arabic writerstranslated his works from Sanskrit into Arabic, they referred it asjiba. However, inArabic writings, vowels are omitted, and it was abbreviated asjb. Later writerssubstituted it withjiab, meaning "cove" or "bay." (In Arabic,jiba is a meaningless word.)Later in the 12th century, whenGherardo of Cremona translated these writings fromArabic into Latin, he replaced the Arabicjiab with its Latin counterpart,sinus, whichmeans "cove" or "bay". And after that, thesinus becamesine in English.[13]

[edit] Indeterminate equations

A problem of great interest to Indian mathematicianssince ancient times has been to findinteger solutions to equations that have the form ax + by = c, a topic that has come to beknown as diophantine equations. This is an example from Bhskara's commentary onAryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as theremainder 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. Ingeneral, diophantine equations, such as this, can be notoriously difficult. They were

discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts mightdate to 800 BCE. Aryabhata's method of solving such problems is called the kuaka() method.Kuttaka means "pulverizing" or "breaking into small pieces", and themethod 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 forsolving first-order diophantine equations and is often referred to as theAryabhataalgorithm.[14]The diophantine equations are of interest in cryptology, and the RSAConference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.

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[edit] Algebra

InAryabhatiya Aryabhata provided elegant results for the summation ofseriesof squaresand cubes:[15]

and

[edit] Astronomy

Aryabhata's system of astronomy was called the audAyaka system, in which days are

reckoned from uday, dawn at lanka or "equator". Some of his later writings onastronomy, which apparently proposed a second model (orardha-rAtrikA, midnight) arelost but can be partly reconstructed from the discussion inBrahmagupta'skhanDakhAdyaka. In some texts, he seems to ascribe the apparent motions of the heavensto the Earth's rotation. He also treated the planet's orbits as ellipticalrather than circular.[16][17]

[edit] Motions of the solar system

Aryabhata appears to have believed that the earth rotates about its axis. This is indicatedin the statement, referring toLanka , 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 movingbackward, just so are the stationary stars seen by the people in Lanka (or on theequator) as moving exactly towards the west." [achalAni bhAnisamapashchimagAni golapAda.9]

But the next verse describes the motion of the stars and planets as real movements: "Thecause of their rising and setting is due to the fact that the circle of the asterisms, togetherwith 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 geocentricmodel of the solar system, in which the Sun and Moonare each carried by epicycles. They in turn revolve around the Earth. In this model, whichis also found in thePaitmahasiddhnta (c. CE 425), the motions of the planets are eachgoverned by two epicycles, a smallermanda (slow) and a largerghra (fast). [18] The

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order of the planets in terms of distance from earth is taken as: theMoon,Mercury,Venus, the Sun, Mars, Jupiter,Saturn, and the asterisms."[3]

The positions and periods of the planets was calculated relative to uniformly movingpoints. In the case of Mercury and Venus, they move around the Earth at the same mean

speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth atspecific speeds, representing each planet's motion through the zodiac. Most historians ofastronomy consider that this two-epicycle model reflects elements of pre-PtolemaicGreek astronomy.[19] Another element in Aryabhata's model, theghrocca, the basicplanetary period in relation to the Sun, is seen by some historians as a sign of anunderlying heliocentric model.[20]

[edit] Eclipses

Aryabhata states that the Moon and planets shine by reflected sunlight. Instead of theprevailing cosmogony in which eclipses were caused by pseudo-planetary nodes Rahu

andKetu, he explains eclipses in terms of shadows cast by and falling on Earth. Thus, thelunar eclipse occurs when the moon enters into the Earth's shadow (verse gola.37). Hediscusses at length the size and extent of the Earth's shadow (verses gola.3848) and thenprovides the computation and the size of the eclipsed part during an eclipse. Later Indianastronomers improved on the calculations, but Aryabhata's methods provided the core.His computational paradigm was so accurate that 18th century scientistGuillaume LeGentil, during a visit to Pondicherry, India, found the Indian computations of the durationof the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (byTobias Mayer, 1752) were long by 68 seconds.[3]

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 approximationwas a significant improvement over the computation byGreek mathematicianEratosthenes(c. 200 BCE), whose exact computation is not known in modern units buthis estimate had an error of around 510%.[21][22]

[edit] Sidereal periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation(the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1seconds; the modern value is 23:56:4.091. Similarly, his value for the length of thesidereal yearat 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 mostother astronomical systems of the time, but this computation was likely the most accurateof the period. [23].

[edit] Heliocentrism

As mentioned, Aryabhata claimed that the Earth turns on its own axis, and some elementsof his planetary epicyclic models rotate at the same speed as the motion of the Earth

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around the Sun. The planetary orbits were also given with respect to the Sun and he alsostates: "Whoever knows thisDasagitika Sutra which describes the movements of theEarth and the planets in the sphere of the asterisms passes through the paths of the planetsand asterisms and goes to the higher Brahman." Thus, it has been suggested thatAryabhata's calculations were based on an underlying heliocentric model, in which the

planets orbit the Sun.[24][25][26]

A detailed rebuttal to this heliocentric interpretation is in areview that describes B. L. van der Waerden's book as "show[ing] a completemisunderstanding of Indian planetary theory [that] is flatly contradicted by every word ofAryabhata's description."[27] However, some concede that Aryabhata's system stems froman earlier heliocentric model, of which he was unaware.[28]ThoughAristarchus of Samos(3rd century BCE) is credited with holding an heliocentric theory, the version ofGreekastronomy known in ancient India as thePaulisa Siddhanta makes no reference to such atheory.

[edit] Legacy

Aryabhata's work was of great influence in the Indian astronomical tradition andinfluenced several neighbouring cultures through translations. The Arabic translationduring the Islamic Golden Age (c. 820 CE), was particularly influential. Some of hisresults are cited by Al-Khwarizmiand in the 10th century Al-Biruni stated thatAryabhata's followers believed that the Earth rotated on its axis.

His definitions ofsine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine(otkram jya) influenced the birth oftrigonometry. He was also the first to specify sine andversine (1 cosx) tables, in 3.75 intervals from 0 to 90, to an accuracy of 4 decimalplaces.

In fact, modern names "sine" and "cosine" are mistranscriptions of the wordsjya andkojya as introduced by Aryabhata. As mentioned, they were translated asjiba and kojibain Arabic and then misunderstood by Gerard of Cremona while translating an Arabicgeometry text to Latin. He assumed thatjiba was the Arabic wordjaib, which means"fold in a garment", L.sinus (c. 1150).[29]

Aryabhata's astronomical calculation methods were also very influential. Along with thetrigonometric tables, they came to be widely used in the Islamic world and used tocompute many Arabic astronomical tables (zijes). In particular, the astronomical tables inthe work of the Arabic Spain scientist Al-Zarqali (11th century) were translated intoLatin as the Tables of Toledo (12th c.) and remained the most accurateephemeris used in

Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have been in continuoususe in India for the practical purposes of fixing the Panchangam(the Hindu calendar). Inthe Islamic world, they formed the basis of theJalali calendarintroduced in 1073 CE by agroup of astronomers including Omar Khayyam,[30] versions of which (modified in 1925)are the national calendars in use in IranandAfghanistantoday. The dates of the Jalalicalendar are based on actual solar transit, as in Aryabhata and earlier Siddhanta

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calendars. This type of calendar requires an ephemeris for calculating dates. Althoughdates were difficult to compute, seasonal errors were less in the Jalali calendar than in theGregorian calendar.

India's first satellite Aryabhataand the lunar craterAryabhata are named in his honour.

An Institute for conducting research in astronomy, astrophysics and atmospheric sciencesis the Aryabhatta Research Institute of Observational Sciences (ARIES) near Nainital,India. The inter-school Aryabhata Maths Competition is also named after him,[31]as isBacillus aryabhata, a species of bacteria discovered byISRO scientists in 2009.[32]

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