# Sutra route to mathematics

Dr. Ramasubramanian   | Photo Credit: Special Arrangement

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## In his lecture in Chennai recently, Dr. Ramasubramanian highlighted enumeration in Vedic literature

Dr. K. Ramasubramanian, Professor IIT Bombay, who has a doctorate in Physics and an M.A. in Sanskrit, gave a series of lectures as part of the “Instructional course on Indian sciences,” organised by Professor K.V. Sarma Research Foundation in Chennai recently. Excerpts:

Mathematics has been defined thus: Ganyate sankkyayate tadganitam. Tatpratipadakatvena tatsamjnam sastram ucyate.

So, calculation and the science that deals with calculation are defined by the word ganita. Nrisimha Daivajna wrote a commentary titled Vasanavarttika for Bhaskaracharya’s (12th century) Vasanabhashya, which in turn is a commentary on Bhaskara’s own work Siddhanta Siromani. Nrisimha Daivajna gives a four-fold classification of ganita: Vyakta, avyakta, graha and gola. Vyakta is arithmetic, which is known to all. Avyakta is that which is not evident. If you say ‘n,’ it could be any value. So, avyakta refers to algebra. Graha ganita is mathematics related to planetary calculations and includes geometry and calculus. Gola ganita is spherical geometry.

There are many kinds of enumeration in Vedic literature, and we also find Rig Vedic passages that use the decimal place value system. In Kalpa sutras, we find a geometric series. Sulba sutras deal with arithmetic, geometry and some algebra. These sutras had to do with the performance of Vedic rituals. When talking of what dakshina to give, the sutras do not explicitly give you a number, but tell you how to arrive at it. For example, in one case, it indicates that a person has to know what 2 raised to the power of 15 is, which is the dakshina he has to give. Talking of decimal notation, Dr. Ramasubramanian quoted mathematician Laplace who said, “It is India that gave us the ingenious method of expressing all numbers by means of ten symbols…’ and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”

Baudhayana sulba sutra (dateable to between 800 BCE to 500 BCE), the oldest among the Sulba Sutras, gives the Pythagorean theorem, in the context of a rectangle (deerghachaturasra), diagonal being denoted by the word akshnayaa rajju. Baudhayana gives different pairs of numbers, the sum of whose squares will equal the square of a third natural number. This is nothing but the Pythagoras theorem. For example, panchadasika ashtika (15 and 8, the sum of the squares of which will give you the square of 17). The sulba sutras also have an approximation for root of two. In

Pingala’s (300 BCE) Chandas sastra was a study of prosody. In the eighth chapter, he gives us some interesting mathematics. Every syllable can be guru (long) or laghu (short). Suppose you want to find out how many rhythms there are which have ‘n’ syllables. Since a syllable can be guru or laghu, you have a choice of 2. The number of rhythms with n syllables is, therefore, 2 raised to the power of n. Pingala gave a solution for this ‘n,’ in his Meruprastara, which is what we now call Pascal’s triangle. The entries in Meruprastara are binomial coefficients.

In Vedic literature, you find only a few chandas, but in classical literature, there is a huge variety. Halayudha wrote a commentary on Chandas Sastra, titled Mritasanjeevini. Aryabhata’s Aryabhatiya written in 499 C.E, is the earliest dateable text which we get fully. Among other things, Aryabhata gives a sine table, an approximate value of pi and an algorithm (kuttaka method) for solving linear indeterminate equations. He also gives a method of representing large numbers. Consonants are given specific integer values, and vowels are represented by powers of ten. So, if one says ‘gi,’ it means 3 multiplied by 10 to the power of 2, because ‘g’ has a value of 3, and ‘i’ is 10 to the power of 2.

While Aryabhata devoted two verses to the sine function, Bhaskara I (7th century) - the commentator for Aryabhatiya, devoted three: Bhaskara II devoted 25 and Nitayananda (17th century) devoted 65 verses. Brahmagupta (6th century C.E.) gave a principle of composition called Bhavana, for solving quadratic indeterminate equations. He gave an equation of this kind, throwing a challenge that if a person could solve it in one year, he would qualify to be called a mathematician (kurvannAvatsarAd gaNakah). Jayadeva (11th century CE), said this was as difficult as making a fly (makshika) remain still during a tornado! Brahmagupta used the Bhavana method to obtain integer solutions to what we know today as Pell’s equation.

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