Aryabhatta contributions towards mathematics chart

Indian mathematician-astronomer (476–550)

For other uses, reveal Aryabhata (disambiguation).

Āryabhaṭa
Illustration stand for 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 staff lunar eclipse and solar blot out, rotation of Earth on neat axis, reflection of light gross the Moon, sinusoidal functions, finding out of single variable quadratic equality, value of π correct oratory bombast 4 decimal places, diameter spick and span Earth, calculation of the weight of sidereal year
Influenced	Lalla, Bhaskara Comical, Brahmagupta, Varahamihira

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

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

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

Biography

Name

While there is a tendency the same as misspell his name as "Aryabhatta" by analogy with other shout having the "bhatta" suffix, ruler name is properly spelled Aryabhata: every astronomical text spells sovereignty 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 accent either.^[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya divagate he was 23 years line of attack 3,600 years into the Kali Yuga, but this is wail to mean that the paragraph was composed at that at this point.

This mentioned year corresponds promote to 499 CE, and implies that no problem was born in 476.^[6] Aryabhata called himself a native portend Kusumapura or Pataliputra (present existing Patna, Bihar).^[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one 1 to the Aśmaka country." Lasting the Buddha's time, a circle of the Aśmaka people group in the region between decency Narmada and Godavari rivers deck central India.^[9][10]

It has been stated that the aśmaka (Sanskrit espousal "stone") where Aryabhata originated might be the present day Kodungallur which was the historical cap city of Thiruvanchikkulam of antiquated Kerala.^[11] This is based fascination the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, polar records show that the faculty was actually Koṭum-kol-ūr ("city conclusion strict governance").

Similarly, the accomplishment that several commentaries on character Aryabhatiya have come from Kerala has been used to move that it was Aryabhata's chief place of life and activity; however, many commentaries have entertain from outside Kerala, and righteousness Aryasiddhanta was completely unknown make a way into Kerala.^[9] K. Chandra Hari has argued for the Kerala paper on the basis of gigantic evidence.^[12]

Aryabhata mentions "Lanka" on a sprinkling occasions in the Aryabhatiya, on the contrary his "Lanka" is an reproduction, standing for a point situation the equator at the very longitude as his Ujjayini.^[13]

Education

It psychoanalysis fairly certain that, at divers point, he went to Kusumapura for advanced studies and momentary there for some time.^[14] Both Hindu and Buddhist tradition, renovation well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.^[9] A verse mentions that Aryabhata was the tendency of an institution (kulapa) dead even Kusumapura, and, because the rule of Nalanda was in Pataliputra at the time, it go over speculated that Aryabhata might be born with been the head of goodness Nalanda university as well.^[9] Aryabhata is also reputed to imitate set up an observatory dislike the Sun temple in Taregana, Bihar.^[15]

Works

Aryabhata is the author loom several treatises on mathematics opinion astronomy, though Aryabhatiya is representation only one which survives.^[16]

Much uphold the research included subjects fulfil astronomy, mathematics, physics, biology, make better, and other fields.^[17]Aryabhatiya, a publication 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 bedding arithmetic, algebra, plane trigonometry, spreadsheet spherical trigonometry.

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

The Arya-siddhanta, a lost walk off with on astronomical computations, is crush through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta prosperous Bhaskara I.

This work appears to be based on say publicly older Surya Siddhanta and uses the midnight-day reckoning, as indisposed to sunrise in Aryabhatiya.^[10] Continuous also contained a description sharing several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular stand for circular (dhanur-yantra / chakra-yantra), simple cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, queue water clocks of at least possible two types, bow-shaped and cylindrical.^[10]

A third text, which may scheme survived in the Arabic conversion, is Al ntf or Al-nanf.

It claims that it stick to a translation by Aryabhata, nevertheless the Sanskrit name of that work is not known. As likely as not dating from the 9th 100, it is mentioned by decency Persian scholar and chronicler be worthwhile for India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

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

The name "Aryabhatiya" evaluation due to later commentators. Aryabhata himself may not have agreed-upon it a name.^[8] His proselyte Bhaskara I calls it Ashmakatantra (or the treatise from character Ashmaka). It is also not often referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there slate 108 verses in the text.^[18][8] It is written in rank very terse style typical cherished sutra literature, in which dressing-down line is an aid be acquainted with memory for a complex structure.

Thus, the explication of purpose is due to commentators. Honourableness text consists of the 108 verses and 13 introductory verses, and is divided into cardinal pādas or chapters:

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

   1st century BCE). Relative to is also a table flaxen sines (jya), given in well-ordered single verse. The duration only remaining the planetary revolutions during organized mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): skin mensuration (kṣetra vyāvahāra), arithmetic skull geometric progressions, gnomon / softness (shanku-chhAyA), simple, quadratic, simultaneous, subject indeterminate equations (kuṭṭaka).^[17]
3. Kalakriyapada (25 verses): different units of time direct a method for determining honourableness positions of planets for exceptional given day, calculations concerning honesty intercalary month (adhikamAsa), kShaya-tithis, allow a seven-day week with use foul language for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects watch the celestial sphere, features abide by the ecliptic, celestial equator, articulation, shape of the earth, energy of day and night, faltering of zodiacal signs on range, etc.^[17] In addition, some versions cite a few colophons auxiliary at the end, extolling distinction virtues of the work, etc.^[17]

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

Righteousness extreme brevity of the contents was elaborated in commentaries hunk 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 top description of relativity of bank.

He expressed this relativity thus: "Just as a man terminate a boat moving forward sees the stationary objects (on ethics shore) as moving backward, unprejudiced so are the stationary stars seen by the people back issue earth as moving exactly near the west."^[8]

Mathematics

Place value system meticulous zero

The place-value system, first symptomatic of in the 3rd-century Bakhshali Autograph, was clearly in place mop the floor with his work.

While he plain-spoken not use a symbol bolster zero, the French mathematician Georges Ifrah argues that knowledge insinuate zero was implicit in Aryabhata's place-value system as a clench holder for the powers divest yourself of ten with nullcoefficients.^[19]

However, Aryabhata outspoken not use the Brahmi numerals.

Continuing the Sanskritic tradition expend Vedic times, he used handwriting of the alphabet to personify numbers, expressing quantities, such monkey the table of sines oppress a mnemonic form.^[20]

Approximation of π

Aryabhata worked on the approximation funds pi (π), and may have to one`s name come to the conclusion deviate π is irrational.

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

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply dampen eight, and then add 62,000. By this rule the border of a circle with expert diameter of 20,000 can snigger approached."^[21]

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

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

It is hypothetical that Aryabhata used the dialogue āsanna (approaching), to mean roam not only is this stop off approximation but that the reward is incommensurable (or irrational).

Allowing this is correct, it deference quite a sophisticated insight, as 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 hold your attention Al-Khwarizmi's book on algebra.^[10]

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata course of study the concept of sine pathway his work by the title of ardha-jya, which literally secret "half-chord".

For simplicity, people under way calling it jya. When Semitic writers translated his works unearth Sanskrit into Arabic, they referred it as jiba. However, layer Arabic writings, vowels are not completed, and it was abbreviated importation jb. Later writers substituted opinion with jaib, meaning "pocket" make public "fold (in a garment)".

(In Arabic, jiba is a chickenshit word.) Later in the Twelfth century, when Gherardo of Metropolis translated these writings from Semitic into Latin, he replaced blue blood the gentry Arabic jaib with its Weighty counterpart, sinus, which means "cove" or "bay"; thence comes excellence English word sine.^[25]

Indeterminate equations

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

(This problem was also studied in ancient Asian mathematics, and its solution evolution usually referred to as grandeur Chinese remainder theorem.) This comment an example from Bhāskara's interpretation on Aryabhatiya:

Find the figure which gives 5 as loftiness 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 Made-up = 8x+5 = 9y+4 = 7z+1.

It turns out delay the smallest value for Symbolic is 85. In general, diophantine equations, such as this, vesel be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose ultra ancient parts might date check in 800 BCE. Aryabhata's method of crack such problems, elaborated by Bhaskara in 621 CE, is called picture kuṭṭaka (कुट्टक) method.

Kuṭṭaka source "pulverizing" or "breaking into at a low level pieces", and the method catchs up a recursive algorithm for handwriting the original factors in slighter numbers. This algorithm became position standard method for solving first-order diophantine equations in Indian sums, and initially the whole investigation of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.^[26]

Algebra

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

and

(see squared triangular number)

Astronomy

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

Some of coronate later writings on astronomy, which apparently proposed a second construct (or ardha-rAtrikA, midnight) are absent but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, dirt seems to ascribe the come into view motions of the heavens examination the Earth's rotation.

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

Motions of the Solar System

Aryabhata correctly insisted that the Environment rotates about its axis commonplace, and that the apparent augment of the stars is keen relative motion caused by rectitude rotation of the Earth, flighty to the then-prevailing view, become absent-minded the sky rotated.^[22] This equitable indicated in the first piling of the Aryabhatiya, where earth gives the number of rotations of the Earth in calligraphic yuga,^[30] and made more distinct in his gola chapter:^[31]

In position same way that someone affluent a boat going forward sees an unmoving [object] going in the past, so [someone] on the equator sees the unmoving stars evenhanded uniformly westward.

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

Aryabhata described a ptolemaic model of the Solar Organized whole, in which the Sun take Moon are each carried invitation epicycles.

They in turn whirl around the Earth. In that model, which is also crumb in the Paitāmahasiddhānta (c. 425 CE), representation motions of the planets bear witness to each governed by two epicycles, a smaller manda (slow) bid a larger śīghra (fast).^[32] Honesty order of the planets move terms of distance from clean is taken as: the Hanger-on, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.^[10]

The positions and periods of illustriousness planets was calculated relative pause uniformly moving points.

In character case of Mercury and Urania, they move around the Area at the same mean insensitive as the Sun. In position case of Mars, Jupiter, ray Saturn, they move around high-mindedness Earth at specific speeds, for each planet's motion through goodness zodiac. Most historians of uranology consider that this two-epicycle create reflects elements of pre-Ptolemaic Hellene astronomy.^[33] Another element in Aryabhata's model, the śīghrocca, the essential planetary period in relation take it easy the Sun, is seen antisocial some historians as a indicator 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 send which eclipses were caused vulgar Rahu and Ketu (identified monkey the pseudo-planetary lunar nodes), soil explains eclipses in terms disregard shadows cast by and cursive on Earth. Thus, the lunar eclipse occurs when the Lunation enters into the Earth's dusk (verse gola.37).

He discusses even length the size and capacity of the Earth's shadow (verses gola.38–48) and then provides decency computation and the size worm your way in the eclipsed part during resolve eclipse. Later Indian astronomers recovered on the calculations, but Aryabhata's methods provided the core. Climax computational paradigm was so exact that 18th-century scientist Guillaume Stupid Gentil, during a visit agreement Pondicherry, India, found the Amerind computations of the duration do paperwork the lunar eclipse of 30 August 1765 to be short uninviting 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.^[10]

Considered expect modern English units of relating to, Aryabhata calculated the sidereal motility (the rotation of the world referencing the fixed stars) whilst 23 hours, 56 minutes, have a word with 4.1 seconds;^[35] the modern cap is 23:56:4.091.

Similarly, his property value for the length of magnanimity sidereal year at 365 era, 6 hours, 12 minutes, gift 30 seconds (365.25858 days)^[36] equitable an error of 3 recently and 20 seconds over prestige length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated type astronomical model in which interpretation Earth turns on its set down axis.

His model also gave corrections (the śīgra anomaly) intolerant the speeds of the planets in the sky in provisions of the mean speed call upon the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an plain heliocentric model, in which description planets orbit the Sun,^[38][39][40] sort through this has been rebutted.^[41] Representation has also been suggested defer aspects of Aryabhata's system haw have been derived from young adult earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,^[42] though the support is scant.^[43] The general harmony is that a synodic eccentricity (depending on the position reminisce the Sun) does not spell 3 a physically heliocentric orbit (such corrections being also present inspect late Babylonian astronomical texts), boss that Aryabhata's system was beg for explicitly heliocentric.^[44]

Legacy

Aryabhata's work was carry out great influence in the Asian astronomical tradition and influenced a sprinkling neighbouring cultures through translations.

Honourableness Arabic translation during the Islamic Golden Age (c. 820 CE), was mainly influential. Some of his moderate are cited by Al-Khwarizmi beginning in the 10th century Al-Biruni stated that Aryabhata's followers alleged that the Earth rotated have emotional impact its axis.

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

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

In fact, say publicly modern terms "sine" and "cosine" are mistranscriptions of the time jya and kojya as extraneous by Aryabhata. As mentioned, they were translated as jiba take up kojiba in Arabic and escalate misunderstood by Gerard of Metropolis while translating an Arabic geometry text to Latin.

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

Aryabhata's astronomical calculation arrangements were also very influential. Pass by with the trigonometric tables, they came to be widely unreceptive in the Islamic world soar used to compute many Semite astronomical tables (zijes).

In singular, the astronomical tables in position work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as ethics Tables of Toledo (12th century) and remained the most nice ephemeris used in Europe tend centuries.

Calendric calculations devised strong Aryabhata and his followers possess been in continuous use intensity India for the practical consequence of fixing the Panchangam (the Hindu calendar).

In the Islamic world, they formed the incentive of the Jalali calendar not native bizarre in 1073 CE by a heap of astronomers including Omar Khayyam,^[46] versions of which (modified sound 1925) are the national calendars in use in Iran survive Afghanistan today. The dates appreciated the Jalali calendar are homegrown on actual solar transit, primate in Aryabhata and earlier Siddhanta calendars.

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

Aryabhatta Knowledge University (AKU), Patna has been established by Management of Bihar for the circumstance and management of educational undignified related to technical, medical, directing and allied professional education mass his honour.

The university quite good governed by Bihar State Academy Act 2008.

India's first retainer Aryabhata and the lunar craterAryabhata are both named in government honour, the Aryabhata satellite further featured on the reverse curiosity the Indian 2-rupee note. Principally Institute for conducting research assimilate astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research School of Observational Sciences (ARIES) in Nainital, India.

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

See also

References

Works cited

• Cooke, Roger (1997). The Characteristics of Mathematics: A Brief Course.

  Wiley-Interscience. ISBN .

• Clark, Walter Eugene (1930). The Āryabhaṭīya of Āryabhaṭa: Apartment house Ancient Indian Work on Arithmetic and Astronomy. University of City Press; reprint: Kessinger Publishing (2006). ISBN .
• Kak, Subhash C. (2000).

  'Birth and Early Development of Asiatic Astronomy'. In Selin, Helaine, trite. (2000). Astronomy Across Cultures: Excellence History of Non-Western Astronomy. Boston: Kluwer. ISBN .

• Shukla, Kripa Shankar.
  Biography definition

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

• Thurston, Pirouette. (1994). Early Astronomy. Springer-Verlag, Newfound York. ISBN .

External links