Snippets from Bharatiya mathematics

In school, one of our first introductions to theorem-proving came from geometry. Sometime in high school, we learnt about Euclid's axioms and several geometric theorems that were proved from the axioms. 

One of the main techniques for proving geometric theorems was by the use of a compass and a (uncalibrated) ruler. For instance, we learnt how to bisect a line segment using a compass and marking intersecting arcs from either ends and connecting the arcs together to cross the line. 

Every method of inquiry has its own advantages and limitations, and embeds within it some deep assumptions about the underlying worldview. So too it is with the way in which Euclid's methods developed. Geometry literally means "earth measurements"-- and Euclid's geometry is deeply rooted in this application area. 

One of the vexing problems with the compass and ruler approach is to make arbitrary divisions of geometric elements. For instance, we used to be challenged by "Maths Olympiad" problems that asked us to trifurcate a line segment into three equal length segments using a compass and a ruler. I vaguely remember having learned the technique from some form of a "brain vita" tutor, but have forgotten it since. 

Recently, I came across Jonathan J. Crabtree-- a mathematics researcher and historian from Australia, who has dedicated himself to the study and popularization of Bharatiya maths dating back to the time of Aryabhata and Brahmagupta. He runs a website called Podometic.in to popularize Bharatiya maths. Crabtree also clarifies that this is not the same as "Vedic Maths". Vedic period preceded advancements in Bharatiya mathematics by a couple of millennia. While there are references to very large and very small numbers in Vedic literature especially based on powers of 10 (resulting in some theories that the notion of zero and infinity was known at that time), there is hardly any literature on formal theorems and mathematical results from that time. Crabtree has even offered a reward of INR 5,00,000/- for anyone unearthing the 16 Sutras in the book ‘Vedic Maths’ in Vedic literature. 

Bharatiya Maths in written form, as we know today, dates back to about 2000 years ago. One of the earliest mathematical treatises is the Brahmasputasiddhanta by Brahmagupta, which is a compilation of several mathematical techniques (thus suggesting that actual mathematics and its application, predated this date). 

Predating the Brahmasputasiddhanta by about 1500 years are the "Shulba Sutras" that provide proofs for several theorems of arithmetic, number theory and geometry. 

The Shulba Sutras around geometry, do not use a ruler and a compass-- but instead, uses a ruler and ropes. And we can see that, just by changing our underlying tools, we can easily address a number of problems that were challenging when tried to be solved using a ruler and a compass. 

For example, Crabtree has a cartoon that shows how to square a circle with rope the Shulba Sutra way, which is impossible the Western way with a compass. 

Let us take the problem of trifurcating a line segment-- or even dividing a line segment into any number of parts. We start as follows. Let us say, we want to divide a line segment into k equal length segments. Start by taking k rulers of equal length (any length). Place them horizontally in a compact sequence one after the other as shown in the figure below. Now, take the line segment that you want to subdivide, and place it in some angle to the horizontal sequence of equal length rulers. Next, take a long ruler and place it such that it connects with the other end of the horizontal line with the other end of the angular line. Now move this rigid ruler inwards and mark off segments from the angular line, whenever a segment boundary is crossed in the horizontal line. 

The following video also illustrates this principle:

 

We can use the same technique to segment a circle into any number of arcs so that their angle at the center is the same. 

For doing this, imagine the angular line in the figure above, as a rope that is stretched out tightly. This rope would have been first wrapped around the circle which we need to segment, and hence represents its circumference. Now use the same technique as earlier, to segment the tightly drawn out rope. Then, wrap this rope again on the circle and mark points on the circumference based on the segments. Connecting these points to the center of the circle, gives us an angular segmentation of the circle into any number of arcs. 

Similarly, the "Pythagoras theorem" was also well known in ancient India centuries before Pythagoras. The above diagram illustrates the proof that was normally used (with altar design being the most widely used application). If we take a square embedded within another larger square such that it segments any edge of the square into two parts: a and b, then we see that the area of the inner square is precisely (a+b)² - 2ab. 

This is illustrated further in the figure above. The right-angle triangle theorem, which may be better termed as the Baudhyana Theorem shows the inner square reorganized into two rectangular and two square regions in the lower figure, essentially dividing c² into a² and b². 

In the talk above, Crabtree also gives several other examples of how Bharatiya thought towards mathematical concepts differed from Western thought, and the implications of same. Note specifically about the concept of negative numbers. In Bharatiya Maths, negative numbers were not considered "lesser than 0" as we are taught in Western maths. They were considered to be an increasing sequence greater than 0, but with opposite semantics of the positive line. This has several implications in interpretation. For example, a wealth of  -7 does not represent a "lesser wealth" than -4, but a "greater debt" than -4. 

The definition of 0 is precisely what results when a negative number is added to a positive number of the same quantity.