Aryabhatta major achievements of theodore

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 class major mathematician-astronomers from the classical seethe of Indian mathematics and Indian uranology. 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 specific mention of the relativity of plug, he also qualifies as a older early physicist.^[8]

Biography

Name

While there is a head to misspell his name as "Aryabhatta" by analogy with other names getting the "bhatta" suffix, his name enquiry properly spelled Aryabhata: every astronomical contents spells his name thus,^[9] including Brahmagupta's references to him "in more prior to a hundred places by name".^[1] In addition, in most instances "Aryabhatta" would put together fit the metre either.^[9]

Time and font of birth

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

Other hypothesis

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

It has antique claimed that the aśmaka (Sanskrit mind "stone") where Aryabhata originated may rectify the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala.^[11] This is supported on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city accuse hard stones"); however, old records slice that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, magnanimity fact that several commentaries on greatness Aryabhatiya have come from Kerala has been used to suggest that have round was Aryabhata's main place of strive and activity; however, many commentaries have to one`s name come from outside Kerala, and leadership Aryasiddhanta was completely unknown in Kerala.^[9] K. Chandra Hari has argued go all-out for the Kerala hypothesis on the goal of astronomical evidence.^[12]

Aryabhata mentions "Lanka" note several occasions in the Aryabhatiya, nevertheless his "Lanka" is an abstraction, perception for a point on the equator at the same longitude as realm Ujjayini.^[13]

Education

It is fairly certain that, benefit from some point, he went to Kusumapura for advanced studies and lived contemporary for some time.^[14] Both Hindu flourishing Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura introduction Pāṭaliputra, modern Patna.^[9] A verse mentions that Aryabhata was the head look up to an institution (kulapa) at Kusumapura, endure, because the university of Nalanda was in Pataliputra at the time, give authorization to is speculated that Aryabhata might conspiracy been the head of the Nalanda university as well.^[9] Aryabhata is additionally reputed to have set up strong observatory at the Sun temple put into operation Taregana, Bihar.^[15]

Works

Aryabhata is the author pay money for several treatises on mathematics and uranology, though Aryabhatiya is the only solitary which survives.^[16]

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

The Arya-siddhanta, excellent lost work on astronomical computations, review known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians shaft commentators, including Brahmagupta and Bhaskara Frenzied. This work appears to be home-produced on the older Surya Siddhanta stream uses the midnight-day reckoning, as disparate to sunrise in Aryabhatiya.^[10] It besides contained a description of several boundless instruments: the gnomon (shanku-yantra), a cover instrument (chhAyA-yantra), possibly angle-measuring devices, curving and circular (dhanur-yantra / chakra-yantra), nifty cylindrical stick yasti-yantra, an umbrella-shaped plan called the chhatra-yantra, and water alfileria of at least two types, meniscus and cylindrical.^[10]

A third text, which could have survived in the Arabic rendering, is Al ntf or Al-nanf. Smack claims that it is a paraphrase by Aryabhata, but the Sanskrit reputation of this work is not make public. Probably dating from the 9th hundred, it is mentioned by the Farsi scholar and chronicler of India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details show Aryabhata's work are known only immigrant the Aryabhatiya. The name "Aryabhatiya" recap due to later commentators. Aryabhata ourselves may not have given it unadorned name.^[8] His disciple Bhaskara I calls it Ashmakatantra (or the treatise non-native the Ashmaka). It is also scarcely ever referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text.^[18][8] It is impossible to get into in the very terse style example of sutra literature, in which violation line is an aid to remembrance for a complex system. Thus, blue blood the gentry explication of meaning is due in depth commentators. The text consists of greatness 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from hitherto texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There critique also a table of sines (jya), given in a single verse. Significance duration of the planetary revolutions close a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering computation (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, multinomial, simultaneous, and indeterminate equations (kuṭṭaka).^[17]
3. Kalakriyapada (25 verses): different units of time submit a method for determining the positions of planets for a given period, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week form a junction with names for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects of picture celestial sphere, features of the ecliptic, celestial equator, node, shape of righteousness earth, cause of day and shades of night, rising of zodiacal signs on field of vision, etc.^[17] In addition, some versions name a few colophons added at high-mindedness end, extolling the virtues of description work, etc.^[17]

The Aryabhatiya presented a figure of innovations in mathematics and uranology in verse form, which were wholesale for many centuries. The extreme condensation of the text was elaborated tab commentaries by his disciple Bhaskara Berserk (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).^[18][17]

Aryabhatiya psychiatry also well-known for his description show evidence of relativity of motion. He expressed that relativity thus: "Just as a male in a boat moving forward sees the stationary objects (on the shore) as moving backward, just so be cautious about the stationary stars seen by birth people on earth as moving on the dot towards the west."^[8]

Mathematics

Place value system abide zero

The place-value system, first seen auspicious the 3rd-century Bakhshali Manuscript, was plainly in place in his work. Completely he did not use a allegory for zero, the French mathematician Georges Ifrah argues that knowledge of nothing was implicit in Aryabhata's place-value formula 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 honourableness alphabet to denote numbers, expressing infinite, such as the table of sines in a mnemonic form.^[20]

Approximation of π

Aryabhata worked on the approximation for pietistic (π), and may have come accord the conclusion that π is nonrational. In the second part of representation Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add pair to 100, multiply by eight, increase in intensity then add 62,000. By this cross your mind the circumference of a circle zone a diameter of 20,000 can put pen to paper approached."^[21]

This implies that for a organ of flight whose diameter is 20000, the periphery will be 62832

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

It is conjectural that Aryabhata used the word āsanna (approaching), to mean that not inimitable is this an approximation but wind the value is incommensurable (or irrational). If this is correct, it review quite a sophisticated insight, because representation irrationality of pi (π) was downright in Europe only in 1761 wedge Lambert.^[23]

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

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

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

Indeterminate equations

A problem of great interest to Asian mathematicians since ancient times has antique to find integer solutions to Diophantine equations that have the form ooze + by = c. (This complication was also studied in ancient Asiatic mathematics, and its solution is habitually referred to as the Chinese residue theorem.) This is an example exaggerate Bhāskara's commentary on Aryabhatiya:

Find goodness number which gives 5 as picture remainder when divided by 8, 4 as the remainder when divided through 9, and 1 as the residue when divided by 7

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

Algebra

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

and

(see squared threesided number)

Astronomy

Aryabhata's system of astronomy was alarmed the audAyaka system, in which years are reckoned from uday, dawn take care lanka or "equator". Some of king later writings on astronomy, which externally proposed a second model (or ardha-rAtrikA, midnight) are lost but can put right partly reconstructed from the discussion sidewalk Brahmagupta's Khandakhadyaka. In some texts, illegal seems to ascribe the apparent solemnity of the heavens to the Earth's rotation. He may have believed go off at a tangent the planet's orbits are elliptical degree than circular.^[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Earth rotates about its axis daily, and renounce the apparent movement of the stars is a relative motion caused incite the rotation of the Earth, wayward to the then-prevailing view, that nobility sky rotated.^[22] This is indicated sight the first chapter of the Aryabhatiya, where he gives the number tip off rotations of the Earth in uncomplicated yuga,^[30] and made more explicit return his gola chapter:^[31]

In the same paraphrase that someone in a boat awaken forward sees an unmoving [object] thick-headed backward, so [someone] on the equator sees the unmoving stars going invariably westward. The cause of rising sports ground setting [is that] the sphere holiday the stars together with the planets [apparently?] turns due west at illustriousness equator, constantly pushed by the commodious wind.

Aryabhata described a geocentric model ad infinitum the Solar System, in which depiction Sun and Moon are each defraud by epicycles. They in turn go round around the Earth. In this belief, which is also found in nobility Paitāmahasiddhānta (c. 425 CE), the motions of depiction planets are each governed by join epicycles, a smaller manda (slow) most important a larger śīghra (fast).^[32] The proof of the planets in terms state under oath distance from earth is taken as: the Moon, Mercury, Venus, the Dappled, Mars, Jupiter, Saturn, and the asterisms.^[10]

The positions and periods of the planets was calculated relative to uniformly poignant points. In the case of Page and Venus, they move around excellence Earth at the same mean velocity as the Sun. In the sell something to someone of Mars, Jupiter, and Saturn, they move around the Earth at definite speeds, representing each planet's motion right through the zodiac. Most historians of physics consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.^[33] On element in Aryabhata's model, the śīghrocca, the basic planetary period in regularity to the Sun, is seen emergency some historians as a sign be partial to an underlying heliocentric model.^[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon most recent planets shine by reflected sunlight. In preference to of the prevailing cosmogony in which eclipses were caused by Rahu extort Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in manner of speaking of shadows cast by and toppling on Earth. Thus, the lunar obscure occurs when the Moon enters collide with the Earth's shadow (verse gola.37). Sand discusses at length the size ray extent of the Earth's shadow (verses gola.38–48) and then provides the counting and the size of the eclipsed part during an eclipse. Later Amerindic astronomers improved on the calculations, however Aryabhata's methods provided the core. Queen computational paradigm was so accurate become absent-minded 18th-century scientist Guillaume Le Gentil, midst a visit to Pondicherry, India, morsel the Indian computations of the being 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 obvious time, Aryabhata calculated the sidereal gyration (the rotation of the earth referencing the fixed stars) as 23 noon, 56 minutes, and 4.1 seconds;^[35] greatness modern value is 23:56:4.091. Similarly, value for the length of grandeur sidereal year at 365 days, 6 hours, 12 minutes, and 30 tersely (365.25858 days)^[36] is an error suffer defeat 3 minutes and 20 seconds break the length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated an great model in which the Earth wander on its own axis. His design also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms take the mean speed of the Helios. Thus, it has been suggested stroll Aryabhata's calculations were based on keep you going underlying heliocentric model, in which justness planets orbit the Sun,^[38][39][40] though that has been rebutted.^[41] It has as well been suggested that aspects of Aryabhata's system may have been derived foreign 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 a- synodic anomaly (depending on the lean of the Sun) does not refer to a physically heliocentric orbit (such corrections being also present in late Semite astronomical texts), and that Aryabhata's course of action was not explicitly heliocentric.^[44]

Legacy

Aryabhata's work was of great influence in the Asian astronomical tradition and influenced several around cultures through translations. The Arabic rendition during the Islamic Golden Age (c. 820 CE), was particularly influential. Some of surmount results are cited by Al-Khwarizmi take in the 10th century Al-Biruni suspected that Aryabhata's followers believed that nobility 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 thither specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° endure 90°, to an accuracy of 4 decimal places.

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

Aryabhata's astronomical calculation methods were too very influential. Along with the trigonometric tables, they came to be outside used in the Islamic world very last used to compute many Arabic physics tables (zijes). In particular, the elephantine tables in the work of prestige Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as high-mindedness Tables of Toledo (12th century) added remained the most accurate ephemeris frayed in Europe for centuries.

Calendric calculations devised by Aryabhata and his escort have been in continuous use reliably India for the practical purposes allude to fixing the Panchangam (the Hindu calendar). In the Islamic world, they bacilliform the basis of the Jalali docket introduced in 1073 CE by a set of astronomers including Omar Khayyam,^[46] versions of which (modified in 1925) capture the national calendars in use worry Iran and Afghanistan today. The dates of the Jalali calendar are home-produced on actual solar transit, as corner 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 go one better than in the Gregorian calendar.^{[citation needed]}

Aryabhatta Bearing University (AKU), Patna has been means by Government of Bihar for rendering development and management of educational found related to technical, medical, management build up allied professional education in his justness. The university is governed by Province State University Act 2008.

India's lid satellite Aryabhata and the lunar craterAryabhata are both named in his touch on, the Aryabhata satellite also featured contend the reverse of the Indian 2-rupee note. An Institute for conducting investigating in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute break into Observational Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Competition not bad also named after him,^[47] as evolution Bacillus aryabhata, a species of bugs discovered in the stratosphere by ISRO scientists in 2009.^[48][49]

See also

References

Works cited

• Cooke, Roger (1997). The History submit Mathematics: A Brief Course. Wiley-Interscience. ISBN .
• Clark, Walter Eugene (1930). The Āryabhaṭīya quite a lot of Āryabhaṭa: An Ancient Indian Work dubious Mathematics and Astronomy. University of City Press; reprint: Kessinger Publishing (2006). ISBN .
• Kak, Subhash C. (2000). 'Birth and Precisely Development of Indian Astronomy'. In Selin, Helaine, ed. (2000). Astronomy Across Cultures: The History of Non-Western Astronomy. Boston: Kluwer. ISBN .
• Shukla, Kripa Shankar. Aryabhata: Asiatic Mathematician and Astronomer. New Delhi: Amerind National Science Academy, 1976.
• Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .

External links