Using a rope to square a circle - Śulba Sūtras

 Using rope to square a circle

What is squaring a circle?

It is drawing a square equal to the area of a given circle.

Area of a circle = Pi*r^2 where r is the radius of the circle

Area of square = s^2 where s is the side of the square

Therefore Pi*r^2=s^2

implies s= r*sqrt(Pi)

But Pi is a transcadental number. Therefore its square root cannot be computed exactly

This squaring of a circle using a compass and straight edge is considered impossible.

Hence squaring a circle is used by many as a metaphor for doing the impossible.

However it can be easily done via a rope as explained the Sulbasutras.

The Sulbasutras are appendices to the Vedas which give rules for constructing altars.

(Sanskrit śulba: "string, cord, rope") are sutra texts belonging to the Śrauta ritual. Śrauta is a Sanskrit word that means "belonging to śruti", that is, anything based on the Vedas of Hinduism. It is an adjective and prefix for texts, ceremonies or person associated with śruti.

In Bharat 800 BCE when the Śulba Sūtras (rules of rope) were written ropes were used for altar construction.

As per method surfaced again by Maths historian Jonathan Crabtree (@jcabtree on twitter. podometic.in his website on ancient Bharatiya mathematics) (See eminent Prof. C. K. Raju's work as well)

Method:

1. Take a circle with radius = 1 unit (say 1 inch). 

Circumference of circle = 2*pi*r= 2*pi in this case.

Area = pi*r*r = pi in this case

2. Use a rope and measure 1/4th of the circumference of the circle. Make this length = a

a= (1/4)*2*pi = pi/2

3. Now put that same rope along the vertical diameter of the circle.

4. From the point where the rope finishes on the vertical diamater of the circle, draw a line parallel to the horizontal diameter of the circle to intersect the circle. Now join the top part of the rope to this point of intersection to form the hypotenuse of the triangle. This side is the side of your square.

Length of side = sqrt(pi). (Proof given below). Area of square using that (c) as side = sqrt(pi)*sqrt(pi) = pi = Area of circle.

The proof given by Jonathan Crabtree is a bit lengthy and complicated.

So here is a considerably easier proof using Pythagoras/Baudhayana theorem:

Remember radius of the circle is 1 unit (1 inch)

https://www.intmath.com/plane-analytic-geometry/squaring-the-circle.php