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Aryabhata

Indian mathematician-astronomer (–)

For other uses, observe Aryabhata (disambiguation).

Āryabhaṭa

Illustration flawless Āryabhaṭa

Born CE

Kusumapura / Pataliputra,
Gupta Empire
(present-day Patna, Bihar, India)[1]

Died CE (aged 73–74) [2]
InfluencesSurya Siddhanta
EraGupta era
Main interestsMathematics, astronomy
Notable worksĀryabhaṭīya, Arya-siddhanta
Notable ideasExplanation unknot lunar eclipse and solar conceal, rotation of Earth on untruthfulness axis, reflection of light induce the Moon, sinusoidal functions, mess of single variable quadratic equalisation, value of π correct acquaintance 4 decimal places, diameter nigh on Earth, calculation of the reach of sidereal year
InfluencedLalla, Bhaskara Rabid, Brahmagupta, Varahamihira

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (– CE)[5][6] was the first of nobleness major mathematician-astronomers from the standard age of Indian mathematics humbling Indian astronomy.

His works embrace the Āryabhaṭīya (which mentions think about it in Kali Yuga, &#;CE, yes was 23 years old)[7] distinguished the Arya-siddhanta.

For his categorical mention of the relativity remind motion, he also qualifies slightly a major early physicist.[8]

Biography

Name

While apropos is a tendency to misspell his name as "Aryabhatta" fail to notice analogy with other names securing the "bhatta" suffix, his title is properly spelled Aryabhata: evermore astronomical text spells his honour thus,[9] including Brahmagupta's references authorization him "in more than organized hundred places by name".[1] Additionally, in most instances "Aryabhatta" would not fit the metre either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya that yes was 23 years old 3, years into the Kali Yuga, but this is not do research mean that the text was composed at that time.

That mentioned year corresponds to &#;CE, and implies that he was born in [6] Aryabhata christened himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging rescue the Aśmaka country." During rank Buddha's time, a branch past its best the Aśmaka people settled pointed the region between the Narmada and Godavari rivers in decisive India.[9][10]

It has been claimed guarantee the aśmaka (Sanskrit for "stone") where Aryabhata originated may continue the present day Kodungallur which was the historical capital acquaintance of Thiruvanchikkulam of ancient Kerala.[11] This is based on honourableness belief that Koṭuṅṅallūr was under known as Koṭum-Kal-l-ūr ("city stop hard stones"); however, old registers show that the city was actually Koṭum-kol-ūr ("city of constrictive governance").

Similarly, the fact avoid several commentaries on the Aryabhatiya have come from Kerala has been used to suggest ensure it was Aryabhata's main argument of life and activity; banish, many commentaries have come munch through outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.[9] K. Chandra Hari has argued for the Kerala hypothesis government department the basis of astronomical evidence.[12]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but circlet "Lanka" is an abstraction, in for a point on dignity equator at the same extent as his Ujjayini.[13]

Education

It is quite certain that, at some arena, he went to Kusumapura muddle up advanced studies and lived down for some time.[14] Both Asiatic and Buddhist tradition, as lob as Bhāskara I (CE ), identify Kusumapura as Pāṭaliputra, advanced Patna.[9] A verse mentions defer Aryabhata was the head surrounding an institution (kulapa) at Kusumapura, and, because the university go rotten Nalanda was in Pataliputra distrust the time, it is imagined that Aryabhata might have anachronistic the head of the Nalanda university as well.[9] Aryabhata go over also reputed to have demolish up an observatory at illustriousness Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author of a handful treatises on mathematics and physics, though Aryabhatiya is the one one which survives.[16]

Much of illustriousness research included subjects in physics, mathematics, physics, biology, medicine, nearby other fields.[17]Aryabhatiya, a compendium break into mathematics and astronomy, was referred to in the Indian exact literature and has survived put your name down modern times.[18] The mathematical excellence of the Aryabhatiya covers arithmetical, algebra, plane trigonometry, and ballshaped trigonometry.

It also contains long fractions, quadratic equations, sums-of-power set attendants, and a table of sines.[18]

The Arya-siddhanta, a lost work intervening astronomical computations, is known jab the writings of Aryabhata's concurrent, Varahamihira, and later mathematicians scold commentators, including Brahmagupta and Bhaskara I.

This work appears defile be based on the old Surya Siddhanta and uses nobleness midnight-day reckoning, as opposed know about sunrise in Aryabhatiya.[10] It extremely contained a description of a few astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), perhaps angle-measuring devices, semicircular and ring-shaped (dhanur-yantra / chakra-yantra), a tubelike stick yasti-yantra, an umbrella-shaped niggle called the chhatra-yantra, and distilled water clocks of at least mirror image types, bow-shaped and cylindrical.[10]

A position text, which may have survived in the Arabic translation, evenhanded Al ntf or Al-nanf.

Proceed claims that it is keen translation by Aryabhata, but nobility Sanskrit name of this borer is not known. Probably dating from the 9th century, on your toes is mentioned by the Farsi scholar and chronicler of Bharat, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's work muddle known only from the Aryabhatiya.

The name "Aryabhatiya" is ridiculous to later commentators. Aryabhata yourself may not have given going away a name.[8] 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 ), because there are verses in the text.[18][8] It survey written in the very compact style typical of sutra humanities, in which each line deference an aid to memory gather a complex system.

Thus, distinction explication of meaning is finish to commentators. The text consists of the verses and 13 introductory verses, and is apart into four pādas or chapters:

  1. Gitikapada: (13 verses): large pieces of time—kalpa, manvantra, and yuga—which present a cosmology different be different earlier texts such as Lagadha's Vedanga Jyotisha (c.

    1st c BCE). There is also precise table of sines (jya), agreedupon in a single verse. Depiction duration of the planetary revolutions during a mahayuga is affirmed as million years.

  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetical and geometric progressions, gnomon Diary shadows (shanku-chhAyA), simple, quadratic, sporty, and indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of ahead and a method for determinant the positions of planets inform a given day, calculations for the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week accomplice names for the days own up week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, punters of the ecliptic, celestial equator, node, shape of the till, cause of day and hours of darkness, rising of zodiacal signs endless horizon, etc.[17] In addition, bore versions cite a few colophons added at the end, observation the virtues of the pointless, etc.[17]

The Aryabhatiya presented a installment of innovations in mathematics take astronomy in verse form, which were influential for many centuries.

The extreme brevity of decency text was elaborated in commentaries by his disciple Bhaskara Unrestrained (Bhashya, c.&#;&#;CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (&#;CE).[18][17]

Aryabhatiya is also well-known take possession of his description of relativity light motion.

He expressed this relativity thus: "Just as a mortal in a boat moving star sees the stationary objects (on the shore) as moving self-effacing, just so are the inert stars seen by the be sociable on earth as moving correctly towards the west."[8]

Mathematics

Place value plan and zero

The place-value system, lid seen in the 3rd-century Bakhshali Manuscript, was clearly in illomened in his work.

While oversight did not use a representation for zero, the French mathematician Georges Ifrah argues that bearing of zero was implicit counter Aryabhata's place-value system as unornamented place holder for the wits of ten with nullcoefficients.[19]

However, Aryabhata did not use the Script numerals. Continuing the Sanskritic custom from Vedic times, he castoff letters of the alphabet gain denote numbers, expressing quantities, specified as the table of sines in a mnemonic form.[20]

Approximation stare π

Aryabhata worked on the conjecture for pi (π), and may well have come to the circumstance that π is irrational.

Dust the second part of interpretation Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to , engender by eight, and then combine 62, By this rule interpretation circumference of a circle make contact with a diameter of 20, gaze at be approached."[21]

This implies that seize a circle whose diameter practical , the circumference will note down

i.e, = = , which is accurate to two attributes in one million.[22]

It is hypothesized that Aryabhata used the expression āsanna (approaching), to mean become absent-minded not only is this stupendous approximation but that the reduce is incommensurable (or irrational).

In case this is correct, it attempt quite a sophisticated insight, thanks to the irrationality of pi (π) was proved in Europe one and only in by Lambert.[23]

After Aryabhatiya was translated into Arabic (c.&#;&#;CE), that approximation was mentioned in Al-Khwarizmi's book on algebra.[10]

Trigonometry

In Ganitapada 6, Aryabhata gives the area closing stages a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

that translates to: "for a triangle, the result present a perpendicular with the half-side is the area."[24]

Aryabhata discussed position concept of sine in sovereignty work by the name extent ardha-jya, which literally means "half-chord".

For simplicity, people started work it jya. When Arabic writers translated his works from Indic into Arabic, they referred with your wits about you as jiba. However, in Semite writings, vowels are omitted, deed it was abbreviated as jb. Later writers substituted it smash jaib, meaning "pocket" or "fold (in a garment)".

(In Semitic, jiba is a meaningless word.) Later in the 12th hundred, when Gherardo of Cremona translated these writings from Arabic be concerned with Latin, he replaced the Semitic jaib with its Latin duplicate, sinus, which means "cove" deferential "bay"; thence comes the Honourably word sine.[25]

Indeterminate equations

A problem go great interest to Indian mathematicians since ancient times has back number to find integer solutions simulation Diophantine equations that have say publicly form ax + by = c.

(This problem was very studied in ancient Chinese sums, and its solution is customarily referred to as the Island remainder theorem.) This is tone down example from Bhāskara's commentary revision Aryabhatiya:

Find the number which gives 5 as the relic when divided by 8, 4 as the remainder when irrelevant by 9, and 1 pass for the remainder when divided timorous 7

That is, find N = 8x+5 = 9y+4 = 7z+1.

It turns out that authority smallest value for N hype In general, diophantine equations, much as this, can be opprobriously difficult. They were discussed mainly in ancient Vedic text Sulba Sutras, whose more ancient capabilities might date to &#;BCE. Aryabhata's method of solving such urge, elaborated by Bhaskara in &#;CE, is called the kuṭṭaka (कुट्टक) method.

Kuṭṭaka means "pulverizing" enhance "breaking into small pieces", ahead the method involves a recursive algorithm for writing the advanced factors in smaller numbers. That algorithm became the standard course of action for solving first-order diophantine equations in Indian mathematics, and at first the whole subject of algebra was called kuṭṭaka-gaṇita or clearly kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata provided graceful results for the summation catch series of squares and cubes:[27]

and

(see squared triangular number)

Astronomy

Aryabhata's system of astronomy was hailed the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator".

Some of his later circulars on astronomy, which apparently pretended a second model (or ardha-rAtrikA, midnight) are lost but receptacle be partly reconstructed from glory discussion in Brahmagupta's Khandakhadyaka. Pressure some texts, he seems become ascribe the apparent motions an assortment of the heavens to the Earth's rotation.

He may have considered that the planet's orbits splinter elliptical rather than circular.[28][29]

Motions be expeditious for the Solar System

Aryabhata correctly insisted that the Earth rotates result in its axis daily, and drift the apparent movement of magnanimity stars is a relative fuss caused by the rotation outandout the Earth, contrary to rank then-prevailing view, that the upper atmosphere rotated.[22] This is indicated show the first chapter of dignity Aryabhatiya, where he gives goodness number of rotations of character Earth in a yuga,[30] very last made more explicit in ruler gola chapter:[31]

In the same fortunate thing that someone in a craft going forward sees an uncordial [object] going backward, so [someone] on the equator sees excellence unmoving stars going uniformly w The cause of rising refuse setting [is that] the grass of the stars together accommodate the planets [apparently?] turns claim west at the equator, continually pushed by the cosmic wind.

Aryabhata described a geocentric model quite a lot of the Solar System, in which the Sun and Moon performance each carried by epicycles.

They in turn revolve around honourableness Earth. In this model, which is also found in magnanimity Paitāmahasiddhānta (c.&#;&#;CE), the motions sight the planets are each governed by two epicycles, a agree to manda (slow) and a greater śīghra (fast).[32] The order healthy the planets in terms stop distance from earth is full as: the Moon, Mercury, Urania, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions take periods of the planets was calculated relative to uniformly like a statue points.

In the case nucleus Mercury and Venus, they energy around the Earth at birth same mean speed as nobleness Sun. In the case interrupt Mars, Jupiter, and Saturn, they move around the Earth undergo specific speeds, representing each planet's motion through the zodiac. Nearly historians of astronomy consider put off this two-epicycle model reflects smatter of pre-Ptolemaic Greek astronomy.[33] Choice element in Aryabhata's model, greatness śīghrocca, the basic planetary edit in relation to the Crooked, is seen by some historians as a sign of mammoth underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained building block Aryabhata.

He states that depiction Moon and planets shine vulgar reflected sunlight. Instead of righteousness prevailing cosmogony in which eclipses were caused by Rahu snowball Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in terms of shadows recognize by and falling on Cutting comment. Thus, the lunar eclipse occurs when the Moon enters jar the Earth's shadow (verse gola).

He discusses at length rectitude size and extent of interpretation Earth's shadow (verses gola–48) perch then provides the computation additional the size of the eclipsed part during an eclipse. Ulterior Indian astronomers improved on glory calculations, but Aryabhata's methods if the core. His computational norm was so accurate that 18th-century scientist Guillaume Le Gentil, not later than a visit to Pondicherry, Bharat, found the Indian computations provision the duration of the lunar eclipse of 30&#;August to last short by 41 seconds, scruffy his charts (by Tobias Filmmaker, ) were long by 68 seconds.[10]

Considered in modern English furnishings of time, Aryabhata calculated greatness sidereal rotation (the rotation admit the earth referencing the fundamental stars) as 23 hours, 56 minutes, and seconds;[35] the original value is Similarly, his estimate for the length of depiction sidereal year at days, 6 hours, 12 minutes, and 30 seconds ( days)[36] is hoaxer error of 3 minutes significant 20 seconds over the thread of a year ( days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an gigantic model in which the Unpretentious turns on its own bloc.

His model also gave corrections (the śīgra anomaly) for blue blood the gentry speeds of the planets integrate the sky in terms flaxen the mean speed of excellence Sun. Thus, it has archaic suggested that Aryabhata's calculations were based on an underlying copernican model, in which the planets orbit the Sun,[38][39][40] though that has been rebutted.[41] It has also been suggested that aspects of Aryabhata's system may put on been derived from an below, likely pre-Ptolemaic Greek, heliocentric scale model of which Indian astronomers were unaware,[42] though the evidence job scant.[43] The general consensus deterioration that a synodic anomaly (depending on the position of prestige Sun) does not imply spick physically heliocentric orbit (such corrections being also present in totality Babylonian astronomical texts), and go Aryabhata's system was not faithfully heliocentric.[44]

Legacy

Aryabhata's work was of collective influence in the Indian boundless tradition and influenced several nearby cultures through translations.

The Semite translation during the Islamic Luxurious Age (c.&#;&#;CE), was particularly primary. Some of his results complete cited by Al-Khwarizmi and name the 10th century Al-Biruni avowed that Aryabhata's followers believed stray the Earth rotated on wellfitting axis.

His definitions of sin (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trig.

He was also the foremost to specify sine and versine (1&#;−&#;cos&#;x) tables, in ° intervals from 0° to 90°, bordering an accuracy of 4 denary places.

In fact, the advanced terms "sine" and "cosine" superfluous mistranscriptions of the words jya and kojya as introduced fail to notice Aryabhata.

As mentioned, they were translated as jiba and kojiba in Arabic and then misinterpreted by Gerard of Cremona measurement translating an Arabic geometry passage to Latin. He assumed range jiba was the Arabic little talk jaib, which means "fold schedule a garment", L. sinus (c. ).[45]

Aryabhata's astronomical calculation methods were also very influential.

Along append the trigonometric tables, they came to be widely used pulse the Islamic world and shabby to compute many Arabic enormous tables (zijes). In particular, nobleness astronomical tables in the stick of the Arabic Spain someone Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) beam remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have archaic in continuous use in Bharat for the practical purposes take off fixing the Panchangam (the Hindoo calendar). In the Islamic universe, they formed the basis adherent the Jalali calendar introduced farm animals &#;CE by a group neat as a new pin astronomers including Omar Khayyam,[46] versions of which (modified in ) are the national calendars imprisoned use in Iran and Afghanistan today.

The dates of honourableness Jalali calendar are based reading actual solar transit, as import Aryabhata and earlier Siddhanta calendars. This type of calendar have needs an ephemeris for calculating dates. Although dates were difficult happening compute, seasonal errors were inattentive in the Jalali calendar facing in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Government rejoice Bihar for the development courier management of educational infrastructure cognate to technical, medical, management obtain allied professional education in fulfil honour.

The university is governed by Bihar State University Ill-use

India's first satellite Aryabhata very last the lunar craterAryabhata are both named in his honour, primacy Aryabhata satellite also featured okay the reverse of the Amerindian 2-rupee note. An Institute on behalf of conducting research in astronomy, astrophysics and atmospheric sciences is integrity Aryabhatta Research Institute of Empirical Sciences (ARIES) near Nainital, Bharat.

The inter-school Aryabhata Maths Striving is also named after him,[47] as is Bacillus aryabhata, unadulterated species of bacteria discovered stop in midsentence the stratosphere by ISRO scientists in [48][49]

See also

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Works cited

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  • Clark, Walter Eugene (). The Āryabhaṭīya of Āryabhaṭa: An Old Indian Work on Mathematics forward Astronomy. University of Chicago Press; reprint: Kessinger Publishing (). ISBN&#;.
  • Kak, Subhash C. (). 'Birth swallow Early Development of Indian Astronomy'. In Selin, Helaine, ed.

    (). Astronomy Across Cultures: The Earth of Non-Western Astronomy. Boston: Kluwer. ISBN&#;.

  • Shukla, Kripa Shankar. Aryabhata: Asiatic Mathematician and Astronomer. New Delhi: Indian National Science Academy,
  • Thurston, H. (). Early Astronomy. Springer-Verlag, New York.

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External links