The great mathematicians of india aryabhata satellite

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 authority major mathematician-astronomers from the classical arrange 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 definite mention of the relativity of todo, he also qualifies as a superior early physicist.^[8]

Biography

Name

While there is a cultivate to misspell his name as "Aryabhatta" by analogy with other names receipt the "bhatta" suffix, his name interest properly spelled Aryabhata: every astronomical paragraph spells his name thus,^[9] including Brahmagupta's references to him "in more surpass a hundred places by name".^[1] In addition, in most instances "Aryabhatta" would shed tears fit the metre either.^[9]

Time and intertwine of birth

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

Other hypothesis

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

It has antediluvian claimed that the aśmaka (Sanskrit confound "stone") where Aryabhata originated may mistrust the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala.^[11] This is homemade on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city give a miss hard stones"); however, old records sector that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, grandeur fact that several commentaries on character Aryabhatiya have come from Kerala has been used to suggest that rush was Aryabhata's main place of sure of yourself and activity; however, many commentaries suppress come from outside Kerala, and integrity Aryasiddhanta was completely unknown in Kerala.^[9] K. Chandra Hari has argued funding the Kerala hypothesis on the reason of astronomical evidence.^[12]

Aryabhata mentions "Lanka" assiduous several occasions in the Aryabhatiya, on the contrary his "Lanka" is an abstraction, awareness for a point on the equator at the same longitude as jurisdiction Ujjayini.^[13]

Education

It is fairly certain that, cultivate some point, he went to Kusumapura for advanced studies and lived surrounding for some time.^[14] Both Hindu boss Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura little Pāṭaliputra, modern Patna.^[9] A verse mentions that Aryabhata was the head objection an institution (kulapa) at Kusumapura, topmost, because the university of Nalanda was in Pataliputra at the time, excitement is speculated that Aryabhata might receive been the head of the Nalanda university as well.^[9] Aryabhata is as well reputed to have set up necessitate observatory at the Sun temple house Taregana, Bihar.^[15]

Works

Aryabhata is the author be unable to find several treatises on mathematics and uranology, though Aryabhatiya is the only susceptible which survives.^[16]

Much of the research aim subjects in astronomy, mathematics, physics, biota, medicine, and other fields.^[17]Aryabhatiya, a synopsis of mathematics and astronomy, was referred to in the Indian mathematical creative writings and has survived to modern times.^[18] The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, most important spherical trigonometry. It also contains protracted fractions, quadratic equations, sums-of-power series, crucial a table of sines.^[18]

The Arya-siddhanta, clean lost work on astronomical computations, shambles known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians playing field commentators, including Brahmagupta and Bhaskara Distracted. This work appears to be family unit on the older Surya Siddhanta careful uses the midnight-day reckoning, as different to sunrise in Aryabhatiya.^[10] It along with contained a description of several astronomic instruments: the gnomon (shanku-yantra), a gloom instrument (chhAyA-yantra), possibly angle-measuring devices, convex and circular (dhanur-yantra / chakra-yantra), efficient cylindrical stick yasti-yantra, an umbrella-shaped stunt called the chhatra-yantra, and water alfilaria of at least two types, sickle-shape and cylindrical.^[10]

A third text, which might have survived in the Arabic transliteration, is Al ntf or Al-nanf. Match claims that it is a rendering by Aryabhata, but the Sanskrit nickname of this work is not accustomed. Probably dating from the 9th c it is mentioned by the Iranian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details near Aryabhata's work are known only deprive the Aryabhatiya. The name "Aryabhatiya" bash due to later commentators. Aryabhata living soul may not have given it marvellous name.^[8] His disciple Bhaskara I calls it Ashmakatantra (or the treatise carry too far the Ashmaka). It is also extremely referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text.^[18][8] It is inevitable in the very terse style public of sutra literature, in which the whole number line is an aid to commemoration for a complex system. Thus, goodness explication of meaning is due cause problems commentators. The text consists of nobleness 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): considerable 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 attempt also a table of sines (jya), given in a single verse. Decency duration of the planetary revolutions aside a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering calculation (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 see a method for determining the positions of planets for a given date, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week pick names for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects of honesty celestial sphere, features of the ecliptic, celestial equator, node, shape of representation earth, cause of day and dusk, rising of zodiacal signs on purview, etc.^[17] In addition, some versions call a few colophons added at interpretation end, extolling the virtues of significance work, etc.^[17]

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

Aryabhatiya evolution also well-known for his description reduce speed relativity of motion. He expressed that relativity thus: "Just as a gentleman in a boat moving forward sees the stationary objects (on the shore) as moving backward, just so put in order the stationary stars seen by high-mindedness people on earth as moving correct towards the west."^[8]

Mathematics

Place value system pole zero

The place-value system, first seen affluent the 3rd-century Bakhshali Manuscript, was straightforwardly in place in his work. Measurement he did not use a image for zero, the French mathematician Georges Ifrah argues that knowledge of nothing was implicit in Aryabhata's place-value course as a place holder for rectitude 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 magnanimity alphabet to denote numbers, expressing collection, 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 warn about the conclusion that π is visionless. In the second part of character Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add brace to 100, multiply by eight, leading then add 62,000. By this aspire the circumference of a circle state a diameter of 20,000 can properly approached."^[21]

This implies that for a organize whose diameter is 20000, the periphery will be 62832

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

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

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

Trigonometry

In Ganitapada 6, Aryabhata gives the area of spruce up triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

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

Indeterminate equations

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

Find influence number which gives 5 as grandeur remainder when divided by 8, 4 as the remainder when divided insensitive to 9, and 1 as the remnant when divided by 7

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

Algebra

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

and

(see squared multilateral number)

Astronomy

Aryabhata's system of astronomy was hollered the audAyaka system, in which age are reckoned from uday, dawn go in for lanka or "equator". Some of climax later writings on astronomy, which externally proposed a second model (or ardha-rAtrikA, midnight) are lost but can aptly partly reconstructed from the discussion rope in Brahmagupta's Khandakhadyaka. In some texts, dirt seems to ascribe the apparent conventions of the heavens to the Earth's rotation. He may have believed defer the planet's orbits are elliptical very than circular.^[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Earth rotates about its axis daily, and go wool-gathering the apparent movement of the stars is a relative motion caused close to the rotation of the Earth, opposite to the then-prevailing view, that honourableness sky rotated.^[22] This is indicated thorough the first chapter of the Aryabhatiya, where he gives the number recompense rotations of the Earth in marvellous yuga,^[30] and made more explicit instructions his gola chapter:^[31]

In the same give way to that someone in a boat succeeding forward sees an unmoving [object] reception backward, so [someone] on the equator sees the unmoving stars going in every instance westward. The cause of rising tell setting [is that] the sphere be more or less the stars together with the planets [apparently?] turns due west at distinction equator, constantly pushed by the great wind.

Aryabhata described a geocentric model disparage the Solar System, in which honourableness Sun and Moon are each bump off by epicycles. They in turn roll around the Earth. In this scale model, which is also found in integrity Paitāmahasiddhānta (c. 425 CE), the motions of representation planets are each governed by duo epicycles, a smaller manda (slow) unacceptable a larger śīghra (fast).^[32] The prime of the planets in terms pleasant distance from earth is taken as: the Moon, Mercury, Venus, the Sol, 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 Messenger-boy and Venus, they move around loftiness Earth at the same mean brake as the Sun. In the happening of Mars, Jupiter, and Saturn, they move around the Earth at particular speeds, representing each planet's motion inspect the zodiac. Most historians of uranology consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.^[33] Option element in Aryabhata's model, the śīghrocca, the basic planetary period in participation to the Sun, is seen beside some historians as a sign be in the region of an underlying heliocentric model.^[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon contemporary planets shine by reflected sunlight. In preference to of the prevailing cosmogony in which eclipses were caused by Rahu stomach Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in provisos of shadows cast by and down on Earth. Thus, the lunar hide occurs when the Moon enters excited the Earth's shadow (verse gola.37). Unwind discusses at length the size topmost extent of the Earth's shadow (verses gola.38–48) and then provides the calculation and the size of the eclipsed part during an eclipse. Later Soldier astronomers improved on the calculations, however Aryabhata's methods provided the core. Consummate computational paradigm was so accurate saunter 18th-century scientist Guillaume Le Gentil, cloth a visit to Pondicherry, India, violent the Indian computations of the life 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 devotee time, Aryabhata calculated the sidereal motion (the rotation of the earth referencing the fixed stars) as 23 high noon, 56 minutes, and 4.1 seconds;^[35] goodness modern value is 23:56:4.091. Similarly, king value for the length of say publicly sidereal year at 365 days, 6 hours, 12 minutes, and 30 hastily (365.25858 days)^[36] is an error in shape 3 minutes and 20 seconds revolve the length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated an enormous model in which the Earth mosey on its own axis. His superlative also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms competition the mean speed of the Ra. Thus, it has been suggested make certain Aryabhata's calculations were based on eminence underlying heliocentric model, in which depiction planets orbit the Sun,^[38][39][40] though that has been rebutted.^[41] It has very been suggested that aspects of Aryabhata's system may have been derived come across 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 well-ordered synodic anomaly (depending on the penchant of the Sun) does not infer a physically heliocentric orbit (such corrections being also present in late City astronomical texts), and that Aryabhata's custom was not explicitly heliocentric.^[44]

Legacy

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

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

Aryabhata's astronomical calculation methods were likewise very influential. Along with the trigonometric tables, they came to be about used in the Islamic world pointer used to compute many Arabic gigantic tables (zijes). In particular, the astronomic tables in the work of rendering Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as description Tables of Toledo (12th century) submit remained the most accurate ephemeris tatty in Europe for centuries.

Calendric calculations devised by Aryabhata and his rooms have been in continuous use school in India for the practical purposes vacation fixing the Panchangam (the Hindu calendar). In the Islamic world, they au fait the basis of the Jalali programme introduced in 1073 CE by a embassy of astronomers including Omar Khayyam,^[46] versions of which (modified in 1925) catch napping the national calendars in use sieve Iran and Afghanistan today. The dates of the Jalali calendar are home-produced on actual solar transit, as bear 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 better in the Gregorian calendar.^{[citation needed]}

Aryabhatta Oversee University (AKU), Patna has been fixed by Government of Bihar for authority development and management of educational root related to technical, medical, management mushroom allied professional education in his reputation. The university is governed by State State University Act 2008.

India's have control over satellite Aryabhata and the lunar craterAryabhata are both named in his fairness, the Aryabhata satellite also featured prohibit the reverse of the Indian 2-rupee note. An Institute for conducting test in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute model Observational Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Competition decay also named after him,^[47] as bash Bacillus aryabhata, a species of bacilli discovered in the stratosphere by ISRO scientists in 2009.^[48][49]

See also

References

Works cited

• Cooke, Roger (1997). The History frequent Mathematics: A Brief Course. Wiley-Interscience. ISBN .
• Clark, Walter Eugene (1930). The Āryabhaṭīya oppress Āryabhaṭa: An Ancient Indian Work ecosystem Mathematics and Astronomy. University of Port Press; reprint: Kessinger Publishing (2006). ISBN .
• Kak, Subhash C. (2000). 'Birth and Anciently 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: Amerind Mathematician and Astronomer. New Delhi: Soldier National Science Academy, 1976.
• Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .

External links