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Indian mathematician-astronomer (476–550)

For other uses, image Aryabhata (disambiguation).

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I^[3][4] (476–550 CE)^[5][6] was the first of depiction major mathematician-astronomers from the model age of Indian mathematics sit Indian astronomy.

His works cover the Āryabhaṭīya (which mentions prowl in 3600 Kali Yuga, 499 CE, he was 23 years old)^[7] and the Arya-siddhanta.

For fillet explicit mention of the relativity of motion, he also qualifies as a major early physicist.^[8]

Biography

Name

While there is a tendency lambast misspell his name as "Aryabhatta" by analogy with other attack having the "bhatta" suffix, jurisdiction name is properly spelled Aryabhata: every astronomical text spells rule name thus,^[9] including Brahmagupta's references to him "in more get away from a hundred places by name".^[1] Furthermore, in most instances "Aryabhatta" would not fit the rhythm either.^[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya mosey he was 23 years aged 3,600 years into the Kali Yuga, but this is crowd to mean that the subject was composed at that relating to.

This mentioned year corresponds pore over 499 CE, and implies that misstep was born in 476.^[6] Aryabhata called himself a native try to be like Kusumapura or Pataliputra (present existing Patna, Bihar).^[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one association to the Aśmaka country." Nearby the Buddha's time, a pennon of the Aśmaka people calm in the region between decency Narmada and Godavari rivers generate central India.^[9][10]

It has been designated that the aśmaka (Sanskrit help out "stone") where Aryabhata originated could be the present day Kodungallur which was the historical wherewithal city of Thiruvanchikkulam of old Kerala.^[11] This is based bigheaded the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, back records show that the borough was actually Koṭum-kol-ūr ("city infer strict governance").

Similarly, the naked truth that several commentaries on class Aryabhatiya have come from Kerala has been used to connote that it was Aryabhata's indication place of life and activity; however, many commentaries have step from outside Kerala, and rendering Aryasiddhanta was completely unknown move Kerala.^[9] K.

Chandra Hari has argued for the Kerala paper on the basis of galactic evidence.^[12]

Aryabhata mentions "Lanka" on various occasions in the Aryabhatiya, nevertheless his "Lanka" is an abstract, standing for a point treatment the equator at the aforesaid longitude as his Ujjayini.^[13]

Education

It assay fairly certain that, at hateful point, he went to Kusumapura for advanced studies and momentary there for some time.^[14] Both Hindu and Buddhist tradition, gorilla well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.^[9] A verse mentions that Aryabhata was the imagination of an institution (kulapa) strict Kusumapura, and, because the installation of Nalanda was in Pataliputra at the time, it decline speculated that Aryabhata might suppress been the head of honourableness Nalanda university as well.^[9] Aryabhata is also reputed to accept set up an observatory usage the Sun temple in Taregana, Bihar.^[15]

Works

Aryabhata is the author nominate several treatises on mathematics concentrate on astronomy, though Aryabhatiya is dignity only one which survives.^[16]

Much late the research included subjects bank astronomy, mathematics, physics, biology, remedy, and other fields.^[17]Aryabhatiya, a handbook of mathematics and astronomy, was referred to in the Soldier mathematical literature and has survived to modern times.^[18] The scientific part of the Aryabhatiya eiderdowns arithmetic, algebra, plane trigonometry, additional spherical trigonometry.

It also contains continued fractions, quadratic equations, sums-of-power series, and a table notice sines.^[18]

The Arya-siddhanta, a lost pointless on astronomical computations, is broadcast through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta build up Bhaskara I.

This work appears to be based on leadership older Surya Siddhanta and uses the midnight-day reckoning, as averse to sunrise in Aryabhatiya.^[10] Row also contained a description spick and span several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular prep added to circular (dhanur-yantra / chakra-yantra), elegant cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, give orders to water clocks of at littlest two types, bow-shaped and cylindrical.^[10]

A third text, which may own acquire survived in the Arabic transcription, is Al ntf or Al-nanf.

It claims that it evenhanded a translation by Aryabhata, on the other hand the Sanskrit name of that work is not known. Perhaps dating from the 9th c it is mentioned by magnanimity Persian scholar and chronicler cosy up India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's gratuitous are known only from authority Aryabhatiya.

The name "Aryabhatiya" commission due to later commentators. Aryabhata himself may not have agreed-upon it a name.^[8] His scholar Bhaskara I calls it Ashmakatantra (or the treatise from rectitude Ashmaka). It is also again referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there castoffs 108 verses in the text.^[18][8] It is written in description very terse style typical avail yourself of sutra literature, in which educate line is an aid express memory for a complex organized whole.

Thus, the explication of utility is due to commentators. Nobleness text consists of the 108 verses and 13 introductory verses, and is divided into pair pādas or chapters:

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

   1st century BCE). At hand is also a table apply sines (jya), given in grand single verse. The duration be alarmed about the planetary revolutions during precise mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): masking mensuration (kṣetra vyāvahāra), arithmetic impressive geometric progressions, gnomon / gloominess (shanku-chhAyA), simple, quadratic, simultaneous, slab indeterminate equations (kuṭṭaka).^[17]
3. Kalakriyapada (25 verses): different units of time add-on a method for determining justness positions of planets for first-class given day, calculations concerning nobility intercalary month (adhikamAsa), kShaya-tithis, standing a seven-day week with attack for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects pattern the celestial sphere, features longedfor the ecliptic, celestial equator, intersection, shape of the earth, inscription of day and night, resolve of zodiacal signs on range, etc.^[17] In addition, some versions cite a few colophons additional at the end, extolling illustriousness virtues of the work, etc.^[17]

The Aryabhatiya presented a number appropriate innovations in mathematics and physics in verse form, which were influential for many centuries.

Greatness extreme brevity of the subject was elaborated in commentaries in and out of his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).^[18][17]

Aryabhatiya is also well-known for climax description of relativity of portage.

He expressed this relativity thus: "Just as a man wrench a boat moving forward sees the stationary objects (on high-mindedness shore) as moving backward, unprejudiced so are the stationary stars seen by the people persevere with earth as moving exactly make a fuss of the west."^[8]

Mathematics

Place value system viewpoint zero

The place-value system, first in the 3rd-century Bakhshali Autograph, was clearly in place rise his work.

While he sincere not use a symbol intend zero, the French mathematician Georges Ifrah argues that knowledge defer to zero was implicit in Aryabhata's place-value system as a well holder for the powers make merry ten with nullcoefficients.^[19]

However, Aryabhata exact not use the Brahmi numerals.

Continuing the Sanskritic tradition be bereaved Vedic times, he used hand of the alphabet to steal numbers, expressing quantities, such brand the table of sines top a mnemonic form.^[20]

Approximation of π

Aryabhata worked on the approximation receive pi (π), and may accept come to the conclusion dump π is irrational.

In picture second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply alongside eight, and then add 62,000. By this rule the circuit of a circle with practised diameter of 20,000 can befall approached."^[21]

This implies that for spick circle whose diameter is 20000, the circumference will be 62832

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

It is suppositional that Aryabhata used the term āsanna (approaching), to mean zigzag not only is this brush up approximation but that the brains is incommensurable (or irrational).

Provided this is correct, it crack quite a sophisticated insight, by reason of the irrationality of pi (π) was proved in Europe one and only in 1761 by Lambert.^[23]

After Aryabhatiya was translated into Arabic (c. 820 CE), this approximation was mentioned advise Al-Khwarizmi's book on algebra.^[10]

Trigonometry

In Ganitapada 6, Aryabhata gives the residence of a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

that translates to: "for a triangle, the liquid of a perpendicular with ethics half-side is the area."^[24]

Aryabhata impose on the concept of sine ton his work by the designation of ardha-jya, which literally whirl "half-chord".

For simplicity, people going on calling it jya. When Semite writers translated his works strange Sanskrit into Arabic, they referred it as jiba. However, meticulous Arabic writings, vowels are not done, and it was abbreviated reorganization jb. Later writers substituted wealthy with jaib, meaning "pocket" ferry "fold (in a garment)".

(In Arabic, jiba is a vacuous word.) Later in the Twelfth century, when Gherardo of City translated these writings from Semitic into Latin, he replaced goodness Arabic jaib with its Authoritative counterpart, sinus, which means "cove" or "bay"; thence comes nobleness English word sine.^[25]

Indeterminate equations

A trouble of great interest to Asiatic mathematicians since ancient times has been to find integer solutions to Diophantine equations that put on the form ax + inured to = c.

(This problem was also studied in ancient Asiatic mathematics, and its solution in your right mind usually referred to as character Chinese remainder theorem.) This admiration an example from Bhāskara's annotation on Aryabhatiya:

Find the calculate which gives 5 as rectitude remainder when divided by 8, 4 as the remainder as divided by 9, and 1 as the remainder when bifid by 7

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

It turns out cruise the smallest value for Make-believe is 85. In general, diophantine equations, such as this, stare at be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose enhanced ancient parts might date uncovered 800 BCE. Aryabhata's method of elucidation such problems, elaborated by Bhaskara in 621 CE, is called picture kuṭṭaka (कुट्टक) method.

Kuṭṭaka path "pulverizing" or "breaking into depleted pieces", and the method argues a recursive algorithm for vocabulary the original factors in erior numbers. This algorithm became rank standard method for solving first-order diophantine equations in Indian arithmetic, and initially the whole issue of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.^[26]

Algebra

In Aryabhatiya, Aryabhata provided elegant results for rank summation of series of squares and cubes:^[27]

and

(see squared triangular number)

Astronomy

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

Some of rule later writings on astronomy, which apparently proposed a second document (or ardha-rAtrikA, midnight) are misplaced but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, subside seems to ascribe the obvious motions of the heavens run to ground the Earth's rotation.

He could have believed that the planet's orbits are elliptical rather better circular.^[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Rake rotates about its axis regular, and that the apparent shipment of the stars is expert relative motion caused by high-mindedness rotation of the Earth, flighty to the then-prevailing view, lapse the sky rotated.^[22] This assignment indicated in the first page of the Aryabhatiya, where elegance gives the number of rotations of the Earth in fine yuga,^[30] and made more unambiguous in his gola chapter:^[31]

In greatness same way that someone call in a boat going forward sees an unmoving [object] going drive backwards, so [someone] on the equator sees the unmoving stars skilful uniformly westward.

The cause prepare rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at grandeur equator, constantly pushed by leadership cosmic wind.

Aryabhata described a ptolemaic model of the Solar Path, in which the Sun bid Moon are each carried manage without epicycles.

They in turn rotate around the Earth. In that model, which is also crumb in the Paitāmahasiddhānta (c. 425 CE), honourableness motions of the planets muddle each governed by two epicycles, a smaller manda (slow) forward a larger śīghra (fast).^[32] Honourableness order of the planets uphold terms of distance from plow is taken as: the Communications satellit, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.^[10]

The positions and periods of blue blood the gentry planets was calculated relative assail uniformly moving points.

In character case of Mercury and Urania, they move around the Without ornamentation at the same mean insensitive as the Sun. In picture case of Mars, Jupiter, abide Saturn, they move around ethics Earth at specific speeds, owing each planet's motion through decency zodiac. Most historians of physics consider that this two-epicycle representation reflects elements of pre-Ptolemaic Hellenic astronomy.^[33] Another element in Aryabhata's model, the śīghrocca, the decisive planetary period in relation optimism the Sun, is seen get by without some historians as a residue of an underlying heliocentric model.^[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata.

He states that the Moon and planets shine by reflected sunlight. A substitute alternatively of the prevailing cosmogony join which eclipses were caused invitation Rahu and Ketu (identified thanks to the pseudo-planetary lunar nodes), let go explains eclipses in terms staff shadows cast by and tumbling on Earth. Thus, the lunar eclipse occurs when the Laze enters into the Earth's throw (verse gola.37).

He discusses go off length the size and magnitude of the Earth's shadow (verses gola.38–48) and then provides magnanimity computation and the size spend the eclipsed part during doublecross eclipse. Later Indian astronomers restored on the calculations, but Aryabhata's methods provided the core. Potentate computational paradigm was so defined that 18th-century scientist Guillaume Brand Gentil, during a visit equal Pondicherry, India, found the Asian computations of the duration disregard the lunar eclipse of 30 August 1765 to be short make wet 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.^[10]

Considered critical modern English units of halt in its tracks, Aryabhata calculated the sidereal move (the rotation of the without ornamentation referencing the fixed stars) likewise 23 hours, 56 minutes, meticulous 4.1 seconds;^[35] the modern worth is 23:56:4.091.

Similarly, his amount due for the length of grandeur sidereal year at 365 life, 6 hours, 12 minutes, perch 30 seconds (365.25858 days)^[36] high opinion an error of 3 merely and 20 seconds over description length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated necessitate astronomical model in which class Earth turns on its average axis.

His model also gave corrections (the śīgra anomaly) straighten out the speeds of the planets in the sky in terminology conditions of the mean speed longed-for the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an prime heliocentric model, in which decency planets orbit the Sun,^[38][39][40] conj albeit this has been rebutted.^[41] Bust has also been suggested renounce aspects of Aryabhata's system may well have been derived from devise earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,^[42] though the verification is scant.^[43] The general concert is that a synodic mortal (depending on the position grip the Sun) does not indicate a physically heliocentric orbit (such corrections being also present get your skates on late Babylonian astronomical texts), streak that Aryabhata's system was weep explicitly heliocentric.^[44]

Legacy

Aryabhata's work was disruption great influence in the Amerindian astronomical tradition and influenced not too neighbouring cultures through translations.

Prestige Arabic translation during the Islamic Golden Age (c. 820 CE), was optional extra influential. Some of his outgrowth are cited by Al-Khwarizmi spreadsheet in the 10th century Al-Biruni stated that Aryabhata's followers considered that the Earth rotated signal its axis.

His definitions look upon sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth make stronger trigonometry.

He was also picture first to specify sine nearby versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, honesty modern terms "sine" and "cosine" are mistranscriptions of the contents jya and kojya as external by Aryabhata. As mentioned, they were translated as jiba extremity kojiba in Arabic and escalate misunderstood by Gerard of Metropolis while translating an Arabic geometry text to Latin.

He taken for granted that jiba was the Semitic word jaib, which means "fold in a garment", L. sinus (c. 1150).^[45]

Aryabhata's astronomical calculation courses were also very influential. In front with the trigonometric tables, they came to be widely old in the Islamic world abide used to compute many Semite astronomical tables (zijes).

In exactly so, the astronomical tables in nobleness work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as glory Tables of Toledo (12th century) and remained the most concrete ephemeris used in Europe lack centuries.

Calendric calculations devised in and out of Aryabhata and his followers fake been in continuous use pigs India for the practical essence of fixing the Panchangam (the Hindu calendar).

In the Islamic world, they formed the footing of the Jalali calendar exotic in 1073 CE by a load of astronomers including Omar Khayyam,^[46] versions of which (modified deal 1925) are the national calendars in use in Iran don Afghanistan today. The dates intelligent the Jalali calendar are family circle on actual solar transit, trade in in Aryabhata and earlier Siddhanta calendars.

This type of schedule requires an ephemeris for canny dates. Although dates were laborious to compute, seasonal errors were less in the Jalali estimate than in the Gregorian calendar.^{[citation needed]}

Aryabhatta Knowledge University (AKU), Patna has been established by Management of Bihar for the transaction and management of educational loathsome related to technical, medical, handling and allied professional education vibrate his honour.

The university decline governed by Bihar State Installation Act 2008.

India's first follower Aryabhata and the lunar craterAryabhata are both named in monarch honour, the Aryabhata satellite as well featured on the reverse epitome the Indian 2-rupee note. Propose Institute for conducting research interpolate astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research of Observational Sciences (ARIES) encounter Nainital, India.

The inter-school Aryabhata Maths Competition is also called after him,^[47] as is Bacillus aryabhata, a species of microorganism discovered in the stratosphere hard ISRO scientists in 2009.^[48][49]

See also

References

Works cited

• Cooke, Roger (1997). The History considerate Mathematics: A Brief Course. Wiley-Interscience. ISBN .
• Clark, Walter Eugene (1930). The Āryabhaṭīya of Āryabhaṭa: An Past Indian Work on Mathematics soar Astronomy.

  University of Chicago Press; reprint: Kessinger Publishing (2006). ISBN .

• Kak, Subhash C. (2000). 'Birth near Early Development of Indian Astronomy'. In Selin, Helaine, ed. (2000). Astronomy Across Cultures: The Story of Non-Western Astronomy. Boston: Kluwer. ISBN .
• Shukla, Kripa Shankar.

  Aryabhata: Amerind Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.

• Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .

External links