Aryabhatiya (IAST:Āryabhaṭīya) orAryabhatiyam (Āryabhaṭīyaṃ), aSanskrit astronomical treatise, is themagnum opus and only known surviving work of the 5th centuryIndian mathematicianAryabhata. Philosopher of astronomy Roger Billard estimates that the book was composed around 510 CE based on historical references it mentions.[1][2]
Aryabhatiya is written inSanskrit and divided into four sections; it covers a total of 121 verses describing different moralitus via a mnemonic writing style typical for such works in India (see definitions below):
It is highly likely that the study of theAryabhatiya 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 theAryabhatiya is the oldest extant Indian work withDevanagari numerals. That is, he used letters of theDevanagari 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.Aryabhata numeration, the Sanskrit numerals.
TheAryabhatiya contains 4 sections, orAdhyāyās. The first section is calledGītīkāpāḍaṃ, containing 13 slokas.Aryabhatiya begins with an introduction called the "Dasageethika" or "Ten Stanzas." This begins by paying tribute toBrahman (not Brāhman), the "Cosmic spirit" in Hinduism. Next, Aryabhata lays out the numeration system used in the work. It includes a listing ofastronomical constants and thesine 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 499 has been calculated for the compilation of theAryabhatiya.[4] The book also contains rules for computing the longitudes of planets usingeccentrics andepicycles.
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 therotation of the Earth on its axis. It further uses thearmillary sphere and details rules relating to problems of trigonometry and the computation of eclipses.
The treatise uses ageocentric model of theSolar System, in which the Sun and Moon are each carried byepicycles which in turn revolve around the Earth. In this model, which is also found in thePaitāmahasiddhānta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smallermanda (slow) epicycle and a largerśīghra (fast) epicycle.[5]
It has been suggested by some commentators, most notablyB. L. van der Waerden, that certain aspects of Aryabhata's geocentric model suggest the influence of an underlyingheliocentric model.[6][7] This view has been contradicted by others and, in particular, strongly criticized byNoel Swerdlow, who characterized it as a direct contradiction of the text.[8][9]
However, despite the work's geocentric approach, theAryabhatiya presents many ideas that are foundational to modern astronomy and mathematics. Aryabhata asserted that the Moon, planets, andasterisms 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 thesidereal year that come very close to modern accepted values.
His value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is only 3 minutes 20 seconds longer than the modern scientific value of 365 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 ≈ 62832/20000 = 3.1416, 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.
The commentaries by the following 12 authors onArya-bhatiya are known, beside some anonymous commentaries:[12]
The estimate of the diameter of the Earth in theTarkīb al-aflāk ofYaqūb ibn Tāriq, of 2,100 farsakhs, appears to be derived from the estimate of the diameter of the Earth in theAryabhatiya of 1,050 yojanas.[13]
The work was translated intoArabic asZij al-Arjabhar (c. 800) by an anonymous author.[12] The work was translated into Arabic around 820 byAl-Khwarizmi,[citation needed] whoseOn the Calculation with Hindu Numerals was in turn influential in the adoption of theHindu-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 thePanchangam (Hindu calendar).
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.
It is based on the assumption of epicycles and eccenters, so it is not heliocentric, but my hypothesis is that it was based on an originally heliocentric theory.
Not only did Aryabhata believe that the earth rotates, but there are glimmerings in his system (and other similar systems) of a possible underlying theory in which the earth (and the planets) orbits the sun, rather than the sun orbiting the earth. The evidence is that the basic planetary periods are relative to the sun.
Such an interpretation, however, shows a complete misunderstanding of Indian planetary theory and is flatly contradicted by every word of Aryabhata's description.
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