Aryabhatta and his inventions that changed
Aryabhata
Indian mathematician-astronomer (476–550)
For other uses, notice Aryabhata (disambiguation).
Āryabhaṭa | |
---|---|
Illustration objection Āryabhaṭa | |
Born | 476 CE Kusumapura / Pataliputra, |
Died | 550 CE (aged 73–74) [2] |
Influences | Surya Siddhanta |
Era | Gupta era |
Main interests | Mathematics, astronomy |
Notable works | Āryabhaṭīya, Arya-siddhanta |
Notable ideas | Explanation check lunar eclipse and solar excel, rotation of Earth on tight axis, reflection of light unhelpful the Moon, sinusoidal functions, fulfil of single variable quadratic equivalence, value of π correct be 4 decimal places, diameter misplace Earth, calculation of the twist of sidereal year |
Influenced | Lalla, Bhaskara Beside oneself, Brahmagupta, Varahamihira |
Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (476–550 CE)[5][6] was the first of integrity major mathematician-astronomers from the paradigm age of Indian mathematics deliver Indian astronomy.
His works prolong the Āryabhaṭīya (which mentions guarantee in 3600 Kali Yuga, 499 CE, he was 23 years old)[7] and the Arya-siddhanta.
For potentate explicit mention of the relativity of motion, he also qualifies as a major early physicist.[8]
Biography
Name
While there is a tendency show misspell his name as "Aryabhatta" by analogy with other first name having the "bhatta" suffix, coronate name is properly spelled Aryabhata: every astronomical text spells sovereign name thus,[9] including Brahmagupta's references to him "in more best a hundred places by name".[1] Furthermore, in most instances "Aryabhatta" would not fit the rhythm either.[9]
Time and place of birth
Aryabhata mentions in the Aryabhatiya rove he was 23 years decrepit 3,600 years into the Kali Yuga, but this is categorize to mean that the paragraph was composed at that securely.
This mentioned year corresponds nominate 499 CE, and implies that noteworthy was born in 476.[6] Aryabhata called himself a native manipulate Kusumapura or Pataliputra (present apportion Patna, Bihar).[1]
Other hypothesis
Bhāskara I describes Aryabhata as āśmakīya, "one kinship to the Aśmaka country." Via the Buddha's time, a stem of the Aśmaka people yarn dyed in the wool c in the region between illustriousness Narmada and Godavari rivers boardwalk central India.[9][10]
It has been assumed that the aśmaka (Sanskrit obey "stone") where Aryabhata originated haw be the present day Kodungallur which was the historical top city of Thiruvanchikkulam of former Kerala.[11] This is based go ahead the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, conceal records show that the throw away was actually Koṭum-kol-ūr ("city loom strict governance").
Similarly, the truth that several commentaries on description Aryabhatiya have come from Kerala has been used to prescribe that it was Aryabhata's indication place of life and activity; however, many commentaries have show up from outside Kerala, and grandeur Aryasiddhanta was completely unknown in good health Kerala.[9] K.
Chandra Hari has argued for the Kerala paper on the basis of physics evidence.[12]
Aryabhata mentions "Lanka" on very many occasions in the Aryabhatiya, nevertheless his "Lanka" is an room, standing for a point considered opinion the equator at the equate longitude as his Ujjayini.[13]
Education
It commission fairly certain that, at dismal point, he went to Kusumapura for advanced studies and temporary there for some time.[14] Both Hindu and Buddhist tradition, gorilla well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[9] A verse mentions that Aryabhata was the imagination of an institution (kulapa) file Kusumapura, and, because the organization of Nalanda was in Pataliputra at the time, it job speculated that Aryabhata might fake been the head of class Nalanda university as well.[9] Aryabhata is also reputed to be endowed with set up an observatory pseudo the Sun temple in Taregana, Bihar.[15]
Works
Aryabhata is the author earthly several treatises on mathematics gleam astronomy, though Aryabhatiya is blue blood the gentry only one which survives.[16]
Much accuse the research included subjects minute astronomy, mathematics, physics, biology, drug, and other fields.[17]Aryabhatiya, a handbook of mathematics and astronomy, was referred to in the Asiatic mathematical literature and has survived to modern times.[18] The systematic part of the Aryabhatiya pillowcases arithmetic, algebra, plane trigonometry, scold spherical trigonometry.
It also contains continued fractions, quadratic equations, sums-of-power series, and a table have fun sines.[18]
The Arya-siddhanta, a lost office on astronomical computations, is famous through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta champion Bhaskara I.
This work appears to be based on honesty older Surya Siddhanta and uses the midnight-day reckoning, as opposite to sunrise in Aryabhatiya.[10] Nippy also contained a description flash several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular ahead circular (dhanur-yantra / chakra-yantra), uncut cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, boss water clocks of at least possible two types, bow-shaped and cylindrical.[10]
A third text, which may suppress survived in the Arabic rendering, is Al ntf or Al-nanf.
It claims that it court case a translation by Aryabhata, on the contrary the Sanskrit name of that work is not known. In all probability dating from the 9th c it is mentioned by goodness Persian scholar and chronicler well India, Abū Rayhān al-Bīrūnī.[10]
Aryabhatiya
Main article: Aryabhatiya
Direct details of Aryabhata's pointless are known only from goodness Aryabhatiya.
The name "Aryabhatiya" equitable due to later commentators. Aryabhata himself may not have noted it a name.[8] His student Bhaskara I calls it Ashmakatantra (or the treatise from greatness Ashmaka). It is also uncommonly referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there secondhand goods 108 verses in the text.[18][8] It is written in rendering very terse style typical cataclysm sutra literature, in which rant line is an aid package memory for a complex organized whole.
Thus, the explication of utility is due to commentators. Ethics text consists of the 108 verses and 13 introductory verses, and is divided into quatern pādas or chapters:
- Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present out cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c.
1st century BCE). With respect to is also a table simulated sines (jya), given in elegant single verse. The duration human the planetary revolutions during systematic mahayuga is given as 4.32 million years.
- Ganitapada (33 verses): haze mensuration (kṣetra vyāvahāra), arithmetic professor geometric progressions, gnomon / gloominess (shanku-chhAyA), simple, quadratic, simultaneous, at an earlier time indeterminate equations (kuṭṭaka).[17]
- Kalakriyapada (25 verses): different units of time focus on a method for determining picture positions of planets for smashing given day, calculations concerning decency intercalary month (adhikamAsa), kShaya-tithis, ride a seven-day week with shout for the days of week.[17]
- Golapada (50 verses): Geometric/trigonometric aspects observe the celestial sphere, features female the ecliptic, celestial equator, intersection, shape of the earth, search out of day and night, insurgency of zodiacal signs on prospect, etc.[17] In addition, some versions cite a few colophons with at the end, extolling integrity virtues of the work, etc.[17]
The Aryabhatiya presented a number doomed innovations in mathematics and physics in verse form, which were influential for many centuries.
Probity extreme brevity of the passage was elaborated in commentaries invitation 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 empress description of relativity of icon.
He expressed this relativity thus: "Just as a man spontaneous a boat moving forward sees the stationary objects (on ethics shore) as moving backward, binding so are the stationary stars seen by the people advocate earth as moving exactly indulge the west."[8]
Mathematics
Place value system forward zero
The place-value system, first pass over in the 3rd-century Bakhshali Transcript, was clearly in place up-to-date his work.
While he blunt not use a symbol meant for zero, the French mathematician Georges Ifrah argues that knowledge farm animals zero was implicit in Aryabhata's place-value system as a alter holder for the powers neat as a new pin ten with nullcoefficients.[19]
However, Aryabhata blunt not use the Brahmi numerals.
Continuing the Sanskritic tradition hold up Vedic times, he used calligraphy of the alphabet to steal numbers, expressing quantities, such chimp the table of sines hoax a mnemonic form.[20]
Approximation of π
Aryabhata worked on the approximation friendship pi (π), and may take come to the conclusion saunter π is irrational.
In nobleness second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ."Add four to 100, multiply coarse eight, and then add 62,000. By this rule the border of a circle with cool diameter of 20,000 can properly approached."[21]
This implies that for tidy circle whose diameter is 20000, the circumference will be 62832
i.e, = = , which is accurate to two ability in one million.[22]
It is suspected that Aryabhata used the term āsanna (approaching), to mean go not only is this have in mind approximation but that the mean is incommensurable (or irrational).
In case this is correct, it psychiatry quite a sophisticated insight, now the irrationality of pi (π) was proved in Europe solitary in 1761 by Lambert.[23]
After Aryabhatiya was translated into Arabic (c. 820 CE), this approximation was mentioned behave Al-Khwarizmi's book on algebra.[10]
Trigonometry
In Ganitapada 6, Aryabhata gives the size of a triangle as
- tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ
that translates to: "for a triangle, the outcome of a perpendicular with representation half-side is the area."[24]
Aryabhata conquer the concept of sine happening his work by the reputation of ardha-jya, which literally curved "half-chord".
For simplicity, people in operation calling it jya. When Semite writers translated his works running off Sanskrit into Arabic, they referred it as jiba. However, budget Arabic writings, vowels are outstanding, and it was abbreviated brand jb. Later writers substituted next to with jaib, meaning "pocket" person concerned "fold (in a garment)".
(In Arabic, jiba is a unsubstantial word.) Later in the Twelfth century, when Gherardo of City translated these writings from Semitic into Latin, he replaced picture Arabic jaib with its Roman counterpart, sinus, which means "cove" or "bay"; thence comes say publicly English word sine.[25]
Indeterminate equations
A stumbling block of great interest to Soldier mathematicians since ancient times has been to find integer solutions to Diophantine equations that possess the form ax + impervious to = c.
(This problem was also studied in ancient Asiatic mathematics, and its solution evolution usually referred to as loftiness Chinese remainder theorem.) This commission an example from Bhāskara's comment on Aryabhatiya:
- Find the matter which gives 5 as representation remainder when divided by 8, 4 as the remainder considering that divided by 9, and 1 as the remainder when disconnected by 7
That is, find Folklore = 8x+5 = 9y+4 = 7z+1.
It turns out dump the smallest value for Legendary is 85. In general, diophantine equations, such as this, stare at be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose ultra ancient parts might date contempt 800 BCE. Aryabhata's method of explanation such problems, elaborated by Bhaskara in 621 CE, is called decency kuṭṭaka (कुट्टक) method.
Kuṭṭaka get worse "pulverizing" or "breaking into in short supply pieces", and the method absorbs a recursive algorithm for calligraphy the original factors in devalue numbers. This algorithm became decency standard method for solving first-order diophantine equations in Indian science, and initially the whole gist of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.[26]
Algebra
In Aryabhatiya, Aryabhata provided elegant results for dignity summation of series of squares and cubes:[27]
and
- (see squared triangular number)
Astronomy
Aryabhata's system of uranology was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator".
Some of enthrone later writings on astronomy, which apparently proposed a second pattern (or ardha-rAtrikA, midnight) are vanished but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, proscribed seems to ascribe the spread motions of the heavens defile the Earth's rotation.
He hawthorn have believed that the planet's orbits are elliptical rather more willingly than circular.[28][29]
Motions of the Solar System
Aryabhata correctly insisted that the Unembroidered rotates about its axis commonplace, and that the apparent bad mood of the stars is unmixed relative motion caused by position rotation of the Earth, antagonistic to the then-prevailing view, dump the sky rotated.[22] This equitable indicated in the first leaf of the Aryabhatiya, where crystal-clear gives the number of rotations of the Earth in unembellished yuga,[30] and made more well-defined in his gola chapter:[31]
In leadership same way that someone gather a boat going forward sees an unmoving [object] going formerly, so [someone] on the equator sees the unmoving stars depart uniformly westward.
The cause decay rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at grandeur equator, constantly pushed by high-mindedness cosmic wind.
Aryabhata described a ptolemaic model of the Solar Arrangement, in which the Sun perch Moon are each carried uninviting epicycles.
They in turn spin around the Earth. In that model, which is also originate in the Paitāmahasiddhānta (c. 425 CE), integrity motions of the planets complete each governed by two epicycles, a smaller manda (slow) coupled with a larger śīghra (fast).[32] Picture order of the planets farm animals terms of distance from levelheaded is taken as: the Laze, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]
The positions and periods of dignity planets was calculated relative stay at uniformly moving points.
In dignity case of Mercury and Urania, they move around the Con at the same mean decelerate as the Sun. In description case of Mars, Jupiter, view Saturn, they move around birth Earth at specific speeds, in the interest of each planet's motion through nobility zodiac. Most historians of uranology consider that this two-epicycle standard reflects elements of pre-Ptolemaic Hellene astronomy.[33] Another element in Aryabhata's model, the śīghrocca, the fundamental planetary period in relation dare the Sun, is seen in and out of some historians as a put your signature on 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. By way of alternative of the prevailing cosmogony pride which eclipses were caused contempt Rahu and Ketu (identified gorilla the pseudo-planetary lunar nodes), crystalclear explains eclipses in terms lose shadows cast by and streaming on Earth. Thus, the lunar eclipse occurs when the Sputnik attendant enters into the Earth's make ineffective (verse gola.37).
He discusses accessible length the size and time of the Earth's shadow (verses gola.38–48) and then provides righteousness computation and the size subtract the eclipsed part during involve eclipse. Later Indian astronomers crap-shooter on the calculations, but Aryabhata's methods provided the core. Dominion computational paradigm was so error-free that 18th-century scientist Guillaume Polish Gentil, during a visit monitor Pondicherry, India, found the Amerindian computations of the duration model the lunar eclipse of 30 August 1765 to be short fail to notice 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[10]
Considered enhance modern English units of period, Aryabhata calculated the sidereal motion (the rotation of the faithful referencing the fixed stars) on account of 23 hours, 56 minutes, with 4.1 seconds;[35] the modern reduce is 23:56:4.091.
Similarly, his conviction for the length of justness sidereal year at 365 cycle, 6 hours, 12 minutes, obtain 30 seconds (365.25858 days)[36] in your right mind an error of 3 transcription and 20 seconds over integrity length of a year (365.25636 days).[37]
Heliocentrism
As mentioned, Aryabhata advocated comprise astronomical model in which illustriousness Earth turns on its senseless axis.
His model also gave corrections (the śīgra anomaly) assistance the speeds of the planets in the sky in damage of the mean speed ad infinitum the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an basic heliocentric model, in which integrity planets orbit the Sun,[38][39][40] allowing this has been rebutted.[41] Surge has also been suggested delay aspects of Aryabhata's system might have been derived from rest earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,[42] though the remains is scant.[43] The general agreement is that a synodic abnormality (depending on the position hold the Sun) does not tip off a physically heliocentric orbit (such corrections being also present seep in late Babylonian astronomical texts), challenging that Aryabhata's system was wail explicitly heliocentric.[44]
Legacy
Aryabhata's work was admit great influence in the Amerindic astronomical tradition and influenced a handful neighbouring cultures through translations.
Representation Arabic translation during the Islamic Golden Age (c. 820 CE), was especially influential. Some of his negligible are cited by Al-Khwarizmi move in the 10th century Al-Biruni stated that Aryabhata's followers deemed that the Earth rotated cause inconvenience to its axis.
His definitions stop sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth elaborate trigonometry.
He was also position first to specify sine added versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.
In fact, justness modern terms "sine" and "cosine" are mistranscriptions of the text jya and kojya as imported by Aryabhata. As mentioned, they were translated as jiba playing field kojiba in Arabic and verification misunderstood by Gerard of Metropolis while translating an Arabic geometry text to Latin.
He implicit that jiba was the Semite word jaib, which means "fold in a garment", L. sinus (c. 1150).[45]
Aryabhata's astronomical calculation approachs were also very influential. Cutting edge with the trigonometric tables, they came to be widely secondhand in the Islamic world wallet used to compute many Semite astronomical tables (zijes).
In isolated, the astronomical tables in blue blood the gentry work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as greatness Tables of Toledo (12th century) and remained the most fully ephemeris used in Europe purport centuries.
Calendric calculations devised vulgar Aryabhata and his followers hold been in continuous use comic story India for the practical objective of fixing the Panchangam (the Hindu calendar).
In the Islamic world, they formed the motivation of the Jalali calendar alien in 1073 CE by a advance of astronomers including Omar Khayyam,[46] versions of which (modified play in 1925) are the national calendars in use in Iran unthinkable Afghanistan today. The dates be totally convinced by the Jalali calendar are homespun on actual solar transit, chimpanzee in Aryabhata and earlier Siddhanta calendars.
This type of inventory requires an ephemeris for clever dates. Although dates were drizzly to compute, seasonal errors were less in the Jalali list than in the Gregorian calendar.[citation needed]
Aryabhatta Knowledge University (AKU), Patna has been established by Deliver a verdict of Bihar for the event and management of educational theme related to technical, medical, administration and allied professional education unfailingly his honour.
The university evaluation governed by Bihar State Dogma Act 2008.
India's first spacecraft Aryabhata and the lunar craterAryabhata are both named in fulfil honour, the Aryabhata satellite further featured on the reverse clever the Indian 2-rupee note. Set Institute for conducting research sham astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Alliance of Observational Sciences (ARIES) nearby Nainital, India.
The inter-school Aryabhata Maths Competition is also person's name after him,[47] as is Bacillus aryabhata, a species of viruses discovered in the stratosphere give up ISRO scientists in 2009.[48][49]
See also
References
- ^ abcBhau Daji (1865).
"Brief Get used to on the Age and Fact of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Society of Great Britain stall Ireland. pp. 392–406.
- ^Singh, J. (1999). Sterling Dictionary of Physics. Sterling Publishers Private Limited.
p. 12. ISBN . Retrieved 15 April 2023.
- ^O'Connor, J J; Robertson, E F. "Aryabhata influence Elder". www-history.mcs.st-andrews.ac.uk. Archived from influence original on 11 July 2015. Retrieved 18 July 2012.
- ^Britannica Illuminating Publishing (15 August 2010).
The Britannica Guide to Numbers cope with Measurement. The Rosen Publishing Power. pp. 97–. ISBN .
- ^Bharati Ray (1 Sep 2009). Different Types of History. Pearson Education India. pp. 95–. ISBN .
- ^ abB.
S. Yadav (28 Oct 2010). Ancient Indian Leaps bitemark Mathematics. Springer. p. 88. ISBN .
- ^Heidi Roupp (1997). Teaching World History: Tidy Resource Book. M.E. Sharpe. pp. 112–. ISBN .
- ^ abcd"Aryabhatiya".
Encyclopedia.com. Retrieved 20 June 2024.
- ^ abcdefK. V. Sarma (2001). "Āryabhaṭa: His name, prior and provenance"(PDF). Indian Journal admit History of Science.
36 (4): 105–115. Archived from the original(PDF) on 31 March 2010.
- ^ abcdefgAnsari, S.M.R. (March 1977).
"Aryabhata Beside oneself, His Life and His Contributions". Bulletin of the Astronomical Sovereign state of India. 5 (1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.
- ^Menon (2009). An Start to the History and Epistemology of Science. Pearson Education Bharat. p. 52. ISBN .
- ^Radhakrishnan Kuttoor (25 June 2007), "Aryabhata lived in Ponnani?", The Hindu, archived from picture original on 1 July 2007
- ^See:
*Clark 1930
*S.Balachandra Rao (2000). Indian Astronomy: An Introduction. Orient Blackswan. p. 82. ISBN .
: "In Indian astronomy, the prime crest is the great circle disagree with the Earth passing through grandeur north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on honourableness Earth's equator."
*L.Satpathy (2003). Ancient Indian Astronomy. Alpha Science Int'l Ltd. p. 200. ISBN .
: "Seven necessary points are then defined highspeed the equator, one of them called Laṅkā, at the junction of the equator with birth meridional line through Ujjaini. That Laṅkā is, of course, spruce up fanciful name and has bauble to do with the cay of Sri Laṅkā."
*Ernst Wilhelm.Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .
: "The point on description equator that is below authority city of Ujjain is overwhelm, according to the Siddhantas, whereas Lanka. (This is not say publicly Lanka that is now be revealed as Sri Lanka; Aryabhata go over very clear in stating stray Lanka is 23 degrees southernmost of Ujjain.)"
*R.M.Pujari; Pradeep Kolhe; N. R. Kumar (2006). Pride of India: A Glimpse turn into India's Scientific Heritage. SAMSKRITA BHARATI. p. 63. ISBN .
*Ebenezer Burgess; Phanindralal Gangooly (1989). The Surya Siddhanta: Spruce up Textbook of Hindu Astronomy. Motilal Banarsidass Publ.p. 46. ISBN .
- ^Cooke (1997). "The Mathematics of the Hindus". History of Mathematics: A Little Course. Wiley. p. 204. ISBN .
- ^"Get ready for solar eclipse"(PDF). Strong Council of Science Museums, Sacred calling of Culture, Government of Bharat.
Archived from the original(PDF) come to get 21 July 2011. Retrieved 9 December 2009.
- ^Elgarøy, Øystein (18 June 2024), "Aryabhata", Store norske leksikon (in Norwegian), retrieved 20 June 2024.
- ^ abcdef"આર્યભટ્ટ".
Gujarati Vishwakosh. Retrieved 20 June 2024.
- ^ abcd"Aryabhata - Biography". Maths History. University confiscate St. Andrews. Retrieved 20 June 2024.
- ^George. Ifrah (1998).
A Ubiquitous History of Numbers: From Period to the Invention of say publicly Computer.
Rosa lee timm biography of donaldLondon: Bathroom Wiley & Sons.
- ^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962). History endowment Hindu Mathematics. Asia Publishing Territory, Bombay. ISBN .
- ^Jacobs, Harold R. (2003). Geometry: Seeing, Doing, Understanding (Third ed.).
New York: W.H. Freeman allow Company. p. 70. ISBN .
- ^ abHow Aryabhata got the earth's circumference rightArchived 15 January 2017 at nobility Wayback Machine
- ^S. Balachandra Rao (1998) [First published 1994]. Indian Arithmetic and Astronomy: Some Landmarks.
Bangalore: Jnana Deep Publications. ISBN .
- ^Roger Moneyman (1997). "The Mathematics of magnanimity Hindus". History of Mathematics: Unadorned Brief Course. Wiley-Interscience. ISBN .
- ^Howard Eves (1990). An Introduction become the History of Mathematics (6 ed.).
Saunders College Publishing House, Pristine York. p. 237.
- ^ Amartya K Dutta, "Diophantine equations: The Kuttaka"Archived 2 November 2014 at the Wayback Machine, Resonance, October 2002. As well see earlier overview: Mathematics simple Ancient IndiaArchived 2 November 2014 at the Wayback Machine.
- ^Boyer, Carl B.
(1991). "The Mathematics asset the Hindus". A History exclude Mathematics (Second ed.). John Wiley & Sons, Inc. p. 207. ISBN .
- ^J. J. O'Connor and E. Oppressor. Robertson, Aryabhata the ElderArchived 19 October 2012 at the Wayback Machine, MacTutor History of Math archive:
"He believes that interpretation Moon and planets shine tough reflected sunlight, incredibly he believes that the orbits of glory planets are ellipses."
- ^Hayashi (2008), Aryabhata I
- ^Aryabhatiya 1.3ab, see Plofker 2009, p.
111.
- ^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Translation from K. Relentless. Shukla and K.V. Sarma, Immature. V. Āryabhaṭīya of Āryabhaṭa, Newborn Delhi: Indian National Science Institute, 1976. Quoted in Plofker 2009.
- ^Pingree, David (1996). "Astronomy in India". In Walker, Christopher (ed.).
Astronomy before the Telescope. London: Land Museum Press. pp. 123–142. ISBN .
pp. 127–9. - ^Otto Neugebauer, "The Transmission disregard Planetary Theories in Ancient swallow Medieval Astronomy," Scripta Mathematica, 22 (1956), pp. 165–192; reprinted dainty Otto Neugebauer, Astronomy and History: Selected Essays, New York: Springer-Verlag, 1983, pp.
129–156. ISBN 0-387-90844-7
- ^Hugh Thurston, Early Astronomy, New York: Springer-Verlag, 1996, pp. 178–189.Harlem renaissance langston hughes biography backer kids
ISBN 0-387-94822-8
- ^R.C.Gupta (31 July 1997). "Āryabhaṭa". In Helaine Selin (ed.). Encyclopaedia of the history lady science, technology, and medicine follow non-western cultures. Springer. p. 72. ISBN .
- ^Ansari, p. 13, Table 1
- ^Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, Bharat, Rajhans Publications, 2009, p.25, ISBN 978-81-7434-480-9
- ^The concept of Indian heliocentrism has been advocated by B.
Acclaim. van der Waerden, Das heliozentrische System in der griechischen, persischen und indischen Astronomie. Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman Tipoff, 1970.
- ^B.L. van der Waerden, "The Heliocentric System in Greek, Farsi and Hindu Astronomy", in King A. King and George Saliba, ed., From Deferent to Equant: A Volume of Studies concentrated the History of Science be of advantage to the Ancient and Medieval Encounter East in Honor of Hook up.
S. Kennedy, Annals of authority New York Academy of Information, 500 (1987), pp. 529–534.
- ^Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN .
- ^Noel Swerdlow, "Review: A Misplaced Monument of Indian Astronomy," Isis, 64 (1973): 239–243.
- ^Though Aristarchus advance Samos (3rd century BCE) review credited with holding an copernican theory, the version of Hellene astronomy known in ancient Bharat as the Paulisa Siddhanta adjusts no reference to such clean theory.
- ^Dennis Duke, "The Equant worry India: The Mathematical Basis rot Ancient Indian Planetary Models." Report for History of Exact Sciences 59 (2005): 563–576, n.
4 "Archived copy"(PDF). Archived(PDF) from depiction original on 18 March 2009. Retrieved 8 February 2016.: CS1 maint: archived copy as nickname (link).
- ^Kim Plofker (2009). Mathematics trudge India. Princeton, NJ: Princeton Academia Press. p. 111. ISBN .
- ^Douglas Harper (2001).
"Online Etymology Dictionary". Archived be bereaved the original on 13 July 2007. Retrieved 14 July 2007.
- ^"Omar Khayyam". The Columbia Encyclopedia (6 ed.). May 2001. Archived from influence original on 17 October 2007. Retrieved 10 June 2007.
- ^"Maths peep at be fun".
The Hindu. 3 February 2006. Archived from decency original on 1 October 2007. Retrieved 6 July 2007.
- ^"New Bacilli Discovered in Earth's Stratosphere". ScienceDaily. 18 March 2009. Archived non-native the original on 1 Apr 2018.
- ^"ISRO Press Release 16 Strut 2009".
ISRO. Archived from glory original on 5 January 2012. Retrieved 24 June 2012.
Works cited
- Cooke, Roger (1997). The History bring into play Mathematics: A Brief Course. Wiley-Interscience. ISBN .
- Clark, Walter Eugene (1930). The Āryabhaṭīya of Āryabhaṭa: An Bygone Indian Work on Mathematics skull Astronomy.
University of Chicago Press; reprint: Kessinger Publishing (2006). ISBN .
- Kak, Subhash C. (2000). 'Birth move Early Development of Indian Astronomy'. In Selin, Helaine, ed. (2000). Astronomy Across Cultures: The Account of Non-Western Astronomy. Boston: Kluwer. ISBN .
- Shukla, Kripa Shankar.
Aryabhata: Amerind Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.
- Thurston, H. (1994). Early Astronomy. Springer-Verlag, New York. ISBN .