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A 3,000-year-old ritual was resurrected at Panjal in Kerala in April 1975. A 12-day Agnicayana, or Atiratra, was performed on a bird-shaped altar of a thousand bricks. The altar was a geometricians' delight.
The area of each layer of the altar, for instance, was seven and a half times a square purusa, the size of the sacrificer or the Yajamana. A fifth of the size of the Yajamana, panchami, was the basic unit of the bricks.
The rules for measurement and construction of sacrificial altars are found in the Sulba Sutras, the earliest documents of geometry in India. Sulba means cord. Of the various Sulba Sutras, those of Baudhayana, Apastamba and Katyayana are best known. Scholars believe the sutras were composed during 800-500 BC.
The mathematical knowledge in the texts comes from the creation of altars or bricks in various shapes-rhombus, isosceles trapezium, square, rectangle, isosceles right-angled triangle or circle. A square-shaped altar sometimes had to become circular without any change in the area or vice-versa. Obviously, the authors of the Sulba texts knew the value of pi, which is the ratio of the circumference to the diameter of a circle.
The theory of right angles is attributed to Greek philosopher Pythagoras (6th century BC). But Baudhayana mentions that the diagonal of a rectangle produces by itself both (the areas) produced separately by its two sides. In simple terms, this means that the square of the diagonal is equal to the sum of the squares of two sides. In the next rule he says that the rectangles for which the theorem is true have the sides as 3 and 4 [32+42=52], 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36. The theorem is given in all the Sulba Sutras.
The relation between the length, breadth and hypotenuse of a rectangle [x2+y2=z2] was discovered by the Babylonians and Egyptians long before Pythagoras. The Chinese followed almost the same algebraic technique. Eminent mathematician A.K. Bag, who has edited Sulba Sutras along with S.N. Sen, has discussed the parallelism in other cultures and ritual geometry in India. He observes that the Egyptian, Indian and Greek methods may have some links at some stages because of the use of cord and peg.
But he says tackling of mathematical and geometrical problems with rational numbers and irrational numbers [such as square-root of 2] was a unique achievement of early Indians. They even had technical terms such as dvikarani, trikarani and panchakarani (for square-roots of 2, 3 and 5) and so on and gave their values to a high degree of approximation. The mathematics in Sulba texts also involves a highly sophisticated brick technology. Ten types of bricks were used to build the altar at Panjal.
Fragments of mathematical works by Jain mathematicians are found in the canonical or non-mathematical texts before the 4th century AD. Sthananga Sutra, a Jain work of the 1st century AD, lists several topics including quadratic equations, algebra and permutations and combinations.
The next mathematical work of significance is the 3rd or 4th century AD Bakshali Manuscript-so called because it was discovered in a village called Bakshali (near Peshawar). The major portion of it deals with fractions, square-roots, progressions, income and expenditure, profit and loss, computation of gold, interest, rule of three and summation of complex series.
The landmark of mathematical work after this is the astronomical work Aryabhatiya of Aryabhatta (b. AD 476). Here we come across geometry. Aryabhatiya geometry moves from the earth to sky.
What the stars foretell
The first formal treatise on astronomy is the Vedanga Jyotisha, dated about 1400 BC. It talks of a five-year yuga (time span) consisting of 67 lunar months, which incorrectly corresponds to 366 days in a year. But a peculiar concept was of the Rahu and Ketu which eclipsed the sun and the moon. This was later identified as two imaginary points where the path of the moon intersects the apparent path of the sun. For an eclipse to occur the moon should be at one of these two points.
The firm historical hand on ancient astronomy is the calendrical information in Asoka's edict (300 BC) and the Mahabharata text (compiled during 400 BC-400 AD). After a grey area from Asoka's period onwards, the major text later is Aryabhatia (499 AD), the Siddhantic or mathematical astronomy text of Aryabhatta. "It is the oldest in whole of Sanskrit literature which is accurately dated," says Rajesh Kochhar, astrophysicist and director of National Institute of Science, Technology and Development Studies in New Delhi.
Aryabhatta taught that the earth spun on its axis and gave the correct explanation of the eclipses. Aryabhatta's genius extends to his development of an alphabetical system of expressing numbers on the decimal place value model and in calculating the most accurate value of pi as 3.1416.
The development of Siddhantic astronomy came as a result of interaction with Greece in the post-Alexandrian period (3rd century BC). "Vedanga Jyotisha does not mention week days or zodiacal signs but in the Siddhantic astronomical texts zodiacal signs are inbuilt," says Kochhar. "There are many new inputs in Aryabhatta's work."
Aryabhatta's follower Varahamihira (c. 505 AD) compiled five siddhantas, two of which bear testimony to outside influence. The most accurate is Surya Siddhanta, which was revised several times.
A significant feature of the siddhantas was the use of time cycles of mahayugas. A mahayuga starts at an epoch when all planets are in conjunction. During a mahayuga they will perform an integral number of revolutions and at the end of a mahayuga they are again in conjunction. A mahayuga is made up of 4,320,000 years and is divided into four: krita, dvapara, treta and kali. Aryabhatta assumed all the yugas to be of equal duration whereas others took it in the ratio of 4:3:2:1. In other words, kali would be 432,000, treta double that, dvapara three and krita four times.
An important name in siddhanta astronomy is Brahmagupta (c. 598 AD). He bitterly criticised Aryabhatta for deviating from tradition, for saying that the earth is not stationary, and for dividing a yuga into four cycles. His books Brahmasphuta Siddhanta and Khandakadhyaya were translated into Arabic in the 8th century. Arab traveller Al-Biruni of the 10th century describes Khandakadhyaya as "the best known of all and preferred by astronomers to all others".
The main occupation of Indian astronomers for the next thousand years was the calculation of planetary orbits. The tradition was alive in Kerala till about 150 years ago. Says Kochhar: "Till German scientist Johannes Kepler's laws in the 17th century, when it became easier to calculate planetary orbits, Indian astronomers were the only ones who could predict eclipses accurately. Kepler's laws are superior to Aryabhatta's calculations."
Calculating planetary orbits led
to many developments in mathematics, the high point of which was the decimal
system. It travelled westwards through 9th century Arab mathematician Al-Khwarizmi.
Aryabhatta also gives tables of astronomical constants and trigonometric
sine tables in the Ganitapada section of his text.
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