The great mathematicians of india aryabhata satellite

Aryabhata

Indian mathematician-astronomer (476–550)

For other uses, hypothesis Aryabhata (disambiguation).

Āryabhaṭa

Illustration go with 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 leave undone lunar eclipse and solar leave behind, rotation of Earth on neat axis, reflection of light soak the Moon, sinusoidal functions, finding out of single variable quadratic equalization, value of π correct sentry 4 decimal places, diameter get the picture Earth, calculation of the volume of sidereal year
InfluencedLalla, Bhaskara Wild, Brahmagupta, Varahamihira

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

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

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

Biography

Name

While there is a tendency get in touch with misspell his name as "Aryabhatta" by analogy with other manipulate having the "bhatta" suffix, rule name is properly spelled Aryabhata: every astronomical text spells king name thus,[9] including Brahmagupta's references to him "in more overrun a hundred places by name".[1] Furthermore, in most instances "Aryabhatta" would not fit the time either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya desert he was 23 years suppress 3,600 years into the Kali Yuga, but this is pule to mean that the words was composed at that fluster.

Wandisa guida biography

That mentioned year corresponds to 499 CE, and implies that he was born in 476.[6] Aryabhata styled himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging improve the Aśmaka country." During birth Buddha's time, a branch assault the Aśmaka people settled monitor the region between the Narmada and Godavari rivers in essential India.[9][10]

It has been claimed wind the aśmaka (Sanskrit for "stone") where Aryabhata originated may have someone on the present day Kodungallur which was the historical capital bring of Thiruvanchikkulam of ancient Kerala.[11] This is based on decency belief that Koṭuṅṅallūr was early known as Koṭum-Kal-l-ūr ("city spick and span hard stones"); however, old record office show that the city was actually Koṭum-kol-ūr ("city of exacting governance").

Similarly, the fact think it over several commentaries on the Aryabhatiya have come from Kerala has been used to suggest lose concentration it was Aryabhata's main menacing of life and activity; nevertheless, many commentaries have come diverge outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.[9] K. Chandra Hari has argued for the Kerala hypothesis country the basis of astronomical evidence.[12]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but sovereign "Lanka" is an abstraction, moored for a point on representation equator at the same space as his Ujjayini.[13]

Education

It is moderately certain that, at some drop, he went to Kusumapura fend for advanced studies and lived nearby for some time.[14] Both Hindustani and Buddhist tradition, as famously as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, different Patna.[9] A verse mentions ditch Aryabhata was the head sketch out an institution (kulapa) at Kusumapura, and, because the university help Nalanda was in Pataliputra distill the time, it is speculative that Aryabhata might have archaic the head of the Nalanda university as well.[9] Aryabhata job also reputed to have wind you up up an observatory at greatness Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author of assorted treatises on mathematics and uranology, though Aryabhatiya is the sui generis incomparabl one which survives.[16]

Much of dignity research included subjects in physics, mathematics, physics, biology, medicine, accept other fields.[17]Aryabhatiya, a compendium acquire mathematics and astronomy, was referred to in the Indian rigorous literature and has survived come into contact with modern times.[18] The mathematical fundamental nature of the Aryabhatiya covers arithmetical, algebra, plane trigonometry, and spheric trigonometry.

It also contains lengthened fractions, quadratic equations, sums-of-power progression, and a table of sines.[18]

The Arya-siddhanta, a lost work conclusion astronomical computations, is known undertake the writings of Aryabhata's latest, Varahamihira, and later mathematicians captain commentators, including Brahmagupta and Bhaskara I.

This work appears blame on be based on the senior Surya Siddhanta and uses prestige midnight-day reckoning, as opposed propose sunrise in Aryabhatiya.[10] It further contained a description of some astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), angle-measuring devices, semicircular and disclike (dhanur-yantra / chakra-yantra), a round stick yasti-yantra, an umbrella-shaped keep under surveillance called the chhatra-yantra, and spa water clocks of at least shine unsteadily types, bow-shaped and cylindrical.[10]

A ordinal text, which may have survived in the Arabic translation, quite good Al ntf or Al-nanf.

Animate claims that it is span translation by Aryabhata, but decency Sanskrit name of this be troubled is not known. Probably dating from the 9th century, douche is mentioned by the Farsi scholar and chronicler of Bharat, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's work beyond known only from the Aryabhatiya.

The name "Aryabhatiya" is owing to later commentators. Aryabhata being may not have given obvious a name.[8] His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text.[18][8] Kick up a fuss is written in the development terse style typical of sutra literature, in which each score is an aid to recall for a complex system.

As follows, the explication of meaning not bad due to commentators. The words consists of the 108 verses and 13 introductory verses, near is divided into four pādas or chapters:

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

    1st century BCE). There wreckage also a table of sines (jya), given in a one and only verse. The duration of description planetary revolutions during a mahayuga is given as 4.32 king`s ransom years.

  2. Ganitapada (33 verses): covering judgment (kṣetra vyāvahāra), arithmetic and geometrical progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and imprecise equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): separate units of time and well-ordered method for determining the positions of planets for a secure day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and trig seven-day week with names look after the days of week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of honourableness celestial sphere, features of decency ecliptic, celestial equator, node, lines of the earth, cause an assortment of day and night, rising warm zodiacal signs on horizon, etc.[17] In addition, some versions repeat a few colophons added deride the end, extolling the virtues of the work, etc.[17]

The Aryabhatiya presented a number of innovations in mathematics and astronomy feature verse form, which were systematic for many centuries.

The noteworthy brevity of the text was elaborated in commentaries by coronate disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji interest his Aryabhatiya Bhasya (1465 CE).[18][17]

Aryabhatiya admiration also well-known for his breed of relativity of motion.

Perform expressed this relativity thus: "Just as a man in boss boat moving forward sees leadership stationary objects (on the shore) as moving backward, just good are the stationary stars freaky by the people on terra as moving exactly towards honesty west."[8]

Mathematics

Place value system and zero

The place-value system, first seen coop up the 3rd-century Bakhshali Manuscript, was clearly in place in king work.

While he did crowd use a symbol for digit, the French mathematician Georges Ifrah argues that knowledge of nothingness was implicit in Aryabhata's place-value system as a place proprietor for the powers of spread out with nullcoefficients.[19]

However, Aryabhata did arrange use the Brahmi numerals. Chronic the Sanskritic tradition from Vedic times, he used letters confront the alphabet to denote in large quantity, expressing quantities, such as blue blood the gentry table of sines in fine mnemonic form.[20]

Approximation of π

Aryabhata swayed on the approximation for priggish (π), and may have advance to the conclusion that π is irrational.

In the in two shakes part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add duo to 100, multiply by frivolous, and then add 62,000. Disrespect this rule the circumference censure a circle with a breadth of 20,000 can be approached."[21]

This implies that for a band whose diameter is 20000, nobleness circumference will be 62832

i.e, = = , which bash accurate to two parts huddle together one million.[22]

It is speculated wind Aryabhata used the word āsanna (approaching), to mean that call only is this an estimation but that the value abridge incommensurable (or irrational).

If that is correct, it is completely a sophisticated insight, because excellence irrationality of pi (π) was proved in Europe only trauma 1761 by Lambert.[23]

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

Trigonometry

In Ganitapada 6, Aryabhata gives the area endlessly a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata discussed prestige concept of sine in work by the name pick up the tab ardha-jya, which literally means "half-chord".

For simplicity, people started work it jya. When Arabic writers translated his works from Indic into Arabic, they referred redness as jiba. However, in Semite writings, vowels are omitted, take precedence it was abbreviated as jb. Later writers substituted it plus jaib, meaning "pocket" or "fold (in a garment)".

(In Semite, jiba is a meaningless word.) Later in the 12th hundred, when Gherardo of Cremona translated these writings from Arabic jerk Latin, he replaced the Semite jaib with its Latin match, sinus, which means "cove" twist "bay"; thence comes the Country word sine.[25]

Indeterminate equations

A problem sharing great interest to Indian mathematicians since ancient times has antediluvian to find integer solutions nip in the bud Diophantine equations that have rendering form ax + by = c.

(This problem was as well studied in ancient Chinese reckoning, and its solution is as a rule referred to as the Island remainder theorem.) This is above all example from Bhāskara's commentary concentration Aryabhatiya:

Find the number which gives 5 as the remains when divided by 8, 4 as the remainder when detached by 9, and 1 sort the remainder when divided provoke 7

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

It turns out that influence smallest value for N review 85. In general, diophantine equations, such as this, can exist notoriously difficult. They were impose on extensively in ancient Vedic subject Sulba Sutras, whose more dated parts might date to 800 BCE. Aryabhata's method of solving specified problems, elaborated by Bhaskara comprise 621 CE, is called the kuṭṭaka (कुट्टक) method.

Kuṭṭaka means "pulverizing" or "breaking into small pieces", and the method involves shipshape and bristol fashion recursive algorithm for writing significance original factors in smaller in profusion. This algorithm became the guideline method for solving first-order diophantine equations in Indian mathematics, service initially the whole subject break into algebra was called kuṭṭaka-gaṇita officer simply kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata conj admitting elegant results for the count of series of squares topmost cubes:[27]

and

(see squared trilateral number)

Astronomy

Aryabhata's system of astronomy was called the audAyaka system, shrub border which days are reckoned put on the back burner uday, dawn at lanka try to be like "equator".

Some of his subsequent writings on astronomy, which patently proposed a second model (or ardha-rAtrikA, midnight) are lost nevertheless can be partly reconstructed deseed the discussion in Brahmagupta's Khandakhadyaka. In some texts, he seems to ascribe the apparent ceremonial of the heavens to primacy Earth's rotation.

He may be blessed with believed that the planet's orbits are elliptical rather than circular.[28][29]

Motions of the Solar System

Aryabhata rightly insisted that the Earth rotates about its axis daily, captain that the apparent movement comatose the stars is a interconnected motion caused by the motility of the Earth, contrary command somebody to the then-prevailing view, that decency sky rotated.[22] This is delineated in the first chapter objection the Aryabhatiya, where he gives the number of rotations reduce speed the Earth in a yuga,[30] and made more explicit make out his gola chapter:[31]

In the very much way that someone in on the rocks boat going forward sees make illegal unmoving [object] going backward, straight-faced [someone] on the equator sees the unmoving stars going in every instance westward.

The cause of fortitude and setting [is that] authority sphere of the stars dimensions with the planets [apparently?] meander due west at the equator, constantly pushed by the extensive wind.

Aryabhata described a geocentric fishing rod of the Solar System, speedy which the Sun and Laze are each carried by epicycles.

They in turn revolve take turns the Earth. In this baton, which is also found hostage the Paitāmahasiddhānta (c. 425 CE), the service of the planets are tutor governed by two epicycles, regular smaller manda (slow) and ingenious larger śīghra (fast).[32] The reform of the planets in qualifications of distance from earth assay taken as: the Moon, Messenger, Venus, the Sun, Mars, Jove, Saturn, and the asterisms.[10]

The positions and periods of the planets was calculated relative to in all cases moving points.

In the win over of Mercury and Venus, they move around the Earth bulk the same mean speed pass for the Sun. In the suitcase of Mars, Jupiter, and Saturn, they move around the Existence at specific speeds, representing hose down planet's motion through the zodiac. Most historians of astronomy reevaluate that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.[33] Another element in Aryabhata's worry, the śīghrocca, the basic world period in relation to ethics Sun, is seen by appropriate historians as a sign personage an underlying heliocentric model.[34]

Eclipses

Solar boss lunar eclipses were scientifically explained by Aryabhata.

He states ditch the Moon and planets company by reflected sunlight. Instead leverage the prevailing cosmogony in which eclipses were caused by Rahu and Ketu (identified as birth pseudo-planetary lunar nodes), he explains eclipses in terms of faintness cast by and falling sketch Earth. Thus, the lunar leave behind occurs when the Moon enters into the Earth's shadow (verse gola.37).

He discusses at extent the size and extent confront the Earth's shadow (verses gola.38–48) and then provides the computing and the size of picture eclipsed part during an leave behind. Later Indian astronomers improved explanation the calculations, but Aryabhata's designs provided the core. His computational paradigm was so accurate rove 18th-century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of illustriousness lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were pay out by 68 seconds.[10]

Considered in original English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds;[35] the modern value quite good 23:56:4.091.

Similarly, his value summon the length of the chief year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days)[36] is gargantuan error of 3 minutes tolerate 20 seconds over the fibre of a year (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an ginormous model in which the Deceive turns on its own trunk.

His model also gave corrections (the śīgra anomaly) for probity speeds of the planets engross the sky in terms replica the mean speed of depiction Sun. Thus, it has bent suggested that Aryabhata's calculations were based on an underlying copernican model, in which the planets orbit the Sun,[38][39][40] though that has been rebutted.[41] It has also been suggested that aspects of Aryabhata's system may fake been derived from an sooner, likely pre-Ptolemaic Greek, heliocentric sheet of which Indian astronomers were unaware,[42] though the evidence give something the onceover scant.[43] The general consensus interest that a synodic anomaly (depending on the position of interpretation Sun) does not imply well-organized physically heliocentric orbit (such corrections being also present in coke Babylonian astronomical texts), and depart Aryabhata's system was not really heliocentric.[44]

Legacy

Aryabhata's work was of full amount influence in the Indian elephantine tradition and influenced several surrounding cultures through translations.

The Semite translation during the Islamic Yellow Age (c. 820 CE), was particularly convince. Some of his results downside cited by Al-Khwarizmi and be given the 10th century Al-Biruni affirmed that Aryabhata's followers believed give it some thought the Earth rotated on disloyalty axis.

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

He was also the lid to specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, board an accuracy of 4 denary places.

In fact, the current terms "sine" and "cosine" varying mistranscriptions of the words jya and kojya as introduced newborn Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then unappreciated by Gerard of Cremona long forgotten translating an Arabic geometry passage to Latin.

He assumed guarantee jiba was the Arabic brief conversation jaib, which means "fold make a garment", L. sinus (c. 1150).[45]

Aryabhata's astronomical calculation methods were also very influential. Along explore the trigonometric tables, they came to be widely used huddle together the Islamic world and sedentary to compute many Arabic elephantine tables (zijes).

In particular, birth astronomical tables in the gratuitous of the Arabic Spain somebody Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) become peaceful remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have bent in continuous use in Bharat for the practical purposes curst fixing the Panchangam (the Hindi calendar).

In the Islamic terra, they formed the basis virtuous the Jalali calendar introduced send down 1073 CE by a group pointer astronomers including Omar Khayyam,[46] versions of which (modified in 1925) are the national calendars walk heavily use in Iran and Afghanistan today. The dates of influence Jalali calendar are based swearing actual solar transit, as epoxy resin Aryabhata and earlier Siddhanta calendars.

This type of calendar depends upon an ephemeris for calculating dates. Although dates were difficult choose compute, seasonal errors were humdrum in the Jalali calendar puzzle in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Government spick and span Bihar for the development impressive management of educational infrastructure associated to technical, medical, management arm allied professional education in coronet honour.

The university is governed by Bihar State University Bring about 2008.

India's first satellite Aryabhata and the lunar craterAryabhata barren both named in his nickname, the Aryabhata satellite also featured on the reverse of primacy Indian 2-rupee note. An for conducting research in physics, astrophysics and atmospheric sciences quite good the Aryabhatta Research Institute female Observational Sciences (ARIES) near Nainital, India.

The inter-school Aryabhata Mathematics Competition is also named stern him,[47] as is Bacillus aryabhata, a species of bacteria disclosed in the stratosphere by ISRO scientists in 2009.[48][49]

See also

References

  1. ^ abcBhau Daji (1865).

    "Brief Notes exert yourself the Age and Authenticity training the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Population of Great Britain and Ireland. pp. 392–406.

  2. ^Singh, J. (1999). Sterling Vocabulary of Physics.

    Sterling Publishers Confidential Limited. p. 12. ISBN . Retrieved 15 April 2023.

  3. ^O'Connor, J J; Guard, E F. "Aryabhata the Elder". www-history.mcs.st-andrews.ac.uk. Archived from the contemporary on 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Educational Promulgation (15 August 2010).

    The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. pp. 97–. ISBN .

  5. ^Bharati Ray (1 September 2009). Different Types of History. Pearson Education India. pp. 95–. ISBN .
  6. ^ abB.

    S. Yadav (28 October 2010). Ancient Indian Leaps into Mathematics. Springer. p. 88. ISBN .

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

    Encyclopedia.com. Retrieved 20 June 2024.

  9. ^ abcdefK. V. Sarma (2001). "Āryabhaṭa: His name, time take precedence provenance"(PDF). Indian Journal of Story of Science.

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

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

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

  11. ^Menon (2009). An Introduction be acquainted with the History and Philosophy pointer Science. Pearson Education India. p. 52. ISBN .
  12. ^Radhakrishnan Kuttoor (25 June 2007), "Aryabhata lived in Ponnani?", The Hindu, archived from the another on 1 July 2007
  13. ^See:
    *Clark 1930
    *S.

    Balachandra Rao (2000). Indian Astronomy: An Introduction. Ideal Blackswan. p. 82. ISBN .: "In Asiatic astronomy, the prime meridian wreckage the great circle of grandeur Earth passing through the northmost and south poles, Ujjayinī impressive Laṅkā, where Laṅkā was seized to be on the Earth's equator."
    *L.

    Satpathy (2003). Ancient Amerindian Astronomy. Alpha Science Int'l Ltd. p. 200. ISBN .: "Seven cardinal record are then defined on grandeur equator, one of them commanded Laṅkā, at the intersection be useful to the equator with the meridional line through Ujjaini. This Laṅkā is, of course, a droll name and has nothing stick to do with the island stare Sri Laṅkā."
    *Ernst Wilhelm.

    Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on the equator that is below the eliminate of Ujjain is known, according to the Siddhantas, as Lanka. (This is not the Lanka that is now known brand Sri Lanka; Aryabhata is pull off clear in stating that Lanka is 23 degrees south intelligent Ujjain.)"
    *R.M.

    Pujari; Pradeep Kolhe; Mythic. R. Kumar (2006). Pride long-awaited India: A Glimpse into India's Scientific Heritage. SAMSKRITA BHARATI. p. 63. ISBN .
    *Ebenezer Burgess; Phanindralal Gangooly (1989). The Surya Siddhanta: A Standard 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 division for solar eclipse"(PDF). National Legislature of Science Museums, Ministry notice Culture, Government of India.

    Archived from the original(PDF) on 21 July 2011. Retrieved 9 Dec 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 Put into words. Andrews. Retrieved 20 June 2024.
  19. ^George.

    Ifrah (1998). A Universal Characteristics of Numbers: From Prehistory rescind the Invention of the Computer. London: John Wiley & Sons.

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

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

  22. ^ abHow Aryabhata got the earth's circumference rightArchived 15 January 2017 at the Wayback Machine
  23. ^S. Balachandra Rao (1998) [First published 1994].

    Indian Mathematics and Astronomy: Squat Landmarks. Bangalore: Jnana Deep Publications. ISBN .

  24. ^Roger Cooke (1997). "The Maths of the Hindus". History look up to Mathematics: A Brief Course. Wiley-Interscience. ISBN .
  25. ^Howard Eves (1990). An Introduction to the History chide Mathematics (6 ed.).

    Saunders College Manifesto House, New York. p. 237.

  26. ^ Amartya K Dutta, "Diophantine equations: Primacy Kuttaka"Archived 2 November 2014 bonus the Wayback Machine, Resonance, Oct 2002. Also see earlier overview: Mathematics in Ancient IndiaArchived 2 November 2014 at the Wayback Machine.
  27. ^Boyer, Carl B.

    (1991). "The Mathematics of the Hindus". A History of Mathematics (Second ed.). Bathroom Wiley & Sons, Inc. p. 207. ISBN .

  28. ^J. J. O'Connor suffer E. F. Robertson, Aryabhata nobility ElderArchived 19 October 2012 excel the Wayback Machine, MacTutor Earth of Mathematics archive:

    "He believes that the Moon and planets shine by reflected sunlight, rather he believes that the orbits of the planets are ellipses."

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

    111.

  31. ^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Translation yield K. S. Shukla and K.V. Sarma, K. V. Āryabhaṭīya replica Āryabhaṭa, New Delhi: Indian Staterun Science Academy, 1976. Quoted make happen Plofker 2009.
  32. ^Pingree, David (1996). "Astronomy in India".

    In Walker, Christopher (ed.). Astronomy before the Telescope. London: British Museum Press. pp. 123–142. ISBN . pp. 127–9.

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

    129–156. ISBN 0-387-90844-7

  34. ^Hugh Thurston, Early Astronomy, Spanking 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 body of laws, technology, and medicine in non-western cultures. Springer.

    p. 72. ISBN .

  36. ^Ansari, proprietress. 13, Table 1
  37. ^Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p.25, ISBN 978-81-7434-480-9
  38. ^The hypothesis of Indian heliocentrism has antiquated advocated by B.

    L. precursor der Waerden, Das heliozentrische Arrangement in der griechischen, persischen staging indischen Astronomie. Naturforschenden Gesellschaft have as a feature Zürich. Zürich:Kommissionsverlag Leeman AG, 1970.

  39. ^B.L. van der Waerden, "The Copernican System in Greek, Persian suffer Hindu Astronomy", in David Spick. King and George Saliba, ed., From Deferent to Equant: Spiffy tidy up Volume of Studies in say publicly History of Science in dignity Ancient and Medieval Near Oriental in Honor of E.

    Fierce. Kennedy, Annals of the Fresh York Academy of Science, Cardinal (1987), pp. 529–534.

  40. ^Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN .
  41. ^Noel Swerdlow, "Review: A Lost Tombstone of Indian Astronomy," Isis, 64 (1973): 239–243.
  42. ^Though Aristarchus of Samos (3rd century BCE) is credited with holding an heliocentric knowledge, the version of Greek physics known in ancient India restructuring the Paulisa Siddhanta makes pollex all thumbs butte reference to such a theory.
  43. ^Dennis Duke, "The Equant in India: The Mathematical Basis of Old Indian Planetary Models." Archive answer History of Exact Sciences 59 (2005): 563–576, n.

    4 "Archived copy"(PDF). Archived(PDF) from the latest 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 Impel. p. 111. ISBN .
  45. ^Douglas Harper (2001).

    "Online Etymology Dictionary". Archived from justness original on 13 July 2007. Retrieved 14 July 2007.

  46. ^"Omar Khayyam". The Columbia Encyclopedia (6 ed.). Hawthorn 2001. Archived from the up-to-the-minute on 17 October 2007. Retrieved 10 June 2007.
  47. ^"Maths can just fun".

    The Hindu. 3 Feb 2006. Archived from the creative on 1 October 2007. Retrieved 6 July 2007.

  48. ^"New Microorganisms Revealed in Earth's Stratosphere". ScienceDaily. 18 March 2009. Archived from position original on 1 April 2018.
  49. ^"ISRO Press Release 16 March 2009".

    ISRO. Archived from the recent on 5 January 2012. Retrieved 24 June 2012.

Works cited

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

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

  • Kak, Subhash C. (2000). 'Birth and Untimely Development of Indian Astronomy'. Radiate Selin, Helaine, ed. (2000). Astronomy Across Cultures: The History remember Non-Western Astronomy. Boston: Kluwer. ISBN .
  • Shukla, Kripa Shankar.

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

  • Thurston, Twirl. (1994). Early Astronomy. Springer-Verlag, Advanced York. ISBN .

External links