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Aryabhatiya

Sanskrit astronomical treatise by the 5th century Indian mathematician Aryabhata

Aryabhatiya (IAST: Āryabhaṭīya) or Aryabhatiyam (Āryabhaṭīyaṃ), a Sanskrit astronomical treatise, is the magnum opus and only known surviving work of the 5th century Indian mathematicianAryabhata.

Philosopher of astronomy Roger Billard estimates that the book was composed around CE based on historical references it mentions.[1][2]

Structure and style

Aryabhatiya is written in Sanskrit and divided into four sections; it covers a total of verses describing different moralitus via a mnemonic writing style typical for such works in India (see definitions below):

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

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    5. 1st century BCE). There is also a table of [sine]s (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as million years, using the same method as in the Surya Siddhanta.[3]

  5. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra); arithmetic and geometric progressions; gnomon/shadows (shanku-chhAyA); and simple, quadratic, simultaneous, and indeterminate equations (Kuṭṭaka).
  6. Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.
  7. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the Earth, cause of day and night, rising of zodiacal signs on horizon, etc.

    In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

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

Indian mathematical works often use word numerals before Aryabhata, but the Aryabhatiya is the oldest extant Indian work with Devanagari numerals. That is, he used letters of the Devanagari alphabet to form number-words, with consonants giving digits and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without it.

At the same time, this system of numeration allows for poetic license even in the author's choice of numbers. Cf.

Write about aryabhatta in sanskrit pdf Motilal Banarsidass Publ. Retrieved 8 February For simplicity, people started calling it jya. Retrieved 20 June

Aryabhata numeration, the Sanskrit numerals.

Contents

The Aryabhatiya contains 4 sections, or Adhyāyās. The first section is called Gītīkāpāḍaṃ, containing 13 slokas. Aryabhatiya begins with an introduction called the "Dasageethika" or "Ten Stanzas." This begins by paying tribute to Brahman (not Brāhman), the "Cosmic spirit" in Hinduism.

Next, Aryabhata lays out the numeration system used in the work. It includes a listing of astronomical constants and the sine table. He then gives an overview of his astronomical findings.

Most of the mathematics is contained in the next section, the "Ganitapada" or "Mathematics."

Following the Ganitapada, the next section is the "Kalakriya" or "The Reckoning of Time." In it, Aryabhata divides up days, months, and years according to the movement of celestial bodies.

He divides up history astronomically; it is from this exposition that a date of AD has been calculated for the compilation of the Aryabhatiya.[4] The book also contains rules for computing the longitudes of planets using eccentrics and epicycles.

Write about aryabhatta in sanskrit language Motilal Banarsidass Publ. Donate icon An illustration of a heart shape "Donate to the archive" User icon An illustration of a person's head and chest. Please see your browser settings for this feature. London: British Museum Press.

In the final section, the "Gola" or "The Sphere," Aryabhata goes into great detail describing the celestial relationship between the Earth and the cosmos. This section is noted for describing the rotation of the Earth on its axis. It further uses the armillary sphere and details rules relating to problems of trigonometry and the computation of eclipses.

Significance

The treatise uses a geocentric model of the Solar System, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. AD ), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle.[5]

It has been suggested by some commentators, most notably B.

L. van der Waerden, that certain aspects of Aryabhata's geocentric model suggest the influence of an underlying heliocentric model.[6][7] This view has been contradicted by others and, in particular, strongly criticized by Noel Swerdlow, who characterized it as a direct contradiction of the text.[8][9]

However, despite the work's geocentric approach, the Aryabhatiya presents many ideas that are foundational to modern astronomy and mathematics.

Aryabhata asserted that the Moon, planets, and asterisms shine by reflected sunlight,[10][11] correctly explained the causes of eclipses of the Sun and the Moon, and calculated values for π and the length of the sidereal year that come very close to modern accepted values.

His value for the length of the sidereal year at days 6 hours 12 minutes 30 seconds is only 3 minutes 20 seconds longer than the modern scientific value of days 6 hours 9 minutes 10 seconds.

A close approximation to π is given as: "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, π ≈ / = , correct to four rounded-off decimal places.

In this book, the day was reckoned from one sunrise to the next, whereas in his "Āryabhata-siddhānta" he took the day from one midnight to another.

There was also difference in some astronomical parameters.

Influence

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

  • Sanskrit language:
    • Prabhakara (c. )
    • Bhaskara I (c.

      )

    • Someshvara (c.

      Write about aryabhatta in sanskrit translation Aryabhata's system of astronomy was called the audAyaka system , in which days are reckoned from uday , dawn at lanka or "equator". Balachandra Rao Retrieved 15 April The inter-school Aryabhata Maths Competition is also named after him, [ 47 ] as is Bacillus aryabhata , a species of bacteria discovered in the stratosphere by ISRO scientists in

      )

    • Surya-deva (born ), Bhata-prakasha
    • Parameshvara (c. ), Bhata-dipika or Bhata-pradipika
    • Nila-kantha (c. )
    • Yallaya (c. )
    • Raghu-natha (c. )
    • Ghati-gopa
    • Bhuti-vishnu
  • Telugu language
    • Virupaksha Suri
    • Kodanda-rama (c.

      )

The estimate of the diameter of the Earth in the Tarkīb al-aflāk of Yaqūb ibn Tāriq, of 2, farsakhs, appears to be derived from the estimate of the diameter of the Earth in the Aryabhatiya of 1, yojanas.[13]

The work was translated into Arabic as Zij al-Arjabhar (c.

) by an anonymous author.[12] The work was translated into Arabic around by Al-Khwarizmi,[citation needed] whose On the Calculation with Hindu Numerals was in turn influential in the adoption of the Hindu-Arabic numeral system in Europe from the 12th century.

Aryabhata's methods of astronomical calculations have been in continuous use for practical purposes of fixing the Panchangam (Hindu calendar).

Errors in Aryabhata's statements

O'Connor and Robertson state:[14] "Aryabhata gives formulae for the areas of a triangle and of a circle which are correct, but the formulae for the volumes of a sphere and of a pyramid are claimed to be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V = Ah/2V=Ah/2 for the volume of a pyramid with height h and triangular base of area AA.

He also appears to give an incorrect expression for the volume of a sphere. However, as is often the case, nothing is as straightforward as it appears and Elfering (see for example [13]) argues that this is not an error but rather the result of an incorrect translation.

This relates to verses 6, 7, and 10 of the second section of the Aryabhatiya Ⓣ and in [13] Elfering produces a translation which yields the correct answer for both the volume of a pyramid and for a sphere.

However, in his translation Elfering translates two technical terms in a different way to the meaning which they usually have.

See also

References

  1. ^Billard, Roger (). Astronomie Indienne.

    Write about aryabhatta in sanskrit word: Open Library American Libraries. Bulletin of the Astronomical Society of India. In Aryabhatiya , Aryabhata provided elegant results for the summation of series of squares and cubes: [ 27 ]. It has been critically edited by Kripa Shankar Shukla.

    Paris: Ecole Française d'Extrême-Orient.

  2. ^Chatterjee, Bita (1 February ). "'Astronomie Indienne', by Roger Billard". Journal for the History of Astronomy. : 65– doi/ S2CID&#;
  3. ^Burgess, Ebenezer (). "Translation of the Surya-Siddhanta, A Text-Book of Hindu Astronomy; With Notes, and an Appendix".

    Journal of the American Oriental Society. 6: doi/ ISSN&#;

  4. ^B. S. Yadav (28 October ). Ancient Indian Leaps Into Mathematics. Springer.

    Write about aryabhatta in sanskrit Narasimhan C. Wikimedia Commons Wikiquote Wikisource Wikidata item. The Columbia Encyclopedia 6 ed. Hamburger icon An icon used to represent a menu that can be toggled by interacting with this icon.

    p.&#; ISBN&#;. Retrieved 24 June

  5. ^David Pingree, "Astronomy in India", in Christopher Walker, ed., Astronomy before the Telescope, (London: British Museum Press, ), pp.
  6. ^van der Waerden, B. L. (June ). "The Heliocentric System in Greek, Persian and Hindu Astronomy". Annals of the New York Academy of Sciences.

    (1): – BibcodeNYASAV. doi/jtbx. S2CID&#;

  7. ^Hugh Thurston (). Early Astronomy. Springer. p.&#; ISBN&#;.
  8. ^Plofker, Kim (). Mathematics in India. Princeton: Princeton University Press. p.&#; ISBN&#;.
  9. ^Swerdlow, Noel (June ). "A Lost Monument of Indian Astronomy".

    Isis. 64 (2): – doi/ S2CID&#;

  10. ^Hayashi (), "Aryabhata I", Encyclopædia Britannica.
  11. ^Gola, 5; p.

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  15. Aryabhatta in sanskrit wikipedia
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    17. 64 in The Aryabhatiya of Aryabhata: An Ancient Indian Work on Mathematics and Astronomy, translated by Walter Eugene Clark (University of Chicago Press, ; reprinted by Kessinger Publishing, ). "Half of the spheres of the Earth, the planets, and the asterisms is darkened by their shadows, and half, being turned toward the Sun, is light (being small or large) according to their size."

  17. ^ abDavid Pingree, ed.

    (). Census of the Exact Sciences in Sanskrit Series A. Vol.&#;1. American Philosophical Society. pp.&#;50–

  18. ^pp. , Pingree, David (). "The Fragments of the Works of Yaʿqūb Ibn Ṭāriq". Journal of Near Eastern Studies. 27 (2): 97– doi/ JSTOR&#; S2CID&#;
  19. ^O'Connor, J J; Robertson, E F.

    "Aryabhata the Elder". Retrieved 26 September

  • William J. Gongol. The Aryabhatiya: Foundations of Indian Mathematics.University of Northern Iowa.
  • Hugh Thurston, "The Astronomy of Āryabhata" in his Early Astronomy, New York: Springer, , pp.&#;– ISBN&#;
  • O'Connor, John J.; Robertson, Edmund F., "Aryabhata", MacTutor History of Mathematics Archive, University of St AndrewsUniversity of St Andrews.

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