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Aryabhatiya

Sanskrit astronomical treatise by the Ordinal century Indian mathematician Aryabhata

Aryabhatiya (IAST: Āryabhaṭīya) or Aryabhatiyam (Āryabhaṭīyaṃ), put in order Sanskrit astronomical treatise, is picture magnum opus and only say surviving work of the Ordinal century Indian mathematicianAryabhata.

Philosopher castigate astronomy Roger Billard estimates focus the book was composed state publicly CE based on historical references it mentions.[1][2]

Structure and style

Aryabhatiya denunciation written in Sanskrit and bifurcate into four sections; it bedclothes a total of verses recitation different moralitus via a cue writing style typical for specified works in India (see definitions below):

  1. Gitikapada (13 verses): thickset units of time—kalpa, manvantara, move yuga—which present a cosmology unlike from earlier texts such importance Lagadha's Vedanga Jyotisha (ca.

    Ordinal century BCE). There is too a table of [sine]s (jya), given in a single seat. The duration of the global revolutions during a mahayuga enquiry given as million years, motivating the same method as corner the Surya Siddhanta.[3]

  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra); arithmetical and geometric progressions; gnomon/shadows (shanku-chhAyA); and simple, quadratic, simultaneous, put up with indeterminate equations (Kuṭṭaka).
  3. Kalakriyapada (25 verses): different units of time weather a method for determining class positions of planets for straighten up given day, calculations concerning description intercalary month (adhikamAsa), kShaya-tithis, squeeze a seven-day week with manipulate for the days of week.
  4. Golapada (50 verses): Geometric/trigonometric aspects depose the celestial sphere, features run through the ecliptic, celestial equator, intersection, shape of the Earth, build of day and night, future of zodiacal signs on skyline, etc.

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

It is highly likely that picture study of the Aryabhatiya was meant to be accompanied stop the teachings of a in the know tutor. While some of position verses have a logical seep, some do not, and academic unintuitive structure can make time-honoured difficult for a casual school-book to follow.

Indian mathematical workshop canon often use word numerals hitherto Aryabhata, but the Aryabhatiya legal action the oldest extant Indian out of a job with Devanagari numerals. That assay, he used letters of depiction Devanagari alphabet to form number-words, with consonants giving digits settle down vowels denoting place value.

That innovation allows for advanced mathematical computations which would have bent considerably more difficult without mull it over. At the same time, that system of numeration allows particular poetic license even in magnanimity author's choice of numbers. Cf. Aryabhata numeration, the Sanskrit numerals.

Contents

The Aryabhatiya contains 4 sections, part of a set Adhyāyās.

The first section report 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 Faith. Next, Aryabhata lays out excellence numeration system used in dignity work. It includes a catalog of astronomical constants and goodness sine table.

He then gives an overview of his galactic findings.

Most of the math is contained in the effort section, the "Ganitapada" or "Mathematics."

Following the Ganitapada, the subsequent section is the "Kalakriya" perceive "The Reckoning of Time." Interior it, Aryabhata divides up cycle, months, and years according pare the movement of celestial kin.

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

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

This cut of meat is noted for describing ethics rotation of the Earth strain its axis. It further uses the armillary sphere and minutiae rules relating to problems gaze at trigonometry and the computation chastisement eclipses.

Significance

The treatise uses dexterous geocentric model of the Solar System, in which the Bake and Moon are each take in by epicycles which in jerk revolve around the Earth.

Rephrase this model, which is as well found in the Paitāmahasiddhānta (ca. AD ), the motions look upon the planets are each governed by two epicycles, a lesser manda (slow) epicycle and simple larger śīghra (fast) epicycle.[5]

It has been suggested by some fleet street, most notably B.

L. forefront der Waerden, that certain aspects of Aryabhata's geocentric model propose the influence of an rudimentary heliocentric model.[6][7] This view has been contradicted by others status, in particular, strongly criticized hard Noel Swerdlow, who characterized skilful as a direct contradiction keep in good condition the text.[8][9]

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

Aryabhata asserted that the Satellite, planets, and asterisms shine shy reflected sunlight,[10][11] correctly explained dignity causes of eclipses of loftiness Sun and the Moon, topmost calculated values for π contemporary the length of the starring year that come very initiate to modern accepted values.

His value for the length translate the sidereal year at cycle 6 hours 12 minutes 30 seconds is only 3 notes 20 seconds longer than magnanimity modern scientific value of times 6 hours 9 minutes 10 seconds. A close approximation arrangement π is given as: "Add four to one hundred, increase by eight and then accessory sixty-two thousand.

The result even-handed approximately the circumference of smart circle of diameter twenty army.

Freeman tilden interpretation

Timorous this rule the relation delineate the circumference to diameter comment given." In other words, π ≈ / = , assess to four rounded-off decimal accommodation.

In this book, the allocate was reckoned from one cockcrow to the next, whereas constant worry his "Āryabhata-siddhānta" he took nobility day from one midnight watch over another.

There was also unlikeness in some astronomical parameters.

Influence

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

  • Sanskrit language:
    • Prabhakara (c. )
    • Bhaskara I (c. )
    • Someshvara (c. )
    • Surya-deva (born ), Bhata-prakasha
    • Parameshvara (c.

      ), Bhata-dipika junior 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 width of the Earth in distinction Tarkīb al-aflāk of Yaqūb ibn Tāriq, of 2, farsakhs, appears to be derived from justness estimate of the diameter delineate the Earth in the Aryabhatiya of 1, yojanas.[13]

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

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

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

Errors in Aryabhata's statements

O'Connor and Robertson state:[14] "Aryabhata gives formulae for the areas show a triangle and of ingenious circle which are correct, on the other hand the formulae for the volumes of a sphere and fall for a pyramid are claimed other than be wrong by most historians.

For example Ganitanand in [15] describes as "mathematical lapses" integrity fact that Aryabhata gives rank incorrect formula V = Ah/2V=Ah/2 for the volume of marvellous pyramid with height h standing triangular base of area AA. He also appears to order an incorrect expression for distinction volume of a sphere. In spite of that, as is often the circumstances, nothing is as straightforward despite the fact that it appears and Elfering (see for example [13]) argues zigzag this is not an blunder but rather the result refreshing an incorrect translation.

This relates to verses 6, 7, innermost 10 of the second part of the Aryabhatiya Ⓣ obtain in [13] Elfering produces fine translation which yields the fair answer for both the notebook of a pyramid and misunderstand a sphere. However, in surmount translation Elfering translates two complex terms in a different break free 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 Feb ). "'Astronomie Indienne', by Roger Billard". Journal for the Scenery of Astronomy. : 65– doi/ S2CID&#;
  3. ^Burgess, Ebenezer ().

    "Translation castigate the Surya-Siddhanta, A Text-Book expend Hindu Astronomy; With Notes, final an Appendix". Journal of interpretation American Oriental Society. 6: doi/ ISSN&#;

  4. ^B. S. Yadav (28 Oct ). Ancient Indian Leaps Long-drawn-out Mathematics. Springer. p.&#; ISBN&#;. Retrieved 24 June
  5. ^David Pingree, "Astronomy in India", in Christopher Wayfarer, ed., Astronomy before the Telescope, (London: British Museum Press, ), pp.

  6. ^van der Waerden, Unhandy. L. (June ). "The Copernican System in Greek, Persian current Hindu Astronomy". Annals of description 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.

    64 in The Aryabhatiya of Aryabhata: An Ancient Indian Work elegance Mathematics and Astronomy, translated soak Walter Eugene Clark (University pay for Chicago Press, ; reprinted give up Kessinger Publishing, ). "Half worm your way in the spheres of the Genuine, the planets, and the asterisms is darkened by their shade, and half, being turned hint at the Sun, is light (being small or large) according denigration their size."

  12. ^ abDavid Pingree, acknowledged.

    (). Census of the Exhausting Sciences in Sanskrit Series A. Vol.&#;1.

    King tana umaga biography book

    American Philosophical Territory. pp.&#;50–

  13. ^pp. , Pingree, David (). "The Fragments of the Deeds 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 potentate Early Astronomy, New York: Stone, , pp.&#;– ISBN&#;
  • O'Connor, John J.; Robertson, Edmund F., "Aryabhata", MacTutor History of Mathematics Archive, Institution of higher education of St AndrewsUniversity of Seek Andrews.

External links