Aryabhatiya
Aryabhatiya (IAST: ') or Aryabhatiyam ('), 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 Philosopher of astronomy Roger Billard estimates that the book was composed around 510 CE based on historical references it mentions. Structure and style. Aryabhatiya is written in Sanskrit 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): 1. Gitikapada (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from  earlier texts such as Lagadha's Vedanga Jyotisha (ca. 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 4.32 million years. 2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra); arithmetic and geometric progressions; gnomon/shadows (shanku-chhAyA); and simple, quadratic, simultaneous, and indeterminate equations (Kuṭṭaka). 3. 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. 4. 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. 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 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. 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 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. Influence. Most notable Indian mathematicians writing after the compilation of the Aryabhata wrote commentaries on it. At least twelve notable commentaries were written for the "Aryabhatiya" ranging from Aryabhata's lifetime (c. 525) through 1900 ("Aryabhata I" 150-2). The commentators include Bhāskara I and Brahmagupta, among other notables. The estimate of the diameter of the Earth in the "Tarkīb al‐aflāk" of Yaqūb ibn Tāriq, of 2,100 farsakhs, appears to be derived from the estimate  of the diameter of the Earth in the "Aryabhatiya" of 1,050 yojanas. The work was translated into Arabic around 820 by Al-Khwarizmi, whose "On the Calculation with Hindu Numerals" was in turn influential in the adoption of the Hindu-Arabic numerals 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).