While the Egyptians, Greeks, Babylonians were busy doing Mathematics and Astronomy, the Indian subcontinent was no less busy. In history, we see the birth and development of various areas in Mathematics and one of the most interesting factors in it is – the motivation to do a certain kind of Mathematics. In India, we find the earliest works on Mathematics in Sulba Sutras written (somewhere between 800 BCE to 200 BCE) by different authors. The earliest available work is said to be of Bodhayana named ‘Baudhayana Sulba Sutra’ followed by many other Sulba Sutrakaras. Sulba Sutras formed part of a bigger treatise called the Kalpam which prescribed instructions to be followed by a citizen for his well-being, his spiritual well-being, societal well-being and so on. In that, Sulba Sutras dealt with construction of fire altars for worshiping. Kalpam also formed a part of a bigger body of knowledge called the Vedanga which had six branches (Shiksha, Kalpam, Vyakaranam, Chandas, Niruktam, Jyothisham)
We restrict this article to some of the significant aspects of Sulba Sutras and Chandas.
The fire altars for worship that were prescribed by the texts involved constructions of geometrical shapes like squares, right triangles, circles, etc. The methods to do these constructions involved the knowledge of the Pythagorean Theorem. Bodhayana’s time was atleast a couple of centuries before Pythagoras and in his work, Bodhayana says in verse 1.12 – the rope corresponding to the diagonal of a rectangle makes whatever is made by the lateral and the vertical sides individually. Literally translated, it might even sound absurd. How can the hypotenuse (what is referred to as diagonal of a rectangle) be equal to the sum of the sides? He doesn’t mention about the squaring part that is involved in the theorem of the right angled triangle (a^2 + b^2 = c^2). One might feel that the theorem is not properly stated. But in the very next verse, Bodhayana lists down a few triplets (Pythagorean triplets) which cannot be enunciated if the property is not known correctly.
One needs to note that the notion of proof in Indian Mathematics is not the same as the Greek tradition which is set up by axioms and definitions. In India, Math was more of an applied pursuit. What makes it clear that Bodhayana knew the theorem well is the methods that he prescribes to construct a square that is sum of/difference of two squares. The methods make it very clear that Bodhayana had to know the theorem and not just a few examples to give out such clear methods of constructions.
Bodhayana also gives a very interesting verse that gives the value of square root of 2 correctly upto 5 or 6 decimal places. Remember, this is 800 BCE that is being talked about. He gives out a very interesting formula to calculate the value of root 2. He says, ‘Add 1 to its third and multiply 1/34 subtracted from 1 to one fourth of one third’. By doing this operation we get 577/408 which gives the value of root 2 approximately correct to 5-6 decimal places. He doesn’t leave it there. Bodhayana goes ahead and adds the word ‘savishesha’ which means, ‘close to’ whereby he says that this value is only close to or approximate value of root 2. This is a very strong claim made by him.
Mathematicians after him have tried to trace Bodhayana’s line of thought and tried to figure out how he might have arrived at the formula.
The literature of the Sulba Sutras, Chandas Shastra, etc are Sutra literature (in the form of aphorisms in prose). This literature style died out later on and what took its place was poetry. That is where Chandas comes into play.
The person who systematised the science of study of metres was Pingala Naga (300 BCE). He too, like Panini, was not the originator of Chandas but the one who systematised it so well that there was no need felt to refer to the earlier works on Chandas. Pingala makes syllables the basic component of classification and study of metres. A syllable is one or more consonants combined with a vowel. A syllable can be a short one (laghu) or a long one (guru) depending on how much time it takes to be pronounced. The time taken cannot be at our liberty, i.e. it doesn’t vary from person to person but it pre-defined for different vowels. E.g. Rama has two syllables and both are Gurus. There are two classification of metres – one based on the nature of the syllable and the other based on how much time it takes to be pronounced. The former is called Varna Vritta and the latter Matra Vritta. In the case of a Varna Vritta, the restriction of a metre is on the number of syllables it contains. So if there are n number of syllables, there are 2^n possibilities to have a metre. For a Matra Vritta, the restriction is on the time taken to say the verse. Laghu is considered as 1 and Guru as 2.
With these basic ideas, Pingala proceeds to give algorithms to list down arrays of syllables, gives formulae to count the number of strings in each array, ways to find the structure of a string in a given row number (and vice-versa), heads into binary mathematics, co-efficient of binomial expansion, so-called Fibonacci numbers, and so on. Here we see the first kind of Computer Science in its origin.
It is fascinating to see how construction of fire altars led to the birth of geometry which led into discussions in irrational numbers, number theory and algebraic nature of treating geometry. One of my favourites is the play of Permutations & Combinations in rules of Poetry and a very clever use of binary Mathematics to work out formulae devised in Pingala’s Chandas Shastra. All this only shows how Mathematics was never taught independent of other subjects but as a very clear inter-disciplinary approach. Can we hope to see such approaches in modern day classrooms?