Aryabhata was an astronomer and the earliest Indian mathematician whose work and history are available to modern scholars. He is also known as Aryabhata I or Aryabhata the Elder to distinguish him from a 10th-century Indian mathematician of the same name.
Background
Aryabhata’s birthplace is uncertain, but it may have been in the area known in ancient texts as Ashmaka, which may have been Maharashtra or Dhaka or in Kusumapura in present-day Patna.
Some archaeological evidence suggests that he came from the present-day Kodungallur, the historical capital city of Thiruvanchikkulam of ancient Kerala - this theory is strengthened by the several commentaries on him having come from Kerala.
Education
Aryabhata went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist traditions, as well as Bhāskara I, the 7th Century mathematician, identify Kusumapura as modern Patna.
Career
Aryabhata’s major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The Aryabhatiya covers arithmetic, algebra, and trigonometry.
A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura. Since, the University of Nalanda was in Pataliputra, and had an astronomical observatory; it is probable that he was its head too. Direct details of his work are known only from the Aryabhatiya. His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka).
The Aryabhatiya is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata’s 108), because there are 108 verses in the text. It also has 13 introductory verses and is divided into four pādas or chapters. Aryabhatiya’s first chapter, Gitikapada, with its large units of time - kalpa, manvantra, and Yuga - introduces a different cosmology. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
Ganitapada, the second chapter of Aryabhatiya has 33 verses covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon or shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations.
Aryabhatiya’s third chapter Kalakriyapada explains different units of time, a method for determining the positions of planets for a given day, and a seven-day week with names for the days of the week.
In the last chapter of the Aryabhatiya, Golapada describes Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, shape of the earth, cause of day and night, and zodiacal signs on the horizon.
He did not use a symbol for zero; its knowledge was implicit in his place-value system as a place holder for the powers of ten with null coefficients. He did not use the Brahmi numerals and continued the Sanskritic tradition from Vedic times of using letters of the alphabet to denote numbers, expressing quantities in a mnemonic form. He worked on the approximation for pi thus - add four to 100, multiply by eight, and then add 62,000, the circumference of a circle with a diameter of 20,000 can be approached.
It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable or irrational. In Ganitapada, he gives the area of a triangle as: "for a triangle, the result of a perpendicular with the half-side is the area." He discussed ‘sine’ by the name of ardha-jya or half-chord.
Like other ancient Indian mathematicians, he too was interested in finding integer solutions to Diophantine equations with the form ax + by = c; he called it the kuṭṭaka (meaning breaking into pieces) method.
His contribution to the study of Algebra is immense. In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and cubes through well-tried formulae.
His system of astronomy was called the audayaka system, in which days are reckoned from uday, dawn at lanka or "equator." His later writings, which apparently proposed the ardha-rAtrikA, or midnight model, are lost. He correctly believed that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, challenging the prevailing view.
In Aryabhatiya, he writes that 'setting and rising of planets' is a perception similar to that of someone in a boat going forward sees an unmoving (object) going backward.
He correctly asserted that the planets shine due to the reflection of sunlight and that the eclipses occur due to the shadows of the moon and earth, and not caused by a demon called "Rahu." He correctly deduced that the orbits of the planets are ellipses; this is another great discovery not credited to him but to Johannes Kepler (a German astronomer, born AD 1571).
Views
In Ganita Aryabhata names the first 10 decimal places and gives algorithms for obtaining square and cubic roots, using the decimal number system. Then he treats geometric measurements - employing 62,832/20,000 (= 3.1416) for π, very close to the actual value 3.14159 - and develops properties of similar right-angled triangles and of two intersecting circles. Using the Pythagorean theorem, he obtained one of the two methods for constructing his table of sines. He also realized that the second-order sine difference is proportional to sine. Mathematical series, quadratic equations, compound interest (involving a quadratic equation), proportions (ratios), and the solution of various linear equations are among the arithmetic and algebraic topics included. Aryabhata’s general solution for linear indeterminate equations, which Bhaskara I called kuttakara ("pulverized"), consisted of breaking the problem down into new problems with successively smaller coefficients - essentially the Euclidean algorithm and related to the method of continued fractions.
With Kala-kriya Aryabhata 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).
Aryabhatiya ends with 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 the 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. Aryabhata also correctly ascribed the luminosity of the Moon and planets to reflected sunlight.
Personality
Quotes from others about the person
Bhaskara I who wrote a commentary on the Aryabhatiya about 100 years later wrote of Aryabhata: "Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world."