Squaring the circle

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Squaring the circle  (1913) 
by Srinivasa Ramanujan

Squaring the circle

(Journal of the Indian Mathematical Society, v, 1913, 138)

Let be a circle with centre , of which a diameter is . Bisect at and let be the point of trisection of nearer . Draw perpendicular to and place the chord .

Join , and draw and parallel to . Place a chord , and draw the tangent . Join , and . Cut off . Draw parallel to , meeting at .

Then the square on will be equal to the circle approximately.

For ,
where is the diameter of the circle.
Therefore .

But and are equal to and respectively.

Therefore , and .
Hence ,
and .
Approximately squaring the circle.svg
But ,
and .
Therefore , very nearly.

Note.—If the area of the circle be square miles, then is greater than the true length by about an inch.

This work was published before January 1, 1926, and is in the public domain worldwide because the author died at least 100 years ago.