Āryabhata the Elder

Indian mathematician and astronomer

• Born: c. 476
• Birthplace: Possibly Ashmaka or Kusumapura (now in India)
• Died: 550
• Place of death: Unknown

ĀryabhatŃa the Elder's treatise, The Aryabhatiya, is the first work of Indian mathematics that has a definite author and date. It indicates what Indian mathematicians had accomplished by the end of the fifth century.

Early Life

ĀryabhatŃa (ahr-yah-BAH-tuh) the Elder is a figure of whom almost nothing is known other than his work. His only surviving book, the Aryabhatiya (499; The Aryabhatiya, 1927), indicates that it was written in 499 and that he was twenty-three years old at the time, which provides the traditional date for his birth. There is a good deal of disagreement about the exact date, as well as about the place of his birth. One of his successors in the field of Indian mathematics refers to ĀryabhatŃa as being from Ashmaka, a region in the northwestern part of modern India. There was a migration from that region to a more southerly part of India, and ĀryabhatŃa is likely to have been born there, as it is close to the only geographic region in which he is known to have worked. He is called ĀryabhatŃa the Elder to distinguish him from a later mathematician who wrote under the same name. The later ĀryabhatŃa may have chosen to use that as the signature for his works as a tribute to his predecessor.

In the course of the mathematical section of The Aryabhatiya, the author refers to the town of Kusumapura. Most scholars believe that this refers to the town where he was doing his academic work rather than his birthplace, but others argue that this town was where the mathematician was born. Kusumapura is frequently identified as the modern-day Patna, the capital of the region of Bihar. Because his work was to prove influential in the Kerala school of astronomy, practiced at the very south tip of India, some scholars have claimed that he spent his career there, but evidence to support that claim is no easier to come by than evidence regarding any other feature of ĀryabhatŃa's life.

The nature of the education that ĀryabhatŃa received is unknown, but from the text of his work it is clear that he studied Greek mathematics and astronomy as well as the Indian traditions in both areas. Because ĀryabhatŃa's work is the first systematic treatment of these subjects that has survived from that part of the world, it may be that he was dependent on odd scraps of knowledge from his predecessors. He would have had plenty of examples of particular sorts of calculation from which to learn. Because he wrote in Sanskrit verse, his training would have been literary as well as scientific. From the religious tone of his invocations, it seems safe to conclude that he was not a Buddhist, as he pays reverence to Brahman and the astronomical bodies.

Life's Work

From later references, it appears that ĀryabhatŃa wrote a book on astronomy as well as The Aryabhatiya. There were periods of time when neither work was available to students, but The Aryabhatiya was subsequently made generally available. The work on astronomy is not known to have survived.

The Aryabhatiya is a Sanskrit poem divided into four sections: an introduction, a mathematical section, a section on time measurement, and a section on celestial spheres. The opening section is written in a different verse form than the others, as though it may have been more like an invocation. There is an invocation at the start of the section on mathematics as well. The entire mathematical section is only thirty-three verses long, which attests to the brevity of his style. As a result, there is no illustration of the techniques and ideas he describes by numerical examples.

By ĀryabhatŃa's time, the use of numerals that were the ancestors of the Hindu-Arabic system was already common. Because ĀryabhatŃa was writing a poem, however, that form of number was not convenient for fitting into the meter. As a result, he introduced a system of words in the first section of the poem to correspond to numbers. His numerical vocabulary therefore is not as wide as the range of numbers already in use in Indian computation but sufficed for the writing of his poem.

After the opening invocation in the section devoted to mathematics (ganita, in Sanskrit), ĀryabhatŃa lists in the second stanza the names for classes of numbers. The third stanza is on the subject of geometry ; it points out that the product of two equal numbers was called by the same name as both the square of a figure and its area. He points that the same consideration applies in three dimensions to the cube and the product of three equal quantities.

The following two stanzas (four and five) present methods for calculating the square root and the cube root of a number. From the form of the instruction, it is clear that ĀryabhatŃa recognized that the square root and cube root were not generally going to terminate in a finite number of places. The geometrical points made in the previous stanzas could have come out of the Greek tradition of theoretical mathematics, but these techniques for calculation could be part of a native Indian tradition. In general, the issue of how far ĀryabhatŃa was indebted to Greek sources is hard to resolve, and the discussion often revolves about political issues as well as mathematical or scholarly ones.

The next four stanzas all take up the area and volume of various two-dimensional and three-dimensional figures. The formulae for the area of a triangle and a circle are correct, as is that for the volume of a pyramid, but the formula for the volume of a sphere is incorrect. If ĀryabhatŃa were drawing on the Greek geometrical tradition throughout, it is hard to see how he could have made this error, which seems to be based on an analogy with the area of a circle. In addition, he gives the area for a trapezoid and plane figures more generally. The more general formulae may be suggested by the kind of averaging involved in handling the trapezoid. It is worth remembering that all the so-called formulae are really expressed in the language of poetry rather than written out as equations.

The tenth stanza includes one of the most interesting features of The Aryabhatiya, an approximation for π (the ratio of the circumference of a circle and its diameter). The specific numerical values given by ĀryabhatŃa lead to a decimal approximation good to four places, and there is plenty of discussion about how this compares with the best possible values that the Greeks obtained. In principle, for example, the method of exhaustion of Archimedes (c. 287-212 b.c.e.) could produce approximations to an arbitrary degree of exactness. What is clear is that ĀryabhatŃa recognized that his value remained only an approximation.

The following two stanzas deal with the calculation of quantities related to the trignometric sine function. The centrality of the sine function in the mathematics of ĀryabhatŃa came from its appearance in the tradition of calculating orbits of planets that goes back to Greek explanations of observations, records of which survive from Babylonian times. The mathematical astronomy of Ptolemy (c. 100-178 c.e.) was probably brought to India in the course of the centuries before ĀryabhatŃa's work, as there were Greek and Egyptian traders in India and Indian traders in the centers of culture of Europe and the Mediterranean. However, there was also a native tradition of astronomy going back many centuries, although it is not clear how mathematical a form it had taken. ĀryabhatŃa's interests seemed to merge those of Greek mathematics and Hindu astronomy.

Greek mathematics took issues of construction as seriously as proofs, and ĀryabhatŃa includes a stanza on the construction of various figures. The next three stanzas take up matters relating to the construction of sundials, not surprising in view of the section following that on mathematics being devoted to time measurement. The seventeenth stanza then turns to the Pythagorean theorem about the relationship between the sides of a right triangle. Historians have found references to the theorem in many mathematical cultures other than the Greek, and it could easily have been part of an Indian tradition as well. Because there is no reference to a proof, it is hard to see any connection with the version given in Euclid (c. 330-c. 270 b.c.e.).

After a stanza on the lengths of chords in a circle, ĀryabhatŃa explains how to calculate the sum of the terms in an arithmetic series (where the difference between successive terms is constant). He also provides the way to calculate the number of terms involved if one has the sum, and this presupposes a knowledge of the quadratic formula, the way to derive the roots of a quadratic (second-degree) equation. The quadratic formula had been used by Babylonian mathematicians much earlier, and it is hard to tell how it fit into the rest of ĀryabhatŃa's algebraic knowledge. The formulae he then proceeds to give for the sum of the squares and cubes of arithmetic series also come without derivation.

A couple of stanzas are devoted to getting the values of individual terms from various ways in which they can be combined. After a stanza (number 25) on interest (in which compounding takes place), ĀryabhatŃa introduces the rule of three, the standard method of dealing with proportions for algebraists for millennia. In subsequent stanzas, he provides a guide to calculating common denominators for fractions and resolving problems by using the inverse of the operations in which the problems are posed. The range of subjects continues to widen, as he gives a way of getting a sum from differences, of calculating the value of an object with an agreed monetary standard, and of calculating the distances between planets.

The height of ĀryabhatŃa's mathematical originality comes in the final two stanzas, in which he presents the general techinique for solving Diophantine equations of the first degree. Diophantus (fl. c. 250 c.e.) had presented solutions of number of special cases, but ĀryabhatŃa clearly developed more general forms. The additional detail given to the method (by taking an extra stanza) and its position at the end of the section suggests that ĀryabhatŃa recognized the extent of what he had accomplished.

The third section of The Aryabhatiya looks at issues relating to time but largely through the medium of astronomy. The final section takes up the celestial sphere, and both sections had an influence on subsequent Indian astronomy as well as being brought subsequently to the West. The work as a whole ends with a couple of stanzas celebrating his own accomplishments and criticizing those who disparage the field of astronomy. It is safe to say that this was in response to critics, but it is not clear what the basis was for their objections.

After ĀryabhatŃa's death, subsequent Indian mathematicians, especially Brahmagupta (c. 598-c. 660), disparaged his accomplishments for not having been presented systematically. On the other hand, his work remained the starting point for Indian mathematics and astronomy for many years. Commentaries were published on The Aryabhatiya for more than a thousand years, and it has also drawn the attention of modern scholars. As historians of mathematics look for sources outside the Greek tradition, ĀryabhatŃa's work is both attractive for its style and worthy of investigation.

Significance

ĀryabhatŃa stands at the head of the Indian tradition of mathematics. There was clearly a stream of predecessors on which he drew, but their anonymity makes their importance a trifle elusive. The work of ĀryabhatŃa provided a spur for a thousand years of mathematical research. The form in which he presented his work also helped make sure that it could be taken as a classic contribution to Indian scholarship, not merely a mathematical thesis.

When India launched its first satellite in 1976, it was the 1500th anniversary of ĀryabhatŃa's birth, and the satellite was named after him. At present, there is an ĀryabhatŃa Forum centered in the United Kingdom working on issues in the history of mathematics. The legacy of ĀryabhatŃa continues to serve as a reminder that not all mathematics can be assumed to have come through the Greek pipeline.

Bibliography

Menon, K. N. Aryabhata: Astronomer, Mathematician. New Delhi, India: Ministry of Information, 1977. Part of the spate of literature produced by the 1500th anniversary of ĀryabhatŃa’s birth, this work is typical in that it claims all sorts of achievements for ĀryabhatŃa at the expense of both other mathematical cultures (like the Greek) and other Indian mathematicians.

Proceedings of the International Seminar and Colloquium on Fifteen Hundred Years of Aryabhateeyam. Kochi, India: Kerala Sastra Sahitya Parishad, 2002. This was sparked by the 1500th anniversary of ĀryabhatŃa’s book rather than his birth. The collection of articles covers a good deal not connected with ĀryabhatŃa, but it does provide evidence for continued disagreements over some of the basic facts of his life.

Velukutty, K. K. Heritages to and from Aryabhatta. Elipara, India: Sahithi, 1997. Another assault on the standard accounts of the details of ĀryabhatŃa’s life. It follows up some other Indian mathematical traditions with attention to details of practice.