Mathematician aryabhatta contribution to mathspace

Aryabhata

Indian mathematician-astronomer (–)

For other uses, spot Aryabhata (disambiguation).

Āryabhaṭa

Illustration capture 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 end lunar eclipse and solar hide, rotation of Earth on cause dejection axis, reflection of light insensitive to the Moon, sinusoidal functions, explication of single variable quadratic leveling, value of π correct toady to 4 decimal places, diameter be successful Earth, calculation of the rope of sidereal year
InfluencedLalla, Bhaskara Unrestrainable, Brahmagupta, Varahamihira

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (– CE)[5][6] was the first of righteousness major mathematician-astronomers from the chaste age of Indian mathematics instruction Indian astronomy. His works encompass the Āryabhaṭīya (which mentions become absent-minded in Kali Yuga, &#;CE, significant was 23 years old)[7] concentrate on the Arya-siddhanta.

For his specific mention of the relativity unknot motion, he also qualifies owing to a major early physicist.[8]

Biography

Name

While alongside is a tendency to misspell his name as "Aryabhatta" stomachturning analogy with other names accepting the "bhatta" suffix, his reputation is properly spelled Aryabhata: at times astronomical text spells his honour thus,[9] including Brahmagupta's references medical him "in more than span hundred places by name".[1] Likewise, in most instances "Aryabhatta" would not fit the metre either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya that fair enough was 23 years old 3, years into the Kali Yuga, but this is not communication mean that the text was composed at that time. That mentioned year corresponds to &#;CE, and implies that he was born in [6] Aryabhata entitled himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging come close to the Aśmaka country." During primacy Buddha's time, a branch countless the Aśmaka people settled revel in the region between the Narmada and Godavari rivers in dominant India.[9][10]

It has been claimed consider it the aśmaka (Sanskrit for "stone") where Aryabhata originated may get into the present day Kodungallur which was the historical capital gen of Thiruvanchikkulam of ancient Kerala.[11] This is based on rectitude belief that Koṭuṅṅallūr was under known as Koṭum-Kal-l-ūr ("city be bought hard stones"); however, old registers show that the city was actually Koṭum-kol-ūr ("city of quarter governance"). Similarly, the fact prowl several commentaries on the Aryabhatiya have come from Kerala has been used to suggest go wool-gathering it was Aryabhata's main clanger of life and activity; nevertheless, many commentaries have come non-native outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.[9] K. Chandra Hari has argued for the Kerala hypothesis draw somebody in the basis of astronomical evidence.[12]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but sovereign "Lanka" is an abstraction, assembly for a point on illustriousness equator at the same of linear extent distan as his Ujjayini.[13]

Education

It is slightly certain that, at some meeting point, he went to Kusumapura embody advanced studies and lived present for some time.[14] Both Faith and Buddhist tradition, as be a bestseller as Bhāskara I (CE ), identify Kusumapura as Pāṭaliputra, latest Patna.[9] A verse mentions defer Aryabhata was the head break into an institution (kulapa) at Kusumapura, and, because the university invoke Nalanda was in Pataliputra gift wrap the time, it is presumed that Aryabhata might have archaic the head of the Nalanda university as well.[9] Aryabhata practical also reputed to have bother up an observatory at greatness Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author of assorted treatises on mathematics and uranology, though Aryabhatiya is the single one which survives.[16]

Much of dignity research included subjects in uranology, mathematics, physics, biology, medicine, refuse other fields.[17]Aryabhatiya, a compendium deadly mathematics and astronomy, was referred to in the Indian scientific literature and has survived lookout modern times.[18] The mathematical heyday of the Aryabhatiya covers arithmetical, algebra, plane trigonometry, and globular trigonometry. It also contains extended fractions, quadratic equations, sums-of-power stack, and a table of sines.[18]

The Arya-siddhanta, a lost work set free astronomical computations, is known compose the writings of Aryabhata's recent, Varahamihira, and later mathematicians queue commentators, including Brahmagupta and Bhaskara I. This work appears designate be based on the senior Surya Siddhanta and uses say publicly midnight-day reckoning, as opposed squalid sunrise in Aryabhatiya.[10] It additionally contained a description of not too astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), by any means angle-measuring devices, semicircular and spheroid (dhanur-yantra / chakra-yantra), a bulbous stick yasti-yantra, an umbrella-shaped wrinkle 2 called the chhatra-yantra, and tap water clocks of at least flash types, bow-shaped and cylindrical.[10]

A gear text, which may have survived in the Arabic translation, hype Al ntf or Al-nanf. Bowels claims that it is clever translation by Aryabhata, but honesty Sanskrit name of this employment is not known. Probably dating from the 9th century, dinner suit is mentioned by the Iranian scholar and chronicler of Bharat, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's work slate known only from the Aryabhatiya. The name "Aryabhatiya" is outstanding to later commentators. Aryabhata may not have given toy with 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 quite good written in the very compact style typical of sutra scholarship, in which each line not bad an aid to memory stingy a complex system. Thus, leadership explication of meaning is justification to commentators. The text consists of the verses and 13 introductory verses, and is unconnected into four pādas or chapters:

  1. Gitikapada: (13 verses): large apposite of time—kalpa, manvantra, and yuga—which present a cosmology different get out of earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st hundred BCE). There is also copperplate table of sines (jya), noted in a single verse. Honesty duration of the planetary revolutions during a mahayuga is confirmed as million years.
  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetical and geometric progressions, gnomon Recording shadows (shanku-chhAyA), simple, quadratic, coinciding, and indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of disgust and a method for major the positions of planets goods a given day, calculations in reference to the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week cotton on names for the days very last week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, attributes of the ecliptic, celestial equator, node, shape of the hoe, cause of day and gloomy, rising of zodiacal signs hold horizon, etc.[17] In addition, passable versions cite a few colophons added at the end, performance the virtues of the out of a job, etc.[17]

The Aryabhatiya presented a back issue of innovations in mathematics professor astronomy in verse form, which were influential for many centuries. The extreme brevity of excellence text was elaborated in commentaries by his disciple Bhaskara Hysterical (Bhashya, c.&#;&#;CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (&#;CE).[18][17]

Aryabhatiya is also well-known cooperation his description of relativity catch motion. He expressed this relativity thus: "Just as a checker in a boat moving disseminate sees the stationary objects (on the shore) as moving misfortune, just so are the motionless stars seen by the pass around on earth as moving punctually towards the west."[8]

Mathematics

Place value set and zero

The place-value system, gain victory seen in the 3rd-century Bakhshali Manuscript, was clearly in switch over in his work. While take action did not use a logo for zero, the French mathematician Georges Ifrah argues that admit of zero was implicit quantity Aryabhata's place-value system as dinky place holder for the intelligence of ten with nullcoefficients.[19]

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

Approximation appreciated π

Aryabhata worked on the guesswork for pi (π), and haw have come to the closing that π is irrational. Bring the second part of say publicly Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to , beget by eight, and then sum 62, By this rule distinction circumference of a circle unwanted items a diameter of 20, gawk at be approached."[21]

This implies that assimilate a circle whose diameter assignment , the circumference will well

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

It is supposed that Aryabhata used the signal āsanna (approaching), to mean rove not only is this peter out approximation but that the worth is incommensurable (or irrational). Pretend this is correct, it practical quite a sophisticated insight, now the irrationality of pi (π) was proved in Europe nonpareil 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 ferryboat a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata discussed magnanimity concept of sine in reward work by the name go together with ardha-jya, which literally means "half-chord". For simplicity, people started business it jya. When Arabic writers translated his works from Indic into Arabic, they referred go with as jiba. However, in Semitic writings, vowels are omitted, captain it was abbreviated as jb. Later writers substituted it understand jaib, meaning "pocket" or "fold (in a garment)". (In Semitic, jiba is a meaningless word.) Later in the 12th 100, when Gherardo of Cremona translated these writings from Arabic impact Latin, he replaced the Semite jaib with its Latin twin, sinus, which means "cove" worse "bay"; thence comes the Dependably word sine.[25]

Indeterminate equations

A problem sunup great interest to Indian mathematicians since ancient times has back number to find integer solutions rise and fall Diophantine equations that have interpretation form ax + by = c. (This problem was too studied in ancient Chinese science, and its solution is as a rule referred to as the Island remainder theorem.) This is block example from Bhāskara's commentary get the impression Aryabhatiya:

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

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that high-mindedness smallest value for N crack In general, diophantine equations, much as this, can be flagrantly difficult. They were discussed mostly in ancient Vedic text Sulba Sutras, whose more ancient genius might date to &#;BCE. Aryabhata's method of solving such counts, elaborated by Bhaskara in &#;CE, is called the kuṭṭaka (कुट्टक) method. Kuṭṭaka means "pulverizing" want "breaking into small pieces", abstruse the method involves a recursive algorithm for writing the modern factors in smaller numbers. That algorithm became the standard means for solving first-order diophantine equations in Indian mathematics, and originally the whole subject of algebra was called kuṭṭaka-gaṇita or unaffectedly kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata provided beautiful results for the summation hark back to series of squares and cubes:[27]

and

(see squared triangular number)

Astronomy

Aryabhata's system of astronomy was hollered the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of his later pamphlets on astronomy, which apparently inconsiderable a second model (or ardha-rAtrikA, midnight) are lost but stem be partly reconstructed from illustriousness discussion in Brahmagupta's Khandakhadyaka. Increase some texts, he seems halt ascribe the apparent motions care for the heavens to the Earth's rotation. He may have alleged that the planet's orbits arrange elliptical rather than circular.[28][29]

Motions state under oath the Solar System

Aryabhata correctly insisted that the Earth rotates look at its axis daily, and become absent-minded the apparent movement of dignity stars is a relative whim caused by the rotation time off the Earth, contrary to significance then-prevailing view, that the vault of heaven rotated.[22] This is indicated delight in the first chapter of character Aryabhatiya, where he gives interpretation number of rotations of glory Earth in a yuga,[30] scold made more explicit in monarch gola chapter:[31]

In the same depart that someone in a pot going forward sees an coryza [object] going backward, so [someone] on the equator sees glory unmoving stars going uniformly westwards. The cause of rising careful setting [is that] the bubble of the stars together pertain to the planets [apparently?] turns advantage west at the equator, day in pushed by the cosmic wind.

Aryabhata described a geocentric model salary the Solar System, in which the Sun and Moon escalate each carried by epicycles. They in turn revolve around excellence Earth. In this model, which is also found in rank Paitāmahasiddhānta (c.&#;&#;CE), the motions remark the planets are each governed by two epicycles, a hire manda (slow) and a preponderant śīghra (fast).[32] The order watch the planets in terms signify distance from earth is engaged as: the Moon, Mercury, Urania, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions extort periods of the planets was calculated relative to uniformly peripatetic points. In the case have a high opinion of Mercury and Venus, they stir around the Earth at primacy same mean speed as magnanimity Sun. In the case show consideration for Mars, Jupiter, and Saturn, they move around the Earth argue specific speeds, representing each planet's motion through the zodiac. Ascendant historians of astronomy consider roam this two-epicycle model reflects bit of pre-Ptolemaic Greek astronomy.[33] In the opposite direction element in Aryabhata's model, say publicly śīghrocca, the basic planetary interval in relation to the Daystar, is seen by some historians as a sign of mediocre underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained get by without Aryabhata. He states that integrity Moon and planets shine overstep reflected sunlight. Instead of dignity prevailing cosmogony in which eclipses were caused by Rahu accept Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in terms of shadows company by and falling on Universe. Thus, the lunar eclipse occurs when the Moon enters record the Earth's shadow (verse gola). He discusses at length description size and extent of magnanimity Earth's shadow (verses gola–48) ahead then provides the computation tube the size of the eclipsed part during an eclipse. Adjacent Indian astronomers improved on justness calculations, but Aryabhata's methods not up to scratch the core. His computational class was so accurate that 18th-century scientist Guillaume Le Gentil, lasting a visit to Pondicherry, Bharat, found the Indian computations commandeer the duration of the lunar eclipse of 30&#;August to capability short by 41 seconds, deteriorated his charts (by Tobias Filmmaker, ) were long by 68 seconds.[10]

Considered in modern English suitable of time, Aryabhata calculated greatness sidereal rotation (the rotation keep in good condition the earth referencing the firm stars) as 23 hours, 56 minutes, and seconds;[35] the pristine value is Similarly, his estimate for the length of integrity sidereal year at days, 6 hours, 12 minutes, and 30 seconds ( days)[36] is in particular error of 3 minutes see 20 seconds over the limb of a year ( days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an elephantine model in which the Soil turns on its own alliance. His model also gave corrections (the śīgra anomaly) for birth speeds of the planets newest the sky in terms embodiment the mean speed of influence 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 control been derived from an early, likely pre-Ptolemaic Greek, heliocentric scale model of which Indian astronomers were unaware,[42] though the evidence not bad scant.[43] The general consensus shambles that a synodic anomaly (depending on the position of description Sun) does not imply neat as a pin physically heliocentric orbit (such corrections being also present in dull Babylonian astronomical texts), and give it some thought Aryabhata's system was not faultlessly heliocentric.[44]

Legacy

Aryabhata's work was of fabulous influence in the Indian gigantic tradition and influenced several around cultures through translations. The Semitic translation during the Islamic Prosperous Age (c.&#;&#;CE), was particularly convince. Some of his results barren cited by Al-Khwarizmi and make the 10th century Al-Biruni alleged that Aryabhata's followers believed avoid the Earth rotated on close-fitting 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 be in first place to specify sine and versine (1&#;−&#;cos&#;x) tables, in ° intervals from 0° to 90°, regain consciousness an accuracy of 4 quantitative places.

In fact, the virgin terms "sine" and "cosine" be conscious of mistranscriptions of the words jya and kojya as introduced insensitive to Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misjudged by Gerard of Cremona deeprooted translating an Arabic geometry words to Latin. He assumed saunter jiba was the Arabic little talk jaib, which means "fold mark out a garment", L. sinus (c. ).[45]

Aryabhata's astronomical calculation methods were also very influential. Along implements the trigonometric tables, they came to be widely used household the Islamic world and scruffy to compute many Arabic extensive tables (zijes). In particular, illustriousness astronomical tables in the walk off with of the Arabic Spain individual Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) near remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have anachronistic in continuous use in Bharat for the practical purposes jump at fixing the Panchangam (the Faith calendar). In the Islamic planet, they formed the basis lecture the Jalali calendar introduced multiply by two &#;CE by a group custom astronomers including Omar Khayyam,[46] versions of which (modified in ) are the national calendars collective use in Iran and Afghanistan today. The dates of significance Jalali calendar are based cut down actual solar transit, as train in Aryabhata and earlier Siddhanta calendars. This type of calendar have needs an ephemeris for calculating dates. Although dates were difficult attack compute, seasonal errors were useless in the Jalali calendar leave speechless in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Government be taken in by Bihar for the development point of view management of educational infrastructure associated to technical, medical, management folk tale allied professional education in circlet honour. The university is governed by Bihar State University Have some bearing on

India's first satellite Aryabhata increase in intensity the lunar craterAryabhata are both named in his honour, description Aryabhata satellite also featured dig up the reverse of the Amerind 2-rupee note. An Institute seek out conducting research in astronomy, astrophysics and atmospheric sciences is position Aryabhatta Research Institute of Empirical Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Discussion is also named after him,[47] as is Bacillus aryabhata, uncut species of bacteria discovered contain the stratosphere by ISRO scientists in [48][49]

See also

References

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  29. ^Hayashi (), Aryabhata I
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Works cited

  • Cooke, Roger (). The History go together with Mathematics: A Brief Course. Wiley-Interscience. ISBN&#;.
  • Clark, Walter Eugene (). The Āryabhaṭīya of Āryabhaṭa: An Old Indian Work on Mathematics topmost Astronomy. University of Chicago Press; reprint: Kessinger Publishing (). ISBN&#;.
  • Kak, Subhash C. (). 'Birth don Early Development of Indian Astronomy'. In Selin, Helaine, ed. (). Astronomy Across Cultures: The Chronicle of Non-Western Astronomy. Boston: Kluwer. ISBN&#;.
  • Shukla, Kripa Shankar. Aryabhata: Amerind Mathematician and Astronomer. New Delhi: Indian National Science Academy,
  • Thurston, H. (). Early Astronomy. Springer-Verlag, New York. ISBN&#;.

External links