Square Roots in the Sulbasutra

by rawemotions Wed Feb 04, 2015 9:46 am

http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html

Excerpts

In this paper I will present a method for finding the numerical value of square roots that was inspired by the Sulbasutra which are Sanskrit texts written by the Vedic Hindu scholars before 600 B.C.. This method works for many numbers and will produce values to any desired degree of accuracy and is more efficient (in the sense of requiring less calculations for the same accuracy) than the divide-and-average method commonly taught today.

Several Sanskrit texts collectively called the Sulbasutra were written by the Vedic Hindus starting before 600 B.C. and are thought² to be compilations of oral wisdom which may go back to 2000 B.C. These texts have prescriptions for building fire altars, or Agni. However, contained in the Sulbasutra are sections which constitute a geometry textbook detailing the geometry necessary for designing and constructing the altars. As far as I have been able to determine these are the oldest geometry (or even mathematics) textbooks in existence. It is apparently the oldest applied geometry text.
It was known in the Sulbasutra (for example, Sutra 52 of Baudhayana's Sulbasutram) that the diagonal of a square is the side of another square with two times the area of the first square as we can see in Figure 1.
Thus, if we consider the side of the original square to be one unit, then the diagonal is the side (or root) of a square of area two, or simply the square root of 2, that is $Square Roots in the Sulbasutra 3835660d$ . The Sanskrit word for this length is dvi-karan[size=16]i or, literally, "that which produces 2".[/size]

The Sulbasutra³ contain the following prescription for finding the length of the diagonal of a square:
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 excess ([size=16]savi´e¸a)⁴.[/size]
Thus the above passage from the Sulbasutram gives the approximation:

$Square Roots in the Sulbasutra 3835660e$ .

I use [size=16]» instead of = indicating that the Vedic Hindus were aware that the length they prescribed is a little too long (savi´e¸a). In fact my calculator gives:[/size]

$Square Roots in the Sulbasutra 3835660f$

and the Sulbasutram's value expressed in decimals is

$Square Roots in the Sulbasutra 38356610$

So the question arises — how did the Vedic Hindus obtain such an accurate numerical value? Unfortunately, there is nothing that survives which records how they arrived at this [size=16]savi´e¸a.[/size]
There have been several speculations⁵ as to how this value was obtained, but no one as far as I can determine has noticed that there is a step-by-step method (based on geometric techniques in the Sulbasutram) that will not only obtain the approximation:

$Square Roots in the Sulbasutra 38356611$ ,

but can also be continued indefinitely to obtain as accurate an approximation as one wishes.

by Vakavaka Pakapaka Wed Feb 04, 2015 10:56 am

The New AP state should build the statues of both Baudhayana and Apasthamba in Godavari near Rajamahendravaram (just like what NTR did for Buddha near tankbund in Hyderabad).

Their contributions were immense.

by Vakavaka Pakapaka Wed Feb 04, 2015 1:44 pm

For traditionists, the most important contribution of Baudhayana is the method he developed to worship Rudra. Rudra worship through Mahanyasam involves the devotee purifying himself, identifying himself with Rudra and performing worship of Rudra by Rudra! The steps involved are full of philosophical thoughts.

Here is Rudram:

https://www.youtube.com/watch?v=vQjBQJqi0Ak

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