The Aryabhatiya: Foundations of Indian Mathematics
William J. Gongol
University of Northern Iowa
December 14, 2003

Aryabhata is the earliest Indian mathematician whom historians know by name. He lived from 476 to 550 C.E. Little else is known about him. There has long been confusion regarding his identity; there was another notable Indian mathematician named Aryabhata who flourished sometime between 950 and 1100 C.E. Often, the latter is referred to as Aryabhata II. Also, the Persian historian al-Biruni believed that there were two famous Indian mathematicians named Aryabhata who lived around 500 C.E. The subsequent confusion from this blunder ensued until it was disproved in 1926 (Suzuki 219-220).

Prior to Aryabhata, Vedic sutras (early Hindu scriptures) had expounded on geometric relationships for religious purposes such as altar construction and keeping track of calendars. Jain mathematicians also excelled at mathematics prior to Aryabhata. In light of this, some scholars suggest that Aryabhata intended for his Aryabhatiya to be a commentary on previous mathematicians and astronomers or possibly a skeletal outline of his small contributions to the canon of knowledge (Srinivasiengar 42).

It is written in the Sanskrit language, the language of the Aryans - the people from Europe who migrated to India around 1500 B.C.E. and melded with the indigenous Indian culture to form Hindu culture (Watson 30). The style of the Aryabhatiya is difficult to describe. It does not read like a practical manual as does the Chinese Nine Chapters nor does it read like a basic set of theoretical proofs like Euclid's Elements. The Aryabhatiya is written in poetic verse - typical of Sanskrit works - and seems to be more like a collection of anecdotes and mnemonic devices to aid in teaching mathematical and astronomical ideas than a traditional text. It is highly likely that the study of the Aryabhatiya would be accompanied by the teachings of a well-versed tutor.

There are 123 stanzas in the Aryabhatiya. Some of them have a logical flow while some seem to come out of nowhere. As regards length, without commentary the Aryabhatiya would barely constitute a pamphlet. As regards coherence, it would be a nightmare for a casual reader to try to understand that text without commentary. Even given a commentary, the logic is explained with a framework that is entirely alien to Western readers.

Indian mathematical works often used word numerals before Aryabhata, but the Aryabhatiya is oldest extant Indian work with alphabet numerals. That is, he used letters of the alphabet to form 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.

The Aryabhatiya begins with an introduction called the Dasagitika or "Ten Giti Stanzas." This begins by paying tribute to Brahman, the high god or "ultimate reality" in Hinduism. Next, Aryabhata lays out the numeration system used in the work (as described above). The book then goes on to give an overview of Aryabhata's astronomical findings. He begins by giving a description of the rotations of the heavens about the Earth. In Aryabhata's time the prevailing view of the cosmos was that the Earth was stationary and the sun, moon, other planets, and everything else in the sky rotated around it. Aryabhata believed that the Earth rotated on its axis while the heavens orbited around it. The belief in a stationary Earth was deeply ingrained in Indian astronomy for centuries to come. In fact, some later commentaries on the Aryabhatiya (by notable mathematicians) attempted to reconcile Aryabhata's findings with their belief in a stationary Earth. The second through ninth stanzas go on to describe the sizes and paths of celestial bodies.

The concluding stanza of the Dasagitika is one of the most notable stanzas in the entire work:

10. The (twenty-four) sines reckoned in minutes of arc are 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7 (Aryabhata 19).
This stanza is cryptic in its form, but an arduous breakdown of its make-up reveals that it is actually analogous to a modern sine table. The figures specifically represent differences between half-chord lengths for a given angle and circle size. Although this stanza may not seem very helpful or practical to a modern day trigonometry student, it surely represents a major advance in the history of mathematics. Even more mind-boggling to modern day readers is the fact that these numbers would have been written out as alpha-numeric words.

Slow to understate the importance of his own work, Aryabhata concludes the Dasagitika with a note that "[w]hoever knows this Dasagitika Sutra...goes on to higher Brahman" (Ibid. 20).

The bulk of the mathematics in the Aryabhatiya is contained in the next part, the Ganitapada or "Mathematics." He begins this section with an invocation and pays lip service to his school, Kusumpara. Some historians speculate that this is also the town of his birth, but there is no consensus on the matter. He then gives the names of ascending powers of ten up to 1 billion. Next, Aryabhata says that the product of two equal quantities, the area of a square, and a square are equivalent and likewise, the product of three quantities and a solid with 12 edges are equivalent. It is worth noting that he makes no distinction between geometric areas, abstract quantities, and volumes. This is a leap of faith that Greek mathematicians were much slower to take.

Aryabhata goes on to provide methods for finding square roots and cube roots (tasks that would be far more difficult prior to the development of a place value system). Aryabhata was not the first Indian mathematician to display that he could find square roots - Jain mathematicians had shown great proficiency at this before him - but the Aryabhatiya is the oldest extant work which provides a method for finding square roots. Aryabhata's method for finding cube roots, though, seems to be original (Srinivasiengar 44).

Strange to Westerners is the appearance in the Aryabhatiya of precise formulas alongside approximations with no distinction between the two. For example:

7. Half of the circumference multiplied by half the diameter is the area of a circle. This area multiplied by its own square root is the exact volume of a sphere (Aryabhata 27).
The first statement is true:

Area of a circle = (C/2) * (d/2)
= (p*d)/2 * r
= pr2
But the second statement is a rough approximation:

Volume of Sphere = pr2 * (pr2)0.5
= p0.5 * pr3
(approximately equals) 1.77pr3
(is not equal to) 4/3 pr3
For most practical purposes in sixth century India, this approximation may have served well, but its inaccuracy glares at modern readers.

Extremely notable in the Aryabhatiya is Aryabhata's estimation of p:

10. Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000 (Aryabhata 28).
This computation yields a value of exactly 3.1416. It should be noted that this value for p is among the most accurate used up to this time. It is more precise than quantities derived by some of the greatest ancient Greek mathematicians. The odd presentation of the value serves a twofold purpose; first, it is written stylistically in poetic verse, and second, it utilizes whole number ratios since there was not yet a concept of decimal fractions.

After giving this value (without derivation or explanation), Aryabhata briefly describes the method by which he derived his sine table (shown above). His explanation has been confusing mathematicians for over 1500 years now (casual readers may take great comfort in the company). Some mathematical historians now believe that the early through present day commentators misunderstood Aryabhata's explanation but 15th century commentator Nilakantha Somasutuan actually understood it properly (Hayashi 397). Needless to say, the explanation is quite cryptic.

Aryabhata gives fairly basic construction definitions such as "[t]he circle is made by turning..." (Aryabhata 30) with the use of a compass implicit. He mentions proportions of triangles with respect to shadows. This could have been applied to the use of shadows on sundials and to find the angle the Earth makes with the sun. Perhaps this is from whence Aryabhata estimated the circumference of the Earth.

The Aryabhatiya goes on to provide formulas for finding the sums of arithmetic progressions as well as sums of progressions of squares and cubes. Similarly, it provides methods for breaking down relationships between principal, interest, and time. Aryabhata's formulas for finding these presuppose knowledge of the quadratic equation.

Next, Aryabhata provides a method for finding common denominators for adding up fractions. He also explains that addition and subtraction are inverses as are multiplication and division for what he calls "inverse method." This is for solving problems in forms like: "What number when subtracted from 6, then multiplied by 15 and divided by 3 is 5?"

Then next section of the Aryabhatiya is the Kalakriya or "The Reckoning of Time." In it, he divides up days, months, and years according to the movement of celestial bodies. He divides up history astrologically - it is from this exposition that historians deduced that the Aryabhatiya was written in 499 C.E. At one point, he notes that: "[t]ime, which has no beginning and no end, is measured by [the movements of] the planets and the asterisms on the sphere" (Aryabhata 55).

In the final section of the Aryabhatiya, the Gola or "The Sphere," Aryabhata goes into great detail describing the celestial relationship between the Earth and the cosmos. In the Aryabhatiya, "the sphere refers to the celestial globe and astronomical terms and calculations" (Srinivasiengar 42). This is the section of the Aryabhatiya that is most highly criticized by later Indian commentators since it delves into the rotation of the Earth in depth.

Contrary to a later popular Western belief that the moon is made of cheese, Aryabhata believed that:

37. The Moon consists of water, the Sun of fire, the Earth of earth, and the Earth's shadow of darkness. The Moon obscures the Sun and the great shadow of the Earth obscures the Moon (Aryabhata 78).
Readers might find a description like this amusing amongst profound discoveries of mathematical relationships.

The Aryabhatiya was an extremely influential work as is exhibited by the fact that most notable Indian mathematicians after Aryabhata wrote commentaries on it. At least twelve notable commentaries were written for the Aryabhatiya ranging from the time he was still alive (c. 525) through 1900 ("Aryabhata I" 150-2). The commentators include Bhaskara and Brahmagupta among other notables. Although the work was influential, there is no definitive English translation. There are a number of translations but many are incomplete. The meanings of certain parts of the work are still disputed to this day.

Works Cited:

Aryabhata. The Aryabhatiya of Aryabhata - An Ancient Work on Mathematics and Astronomy. Walter E. Clark, Trans. and Ed. University of Chicago Press: Chicago, Illinois, 1930.

"Aryabhata I." Biographical Dictionary of Mathematicians: Reference Biographies from the Dictionary of Scientific Biography. Vol. 1, (150-152). Charles Scribner and Sons: New York, 1991.

Hayashi, Takao, "Aryabhata's Rule and Table for Sine-Differences," Historia Mathematica, vol. 24 (1997), pp. 396-406.

Srinivasiengar, C. N. The History of Ancient Indian Mathematics. The World Press Private LTD: Calcutta, 1967.

Suzuki, Jeff. A History of Mathematics. Prentice Hall: New Jersey, 2002.

Watson, Francis. A Concise History of India. Thames and Hudson: London, 1974.