The Sulbasutras

Around 1500 B.C., a "tribe" of people known as the Aryans, invaded an ancient civilization called the Harappans, who lived around the Indus Valley in what is now Southeast Asia. With them, the Aryans brought to India the earliest known Indian literary record, the language of Sanskrit, and most of all the Vedic religion. The Vedic religion is centered around religious scriptures known as the Vedas. The Vedas not only contain religious text like the Bible does, but they also contain discoveries and beliefs on math. Through the Vedas, we are able to find the earliest forms of Ancient Indian math, since everything was passed down orally before the Vedas. This paper will be about appendices of the Vedas called the Sulbasutras. So far, there have been 4 Sulbasutras found, the earliest one dating to around 800 B.C., composed by a man of the name of Baudhayana.

The Sulbasutras were made mainly for the purpose of construction of religious items such as statues and altars. The Aryans believed that since god is perfect, anything and everything made to worship god should also be perfect. They saw math in a religious sense, for it provided them with a way to attain the perfection needed in their construction. Since the Aryans made the Sulbasutras mostly as a guide for construction, they contain mainly geometric knowledge such as constructions, triangle, and so on. First, I will explain some of the knowledge the Aryans had on triangles that was found in the Sulbasutras.

The first thing on triangles that draws attention in the Sulbasutras was a theory made by the Aryans that was the same as Pythagoras' theorem, only it looked at a triangle in a more practical way that could be used for construction. The most detailed description of this theory is found in the fourth Sulbasutra written by Katyayana. As Katyayana put it in his Sulbasutra, "The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together." Using this theorem, the Sulbasatras also made a list of Pythagorean triplets, such as (5, 12, 13), (12, 16, 20), (8, 15, 17), (15, 20, 25), (12, 35, 37), (15, 36, 39), (⁵/₂ , 6, ¹³/₂), and (¹⁵/₂ , 10, ²⁵/₂).

The next major mathematical "feat" found in the Sulbasutras is the types of equations that are shown how to be solved in it. The sulbasutras effectively described how to solve everything from simple equations to quadratic equations. The Sulbasutras gave the standard form of a quadratic equation as ax(squared) + bx=c, and ax(squared)=c. The sulbasutras also give a number of approximations for pi, the closest one being 3.225. Though this number isn't particularly accurate, it served the purposes needed for the construction it was being used for.

Another amazing thing is the calculation for the square root of two in the Sulbasutras. In the Sulbasutras, the square root of 2 was found to be 1.414215686, which we now know is correct to 5 decimal places. How Baudhayana, the writer of the first Sulbasutra got this answer is shown below.

2 = 1 + ¹/₃ + ¹/_{(3 4)} - ¹/_{(3 4 34)} = ⁵⁷⁷/₄₀₈

But how Baudhayana got the numbers above is, sadly, unknown (though there are many theories), seeing that none of the authors of any of the sulbasutras showed there work and none of them cared to write proofs on there calculations and theories.

As seen by some of the processes above, the math of the Ancient Indians was very comprehensive and it has helped contribute much to the math we use today. Though the Sulbasutras have a lot of mathematical knowledge in them themselves, they contain only a tiny bit of what the Ancient Indians were discovering about the world.

Bibliography

1. (Book Source) Orientalism and Race. Tony Ballantyne. Volume Number 1. Page Number 6-13.
2. (Book Source) Hinduism and Math. Thiagu S. Garg. Volume Number 1. Pages 29-45.
3. (Internet Source) The Indian Sulbasutras. J.J. O'Connor and E.F. Robertson. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html.