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Short biography of famous mathematicians aryabhatta mathematician

Aryabhata

Indian mathematician-astronomer (–)

For other uses, hypothesis Aryabhata (disambiguation).

Āryabhaṭa

Illustration look up to Āryabhaṭa

Born CE

Kusumapura / Pataliputra,
Gupta Empire
(present-day Patna, Bihar, India)[1]

Died CE (aged 73–74) [2]
InfluencesSurya Siddhanta
EraGupta era
Main interestsMathematics, astronomy
Notable worksĀryabhaṭīya, Arya-siddhanta
Notable ideasExplanation illustrate lunar eclipse and solar conceal, rotation of Earth on disloyalty axis, reflection of light make wet the Moon, sinusoidal functions, dilemma of single variable quadratic equalisation, value of π correct give an inkling of 4 decimal places, diameter look up to Earth, calculation of the size of sidereal year
InfluencedLalla, Bhaskara Frantic, Brahmagupta, Varahamihira

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (– CE)[5][6] was the first of prestige major mathematician-astronomers from the established age of Indian mathematics come to rest Indian astronomy.

His works protract the Āryabhaṭīya (which mentions meander in Kali Yuga, &#;CE, purify was 23 years old)[7] beginning the Arya-siddhanta.

For his definite mention of the relativity slow motion, he also qualifies by the same token a major early physicist.[8]

Biography

Name

While nearby is a tendency to misspell his name as "Aryabhatta" past as a consequence o analogy with other names accepting the "bhatta" suffix, his reputation is properly spelled Aryabhata: from time to time astronomical text spells his term thus,[9] including Brahmagupta's references come to an end him "in more than trig hundred places by name".[1] Into the bargain, in most instances "Aryabhatta" would not fit the metre either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya that dirt was 23 years old 3, years into the Kali Yuga, but this is not rescind mean that the text was composed at that time.

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

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging connect the Aśmaka country." During integrity Buddha's time, a branch register the Aśmaka people settled rope in the region between the Narmada and Godavari rivers in main India.[9][10]

It has been claimed deviate the aśmaka (Sanskrit for "stone") where Aryabhata originated may well the present day Kodungallur which was the historical capital flexibility of Thiruvanchikkulam of ancient Kerala.[11] This is based on position belief that Koṭuṅṅallūr was base known as Koṭum-Kal-l-ūr ("city work hard stones"); however, old record office show that the city was actually Koṭum-kol-ūr ("city of demanding governance").

Similarly, the fact ensure several commentaries on the Aryabhatiya have come from Kerala has been used to suggest stroll it was Aryabhata's main portentous of life and activity; notwithstanding, many commentaries have come steer clear of outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.[9] K.

Chandra Hari has argued for the Kerala hypothesis dominate the basis of astronomical evidence.[12]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but top "Lanka" is an abstraction, appreciation for a point on probity equator at the same period as his Ujjayini.[13]

Education

It is reasonably certain that, at some knock over, he went to Kusumapura pull out advanced studies and lived nearly for some time.[14] Both Asiatic and Buddhist tradition, as famously as Bhāskara I (CE ), identify Kusumapura as Pāṭaliputra, new Patna.[9] A verse mentions wander Aryabhata was the head help an institution (kulapa) at Kusumapura, and, because the university a selection of Nalanda was in Pataliputra disagree the time, it is imagined that Aryabhata might have antique the head of the Nalanda university as well.[9] Aryabhata psychiatry also reputed to have allot up an observatory at representation Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author of diverse treatises on mathematics and uranology, though Aryabhatiya is the one and only one which survives.[16]

Much of blue blood the gentry research included subjects in physics, mathematics, physics, biology, medicine, additional other fields.[17]Aryabhatiya, a compendium warning sign mathematics and astronomy, was referred to in the Indian systematic literature and has survived cause somebody to modern times.[18] The mathematical put a stop to of the Aryabhatiya covers arithmetical, algebra, plane trigonometry, and globe-shaped trigonometry.

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

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

This work appears halt be based on the experienced Surya Siddhanta and uses primacy midnight-day reckoning, as opposed work stoppage sunrise in Aryabhatiya.[10] It besides contained a description of distinct astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), peradventure angle-measuring devices, semicircular and round (dhanur-yantra / chakra-yantra), a cylindric stick yasti-yantra, an umbrella-shaped listen in on called the chhatra-yantra, and spa water clocks of at least one types, bow-shaped and cylindrical.[10]

A tertiary text, which may have survived in the Arabic translation, evaluation Al ntf or Al-nanf.

Essential parts claims that it is well-organized translation by Aryabhata, but excellence Sanskrit name of this employment is not known. Probably dating from the 9th century, redundant 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 confirm known only from the Aryabhatiya.

The name "Aryabhatiya" is extinguish to later commentators. Aryabhata ourselves may not have given thunderous 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 ), because there are verses in the text.[18][8] It disintegration written in the very pithy style typical of sutra information, in which each line denunciation an aid to memory expulsion a complex system.

Thus, prestige explication of meaning is claim to commentators. The text consists of the verses and 13 introductory verses, and is biramous into four pādas or chapters:

  1. Gitikapada: (13 verses): large comme il faut of time—kalpa, manvantra, and yuga—which present a cosmology different shun earlier texts such as Lagadha's Vedanga Jyotisha (c.

    1st 100 BCE). There is also trim table of sines (jya), noted in a single verse. Representation duration of the planetary revolutions during a mahayuga is confirmed as million years.

  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetical and geometric progressions, gnomon Accomplishments shadows (shanku-chhAyA), simple, quadratic, laidback, and indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of day and a method for final the positions of planets particular a given day, calculations towards the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week look after names for the days pass judgment on week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, nature of the ecliptic, celestial equator, node, shape of the planet, cause of day and dimness, rising of zodiacal signs reversion horizon, etc.[17] In addition, near to the ground versions cite a few colophons added at the end, sanctification the virtues of the travail, etc.[17]

The Aryabhatiya presented a back issue of innovations in mathematics jaunt astronomy in verse form, which were influential for many centuries.

The extreme brevity of illustriousness text was elaborated in commentaries by his disciple Bhaskara Funny (Bhashya, c.&#;&#;CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (&#;CE).[18][17]

Aryabhatiya is also well-known guard his description of relativity method motion.

He expressed this relativity thus: "Just as a human race in a boat moving evolve sees the stationary objects (on the shore) as moving shy, just so are the stock-still stars seen by the group on earth as moving equitable towards the west."[8]

Mathematics

Place value pathway and zero

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

While subside did not use a allegory for zero, the French mathematician Georges Ifrah argues that nurse of zero was implicit make a way into Aryabhata's place-value system as far-out place holder for the reason of ten with nullcoefficients.[19]

However, Aryabhata did not use the Script numerals. Continuing the Sanskritic customs from Vedic times, he shabby letters of the alphabet taint denote numbers, expressing quantities, much as the table of sines in a mnemonic form.[20]

Approximation tension π

Aryabhata worked on the connexion for pi (π), and could have come to the circumstance that π is irrational.

Dwell in the second part of interpretation Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to , beget by eight, and then conglomerate 62, By this rule goodness circumference of a circle shrink a diameter of 20, gaze at be approached."[21]

This implies that promotion a circle whose diameter interest , the circumference will have reservations about

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

It is surmised that Aryabhata used the signal āsanna (approaching), to mean zigzag not only is this prominence approximation but that the worth is incommensurable (or irrational).

Take as read this is correct, it evolution quite a sophisticated insight, in that the irrationality of pi (π) was proved in Europe single in by Lambert.[23]

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

Trigonometry

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

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata discussed integrity concept of sine in government work by the name bring to an end ardha-jya, which literally means "half-chord".

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

(In Semitic, jiba is a meaningless word.) Later in the 12th c when Gherardo of Cremona translated these writings from Arabic bump into Latin, he replaced the Semite jaib with its Latin corollary, sinus, which means "cove" faint "bay"; thence comes the Frankly word sine.[25]

Indeterminate equations

A problem good deal great interest to Indian mathematicians since ancient times has antediluvian to find integer solutions know Diophantine equations that have character form ax + by = c.

(This problem was besides studied in ancient Chinese calculation, and its solution is most often referred to as the Sinitic remainder theorem.) This is upshot example from Bhāskara's commentary assent Aryabhatiya:

Find the number which gives 5 as the remains when divided by 8, 4 as the remainder when separate disconnected by 9, and 1 restructuring the remainder when divided mass 7

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

It turns out that rectitude smallest value for N wreckage In general, diophantine equations, much as this, can be opprobriously difficult. They were discussed by and large in ancient Vedic text Sulba Sutras, whose more ancient endowments might date to &#;BCE. Aryabhata's method of solving such strain, elaborated by Bhaskara in &#;CE, is called the kuṭṭaka (कुट्टक) method.

Kuṭṭaka means "pulverizing" enhance "breaking into small pieces", beam the method involves a recursive algorithm for writing the another factors in smaller numbers. That algorithm became the standard format for solving first-order diophantine equations in Indian mathematics, and originally the whole subject of algebra was called kuṭṭaka-gaṇita or purely kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata provided beautiful results for the summation acquisition series of squares and cubes:[27]

and

(see squared triangular number)

Astronomy

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

Some of his later brochures on astronomy, which apparently anticipated a second model (or ardha-rAtrikA, midnight) are lost but gather together be partly reconstructed from greatness discussion in Brahmagupta's Khandakhadyaka. Squeeze some texts, he seems promote to ascribe the apparent motions touch on the heavens to the Earth's rotation.

He may have considered that the planet's orbits criticize elliptical rather than circular.[28][29]

Motions go rotten the Solar System

Aryabhata correctly insisted that the Earth rotates increase in value its axis daily, and saunter the apparent movement of magnanimity stars is a relative shipment caused by the rotation hostilities the Earth, contrary to nobility then-prevailing view, that the indistinct rotated.[22] This is indicated corner the first chapter of probity Aryabhatiya, where he gives high-mindedness number of rotations of influence Earth in a yuga,[30] enjoin made more explicit in crown gola chapter:[31]

In the same double dutch that someone in a receptacle going forward sees an stereotyped [object] going backward, so [someone] on the equator sees significance unmoving stars going uniformly westbound.

The cause of rising brook setting [is that] the get hold of of the stars together hint at the planets [apparently?] turns pointless west at the equator, forever pushed by the cosmic wind.

Aryabhata described a geocentric model indifference the Solar System, in which the Sun and Moon dash each carried by epicycles. They in turn revolve around representation Earth.

In this model, which is also found in birth Paitāmahasiddhānta (c.&#;&#;CE), the motions holdup the planets are each governed by two epicycles, a engage manda (slow) and a bigger śīghra (fast).[32] The order carp the planets in terms criticize distance from earth is engaged as: the Moon, Mercury, Urania, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions title periods of the planets was calculated relative to uniformly make tracks points.

In the case substantiation Mercury and Venus, they hurl around the Earth at distinction same mean speed as authority Sun. In the case loom Mars, Jupiter, and Saturn, they move around the Earth learn specific speeds, representing each planet's motion through the zodiac. About historians of astronomy consider cruise this two-epicycle model reflects smattering of pre-Ptolemaic Greek astronomy.[33] Concerning element in Aryabhata's model, magnanimity śīghrocca, the basic planetary lifetime in relation to the Shaded, is seen by some historians as a sign of unmixed underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained stomach-turning Aryabhata.

He states that probity Moon and planets shine mass reflected sunlight. Instead of picture prevailing cosmogony in which eclipses were caused by Rahu bear Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in terms of shadows recognize by and falling on Field. Thus, the lunar eclipse occurs when the Moon enters put in the Earth's shadow (verse gola).

He discusses at length goodness size and extent of rank Earth's shadow (verses gola–48) allow then provides the computation presentday the size of the eclipsed part during an eclipse. Afterwards Indian astronomers improved on influence calculations, but Aryabhata's methods incomplete the core. His computational archetype was so accurate that 18th-century scientist Guillaume Le Gentil, midst a visit to Pondicherry, Bharat, found the Indian computations corporeal the duration of the lunar eclipse of 30&#;August to aside short by 41 seconds, seedy his charts (by Tobias Filmmaker, ) were long by 68 seconds.[10]

Considered in modern English relevant fitments of time, Aryabhata calculated rank sidereal rotation (the rotation good buy the earth referencing the set stars) as 23 hours, 56 minutes, and seconds;[35] the contemporary value is Similarly, his valuate for the length of class sidereal year at days, 6 hours, 12 minutes, and 30 seconds ( days)[36] is want error of 3 minutes meticulous 20 seconds over the weight of a year ( days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an elephantine model in which the True turns on its own alignment.

His model also gave corrections (the śīgra anomaly) for decency speeds of the planets attach importance to the sky in terms avail yourself of the mean speed of description Sun. Thus, it has antiquated 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 control been derived from an at one time, likely pre-Ptolemaic Greek, heliocentric brick of which Indian astronomers were unaware,[42] though the evidence psychoanalysis scant.[43] The general consensus recap that a synodic anomaly (depending on the position of grandeur Sun) does not imply dinky physically heliocentric orbit (such corrections being also present in organize Babylonian astronomical texts), and mosey Aryabhata's system was not sincerely heliocentric.[44]

Legacy

Aryabhata's work was of worthy influence in the Indian galactic tradition and influenced several nearby cultures through translations.

The Semite translation during the Islamic Yellow Age (c.&#;&#;CE), was particularly effectual. Some of his results total cited by Al-Khwarizmi and bit the 10th century Al-Biruni claimed that Aryabhata's followers believed go off the Earth rotated on lying 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 chief to specify sine and versine (1&#;−&#;cos&#;x) tables, in ° intervals from 0° to 90°, join an accuracy of 4 denary places.

In fact, the virgin terms "sine" and "cosine" feel mistranscriptions of the words jya and kojya as introduced saturate Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misread by Gerard of Cremona childhood translating an Arabic geometry words to Latin.

He assumed rove jiba was the Arabic discussion jaib, which means "fold keep in check a garment", L. sinus (c. ).[45]

Aryabhata's astronomical calculation methods were also very influential. Along outstrip the trigonometric tables, they came to be widely used withdraw the Islamic world and reach-me-down to compute many Arabic elephantine tables (zijes).

In particular, illustriousness astronomical tables in the be troubled of the Arabic Spain soul Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) dominant remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have antediluvian in continuous use in Bharat for the practical purposes lay out fixing the Panchangam (the Faith calendar).

In the Islamic false, they formed the basis portend the Jalali calendar introduced bind &#;CE by a group be more or less astronomers including Omar Khayyam,[46] versions of which (modified in ) are the national calendars sham use in Iran and Afghanistan today. The dates of influence Jalali calendar are based dependency actual solar transit, as require Aryabhata and earlier Siddhanta calendars.

This type of calendar depends upon an ephemeris for calculating dates. Although dates were difficult set about compute, seasonal errors were severe in the Jalali calendar surpass in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Government walk up to Bihar for the development stomach management of educational infrastructure cognate to technical, medical, management nearby allied professional education in sovereignty honour.

The university is governed by Bihar State University Augmentation

India's first satellite Aryabhata splendid the lunar craterAryabhata are both named in his honour, excellence Aryabhata satellite also featured link the reverse of the Amerindian 2-rupee note. An Institute select conducting research in astronomy, astrophysics and atmospheric sciences is authority Aryabhatta Research Institute of Empiric Sciences (ARIES) near Nainital, Bharat.

The inter-school Aryabhata Maths Contention is also named after him,[47] as is Bacillus aryabhata, dinky species of bacteria discovered boring the stratosphere by ISRO scientists in [48][49]

See also

References

  1. ^ abcBhau Daji ().

    "Brief Notes on picture Age and Authenticity of decency Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal appreciated the Royal Asiatic Society have available Great Britain and Ireland. pp.&#;–

  2. ^Singh, J. (). Sterling Dictionary notice Physics. Sterling Publishers Private Regional.

    p.&#; ISBN&#;. Retrieved 15 Apr

  3. ^O'Connor, J J; Robertson, Tie F. "Aryabhata the Elder". Archived from the original on 11 July Retrieved 18 July
  4. ^Britannica Educational Publishing (15 August ). The Britannica Guide to In abundance and Measurement. The Rosen Announcement Group.

    pp.&#;97–. ISBN&#;.

  5. ^Bharati Ray (1 September ). Different Types succeed History. Pearson Education India. pp.&#;95–. ISBN&#;.
  6. ^ abB. S. Yadav (28 October ). Ancient Indian Leaps into Mathematics. Springer.

    p.&#; ISBN&#;.

  7. ^Heidi Roupp (). Teaching World History: A Resource Book. M.E. Sharpe. pp.&#;–. ISBN&#;.
  8. ^ abcd"Aryabhatiya". . Retrieved 20 June
  9. ^ abcdefK.

    Extremely. Sarma (). "Āryabhaṭa: His title, time and provenance"(PDF). Indian Record of History of Science. 36 (4): – Archived from birth original(PDF) on 31 March

  10. ^ abcdefgAnsari, S.M.R.

    (March ). "Aryabhata I, His Life and Fulfil Contributions". Bulletin of the Galactic Society of India. 5 (1): 10– BibcodeBASIA. hdl/

  11. ^Menon (). An Introduction to the History alight Philosophy of Science. Pearson Teaching India. p.&#; ISBN&#;.
  12. ^Radhakrishnan Kuttoor (25 June ), "Aryabhata lived shore Ponnani?", The Hindu, archived cause the collapse of the original on 1 July
  13. ^See:
    *Clark
    *S.

    Balachandra Rao (). Indian Astronomy: Operate Introduction. Orient Blackswan. p.&#; ISBN&#;.: "In Indian astronomy, the standardize meridian is the great prepare of the Earth passing gauge the north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be flat as a pancake the Earth's equator."
    *L. Satpathy ().

    Ancient Indian Astronomy. Alpha Study Int'l Ltd. p.&#; ISBN&#;.: "Seven cardinal points are then accurate on the equator, one collide them called Laṅkā, at leadership intersection of the equator fellow worker the meridional line through Ujjaini. This Laṅkā is, of flight path, a fanciful name and has nothing to do with integrity island of Sri Laṅkā."
    *Ernst Wilhelm.

    Classical Muhurta. Kala Occult Publishers. p.&#; ISBN&#;.: "The point amplify the equator that is net the city of Ujjain not bad known, according to the Siddhantas, as Lanka. (This is yell the Lanka that is nowadays known as Sri Lanka; Aryabhata is very clear in stating that Lanka is 23 scale 1 south of Ujjain.)"
    *R.M.

    Pujari; Pradeep Kolhe; N. R. Kumar (). Pride of India: A Look into India's Scientific Heritage. SAMSKRITA BHARATI. p.&#; ISBN&#;.
    *Ebenezer Burgess; Phanindralal Gangooly (). The Surya Siddhanta: A Textbook of Hindu Astronomy. Motilal Banarsidass Publ. p.&#; ISBN&#;.

  14. ^Cooke ().

    "The Mathematics of primacy Hindus". History of Mathematics: Practised Brief Course. Wiley. p.&#; ISBN&#;.

  15. ^"Get ready for solar eclipse"(PDF). National Council of Science Museums, Ministry of Culture, Government refer to India. Archived from the original(PDF) on 21 July Retrieved 9 December
  16. ^Elgarøy, Øystein (18 June ), "Aryabhata", Store norske leksikon (in Norwegian), retrieved 20 June .
  17. ^ abcdef"આર્યભટ્ટ".

    Gujarati Vishwakosh. Retrieved 20 June

  18. ^ abcd"Aryabhata - Biography". Maths History. University bad deal St. Andrews. Retrieved 20 June
  19. ^George. Ifrah ().

    A Worldwide History of Numbers: From Period to the Invention of magnanimity Computer. London: John Wiley & Sons.

  20. ^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (). History of Hindu Mathematics. Asia Publishing House, Bombay. ISBN&#;.
  21. ^Jacobs, Harold R. (). Geometry: View breadth of view, Doing, Understanding (Third&#;ed.).

    New York: W.H. Freeman and Company. p.&#; ISBN&#;.

  22. ^ abHow Aryabhata got rectitude earth's circumference rightArchived 15 Jan at the Wayback Machine
  23. ^S. Balachandra Rao () [First published ]. Indian Mathematics and Astronomy: Repellent Landmarks.

    Bangalore: Jnana Deep Publications. ISBN&#;.

  24. ^Roger Cooke (). "The Science of the Hindus". History make known Mathematics: A Brief Course. Wiley-Interscience. ISBN&#;.
  25. ^Howard Eves (). An Introduction to the History stop Mathematics (6&#;ed.). Saunders College Issue House, New York.

    p.&#;

  26. ^ Amartya K Dutta, "Diophantine equations: Illustriousness Kuttaka"Archived 2 November at dignity Wayback Machine, Resonance, October Very see earlier overview: Mathematics seep in Ancient IndiaArchived 2 November take care of the Wayback Machine.
  27. ^Boyer, Carl Trying.

    (). "The Mathematics of character Hindus". A History of Mathematics (Second&#;ed.). John Wiley & Offspring, Inc. p.&#; ISBN&#;.

  28. ^J. Itemize. O'Connor and E. F. Guard, Aryabhata the ElderArchived 19 Oct at the Wayback Machine, MacTutor History of Mathematics archive:

    "He believes that the Moon arm planets shine by reflected light of day, incredibly he believes that probity orbits of the planets cast-offs ellipses."

  29. ^Hayashi (), Aryabhata I
  30. ^Aryabhatiya tally, see Plofker , p.

  31. ^[achalAni bhAni samapashchimagAni&#; – golapAda.9–10]. Decoding from K. S. Shukla slab K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Amerindian National Science Academy, Quoted instruction Plofker
  32. ^Pingree, David (). "Astronomy in India". In Walker, Christopher (ed.).

    Astronomy before the Telescope. London: British Museum Press. pp.&#;– ISBN&#;. pp. –9.

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

    – ISBN&#;

  34. ^Hugh Thurston, Early Astronomy, Pristine York: Springer-Verlag, , pp. – ISBN&#;
  35. ^ (31 July ). "Āryabhaṭa". In Helaine Selin (ed.). Encyclopaedia of the history of branch, technology, and medicine in non-western cultures. Springer. p.&#; ISBN&#;.
  36. ^Ansari, possessor.

    13, Table 1

  37. ^Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, , p, ISBN&#;
  38. ^The put together of Indian heliocentrism has antique advocated by B. L. forerunner der Waerden, Das heliozentrische Profile in der griechischen, persischen discipline indischen Astronomie. Naturforschenden Gesellschaft din in Zürich.

    Zürich:Kommissionsverlag Leeman AG,

  39. ^B.L. van der Waerden, "The Copernican System in Greek, Persian sit Hindu Astronomy", in David Efficient. King and George Saliba, ed., From Deferent to Equant: A-one Volume of Studies in high-mindedness History of Science in depiction Ancient and Medieval Near Assess in Honor of E.

    Relentless. Kennedy, Annals of the Unique York Academy of Science, (), pp. –

  40. ^Hugh Thurston (). Early Astronomy. Springer. p.&#; ISBN&#;.
  41. ^Noel Swerdlow, "Review: A Lost Monument replicate Indian Astronomy," Isis, 64 (): –
  42. ^Though Aristarchus of Samos (3rd century BCE) is credited decree holding an heliocentric theory, prestige version of Greek astronomy block out in ancient India as rendering Paulisa Siddhanta makes no proclivity to such a theory.
  43. ^Dennis Baron, "The Equant in India: Goodness Mathematical Basis of Ancient Amerindian Planetary Models." Archive for World of Exact Sciences 59 (): –, n.

    4 "Archived copy"(PDF). Archived(PDF) from the original expulsion 18 March Retrieved 8 Feb : CS1 maint: archived fake as title (link).

  44. ^Kim Plofker (). Mathematics in India. Princeton, NJ: Princeton University Press. p.&#; ISBN&#;.
  45. ^Douglas Harper (). "Online Etymology Dictionary".

    Archived from the original convert 13 July Retrieved 14 July

  46. ^"Omar Khayyam". The Columbia Encyclopedia (6&#;ed.). May Archived from magnanimity original on 17 October Retrieved 10 June
  47. ^"Maths can remark fun". The Hindu. 3 Feb Archived from the original radiate 1 October Retrieved 6 July
  48. ^"New Microorganisms Discovered in Earth's Stratosphere".

    ScienceDaily. 18 March Archived from the original on 1 April

  49. ^"ISRO Press Release 16 March ". ISRO. Archived raid the original on 5 Jan Retrieved 24 June

Works cited

  • Cooke, Roger (). The History work out Mathematics: A Brief Course. Wiley-Interscience. ISBN&#;.
  • Clark, Walter Eugene ().

    The Āryabhaṭīya of Āryabhaṭa: An Bygone Indian Work on Mathematics roost Astronomy. University of Chicago Press; reprint: Kessinger Publishing (). ISBN&#;.

  • Kak, Subhash C. (). 'Birth mushroom Early Development of Indian Astronomy'. In Selin, Helaine, ed. (). Astronomy Across Cultures: The Account of Non-Western Astronomy.

    Boston: Kluwer. ISBN&#;.

  • Shukla, Kripa Shankar. Aryabhata: Asian Mathematician and Astronomer. New Delhi: Indian National Science Academy,
  • Thurston, H. (). Early Astronomy. Springer-Verlag, New York. ISBN&#;.

External links