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Algorithms in Indian Astronomy K.Ramasubramanian Cell for Indian Science and Technology in Sanskrit Department of HSS, IIT Bombay Powai, Mumbai 400 076, India [email protected]

Abstract Indian Astronomy is rich in algorithms. The algorithms presented in the Indian astronomical texts have varying degrees of complexities starting from the simple trair¯ a´sika rule, to the treatment of parallax in a solar eclipse or the computation of the elevation of lunar cusps. In the present article we will discuss a few algorithms that are representative of the ingenuity and continuity of the Indian astronomical tradition. We start with the interpolation formula presented by Brahmagupta (c.665 AD) and then proceed to describe a select few algorithms from Tantrasangraha ˙ of N¯ilakan.t.ha composed in 1500 AD. Here we present the algorithm for the calculation of time from shadow measurements and the exact algorithm for the computation of lagna and the time for the duration of an eclipse. We also comment on the iterative process known as avi´ses.akarma which aims at circumventing the problem of interdependencies among several variables.

1

Introduction

It is not uncommon to find words which originate with a different connotation and in due course pick up a completely different connotation. The word algorithm forms a good example of this. Its origin can be traced back to the Persian mathematician, al-Khw¯arazm¯i (800-847 AD). It is quite interesting to note the observations made by D.E. Knuth in this context [1]: 183

184

K.Ramasubramanian

In the middle ages, abacists computed on the abacus and algorists computed by algorism. Following the middle ages, the origin of this word was in doubt, and early linguists attempted to guess at its derivation by making combinations like algiros [painful] + arithmos [number]; others said no, the word comes from “King Algor of Castile”. Finally, historians of mathematics found the true origin of the word algorism: it comes from the name of a famous Persian textbook author, Abu Ja‘far Mohammed ibn Musa al-Khw¯arizm¯i (∼ 825 AD) - literally, “Father of Jafar, Mohammed, son of Moses, native of Khw¯arizm.” Khw¯arizm is today the small Soviet city of Khiva. al-Khw¯arizm¯i wrote the celebrated book Kitab al jabr w’al-muqabala (“Rules of restoration and reduction”); another word, “algebra”, stems from the title of his book, although the book wasn’t really very algebraic. Gradually the form and meaning of “algorism” got distorted; The change from “algorism” to “algorithm” is any body’s guess. The remarks by the well-known historian C.B.Boyer in this context are also noteworthy [2]. . . . . . . when subsequently Latin translations of his (al Khw¯arizm¯i’s) work appeared in Europe, careless readers began to attribute not only the book but also the numeration to the author. The new notation came to be known as that of al-Khw¯arizm¯i, or more carelessly, algorismi; ultimately the scheme of numeration making use of Hindu numerals came to be called simply algorism or algorithm. The terms process, method, technique, procedure, routine, and so on all essentially refer to a sequence of operations to be carried out to accomplish a given task. The word “algorithm” though similar, connotes something more. For a procedure to be termed algorithm it must terminate after n steps and upon termination it must yield a sensible result. However, this is not true of all procedures. In this sense, all algorithms are procedures; but all procedures are not algorithms. Indian astronomy is essentially algorithmic in nature. The algorithms presented are precise and fairly sophisticated [3]. Some of them are

Algorithms in Indian Astronomy

185

amazingly accurate [4]. We shall illustrate these points by considering a few examples.

2

Brahmagupta’s interpolation formula

Interpolation is the art of reading between the lines in a table. The rule of trair¯ a´sika [5,6] employed in Indian astronomy is close to what is known as the first-order interpolation in modern parlance. This technique has been extensively applied to solve a variety of mathematical and astronomical problems, beginning from the evaluation of sine func¯ tion to the calculation of eclipses, at least from the time of Aryabhat .¯ıya (c.499 AD). It is quite interesting to note that Brahmagupta introduced the second order interpolation formula to determine more precise value of the sine function, called jy¯ a , for an arbitrary angle, from the set of tabulated values of sine given at fixed intervals. The following verse from his famous work Khan.d.akh¯ adyaka (c.665 AD) explains the algorithm [7]: ! " # %$ & '( ) +* ( ,. - 0 213 1 45 ) 0 67 819: ) ;

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Abstract Indian Astronomy is rich in algorithms. The algorithms presented in the Indian astronomical texts have varying degrees of complexities starting from the simple trair¯ a´sika rule, to the treatment of parallax in a solar eclipse or the computation of the elevation of lunar cusps. In the present article we will discuss a few algorithms that are representative of the ingenuity and continuity of the Indian astronomical tradition. We start with the interpolation formula presented by Brahmagupta (c.665 AD) and then proceed to describe a select few algorithms from Tantrasangraha ˙ of N¯ilakan.t.ha composed in 1500 AD. Here we present the algorithm for the calculation of time from shadow measurements and the exact algorithm for the computation of lagna and the time for the duration of an eclipse. We also comment on the iterative process known as avi´ses.akarma which aims at circumventing the problem of interdependencies among several variables.

1

Introduction

It is not uncommon to find words which originate with a different connotation and in due course pick up a completely different connotation. The word algorithm forms a good example of this. Its origin can be traced back to the Persian mathematician, al-Khw¯arazm¯i (800-847 AD). It is quite interesting to note the observations made by D.E. Knuth in this context [1]: 183

184

K.Ramasubramanian

In the middle ages, abacists computed on the abacus and algorists computed by algorism. Following the middle ages, the origin of this word was in doubt, and early linguists attempted to guess at its derivation by making combinations like algiros [painful] + arithmos [number]; others said no, the word comes from “King Algor of Castile”. Finally, historians of mathematics found the true origin of the word algorism: it comes from the name of a famous Persian textbook author, Abu Ja‘far Mohammed ibn Musa al-Khw¯arizm¯i (∼ 825 AD) - literally, “Father of Jafar, Mohammed, son of Moses, native of Khw¯arizm.” Khw¯arizm is today the small Soviet city of Khiva. al-Khw¯arizm¯i wrote the celebrated book Kitab al jabr w’al-muqabala (“Rules of restoration and reduction”); another word, “algebra”, stems from the title of his book, although the book wasn’t really very algebraic. Gradually the form and meaning of “algorism” got distorted; The change from “algorism” to “algorithm” is any body’s guess. The remarks by the well-known historian C.B.Boyer in this context are also noteworthy [2]. . . . . . . when subsequently Latin translations of his (al Khw¯arizm¯i’s) work appeared in Europe, careless readers began to attribute not only the book but also the numeration to the author. The new notation came to be known as that of al-Khw¯arizm¯i, or more carelessly, algorismi; ultimately the scheme of numeration making use of Hindu numerals came to be called simply algorism or algorithm. The terms process, method, technique, procedure, routine, and so on all essentially refer to a sequence of operations to be carried out to accomplish a given task. The word “algorithm” though similar, connotes something more. For a procedure to be termed algorithm it must terminate after n steps and upon termination it must yield a sensible result. However, this is not true of all procedures. In this sense, all algorithms are procedures; but all procedures are not algorithms. Indian astronomy is essentially algorithmic in nature. The algorithms presented are precise and fairly sophisticated [3]. Some of them are

Algorithms in Indian Astronomy

185

amazingly accurate [4]. We shall illustrate these points by considering a few examples.

2

Brahmagupta’s interpolation formula

Interpolation is the art of reading between the lines in a table. The rule of trair¯ a´sika [5,6] employed in Indian astronomy is close to what is known as the first-order interpolation in modern parlance. This technique has been extensively applied to solve a variety of mathematical and astronomical problems, beginning from the evaluation of sine func¯ tion to the calculation of eclipses, at least from the time of Aryabhat .¯ıya (c.499 AD). It is quite interesting to note that Brahmagupta introduced the second order interpolation formula to determine more precise value of the sine function, called jy¯ a , for an arbitrary angle, from the set of tabulated values of sine given at fixed intervals. The following verse from his famous work Khan.d.akh¯ adyaka (c.665 AD) explains the algorithm [7]: ! " # %$ & '( ) +* ( ,. - 0 213 1 45 ) 0 67 819: ) ;

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