Utility in Ancient Indian Texts

A cartoon that appeared in today's New Indian Express. This clearly shows the bias of the cartoonist. Anything that is Indian is a burden whereas learning things in a western context is the "correct" way?

I will comment on Ancient Indian Mathematics because that is one area where I have made a comparative study. I will just highlight one topic in mathematics and its utility in teaching a complex topic using the works of Indian mathematicians.

Quadratic indeterminate equations (equations of the form x^2-Dy^2=1) is a topic of advanced number theory currently introduced at MSc part I because the methods used to find integer solution to these equations use the continued fractions method of Lagrange (1766). Bhaskara II (1140 CE) gave a Chakravala (cyclic) method which uses elementary algebra to solve such equations. Hermann Hankel calls the chakravala method "the finest thing achieved in the theory of numbers before Lagrange" (Kaye 1919 Pg 337). Hankel worked with great mathematicians like Mobius, Reimann, Weierstrass, and Kronecker. In fact, the impetus for Bhaskara II was provided by Brahmagupta (628 CE) in his book Brahmasputtasiddhanta where he discusses how to find infinitely many integer solutions from a given integer solution. I am not glorifying the contribution but one needs to understand that a study of Brahmagupta's work can help us introduce these topics at the high school level. How can we expect people to read these fantastic works of eminent mathematicians if the media is going to ridicule the government for introducing anything that is ancient? Whose loss is it? We need to evaluate these things and not just analyze from a political angle only.

A question that arises how can I argue that such topics can be introduced at the high school level? Simple, at Raising a Mathematician Training Program (RAMTP 2017, 2018) we introduced Quadratic Indeterminate Equations in RAMTP 2017 using Brahmagupta's approach and easily grasped by the students. In 2018, we provided students with the reading material and they self-studied the topic and solved problems without being introduced to the topic in a formal way. This helped us introduce Bhaskara's Chakravala method to find an integer solution to these equations. Thus, school students were exposed to a topic which they might have never learned and this was possible mainly because we used Brahmagupta's and Bhaskara's elementary algebra methods.

Conclusion:
1) Such cartoons only create low self-esteem among the countrymen of its nation and devoid them of an ocean of knowledge hidden in the texts written by these great ancient scholars.
2) Learning Pythagoras theorem is not glorifying the ancient Greeks.
3) If the pedagogical goals can be served in a better way by using these ancient texts, we should accept it with open arms.
4) I am only commenting on an aspect that I have researched on and abstaining from commenting on other aspects.

What is special about square root of 2?

Mathematics of ancient India can be very useful to understand the motivation behind the development of Mathematics and can help the students understand how applied Mathematics can be. The interesting questions that can be asked what is so special about √2. Why did Boudhayana specifically provide its value? The length of the diagonal of a square is √2 times the side of the square. Why should I know the diagonal of a square? Squares were used for geometrical constructions and had applications in designing sacrificial altars.

For example: Boudhayana's sulbasutra (800 BCE) provides the value of √2

Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6) gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:¹

samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet
tac caturthenātmacatustriṃśonena saviśeṣaḥ

samasya problem or puzzle to be solved.

dvikaraṇī. means “that which produces 2” or diagonal of a square

pramāṇaṃ means proof or first term in a rule of three sum

vardhayet means increase

saviśeṣaḥ one that remains

The last word is the most important part of the sutra because it tells us that Boudhayana knew that the result will be an approximation and not the exact value of √2.

The prescription in sulbasutra can be translated as:

Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. The increased length is a small amount in excess²

Thus the above passage from the Sulbasutram gives the approximation:

The symbol ≈ instead of = is because of the word saviśeṣaḥ

In fact this method will in one more step obtain: ³

 ,

Where the only numerical computation needed is 1154 = 2[(34) (17) -1] and, moreover, the method shows that the square of this approximation is less than 2 by exactly

This approximation is accurate to eleven decimal places.

Another thing that I observe is how thin were the compartmentalisation of Mathematics. For example root 2 by Boudhayana uses Geometry. However, value of √2 can be also found using Brahmagupta's varga-prakrti (Brahma-sputa-siddhanata). Brahmagupta (628 CE) uses algebra to find √2 by solving the equation  using “the principle of composition”⁴ or bhavana and finding the ratio  for large values of x & y. 

As can be observed the first positive integer solution for (x, y) is (3, 2) and there are infinitely many solutions to this problem. Although, Brahmagupta couldn’t give the solution to all types of this problem. It sowed the seeds for Acharya Jayadeva and Bhaskara II to design Chakravala process to get a general solution. Another solution to the above equation is (17, 12) which corresponds to Boudhayana’s value of 17/12 if we take first 3 terms  (1+1/3+1/12). Another solution is (99, 70) or (577,408). 

1) Can you figure out how to obtain 577/408 in the Boudhayana’s value of √2? 

2) Can you tell why does x/y gives the approximate value of √2? 

3) Why does the approximation improve for larger values of x & y?

It is needless to say that finding √2 is technically a mundane arithmetic exercise and is not interesting.

Learning this way, we have given an altogether different meaning to √2 and also shown how there can be multiple ways to tackle a same problem in Mathematics. 

Conclusion:

This is just one of the many examples where Ancient Indian Mathematics can be used to motivate higher order thinking among school students and break the barriers of conditional thinking and such inter-linkages will be very useful in contemporary times when knowledge is becoming more and more inter-disciplinary. It is very interesting that the ancient knowledge especially in India was multi-disciplinary. 

References:

1 http://self.gutenberg.org/articles/baudhayana_sulba_sutra#Square_root_of_2

2 This last word is translated by some authors as "The increased length is called savi´e¸a". I follow the translation of "savi´e¸a" given by B. Datta on pp. 196-202 in The Science of the Sulba, University of Calcutta, 1932; see also G. Joseph (The Crest of the Peacock, I.B. Taurus, London, 1991) who translates the word as "a special quantity in excess" and also based on a presentation by Prof Kannan at International Conference on History and Development of Mathematics, Jaipur 2013

3 Square roots in the Sulbasutra – David W Henderson http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html#FOOTNOTE

4 Rationale of the Chakravala process of Jayadeva and Bhaskara II (Historia Mathematica, Page 168)