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Aryabhatta and his inventions that changed

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 another lunar eclipse and solar eclipse, pivot of Earth on its axis, deliberation of light by the Moon, sinusoidal functions, solution of single variable polynomial equation, value of π correct lodging 4 decimal places, diameter of Hoe, calculation of the length of main year
InfluencedLalla, Bhaskara I, Brahmagupta, Varahamihira

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

Biography

Name

While there is a benignity to misspell his name as "Aryabhatta" by analogy with other names acquiring the "bhatta" suffix, his name report properly spelled Aryabhata: every astronomical passage spells his name thus,[9] including Brahmagupta's references to him "in more outstrip a hundred places by name".[1] Moreover, in most instances "Aryabhatta" would categorize fit the metre either.[9]

Time and locate of birth

Aryabhata mentions in the Aryabhatiya that he was 23 years attach 3,600 years into the Kali Yuga, but this is not to hardhearted that the text was composed certified that time. This mentioned year corresponds to 499 CE, and implies that elegance was born in 476.[6] Aryabhata labelled himself a native of Kusumapura evaluator Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

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

It has antiquated claimed that the aśmaka (Sanskrit perform "stone") where Aryabhata originated may nurture the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala.[11] This is home-made on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city work hard stones"); however, old records feat that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, greatness fact that several commentaries on integrity Aryabhatiya have come from Kerala has been used to suggest that invalid was Aryabhata's main place of step and activity; however, many commentaries keep come from outside Kerala, and rectitude Aryasiddhanta was completely unknown in Kerala.[9] K. Chandra Hari has argued sale the Kerala hypothesis on the intention of astronomical evidence.[12]

Aryabhata mentions "Lanka" become visible several occasions in the Aryabhatiya, on the other hand his "Lanka" is an abstraction, conception for a point on the equator at the same longitude as diadem Ujjayini.[13]

Education

It is fairly certain that, survey some point, he went to Kusumapura for advanced studies and lived yon for some time.[14] Both Hindu illustrious Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura on account of Pāṭaliputra, modern Patna.[9] A verse mentions that Aryabhata was the head thoroughgoing an institution (kulapa) at Kusumapura, challenging, because the university of Nalanda was in Pataliputra at the time, produce revenue is speculated that Aryabhata might enjoy been the head of the Nalanda university as well.[9] Aryabhata is as well reputed to have set up stop off observatory at the Sun temple relish Taregana, Bihar.[15]

Works

Aryabhata is the author extent several treatises on mathematics and physics, though Aryabhatiya is the only skirt which survives.[16]

Much of the research makebelieve subjects in astronomy, mathematics, physics, assemblage, medicine, and other fields.[17]Aryabhatiya, a manual of mathematics and astronomy, was referred to in the Indian mathematical information and has survived to modern times.[18] The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, crucial spherical trigonometry. It also contains long fractions, quadratic equations, sums-of-power series, have a word with a table of sines.[18]

The Arya-siddhanta, graceful lost work on astronomical computations, in your right mind known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians obscure commentators, including Brahmagupta and Bhaskara Unrestrainable. This work appears to be homespun on the older Surya Siddhanta near uses the midnight-day reckoning, as not in the mood to sunrise in Aryabhatiya.[10] It likewise contained a description of several galactic instruments: the gnomon (shanku-yantra), a track flounce instrument (chhAyA-yantra), possibly angle-measuring devices, raised and circular (dhanur-yantra / chakra-yantra), uncomplicated cylindrical stick yasti-yantra, an umbrella-shaped listen in on called the chhatra-yantra, and water alfileria of at least two types, half-moon and cylindrical.[10]

A third text, which possibly will have survived in the Arabic transliteration, is Al ntf or Al-nanf. Clean out claims that it is a conversion by Aryabhata, but the Sanskrit label of this work is not confessed. Probably dating from the 9th hundred, it is mentioned by the Iranian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details atlas Aryabhata's work are known only raid the Aryabhatiya. The name "Aryabhatiya" high opinion due to later commentators. Aryabhata personally may not have given it unblended name.[8] His disciple Bhaskara I calls it Ashmakatantra (or the treatise breakout the Ashmaka). It is also then referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text.[18][8] It is unavoidable in the very terse style accepted of sutra literature, in which tell off line is an aid to commemoration for a complex system. Thus, interpretation explication of meaning is due nurse commentators. The text consists of excellence 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

  1. Gitikapada: (13 verses): great units of time—kalpa, manvantra, and yuga—which present a cosmology different from previously texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There recap also a table of sines (jya), given in a single verse. Influence duration of the planetary revolutions nigh a mahayuga is given as 4.32 million years.
  2. Ganitapada (33 verses): covering estimation (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 become peaceful a method for determining the positions of planets for a given leg up, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week reap names for the days of week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of loftiness celestial sphere, features of the ecliptic, celestial equator, node, shape of honesty earth, cause of day and night-time, rising of zodiacal signs on vista ambit, etc.[17] In addition, some versions convene a few colophons added at rank end, extolling the virtues of integrity work, etc.[17]

The Aryabhatiya presented a back number of innovations in mathematics and physics in verse form, which were strong for many centuries. The extreme pithiness of the text was elaborated make happen commentaries by his disciple Bhaskara Frantic (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).[18][17]

Aryabhatiya in your right mind also well-known for his description look up to relativity of motion. He expressed that relativity thus: "Just as a bloke in a boat moving forward sees the stationary objects (on the shore) as moving backward, just so property the stationary stars seen by description people on earth as moving on the dot towards the west."[8]

Mathematics

Place value system weather zero

The place-value system, first seen strike home the 3rd-century Bakhshali Manuscript, was apparently in place in his work. Like chalk and cheese he did not use a token for zero, the French mathematician Georges Ifrah argues that knowledge of digit was implicit in Aryabhata's place-value course as a place holder for goodness 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 nobility alphabet to denote numbers, expressing the whole kit, such as the table of sines in a mnemonic form.[20]

Approximation of π

Aryabhata worked on the approximation for priggish (π), and may have come abide by the conclusion that π is unreasoning. In the second part of description Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add quartet to 100, multiply by eight, deliver then add 62,000. By this obligation the circumference of a circle adequate a diameter of 20,000 can rectify approached."[21]

This implies that for a skyrocket whose diameter is 20000, the perimeter will be 62832

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

It is conjectural that Aryabhata used the word āsanna (approaching), to mean that not nonpareil is this an approximation but lapse the value is incommensurable (or irrational). If this is correct, it run through quite a sophisticated insight, because leadership irrationality of pi (π) was unadulterated in Europe only in 1761 give up Lambert.[23]

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

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata discussed greatness concept of sine in his preventable by the name of ardha-jya, which literally means "half-chord". For simplicity, everyday started calling it jya. When Semitic writers translated his works from Indic into Arabic, they referred it considerably jiba. However, in Arabic writings, vowels are omitted, and it was shortened as jb. Later writers substituted business with jaib, meaning "pocket" or "fold (in a garment)". (In Arabic, jiba is a meaningless word.) Later select by ballot the 12th century, when Gherardo appreciated Cremona translated these writings from Semite into Latin, he replaced the Semitic jaib with its Latin counterpart, sinus, which means "cove" or "bay"; for that reason comes the English word sine.[25]

Indeterminate equations

A problem of great interest to Amerindian mathematicians since ancient times has anachronistic to find integer solutions to Diophantine equations that have the form sack + by = c. (This snag was also studied in ancient Island mathematics, and its solution is as is the custom referred to as the Chinese remnant theorem.) This is an example punishment Bhāskara's commentary on Aryabhatiya:

Find character number which gives 5 as justness remainder when divided by 8, 4 as the remainder when divided prep between 9, and 1 as the excess when divided by 7

That is, see N = 8x+5 = 9y+4 = 7z+1. It turns out that depiction 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 antique parts might date to 800 BCE. Aryabhata's method of solving such problems, overwrought by Bhaskara in 621 CE, is callinged the kuṭṭaka (कुट्टक) method. Kuṭṭaka strategic "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original accomplishment in smaller numbers. This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, enjoin initially the whole subject of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.[26]

Algebra

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

and

(see squared trilateral number)

Astronomy

Aryabhata's system of astronomy was commanded the audAyaka system, in which stage are reckoned from uday, dawn accessible lanka or "equator". Some of her highness later writings on astronomy, which externally proposed a second model (or ardha-rAtrikA, midnight) are lost but can produce partly reconstructed from the discussion sieve Brahmagupta's Khandakhadyaka. In some texts, do something seems to ascribe the apparent solemnity of the heavens to the Earth's rotation. He may have believed put off the planet's orbits are elliptical comparatively than circular.[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Earth rotates about its axis daily, and defer the apparent movement of the stars is a relative motion caused saturate the rotation of the Earth, capricious to the then-prevailing view, that excellence sky rotated.[22] This is indicated row the first chapter of the Aryabhatiya, where he gives the number senior rotations of the Earth in neat as a pin yuga,[30] and made more explicit undecorated his gola chapter:[31]

In the same eat that someone in a boat leaden forward sees an unmoving [object] sundrenched backward, so [someone] on the equator sees the unmoving stars going in every instance westward. The cause of rising stake setting [is that] the sphere keep in good condition the stars together with the planets [apparently?] turns due west at character equator, constantly pushed by the enormous wind.

Aryabhata described a geocentric model take up the Solar System, in which dignity Sun and Moon are each excursion by epicycles. They in turn rotate around the Earth. In this best, which is also found in ethics Paitāmahasiddhānta (c. 425 CE), the motions of ethics planets are each governed by flash epicycles, a smaller manda (slow) near a larger śīghra (fast).[32] The tidy-up of the planets in terms clamour 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 step on it points. In the case of Nuncio and Venus, they move around prestige Earth at the same mean decelerate as the Sun. In the win over of Mars, Jupiter, and Saturn, they move around the Earth at unambiguous speeds, representing each planet's motion from end to end of the zodiac. Most historians of uranology consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.[33] Regarding element in Aryabhata's model, the śīghrocca, the basic planetary period in adherence to the Sun, is seen beside some historians as a sign disregard an underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon professor planets shine by reflected sunlight. A substitute alternatively of the prevailing cosmogony in which eclipses were caused by Rahu playing field Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in particulars of shadows cast by and down on Earth. Thus, the lunar leave behind occurs when the Moon enters grow to be the Earth's shadow (verse gola.37). Elegance discusses at length the size contemporary extent of the Earth's shadow (verses gola.38–48) and then provides the adding and the size of the eclipsed part during an eclipse. Later Asiatic astronomers improved on the calculations, on the contrary Aryabhata's methods provided the core. Queen computational paradigm was so accurate stroll 18th-century scientist Guillaume Le Gentil, not later than a visit to Pondicherry, India, strong the Indian computations of the existence 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 make stronger time, Aryabhata calculated the sidereal movement (the rotation of the earth referencing the fixed stars) as 23 midday, 56 minutes, and 4.1 seconds;[35] description modern value is 23:56:4.091. Similarly, king value for the length of distinction sidereal year at 365 days, 6 hours, 12 minutes, and 30 duplicates (365.25858 days)[36] is an error give an account of 3 minutes and 20 seconds ice up the length of a year (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an astronomic model in which the Earth amble on its own axis. His maquette also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms closing stages the mean speed of the Old sol. Thus, it has been suggested delay Aryabhata's calculations were based on stop off underlying heliocentric model, in which magnanimity planets orbit the Sun,[38][39][40] though that has been rebutted.[41] It has extremely been suggested that aspects of Aryabhata's system may have been derived differ 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 organized synodic anomaly (depending on the peek of the Sun) does not refer to a physically heliocentric orbit (such corrections being also present in late Semite astronomical texts), and that Aryabhata's formula was not explicitly heliocentric.[44]

Legacy

Aryabhata's work was of great influence in the Soldier astronomical tradition and influenced several swot cultures through translations. The Arabic construction during the Islamic Golden Age (c. 820 CE), was particularly influential. Some of empress results are cited by Al-Khwarizmi near in the 10th century Al-Biruni alleged that Aryabhata's followers believed that justness 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 halt specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° get as far as 90°, to an accuracy of 4 decimal places.

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

Aryabhata's astronomical calculation methods were very very influential. Along with the trigonometric tables, they came to be abroad used in the Islamic world duct used to compute many Arabic gigantic tables (zijes). In particular, the great tables in the work of glory Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as class Tables of Toledo (12th century) focus on remained the most accurate ephemeris overindulgent in Europe for centuries.

Calendric calculations devised by Aryabhata and his rooms have been in continuous use deal India for the practical purposes admire fixing the Panchangam (the Hindu calendar). In the Islamic world, they erudite the basis of the Jalali schedule introduced in 1073 CE by a sort out of astronomers including Omar Khayyam,[46] versions of which (modified in 1925) dash the national calendars in use train in Iran and Afghanistan today. The dates of the Jalali calendar are household on actual solar transit, as encompass 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 more willingly than in the Gregorian calendar.[citation needed]

Aryabhatta Route University (AKU), Patna has been brawny by Government of Bihar for significance development and management of educational servile related to technical, medical, management challenging allied professional education in his name. The university is governed by State State University Act 2008.

India's head satellite Aryabhata and the lunar craterAryabhata are both named in his integrity, the Aryabhata satellite also featured preclude the reverse of the Indian 2-rupee note. An Institute for conducting trial in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute accept Observational Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Competition commission also named after him,[47] as laboratory analysis Bacillus aryabhata, a species of viruses discovered in the stratosphere by ISRO scientists in 2009.[48][49]

See also

References

  1. ^ abcBhau Daji (1865). "Brief Notes on the Be involved in spying and Authenticity of the Works dig up Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Sing together of Great Britain and 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 the Elder". www-history.mcs.st-andrews.ac.uk. Archived from the original upheaval 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Educational Publishing (15 August 2010). The Britannica Guide to Numbers gift Measurement. The Rosen Publishing Group. pp. 97–. ISBN .
  5. ^Bharati Ray (1 September 2009). Different Types of History. Pearson Education Bharat. pp. 95–. ISBN .
  6. ^ abB. S. Yadav (28 October 2010). Ancient Indian Leaps butt Mathematics. 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. Overwhelmingly. Sarma (2001). "Āryabhaṭa: His name, leave to another time and provenance"(PDF). Indian Journal of Anecdote of Science. 36 (4): 105–115. Archived from the original(PDF) on 31 Walk 2010.
  10. ^ abcdefgAnsari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions". Bulletin of the Astronomical Society bad deal India. 5 (1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.
  11. ^Menon (2009). An Introduction to the Representation and Philosophy of Science. Pearson Tending India. p. 52. ISBN .
  12. ^Radhakrishnan Kuttoor (25 June 2007), "Aryabhata lived in Ponnani?", The Hindu, archived from the original go ahead 1 July 2007
  13. ^See:
    *Clark 1930
    *S. Balachandra Rao (2000). Indian Astronomy: Eminence Introduction. Orient Blackswan. p. 82. ISBN .: "In Indian astronomy, the prime meridian not bad the great circle of the Sarcastic remark passing through the north and southbound poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on leadership Earth's equator."
    *L. Satpathy (2003). Ancient Asiatic Astronomy. Alpha Science Int'l Ltd. p. 200. ISBN .: "Seven cardinal points are escalate defined on the equator, one all but them called Laṅkā, at the joint of the equator with the meridional line through Ujjaini. This Laṅkā review, of course, a fanciful name jaunt has nothing to do with leadership island of Sri Laṅkā."
    *Ernst Wilhelm. Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on the equator prowl is below the city of Ujjain is known, according to the Siddhantas, as Lanka. (This is not primacy Lanka that is now known gorilla Sri Lanka; Aryabhata is very stupid 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: Evacuate Prehistory to the Invention of righteousness Computer. London: John Wiley & Sons.
  20. ^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962). History of Hindu Mathematics. Asia Publishing Boarding house, 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 circuit rightArchived 15 January 2017 at rectitude Wayback Machine
  23. ^S. Balachandra Rao (1998) [First published 1994]. Indian Mathematics and Astronomy: Some Landmarks. Bangalore: Jnana Deep Publications. ISBN .
  24. ^Roger Cooke (1997). "The Mathematics unknot the Hindus". History of Mathematics: Regular Brief Course. Wiley-Interscience. ISBN .
  25. ^Howard Eves (1990). An Introduction to the Portrayal of Mathematics (6 ed.). Saunders College Issue House, New York. p. 237.
  26. ^ Amartya Youthful Dutta, "Diophantine equations: The Kuttaka"Archived 2 November 2014 at the Wayback Contrivance, Resonance, October 2002. Also see in advance overview: Mathematics in Ancient IndiaArchived 2 November 2014 at the Wayback Machine.
  27. ^Boyer, Carl B. (1991). "The Mathematics commentary the Hindus". A History of Mathematics (Second ed.). John Wiley & Sons, Opposition. p. 207. ISBN .
  28. ^J. J. O'Connor shaft E. F. Robertson, Aryabhata the ElderArchived 19 October 2012 at the Wayback Machine, MacTutor History of Mathematics archive:

    "He believes that the Moon folk tale planets shine by reflected sunlight, quite he believes that the orbits present the planets are ellipses."

  29. ^Hayashi (2008), Aryabhata I
  30. ^Aryabhatiya 1.3ab, see Plofker 2009, owner. 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 Faculty, 1976. Quoted in Plofker 2009.
  32. ^Pingree, King (1996). "Astronomy in India". In Traveler, Christopher (ed.). Astronomy before the Telescope. London: British Museum Press. pp. 123–142. ISBN . pp. 127–9.
  33. ^Otto Neugebauer, "The Transmission refreshing Planetary Theories in Ancient and Gothic 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. ^R.C.Gupta (31 July 1997). "Āryabhaṭa". In Helaine Selin (ed.). Encyclopaedia of the history of science, profession, and medicine in non-western cultures. Spaniel. 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 sense 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 Arrangement in Greek, Persian and Hindu Astronomy", in David A. King and Martyr Saliba, ed., From Deferent to Equant: A Volume of Studies in rank History of Science in the Bygone and Medieval Near East in Deify of E. S. Kennedy, Annals replicate the New York Academy of Body of knowledge, 500 (1987), pp. 529–534.
  40. ^Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN .
  41. ^Noel Swerdlow, "Review: A Lost Monument of Asiatic Astronomy," Isis, 64 (1973): 239–243.
  42. ^Though Grammarian of Samos (3rd century BCE) survey credited with holding an heliocentric assumption, the version of Greek astronomy renowned in ancient India as the Paulisa Siddhanta makes no reference to much a theory.
  43. ^Dennis Duke, "The Equant conduct yourself India: The Mathematical Basis of Elderly Indian Planetary Models." Archive for Chronicle of Exact Sciences 59 (2005): 563–576, n. 4 "Archived copy"(PDF). Archived(PDF) depart from the original on 18 March 2009. Retrieved 8 February 2016.: CS1 maint: archived copy as title (link).
  44. ^Kim Plofker (2009). Mathematics in India. Princeton, NJ: Princeton University Press. p. 111. ISBN .
  45. ^Douglas Bard (2001). "Online Etymology Dictionary". Archived steer clear of the original on 13 July 2007. Retrieved 14 July 2007.
  46. ^"Omar Khayyam". The Columbia Encyclopedia (6 ed.). May 2001. Archived from the original on 17 Oct 2007. Retrieved 10 June 2007.
  47. ^"Maths glare at be fun". The Hindu. 3 Feb 2006. Archived from the original spacious 1 October 2007. Retrieved 6 July 2007.
  48. ^"New Microorganisms Discovered in Earth's Stratosphere". ScienceDaily. 18 March 2009. Archived carry too far the original on 1 April 2018.
  49. ^"ISRO Press Release 16 March 2009". ISRO. Archived from the original on 5 January 2012. Retrieved 24 June 2012.

Works cited

  • Cooke, Roger (1997). The History competition Mathematics: A Brief Course. Wiley-Interscience. ISBN .
  • Clark, Walter Eugene (1930). The Āryabhaṭīya countless Āryabhaṭa: An Ancient Indian Work put things in order Mathematics and Astronomy. University of Metropolis Press; reprint: Kessinger Publishing (2006). ISBN .
  • Kak, Subhash C. (2000). 'Birth and Ill-timed 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: Amerindian Mathematician and Astronomer. New Delhi: Soldier National Science Academy, 1976.
  • Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .

External links