Source on Aryabhata

“The greatest of Hindu astronomers and mathematicians, Aryabhata, discussed in verse such poetic subjects as quadratic equations, sines, and the value of π; he explained eclipses, solstices and equinoxes, announced the sphericity of the earth and its diurnal revolution on its axis, and wrote, in daring anticipation of Renaissance science: “The sphere of the stars is stationary, and the earth, by its revolution, produces the daily rising and setting of planets and stars.””

“He was a celebrated astronomer and mathematician of the classical period of the Gupta Dynasty...He played an important role in shaping scientific astronomy in India. He is designated as Arayabhata I to differentiate him from Arya Bhata II who flourished much later (ca. AD 950-1100) and who wrote the Mahasiddhanta.”

“His fame rests mainly on his Aryabhatiya, but from the writings of Varahamihira (Sixth century AD), Bhaskara I, and Brahmagupta (seventh century) it is clear that earlier he composed the Aryabhata Siddhantha (voluminous) is not extant. It is also called Ardharatrika Siddhanta, because in it the civil days were reckoned from one midnight to the next; 34 verses on astronomical instruments from this have been quoted by Ramakrishna Aradya.”

“Aryabhatiya, an improved work, is product of mature intellect, which he wrote when he was 23 years old. Unlike in the Aryabhata siddhanta, the civil days are reckoned from one sunrise to the next, a practice which is still prevalent among the followers of Hindu calendar.”

“The Aryabhatiya consists of four sections:1.Dasagitika (10+3 couplets in Giti meter); 2.Ganitapada (33 verses on mathematics); 3.Kala-kriyapada 25 verses on time-reckoning), and 4. Golapada (50 verses on spherical astronomy)”

“Use of better planetary parameters, the innovations in astronomical methods, and the concise style of exposition of Aryabhatiya makes it an excellent text book on Astronomy. As opposed to the geostationary theory, Aryahabata held the view that the earth rotates on its axis. His estimate of the period of the sidereal rotation of earth was 23 hours 56 min, and 4.1 s is close to the actual value.”

“He was the father of the Indian epicyclic astronomy which resulted in the planetary theory that determines more accurately the true positions and distances of the planets (including the Sun and the Moon)...was also the first to produce celestial latitudes...proposed the scientific cause of eclipses as against the mythological demon Rahu [Moon's node]. His ideas resulted in the new school of Indian Astronomy – the Āryabhata School Āryapakșa based on the text of Āryabhatīțya.”

“The peculiar system of alphabetic numerals evolved by him with 33 consonants of the Sanskrit alphabet (Nagari script) denoted various numbers in conjunction with vowels which themselves did not represent any numerical value. For example khyughr (=khu+yu+ghr) is denoted by 2x100^2 +30x100^2+4x10^3 =4,300,000 which is the number of revolutions of the Sun in a yuga (epoch)”

“The development of Indian trigonometry, based on sine as against chord of the Greeks, a necessity for astronomical calculations with his own concise notation which expresses the full sine table in just one couplet for easy remembrance. One of the two methods suggested by him for the sine table is based on the property that the second order sine differences were proportional to sines themselves.”

“In geometry his greatest achievement was an accurate value of π. His rule is stated as: dn^2+(2a-d)n=2s, which implies the approximation 3.1416 which is correct to the last decimal place.”

“...he flourished in Kusumapura—near Patalipurta (Patna), then the capital of the Gupta dynasty — where he composed at least two works, Aryabhatiya (c. 499) and the now lost Aryabhatasiddhanta. Aryabhatasiddhanta circulated mainly in the northwest of India and, through the Sāsānian dynasty (224–651) of Iran, had a profound influence on the development of Islamic astronomy. Its contents are preserved to some extent in the works of Varahamihira (flourished c. 550), Bhaskara I (flourished c. 629), Brahmagupta (598–c. 665), and others. It is one of the earliest astronomical works to assign the start of each day to midnight.”

“Aryabhatiya...written in verse couplets ...contains astronomical tables and Aryabhata’s system of phonemic number notation, the work is characteristically divided into three sections: Ganita (“Mathematics”), Kala-kriya (“Time Calculations”), and Gola (“Sphere”).”

“In Ganita, he names the first 10 decimal places and gives algorithms for obtaining square and cubic roots, utilizing the decimal number system. Then he treats geometric measurements — employing 62,832/20,000 (= 3.1416) for π—and develops properties of similar right-angled triangles and of two intersecting circles.”

“With Kala-kriya he turned to astronomy — in particular, treating planetary motion along the ecliptic. The topics include definitions of various units of time, eccentric and epicyclic models of planetary motion (see Hipparchus for earlier Greek models), planetary longitude corrections for different terrestrial locations, and a theory of “lords of the hours and days” (an astrological concept used for determining propitious times for action).”

“...spherical astronomy in Gola, where he applied plane trigonometry to spherical geometry by projecting points and lines on the surface of a sphere onto appropriate planes. Topics include prediction of solar and lunar eclipses and an explicit statement that the apparent westward motion of the stars is due to the spherical Earth’s rotation about its axis. He also correctly ascribed the luminosity of the Moon and planets to reflected sunlight.”