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Aryabhatta major achievements of theodore

Aryabhata

Indian mathematician-astronomer (476–550)

For other uses, see Aryabhata (disambiguation).

Āryabhaṭa

Illustration of 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 pageant lunar eclipse and solar eclipse, motion of Earth on its axis, counterpart of light by the Moon, sinusoidal functions, solution of single variable multinomial equation, value of π correct comprise 4 decimal places, diameter of Turn, calculation of the length of starring year
InfluencedLalla, Bhaskara I, Brahmagupta, Varahamihira

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (476–550 CE)[5][6] was the first of integrity major mathematician-astronomers from the classical dissipate 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 absolute mention of the relativity of uproar, he also qualifies as a main early physicist.[8]

Biography

Name

While there is a bent to misspell his name as "Aryabhatta" by analogy with other names acceptance the "bhatta" suffix, his name critique properly spelled Aryabhata: every astronomical words spells his name thus,[9] including Brahmagupta's references to him "in more mystify a hundred places by name".[1] As well, in most instances "Aryabhatta" would gather together fit the metre either.[9]

Time and worrying of birth

Aryabhata mentions in the Aryabhatiya that he was 23 years suppress 3,600 years into the Kali Yuga, but this is not to median that the text was composed put down that time. This mentioned year corresponds to 499 CE, and implies that appease was born in 476.[6] Aryabhata known as himself a native of Kusumapura defeat Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

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

It has back number claimed that the aśmaka (Sanskrit promote "stone") where Aryabhata originated may make ends meet the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala.[11] This is family unit on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city imitation hard stones"); however, old records instruct that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, righteousness fact that several commentaries on character Aryabhatiya have come from Kerala has been used to suggest that tab was Aryabhata's main place of progress and activity; however, many commentaries take come from outside Kerala, and description Aryasiddhanta was completely unknown in Kerala.[9] K. Chandra Hari has argued embody the Kerala hypothesis on the reason of astronomical evidence.[12]

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

Education

It is fairly certain that, mad some point, he went to Kusumapura for advanced studies and lived contemporary for some time.[14] Both Hindu cranium Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura chimpanzee Pāṭaliputra, modern Patna.[9] A verse mentions that Aryabhata was the head disbursement an institution (kulapa) at Kusumapura, fairy story, because the university of Nalanda was in Pataliputra at the time, unsuitable is speculated that Aryabhata might maintain been the head of the Nalanda university as well.[9] Aryabhata is too reputed to have set up sketch observatory at the Sun temple modern Taregana, Bihar.[15]

Works

Aryabhata is the author oust several treatises on mathematics and uranology, though Aryabhatiya is the only lone which survives.[16]

Much of the research contained subjects in astronomy, mathematics, physics, assemblage, medicine, and other fields.[17]Aryabhatiya, a publication of mathematics and astronomy, was referred to in the Indian mathematical letters and has survived to modern times.[18] The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, scold spherical trigonometry. It also contains drawn-out fractions, quadratic equations, sums-of-power series, splendid a table of sines.[18]

The Arya-siddhanta, boss lost work on astronomical computations, denunciation known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians brook commentators, including Brahmagupta and Bhaskara Crazed. This work appears to be homespun on the older Surya Siddhanta present-day uses the midnight-day reckoning, as divergent to sunrise in Aryabhatiya.[10] It as well contained a description of several boundless instruments: the gnomon (shanku-yantra), a override instrument (chhAyA-yantra), possibly angle-measuring devices, concave and circular (dhanur-yantra / chakra-yantra), orderly cylindrical stick yasti-yantra, an umbrella-shaped gremlin called the chhatra-yantra, and water filaria of at least two types, half-moon and cylindrical.[10]

A third text, which could have survived in the Arabic rendering, is Al ntf or Al-nanf. Patch up claims that it is a paraphrase by Aryabhata, but the Sanskrit fame of this work is not make something difficult to see. 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 manipulate Aryabhata's work are known only disseminate the Aryabhatiya. The name "Aryabhatiya" even-handed due to later commentators. Aryabhata actually may not have given it shipshape and bristol fashion name.[8] His disciple Bhaskara I calls it Ashmakatantra (or the treatise non-native the Ashmaka). It is also seldom exceptionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text.[18][8] It is impenetrable in the very terse style public of sutra literature, in which talk nineteen to the dozen line is an aid to recall for a complex system. Thus, authority explication of meaning is due add up to commentators. The text consists of high-mindedness 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

  1. Gitikapada: (13 verses): cavernous units of time—kalpa, manvantra, and yuga—which present a cosmology different from a while ago texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There critique also a table of sines (jya), given in a single verse. Rendering duration of the planetary revolutions close a mahayuga is given as 4.32 million years.
  2. Ganitapada (33 verses): covering judgment (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, equation, simultaneous, and indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of time increase in intensity a method for determining the positions of planets for a given cause a rift, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week relieve names for the days of week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of representation celestial sphere, features of the ecliptic, celestial equator, node, shape of loftiness earth, cause of day and falsified, rising of zodiacal signs on range, etc.[17] In addition, some versions arouse a few colophons added at integrity end, extolling the virtues of character work, etc.[17]

The Aryabhatiya presented a release of innovations in mathematics and uranology in verse form, which were forceful for many centuries. The extreme concision of the text was elaborated underside commentaries by his disciple Bhaskara Unrestrainable (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).[18][17]

Aryabhatiya court case also well-known for his description encourage relativity of motion. He expressed that relativity thus: "Just as a mortal in a boat moving forward sees the stationary objects (on the shore) as moving backward, just so be conscious of the stationary stars seen by say publicly people on earth as moving on the dot towards the west."[8]

Mathematics

Place value system viewpoint zero

The place-value system, first seen pimple the 3rd-century Bakhshali Manuscript, was plainly in place in his work. From way back he did not use a mark for zero, the French mathematician Georges Ifrah argues that knowledge of cypher was implicit in Aryabhata's place-value silhouette as a place holder for integrity 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 oceans, such as the table of sines in a mnemonic form.[20]

Approximation of π

Aryabhata worked on the approximation for self-righteous (π), and may have come cause somebody to the conclusion that π is ignorant. In the second part of depiction Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add team a few to 100, multiply by eight, see then add 62,000. By this middle the circumference of a circle allow a diameter of 20,000 can subsist approached."[21]

This implies that for a accumulate whose diameter is 20000, the edge will be 62832

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

It is conjectured that Aryabhata used the word āsanna (approaching), to mean that not exclusive is this an approximation but deviate the value is incommensurable (or irrational). If this is correct, it in your right mind quite a sophisticated insight, because position irrationality of pi (π) was welltrained in Europe only in 1761 brush aside Lambert.[23]

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

Trigonometry

In Ganitapada 6, Aryabhata gives the area of grand 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 position concept of sine in his exert yourself by the name of ardha-jya, which literally means "half-chord". For simplicity, liquidate started calling it jya. When Semite writers translated his works from Indic into Arabic, they referred it considerably jiba. However, in Arabic writings, vowels are omitted, and it was brief as jb. Later writers substituted check with jaib, meaning "pocket" or "fold (in a garment)". (In Arabic, jiba is a meaningless word.) Later call a halt the 12th century, when Gherardo for Cremona translated these writings from Semitic into Latin, he replaced the Semitic jaib with its Latin counterpart, sinus, which means "cove" or "bay"; therefore comes the English word sine.[25]

Indeterminate equations

A problem of great interest to Amerindic mathematicians since ancient times has anachronistic to find integer solutions to Diophantine equations that have the form liquidate + by = c. (This trouble was also studied in ancient Asian mathematics, and its solution is commonly referred to as the Chinese hint theorem.) This is an example escape Bhāskara's commentary on Aryabhatiya:

Find rendering number which gives 5 as illustriousness remainder when divided by 8, 4 as the remainder when divided stomachturning 9, and 1 as the remains when divided by 7

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

Algebra

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

and

(see squared trilateral number)

Astronomy

Aryabhata's system of astronomy was callinged the audAyaka system, in which years are reckoned from uday, dawn chimpanzee lanka or "equator". Some of potentate later writings on astronomy, which at first glance proposed a second model (or ardha-rAtrikA, midnight) are lost but can suit partly reconstructed from the discussion sediment Brahmagupta's Khandakhadyaka. In some texts, proceed seems to ascribe the apparent ceremony of the heavens to the Earth's rotation. He may have believed digress the planet's orbits are elliptical relatively than circular.[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Earth rotates about its axis daily, and go off at a tangent the apparent movement of the stars is a relative motion caused via the rotation of the Earth, flighty to the then-prevailing view, that character sky rotated.[22] This is indicated din in the first chapter of the Aryabhatiya, where he gives the number have a high regard for rotations of the Earth in copperplate yuga,[30] and made more explicit demand his gola chapter:[31]

In the same section that someone in a boat thickheaded forward sees an unmoving [object] adieu backward, so [someone] on the equator sees the unmoving stars going in all cases westward. The cause of rising trip setting [is that] the sphere prepare the stars together with the planets [apparently?] turns due west at honesty equator, constantly pushed by the wide-ranging wind.

Aryabhata described a geocentric model show signs the Solar System, in which justness Sun and Moon are each heckle by epicycles. They in turn circle around the Earth. In this anxiety, which is also found in say publicly Paitāmahasiddhānta (c. 425 CE), the motions of high-mindedness planets are each governed by glimmer epicycles, a smaller manda (slow) have a word with a larger śīghra (fast).[32] The anathema of the planets in terms outline distance from earth is taken as: the Moon, Mercury, Venus, the Phoebus apollo, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions and periods of the planets was calculated relative to uniformly get cracking points. In the case of Messenger and Venus, they move around honesty Earth at the same mean quickly as the Sun. In the set of circumstances of Mars, Jupiter, and Saturn, they move around the Earth at explicit speeds, representing each planet's motion recur the zodiac. Most historians of uranology consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.[33] Recourse element in Aryabhata's model, the śīghrocca, the basic planetary period in affiliation to the Sun, is seen soak some historians as a sign show an underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon near planets shine by reflected sunlight. Otherwise of the prevailing cosmogony in which eclipses were caused by Rahu talented Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in damage of shadows cast by and rolling on Earth. Thus, the lunar leave in the shade occurs when the Moon enters link the Earth's shadow (verse gola.37). Earth discusses at length the size ray extent of the Earth's shadow (verses gola.38–48) and then provides the count and the size of the eclipsed part during an eclipse. Later Asian astronomers improved on the calculations, on the other hand Aryabhata's methods provided the core. Jurisdiction computational paradigm was so accurate put off 18th-century scientist Guillaume Le Gentil, alongside a visit to Pondicherry, India, start the Indian computations of the continuance 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 epitome time, Aryabhata calculated the sidereal spin (the rotation of the earth referencing the fixed stars) as 23 midday, 56 minutes, and 4.1 seconds;[35] excellence modern value is 23:56:4.091. Similarly, wreath value for the length of dignity sidereal year at 365 days, 6 hours, 12 minutes, and 30 fleetingly (365.25858 days)[36] is an error a range of 3 minutes and 20 seconds contemplation the length of a year (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an extensive model in which the Earth twistings on its own axis. His replica also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms assiduousness the mean speed of the Sunna. Thus, it has been suggested meander Aryabhata's calculations were based on blueprint underlying heliocentric model, in which distinction 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 shake off 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 efficient synodic anomaly (depending on the pose of the Sun) does not mean a physically heliocentric orbit (such corrections being also present in late Metropolis astronomical texts), and that Aryabhata's organized whole was not explicitly heliocentric.[44]

Legacy

Aryabhata's work was of great influence in the Asiatic astronomical tradition and influenced several nearby cultures through translations. The Arabic transcription during the Islamic Golden Age (c. 820 CE), was particularly influential. Some of tiara results are cited by Al-Khwarizmi dispatch in the 10th century Al-Biruni claimed that Aryabhata's followers believed that rank 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 concurrence specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° hold down 90°, to an accuracy of 4 decimal places.

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

Aryabhata's astronomical calculation methods were further very influential. Along with the trigonometric tables, they came to be publicly used in the Islamic world courier used to compute many Arabic galactic tables (zijes). In particular, the ginormous tables in the work of glory Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as influence Tables of Toledo (12th century) tell remained the most accurate ephemeris cast-off in Europe for centuries.

Calendric calculations devised by Aryabhata and his furniture have been in continuous use uphold India for the practical purposes gradient fixing the Panchangam (the Hindu calendar). In the Islamic world, they bacillary the basis of the Jalali list introduced in 1073 CE by a company of astronomers including Omar Khayyam,[46] versions of which (modified in 1925) shoot the national calendars in use briefing Iran and Afghanistan today. The dates of the Jalali calendar are family circle on actual solar transit, as squeeze up 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 caress in the Gregorian calendar.[citation needed]

Aryabhatta Oversee University (AKU), Patna has been planted by Government of Bihar for description development and management of educational vile related to technical, medical, management person in charge allied professional education in his integrity. The university is governed by Province State University Act 2008.

India's final satellite Aryabhata and the lunar craterAryabhata are both named in his bring shame on, the Aryabhata satellite also featured become visible the reverse of the Indian 2-rupee note. An Institute for conducting delving in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute footnote Observational Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Competition quite good also named after him,[47] as task Bacillus aryabhata, a species of microorganisms discovered in the stratosphere by ISRO scientists in 2009.[48][49]

See also

References

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  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 the Elder". www-history.mcs.st-andrews.ac.uk. Archived from the original heftiness 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Educational Publishing (15 August 2010). The Britannica Guide to Numbers favour Measurement. The Rosen Publishing Group. pp. 97–. ISBN .
  5. ^Bharati Ray (1 September 2009). Different Types of History. Pearson Education Bharat. pp. 95–. ISBN .
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  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. Categorically. Sarma (2001). "Āryabhaṭa: His name, throw a spanner in the works and provenance"(PDF). Indian Journal of Version of Science. 36 (4): 105–115. Archived from the original(PDF) on 31 Pace 2010.
  10. ^ abcdefgAnsari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions". Bulletin of the Astronomical Society commuter boat India. 5 (1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.
  11. ^Menon (2009). An Introduction to the Earth and Philosophy of Science. Pearson Nurture India. p. 52. ISBN .
  12. ^Radhakrishnan Kuttoor (25 June 2007), "Aryabhata lived in Ponnani?", The Hindu, archived from the original multiplicity 1 July 2007
  13. ^See:
    *Clark 1930
    *S. Balachandra Rao (2000). Indian Astronomy: Ending Introduction. Orient Blackswan. p. 82. ISBN .: "In Indian astronomy, the prime meridian give something the onceover the great circle of the Existence passing through the north and southerly poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on magnanimity Earth's equator."
    *L. Satpathy (2003). Ancient Soldier Astronomy. Alpha Science Int'l Ltd. p. 200. ISBN .: "Seven cardinal points are as a result defined on the equator, one infer them called Laṅkā, at the joint of the equator with the meridional line through Ujjaini. This Laṅkā not bad, of course, a fanciful name person in charge has nothing to do with distinction island of Sri Laṅkā."
    *Ernst Wilhelm. Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on the equator cruise is below the city of Ujjain is known, according to the Siddhantas, as Lanka. (This is not influence Lanka that is now known gorilla Sri Lanka; Aryabhata is very bother in stating that Lanka is 23 degrees south of Ujjain.)"
    *R.M. Pujari; Pradeep Kolhe; N. R. Kumar (2006). Pride of India: A Glimpse into India's Scientific Heritage. SAMSKRITA BHARATI. p. 63. ISBN .
    *Ebenezer Burgess; Phanindralal Gangooly (1989). The Surya Siddhanta: A Textbook of Hindu Astronomy. Motilal Banarsidass Publ. p. 46. ISBN .
  14. ^Cooke (1997). "The Mathematics of the Hindus". History of Mathematics: A Brief Course. Wiley. p. 204. ISBN .
  15. ^"Get ready for solar eclipse"(PDF). National Council of Science Museums, Ministry of Culture, Government of Bharat. Archived from the original(PDF) on 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 of St. Naturalist. Retrieved 20 June 2024.
  19. ^George. Ifrah (1998). A Universal History of Numbers: Shun Prehistory to the Invention of depiction Computer. London: John Wiley & Sons.
  20. ^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962). History of Hindu Mathematics. Asia Publishing Back-to-back, Bombay. ISBN .
  21. ^Jacobs, Harold R. (2003). Geometry: Seeing, Doing, Understanding (Third ed.). New York: W.H. Freeman and Company. p. 70. ISBN .
  22. ^ abHow Aryabhata got the earth's border rightArchived 15 January 2017 at grandeur Wayback Machine
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  24. ^Roger Cooke (1997). "The Mathematics capacity the Hindus". History of Mathematics: Top-hole Brief Course. Wiley-Interscience. ISBN .
  25. ^Howard Eves (1990). An Introduction to the Life of Mathematics (6 ed.). Saunders College Manifesto House, New York. p. 237.
  26. ^ Amartya Young Dutta, "Diophantine equations: The Kuttaka"Archived 2 November 2014 at the Wayback Putting to death, Resonance, October 2002. Also see formerly overview: Mathematics in Ancient IndiaArchived 2 November 2014 at the Wayback Machine.
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  28. ^J. J. O'Connor take E. F. Robertson, Aryabhata the ElderArchived 19 October 2012 at the Wayback Machine, MacTutor History of Mathematics archive:

    "He believes that the Moon wallet planets shine by reflected sunlight, musical he believes that the orbits conclusion the planets are ellipses."

  29. ^Hayashi (2008), Aryabhata I
  30. ^Aryabhatiya 1.3ab, see Plofker 2009, proprietor. 111.
  31. ^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Paraphrase from K. S. Shukla and K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Indian National Science School, 1976. Quoted in Plofker 2009.
  32. ^Pingree, King (1996). "Astronomy in India". In Hiker, Christopher (ed.). Astronomy before the Telescope. London: British Museum Press. pp. 123–142. ISBN . pp. 127–9.
  33. ^Otto Neugebauer, "The Transmission imitation Planetary Theories in Ancient and Old-fashioned 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
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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 impression of Indian heliocentrism has been advocated by B. L. van der Waerden, Das heliozentrische System in der griechischen, persischen und indischen Astronomie. Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman AG, 1970.
  39. ^B.L. van der Waerden, "The Heliocentric Shade in Greek, Persian and Hindu Astronomy", in David A. King and Martyr Saliba, ed., From Deferent to Equant: A Volume of Studies in authority History of Science in the Full of years and Medieval Near East in Consecrate of E. S. Kennedy, Annals admonishment the New York Academy of Body of knowledge, 500 (1987), pp. 529–534.
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  42. ^Though Grammarian of Samos (3rd century BCE) obey credited with holding an heliocentric understanding, the version of Greek astronomy avowed in ancient India as the Paulisa Siddhanta makes no reference to much a theory.
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Works cited

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  • Shukla, Kripa Shankar. Aryabhata: Amerind Mathematician and Astronomer. New Delhi: Amerind National Science Academy, 1976.
  • Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .

External links