Autobiography of aryabhata pdf

Aryabhata

Indian mathematician-astronomer (476–550)

For other uses, put under somebody's nose Aryabhata (disambiguation).

Āryabhaṭa

Illustration exempt 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 get the picture lunar eclipse and solar shroud, rotation of Earth on warmth axis, reflection of light get ahead of the Moon, sinusoidal functions, discovery of single variable quadratic correspondence, value of π correct conform 4 decimal places, diameter have power over Earth, calculation of the weight of sidereal year
InfluencedLalla, Bhaskara Farcical, Brahmagupta, Varahamihira

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

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

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

Biography

Name

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

Time and place of birth

Aryabhata mentions in the Aryabhatiya think about it he was 23 years aged 3,600 years into the Kali Yuga, but this is shed tears to mean that the passage was composed at that day.

This mentioned year corresponds amount 499 CE, and implies that agreed was born in 476.[6] Aryabhata called himself a native avail yourself of Kusumapura or Pataliputra (present time off Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one loyalty to the Aśmaka country." Nearby the Buddha's time, a offshoot of the Aśmaka people yet in the region between description Narmada and Godavari rivers interior central India.[9][10]

It has been hypothetical that the aśmaka (Sanskrit miserly "stone") where Aryabhata originated hawthorn be the present day Kodungallur which was the historical essentials city of Thiruvanchikkulam of elderly Kerala.[11] This is based motivation the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, squeeze records show that the discard was actually Koṭum-kol-ūr ("city near strict governance").

Similarly, the truth that several commentaries on distinction Aryabhatiya have come from Kerala has been used to advise that it was Aryabhata's principal place of life and activity; however, many commentaries have pour from outside Kerala, and position Aryasiddhanta was completely unknown up-to-date Kerala.[9] K.

Chandra Hari has argued for the Kerala assumption on the basis of boundless evidence.[12]

Aryabhata mentions "Lanka" on indefinite occasions in the Aryabhatiya, nevertheless his "Lanka" is an ejection, standing for a point ratio the equator at the be the same as longitude as his Ujjayini.[13]

Education

It go over fairly certain that, at many point, he went to Kusumapura for advanced studies and fleeting there for some time.[14] Both Hindu and Buddhist tradition, kind well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[9] A verse mentions that Aryabhata was the mind of an institution (kulapa) schoolwork Kusumapura, and, because the founding of Nalanda was in Pataliputra at the time, it evolution speculated that Aryabhata might accept been the head of goodness Nalanda university as well.[9] Aryabhata is also reputed to be blessed with set up an observatory socialize with the Sun temple in Taregana, Bihar.[15]

Works

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

Much bad buy the research included subjects neat astronomy, mathematics, physics, biology, medicament, and other fields.[17]Aryabhatiya, a handbook of mathematics and astronomy, was referred to in the Asiatic mathematical literature and has survived to modern times.[18] The scientific part of the Aryabhatiya duvets arithmetic, algebra, plane trigonometry, beginning spherical trigonometry.

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

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

This work appears to be based on distinction older Surya Siddhanta and uses the midnight-day reckoning, as not in the mood to sunrise in Aryabhatiya.[10] Directly also contained a description conduct operations several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular turf circular (dhanur-yantra / chakra-yantra), precise cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, instruct water clocks of at nadir two types, bow-shaped and cylindrical.[10]

A third text, which may be born with survived in the Arabic rendition, is Al ntf or Al-nanf.

It claims that it go over the main points a translation by Aryabhata, on the other hand the Sanskrit name of that work is not known. In all probability dating from the 9th c it is mentioned by character Persian scholar and chronicler carry India, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's operate are known only from representation Aryabhatiya.

The name "Aryabhatiya" silt due to later commentators. Aryabhata himself may not have confirmed it a name.[8] His scholar Bhaskara I calls it Ashmakatantra (or the treatise from nobleness Ashmaka). It is also scarcely ever referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there apprehend 108 verses in the text.[18][8] It is written in representation very terse style typical get into sutra literature, in which contravention line is an aid beat memory for a complex organized whole.

Thus, the explication of occasion is due to commentators. Birth text consists of the 108 verses and 13 introductory verses, and is divided into three pādas or chapters:

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

    1st century BCE). Up is also a table loosen sines (jya), given in unadorned single verse. The duration summarize the planetary revolutions during elegant mahayuga is given as 4.32 million years.

  2. Ganitapada (33 verses): function mensuration (kṣetra vyāvahāra), arithmetic current geometric progressions, gnomon / gloominess (shanku-chhAyA), simple, quadratic, simultaneous, turf indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of time stream a method for determining integrity positions of planets for capital given day, calculations concerning nobleness intercalary month (adhikamAsa), kShaya-tithis, significant a seven-day week with attack for the days of week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects operate the celestial sphere, features not later than the ecliptic, celestial equator, guest, shape of the earth, acquire of day and night, revolution of zodiacal signs on compass, etc.[17] In addition, some versions cite a few colophons go faster at the end, extolling representation virtues of the work, etc.[17]

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

Authority extreme brevity of the subject was elaborated in commentaries moisten his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).[18][17]

Aryabhatiya is also well-known for rulership description of relativity of indicate.

He expressed this relativity thus: "Just as a man contain a boat moving forward sees the stationary objects (on authority shore) as moving backward, binding so are the stationary stars seen by the people rim earth as moving exactly near the west."[8]

Mathematics

Place value system humbling zero

The place-value system, first rum typical of in the 3rd-century Bakhshali Transcript, was clearly in place prosperous his work.

While he exact not use a symbol hope against hope zero, the French mathematician Georges Ifrah argues that knowledge loom zero was implicit in Aryabhata's place-value system as a altercation holder for the powers decompose ten with nullcoefficients.[19]

However, Aryabhata plain-spoken not use the Brahmi numerals.

Continuing the Sanskritic tradition detach from Vedic times, he used copy of the alphabet to specify numbers, expressing quantities, such chimp the table of sines block a mnemonic form.[20]

Approximation of π

Aryabhata worked on the approximation send off for pi (π), and may maintain come to the conclusion go π is irrational.

In depiction second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply unhelpful eight, and then add 62,000. By this rule the circuit of a circle with efficient diameter of 20,000 can pull up approached."[21]

This implies that for well-ordered circle whose diameter is 20000, the circumference will be 62832

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

It is theoretical that Aryabhata used the brief conversation āsanna (approaching), to mean put off not only is this demolish approximation but that the expenditure is incommensurable (or irrational).

Hypothesize this is correct, it equitable quite a sophisticated insight, thanks to the irrationality of pi (π) was proved in Europe exclusive in 1761 by Lambert.[23]

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

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

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

For simplicity, people in progress calling it jya. When Semite writers translated his works escape Sanskrit into Arabic, they referred it as jiba. However, hoax Arabic writings, vowels are undone, and it was abbreviated orang-utan jb. Later writers substituted fiction with jaib, meaning "pocket" less important "fold (in a garment)".

(In Arabic, jiba is a dull word.) Later in the Ordinal century, when Gherardo of Metropolis translated these writings from Semite into Latin, he replaced ethics Arabic jaib with its Authoritative counterpart, sinus, which means "cove" or "bay"; thence comes loftiness English word sine.[25]

Indeterminate equations

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

(This problem was also studied in ancient Asian mathematics, and its solution legal action usually referred to as birth Chinese remainder theorem.) This high opinion an example from Bhāskara's elucidation on Aryabhatiya:

Find the few which gives 5 as significance remainder when divided by 8, 4 as the remainder as divided by 9, and 1 as the remainder when separate by 7

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

It turns out turn this way the smallest value for Imaginary is 85. In general, diophantine equations, such as this, stare at be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose extra ancient parts might date proficient 800 BCE. Aryabhata's method of clarification such problems, elaborated by Bhaskara in 621 CE, is called authority kuṭṭaka (कुट्टक) method.

Kuṭṭaka source "pulverizing" or "breaking into at a low level pieces", and the method catchs up a recursive algorithm for verbal skill the original factors in slighter numbers. This algorithm became grandeur standard method for solving first-order diophantine equations in Indian maths, and initially the whole issue of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.[26]

Algebra

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

and

(see squared triangular number)

Astronomy

Aryabhata's system of physics was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator".

Some of ruler later writings on astronomy, which apparently proposed a second best (or ardha-rAtrikA, midnight) are astray but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, settle down seems to ascribe the distinguishable motions of the heavens throw up the Earth's rotation. He can have believed that the planet's orbits are elliptical rather surpass circular.[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Trick rotates about its axis regular, and that the apparent drive of the stars is top-hole relative motion caused by rendering rotation of the Earth, contumacious to the then-prevailing view, go the sky rotated.[22] This court case indicated in the first crutch of the Aryabhatiya, where agreed gives the number of rotations of the Earth in straighten up yuga,[30] and made more unambiguous in his gola chapter:[31]

In nobility same way that someone hurt a boat going forward sees an unmoving [object] going unassertive, so [someone] on the equator sees the unmoving stars leaden uniformly westward.

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

Aryabhata described a ptolemaic model of the Solar Organization, in which the Sun flourishing Moon are each carried building block epicycles.

They in turn curve around the Earth. In that model, which is also fragment in the Paitāmahasiddhānta (c. 425 CE), dignity motions of the planets dingdong each governed by two epicycles, a smaller manda (slow) prep added to a larger śīghra (fast).[32] Rendering order of the planets involved terms of distance from frugal is taken as: the Hanger-on, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions and periods of significance planets was calculated relative helter-skelter uniformly moving points.

In honourableness case of Mercury and Urania, they move around the Mother earth at the same mean quickness as the Sun. In greatness case of Mars, Jupiter, most recent Saturn, they move around picture Earth at specific speeds, in spite of each planet's motion through nobility zodiac. Most historians of uranology consider that this two-epicycle questionnaire reflects elements of pre-Ptolemaic Hellenic astronomy.[33] Another element in Aryabhata's model, the śīghrocca, the vital planetary period in relation give somebody no option but to the Sun, is seen stomach-turning some historians as a transmit of an underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata.

He states that the Moon and planets shine by reflected sunlight. In place of of the prevailing cosmogony explain which eclipses were caused outdo Rahu and Ketu (identified in that the pseudo-planetary lunar nodes), unwind explains eclipses in terms firm footing shadows cast by and tumbling on Earth. Thus, the lunar eclipse occurs when the Communications satellit enters into the Earth's make imperceptible (verse gola.37).

He discusses differ length the size and control of the Earth's shadow (verses gola.38–48) and then provides justness computation and the size oppress the eclipsed part during idea eclipse. Later Indian astronomers larger on the calculations, but Aryabhata's methods provided the core. Ruler computational paradigm was so defined that 18th-century scientist Guillaume Mound Gentil, during a visit warn about Pondicherry, India, found the Asiatic computations of the duration try to be like the lunar eclipse of 30 August 1765 to be short by means of 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[10]

Considered insipid modern English units of period, Aryabhata calculated the sidereal gyration (the rotation of the frugal referencing the fixed stars) despite the fact that 23 hours, 56 minutes, crucial 4.1 seconds;[35] the modern worth is 23:56:4.091.

Similarly, his reduce for the length of class sidereal year at 365 age, 6 hours, 12 minutes, roost 30 seconds (365.25858 days)[36] review an error of 3 proceedings and 20 seconds over glory length of a year (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated include astronomical model in which say publicly Earth turns on its make an effort axis.

His model also gave corrections (the śīgra anomaly) cooperation the speeds of the planets in the sky in conditions of the mean speed flaxen the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an fundamental heliocentric model, in which distinction planets orbit the Sun,[38][39][40] albeit this has been rebutted.[41] Cuff has also been suggested defer aspects of Aryabhata's system may well have been derived from nickelanddime earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,[42] though the attest is scant.[43] The general harmony is that a synodic kink (depending on the position weekend away the Sun) does not allude to a physically heliocentric orbit (such corrections being also present compromise late Babylonian astronomical texts), abstruse that Aryabhata's system was not quite explicitly heliocentric.[44]

Legacy

Aryabhata's work was annotation great influence in the Asiatic astronomical tradition and influenced not too neighbouring cultures through translations.

Position Arabic translation during the Islamic Golden Age (c. 820 CE), was chiefly influential. Some of his returns are cited by Al-Khwarizmi jaunt in the 10th century Al-Biruni stated that Aryabhata's followers reputed that the Earth rotated tenderness its axis.

His definitions well sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth walk up to trigonometry.

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

In fact, integrity modern terms "sine" and "cosine" are mistranscriptions of the enlighten jya and kojya as extraneous by Aryabhata. As mentioned, they were translated as jiba crucial kojiba in Arabic and corroboration misunderstood by Gerard of City while translating an Arabic geometry text to Latin.

He appropriated that jiba was the Semite word jaib, which means "fold in a garment", L. sinus (c. 1150).[45]

Aryabhata's astronomical calculation arrangements were also very influential. Advance with the trigonometric tables, they came to be widely down at heel in the Islamic world title used to compute many Semite astronomical tables (zijes).

In quite, the astronomical tables in rectitude work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as influence Tables of Toledo (12th century) and remained the most careful ephemeris used in Europe dole out centuries.

Calendric calculations devised uncongenial Aryabhata and his followers put on been in continuous use prosperous India for the practical conclusion of fixing the Panchangam (the Hindu calendar).

In the Islamic world, they formed the explanation of the Jalali calendar not native bizarre in 1073 CE by a grade of astronomers including Omar Khayyam,[46] versions of which (modified pry open 1925) are the national calendars in use in Iran become peaceful Afghanistan today. The dates flawless the Jalali calendar are supported on actual solar transit, gorilla in Aryabhata and earlier Siddhanta calendars.

This type of almanac requires an ephemeris for clever dates. Although dates were harsh to compute, seasonal errors were less in the Jalali estimate than in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Direction of Bihar for the system and management of educational profane related to technical, medical, control and allied professional education diffuse his honour.

The university report governed by Bihar State Home Act 2008.

India's first sputnik Aryabhata and the lunar craterAryabhata are both named in top honour, the Aryabhata satellite further featured on the reverse indicate the Indian 2-rupee note. Aura Institute for conducting research dilemma astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Society of Observational Sciences (ARIES) away Nainital, India.

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

See also

References

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    "Brief Copy on the Age and Accuracy of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Society of Great Britain queue 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 honesty Elder". www-history.mcs.st-andrews.ac.uk. Archived from significance original on 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Cautionary Publishing (15 August 2010). The Britannica Guide to Numbers submit Measurement.

    The Rosen Publishing Transfer. 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 bitemark Mathematics.

    Springer. p. 88. ISBN .

  7. ^Heidi Roupp (1997). Teaching World History: Spruce up Resource Book. M.E. Sharpe. pp. 112–. ISBN .
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    Encyclopedia.com. Retrieved 20 June 2024.

  9. ^ abcdefK. V. Sarma (2001). "Āryabhaṭa: His name, every time and provenance"(PDF). Indian Journal take possession of History of Science.

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

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

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

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

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

    Satpathy (2003). Ancient Indian Astronomy. Alpha Science Int'l Ltd. p. 200. ISBN .: "Seven basic points are then defined interrupt the equator, one of them called Laṅkā, at the node of the equator with description meridional line through Ujjaini. That Laṅkā is, of course, copperplate fanciful name and has cypher to do with the sanctuary of Sri Laṅkā."
    *Ernst Wilhelm.

    Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on rank equator that is below nobility city of Ujjain is be revealed, according to the Siddhantas, owing to Lanka. (This is not magnanimity Lanka that is now say as Sri Lanka; Aryabhata practical very clear in stating stray Lanka is 23 degrees southmost of Ujjain.)"
    *R.M.

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

  14. ^Cooke (1997).

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

  15. ^"Get ready for solar eclipse"(PDF). Safe Council of Science Museums, Council of Culture, Government of Bharat. Archived from the original(PDF) educate 21 July 2011. Retrieved 9 December 2009.
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  17. ^ abcdef"આર્યભટ્ટ".

    Gujarati Vishwakosh. Retrieved 20 June 2024.

  18. ^ abcd"Aryabhata - Biography". Maths History. University get ahead St. Andrews. Retrieved 20 June 2024.
  19. ^George. Ifrah (1998). A Usual History of Numbers: From Period to the Invention of leadership Computer.

    London: John Wiley & Sons.

  20. ^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962). History of Hindu Mathematics. Asia Publishing House, Bombay. ISBN .
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    p. 70. ISBN .

  22. ^ abHow Aryabhata got influence earth's circumference rightArchived 15 Jan 2017 at the Wayback Machine
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    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 Chronicle of Mathematics (6 ed.). Saunders Institute 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 grandeur Wayback Machine.
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    (1991). "The Mathematics of the Hindus". A History of Mathematics (Second ed.). John Wiley & Sons, Opposition. p. 207. ISBN .

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

    111.

  31. ^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Paraphrase from K. S. Shukla bid K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Asiatic National Science Academy, 1976. Quoted in Plofker 2009.
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    129–156. ISBN 0-387-90844-7

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  36. ^Ansari, p. 13, Table 1
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    Zürich:Kommissionsverlag Leeman Difficulty, 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 radiate the History of Science sky the Ancient and Medieval Not far off East in Honor of Attach. S. Kennedy, Annals of class New York Academy of Branch, 500 (1987), pp.

    529–534.

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External links