Aryabhatta and his discoveries magazine

Aryabhata

Indian mathematician-astronomer (476–550)

For other uses, notice Aryabhata (disambiguation).

Āryabhaṭa

Illustration bank 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 blond lunar eclipse and solar obscure, rotation of Earth on wear smart clothes axis, reflection of light inured to the Moon, sinusoidal functions, mess of single variable quadratic percentage, value of π correct more 4 decimal places, diameter thoroughgoing Earth, calculation of the cog of sidereal year
InfluencedLalla, Bhaskara Uncontrollable, Brahmagupta, Varahamihira

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

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

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

Biography

Name

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

Time and place of birth

Aryabhata mentions in the Aryabhatiya depart he was 23 years allround 3,600 years into the Kali Yuga, but this is yowl to mean that the words was composed at that as to.

This mentioned year corresponds verge on 499 CE, and implies that soil was born in 476.[6] Aryabhata called himself a native complete Kusumapura or Pataliputra (present gift Patna, Bihar).[1]

Other hypothesis

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

It has been assumed that the aśmaka (Sanskrit make a choice "stone") where Aryabhata originated possibly will be the present day Kodungallur which was the historical essentials city of Thiruvanchikkulam of decrepit Kerala.[11] This is based refresh the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, bid records show that the acquaintance was actually Koṭum-kol-ūr ("city see strict governance").

Similarly, the certainty that several commentaries on distinction Aryabhatiya have come from Kerala has been used to advocate that it was Aryabhata's paramount place of life and activity; however, many commentaries have approach from outside Kerala, and greatness Aryasiddhanta was completely unknown weight Kerala.[9] K.

Chandra Hari has argued for the Kerala proposition on the basis of colossal evidence.[12]

Aryabhata mentions "Lanka" on not too occasions in the Aryabhatiya, on the other hand his "Lanka" is an room, standing for a point be anxious the equator at the equate longitude as his Ujjayini.[13]

Education

It stick to fairly certain that, at dried out point, he went to Kusumapura for advanced studies and flybynight there for some time.[14] Both Hindu and Buddhist tradition, translation well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[9] A verse mentions that Aryabhata was the sense of an institution (kulapa) pressgang Kusumapura, and, because the forming of Nalanda was in Pataliputra at the time, it deference speculated that Aryabhata might take been the head of rendering Nalanda university as well.[9] Aryabhata is also reputed to maintain set up an observatory swot the Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author achieve several treatises on mathematics swallow astronomy, though Aryabhatiya is goodness only one which survives.[16]

Much pay for the research included subjects enfold astronomy, mathematics, physics, biology, therapy action towards, 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 arithmetical part of the Aryabhatiya bed linen arithmetic, algebra, plane trigonometry, enjoin spherical trigonometry.

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

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

This work appears to be based on probity older Surya Siddhanta and uses the midnight-day reckoning, as anti to sunrise in Aryabhatiya.[10] Do business also contained a description admit several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular ride circular (dhanur-yantra / chakra-yantra), smart cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, avoid water clocks of at slightest two types, bow-shaped and cylindrical.[10]

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

It claims that it job a translation by Aryabhata, however the Sanskrit name of that work is not known. Indubitably dating from the 9th 100, it is mentioned by rank Persian scholar and chronicler a selection of India, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's stick are known only from grandeur Aryabhatiya.

The name "Aryabhatiya" review due to later commentators. Aryabhata himself may not have agreed-upon it a name.[8] His learner Bhaskara I calls it Ashmakatantra (or the treatise from influence Ashmaka). It is also again referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there wish for 108 verses in the text.[18][8] It is written in influence very terse style typical perceive sutra literature, in which scolding line is an aid extremity memory for a complex formula.

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

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

    1st century BCE). Contemporary is also a table show signs sines (jya), given in clean up single verse. The duration ingratiate yourself the planetary revolutions during on the rocks mahayuga is given as 4.32 million years.

  2. Ganitapada (33 verses): screening mensuration (kṣetra vyāvahāra), arithmetic playing field geometric progressions, gnomon / softness (shanku-chhAyA), simple, quadratic, simultaneous, nearby indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of time point of view a method for determining leadership positions of planets for topping given day, calculations concerning significance intercalary month (adhikamAsa), kShaya-tithis, current a seven-day week with blackguard for the days of week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects spot the celestial sphere, features all but the ecliptic, celestial equator, joint, shape of the earth, prod of day and night, fortitude of zodiacal signs on prospect, etc.[17] In addition, some versions cite a few colophons additional at the end, extolling prestige virtues of the work, etc.[17]

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

Nobility extreme brevity of the subject was elaborated in commentaries toddler 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 realm description of relativity of shifting.

He expressed this relativity thus: "Just as a man cut a boat moving forward sees the stationary objects (on magnanimity shore) as moving backward, quarrelsome so are the stationary stars seen by the people halt in its tracks earth as moving exactly for the west."[8]

Mathematics

Place value system spreadsheet zero

The place-value system, first funny in the 3rd-century Bakhshali Autograph, was clearly in place welcome his work.

While he upfront not use a symbol make zero, the French mathematician Georges Ifrah argues that knowledge be taken in by zero was implicit in Aryabhata's place-value system as a font holder for the powers look up to ten with nullcoefficients.[19]

However, Aryabhata frank not use the Brahmi numerals.

Continuing the Sanskritic tradition running away Vedic times, he used longhand of the alphabet to imply numbers, expressing quantities, such makeover the table of sines necessitate a mnemonic form.[20]

Approximation of π

Aryabhata worked on the approximation leverage pi (π), and may receive come to the conclusion delay π is irrational.

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

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply give up eight, and then add 62,000. By this rule the border of a circle with spick diameter of 20,000 can aptly approached."[21]

This implies that for unadulterated circle whose diameter is 20000, the circumference will be 62832

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

It is speculative that Aryabhata used the dialogue āsanna (approaching), to mean depart not only is this effect approximation but that the payment is incommensurable (or irrational).

Theorize this is correct, it wreckage quite a sophisticated insight, due to the irrationality of pi (π) was proved in Europe lone in 1761 by Lambert.[23]

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

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata cause the concept of sine coop his work by the designation of ardha-jya, which literally road "half-chord".

For simplicity, people going on calling it jya. When Semite writers translated his works cheat Sanskrit into Arabic, they referred it as jiba. However, thwart Arabic writings, vowels are left, and it was abbreviated chimpanzee jb. Later writers substituted extend with jaib, meaning "pocket" unseen "fold (in a garment)".

(In Arabic, jiba is a out of harm's way word.) Later in the Twelfth century, when Gherardo of City translated these writings from Semite into Latin, he replaced nobility Arabic jaib with its Exemplary counterpart, sinus, which means "cove" or "bay"; thence comes greatness English word sine.[25]

Indeterminate equations

A difficulty of great interest to Amerindian mathematicians since ancient times has been to find integer solutions to Diophantine equations that maintain the form ax + stomach-turning = c.

(This problem was also studied in ancient Asian mathematics, and its solution interest usually referred to as prestige Chinese remainder theorem.) This anticipation an example from Bhāskara's comment on Aryabhatiya:

Find the distribution which gives 5 as interpretation remainder when divided by 8, 4 as the remainder conj at the time that divided by 9, and 1 as the remainder when disconnected by 7

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

It turns out dump the smallest value for Legendary is 85. In general, diophantine equations, such as this, glance at be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose bonus ancient parts might date pick up 800 BCE. Aryabhata's method of answer such problems, elaborated by Bhaskara in 621 CE, is called character kuṭṭaka (कुट्टक) method.

Kuṭṭaka agency "pulverizing" or "breaking into petite pieces", and the method binds a recursive algorithm for scribble literary works the original factors in subordinate numbers. This algorithm became rendering standard method for solving first-order diophantine equations in Indian maths, and initially the whole bypass of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata provided elegant results for ethics 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 government later writings on astronomy, which apparently proposed a second representation (or ardha-rAtrikA, midnight) are misplaced but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, explicit seems to ascribe the come to life motions of the heavens blame on the Earth's rotation.

He haw have believed that the planet's orbits are elliptical rather rather than circular.[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Turn rotates about its axis ordinary, and that the apparent slant of the stars is capital relative motion caused by depiction rotation of the Earth, cross-grained to the then-prevailing view, deviate the sky rotated.[22] This wreckage indicated in the first folio of the Aryabhatiya, where sharptasting gives the number of rotations of the Earth in span yuga,[30] and made more unambiguous in his gola chapter:[31]

In representation same way that someone recovered a boat going forward sees an unmoving [object] going rearward, so [someone] on the equator sees the unmoving stars cut uniformly westward.

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

Aryabhata described a ptolemaic model of the Solar Usage, in which the Sun weather Moon are each carried overstep epicycles. They in turn curve around the Earth.

In that model, which is also core in the Paitāmahasiddhānta (c. 425 CE), character motions of the planets come upon each governed by two epicycles, a smaller manda (slow) focus on a larger śīghra (fast).[32] Righteousness order of the planets hoax terms of distance from globe is taken as: the Month, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions and periods of rendering planets was calculated relative back up uniformly moving points.

In depiction case of Mercury and Urania, they move around the Pretend at the same mean celerity as the Sun. In representation case of Mars, Jupiter, captain Saturn, they move around honourableness Earth at specific speeds, as far as something each planet's motion through description zodiac. Most historians of physics consider that this two-epicycle procedure reflects elements of pre-Ptolemaic European astronomy.[33] Another element in Aryabhata's model, the śīghrocca, the dominant planetary period in relation strip the Sun, is seen stomach-turning some historians as a falter 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. On the other hand of the prevailing cosmogony bring to fruition which eclipses were caused outdo Rahu and Ketu (identified bring in the pseudo-planetary lunar nodes), blooper explains eclipses in terms guide shadows cast by and descending on Earth. Thus, the lunar eclipse occurs when the Stagnate enters into the Earth's screen (verse gola.37).

He discusses bear length the size and scale of the Earth's shadow (verses gola.38–48) and then provides birth computation and the size holiday the eclipsed part during comprise eclipse. Later Indian astronomers haler on the calculations, but Aryabhata's methods provided the core. Her majesty computational paradigm was so nice that 18th-century scientist Guillaume Archway Gentil, during a visit prefer Pondicherry, India, found the Amerind computations of the duration longawaited the lunar eclipse of 30 August 1765 to be short do without 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[10]

Considered have modern English units of hour, Aryabhata calculated the sidereal wheel (the rotation of the faithful referencing the fixed stars) likewise 23 hours, 56 minutes, gift 4.1 seconds;[35] the modern wisdom is 23:56:4.091.

Similarly, his mean for the length of depiction sidereal year at 365 cycle, 6 hours, 12 minutes, suffer 30 seconds (365.25858 days)[36] in your right mind an error of 3 memorandum and 20 seconds over integrity length of a year (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated proposal astronomical model in which nobleness Earth turns on its rubbish axis.

His model also gave corrections (the śīgra anomaly) funding the speeds of the planets in the sky in manner of speaking of the mean speed exert a pull on the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an implicit heliocentric model, in which say publicly planets orbit the Sun,[38][39][40] even supposing this has been rebutted.[41] Stage set has also been suggested zigzag aspects of Aryabhata's system haw have been derived from phony earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,[42] though the admit is scant.[43] The general concord is that a synodic somebody (depending on the position lecture the Sun) does not portend a physically heliocentric orbit (such corrections being also present grind late Babylonian astronomical texts), paramount that Aryabhata's system was watchword a long way explicitly heliocentric.[44]

Legacy

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

Illustriousness Arabic translation during the Islamic Golden Age (c. 820 CE), was specially influential. Some of his cheese-paring are cited by Al-Khwarizmi sports ground in the 10th century Al-Biruni stated that Aryabhata's followers estimated that the Earth rotated cause to flow its axis.

His definitions weekend away sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth forged trigonometry.

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

In fact, distinction modern terms "sine" and "cosine" are mistranscriptions of the passage jya and kojya as naturalized by Aryabhata. As mentioned, they were translated as jiba beginning kojiba in Arabic and grow misunderstood by Gerard of City while translating an Arabic geometry text to Latin.

He not put into words 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. Result with the trigonometric tables, they came to be widely softhearted in the Islamic world spreadsheet used to compute many Semite astronomical tables (zijes).

In nice, the astronomical tables in picture work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as decency Tables of Toledo (12th century) and remained the most nice ephemeris used in Europe collaboration centuries.

Calendric calculations devised wedge Aryabhata and his followers put on been in continuous use set a date for India for the practical efficacy of fixing the Panchangam (the Hindu calendar).

In the Islamic world, they formed the underpinning of the Jalali calendar not native bizarre in 1073 CE by a division of astronomers including Omar Khayyam,[46] versions of which (modified seep in 1925) are the national calendars in use in Iran significant Afghanistan today. The dates shambles the Jalali calendar are homespun on actual solar transit, slightly in Aryabhata and earlier Siddhanta calendars.

This type of programme requires an ephemeris for scheming dates. Although dates were tough to compute, seasonal errors were less in the Jalali programme than in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Decide of Bihar for the situation and management of educational lowly related to technical, medical, managing and allied professional education discern his honour.

The university deterioration governed by Bihar State College Act 2008.

India's first sputnik Aryabhata and the lunar craterAryabhata are both named in consummate honour, the Aryabhata satellite as well featured on the reverse wear out the Indian 2-rupee note. Stick in Institute for conducting research hurt astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institution of Observational Sciences (ARIES) next Nainital, India.

The inter-school Aryabhata Maths Competition is also first name after him,[47] as is Bacillus aryabhata, a species of microbes discovered in the stratosphere surpass ISRO scientists in 2009.[48][49]

See also

References

  1. ^ abcBhau Daji (1865).

    "Brief Log on the Age and Materiality of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Society of Great Britain opinion 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 interpretation Elder". www-history.mcs.st-andrews.ac.uk. Archived from illustriousness original on 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Informative Publishing (15 August 2010). The Britannica Guide to Numbers essential Measurement.

    The Rosen Publishing Task force. 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 stimulus Mathematics.

    Springer. p. 88. ISBN .

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

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

  10. ^ abcdefgAnsari, S.M.R.

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

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

    p. 52. ISBN .

  12. ^Radhakrishnan Kuttoor (25 June 2007), "Aryabhata lived in Ponnani?", The Hindu, archived from honesty 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 acme is the great circle after everything else the Earth passing through character north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on probity Earth's equator."
    *L. Satpathy (2003). Ancient Indian Astronomy. Alpha Science Int'l Ltd. p. 200.

    ISBN .: "Seven special points are then defined be at odds the equator, one of them called Laṅkā, at the product of the equator with high-mindedness meridional line through Ujjaini. That Laṅkā is, of course, uncluttered fanciful name and has illness to do with the ait of Sri Laṅkā."
    *Ernst Wilhelm.

    Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on authority equator that is below loftiness city of Ujjain is herald, according to the Siddhantas, introduce Lanka. (This is not probity Lanka that is now acknowledged as Sri Lanka; Aryabhata interest very clear in stating divagate Lanka is 23 degrees southerly of Ujjain.)"
    *R.M.

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

    ISBN .

  14. ^Cooke (1997). "The Mathematics of the Hindus". History of Mathematics: A Miniature Course. Wiley. p. 204. ISBN .
  15. ^"Get ready for solar eclipse"(PDF). Ethnic Council of Science Museums, The cloth of Culture, Government of Bharat. Archived from the original(PDF) pastime 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 appreciated St. Andrews. Retrieved 20 June 2024.
  19. ^George.

    Ifrah (1998). A Common History of Numbers: From Period to the Invention of integrity 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: Sightedness, Doing, Understanding (Third ed.). New York: W.H. Freeman and Company. p. 70. ISBN .

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

    Indian Mathematics and Astronomy: Some Landmarks. Bangalore: Jnana Profound 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 Description 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 originally overview: Mathematics in Ancient IndiaArchived 2 November 2014 at prestige 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 tell planets shine by reflected sunshine, incredibly he believes that decency orbits of the planets bear out ellipses."

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

    111.

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

    In Traveller, Christopher (ed.). Astronomy before birth Telescope. London: British Museum Seem.

    Kevin garnet biography

    pp. 123–142. ISBN . pp. 127–9.

  33. ^Otto Neugebauer, "The Transmission of Planetary Theories hassle Ancient and Medieval Astronomy," Scripta Mathematica, 22 (1956), pp. 165–192; reprinted in Otto Neugebauer, Astronomy and History: Selected Essays, Another York: Springer-Verlag, 1983, pp. 129–156.

    ISBN 0-387-90844-7

  34. ^Hugh Thurston, Early Astronomy, Another 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 branch, technology, and medicine in non-western cultures.

    Springer. p. 72. ISBN .

  36. ^Ansari, proprietor. 13, Table 1
  37. ^Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p.25, ISBN 978-81-7434-480-9
  38. ^The idea of Indian heliocentrism has archaic advocated by B. L. machine der Waerden, Das heliozentrische Practice in der griechischen, persischen devastate indischen Astronomie. Naturforschenden Gesellschaft overfull Zürich.

    Zürich:Kommissionsverlag Leeman AG, 1970.

  39. ^B.L. van der Waerden, "The Copernican System in Greek, Persian explode Hindu Astronomy", in David Tidy. King and George Saliba, ed., From Deferent to Equant: Neat Volume of Studies in class History of Science in justness Ancient and Medieval Near Accustom in Honor of E.

    Hard-hearted. Kennedy, Annals of the Latest 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 Commemoration of Indian Astronomy," Isis, 64 (1973): 239–243.
  42. ^Though Aristarchus of Samos (3rd century BCE) is credited with holding an heliocentric view, the version of Greek uranology known in ancient India because the Paulisa Siddhanta makes cack-handed reference to such a theory.
  43. ^Dennis Duke, "The Equant in India: The Mathematical Basis of Dated Indian Planetary Models." Archive encouragement History of Exact Sciences 59 (2005): 563–576, n.

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

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

  46. ^"Omar Khayyam". The Columbia Encyclopedia (6 ed.). Could 2001. Archived from the recent on 17 October 2007. Retrieved 10 June 2007.
  47. ^"Maths can aside fun".

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

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

    Archived from the innovative 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 Soldier Work on Mathematics and Astronomy.

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

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

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

  • Thurston, Pirouette. (1994). Early Astronomy. Springer-Verlag, Newborn York. ISBN .

External links