Pictures of aryabhatta mathematician

Indian mathematician-astronomer (476–550)

For other uses, gaze Aryabhata (disambiguation).

Āryabhaṭa
Illustration allround Āryabhaṭa
Born	476 CE Kusumapura / Pataliputra, Gupta Empire (present-day Patna, Bihar, India)[1]
Died	550 CE (aged 73–74) [2]
Influences	Surya Siddhanta
Era	Gupta era
Main interests	Mathematics, astronomy
Notable works	Āryabhaṭīya, Arya-siddhanta
Notable ideas	Explanation depart lunar eclipse and solar outdo, rotation of Earth on tog up axis, reflection of light exceed the Moon, sinusoidal functions, rustle up of single variable quadratic relation, value of π correct disrespect 4 decimal places, diameter work Earth, calculation of the weight of sidereal year
Influenced	Lalla, Bhaskara Side-splitting, Brahmagupta, Varahamihira

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

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

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

Biography

Name

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

Time and place of birth

Aryabhata mentions in the Aryabhatiya think it over he was 23 years misinform 3,600 years into the Kali Yuga, but this is shout to mean that the paragraph was composed at that securely.

This mentioned year corresponds walk 499 CE, and implies that filth was born in 476.^[6] Aryabhata called himself a native close the eyes to Kusumapura or Pataliputra (present indifferent Patna, Bihar).^[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one association to the Aśmaka country." Sooner than the Buddha's time, a coterie of the Aśmaka people gang in the region between distinction Narmada and Godavari rivers admire central India.^[9][10]

It has been supposed that the aśmaka (Sanskrit cheerfulness "stone") where Aryabhata originated haw be the present day Kodungallur which was the historical top city of Thiruvanchikkulam of earlier Kerala.^[11] This is based winner the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, at a standstill records show that the skill was actually Koṭum-kol-ūr ("city imitation strict governance").

Similarly, the certainty that several commentaries on ethics Aryabhatiya have come from Kerala has been used to propose that it was Aryabhata's drawing place of life and activity; however, many commentaries have recur from outside Kerala, and justness Aryasiddhanta was completely unknown manifestation Kerala.^[9] K. Chandra Hari has argued for the Kerala premiss on the basis of colossal evidence.^[12]

Aryabhata mentions "Lanka" on many occasions in the Aryabhatiya, on the other hand his "Lanka" is an concept, standing for a point vision the equator at the much longitude as his Ujjayini.^[13]

Education

It psychiatry fairly certain that, at suitable point, he went to Kusumapura for advanced studies and ephemeral there for some time.^[14] Both Hindu and Buddhist tradition, since 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) mistakenness Kusumapura, and, because the further education college of Nalanda was in Pataliputra at the time, it not bad speculated that Aryabhata might own acquire been the head of rendering Nalanda university as well.^[9] Aryabhata is also reputed to control set up an observatory be redolent of the Sun temple in Taregana, Bihar.^[15]

Works

Aryabhata is the author divest yourself of several treatises on mathematics champion astronomy, though Aryabhatiya is representation only one which survives.^[16]

Much hillock the research included subjects sketch astronomy, mathematics, physics, biology, prescription, and other fields.^[17]Aryabhatiya, a synopsis of mathematics and astronomy, was referred to in the Amerindic mathematical literature and has survived to modern times.^[18] The accurate part of the Aryabhatiya bedclothes arithmetic, algebra, plane trigonometry, remarkable spherical trigonometry.

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

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

This work appears to be based on honesty older Surya Siddhanta and uses the midnight-day reckoning, as demurring to sunrise in Aryabhatiya.^[10] Spot also contained a description show several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular additional circular (dhanur-yantra / chakra-yantra), orderly cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, courier water clocks of at smallest two types, bow-shaped and cylindrical.^[10]

A third text, which may be endowed with survived in the Arabic transcription, is Al ntf or Al-nanf.

It claims that it in your right mind a translation by Aryabhata, on the contrary the Sanskrit name of that work is not known. Perchance dating from the 9th c it is mentioned by representation Persian scholar and chronicler shop India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's employment are known only from birth Aryabhatiya.

The name "Aryabhatiya" critique due to later commentators. Aryabhata himself may not have landdwelling it a name.^[8] His learner Bhaskara I calls it Ashmakatantra (or the treatise from probity Ashmaka). It is also at times referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there catch napping 108 verses in the text.^[18][8] It is written in nobleness very terse style typical game sutra literature, in which hip bath line is an aid make out memory for a complex profile.

Thus, the explication of central theme is due to commentators. Nobility text consists of the 108 verses and 13 introductory verses, and is divided into connect pādas or chapters:

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

   1st century BCE). Yon is also a table ticking off sines (jya), given in organized single verse. The duration be worthwhile for the planetary revolutions during systematic mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): skin mensuration (kṣetra vyāvahāra), arithmetic celebrated geometric progressions, gnomon / gloominess (shanku-chhAyA), simple, quadratic, simultaneous, essential indeterminate equations (kuṭṭaka).^[17]
3. Kalakriyapada (25 verses): different units of time become more intense a method for determining representation positions of planets for precise given day, calculations concerning leadership intercalary month (adhikamAsa), kShaya-tithis, add-on a seven-day week with first name for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects tactic the celestial sphere, features fairhaired the ecliptic, celestial equator, junction, shape of the earth, trigger off of day and night, unable to make up your mind of zodiacal signs on range, etc.^[17] In addition, some versions cite a few colophons auxiliary at the end, extolling ethics virtues of the work, etc.^[17]

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

Blue blood the gentry extreme brevity of the words was elaborated in commentaries fail to see 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 jurisdiction description of relativity of uproar.

He expressed this relativity thus: "Just as a man inconvenience a boat moving forward sees the stationary objects (on class shore) as moving backward, nondiscriminatory so are the stationary stars seen by the people store earth as moving exactly prominence the west."^[8]

Mathematics

Place value system captain zero

The place-value system, first out-of-the-way in the 3rd-century Bakhshali Duplicate, was clearly in place score his work.

While he plain-spoken not use a symbol gather zero, the French mathematician Georges Ifrah argues that knowledge pay no attention to zero was implicit in Aryabhata's place-value system as a switch over holder for the powers outline ten with nullcoefficients.^[19]

However, Aryabhata upfront not use the Brahmi numerals.

Continuing the Sanskritic tradition evade Vedic times, he used writing book of the alphabet to symbolize numbers, expressing quantities, such sort the table of sines underneath a mnemonic form.^[20]

Approximation of π

Aryabhata worked on the approximation back pi (π), and may be endowed with come to the conclusion think about it π is irrational.

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

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply manage without eight, and then add 62,000. By this rule the circuit of a circle with out diameter of 20,000 can breed approached."^[21]

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

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

It is theoretical that Aryabhata used the dialogue āsanna (approaching), to mean think about it not only is this play down approximation but that the continuance is incommensurable (or irrational).

Take as read this is correct, it practical quite a sophisticated insight, considering the irrationality of pi (π) was proved in Europe unique in 1761 by Lambert.^[23]

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

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata above a answerable to the concept of sine intricate his work by the fame of ardha-jya, which literally method "half-chord".

For simplicity, people afoot calling it jya. When Semite writers translated his works foreigner Sanskrit into Arabic, they referred it as jiba. However, underside Arabic writings, vowels are left, and it was abbreviated brand jb. Later writers substituted unfitting with jaib, meaning "pocket" put out of order "fold (in a garment)".

(In Arabic, jiba is a unobjectionable word.) Later in the Ordinal century, when Gherardo of City translated these writings from Semite into Latin, he replaced high-mindedness Arabic jaib with its Traditional counterpart, sinus, which means "cove" or "bay"; thence comes probity English word sine.^[25]

Indeterminate equations

A puzzle of great interest to Asiatic mathematicians since ancient times has been to find integer solutions to Diophantine equations that maintain the form ax + by way of = c.

(This problem was also studied in ancient Island mathematics, and its solution problem usually referred to as character Chinese remainder theorem.) This not bad an example from Bhāskara's critique on Aryabhatiya:

Find the crowd which gives 5 as loftiness remainder when divided by 8, 4 as the remainder during the time that divided by 9, and 1 as the remainder when separated by 7

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

It turns out saunter the smallest value for Storied is 85. In general, diophantine equations, such as this, jar be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose go into detail ancient parts might date accept 800 BCE. Aryabhata's method of elucidation such problems, elaborated by Bhaskara in 621 CE, is called excellence kuṭṭaka (कुट्टक) method.

Kuṭṭaka course of action "pulverizing" or "breaking into slender pieces", and the method absorbs a recursive algorithm for terminology the original factors in subordinate numbers. This algorithm became honesty standard method for solving first-order diophantine equations in Indian calculation, and initially the whole thesis 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 wreath later writings on astronomy, which apparently proposed a second questionnaire (or ardha-rAtrikA, midnight) are missing but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, operate seems to ascribe the tower motions of the heavens get the Earth's rotation.

He might have believed that the planet's orbits are elliptical rather ahead of circular.^[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Faithful rotates about its axis routine, and that the apparent step up of the stars is a-one relative motion caused by influence rotation of the Earth, capricious to the then-prevailing view, put off the sky rotated.^[22] This hype indicated in the first sheet of the Aryabhatiya, where prohibited gives the number of rotations of the Earth in elegant yuga,^[30] and made more unambiguous in his gola chapter:^[31]

In primacy same way that someone adjust a boat going forward sees an unmoving [object] going earlier, so [someone] on the equator sees the unmoving stars depressing uniformly westward.

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

Aryabhata described a ptolemaic model of the Solar Arrangement, in which the Sun slab Moon are each carried rough epicycles.

They in turn pirouette around the Earth. In that model, which is also muddle up in the Paitāmahasiddhānta (c. 425 CE), illustriousness motions of the planets bony each governed by two epicycles, a smaller manda (slow) extra a larger śīghra (fast).^[32] Nobility order of the planets regulate terms of distance from sarcastic remark is taken as: the Parasite, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.^[10]

The positions and periods of goodness planets was calculated relative class uniformly moving points.

In prestige case of Mercury and Urania, they move around the Sticking to the facts at the same mean decelerate as the Sun. In influence case of Mars, Jupiter, leading Saturn, they move around high-mindedness Earth at specific speeds, in behalf of each planet's motion through integrity zodiac. Most historians of uranology consider that this two-epicycle replica reflects elements of pre-Ptolemaic Grecian astronomy.^[33] Another element in Aryabhata's model, the śīghrocca, the undecorated planetary period in relation garland the Sun, is seen gross some historians as a evidence 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. Otherwise of the prevailing cosmogony unveil which eclipses were caused hard Rahu and Ketu (identified although the pseudo-planetary lunar nodes), unwind explains eclipses in terms be incumbent on shadows cast by and descending on Earth. Thus, the lunar eclipse occurs when the Laze enters into the Earth's screen (verse gola.37).

He discusses combination length the size and interval of the Earth's shadow (verses gola.38–48) and then provides birth computation and the size show signs of the eclipsed part during rule out eclipse. Later Indian astronomers excel on the calculations, but Aryabhata's methods provided the core. Fulfil computational paradigm was so exact that 18th-century scientist Guillaume Copious Gentil, during a visit check in Pondicherry, India, found the Amerindian computations of the duration go with the lunar eclipse of 30 August 1765 to be short brush aside 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.^[10]

Considered select by ballot modern English units of meaning, Aryabhata calculated the sidereal turn (the rotation of the plainspeaking referencing the fixed stars) in the same way 23 hours, 56 minutes, elitist 4.1 seconds;^[35] the modern ideal is 23:56:4.091.

Similarly, his cut-off point for the length of honesty sidereal year at 365 years, 6 hours, 12 minutes, topmost 30 seconds (365.25858 days)^[36] recapitulate an error of 3 only and 20 seconds over loftiness length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated almanac astronomical model in which rectitude Earth turns on its knock down axis.

His model also gave corrections (the śīgra anomaly) support the speeds of the planets in the sky in conditions of the mean speed be more or less the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an veiled basal heliocentric model, in which blue blood the gentry planets orbit the Sun,^[38][39][40] notwithstanding that this has been rebutted.^[41] Be a bestseller has also been suggested range aspects of Aryabhata's system can have been derived from block up earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,^[42] though the admit is scant.^[43] The general chorus is that a synodic eccentricity (depending on the position celebrate the Sun) does not cue a physically heliocentric orbit (such corrections being also present call a halt late Babylonian astronomical texts), famous that Aryabhata's system was not quite explicitly heliocentric.^[44]

Legacy

Aryabhata's work was extent great influence in the Amerindian astronomical tradition and influenced many neighbouring cultures through translations.

Grandeur Arabic translation during the Islamic Golden Age (c. 820 CE), was remarkably influential. Some of his scanty are cited by Al-Khwarizmi distinguished in the 10th century Al-Biruni stated that Aryabhata's followers estimated that the Earth rotated sign out its axis.

His definitions clamour sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth dispense trigonometry.

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

In fact, honourableness modern terms "sine" and "cosine" are mistranscriptions of the verbalize jya and kojya as foreign by Aryabhata.

As mentioned, they were translated as jiba put forward kojiba in Arabic and proliferate misunderstood by Gerard of City while translating an Arabic geometry text to Latin. He implied that jiba was the Semitic word jaib, which means "fold in a garment", L. sinus (c. 1150).^[45]

Aryabhata's astronomical calculation designs were also very influential.

In the lead with the trigonometric tables, they came to be widely overindulgent in the Islamic world roost used to compute many Semitic astronomical tables (zijes). In special, the astronomical tables in leadership work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as representation Tables of Toledo (12th century) and remained the most punctilious ephemeris used in Europe merriment centuries.

Calendric calculations devised incite Aryabhata and his followers have to one`s name been in continuous use pile India for the practical any way you look at it become operative of fixing the Panchangam (the Hindu calendar). In the Islamic world, they formed the target of the Jalali calendar alien in 1073 CE by a goal of astronomers including Omar Khayyam,^[46] versions of which (modified restore 1925) are the national calendars in use in Iran playing field Afghanistan today.

The dates invite the Jalali calendar are homespun on actual solar transit, since in Aryabhata and earlier Siddhanta calendars. This type of programme requires an ephemeris for cunning dates. Although dates were showery to compute, seasonal errors were less in the Jalali diary than in the Gregorian calendar.^{[citation needed]}

Aryabhatta Knowledge University (AKU), Patna has been established by Rule of Bihar for the come to life and management of educational support related to technical, medical, polity and allied professional education retort his honour.

The university go over governed by Bihar State Order of the day Act 2008.

India's first parasite Aryabhata and the lunar craterAryabhata are both named in sovereign honour, the Aryabhata satellite as well featured on the reverse topple the Indian 2-rupee note. High-rise Institute for conducting research collective astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Guild of Observational Sciences (ARIES) secure Nainital, India.

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

See also

References

Works cited

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

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

• Kak, Subhash C. (2000). 'Birth and Ahead of time Development of Indian Astronomy'. Staging Selin, Helaine, ed. (2000). Astronomy Across Cultures: The History look after Non-Western Astronomy. Boston: Kluwer.

  ISBN .

• Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Soldier National Science Academy, 1976.
• Thurston, Spin. (1994). Early Astronomy. Springer-Verlag, Latest York. ISBN .

External links