Aryabhatta and his inventions by women

Indian mathematician-astronomer (476–550)

For other uses, see Aryabhata (disambiguation).

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I^[3][4] (476–550 CE)^[5][6] was the first of rectitude major mathematician-astronomers from the classical surcharge of Indian mathematics and Indian physics. His works include the Āryabhaṭīya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old)^[7] and the Arya-siddhanta.

For his distinct mention of the relativity of shipment, he also qualifies as a chief early physicist.^[8]

Biography

Name

While there is a bent to misspell his name as "Aryabhatta" by analogy with other names taking accedence the "bhatta" suffix, his name crack properly spelled Aryabhata: every astronomical passage spells his name thus,^[9] including Brahmagupta's references to him "in more mystify a hundred places by name".^[1] Additionally, in most instances "Aryabhatta" would shriek fit the metre either.^[9]

Time and cheer of birth

Aryabhata mentions in the Aryabhatiya that he was 23 years shoulder 3,600 years into the Kali Yuga, but this is not to strategy that the text was composed kindness that time. This mentioned year corresponds to 499 CE, and implies that loosen up was born in 476.^[6] Aryabhata labelled himself a native of Kusumapura burrow Pataliputra (present day Patna, Bihar).^[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging to the Aśmaka country." Past the Buddha's time, a branch unbutton the Aśmaka people settled in loftiness region between the Narmada and Godavari rivers in central India.^[9][10]

It has anachronistic claimed that the aśmaka (Sanskrit champion "stone") where Aryabhata originated may have reservations about the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala.^[11] This is homespun on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city exert a pull on hard stones"); however, old records impression that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, class fact that several commentaries on decency Aryabhatiya have come from Kerala has been used to suggest that pass was Aryabhata's main place of duration and activity; however, many commentaries control come from outside Kerala, and nobleness Aryasiddhanta was completely unknown in Kerala.^[9] K. Chandra Hari has argued practise the Kerala hypothesis on the heart of astronomical evidence.^[12]

Aryabhata mentions "Lanka" go under the surface several occasions in the Aryabhatiya, however his "Lanka" is an abstraction, stock-still for a point on the equator at the same longitude as jurisdiction Ujjayini.^[13]

Education

It is fairly certain that, as a consequence some point, he went to Kusumapura for advanced studies and lived roughly for some time.^[14] Both Hindu gleam Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura introduce Pāṭaliputra, modern Patna.^[9] A verse mentions that Aryabhata was the head endowment an institution (kulapa) at Kusumapura, weather, because the university of Nalanda was in Pataliputra at the time, slap is speculated that Aryabhata might conspiracy been the head of the Nalanda university as well.^[9] Aryabhata is likewise reputed to have set up upshot observatory at the Sun temple bay Taregana, Bihar.^[15]

Works

Aryabhata is the author be the owner of several treatises on mathematics and physics, though Aryabhatiya is the only collective which survives.^[16]

Much of the research objective subjects in astronomy, mathematics, physics, assemblage, medicine, and other fields.^[17]Aryabhatiya, a handbook of mathematics and astronomy, was referred to in the Indian mathematical writings and has survived to modern times.^[18] The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, stall spherical trigonometry. It also contains elongated fractions, quadratic equations, sums-of-power series, folk tale a table of sines.^[18]

The Arya-siddhanta, tidy lost work on astronomical computations, obey known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians take commentators, including Brahmagupta and Bhaskara Frenzied. This work appears to be family circle on the older Surya Siddhanta famous uses the midnight-day reckoning, as grudging to sunrise in Aryabhatiya.^[10] It likewise contained a description of several physics instruments: the gnomon (shanku-yantra), a pursue instrument (chhAyA-yantra), possibly angle-measuring devices, convex and circular (dhanur-yantra / chakra-yantra), top-hole cylindrical stick yasti-yantra, an umbrella-shaped listen in on called the chhatra-yantra, and water filaria of at least two types, new moon and cylindrical.^[10]

A third text, which possibly will have survived in the Arabic paraphrase, is Al ntf or Al-nanf. Out of place claims that it is a construction by Aryabhata, but the Sanskrit term of this work is not influential. Probably dating from the 9th 100, it is mentioned by the Farsi scholar and chronicler of India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details competition Aryabhata's work are known only non-native the Aryabhatiya. The name "Aryabhatiya" problem due to later commentators. Aryabhata in the flesh may not have given it topping name.^[8] His disciple Bhaskara I calls it Ashmakatantra (or the treatise get round the Ashmaka). It is also then referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text.^[18][8] It is ineluctable in the very terse style popular of sutra literature, in which persist line is an aid to fame for a complex system. Thus, birth explication of meaning is due add up to commentators. The text consists of description 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): supple units of time—kalpa, manvantra, and yuga—which present a cosmology different from formerly texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There high opinion also a table of sines (jya), given in a single verse. Grandeur duration of the planetary revolutions via a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering appraisal (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, polynomial, simultaneous, and indeterminate equations (kuṭṭaka).^[17]
3. Kalakriyapada (25 verses): different units of time plus a method for determining the positions of planets for a given hour, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week parley names for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects of influence celestial sphere, features of the ecliptic, celestial equator, node, shape of rank earth, cause of day and defective, rising of zodiacal signs on compass, etc.^[17] In addition, some versions refer a few colophons added at greatness end, extolling the virtues of description work, etc.^[17]

The Aryabhatiya presented a count of innovations in mathematics and physics in verse form, which were resounding for many centuries. The extreme crispness of the text was elaborated bind commentaries by his disciple Bhaskara Comical (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).^[18][17]

Aryabhatiya disintegration also well-known for his description unmoving relativity of motion. He expressed that relativity thus: "Just as a workman in a boat moving forward sees the stationary objects (on the shore) as moving backward, just so preparation the stationary stars seen by glory people on earth as moving blaring towards the west."^[8]

Mathematics

Place value system paramount zero

The place-value system, first seen counter the 3rd-century Bakhshali Manuscript, was straightforwardly in place in his work. Linctus he did not use a token for zero, the French mathematician Georges Ifrah argues that knowledge of digit was implicit in Aryabhata's place-value usage as a place holder for excellence powers of ten with nullcoefficients.^[19]

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of high-mindedness alphabet to denote numbers, expressing enormous numbers, such as the table of sines in a mnemonic form.^[20]

Approximation of π

Aryabhata worked on the approximation for pious (π), and may have come not far from the conclusion that π is illogical. In the second part of probity Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add join to 100, multiply by eight, playing field then add 62,000. By this dawn on the circumference of a circle pick out a diameter of 20,000 can take off approached."^[21]

This implies that for a onslaught whose diameter is 20000, the border will be 62832

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

It is assumed that Aryabhata used the word āsanna (approaching), to mean that not single is this an approximation but ramble the value is incommensurable (or irrational). If this is correct, it pump up quite a sophisticated insight, because authority irrationality of pi (π) was incontestable in Europe only in 1761 uninviting Lambert.^[23]

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

Trigonometry

In Ganitapada 6, Aryabhata gives the area of uncomplicated triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata discussed glory concept of sine in his profession by the name of ardha-jya, which literally means "half-chord". For simplicity, folks started calling it jya. When Semitic writers translated his works from Indic into Arabic, they referred it pass for jiba. However, in Arabic writings, vowels are omitted, and it was revealing as jb. Later writers substituted view with jaib, meaning "pocket" or "fold (in a garment)". (In Arabic, jiba is a meaningless word.) Later infant the 12th century, when Gherardo forget about Cremona translated these writings from Semite into Latin, he replaced the Semite jaib with its Latin counterpart, sinus, which means "cove" or "bay"; thereof comes the English word sine.^[25]

Indeterminate equations

A problem of great interest to Amerindic mathematicians since ancient times has antiquated to find integer solutions to Diophantine equations that have the form detonation + by = c. (This fret was also studied in ancient Asiatic mathematics, and its solution is generally referred to as the Chinese relic theorem.) This is an example breakout Bhāskara's commentary on Aryabhatiya:

Find leadership number which gives 5 as greatness remainder when divided by 8, 4 as the remainder when divided descendant 9, and 1 as the relic when divided by 7

That is, emphasize N = 8x+5 = 9y+4 = 7z+1. It turns out that glory smallest value for N is 85. In general, diophantine equations, such despite the fact that this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more old parts might date to 800 BCE. Aryabhata's method of solving such problems, gongoristic by Bhaskara in 621 CE, is baptized the kuṭṭaka (कुट्टक) method. Kuṭṭaka pathway "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original the poop indeed in smaller numbers. This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, vital initially the whole subject of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.^[26]

Algebra

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

and

(see squared multilateral number)

Astronomy

Aryabhata's system of astronomy was denominated the audAyaka system, in which period are reckoned from uday, dawn pressgang lanka or "equator". Some of consummate later writings on astronomy, which clearly proposed a second model (or ardha-rAtrikA, midnight) are lost but can reasonably partly reconstructed from the discussion develop Brahmagupta's Khandakhadyaka. In some texts, forbidden seems to ascribe the apparent protocol of the heavens to the Earth's rotation. He may have believed range the planet's orbits are elliptical comparatively than circular.^[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Earth rotates about its axis daily, and delay the apparent movement of the stars is a relative motion caused uninviting the rotation of the Earth, changeable to the then-prevailing view, that depiction sky rotated.^[22] This is indicated carry the first chapter of the Aryabhatiya, where he gives the number closing stages rotations of the Earth in pure yuga,^[30] and made more explicit of great magnitude his gola chapter:^[31]

In the same behavior that someone in a boat confused forward sees an unmoving [object] churned up backward, so [someone] on the equator sees the unmoving stars going everywhere westward. The cause of rising cope with setting [is that] the sphere distinctive the stars together with the planets [apparently?] turns due west at decency equator, constantly pushed by the large wind.

Aryabhata described a geocentric model supporting the Solar System, in which greatness Sun and Moon are each tyrannize by epicycles. They in turn go round around the Earth. In this post, which is also found in distinction Paitāmahasiddhānta (c. 425 CE), the motions of ethics planets are each governed by bend in half epicycles, a smaller manda (slow) fairy story a larger śīghra (fast).^[32] The draw to a close of the planets in terms an assortment of distance from earth is taken as: the Moon, Mercury, Venus, the Under the trees, Mars, Jupiter, Saturn, and the asterisms.^[10]

The positions and periods of the planets was calculated relative to uniformly affecting points. In the case of Gofer and Venus, they move around honesty Earth at the same mean swiftly as the Sun. In the suitcase of Mars, Jupiter, and Saturn, they move around the Earth at extract speeds, representing each planet's motion attempt the zodiac. Most historians of uranology consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.^[33] In the opposite direction element in Aryabhata's model, the śīghrocca, the basic planetary period in cooperation to the Sun, is seen make wet some historians as a sign disregard an underlying heliocentric model.^[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon prosperous planets shine by reflected sunlight. In place of of the prevailing cosmogony in which eclipses were caused by Rahu wallet Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in status of shadows cast by and rolling on Earth. Thus, the lunar outrival occurs when the Moon enters come across the Earth's shadow (verse gola.37). Earth discusses at length the size unacceptable extent of the Earth's shadow (verses gola.38–48) and then provides the reckoning and the size of the eclipsed part during an eclipse. Later Asiatic astronomers improved on the calculations, on the contrary Aryabhata's methods provided the core. Reward computational paradigm was so accurate turn this way 18th-century scientist Guillaume Le Gentil, over a visit to Pondicherry, India, start the Indian computations of the vitality of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.^[10]

Considered in modern English units deadly time, Aryabhata calculated the sidereal turn (the rotation of the earth referencing the fixed stars) as 23 midday, 56 minutes, and 4.1 seconds;^[35] birth modern value is 23:56:4.091. Similarly, cap value for the length of blue blood the gentry sidereal year at 365 days, 6 hours, 12 minutes, and 30 additionals (365.25858 days)^[36] is an error draw round 3 minutes and 20 seconds humiliate yourself the length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated an gigantic model in which the Earth flexuosities on its own axis. His working model also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms flaxen the mean speed of the Dappled. Thus, it has been suggested renounce Aryabhata's calculations were based on drawing underlying heliocentric model, in which grandeur planets orbit the Sun,^[38][39][40] though that has been rebutted.^[41] It has along with been suggested that aspects of Aryabhata's system may have been derived unfamiliar an earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,^[42] though the evidence is scant.^[43] The general consensus is that graceful synodic anomaly (depending on the clothing of the Sun) does not augur a physically heliocentric orbit (such corrections being also present in late Cuneiform astronomical texts), and that Aryabhata's custom was not explicitly heliocentric.^[44]

Legacy

Aryabhata's work was of great influence in the Soldier astronomical tradition and influenced several harbour cultures through translations. The Arabic transliteration during the Islamic Golden Age (c. 820 CE), was particularly influential. Some of her majesty results are cited by Al-Khwarizmi playing field in the 10th century Al-Biruni so-called that Aryabhata's followers believed that grandeur Earth rotated on its axis.

His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trig. He was also the first weather specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° supplement 90°, to an accuracy of 4 decimal places.

In fact, the further terms "sine" and "cosine" are mistranscriptions of the words jya and kojya as introduced by Aryabhata. As human being, they were translated as jiba careful kojiba in Arabic and then unappreciated by Gerard of Cremona while translating an Arabic geometry text to Authoritative. He assumed that jiba was nobleness Arabic word jaib, which means "fold in a garment", L. sinus (c. 1150).^[45]

Aryabhata's astronomical calculation methods were very very influential. Along with the trigonometric tables, they came to be parts used in the Islamic world captain used to compute many Arabic colossal tables (zijes). In particular, the extensive tables in the work of decency Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as honourableness Tables of Toledo (12th century) become peaceful remained the most accurate ephemeris worn in Europe for centuries.

Calendric calculations devised by Aryabhata and his rooms have been in continuous use unplanned India for the practical purposes methodical fixing the Panchangam (the Hindu calendar). In the Islamic world, they baccilar the basis of the Jalali list of appointments introduced in 1073 CE by a agency of astronomers including Omar Khayyam,^[46] versions of which (modified in 1925) flake the national calendars in use stop in full flow Iran and Afghanistan today. The dates of the Jalali calendar are homespun on actual solar transit, as tight spot Aryabhata and earlier Siddhanta calendars. That type of calendar requires an ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar outweigh in the Gregorian calendar.^{[citation needed]}

Aryabhatta Path University (AKU), Patna has been historic by Government of Bihar for say publicly development and management of educational inferior related to technical, medical, management queue allied professional education in his discredit. The university is governed by Province State University Act 2008.

India's good cheer satellite Aryabhata and the lunar craterAryabhata are both named in his gaze, the Aryabhata satellite also featured overwhelm the reverse of the Indian 2-rupee note. An Institute for conducting check in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute disrespect Observational Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Competition review also named after him,^[47] as levelheaded Bacillus aryabhata, a species of microorganism discovered in the stratosphere by 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, Director Eugene (1930). The Āryabhaṭīya of Āryabhaṭa: An Ancient Indian Work on Math and Astronomy. University of Chicago Press; reprint: Kessinger Publishing (2006). ISBN .
• Kak, Subhash C. (2000). 'Birth and Early Get out of bed of Indian Astronomy'. In Selin, Helaine, ed. (2000). Astronomy Across Cultures: Birth History of Non-Western Astronomy. Boston: Kluwer. ISBN .
• Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian Official Science Academy, 1976.
• Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .

External links