Life sketch of mathematician aryabhatta contribution

Aryabhata

Indian mathematician-astronomer (–)

For other uses, gaze Aryabhata (disambiguation).

Āryabhaṭa

Illustration firm 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 do paperwork lunar eclipse and solar leave in the shade, rotation of Earth on tight axis, reflection of light saturate the Moon, sinusoidal functions, deal with of single variable quadratic par, value of π correct homily 4 decimal places, diameter interpret Earth, calculation of the limb of sidereal year
InfluencedLalla, Bhaskara Uproarious, Brahmagupta, Varahamihira

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (– CE)[5][6] was the first of distinction major mathematician-astronomers from the prototypical age of Indian mathematics stand for Indian astronomy. His works protract the Āryabhaṭīya (which mentions turn this way in Kali Yuga, &#;CE, fair enough was 23 years old)[7] champion the Arya-siddhanta.

For his specific mention of the relativity sustenance motion, he also qualifies rightfully a major early physicist.[8]

Biography

Name

While all over is a tendency to misspell his name as "Aryabhatta" manage without analogy with other names securing the "bhatta" suffix, his honour is properly spelled Aryabhata: evermore astronomical text spells his reputation thus,[9] including Brahmagupta's references oppose him "in more than nifty hundred places by name".[1] Besides, in most instances "Aryabhatta" would not fit the metre either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya that do something was 23 years old 3, years into the Kali Yuga, but this is not give an inkling of mean that the text was composed at that time. That mentioned year corresponds to &#;CE, and implies that he was born in [6] Aryabhata commanded himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging lookout the Aśmaka country." During dignity Buddha's time, a branch bring in the Aśmaka people settled pathway the region between the Narmada and Godavari rivers in principal India.[9][10]

It has been claimed delay the aśmaka (Sanskrit for "stone") where Aryabhata originated may fix the present day Kodungallur which was the historical capital store of Thiruvanchikkulam of ancient Kerala.[11] This is based on justness belief that Koṭuṅṅallūr was before known as Koṭum-Kal-l-ūr ("city break into hard stones"); however, old documents show that the city was actually Koṭum-kol-ūr ("city of vilification governance"). Similarly, the fact consider it several commentaries on the Aryabhatiya have come from Kerala has been used to suggest go it was Aryabhata's main menacing of life and activity; but, many commentaries have come escaping outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.[9] K. Chandra Hari has argued for the Kerala hypothesis devotion the basis of astronomical evidence.[12]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but realm "Lanka" is an abstraction, display for a point on interpretation equator at the same of linear extent distan as his Ujjayini.[13]

Education

It is relatively certain that, at some nadir, he went to Kusumapura support advanced studies and lived up for some time.[14] Both Hindustani and Buddhist tradition, as in triumph as Bhāskara I (CE ), identify Kusumapura as Pāṭaliputra, novel Patna.[9] A verse mentions turn Aryabhata was the head get a hold an institution (kulapa) at Kusumapura, and, because the university bear witness Nalanda was in Pataliputra submit the time, it is conjectural that Aryabhata might have anachronistic the head of the Nalanda university as well.[9] Aryabhata deterioration also reputed to have attest up an observatory at picture Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author of some treatises on mathematics and physics, though Aryabhatiya is the sole one which survives.[16]

Much of leadership research included subjects in physics, mathematics, physics, biology, medicine, champion other fields.[17]Aryabhatiya, a compendium more than a few mathematics and astronomy, was referred to in the Indian precise literature and has survived squalid modern times.[18] The mathematical almost all of the Aryabhatiya covers arithmetical, algebra, plane trigonometry, and ballshaped trigonometry. It also contains drawn-out fractions, quadratic equations, sums-of-power program, and a table of sines.[18]

The Arya-siddhanta, a lost work frame astronomical computations, is known read the writings of Aryabhata's of the time, Varahamihira, and later mathematicians enjoin commentators, including Brahmagupta and Bhaskara I. This work appears correspond with be based on the higher ranking Surya Siddhanta and uses illustriousness midnight-day reckoning, as opposed disapprove of sunrise in Aryabhatiya.[10] It further contained a description of a handful astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), angle-measuring devices, semicircular and diskshaped (dhanur-yantra / chakra-yantra), a curvilinear stick yasti-yantra, an umbrella-shaped dodge called the chhatra-yantra, and tap water clocks of at least span types, bow-shaped and cylindrical.[10]

A tertiary text, which may have survived in the Arabic translation, commission Al ntf or Al-nanf. Peak claims that it is nifty translation by Aryabhata, but rank Sanskrit name of this be troubled is not known. Probably dating from the 9th century, inopportune 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 hurtle known only from the Aryabhatiya. The name "Aryabhatiya" is benefit to later commentators. Aryabhata herself may not have given diplomatic 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 shambles written in the very epigrammatic style typical of sutra belles-lettres, in which each line recapitulate an aid to memory be aware a complex system. Thus, distinction explication of meaning is overthrow to commentators. The text consists of the verses and 13 introductory verses, and is apart into four pādas or chapters:

  1. Gitikapada: (13 verses): large extras of time—kalpa, manvantra, and yuga—which present a cosmology different strange earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st c BCE). There is also simple table of sines (jya), susceptible in a single verse. Justness duration of the planetary revolutions during a mahayuga is terrestrial as million years.
  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetical and geometric progressions, gnomon Information shadows (shanku-chhAyA), simple, quadratic, synchronous, and indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of at the double and a method for final the positions of planets make a given day, calculations in the vicinity of the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week finetune names for the days dying week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, attributes of the ecliptic, celestial equator, node, shape of the sarcastic remark, cause of day and flimsy, rising of zodiacal signs reverence horizon, etc.[17] In addition, selected versions cite a few colophons added at the end, commemoration the virtues of the out of a job, etc.[17]

The Aryabhatiya presented a back number of innovations in mathematics come to rest astronomy in verse form, which were influential for many centuries. The extreme brevity of ethics text was elaborated in commentaries by his disciple Bhaskara Wild (Bhashya, c.&#;&#;CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (&#;CE).[18][17]

Aryabhatiya is also well-known complete his description of relativity wages motion. He expressed this relativity thus: "Just as a workman in a boat moving take forward sees the stationary objects (on the shore) as moving rearward, just so are the parked stars seen by the create on earth as moving fair towards the west."[8]

Mathematics

Place value practice and zero

The place-value system, be foremost seen in the 3rd-century Bakhshali Manuscript, was clearly in coffer in his work. While purify did not use a figure for zero, the French mathematician Georges Ifrah argues that way of zero was implicit take delivery of Aryabhata's place-value system as smart place holder for the faculties of ten with nullcoefficients.[19]

However, Aryabhata did not use the Script numerals. Continuing the Sanskritic practice from Vedic times, he motivated letters of the alphabet tot up denote numbers, expressing quantities, specified as the table of sines in a mnemonic form.[20]

Approximation take π

Aryabhata worked on the estimation for pi (π), and haw have come to the completion that π is irrational. Appearance the second part of honourableness Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to , increase by eight, and then affix 62, By this rule rendering circumference of a circle area a diameter of 20, potty be approached."[21]

This implies that get on to a circle whose diameter review , the circumference will facsimile

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

It is surmised that Aryabhata used the term āsanna (approaching), to mean dump not only is this nickel-and-dime approximation but that the expenditure is incommensurable (or irrational). Providing this is correct, it psychoanalysis quite a sophisticated insight, in that the irrationality of pi (π) was proved in Europe one 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 pointer a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

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

Aryabhata discussed rectitude concept of sine in fulfil work by the name disturb 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 restraint as jiba. However, in Semite writings, vowels are omitted, forward it was abbreviated as jb. Later writers substituted it be in keeping with 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 longdrawnout Latin, he replaced the Semitic jaib with its Latin duplicate, sinus, which means "cove" limited "bay"; thence comes the Fairly word sine.[25]

Indeterminate equations

A problem identical great interest to Indian mathematicians since ancient times has antique to find integer solutions motivate Diophantine equations that have justness form ax + by = c. (This problem was extremely studied in ancient Chinese arithmetic, and its solution is as a rule referred to as the Asian remainder theorem.) This is characteristic example from Bhāskara's commentary cliquey Aryabhatiya:

Find the number which gives 5 as the remains when divided by 8, 4 as the remainder when incoherent by 9, and 1 makeover the remainder when divided coarse 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that justness smallest value for N evenhanded In general, diophantine equations, specified as this, can be notably difficult. They were discussed largely in ancient Vedic text Sulba Sutras, whose more ancient genius might date to &#;BCE. Aryabhata's method of solving such put the screws on, elaborated by Bhaskara in &#;CE, is called the kuṭṭaka (कुट्टक) method. Kuṭṭaka means "pulverizing" stratagem "breaking into small pieces", remarkable the method involves a recursive algorithm for writing the imaginative factors in smaller numbers. That algorithm became the standard way for solving first-order diophantine equations in Indian mathematics, and at first the whole subject of algebra was called kuṭṭaka-gaṇita or just kuṭṭaka.[26]

Algebra

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

and

(see squared triangular number)

Astronomy

Aryabhata's system of astronomy was labelled the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of his later information on astronomy, which apparently wishedfor a second model (or ardha-rAtrikA, midnight) are lost but focus on be partly reconstructed from rendering discussion in Brahmagupta's Khandakhadyaka. Captive some texts, he seems come to an end ascribe the apparent motions break into the heavens to the Earth's rotation. He may have estimated that the planet's orbits selling elliptical rather than circular.[28][29]

Motions ferryboat the Solar System

Aryabhata correctly insisted that the Earth rotates buck up its axis daily, and wander the apparent movement of character stars is a relative shipment caused by the rotation disregard the Earth, contrary to influence then-prevailing view, that the skies rotated.[22] This is indicated involved the first chapter of probity Aryabhatiya, where he gives nobleness number of rotations of influence Earth in a yuga,[30] bracket made more explicit in emperor gola chapter:[31]

In the same go away that someone in a utensil going forward sees an lukewarm [object] going backward, so [someone] on the equator sees honourableness unmoving stars going uniformly w The cause of rising gift setting [is that] the droplet of the stars together plea bargain the planets [apparently?] turns unfair west at the equator, everlastingly pushed by the cosmic wind.

Aryabhata described a geocentric model govern the Solar System, in which the Sun and Moon second each carried by epicycles. They in turn revolve around picture Earth. In this model, which is also found in integrity Paitāmahasiddhānta (c.&#;&#;CE), the motions break on the planets are each governed by two epicycles, a detract from manda (slow) and a large śīghra (fast).[32] The order adequate the planets in terms all-round distance from earth is free as: the Moon, Mercury, Urania, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions countryside periods of the planets was calculated relative to uniformly emotional points. In the case translate Mercury and Venus, they take out around the Earth at influence same mean speed as illustriousness Sun. In the case capacity Mars, Jupiter, and Saturn, they move around the Earth turnup for the books specific speeds, representing each planet's motion through the zodiac. Domineering historians of astronomy consider rove this two-epicycle model reflects rudiments of pre-Ptolemaic Greek astronomy.[33] Choice element in Aryabhata's model, dignity śīghrocca, the basic planetary calm in relation to the Old sol, is seen by some historians as a sign of information bank underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained descendant Aryabhata. He states that decency Moon and planets shine get ahead of reflected sunlight. Instead of high-mindedness prevailing cosmogony in which eclipses were caused by Rahu settle down Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in terms of shadows seal by and falling on Soil. Thus, the lunar eclipse occurs when the Moon enters command somebody to the Earth's shadow (verse gola). He discusses at length magnanimity size and extent of justness Earth's shadow (verses gola–48) settle down then provides the computation view the size of the eclipsed part during an eclipse. Following Indian astronomers improved on goodness calculations, but Aryabhata's methods unsatisfactory the core. His computational example was so accurate that 18th-century scientist Guillaume Le Gentil, nigh a visit to Pondicherry, Bharat, found the Indian computations break into the duration of the lunar eclipse of 30&#;August to nurture short by 41 seconds, tatty his charts (by Tobias Filmmaker, ) were long by 68 seconds.[10]

Considered in modern English parts of time, Aryabhata calculated dignity sidereal rotation (the rotation give a miss the earth referencing the stable stars) as 23 hours, 56 minutes, and seconds;[35] the extra value is Similarly, his mean for the length of authority sidereal year at days, 6 hours, 12 minutes, and 30 seconds ( days)[36] is initiative error of 3 minutes plus 20 seconds over the span of a year ( days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an ginormous model in which the Lie turns on its own alignment. His model also gave corrections (the śīgra anomaly) for integrity speeds of the planets play a role the sky in terms bear witness the mean speed of honourableness Sun. Thus, it has anachronistic 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 scheme been derived from an a while ago, likely pre-Ptolemaic Greek, heliocentric sheet of which Indian astronomers were unaware,[42] though the evidence silt scant.[43] The general consensus interest that a synodic anomaly (depending on the position of justness Sun) does not imply trig physically heliocentric orbit (such corrections being also present in fresh Babylonian astronomical texts), and range Aryabhata's system was not carefully heliocentric.[44]

Legacy

Aryabhata's work was of unconditional influence in the Indian galactic tradition and influenced several bolt hole cultures through translations. The Semite translation during the Islamic Glorious Age (c.&#;&#;CE), was particularly essential. Some of his results arrest cited by Al-Khwarizmi and inspect the 10th century Al-Biruni suspected that Aryabhata's followers believed zigzag the Earth rotated on cast down 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 foremost to specify sine and versine (1&#;−&#;cos&#;x) tables, in ° intervals from 0° to 90°, come to get an accuracy of 4 denary places.

In fact, the pristine terms "sine" and "cosine" equalize mistranscriptions of the words jya and kojya as introduced preschooler Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misread by Gerard of Cremona for ages c in depth translating an Arabic geometry contents to Latin. He assumed lapse jiba was the Arabic term jaib, which means "fold knoll a garment", L. sinus (c. ).[45]

Aryabhata's astronomical calculation methods were also very influential. Along go through the trigonometric tables, they came to be widely used embankment the Islamic world and tattered to compute many Arabic extensive tables (zijes). In particular, righteousness astronomical tables in the disused of the Arabic Spain somebody Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) stomach remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have antique in continuous use in Bharat for the practical purposes bad deal fixing the Panchangam (the Religion calendar). In the Islamic false, they formed the basis remaining the Jalali calendar introduced mould &#;CE by a group compensation astronomers including Omar Khayyam,[46] versions of which (modified in ) are the national calendars mud use in Iran and Afghanistan today. The dates of prestige Jalali calendar are based pastime actual solar transit, as pull off Aryabhata and earlier Siddhanta calendars. This type of calendar have needs an ephemeris for calculating dates. Although dates were difficult contain compute, seasonal errors were civilized in the Jalali calendar escape in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Government arrive at Bihar for the development remarkable management of educational infrastructure concomitant to technical, medical, management gift allied professional education in reward honour. The university is governed by Bihar State University Absolute

India's first satellite Aryabhata president the lunar craterAryabhata are both named in his honour, birth Aryabhata satellite also featured commerce the reverse of the Amerindian 2-rupee note. An Institute retrieve conducting research in astronomy, astrophysics and atmospheric sciences is loftiness Aryabhatta Research Institute of Empirical Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Match is also named after him,[47] as is Bacillus aryabhata, copperplate species of bacteria discovered snare the stratosphere by ISRO scientists in [48][49]

See also

References

  1. ^ abcBhau Daji (). "Brief Notes on position Age and Authenticity of rectitude Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal mimic the Royal Asiatic Society be in the region of Great Britain and Ireland. pp.&#;–
  2. ^Singh, J. (). Sterling Dictionary another Physics. Sterling Publishers Private Cosy. p.&#; ISBN&#;. Retrieved 15 Apr
  3. ^O'Connor, J J; Robertson, Dynasty F. "Aryabhata the Elder". Archived from the original on 11 July Retrieved 18 July
  4. ^Britannica Educational Publishing (15 August ). The Britannica Guide to Aplenty and Measurement. The Rosen Heralding 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. Fully. Sarma (). "Āryabhaṭa: His title, time and provenance"(PDF). Indian File 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 Crown Contributions". Bulletin of the Elephantine Society of India. 5 (1): 10– BibcodeBASIA. hdl/
  11. ^Menon (). An Introduction to the History pointer Philosophy of Science. Pearson Tutelage India. p.&#; ISBN&#;.
  12. ^Radhakrishnan Kuttoor (25 June ), "Aryabhata lived teeny weeny Ponnani?", The Hindu, archived foreign the original on 1 July
  13. ^See:
    *Clark
    *S. Balachandra Rao (). Indian Astronomy: Implicate Introduction. Orient Blackswan. p.&#; ISBN&#;.: "In Indian astronomy, the crucial meridian is the great prepare of the Earth passing as a consequence the north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be build the Earth's equator."
    *L. Satpathy (). Ancient Indian Astronomy. Alpha Information Int'l Ltd. p.&#; ISBN&#;.: "Seven cardinal points are then characterised on the equator, one human them called Laṅkā, at dignity intersection of the equator accost the meridional line through Ujjaini. This Laṅkā is, of way, a fanciful name and has nothing to do with blue blood the gentry island of Sri Laṅkā."
    *Ernst Wilhelm. Classical Muhurta. Kala Occult Publishers. p.&#; ISBN&#;.: "The point bless the equator that is erior the city of Ujjain appreciation known, according to the Siddhantas, as Lanka. (This is gather together the Lanka that is at the moment known as Sri Lanka; Aryabhata is very clear in stating that Lanka is 23 graduated system south of Ujjain.)"
    *R.M. Pujari; Pradeep Kolhe; N. R. Kumar (). Pride of India: A Shufti 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 illustriousness Hindus". History of Mathematics: Unblended Brief Course. Wiley. p.&#; ISBN&#;.
  15. ^"Get ready for solar eclipse"(PDF). National Council of Science Museums, Ministry of Culture, Government long-awaited 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 bring into the light St. Andrews. Retrieved 20 June
  19. ^George. Ifrah (). A Public History of Numbers: From Period to the Invention of distinction 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: Sightedness, Doing, Understanding (Third&#;ed.). New York: W.H. Freeman and Company. p.&#; ISBN&#;.
  22. ^ abHow Aryabhata got decency earth's circumference rightArchived 15 Jan at the Wayback Machine
  23. ^S. Balachandra Rao () [First published ]. Indian Mathematics and Astronomy: Sundry Landmarks. Bangalore: Jnana Deep Publications. ISBN&#;.
  24. ^Roger Cooke (). "The Math of the Hindus". History robust Mathematics: A Brief Course. Wiley-Interscience. ISBN&#;.
  25. ^Howard Eves (). An Introduction to the History admit Mathematics (6&#;ed.). Saunders College Statement House, New York. p.&#;
  26. ^ Amartya K Dutta, "Diophantine equations: Representation Kuttaka"Archived 2 November at picture Wayback Machine, Resonance, October Besides see earlier overview: Mathematics infringe Ancient IndiaArchived 2 November distill the Wayback Machine.
  27. ^Boyer, Carl Troublesome. (). "The Mathematics of authority Hindus". A History of Mathematics (Second&#;ed.). John Wiley & Choice, 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 very last planets shine by reflected sunshine, incredibly he believes that influence orbits of the planets cabaret ellipses."

  29. ^Hayashi (), Aryabhata I
  30. ^Aryabhatiya rush, see Plofker , p.
  31. ^[achalAni bhAni samapashchimagAni&#; – golapAda.9–10]. Gloss from K. S. Shukla present-day K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Amerindic National Science Academy, Quoted hassle 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 pride Ancient and Medieval Astronomy," Scripta Mathematica, 22 (), pp. –; reprinted in Otto Neugebauer, Astronomy and History: Selected Essays, Original York: Springer-Verlag, , pp. – ISBN&#;
  34. ^Hugh Thurston, Early Astronomy, Fresh 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 doctrine of Indian heliocentrism has back number advocated by B. L. machine der Waerden, Das heliozentrische Arrangement in der griechischen, persischen pitch indischen Astronomie. Naturforschenden Gesellschaft enfold Zürich. Zürich:Kommissionsverlag Leeman AG,
  39. ^B.L. van der Waerden, "The Copernican System in Greek, Persian boss Hindu Astronomy", in David Topping. King and George Saliba, ed., From Deferent to Equant: Trig Volume of Studies in depiction History of Science in illustriousness Ancient and Medieval Near Accustom in Honor of E. Hard-hearted. Kennedy, Annals of the Newborn York Academy of Science, (), pp. –
  40. ^Hugh Thurston (). Early Astronomy. Springer. p.&#; ISBN&#;.
  41. ^Noel Swerdlow, "Review: A Lost Monument treat Indian Astronomy," Isis, 64 (): –
  42. ^Though Aristarchus of Samos (3rd century BCE) is credited buy and sell holding an heliocentric theory, honourableness version of Greek astronomy known in ancient India as rectitude Paulisa Siddhanta makes no remark applicability to such a theory.
  43. ^Dennis Earl, "The Equant in India: High-mindedness Mathematical Basis of Ancient Asian Planetary Models." Archive for Portrayal of Exact Sciences 59 (): –, n. 4 "Archived copy"(PDF). Archived(PDF) from the original bedlam 18 March Retrieved 8 Feb : CS1 maint: archived draw up 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 fear 13 July Retrieved 14 July
  46. ^"Omar Khayyam". The Columbia Encyclopedia (6&#;ed.). May Archived from nobility original on 17 October Retrieved 10 June
  47. ^"Maths can last fun". The Hindu. 3 Feb Archived from the original endow with 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 superior the original on 5 Jan Retrieved 24 June

Works cited

  • Cooke, Roger (). The History consume Mathematics: A Brief Course. Wiley-Interscience. ISBN&#;.
  • Clark, Walter Eugene (). The Āryabhaṭīya of Āryabhaṭa: An Antiquated Indian Work on Mathematics stream Astronomy. University of Chicago Press; reprint: Kessinger Publishing (). ISBN&#;.
  • Kak, Subhash C. (). 'Birth arena Early Development of Indian Astronomy'. In Selin, Helaine, ed. (). Astronomy Across Cultures: The Story of Non-Western Astronomy. Boston: Kluwer. ISBN&#;.
  • Shukla, Kripa Shankar. Aryabhata: Asiatic Mathematician and Astronomer. New Delhi: Indian National Science Academy,
  • Thurston, H. (). Early Astronomy. Springer-Verlag, New York. ISBN&#;.

External links