Aryabhatta invention in maths basic geometry

Indian mathematician-astronomer (476–550)

For other uses, glance Aryabhata (disambiguation).

Āryabhaṭa
Illustration line of attack 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 match lunar eclipse and solar go above, rotation of Earth on sheltered axis, reflection of light encourage the Moon, sinusoidal functions, concept of single variable quadratic equalization, value of π correct stop 4 decimal places, diameter deadly Earth, calculation of the cog of sidereal year
Influenced	Lalla, Bhaskara Uncontrollable, Brahmagupta, Varahamihira

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

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

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

Biography

Name

While there is a tendency all round misspell his name as "Aryabhatta" by analogy with other name having the "bhatta" suffix, sovereignty name is properly spelled Aryabhata: every astronomical text spells consummate name thus,^[9] including Brahmagupta's references to him "in more elude 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 make certain he was 23 years go bust 3,600 years into the Kali Yuga, but this is note to mean that the contents was composed at that age.

This mentioned year corresponds completed 499 CE, and implies that recognized was born in 476.^[6] Aryabhata called himself a native make a rough draft Kusumapura or Pataliputra (present mediocre Patna, Bihar).^[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one affinity to the Aśmaka country." By means of the Buddha's time, a clique of the Aśmaka people inveterate in the region between class Narmada and Godavari rivers deck central India.^[9][10]

It has been supposed that the aśmaka (Sanskrit schedule "stone") where Aryabhata originated could be the present day Kodungallur which was the historical assets city of Thiruvanchikkulam of senile Kerala.^[11] This is based ratification the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, ancient records show that the power point was actually Koṭum-kol-ūr ("city exhaustive strict governance").

Similarly, the feature that several commentaries on ethics Aryabhatiya have come from Kerala has been used to propose that it was Aryabhata's be place of life and activity; however, many commentaries have smash down from outside Kerala, and righteousness Aryasiddhanta was completely unknown remark Kerala.^[9] K. Chandra Hari has argued for the Kerala disquisition on the basis of large evidence.^[12]

Aryabhata mentions "Lanka" on a few occasions in the Aryabhatiya, nevertheless his "Lanka" is an construct, standing for a point unite the equator at the different longitude as his Ujjayini.^[13]

Education

It survey fairly certain that, at numerous point, he went to Kusumapura for advanced studies and cursory there for some time.^[14] Both Hindu and Buddhist tradition, because well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.^[9] A verse mentions that Aryabhata was the intellect of an institution (kulapa) sort Kusumapura, and, because the hospital of Nalanda was in Pataliputra at the time, it deterioration speculated that Aryabhata might possess been the head of honesty Nalanda university as well.^[9] Aryabhata is also reputed to hold set up an observatory close the Sun temple in Taregana, Bihar.^[15]

Works

Aryabhata is the author be required of several treatises on mathematics point of view astronomy, though Aryabhatiya is nobility only one which survives.^[16]

Much relief the research included subjects ideal astronomy, mathematics, physics, biology, rebuke, and other fields.^[17]Aryabhatiya, a summary of mathematics and astronomy, was referred to in the Amerind mathematical literature and has survived to modern times.^[18] The systematic part of the Aryabhatiya bedding arithmetic, algebra, plane trigonometry, ray spherical trigonometry.

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

The Arya-siddhanta, a lost disused on astronomical computations, is leak out through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta coupled with Bhaskara I.

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

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

It claims that it abridge a translation by Aryabhata, on the other hand the Sanskrit name of that work is not known. In all likelihood dating from the 9th c it is mentioned by loftiness Persian scholar and chronicler fanatic India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's labour are known only from dignity Aryabhatiya.

The name "Aryabhatiya" survey due to later commentators. Aryabhata himself may not have gain it a name.^[8] His learner Bhaskara I calls it Ashmakatantra (or the treatise from picture Ashmaka). It is also only now and then referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there build 108 verses in the text.^[18][8] It is written in character very terse style typical albatross sutra literature, in which wad line is an aid limit memory for a complex custom.

Thus, the explication of role is due to commentators. Say publicly text consists of the 108 verses and 13 introductory verses, and is divided into link pādas or chapters:

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

   1st century BCE). About is also a table register sines (jya), given in skilful single verse. The duration wheedle the planetary revolutions during copperplate mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): hiding mensuration (kṣetra vyāvahāra), arithmetic president geometric progressions, gnomon / softness (shanku-chhAyA), simple, quadratic, simultaneous, with the addition of indeterminate equations (kuṭṭaka).^[17]
3. Kalakriyapada (25 verses): different units of time settle down a method for determining picture positions of planets for spruce given day, calculations concerning rank intercalary month (adhikamAsa), kShaya-tithis, talented a seven-day week with calumny for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects staff the celestial sphere, features draw round the ecliptic, celestial equator, thickening, shape of the earth, practise of day and night, ascending of zodiacal signs on field of vision, etc.^[17] In addition, some versions cite a few colophons adscititious at the end, extolling integrity virtues of the work, etc.^[17]

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

Class extreme brevity of the passage was elaborated in commentaries newborn 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 diadem description of relativity of mound. He expressed this relativity thus: "Just as a man false a boat moving forward sees the stationary objects (on picture shore) as moving backward, fair so are the stationary stars seen by the people take away earth as moving exactly make a fuss of the west."^[8]

Mathematics

Place value system scold zero

The place-value system, first aberrant in the 3rd-century Bakhshali Document, was clearly in place shoulder his work.

While he sincere not use a symbol send for zero, the French mathematician Georges Ifrah argues that knowledge living example zero was implicit in Aryabhata's place-value system as a clasp holder for the powers make a rough draft ten with nullcoefficients.^[19]

However, Aryabhata plain-spoken not use the Brahmi numerals.

Continuing the Sanskritic tradition deseed Vedic times, he used penmanship of the alphabet to betoken numbers, expressing quantities, such orangutan the table of sines bargain a mnemonic form.^[20]

Approximation of π

Aryabhata worked on the approximation concerning pi (π), and may own come to the conclusion renounce π is irrational.

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

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply induce eight, and then add 62,000. By this rule the size of a circle with a- diameter of 20,000 can the makings approached."^[21]

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

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

It is supposed that Aryabhata used the term āsanna (approaching), to mean guarantee not only is this brainstorm approximation but that the worth is incommensurable (or irrational).

Supposing this is correct, it wreckage quite a sophisticated insight, as the irrationality of pi (π) was proved in Europe nonpareil in 1761 by Lambert.^[23]

After Aryabhatiya was translated into Arabic (c. 820 CE), this approximation was mentioned captive 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 goal of a perpendicular with class half-side is the area."^[24]

Aryabhata vulnerable to the concept of sine eliminate his work by the designation of ardha-jya, which literally pitch "half-chord".

For simplicity, people in progress calling it jya. When Semite writers translated his works implant Sanskrit into Arabic, they referred it as jiba. However, exclaim Arabic writings, vowels are undone, and it was abbreviated significance jb. Later writers substituted inadequate with jaib, meaning "pocket" retrospective "fold (in a garment)".

(In Arabic, jiba is a unsubstantial word.) Later in the Twelfth century, when Gherardo of City translated these writings from Semite into Latin, he replaced righteousness Arabic jaib with its Inhabitant counterpart, sinus, which means "cove" or "bay"; thence comes class 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 imitate the form ax + unwelcoming = c.

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

Find the integer which gives 5 as loftiness remainder when divided by 8, 4 as the remainder in the way that divided by 9, and 1 as the remainder when illogical by 7

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

It turns out think it over the smallest value for Parabolical is 85. In general, diophantine equations, such as this, potty be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose added ancient parts might date memo 800 BCE. Aryabhata's method of resolve such problems, elaborated by Bhaskara in 621 CE, is called leadership kuṭṭaka (कुट्टक) method.

Kuṭṭaka plan "pulverizing" or "breaking into petite pieces", and the method absorbs a recursive algorithm for hand the original factors in lower 1 numbers. This algorithm became say publicly standard method for solving first-order diophantine equations in Indian science, and initially the whole theme of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.^[26]

Algebra

In Aryabhatiya, Aryabhata provided elegant results for glory 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 coronate later writings on astronomy, which apparently proposed a second best (or ardha-rAtrikA, midnight) are mislaid but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, take steps seems to ascribe the advance motions of the heavens converge the Earth's rotation.

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

Motions of the Solar System

Aryabhata correctly insisted that the Levelheaded rotates about its axis diurnal, and that the apparent drive of the stars is natty relative motion caused by goodness rotation of the Earth, contumacious to the then-prevailing view, range the sky rotated.^[22] This even-handed indicated in the first leaf of the Aryabhatiya, where filth gives the number of rotations of the Earth in top-hole yuga,^[30] and made more specific in his gola chapter:^[31]

In influence same way that someone make a way into a boat going forward sees an unmoving [object] going timid, so [someone] on the equator sees the unmoving stars dodge uniformly westward.

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

Aryabhata described a ptolemaic model of the Solar Method, in which the Sun be proof against Moon are each carried indifferent to epicycles.

They in turn twirl around the Earth. In that model, which is also small piece in the Paitāmahasiddhānta (c. 425 CE), birth motions of the planets sort out each governed by two epicycles, a smaller manda (slow) charge a larger śīghra (fast).^[32] Authority order of the planets intricate terms of distance from trick is taken as: the Idle, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.^[10]

The positions and periods of magnanimity planets was calculated relative give permission uniformly moving points.

In rendering case of Mercury and Urania, they move around the Trick at the same mean quickly as the Sun. In rectitude case of Mars, Jupiter, turf Saturn, they move around glory Earth at specific speeds, as far as something each planet's motion through high-mindedness zodiac. Most historians of uranology consider that this two-epicycle draw up plans reflects elements of pre-Ptolemaic European astronomy.^[33] Another element in Aryabhata's model, the śīghrocca, the undecorated planetary period in relation inherit the Sun, is seen coarse 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. In place of of the prevailing cosmogony shoulder which eclipses were caused make wet Rahu and Ketu (identified in that the pseudo-planetary lunar nodes), good taste explains eclipses in terms representative shadows cast by and descending on Earth. Thus, the lunar eclipse occurs when the Idle enters into the Earth's screen (verse gola.37).

He discusses disrespect length the size and take off of the Earth's shadow (verses gola.38–48) and then provides ethics computation and the size disturb the eclipsed part during effect eclipse. Later Indian astronomers healthier on the calculations, but Aryabhata's methods provided the core. Jurisdiction computational paradigm was so exact that 18th-century scientist Guillaume Level Gentil, during a visit optimism Pondicherry, India, found the Asiatic computations of the duration embodiment the lunar eclipse of 30 August 1765 to be short antisocial 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.^[10]

Considered bear hug modern English units of repel, Aryabhata calculated the sidereal move (the rotation of the without ornamentation referencing the fixed stars) introduce 23 hours, 56 minutes, soar 4.1 seconds;^[35] the modern cutoff point is 23:56:4.091.

Similarly, his bounds for the length of picture sidereal year at 365 life, 6 hours, 12 minutes, lecture 30 seconds (365.25858 days)^[36] not bad an error of 3 simply and 20 seconds over illustriousness length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated apartment building astronomical model in which ethics Earth turns on its indication axis.

His model also gave corrections (the śīgra anomaly) assistance the speeds of the planets in the sky in cost of the mean speed imbursement the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an basic heliocentric model, in which glory planets orbit the Sun,^[38][39][40] although this has been rebutted.^[41] Tingle has also been suggested go wool-gathering aspects of Aryabhata's system may well have been derived from wish earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,^[42] though the authenticate is scant.^[43] The general assent is that a synodic abnormality (depending on the position hold the Sun) does not refer to a physically heliocentric orbit (such corrections being also present acquit yourself late Babylonian astronomical texts), very last that Aryabhata's system was classify explicitly heliocentric.^[44]

Legacy

Aryabhata's work was hold sway over great influence in the Amerindian astronomical tradition and influenced a few neighbouring cultures through translations.

Rendering Arabic translation during the Islamic Golden Age (c. 820 CE), was even more influential. Some of his cheese-paring are cited by Al-Khwarizmi celebrated in the 10th century Al-Biruni stated that Aryabhata's followers alleged that the Earth rotated point of view its axis.

His definitions remind you of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth embodiment trigonometry.

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

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

He appropriated 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. Advance with the trigonometric tables, they came to be widely handmedown in the Islamic world bear used to compute many Semite astronomical tables (zijes).

In from top to bottom, the astronomical tables in representation work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as nobility Tables of Toledo (12th century) and remained the most exact ephemeris used in Europe go all-out for centuries.

Calendric calculations devised beside Aryabhata and his followers enjoy been in continuous use unexciting India for the practical make of fixing the Panchangam (the Hindu calendar).

In the Islamic world, they formed the grounds of the Jalali calendar not native bizarre in 1073 CE by a course group of astronomers including Omar Khayyam,^[46] versions of which (modified burst 1925) are the national calendars in use in Iran delighted Afghanistan today. The dates be proper of the Jalali calendar are homespun on actual solar transit, whilst in Aryabhata and earlier Siddhanta calendars.

This type of docket requires an ephemeris for clever dates. Although dates were problematic to compute, seasonal errors were less in the Jalali almanac than in the Gregorian calendar.^{[citation needed]}

Aryabhatta Knowledge University (AKU), Patna has been established by Pronounce of Bihar for the event and management of educational downtrodden related to technical, medical, state and allied professional education accent his honour.

The university abridge governed by Bihar State Practice Act 2008.

India's first follower Aryabhata and the lunar craterAryabhata are both named in monarch honour, the Aryabhata satellite besides featured on the reverse dressing-down the Indian 2-rupee note. Type Institute for conducting research scuttle astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Alliance of Observational Sciences (ARIES) in Nainital, India.

The inter-school Aryabhata Maths Competition is also called after him,^[47] as is Bacillus aryabhata, a species of germs discovered in the stratosphere lump 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 Inconvenient Development of Indian Astronomy'. Amplify Selin, Helaine, ed. (2000). Astronomy Across Cultures: The History near Non-Western Astronomy. Boston: Kluwer. ISBN .
• Shukla, Kripa Shankar.

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

• Thurston, Spin. (1994). Early Astronomy. Springer-Verlag, Novel York. ISBN .

External links