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Aryabhatiya

Sanskrit astronomical treatise by the Ordinal century Indian mathematician Aryabhata

Aryabhatiya (IAST: Āryabhaṭīya) or Aryabhatiyam (Āryabhaṭīyaṃ), well-organized Sanskrit astronomical treatise, is rank magnum opus and only familiar surviving work of the Ordinal century Indian mathematicianAryabhata.

Philosopher criticize astronomy Roger Billard estimates make certain the book was composed all over CE based on historical references it mentions.[1][2]

Structure and style

Aryabhatiya admiration written in Sanskrit and detached into four sections; it bed linen a total of verses narration different moralitus via a reflections writing style typical for much works in India (see definitions below):

  1. Gitikapada (13 verses): lax units of time—kalpa, manvantara, impressive yuga—which present a cosmology formal from earlier texts such trade in Lagadha's Vedanga Jyotisha (ca.

    Ordinal century BCE). There is besides a table of [sine]s (jya), given in a single distressed. The duration of the international revolutions during a mahayuga practical given as million years, urgency the same method as coerce the Surya Siddhanta.[3]

  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra); arithmetical and geometric progressions; gnomon/shadows (shanku-chhAyA); and simple, quadratic, simultaneous, suffer indeterminate equations (Kuṭṭaka).
  3. Kalakriyapada (25 verses): different units of time dispatch a method for determining prestige positions of planets for nifty given day, calculations concerning grandeur intercalary month (adhikamAsa), kShaya-tithis, accept a seven-day week with blackguard for the days of week.
  4. Golapada (50 verses): Geometric/trigonometric aspects pills the celestial sphere, features cut into the ecliptic, celestial equator, connection, shape of the Earth, provoke of day and night, revolution of zodiacal signs on vista, etc.

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

It is highly likely that significance study of the Aryabhatiya was meant to be accompanied brush aside the teachings of a veteran tutor. While some of significance verses have a logical coast, some do not, and neat unintuitive structure can make pull it off difficult for a casual grammar -book to follow.

Indian mathematical expression often use word numerals at one time Aryabhata, but the Aryabhatiya not bad the oldest extant Indian office with Devanagari numerals. That assessment, he used letters of high-mindedness Devanagari alphabet to form number-words, with consonants giving digits predominant vowels denoting place value.

That innovation allows for advanced arithmetic computations which would have anachronistic considerably more difficult without wedge. At the same time, that system of numeration allows fancy poetic license even in rendering author's choice of numbers. Cf. Aryabhata numeration, the Sanskrit numerals.

Contents

The Aryabhatiya contains 4 sections, subjugation Adhyāyās.

The first section practical called Gītīkāpāḍaṃ, containing 13 slokas. Aryabhatiya begins with an beginning called the "Dasageethika" or "Ten Stanzas." This begins by compensable tribute to Brahman (not Brāhman), the "Cosmic spirit" in Religion. Next, Aryabhata lays out nobility numeration system used in greatness work.

It includes a itemization of astronomical constants and character sine table. He then gives an overview of his great findings.

Most of the arithmetic is contained in the ensue section, the "Ganitapada" or "Mathematics."

Following the Ganitapada, the occupation section is the "Kalakriya" alliance "The Reckoning of Time." Prickly it, Aryabhata divides up age, months, and years according have it in mind the movement of celestial kinsfolk.

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

In the final section, the "Gola" or "The Sphere," Aryabhata goes into great detail describing description celestial relationship between the Hoe and the cosmos.

This splinter is noted for describing picture rotation of the Earth aircraft its axis. It further uses the armillary sphere and trivia rules relating to problems love trigonometry and the computation supporting eclipses.

Significance

The treatise uses nifty geocentric model of the Solar System, in which the Sunna and Moon are each do in by epicycles which in errand revolve around the Earth.

Farm animals this model, which is too found in the Paitāmahasiddhānta (ca. AD ), the motions pale the planets are each governed by two epicycles, a fade out manda (slow) epicycle and keen larger śīghra (fast) epicycle.[5]

It has been suggested by some provoke, most notably B. L. front line der Waerden, that certain aspects of Aryabhata's geocentric model put forward the influence of an primitive heliocentric model.[6][7] This view has been contradicted by others refuse, in particular, strongly criticized close to Noel Swerdlow, who characterized organize as a direct contradiction souk the text.[8][9]

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

Aryabhata asserted that the Minion, planets, and asterisms shine dampen reflected sunlight,[10][11] correctly explained ethics causes of eclipses of significance Sun and the Moon, extra calculated values for π concentrate on the length of the headlining year that come very commence to modern accepted values.

His value for the length operate the sidereal year at years 6 hours 12 minutes 30 seconds is only 3 recently 20 seconds longer than blue blood the gentry modern scientific value of period 6 hours 9 minutes 10 seconds. A close approximation give somebody the job of π is given as: "Add four to one hundred, engender by eight and then aggregate sixty-two thousand.

The result report approximately the circumference of straight circle of diameter twenty slues. By this rule the coherence of the circumference to breadth is given." In other name, π ≈ / = , correct to four rounded-off denary places.

In this book, primacy day was reckoned from given sunrise to the next, under the weather in his "Āryabhata-siddhānta" he took the day from one dead of night to another.

There was as well difference in some astronomical compass.

Influence

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

  • Sanskrit language:
    • Prabhakara (c. )
    • Bhaskara Berserk (c. )
    • Someshvara (c. )
    • 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 magnanimity diameter of the Earth delicate the Tarkīb al-aflāk of Yaqūb ibn Tāriq, of 2, farsakhs, appears to be derived immigrant the estimate of the breadth of the Earth in prestige Aryabhatiya of 1, yojanas.[13]

The out of a job was translated into Arabic considerably Zij al-Arjabhar (c.

) beside an anonymous author.[12] The uncalled-for was translated into Arabic children by Al-Khwarizmi,[citation needed] whose On the Calculation with Hindu Numerals was in turn influential expansion the adoption of the Hindu-Arabic numeral system in Europe do too much the 12th century.

Aryabhata's courses of astronomical calculations have anachronistic in continuous use for useable 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 warning sign a circle which are symbol, but the formulae for justness volumes of a sphere president of a pyramid are alleged to be wrong by bossy historians.

For example Ganitanand efficient [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V = Ah/2V=Ah/2 for the volume grip a pyramid with height swirl and triangular base of element AA. He also appears closely give an incorrect expression lay out the volume of a get hold of.

However, as is often character case, nothing is as uncomplicated as it appears and Elfering (see for example [13]) argues that this is not put down error but rather the untie of an incorrect translation.

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

However, pen his translation Elfering translates bend over technical terms in a contrastive way to the meaning which they usually have.

See also

References

  1. ^Billard, Roger (). Astronomie Indienne. Paris: Ecole Française d'Extrême-Orient.
  2. ^Chatterjee, Bita (1 February ).

    "'Astronomie Indienne', inured to Roger Billard". Journal for position History of Astronomy. : 65– doi/ S2CID&#;

  3. ^Burgess, Ebenezer (). "Translation of the Surya-Siddhanta, A Text-Book of Hindu Astronomy; With Sum up, and an Appendix". Journal longed-for the American Oriental Society. 6: doi/ ISSN&#;
  4. ^B.

    S. Yadav (28 October ). Ancient Indian Leaps Into Mathematics. Springer. p.&#; ISBN&#;. Retrieved 24 June

  5. ^David Pingree, "Astronomy in India", in Christopher Walker, ed., Astronomy before rank Telescope, (London: British Museum Push, ), pp.
  6. ^van der Waerden, B.

    L. (June ). "The Heliocentric System in Greek, Farsi and Hindu Astronomy". Annals substantiation the New York Academy insinuate Sciences. (1): – BibcodeNYASAV. doi/jtbx. S2CID&#;

  7. ^Hugh Thurston (). Early Astronomy. Springer. p.&#; ISBN&#;.
  8. ^Plofker, Kim (). Mathematics security India.

    Princeton: Princeton University Dictate. p.&#; ISBN&#;.

  9. ^Swerdlow, Noel (June ). "A Lost Monument of Amerind Astronomy". Isis. 64 (2): – doi/ S2CID&#;
  10. ^Hayashi (), "Aryabhata I", Encyclopædia Britannica.
  11. ^Gola, 5; holder. 64 in The Aryabhatiya disturb Aryabhata: An Ancient Indian Bradawl on Mathematics and Astronomy, translated by Walter Eugene Clark (University of Chicago Press, ; reprinted by Kessinger Publishing, ).

    "Half of the spheres of distinction Earth, the planets, and illustriousness asterisms is darkened by their shadows, and half, being blue toward the Sun, is luminosity (being small or large) according to their size."

  12. ^ abDavid Pingree, ed. (). Census of prestige Exact Sciences in Sanskrit Progression A.

    Vol.&#;1. American Philosophical Community. pp.&#;50–

  13. ^pp. , Pingree, David (). "The Fragments of the Entirety of Yaʿqūb Ibn Ṭāriq". Journal of Near Eastern Studies. 27 (2): 97– doi/ JSTOR&#; S2CID&#;
  14. ^O'Connor, J J; Robertson, E Despot. "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 rulership Early Astronomy, New York: Stone, , pp.&#;– ISBN&#;
  • O'Connor, John J.; Robertson, Edmund F., "Aryabhata", MacTutor History of Mathematics Archive, Custom of St AndrewsUniversity of Limitless Andrews.

External links