The Indian mathematician Aryabhata (476–c. 550) made important discoveries in many fields of mathematics and astronomy, asserting solutions to problems that were not proven correct until centuries later. Aryabhata's works were translated into Arabic, where they influenced the development of mathematics in the Arab world and later in the West.

Aryabhata's birthplace is uncertain. He states in the *Aryabhatiya* that he hailed from Kusumapura, which is thought to have been near the present-day city of Patna, India, but he may have been born elsewhere, perhaps as far south as the present-day state of Kerala, where several later commentaries on his work were written. If not born there, he likely moved to Kusumapura during his studies.

Kusumapura (also known by other names, such as Pataliputra) was located on the banks of the Ganges river and served India as a major capital of trade and learning. A cosmopolitan city, it maintained communications with other world cities due to the travelers that passed through it and likely provided a stimulating atmosphere for a young intellectual. Several centuries prior to Aryabhata's career, a Greek ambassador to Pataliputra described Kusumapura as the greatest city on earth. More recent historians have indicated that Aryabhata was the head of an educational institution located in the city, and this may have been the University of Nalanda, which boasted its own observatory and could have helped the mathematician make his strikingly accurate astronomical calculations.

Aryabhata's fame rests on the *Aryabhatiya*, the name given to his manuscript by later commentators. The manuscript of the *Aryabhatiya* contains 118 or 121 verses (scholars disagree over whether the three-verse introduction was part of the original work or added later). These verses are divided into four sections: an introduction with ten (or 13) verses, a mathematics section with 33 verses, a section on astronomical time with 25 verses, and finally a section of 50 verses discussing spheres and eclipses. Written in the Sanskrit language of ancient India, it compiled that civilization's mathematical knowledge up to that time, with Aryabhata adding his contributions. The introduction of the *Aryabhatiya* includes religious text and also specifies the large units of astronomical time in Hindu cosmology. One verse presents a table of sines.

Aryabhata's mathematical accomplishments were groundbreaking. His system of numbering is interesting in that he seems to have been aware of the functions that would define the numeral system developing in India in later years. This system would be the basis for modern Arabic numerals, called in Arabic *rakam al-Hind*, meaning “Hindu numerals.” Aryabhata did not use numerals, however; instead, he followed the older Indian system of employing letters from the Sanskrit alphabet to represent numbers. With the 33 consonants in Sanskrit, he could represent numbers up to 100, and he then added a vowel for larger values. This enabled him to represents numbers as large as ten to the 18th power, an ambitious scope for the time. The implications of Aryabhata's system were even more innovative, however: His method of using consonants and vowels is akin to what is now known as the positional system or place-value system.

The positional system is used in modern mathematics: The same symbol (in our case, a numeral) can be used in different positions within a series of consecutive symbols to indicate a declining order of magnitude, typically powers of ten. Aryabhata was not the first mathematician to use the positional system, the use of which dates as far back as ancient Babylon, but the continued use of the system today may be traced to his efforts. Compared with Roman numerals, the predominant system through much of the ancient world, the positional system greatly facilitated basic arithmetic. Aryabhata, who did not use numerals, had no symbol for zero; that all-important discovery would be made by another Indian mathematician, Brahmagupta (c. 598–665). In undertaking computations involving square and cube roots, Aryabhata seemed to be aware of the concept of zero, and his work may well have influenced that of Brahmagupta.

Aryabhata's treatment of what would soon be called *al-jabr* or algebra by the Arabs also involved novel ideas, principally in the solution of so-called Diophantine equations: equations of the form *by = ax + c* (or *by = ax — c*), where *a, b*, and *c* are integers. Aryabhata called his method *kuttaka*, meaning “to pulverize” or “to break into pieces”; it involved progressively reducing the equation to its simplest form. Aryabhata also devised ways of calculating the area of a triangle and of a circle; his formulae for the volumes of a sphere and a pyramid are thought to be incorrect, but the problem may have resulted from incorrect translation of his Sanskrit manuscript. He provided a table of sines and introduced the trigonometric function called the versine.

Aryabhata's discoveries in the realm of astronomy were no less progressive than those he made in pure mathematics. He correctly understood that solar and lunar eclipses were caused by the interposition of the moon between Earth and its sun, and Earth between its moon and sun, respectively. In this he was preceded by Greek and Roman astronomers, although previous generations of Indian astronomers held that eclipses were actually caused by a demon named Rahu. Aryabhata also calculated the circumference of Earth, deeming it to be 24,902 miles, which is remarkably close to the actual circumference of 24,901 miles.

Aryabhata believed that the earth both rotated on its axis and rotated around the sun, and these suppositions were far from universal during his time. Some Greek thinkers had guessed that the former proposition was true, but they were overruled by the great (but mistaken) Aristotle, who reasoned that the stars seemed to revolve around the earth because the heavens were a sphere that rotated around a stationary and round planetary object. Later Indian astronomers argued that Aryabhata was mistaken, and his view did not gain prominence until it was supported by Arab astronomers in the tenth century. Aryabhata correctly concluded that the moon and the other planets are visible because they reflect sunlight, and he argued that planetary orbits were elliptical in shape, a fact that was not generally accepted until the observations of German astronomer Johannes Kepler (1571–1630).

Aryabhata calculated the length of the year as 365 days, six hours, 12 minutes, and 30 seconds, an estimate that was more than 12 minutes too long. Although he may have corrected his various errors in a second volume, the *Aryabhata-Siddhanta*, this work has been lost; it is known to have existed through commentaries by Brahmagupta and other more-recent Indian writers. The latter part of Aryabhata's life was sparsely recorded; apparently active as a teacher, he died around the year 550.

Even compared with those of Brahmagupta, Aryabhata's contributions have been incompletely appreciated in the history of mathematics. Most studies of individual aspects of his work have been published in Indian periodicals rarely read or referenced in the West. In contrast, the *Aryabhatiya* was translated into Arabic under the title *Zij al-Arjabhar* and directly influenced the flowering of mathematics and astronomy in the Arab world, which in turn directly shaped the emergence of higher learning in Europe.

In India itself Aryabhata's influence was even greater. The commentator Bhaskara I, who lived a century after the mathematician/astronomer, wrote (according to Dick Teresi in *Lost Discoveries: The Ancient Roots of Modern Science, from the Babylonians to the Maya*): “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.”

Boyer, Carl, *A History of Mathematics*, Wiley, 1968.

*Complete Dictionary of Scientific Biography*, Scribner, 2008.

Teresi, Dick, *Lost Discoveries: The Ancient Roots of Modern Science, from the Babylonians to the Maya*, Simon & Schuster, 2002.

*Famous Mathematicians website*,
http://www.famous-mathematicians.com/
(January 20, 2017), “Aryabhata.”

*MacTutor History of Mathematics/St. Andrews University website*,
http://www-history.mcs.st-andrews.ac.uk/
(January 20, 2017), J.J. O'Connor and E.F. Robertson, “Aryabhata the Elder.”❑

(MLA 8^{th} Edition)