Value of $\pi$ by Aryabhata

Question

Aryabhata gave accurate approximate value of $\pi$.

He wrote in Aryabhatiya following:

add 4 to 100, multiply by 8 and then add 62,000. The result is approximately the circumference of circle of diameter twenty thousand.

By this rule the relation of the circumference to diameter is given.

This gives $\pi= \frac{62832}{20000}=3.1416$ which is approximately 3.14159265...(correct value of $\pi$)

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From where did Aryabhata got all the above values(4, 100, 8, 62000)?

Answer 1

I am not a specialist in that history, but it seems unlikely that this formula be related to a computational algorithm he used. The constants $100$ and $62000$ do no remind anything that appears in the early computations of Pi. (At the time, inscribed polygons was the main approach.)

It really looks like this is a mnemonic way to express the figures, noting that $32$ is a multiple of $8$.

Using a diameter of $20000$ rather than $10000$ is another mystery, as it doesn't bring any simplification. Maybe Aryabhata needed $2\pi$ more frequently than $\pi$.

This is pure speculation from me.

Answer 2

Kim Plofker says that it probably was by computing the circumference of an inscribed polygon with 384 sides. She gives Sarasvati Amma as a source; I have also seen a reference to Florian Cajori with the same claim.