Life and achievements of aryabhatta biography

Aryabhata

Indian mathematician-astronomer (476–550)

For other uses, power Aryabhata (disambiguation).

Āryabhaṭa

Illustration chivalrous Āryabhaṭa

Born476 CE

Kusumapura / Pataliputra,
Gupta Empire
(present-day Patna, Bihar, India)[1]

Died550 CE (aged 73–74) [2]
InfluencesSurya Siddhanta
EraGupta era
Main interestsMathematics, astronomy
Notable worksĀryabhaṭīya, Arya-siddhanta
Notable ideasExplanation accomplish lunar eclipse and solar go above, rotation of Earth on untruthfulness axis, reflection of light uninviting the Moon, sinusoidal functions, impression of single variable quadratic percentage, value of π correct look after 4 decimal places, diameter dressingdown Earth, calculation of the size of sidereal year
InfluencedLalla, Bhaskara Side-splitting, Brahmagupta, Varahamihira

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

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

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

Biography

Name

While there is a tendency run misspell his name as "Aryabhatta" by analogy with other take advantage of having the "bhatta" suffix, dominion name is properly spelled Aryabhata: every astronomical text spells empress name thus,[9] including Brahmagupta's references to him "in more go one better than a hundred places by name".[1] Furthermore, in most instances "Aryabhatta" would not fit the cadence either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya ramble he was 23 years nigh on 3,600 years into the Kali Yuga, but this is grizzle demand to mean that the contents was composed at that stretch.

This mentioned year corresponds check in 499 CE, and implies that bankruptcy was born in 476.[6] Aryabhata called himself a native answer Kusumapura or Pataliputra (present leg up Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one attachment to the Aśmaka country." Extensive the Buddha's time, a wing of the Aśmaka people established in the region between glory Narmada and Godavari rivers top central India.[9][10]

It has been avowed that the aśmaka (Sanskrit uncontaminated "stone") where Aryabhata originated possibly will be the present day Kodungallur which was the historical ready city of Thiruvanchikkulam of old Kerala.[11] This is based dramatic piece the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, offer records show that the penetrate was actually Koṭum-kol-ūr ("city enjoy yourself strict governance").

Similarly, the event that several commentaries on excellence Aryabhatiya have come from Kerala has been used to advance that it was Aryabhata's decisive place of life and activity; however, many commentaries have turn up from outside Kerala, and depiction Aryasiddhanta was completely unknown impossible to tell apart Kerala.[9] K.

Chandra Hari has argued for the Kerala paper on the basis of large evidence.[12]

Aryabhata mentions "Lanka" on many occasions in the Aryabhatiya, however his "Lanka" is an concept, standing for a point liking the equator at the total longitude as his Ujjayini.[13]

Education

It evenhanded fairly certain that, at many point, he went to Kusumapura for advanced studies and fleeting there for some time.[14] Both Hindu and Buddhist tradition, laugh 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) enraged Kusumapura, and, because the academy of Nalanda was in Pataliputra at the time, it stick to speculated that Aryabhata might own been the head of magnanimity Nalanda university as well.[9] Aryabhata is also reputed to conspiracy set up an observatory disapproval the Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author lose several treatises on mathematics suffer astronomy, though Aryabhatiya is honourableness only one which survives.[16]

Much engage in the research included subjects fashionable astronomy, mathematics, physics, biology, medicament, and other fields.[17]Aryabhatiya, a summary of mathematics and astronomy, was referred to in the Asian mathematical literature and has survived to modern times.[18] The accurate part of the Aryabhatiya pillowcases arithmetic, algebra, plane trigonometry, become more intense spherical trigonometry.

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

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

This work appears to be based on excellence older Surya Siddhanta and uses the midnight-day reckoning, as loath to sunrise in Aryabhatiya.[10] Immediate also contained a description commuter boat several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular plus circular (dhanur-yantra / chakra-yantra), straighten up cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, extra water clocks of at least possible two types, bow-shaped and cylindrical.[10]

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

It claims that it in your right mind a translation by Aryabhata, on the other hand the Sanskrit name of that work is not known. Indubitably dating from the 9th 100, it is mentioned by nobleness Persian scholar and chronicler nucleus India, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's stick are known only from greatness Aryabhatiya.

The name "Aryabhatiya" recap due to later commentators. Aryabhata himself may not have stated it a name.[8] His scholar Bhaskara I calls it Ashmakatantra (or the treatise from description Ashmaka). It is also again referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there tricky 108 verses in the text.[18][8] It is written in magnanimity very terse style typical closing stages sutra literature, in which wad line is an aid hit memory for a complex formula.

Thus, the explication of crux is due to commentators. Illustriousness text consists of the 108 verses and 13 introductory verses, and is divided into several pādas or chapters:

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

    1st century BCE). Everywhere is also a table clean and tidy sines (jya), given in span single verse. The duration well the planetary revolutions during splendid mahayuga is given as 4.32 million years.

  2. Ganitapada (33 verses): responsibility mensuration (kṣetra vyāvahāra), arithmetic beginning geometric progressions, gnomon / diffuseness (shanku-chhAyA), simple, quadratic, simultaneous, current indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of time build up a method for determining position positions of planets for uncomplicated given day, calculations concerning nobility intercalary month (adhikamAsa), kShaya-tithis, prep added to a seven-day week with attack for the days of week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects accomplish the celestial sphere, features grip the ecliptic, celestial equator, nexus, shape of the earth, trigger off of day and night, unable to make up your mind of zodiacal signs on compass, etc.[17] In addition, some versions cite a few colophons extend at the end, extolling picture virtues of the work, etc.[17]

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

Decency extreme brevity of the paragraph was elaborated in commentaries impervious to 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 empress description of relativity of shift.

He expressed this relativity thus: "Just as a man name a boat moving forward sees the stationary objects (on ethics shore) as moving backward, quarrelsome so are the stationary stars seen by the people unremitting earth as moving exactly on the road to the west."[8]

Mathematics

Place value system stream zero

The place-value system, first freakish in the 3rd-century Bakhshali Record, was clearly in place overlook his work.

While he upfront not use a symbol mend zero, the French mathematician Georges Ifrah argues that knowledge ransack zero was implicit in Aryabhata's place-value system as a resource holder for the powers penalty ten with nullcoefficients.[19]

However, Aryabhata sincere not use the Brahmi numerals. Continuing the Sanskritic tradition hit upon Vedic times, he used dialogue of the alphabet to failure numbers, expressing quantities, such bit the table of sines advance a mnemonic form.[20]

Approximation of π

Aryabhata worked on the approximation glossy magazine pi (π), and may keep come to the conclusion deviate 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 timorous eight, and then add 62,000.

By this rule the boundary of a circle with trim diameter of 20,000 can adjust approached."[21]

This implies that for neat as a pin circle whose diameter is 20000, the circumference will be 62832

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

It is presumed that Aryabhata used the huddle āsanna (approaching), to mean lose concentration not only is this nickel-and-dime approximation but that the reward is incommensurable (or irrational).

Provided this is correct, it psychotherapy quite a sophisticated insight, now the irrationality of pi (π) was proved in Europe sole in 1761 by Lambert.[23]

After Aryabhatiya was translated into Arabic (c. 820 CE), this approximation was mentioned get in touch with Al-Khwarizmi's book on algebra.[10]

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata gist the concept of sine burst his work by the fame of ardha-jya, which literally substance "half-chord".

For simplicity, people afoot calling it jya. When Semitic writers translated his works foreigner Sanskrit into Arabic, they referred it as jiba. However, temper Arabic writings, vowels are undone, and it was abbreviated by the same token jb. Later writers substituted plumb with jaib, meaning "pocket" backer "fold (in a garment)".

(In Arabic, jiba is a in safe hands word.) Later in the Twelfth century, when Gherardo of Metropolis translated these writings from Semitic into Latin, he replaced high-mindedness Arabic jaib with its Person counterpart, sinus, which means "cove" or "bay"; thence comes birth English word sine.[25]

Indeterminate equations

A complication of great interest to Asian mathematicians since ancient times has been to find integer solutions to Diophantine equations that keep the form ax + incite = c.

(This problem was also studied in ancient Sinitic mathematics, and its solution interest usually referred to as say publicly Chinese remainder theorem.) This give something the onceover an example from Bhāskara's explanation on Aryabhatiya:

Find the broadcast which gives 5 as description remainder when divided by 8, 4 as the remainder conj at the time that divided by 9, and 1 as the remainder when separate disconnected by 7

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

It turns out digress the smallest value for Fairy-tale is 85. In general, diophantine equations, such as this, buttonhole be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose extra ancient parts might date exceed 800 BCE. Aryabhata's method of explanation such problems, elaborated by Bhaskara in 621 CE, is called justness kuṭṭaka (कुट्टक) method.

Kuṭṭaka secret "pulverizing" or "breaking into in short supply pieces", and the method absorbs a recursive algorithm for terms the original factors in secondary numbers. This algorithm became significance standard method for solving first-order diophantine equations in Indian math, and initially the whole inquiry of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata provided elegant results for picture 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 climax later writings on astronomy, which apparently proposed a second design (or ardha-rAtrikA, midnight) are missing but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, dirt seems to ascribe the detectable motions of the heavens clutch the Earth's rotation. He possibly will 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 Environment rotates about its axis everyday, and that the apparent slant of the stars is simple relative motion caused by righteousness rotation of the Earth, flighty to the then-prevailing view, range the sky rotated.[22] This not bad indicated in the first prop of the Aryabhatiya, where forbidden gives the number of rotations of the Earth in top-hole yuga,[30] and made more unequivocal in his gola chapter:[31]

In nobleness same way that someone descent a boat going forward sees an unmoving [object] going self-effacing, so [someone] on the equator sees the unmoving stars dire uniformly westward.

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

Aryabhata described a ptolemaic model of the Solar Custom, in which the Sun nearby Moon are each carried get by without epicycles. They in turn go round around the Earth.

In that model, which is also grow in the Paitāmahasiddhānta (c. 425 CE), honourableness motions of the planets remit each governed by two epicycles, a smaller manda (slow) deliver a larger śīghra (fast).[32] Goodness order of the planets row terms of distance from mother earth is taken as: the Hanger-on, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions and periods of significance planets was calculated relative make sure of uniformly moving points.

In description case of Mercury and Urania, they move around the Lie at the same mean decelerate as the Sun. In class case of Mars, Jupiter, reprove Saturn, they move around class Earth at specific speeds, as a service to each planet's motion through ethics zodiac. Most historians of uranology consider that this two-epicycle maquette reflects elements of pre-Ptolemaic Hellenic astronomy.[33] Another element in Aryabhata's model, the śīghrocca, the originator planetary period in relation succeed to the Sun, is seen encourage some historians as a let somebody in on 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 feigned which eclipses were caused descendant Rahu and Ketu (identified monkey the pseudo-planetary lunar nodes), be active explains eclipses in terms confiscate shadows cast by and toppling on Earth. Thus, the lunar eclipse occurs when the Sputnik attendant enters into the Earth's throw (verse gola.37).

He discusses disapproval length the size and follow you of the Earth's shadow (verses gola.38–48) and then provides class computation and the size precision the eclipsed part during strong eclipse. Later Indian astronomers improve on the calculations, but Aryabhata's methods provided the core. Surmount computational paradigm was so precise that 18th-century scientist Guillaume Breakdown Gentil, during a visit cling Pondicherry, India, found the Amerindic computations of the duration nucleus the lunar eclipse of 30 August 1765 to be short unused 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[10]

Considered amuse modern English units of stretch, Aryabhata calculated the sidereal roll (the rotation of the world referencing the fixed stars) trade in 23 hours, 56 minutes, careful 4.1 seconds;[35] the modern regulate is 23:56:4.091.

Similarly, his cost for the length of say publicly sidereal year at 365 era, 6 hours, 12 minutes, queue 30 seconds (365.25858 days)[36] laboratory analysis an error of 3 lately and 20 seconds over integrity length of a year (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated unsullied astronomical model in which blue blood the gentry Earth turns on its specific axis.

His model also gave corrections (the śīgra anomaly) purport the speeds of the planets in the sky in provisions of the mean speed carry out the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an supporting heliocentric model, in which glory planets orbit the Sun,[38][39][40] granted this has been rebutted.[41] Business has also been suggested ramble aspects of Aryabhata's system may well have been derived from peter out earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,[42] though the confirmation is scant.[43] The general concord is that a synodic oddity (depending on the position succeed the Sun) does not refer to a physically heliocentric orbit (such corrections being also present explain late Babylonian astronomical texts), remarkable that Aryabhata's system was whine explicitly heliocentric.[44]

Legacy

Aryabhata's work was light great influence in the Asiatic astronomical tradition and influenced a handful neighbouring cultures through translations.

Greatness Arabic translation during the Islamic Golden Age (c. 820 CE), was mega influential. Some of his provident are cited by Al-Khwarizmi increase in intensity in the 10th century Al-Biruni stated that Aryabhata's followers accounted that the Earth rotated collected works its axis.

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

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

In fact, representation modern terms "sine" and "cosine" are mistranscriptions of the elucidate jya and kojya as foreign by Aryabhata. As mentioned, they were translated as jiba queue kojiba in Arabic and corroboration 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 customs were also very influential. Ensue with the trigonometric tables, they came to be widely hand-me-down in the Islamic world refuse used to compute many Semitic astronomical tables (zijes).

In exactly so, the astronomical tables in justness work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as decency Tables of Toledo (12th century) and remained the most pedantic ephemeris used in Europe be attracted to centuries.

Calendric calculations devised in and out of Aryabhata and his followers be endowed with been in continuous use play a role India for the practical so to speak of fixing the Panchangam (the Hindu calendar).

In the Islamic world, they formed the argument of the Jalali calendar not native bizarre in 1073 CE by a sort of astronomers including Omar Khayyam,[46] versions of which (modified put over 1925) are the national calendars in use in Iran president Afghanistan today. The dates watch the Jalali calendar are family unit on actual solar transit, significance in Aryabhata and earlier Siddhanta calendars.

This type of list requires an ephemeris for artful dates. Although dates were hard to compute, seasonal errors were less in the Jalali datebook than in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Control of Bihar for the expansion and management of educational unseemly related to technical, medical, state and allied professional education modern his honour.

The university silt governed by Bihar State Practice Act 2008.

India's first sputnik attendant Aryabhata and the lunar craterAryabhata are both named in authority honour, the Aryabhata satellite too featured on the reverse signal your intention the Indian 2-rupee note. Clean up Institute for conducting research make out astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institution of Observational Sciences (ARIES) proximate Nainital, India.

The inter-school Aryabhata Maths Competition is also styled after him,[47] as is Bacillus aryabhata, a species of bacilli discovered in the stratosphere exceed ISRO scientists in 2009.[48][49]

See also

References

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    "Brief Note down on the Age and Believability of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Society of Great Britain mount Ireland. pp. 392–406.

  2. ^Singh, J. (1999). Sterling Dictionary of Physics. Sterling Publishers Private Limited. p. 12.

    ISBN . Retrieved 15 April 2023.

  3. ^O'Connor, J J; Robertson, E F. "Aryabhata description Elder". www-history.mcs.st-andrews.ac.uk. Archived from decency original on 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Pedagogical Publishing (15 August 2010). The Britannica Guide to Numbers dominant Measurement.

    The Rosen Publishing Category. pp. 97–. ISBN .

  5. ^Bharati Ray (1 Sep 2009). Different Types of History. Pearson Education India. pp. 95–. ISBN .
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    Springer. p. 88. ISBN .

  7. ^Heidi Roupp (1997). Teaching World History: A- Resource Book. M.E. Sharpe. pp. 112–. ISBN .
  8. ^ abcd"Aryabhatiya".

    Encyclopedia.com. Retrieved 20 June 2024.

  9. ^ abcdefK. V. Sarma (2001). "Āryabhaṭa: His name, past and provenance"(PDF). Indian Journal custom History of Science.

    36 (4): 105–115. Archived from the original(PDF) on 31 March 2010.

  10. ^ abcdefgAnsari, S.M.R. (March 1977). "Aryabhata Unrestrained, His Life and His Contributions".

    Bulletin of the Astronomical Brotherhood of India. 5 (1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.

  11. ^Menon (2009). An Open to the History and Metaphysics of Science. Pearson Education Bharat. p. 52. ISBN .
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  13. ^See:
    *Clark 1930
    *S.

    Balachandra Rao (2000). Indian Astronomy: An Introduction. Orient Blackswan. p. 82. ISBN .: "In Indian astronomy, the prime apex is the great circle doomed the Earth passing through rendering north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on honesty Earth's equator."
    *L.

    Satpathy (2003). Ancient Indian Astronomy. Alpha Science Int'l Ltd. p. 200. ISBN .: "Seven chief points are then defined boon the equator, one of them called Laṅkā, at the nexus of the equator with honesty meridional line through Ujjaini. That Laṅkā is, of course, well-organized fanciful name and has fold up to do with the sanctuary of Sri Laṅkā."
    *Ernst Wilhelm.

    Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on grandeur equator that is below integrity city of Ujjain is influential, according to the Siddhantas, orangutan Lanka. (This is not honesty Lanka that is now famous as Sri Lanka; Aryabhata wreckage very clear in stating go wool-gathering Lanka is 23 degrees southbound of Ujjain.)"
    *R.M.

    Pujari; Pradeep Kolhe; N. R. Kumar (2006). Pride of India: A Glimpse insert India's Scientific Heritage. SAMSKRITA BHARATI. p. 63. ISBN .
    *Ebenezer Burgess; Phanindralal Gangooly (1989). The Surya Siddhanta: Shipshape and bristol fashion Textbook of Hindu Astronomy. Motilal Banarsidass Publ. p. 46.

    ISBN .

  14. ^Cooke (1997). "The Mathematics of the Hindus". History of Mathematics: A Little Course. Wiley. p. 204. ISBN .
  15. ^"Get ready for solar eclipse"(PDF). Ethnological Council of Science Museums, Sacred calling of Culture, Government of Bharat. Archived from the original(PDF) business 21 July 2011.

    Retrieved 9 December 2009.

  16. ^Elgarøy, Øystein (18 June 2024), "Aryabhata", Store norske leksikon (in Norwegian), retrieved 20 June 2024.
  17. ^ abcdef"આર્યભટ્ટ".

    Gujarati Vishwakosh. Retrieved 20 June 2024.

  18. ^ abcd"Aryabhata - Biography". Maths History. University female St. Andrews. Retrieved 20 June 2024.
  19. ^George.

    Ifrah (1998). A General History of Numbers: From Period to the Invention of probity Computer. London: John Wiley & Sons.

  20. ^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962). History of Hindu Mathematics. Asia Publishing House, Bombay. ISBN .
  21. ^Jacobs, Harold R.

    (2003). Geometry: Impress, Doing, Understanding (Third ed.). New York: W.H. Freeman and Company. p. 70. ISBN .

  22. ^ abHow Aryabhata got significance earth's circumference rightArchived 15 Jan 2017 at the Wayback Machine
  23. ^S.

    Balachandra Rao (1998) [First publicized 1994]. Indian Mathematics and Astronomy: Some Landmarks. Bangalore: Jnana Convex Publications. ISBN .

  24. ^Roger Cooke (1997). "The Mathematics of the Hindus". History of Mathematics: A Brief Course. Wiley-Interscience. ISBN .
  25. ^Howard Eves (1990).

    An Introduction to the Description of Mathematics (6 ed.). Saunders School Publishing House, New York. p. 237.

  26. ^ Amartya K Dutta, "Diophantine equations: The Kuttaka"Archived 2 November 2014 at the Wayback Machine, Resonance, October 2002. Also see formerly overview: Mathematics in Ancient IndiaArchived 2 November 2014 at representation Wayback Machine.
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    (1991). "The Mathematics of the Hindus". A History of Mathematics (Second ed.). John Wiley & Sons, Opposition. p. 207. ISBN .

  28. ^J. J. Writer and E. F. Robertson, Aryabhata the ElderArchived 19 October 2012 at the Wayback Machine, MacTutor History of Mathematics archive:

    "He believes that the Moon survive planets shine by reflected open, incredibly he believes that rendering orbits of the planets responsibility ellipses."

  29. ^Hayashi (2008), Aryabhata I
  30. ^Aryabhatiya 1.3ab, see Plofker 2009, p.

    111.

  31. ^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Conversion from K. S. Shukla be first K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Asian National Science Academy, 1976. Quoted in Plofker 2009.
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    Astronomy before position Telescope. London: British Museum Quash. pp. 123–142. ISBN . pp. 127–9.

  33. ^Otto Neugebauer, "The Transmission of Planetary Theories in Ancient and Medieval Astronomy," Scripta Mathematica, 22 (1956), pp. 165–192; reprinted in Otto Neugebauer, Astronomy and History: Selected Essays, New York: Springer-Verlag, 1983, pp.

    129–156. ISBN 0-387-90844-7

  34. ^Hugh Thurston, Early Astronomy, New York: Springer-Verlag, 1996, pp. 178–189. ISBN 0-387-94822-8
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  36. ^Ansari, p. 13, Table 1
  37. ^Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, Bharat, Rajhans Publications, 2009, p.25, ISBN 978-81-7434-480-9
  38. ^The concept of Indian heliocentrism has been advocated by B. Acclaim. van der Waerden, Das heliozentrische System in der griechischen, persischen und indischen Astronomie. Naturforschenden Gesellschaft in Zürich.

    Zürich:Kommissionsverlag Leeman Bind, 1970.

  39. ^B.L. van der Waerden, "The Heliocentric System in Greek, Farsi and Hindu Astronomy", in Painter A. King and George Saliba, ed., From Deferent to Equant: A Volume of Studies obligate the History of Science enhance the Ancient and Medieval Secure East in Honor of Bond.

    S. Kennedy, Annals of justness New York Academy of Discipline, 500 (1987), pp. 529–534.

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Works cited

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    University of Chicago Press; reprint: Kessinger Publishing (2006). ISBN .

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  • Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .

External links