Арьябхатия: Үнді математигінің астрономиялық трактаты

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 mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that the book was composed around 510 CE based on historical references it mentions.

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 499 has been calculated for the compilation of the Aryabhatiya. The book also contains rules for computing the longitudes of planets using eccentrics and epicycles. 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.

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 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle. 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. 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. 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, 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 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.