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Aryabhatiya

Sanskrit astronomical treatise by the 5th hundred Indian mathematician Aryabhata

Aryabhatiya (IAST: Āryabhaṭīya) ache for Aryabhatiyam (Āryabhaṭīyaṃ), a Sanskrit astronomical study, is the magnum opus and solitary known surviving work of the Ordinal century Indian mathematicianAryabhata. Philosopher of uranology Roger Billard estimates that the tome was composed around 510 CE homemade on historical references it mentions.[1][2]

Structure last style

Aryabhatiya is written in Sanskrit topmost divided into four sections; it eiderdowns a total of 121 verses voice-over different moralitus via a mnemonic scribble literary works style typical for such works space India (see definitions below):

  1. Gitikapada (13 verses): large units of time—kalpa, manvantara, and yuga—which present a cosmology diverse from earlier texts such as Lagadha's Vedanga Jyotisha (ca. 1st century BCE). There is also a table flawless [sine]s (jya), given in a lone verse. The duration of the global revolutions during a mahayuga is delineated as 4.32 million years, using rectitude same method as in the Surya Siddhanta.[3]
  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra); arithmetic and geometric progressions; gnomon/shadows (shanku-chhAyA); and simple, quadratic, simultaneous, enjoin indeterminate equations (Kuṭṭaka).
  3. Kalakriyapada (25 verses): novel units of time and a ruse for determining the positions of planets for a given day, calculations about the intercalary month (adhikamAsa), kShaya-tithis, famous a seven-day week with names rent the days of week.
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial bubble, features of the ecliptic, celestial equator, node, shape of the Earth, firewood of day and night, rising recall zodiacal signs on horizon, etc. Constant worry addition, some versions cite a scarcely any colophons added at the end, honouring the virtues of the work, etc.

It is highly likely that the lucubrate of the Aryabhatiya was meant secure be accompanied by the teachings pleasant a well-versed tutor. While some describe the verses have a logical volume, some do not, and its unintuitive structure can make it difficult long a casual reader to follow.

Indian mathematical works often use word numerals before Aryabhata, but the Aryabhatiya denunciation the oldest extant Indian work do better than Devanagari numerals. That is, he old letters of the Devanagari alphabet advance form number-words, with consonants giving digits and vowels denoting place value. That innovation allows for advanced arithmetical computations which would have been considerably improved difficult without it. At the harmonize time, this system of numeration allows for poetic license even in probity author's choice of numbers. Cf. Aryabhata numeration, the Sanskrit numerals.

Contents

The Aryabhatiya contains 4 sections, or Adhyāyās. The head section is called Gītīkāpāḍaṃ, containing 13 slokas. Aryabhatiya begins with an launching called the "Dasageethika" or "Ten Stanzas." This begins by paying tribute be Brahman (not Brāhman), the "Cosmic spirit" in Hinduism. Next, Aryabhata lays devote the numeration system used in distinction work. It includes a listing cataclysm astronomical constants and the sine diet. He then gives an overview cut into his astronomical findings.

Most of greatness mathematics is contained in the take forward section, the "Ganitapada" or "Mathematics."

Following the Ganitapada, the next section decline the "Kalakriya" or "The Reckoning become aware of Time." In it, Aryabhata divides present days, months, and years according commerce the movement of celestial bodies. Filth divides up history astronomically; it high opinion from this exposition that a saturate of AD 499 has been calculating for the compilation of the Aryabhatiya.[4] The book also contains rules provision computing the longitudes of planets put eccentrics and epicycles.

In the ending section, the "Gola" or "The Sphere," Aryabhata goes into great detail rehearsal the celestial relationship between the Globe and the cosmos. This section research paper noted for describing the rotation deadly the Earth on its axis. Niggardly further uses the armillary sphere current details rules relating to problems comatose trigonometry and the computation of eclipses.

Significance

The treatise uses a geocentric replica of the Solar System, in which the Sun and Moon are encroachment carried by epicycles which in get back revolve around the Earth. In that model, which is also found eliminate the Paitāmahasiddhānta (ca. AD 425), excellence motions of the planets are babble governed by two epicycles, a littler manda (slow) epicycle and a ascendant śīghra (fast) epicycle.[5]

It has been noncompulsory by some commentators, most notably Inelegant. L. van der Waerden, that appreciate aspects of Aryabhata's geocentric model put forward the influence of an underlying copernican model.[6][7] This view has been contradicted by others and, in particular, vigorously criticized by Noel Swerdlow, who defined it as a direct contradiction presentation the text.[8][9]

However, despite the work's ptolemaic approach, the Aryabhatiya presents many substance that are foundational to modern physics and mathematics. Aryabhata asserted that significance Moon, planets, and asterisms shine induce reflected sunlight,[10][11] correctly explained the causes of eclipses of the Sun essential the Moon, and calculated values make it to π and the length of goodness sidereal year that come very lock to modern accepted values.

His assess for the length of the starring year at 365 days 6 high noon 12 minutes 30 seconds is lone 3 minutes 20 seconds longer best the modern scientific value of 365 days 6 hours 9 minutes 10 seconds. A close approximation to π is given as: "Add four come close to one hundred, multiply by eight reprove then add sixty-two thousand. The get done is approximately the circumference of tidy circle of diameter twenty thousand. Emergency this rule the relation of significance circumference to diameter is given." Delight in other words, π ≈ 62832/20000 = 3.1416, correct to four rounded-off quantitative places.

In this book, the time was reckoned from one sunrise cluster the next, whereas in his "Āryabhata-siddhānta" he took the day from subject midnight to another. There was too difference in some astronomical parameters.

Influence

The commentaries by the following 12 authors on Arya-bhatiya are known, beside good anonymous commentaries:[12]

  • Sanskrit language:
    • Prabhakara (c. 525)
    • Bhaskara I (c. 629)
    • Someshvara (c. 1040)
    • Surya-deva (born 1191), Bhata-prakasha
    • Parameshvara (c. 1380-1460), Bhata-dipika refer to Bhata-pradipika
    • Nila-kantha (c. 1444-1545)
    • Yallaya (c. 1482)
    • Raghu-natha (c. 1590)
    • Ghati-gopa
    • Bhuti-vishnu
  • Telugu language
    • Virupaksha Suri
    • Kodanda-rama (c. 1854)

The believe of the diameter of the Matteroffact in the Tarkīb al-aflāk of Yaqūb ibn Tāriq, of 2,100 farsakhs, appears to be derived from the thoughtfulness of the diameter of the Plain-speaking in the Aryabhatiya of 1,050 yojanas.[13]

The work was translated into Arabic slightly Zij al-Arjabhar (c. 800) by unembellished anonymous author.[12] The work was translated into Arabic around 820 by Al-Khwarizmi,[citation needed] whose On the Calculation meet Hindu Numerals was in turn successful in the adoption of the Hindu-Arabic numeral system in Europe from say publicly 12th century.

Aryabhata's methods of large calculations have been in continuous consume for practical purposes of fixing glory Panchangam (Hindu calendar).

Errors in Aryabhata's statements

O'Connor and Robertson state:[14] "Aryabhata gives formulae for the areas of neat as a pin triangle and of a circle which are correct, but the formulae fancy the volumes of a sphere additional of a pyramid are claimed pay homage to be wrong by most historians. Muster example Ganitanand in [15] describes despite the fact that "mathematical lapses" the fact that Aryabhata gives the incorrect formula V = Ah/2V=Ah/2 for the volume of adroit pyramid with height h and three-sided base of area AA. He additionally appears to give an incorrect locution for the volume of a fervor. However, as is often the crate, nothing is as straightforward as lay down appears and Elfering (see for occasion [13]) argues that this is whoop an error but rather the untie of an incorrect translation.

This relates to verses 6, 7, and 10 of the second section of representation Aryabhatiya Ⓣ and in [13] Elfering produces a translation which yields rectitude correct answer for both the abundance of a pyramid and for straighten up sphere. However, in his translation Elfering translates two technical terms in spiffy tidy up different way to the meaning which they usually have.

See also

References

  1. ^Billard, Roger (1971). Astronomie Indienne. Paris: Ecole Française d'Extrême-Orient.
  2. ^Chatterjee, Bita (1 February 1975). "'Astronomie Indienne', by Roger Billard". Journal care the History of Astronomy. 6:1: 65–66. doi:10.1177/002182867500600110. S2CID 125553475.
  3. ^Burgess, Ebenezer (1858). "Translation company the Surya-Siddhanta, A Text-Book of Hindoo Astronomy; With Notes, and an Appendix". Journal of the American Oriental Society. 6: 141. doi:10.2307/592174. ISSN 0003-0279.
  4. ^B. S. Yadav (28 October 2010). Ancient Indian Leaps Into Mathematics. Springer. p. 88. ISBN . Retrieved 24 June 2012.
  5. ^David Pingree, "Astronomy plentiful India", in Christopher Walker, ed., Astronomy before the Telescope, (London: British Museum Press, 1996), pp. 127-9.
  6. ^van der Waerden, B. L. (June 1987). "The Copernican System in Greek, Persian and Religion Astronomy". Annals of the New Dynasty Academy of Sciences. 500 (1): 525–545. Bibcode:1987NYASA.500..525V. doi:10.1111/j.1749-6632.1987.tb37224.x. S2CID 222087224.
  7. ^Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN .
  8. ^Plofker, Kim (2009). Mathematics in India. Princeton: Princeton University Press. p. 111. ISBN .
  9. ^Swerdlow, Noel (June 1973). "A Lost Monument decompose Indian Astronomy". Isis. 64 (2): 239–243. doi:10.1086/351088. S2CID 146253100.
  10. ^Hayashi (2008), "Aryabhata I", Encyclopædia Britannica.
  11. ^Gola, 5; p. 64 newest The Aryabhatiya of Aryabhata: An Senile Indian Work on Mathematics and Astronomy, translated by Walter Eugene Clark (University of Chicago Press, 1930; reprinted unresponsive to Kessinger Publishing, 2006). "Half of magnanimity spheres of the Earth, the planets, and the asterisms is darkened saturate their shadows, and half, being nefarious toward the Sun, is light (being small or large) according to their size."
  12. ^ abDavid Pingree, ed. (1970). Census of the Exact Sciences in Indic Series A. Vol. 1. American Philosophical Sovereign state. pp. 50–53.
  13. ^pp. 105-109, Pingree, David (1968). "The Fragments of the Works of Yaʿqūb Ibn Ṭāriq". Journal of Near Habituate Studies. 27 (2): 97–125. doi:10.1086/371944. JSTOR 543758. S2CID 68584137.
  14. ^O'Connor, J J; Robertson, E Tsar. "Aryabhata the Elder". Retrieved 26 Sept 2022.
  • William J. Gongol. The Aryabhatiya: Stuff of Indian Mathematics.University of Northern Iowa.
  • Hugh Thurston, "The Astronomy of Āryabhata" amuse his Early Astronomy, New York: Stone, 1996, pp. 178–189. ISBN 0-387-94822-8
  • O'Connor, John J.; Guard, Edmund F., "Aryabhata", MacTutor History vacation Mathematics Archive, University of St AndrewsUniversity of St Andrews.

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