When was $\pi$ first suggested to be irrational?

Question

When was $\pi$ first suggested to be irrational?

According to Wikipedia, this was proved in the 18th century.

Who first claimed / suggested (but not necessarily proved) that $\pi$ is irrational?

I found a passage in Maimonides's Mishna commentary (written circa 1168, Eiruvin 1:5) in which he seems to claim that $\pi$ is irrational. Is this the first mention?

Answer 1

There is a claim on the wikipedia article on irrational numbers that Aryabhata wrote that pi was incommensurable (5th century) but the question had to be asked as soon someone realized there was such numbers... (that was 5th century Before Christ)

Answer 2

I don't know about the first suggestion but as far as I know the first proof was in 1761 by Johann Heinrich Lambert (Wikipedia link).

Answer 3

From a non-wiki source:

Archimedes [1], in the third century B.C. used regular polygons inscribed and circumscribed to a circle to approximate : the more sides a polygon has, the closer to the circle it becomes and therefore the ratio between the polygon's area between the square of the radius yields approximations to $\pi$. Using this method he showed that $223/71<\pi<22/7$.

Found on PlanetMath: Pi, with a reference to Archimedes' work...

Answer 4

Bhaskara I (the less famous of the two Bhaskara) wrote a comment on Aryabhata in 629, where he gives Aryabhata's approximation $\pi\approx {62832\over 20000}=3.1416$, and states that a nonapproximate value for this ratio is impossible. See Kim Plofker: Mathematics in India, p. 140.

On the surface this would seem like a clear statement equivalent to the irrationality of $\pi$, but perhaps it is not quite so easy. The "reason" Bhaskara gives is that "surds (square roots of nonsquare numbers) do not have a statable size". It is believed that Aryabhata's method of approximating $\pi$ is essentially the same as Archimedes', computing the circumference of an inscribed regular polygon in a circle with 384 sides. This requires the computation of many surds, so perhaps Bhaskara only meant that he could not get an exact value of the circumference of the inscribed polygon.