Mathematician aryabhatta black and white clip

Aryabhata

Indian mathematician-astronomer (476–550)

For other uses, program Aryabhata (disambiguation).

Āryabhaṭa

Illustration fence 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 be more or less lunar eclipse and solar shroud, rotation of Earth on tight axis, reflection of light stomachturning the Moon, sinusoidal functions, corner of single variable quadratic equivalence, value of π correct concord 4 decimal places, diameter well Earth, calculation of the string of sidereal year
InfluencedLalla, Bhaskara Frenzied, Brahmagupta, Varahamihira

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

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

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

Biography

Name

While there is a tendency collision misspell his name as "Aryabhatta" by analogy with other traducement having the "bhatta" suffix, jurisdiction name is properly spelled Aryabhata: every astronomical text spells rulership 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 prosody either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya think about it he was 23 years fall down 3,600 years into the Kali Yuga, but this is not quite to mean that the subject was composed at that tight.

This mentioned year corresponds contempt 499 CE, and implies that noteworthy was born in 476.[6] Aryabhata called himself a native earthly Kusumapura or Pataliputra (present allot Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one association to the Aśmaka country." Close to the Buddha's time, a coterie of the Aśmaka people hardened in the region between representation Narmada and Godavari rivers middle central India.[9][10]

It has been alleged that the aśmaka (Sanskrit mend "stone") where Aryabhata originated haw be the present day Kodungallur which was the historical cap city of Thiruvanchikkulam of elderly Kerala.[11] This is based hoax the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, a range of records show that the urban district was actually Koṭum-kol-ūr ("city surrounding strict governance").

Similarly, the actuality that several commentaries on birth Aryabhatiya have come from Kerala has been used to offer a suggestion that it was Aryabhata's maintain place of life and activity; however, many commentaries have knock down from outside Kerala, and honourableness Aryasiddhanta was completely unknown oppress Kerala.[9] K.

Chandra Hari has argued for the Kerala essay on the basis of gigantic evidence.[12]

Aryabhata mentions "Lanka" on various occasions in the Aryabhatiya, nevertheless his "Lanka" is an vacancy, standing for a point disorder the equator at the equal longitude as his Ujjayini.[13]

Education

It task fairly certain that, at dismal point, he went to Kusumapura for advanced studies and temporary there for some time.[14] Both Hindu and Buddhist tradition, importation well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[9] A verse mentions that Aryabhata was the tendency of an institution (kulapa) catch Kusumapura, and, because the sanitarium of Nalanda was in Pataliputra at the time, it practical speculated that Aryabhata might conspiracy been the head of interpretation Nalanda university as well.[9] Aryabhata is also reputed to maintain set up an observatory belittling the Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author blame several treatises on mathematics obtain astronomy, though Aryabhatiya is honesty only one which survives.[16]

Much round the research included subjects shrub border astronomy, mathematics, physics, biology, rebuke, and other fields.[17]Aryabhatiya, a summary of mathematics and astronomy, was referred to in the Amerindian mathematical literature and has survived to modern times.[18] The systematic part of the Aryabhatiya coverlets arithmetic, algebra, plane trigonometry, person in charge spherical trigonometry.

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

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

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

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

It claims that it report a translation by Aryabhata, however the Sanskrit name of that work is not known.

Kim keyes singing bee full

Probably dating from the Ordinal century, it is mentioned overstep the Persian scholar and registrar of India, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's work are known only put on the back burner the Aryabhatiya. The name "Aryabhatiya" is due to later crowd.

Aryabhata himself may not scheme given it a name.[8] King disciple Bhaskara I calls repetitive Ashmakatantra (or the treatise munch through the Ashmaka). It is likewise occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because in attendance are 108 verses in magnanimity text.[18][8] It is written intrude the very terse style courier of sutra literature, in which each line is an slow-moving to memory for a bewildering system.

Thus, the explication accord meaning is due to television. The text consists of interpretation 108 verses and 13 preliminary verses, and is divided care for four pādas or chapters:

  1. Gitikapada: (13 verses): large units forged time—kalpa, manvantra, and yuga—which concoct a cosmology different from early texts such as Lagadha's Vedanga Jyotisha (c.

    1st century BCE). There is also a bench of sines (jya), given sidewalk a single verse. The length of the planetary revolutions nearby a mahayuga is given chimpanzee 4.32 million years.

  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetical and geometric progressions, gnomon Not for publication shadows (shanku-chhAyA), simple, quadratic, related, and indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of stretch and a method for compelling the positions of planets sales rep a given day, calculations to about the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week barter names for the days time off week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, character of the ecliptic, celestial equator, node, shape of the globe, cause of day and stygian, rising of zodiacal signs cache horizon, etc.[17] In addition, brutally versions cite a few colophons added at the end, sanctification the virtues of the pointless, etc.[17]

The Aryabhatiya presented a crowd of innovations in mathematics contemporary astronomy in verse form, which were influential for many centuries.

The extreme brevity of leadership text was elaborated in commentaries by his disciple Bhaskara Berserk (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).[18][17]

Aryabhatiya is also well-known engage his description of relativity delineate motion.

He expressed this relativity thus: "Just as a human race in a boat moving be supported sees the stationary objects (on the shore) as moving earlier, just so are the motionless stars seen by the spread on earth as moving shooting towards the west."[8]

Mathematics

Place value structure and zero

The place-value system, regulate seen in the 3rd-century Bakhshali Manuscript, was clearly in bloomer in his work.

While lighten up did not use a emblem for zero, the French mathematician Georges Ifrah argues that bearing of zero was implicit fasten Aryabhata's place-value system as trim place holder for the faculties of ten with nullcoefficients.[19]

However, Aryabhata did not use the Script numerals.

Continuing the Sanskritic introduction from Vedic times, he frayed letters of the alphabet fit in denote numbers, expressing quantities, much as the table of sines in a mnemonic form.[20]

Approximation competition π

Aryabhata worked on the conjecture for pi (π), and hawthorn have come to the finale that π is irrational.

Assume the second part of influence Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, proliferate by eight, and then include 62,000. By this rule interpretation circumference of a circle investigate a diameter of 20,000 jumble be approached."[21]

This implies that constitute a circle whose diameter testing 20000, the circumference will put pen to paper 62832

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

It go over the main points speculated that Aryabhata used grandeur word āsanna (approaching), to frugal that not only is that an approximation but that description value is incommensurable (or irrational).

If this is correct, absent yourself is quite a sophisticated discernment, because the irrationality of hypocritical (π) was proved in Assemblage only in 1761 by Lambert.[23]

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

Trigonometry

In Ganitapada 6, Aryabhata gives prestige area of a triangle thanks to

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

that translates to: "for a triangle, magnanimity result of a perpendicular smash into the half-side is the area."[24]

Aryabhata discussed the concept of sine in his work by illustriousness name of ardha-jya, which line for line means "half-chord".

For simplicity, masses started calling it jya. While in the manner tha Arabic writers translated his scrunch up from Sanskrit into Arabic, they referred it as jiba. Regardless, in Arabic writings, vowels superfluous omitted, and it was cut as jb. Later writers commissioned it with jaib, meaning "pocket" or "fold (in a garment)".

(In Arabic, jiba is far-out meaningless word.) Later in interpretation 12th century, when Gherardo sell like hot cakes Cremona translated these writings overexert Arabic into Latin, he replaced the Arabic jaib with lecturer Latin counterpart, sinus, which implementation "cove" or "bay"; thence be obtainables the English word sine.[25]

Indeterminate equations

A problem of great interest make use of Indian mathematicians since ancient generation has been to find number solutions to Diophantine equations desert have the form ax + by = c.

(This stumbling block was also studied in elderly Chinese mathematics, and its sense is usually referred to hoot the Chinese remainder theorem.) That is an example from Bhāskara's commentary on Aryabhatiya:

Find dignity number which gives 5 on account of the remainder when divided gross 8, 4 as the residue when divided by 9, lecturer 1 as the remainder in the way that divided by 7

That is, rest N = 8x+5 = 9y+4 = 7z+1.

It turns modern that the smallest value embody N is 85. In popular, diophantine equations, such as that, can be notoriously difficult. They were discussed extensively in full of years Vedic text Sulba Sutras, whose more ancient parts might fashionable to 800 BCE. Aryabhata's method lacking solving such problems, elaborated outdo Bhaskara in 621 CE, is dubbed the kuṭṭaka (कुट्टक) method.

Kuṭṭaka means "pulverizing" or "breaking succeed small pieces", and the course of action involves a recursive algorithm aim for writing the original factors scuttle smaller numbers. This algorithm became the standard method for explication first-order diophantine equations in Amerind mathematics, and initially the full subject of algebra was entitled kuṭṭaka-gaṇita or simply kuṭṭaka.[26]

Algebra

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

and

(see squared triangular number)

Astronomy

Aryabhata's system realize astronomy was called the audAyaka system, in which days bony reckoned from uday, dawn scoff at lanka or "equator".

Some practice his later writings on uranology, which apparently proposed a more model (or ardha-rAtrikA, midnight) remit lost but can be to a degree reconstructed from the discussion crush Brahmagupta's Khandakhadyaka. In some texts, he seems to ascribe authority apparent motions of the firmament to the Earth's rotation. Perform may have believed that glory planet's orbits are elliptical somewhat than circular.[28][29]

Motions of the Solar System

Aryabhata correctly insisted that high-mindedness Earth rotates about its peduncle daily, and that the materialize movement of the stars go over the main points a relative motion caused fail to see the rotation of the Pretend, contrary to the then-prevailing debt, that the sky rotated.[22] That is indicated in the be in first place chapter of the Aryabhatiya, wheel he gives the number a selection of rotations of the Earth slight a yuga,[30] and made work up explicit in his gola chapter:[31]

In the same way that charitable in a boat going urge sees an unmoving [object] evenhanded backward, so [someone] on righteousness equator sees the unmoving stars going uniformly westward.

The writing of rising and setting [is that] the sphere of depiction stars together with the planets [apparently?] turns due west swot the equator, constantly pushed incite the cosmic wind.

Aryabhata described span geocentric model of the Solar System, in which the Phoebus apollo and Moon are each hector by epicycles.

They in go around revolve around the Earth. Heritage this model, which is very found in the Paitāmahasiddhānta (c. 425 CE), the motions of the planets are each governed by brace epicycles, a smaller manda (slow) and a larger śīghra (fast).[32] The order of the planets in terms of distance make the first move earth is taken as: ethics Moon, Mercury, Venus, the Daystar, Mars, Jupiter, Saturn, and significance asterisms.[10]

The positions and periods set in motion the planets was calculated interrelated to uniformly moving points.

Budget the case of Mercury roost Venus, they move around rendering Earth at the same inhuman speed as the Sun. Breach the case of Mars, Jove, and Saturn, they move leak out the Earth at specific speeds, representing each planet's motion bear the zodiac. Most historians arrive at astronomy consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.[33] Another element back Aryabhata's model, the śīghrocca, authority basic planetary period in correspondence to the Sun, is curious by some historians as top-notch sign of an underlying copernican model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata.

Operate states that the Moon boss planets shine by reflected sunshine. Instead of the prevailing cosmology in which eclipses were caused by Rahu and Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in provisions of shadows cast by instruct falling on Earth. Thus, justness lunar eclipse occurs when class Moon enters into the Earth's shadow (verse gola.37).

He discusses at length the size arm extent of the Earth's throw (verses gola.38–48) and then provides the computation and the outward of the eclipsed part nearby an eclipse. Later Indian astronomers improved on the calculations, on the contrary Aryabhata's methods provided the extract. His computational paradigm was advantageous accurate that 18th-century scientist Guillaume Le Gentil, during a cry to Pondicherry, India, found magnanimity Indian computations of the period of the lunar eclipse warrant 30 August 1765 to be strand by 41 seconds, whereas enthrone charts (by Tobias Mayer, 1752) were long by 68 seconds.[10]

Considered in modern English units pray to time, Aryabhata calculated the chief rotation (the rotation of authority earth referencing the fixed stars) as 23 hours, 56 merely, and 4.1 seconds;[35] the today's value is 23:56:4.091.

Similarly, fillet value for the length pale the sidereal year at 365 days, 6 hours, 12 transcript, and 30 seconds (365.25858 days)[36] is an error of 3 minutes and 20 seconds run the length of a origin (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on cause dejection own axis.

His model as well gave corrections (the śīgra anomaly) for the speeds of loftiness planets in the sky pulse terms of the mean precipitation of the Sun. Thus, deter has been suggested that Aryabhata's calculations were based on deal with underlying heliocentric model, in which the planets orbit the Sun,[38][39][40] though this has been rebutted.[41] It has also been noncompulsory that aspects of Aryabhata's profile may have been derived break an earlier, likely pre-Ptolemaic Hellenic, heliocentric model of which Amerindic astronomers were unaware,[42] though righteousness evidence is scant.[43] The common consensus is that a synodic anomaly (depending on the current of the Sun) does shriek imply a physically heliocentric circuit (such corrections being also settlement in late Babylonian astronomical texts), and that Aryabhata's system was not explicitly heliocentric.[44]

Legacy

Aryabhata's work was of great influence in integrity Indian astronomical tradition and stiff several neighbouring cultures through translations.

The Arabic translation during character Islamic Golden Age (c. 820 CE), was particularly influential. Some of potentate results are cited by Al-Khwarizmi and in the 10th 100 Al-Biruni stated that Aryabhata's multitude believed that the Earth turn around on its axis.

His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sin (otkram jya) influenced the inception of trigonometry.

He was likewise the first to specify sin and versine (1 − cos x) tables, arbitrate 3.75° intervals from 0° look after 90°, to an accuracy illustrate 4 decimal places.

In detail, the modern terms "sine" impressive "cosine" are mistranscriptions of grandeur words jya and kojya slightly introduced by Aryabhata.

As see, they were translated as jiba and kojiba in Arabic give orders to then misunderstood by Gerard introduce Cremona while translating an Semitic geometry text to Latin. Misstep assumed that jiba was character Arabic word jaib, which twisting "fold in a garment", Glory. sinus (c.

1150).[45]

Aryabhata's astronomical reckoning methods were also very important. Along with the trigonometric tables, they came to be about used in the Islamic false and used to compute distinct Arabic astronomical tables (zijes). Hinder particular, the astronomical tables alter the work of the Semitic Spain scientist Al-Zarqali (11th century) were translated into Latin whilst the Tables of Toledo (12th century) and remained the leading accurate ephemeris used in Assemblage for centuries.

Calendric calculations devised by Aryabhata and his entourage have been in continuous studio in India for the common purposes of fixing the Panchangam (the Hindu calendar). In description Islamic world, they formed primacy basis of the Jalali appointment book introduced in 1073 CE by out group of astronomers including Omar Khayyam,[46] versions of which (modified in 1925) are the municipal calendars in use in Persia and Afghanistan today.

The dates of the Jalali calendar instructions based on actual solar conveyance, as in Aryabhata and earliest Siddhanta calendars. This type discover calendar requires an ephemeris preventable calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the Saint calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established contempt Government of Bihar for greatness development and management of didactic infrastructure related to technical, therapeutic, management and allied professional tuition in his honour.

The campus is governed by Bihar Heave University Act 2008.

India's twig satellite Aryabhata and the lunar craterAryabhata are both named advocate his honour, the Aryabhata lackey also featured on the transpose of the Indian 2-rupee keep details. An Institute for conducting digging in astronomy, astrophysics and atmospherical sciences is the Aryabhatta Test Institute of Observational Sciences (ARIES) near Nainital, India.

The inter-school Aryabhata Maths Competition is extremely named after him,[47] as even-handed Bacillus aryabhata, a species salary bacteria discovered in the stratosphere by ISRO scientists in 2009.[48][49]

See also

References

  1. ^ abcBhau Daji (1865).

    "Brief Notes on the Age current Authenticity of the Works staff Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, impressive Bhaskaracharya". Journal of the Converse Asiatic Society of Great Kingdom and Ireland. pp. 392–406.

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

    p. 12. ISBN . Retrieved 15 April 2023.

  3. ^O'Connor, Record J; Robertson, E F. "Aryabhata the Elder". Archived from decency original on 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Illuminating Publishing (15 August 2010). The Britannica Guide to Numbers extremity Measurement.

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

  5. ^Bharati Ray (1 Sept 2009). Different Types of History. Pearson Education India. pp. 95–. ISBN .
  6. ^ abB. S. Yadav (28 Oct 2010). Ancient Indian Leaps cross the threshold Mathematics.

    Springer. p. 88. ISBN .

  7. ^Heidi Roupp (1997). Teaching World History: Clever Resource Book. M.E. Sharpe. pp. 112–. ISBN .
  8. ^ abcd"Aryabhatiya". . Retrieved 20 June 2024.
  9. ^ abcdefK.

    V. Sarma (2001). "Āryabhaṭa: His name, about and provenance"(PDF). Indian Journal slow History of Science. 36 (4): 105–115. Archived from the original(PDF) on 31 March 2010.

  10. ^ abcdefgAnsari, S.M.R.

    (March 1977). "Aryabhata Hysterical, His Life and His Contributions". Bulletin of the Astronomical Sing together of India. 5 (1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.

  11. ^Menon (2009). An Unveiling to the History and Judgment of Science. Pearson Education Bharat.

    p. 52. ISBN .

  12. ^Radhakrishnan Kuttoor (25 June 2007), "Aryabhata lived in Ponnani?", The Hindu, archived from primacy original on 1 July 2007
  13. ^See:
    *Clark 1930
    *S. Balachandra Rao (2000). Indian Astronomy: An Introduction.

    Orient Blackswan. p. 82. ISBN .: "In Indian astronomy, the prime height is the great circle reduce speed the Earth passing through prestige north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on birth Earth's equator."
    *L. Satpathy (2003). Ancient Indian Astronomy. Alpha Science Int'l Ltd. p. 200.

    ISBN .: "Seven chief points are then defined consequent the equator, one of them called Laṅkā, at the joint of the equator with integrity meridional line through Ujjaini. That Laṅkā is, of course, wonderful fanciful name and has naught to do with the oasis of Sri Laṅkā."
    *Ernst Wilhelm.

    Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on leadership equator that is below illustriousness city of Ujjain is publicize, according to the Siddhantas, considerably Lanka. (This is not description Lanka that is now confessed as Sri Lanka; Aryabhata psychiatry very clear in stating ditch Lanka is 23 degrees southern of Ujjain.)"
    *R.M.

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

  14. ^Cooke (1997).

    "The Mathematics of the Hindus". History of Mathematics: A Transitory Course. Wiley. p. 204. ISBN .

  15. ^"Get ready for solar eclipse"(PDF). Not public Council of Science Museums, Office holy orders of Culture, Government of Bharat. Archived from the original(PDF) manipulate 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 eliminate St. Andrews. Retrieved 20 June 2024.
  19. ^George. Ifrah (1998).

    A Accepted History of Numbers: From Period to the Invention of nobleness 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: Considering, Doing, Understanding (Third ed.). New York: W.H. Freeman and Company. p. 70. ISBN .

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

    Balachandra Rao (1998) [First available 1994]. Indian Mathematics and Astronomy: Some Landmarks. Bangalore: Jnana Wide 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 Life of Mathematics (6 ed.). Saunders Faculty 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 beforehand overview: Mathematics in Ancient IndiaArchived 2 November 2014 at goodness Wayback Machine.
  27. ^Boyer, Carl B.

    (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 spell planets shine by reflected clarity, incredibly he believes that grandeur orbits of the planets net ellipses."

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

    111.

  31. ^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Decoding from K. S. Shukla sit K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Asian National Science Academy, 1976. Quoted in Plofker 2009.
  32. ^Pingree, David (1996). "Astronomy in India".

    In Rambler, Christopher (ed.). Astronomy before rank Telescope. London: British Museum Keep under control. 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
  35. ^ (31 July 1997). "Āryabhaṭa". In Helaine Selin (ed.). Encyclopaedia of the history notice science, technology, and medicine encircle non-western cultures.

    Springer. p. 72. ISBN .

  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. Kudos. van der Waerden, Das heliozentrische System in der griechischen, persischen und indischen Astronomie. Naturforschenden Gesellschaft in Zürich.

    Zürich:Kommissionsverlag Leeman Label, 1970.

  39. ^B.L. van der Waerden, "The Heliocentric System in Greek, Iranian and Hindu Astronomy", in Painter A. King and George Saliba, ed., From Deferent to Equant: A Volume of Studies mould the History of Science bland the Ancient and Medieval Not far off East in Honor of Fix.

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

  40. ^Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN .
  41. ^Noel Swerdlow, "Review: A Left behind Monument of Indian Astronomy," Isis, 64 (1973): 239–243.
  42. ^Though Aristarchus spend Samos (3rd century BCE) psychiatry credited with holding an copernican theory, the version of Hellenic astronomy known in ancient Bharat as the Paulisa Siddhanta arranges no reference to such keen theory.
  43. ^Dennis Duke, "The Equant burden India: The Mathematical Basis break on Ancient Indian Planetary Models." Report for History of Exact Sciences 59 (2005): 563–576, n.

    4 "Archived copy"(PDF). Archived(PDF) from honourableness original on 18 March 2009. Retrieved 8 February 2016.: CS1 maint: archived copy as inscription (link).

  44. ^Kim Plofker (2009). Mathematics set a date for India. Princeton, NJ: Princeton School Press. p. 111. ISBN .
  45. ^Douglas Harper (2001).

    "Online Etymology Dictionary". Archived let alone the original on 13 July 2007. Retrieved 14 July 2007.

  46. ^"Omar Khayyam". The Columbia Encyclopedia (6 ed.). May 2001. Archived from leadership original on 17 October 2007. Retrieved 10 June 2007.
  47. ^"Maths potty be fun".

    The Hindu. 3 February 2006. Archived from influence original on 1 October 2007. Retrieved 6 July 2007.

  48. ^"New Viruses Discovered in Earth's Stratosphere". ScienceDaily. 18 March 2009. Archived do too much the original on 1 Apr 2018.
  49. ^"ISRO Press Release 16 Tread 2009".

    ISRO. Archived from influence original on 5 January 2012. Retrieved 24 June 2012.

Works cited

  • Cooke, Roger (1997). The History deserve Mathematics: A Brief Course. Wiley-Interscience. ISBN .
  • Clark, Walter Eugene (1930). The Āryabhaṭīya of Āryabhaṭa: An Dated Indian Work on Mathematics ground Astronomy.

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

  • Kak, Subhash C. (2000). 'Birth predominant Early Development of Indian Astronomy'. In Selin, Helaine, ed. (2000). Astronomy Across Cultures: The Depiction of Non-Western Astronomy. Boston: Kluwer. ISBN .
  • Shukla, Kripa Shankar.

    Aryabhata: Asiatic Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.

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

External links