Ancient Indian Geometry: Math Mazes
Any math scholar would say they know everything about geometry, integral calculus, trigonometry or circular quadrilateral. But would they be able to say what is Shulvasutra, or Kriyakramakari? Well, Shulvasutra is ancient Indian geometry; and Kriyakramakari spells out the infinite series of Pi (π). There is need to go into these to come to a stand on how to take our sciences forward
Since there is evidence that India first invented Zero as an integer, Indians deserve to get paid thousands of dollars for that. But this is just a happenstance in the history of science. Just as the Archimedes Principle is a happenstance in which a Greek thinker created a massive leap in science does not entitle every Greek today to get paid thousands of dollars for inventing that! So let us proceed with the reality of Indian science discoveries and the geometry from those times.
Indian Geometry: Preceding Pythagoras
Now let us step back by a few centuries, to say around 800 to 1000 BCE. And here I quote directly from Sattrajeet Mukherjee’s seminal paper. (Ref: For Mukherjee’s short CV read previous article in this series.)
To understand, grasp and act according to the tenets of the four Vedas, we need to take recourse to six other scriptures, which are jointly called the “Shada Vedanga”, or the Six Vedangas, of which the primary is the Kalpa.
And Kalpa itself is divided into four parts:
- Shroutrasutra, which describes and lays down the systems of conducting the various yajnas, or rituals involving the Fire
- Grihyasutra, or the system of living a family life
- Dharmasutra, or the path of righteous living, and
- Shulvasutra, or the measurements of the various yajna altars.
In modern parlance, this Shulvasutra is called Geometry. The three main persons who scripted this Shulvasutra are Katyayana, Baudhyana and Aapastamba.
According to historian RC Mazumdar, this Shulvasutra is estimated to have been created circa 800 to 1000 BCE.
One of the slokas of this sutra mentions “Deerghachaturasra”, that is the Rectangle, and it describes it as having a “Parshvamani,” or long hand, or the base, and a “Teerjatmani” or a short hand, that is, the perpendicular side. And the sloka says that (base)^2 + (height) ^2 = diagonal ^2, where the diagonal is the hypotenuse! (Baudhayan – 1/48; Katyayana – 2/11; and Aapastamba – 1/7)”/
Aryabhattiya is the treatise that first brought to fore the Theorem of Circular Quadrilateral. Calculating the area of a circle and the area of a trapezium are also Aryabhatta’s contribution. In fact, it is Aryabhatta who established the method of calculating the area of any quadrilateral
However, it is possible that the ancient Babylonians may have had a sense of this theorem too, for Roy shows a clay tablet excavated from ancient Babylonian sites, particularly Tablet YBC 7289, that has three chess-board like boxes on the three sides of a right-angled triangle and that seems to suggest the same theorem. (Ref: For Roy’s short CV read previous article in this series.)
Circular Quadrilateral and Integral Calculus
Aryabhattiya is the treatise that first brought to fore the Theorem of Circular Quadrilateral. Calculating the area of a circle and the area of a trapezium are also Aryabhatta’s contribution.
In fact, it is Aryabhatta who established the method of calculating the area of any quadrilateral, which was an ancient form of what we now call the Trapezoidal Rule of Integral Calculus.
Arithmetic Mean and Arithmetic Series finds mention in the 19th sloka of the section on Mathematics in his Aryabhattiya, followed by the theory on the value of the sum total of Arithmetic Progression Series (sloka 20 of the same section).
In fact, the French-Sanskrit scholar Léon Rodet (1850 to 1895) – among the first to work on Aryabhatta ‑ says that this would have required Aryabhatta “having knowledge of solving Quadratic Equations”.
So, is it possible that even five hundred years before the fabled Shridhar Acharya, who specialised in Quadratic Equations, India had discovered how to solve that problem? Seems so, at least by inference.
Even Trigonometry, the foundation of modern mathematics, was a contribution of Aryabhatta. In Sanskrit, ‘sine’ was termed as “jya” and “cosine” was “koti-jya”.
Lest we forget, let me mention that in ancient Indian sciences, we find the first ever mention of Gravitation in the treatise titled Siddhanta Shiromani (1150 CE) by Bhaskaracharya. He wrote: “Each celestial object is being pulled by the earth towards its own centre.
Bhaskar also posits the method of calculating the volume of a sphere, and unwittingly dwells upon what is now called Integral Calculus. And then, we come to another scientific genius, Madhavacharya, born in Sangamgram, Kerala in 1340.
Towards the end of the 13th century, however, all this collapsed, with Nalanda, Vikramashila and other internationally known centres of learning crumbling, and the glorious past of Aryabhatta, Brahmagupta, Madhav, Bhaskar or Shridhar got buried under the rubbles of history.”
He, in fact, worked out all the basics of calculus. His genius can be realised if we see how he had worked out the “Infinite Series of Sine-Cosine Function” of trigonometry, and also Power Series and Rational Approximation of Infinite Series, the Infinite Series of π (pi) which is often referred to as the Madhav-Leibniz Series.
The unique thing about the series given by Madhava is that he gave this series in form of a beautiful verse.
Infinite series for pi — as given in Kriyakramakari, means: “The diameter multiplied by four and divided by unity (is found and stored). Again, the products of the diameter and four are divided by the odd numbers like 3,5 etc., and the results are subtracted and added in order (to earlier stored result).”
En passant, in this long history of ancient Indian scientists, we may have left out the name of one such sterling name: Shridhar Acharya, born in Bengal in 870 CE. He resolved the problem of the Quadratic Equation as also dealt with formulation of Equated Monthly Installment (EMI) and Derivative Banking Product. But that is a story better told in greater detail on some other day.
Towards the end of the 13th century, however, all this collapsed, with Nalanda, Vikramashila and other internationally known centres of learning crumbling, and the glorious past of Aryabhatta, Brahmagupta, Madhav, Bhaskar or Shridhar got buried under the rubbles of history.”
So, there is need to study a whole gamut or researches, especially by foreign scholars, to avoid any clash of interest to see what the true extent of Indian mathematics was, without robbing claims of other civilisations to have come to similar ides, or close approximates of them, either simultaneously and independently, or through sharing via traders from the respective countries.
This dispassionate attitude in dealing with such a vast and complex issue, without getting into semantics alone can lead us towards a scientific delineation of a policy framework
(Coming up: Indian trigonometry and astronomy)
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