Was calculus invented in India?

Bressoud, David

Introduction

No. Calculus was not invented in India. But two hundred years before Newton or Leibniz, Indian astronomers came very close to creating what we would call calculus. Sometime before 1500, they had advanced to the point where they could apply ideas from both integral and differential calculus to derive the infinite series expansions of the sine, cosine, and arctangent functions:

Roy [13] and Katz [7, 8] have given excellent expositions of the Indian derivation of these infinite summations. I will give a slightly different explanation of how Indian astronomers obtained the sine and cosine expansions, with an emphasis on the succession of problems and insights that ultimately led to these series.

This story provides illuminations of calculus that may have pedagogical implications. The traditional introduction of calculus is as a collection of algebraic techniques that solve essentially geometric problems: calculation of areas and construction of tangents. This was not the case in India. There, ideas of calculus were discovered as solutions to essentially algebraic problems: evaluating sums and interpolating tables of sines.

Geometry was well-developed in pre-1500 India. As we will see, it played a role. But it was, at best, a bit player. The story of calculus in India shows us how calculus can emerge in the absence of the traditional geometric context. This story should also serve as a cautionary tale, for what did emerge was sterile. These mathematical discoveries led nowhere. Ultimately, they were forgotten, saved from oblivion only by modern scholars.

Greek origins of trigonometry

Trigonometry arose from, and for over fifteen hundred years was used exclusively for, the study of astronomy/astrology. Hipparchus of Nicaea (ca. 161-126 BC) is considered the greatest astronomer of antiquity and the originator of trigonometry. Trigonometry was born in response to a scientific crisis. The Greek attempt to caste astronomy in the language of geometry was running up against the disturbing fact that the heavens are lop-sided. New tools were needed for analyzing astronomical phenomena.

Let me paint the background to this crisis. It begins with the assumption that the earth is stationary. While this was debated in early Greek science-does the earth go around the sun or the sun around the earth?-the simple fact that we perceive no sense of motion is a powerful indication that the earth does not move. In fact, when in the early seventeenth century it became clear that the earth revolves about the sun, it created a tremendous problem for scientists: How to explain how this was possible? How could it be that we were spinning at thousands of miles per hour and hurtling through space at even greater speeds without experiencing any of this? Surely if the earth did move, we would have been flung off long ago. Newton’s great accomplishment in the Principia was to solve this problem. He created inertial mechanics for this purpose, building it with the then nascent tools of calculus.

So we begin with a fixed and immovable earth. Above it is the great dome of the night sky, rotating once in every 24 hours. In far antiquity it was realized that the stars do not actually disappear during the day. They are present, but impossible to see against the glare of the sun. The position of the sun in this dome is not fixed. During the year, it travels in its own circle, called the ecliptic, through the constellations. One can tell the season by locating the position of the sun in its annual journey around this great circle. This is what the zodiac does. The sign of the zodiac describes the location of the sun by pinpointing the constellation in which it is located (see the cover for a medieval rendering of the zodiac).

Most stars are fixed in the rotating dome of the sky, but a few, called the wanderers or, in Greek, the planetes (hence our word planets), also move across the dome following this same ecliptic circle. If the position of the sun is so important in determining seasons of heat and cold, rain and drought, it appears self-evident that the positions of the wanderers should have important-if more subtle-influences on our lives. Astronomy/astrology was born.

Aristotle, in the 4th century BC, inherited a world-view that saw the earth as the fixed center of the universe with the moon, sun, and planets embedded in concentric, ethereal spheres that rotated with perfect regularity around us. It became the basis for a comprehensive world-view that was tight and consistent and would last for almost two millenia. But its first cracks appeared in less than two hundred years.

The four cardinal points of the great circle traversed by the sun mark the boundaries of the seasons: winter solstice, spring equinox, summer solstice, and autumnal equinox. If the sun travels the ecliptic at constant speed, the four seasons should be of equal length. They are not (see Figure 1). Winter solstice to spring equinox is a short 89 days. Spring is almost 90 days. Summer, the longest season, is over 93 days. And fall comes close to 93 days. If, in fact, the sun moves at a constant speed, this can only mean that the earth is off-center. Hipparchus tackled the problem of calculating the position of the earth.

The basic problem of trigonometry as understood by Hipparchus and his contemporaries is the following: Given an arc of a circle, find the length of the chord that connects the endpoints of that arc (see Figure 2). This chord length depends on both the length of the arc and the radius of the circle. For the Greeks, as for all scientists right through Newton, 90 deg was not the measure of a right angle, but of the distance around one quarter of the circumference of a circle. Degrees were a measure of distance. Given a circle of circumference 360 deg, it would be natural to take the radius to be 360/2pi = 57.2957795… For greater accuracy, the circumference of this standard circle could be measured in minutes. The circumference is then 21,600 minutes and the radius is 3437.74677 … It would become common in Indian trigonometry to use a radius of 3438. There is some evidence that Hipparchus, whose trigonometric tables no longer survive, also may have used a radius of 3438.

Hipparchus was probably the first to construct a table of values of the length of the chord for a given arc, what is sometimes called crd a. In modern trigonometric notation, the chord is twice the sine of half the angle, multiplied by the radius of the circle which we will take to be 3438 (see Figure 2):

References

1. D. M. Bose, S. N. Sen, and B. V. Subbarayappa, A Concise History of Science in India, Indian National Science Academy, 1971.

2. B. Datta and A. N. Singh, revised by K. S. Shukla, Hindu geometry, Indian Journal of History of Science, 15 (1980) 121-188.

3. -, Hindu trigonometry, Indian Journal of History of Science, 18 (1983) 39-108.

4. D. Gold and D. Pingree, A hitherto unknown Sanskrit work concerning Madhava’s derivation of the power series for sine and cosine, Historia Scientiarum, 42 (1991) 49-65.

5. R. C. Gupta, An Indian form of third order Taylor series approximation of the sine, Historia Mathematica, 1 (1974)287-289.

6. T. Heath, A History of Greek Mathematics, reprint of Oxford edition of 1921, Dover, 1981. 7. V. J. Katz, A History of Mathematics: an Introduction, 2nd edition, Addison-Wesley, 1998. 8. -, Ideas of calculus in Islam and India, Math. Magazine, 68 (1995) 163-174.

9. 0. Neugebauer and D. Pingree, The Pancasiddhantika of Varahamihira, Det Kongelige Danske Videnskabernes Selskab, Historisk-Filosofiske Skrifter, Vol. 6, Nos. I & 2, 1970.

10. D. Pingree, Jyotihsastra, Astral and Mathematical Literature, A History of Indian Literature, Vol. 6, Otto Harrassowitz, Weisbaden, 1981.

11. C. T. Rajagopal and M. S. Rangachari, On an untapped source of medieval Keralese mathematics, Archive for History of Exact Sciences, 18 (1978) 89-102.

12. C. T. Rajagopal and A. Venkataram, The sine and cosine power-series in Hindu mathematics, J. Royal Asiatic Society of Bengal, Science, 15 (1949) 1-13.

13. R. Roy, The discovery of the series formula for pi by Leibniz, Gregory and Nilakantha, Math. Magazine, 63 (1990)291-306.

14. T. A. Saraswathi, The development of mathematical series in India after Bhaskara II, Bulletin of the National Institute of Sciences, 21 (1963) 320-343.

15. K. V. Sarma, A History of the Kerala School of Hindu Astronomy, Vishveshvaranand Institute, Hoshiarpur, 1972.

David Bressoud (www.macalester.edu/-bressoud) is fascinated by the stories and histories of mathematics and enjoys weaving them through his books and the classes he teaches at Macalester College. He also writes and reads for the AP Calculus exams. He received an MAA Award for Distinguished Teaching in 1994 and its Beckenbach Book Prize for Proofs and Confirmations in 2000.

Copyright Mathematical Association Of America Jan 2002

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