# What was the best approximation of π known to ancient Babylonians?

Wikipedia's Babylonian mathematics says that the ancient Babylonians usually used a round value for $\pi$ (3). But they knew a more precise value:

Babylonian texts usually approximated π≈3, sufficient for the architectural projects of the time. The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of π as 25/8=3.125, about 0.5 percent below the exact value.

My question is: is this tablet the only one mentioning a better value for π than 3, or are there other known Babylonian sources with more precise values before the Hellenistic conquest in 4th century BCE?

• In 1950, H.C. Schepler, "The Chronology of PI" (as reprinted in Berggren, Borwein, and Borwein, "Pi: A Source Book 3rd ed.") stated: "The Babylonians, Hindus, and Chinese used the value 3. [...] No definite statement for the value of $\pi$ has yet been found on the Babylonian cylinders (1600 to 2300 B. C.). Petr Beckmann, "A History of $\pi$" (1970) mentions that a translation for the tablet found in 1936 was not published until 1950, and that it provides the value $\frac{3}{\pi}=\frac{57}{60}+\frac{36}{(60)^{2}}$, thus $\pi = 3\frac{1}{8}$, based on a circumscribed hexagon. – njuffa Nov 30 '16 at 18:54
• Alfred S. Posamentier and Ingmar Lehman, "$\pi$ A Biography of the World's Most Mysterious Number" (2004) also mentions $\pi = 3.125$ as the best Babylonian approximation to $\pi$, which may be a good indication that no new information has come to light in recent times. – njuffa Nov 30 '16 at 19:03
• Are you talking about 1-2 century bc or 1-2 century ad? The problem is that there was no such state as Babylon in 2bc-2ad. This area was conquerred by Alexander, and Hellenistic Greeks certainly knew much better approximation of pi. So it is not clear what you are asking about. – Alexandre Eremenko Nov 30 '16 at 21:18
• @AlexandreEremenko. Babylonia was not a "state" but it was a well-defined geographic and cultural area. Perhaps you have heard of the Babylonian Talmud, written not before the 4th century AD. – fdb Dec 1 '16 at 23:48
• Better Greek values go back to Archimedes, and there was much interaction between Babylonian and Greek astronomy since early 3rd century BCE, see en.wikipedia.org/wiki/Berossus – Conifold Dec 2 '16 at 17:00

No additional or more precise approximations to $$\pi$$ seem to have been found in Babylonian records up till now.

Herman C. Schepler, "The Chronology of PI", Mathematics Magazine, Vol. 23, No. 4, Mar. / Apr. 1950, pp. 216-228 (as reprinted in L. Berggren, J. Borwein, and P. Borwein, "Pi: A Source Book, 3rd Edition", Springer 2003) summed up the extent of our knowledge of Babylonian $$\pi$$-approximations at that time:

The Babylonians, Hindus, and Chinese used the value 3. It is probable that the Hebrews adopted this value from the Semites (Babylonian predecessors). No definite statement for the value of $$\pi$$ has yet been found on the Babylonian cylinders (1600 to 2300 B. C.).

Petr Beckmann, "A History of $$\pi$$", New York: St. Martin's Press 1971, states (p. 21) that a translation of the tablet found in 1936 about 200 miles from Babylon was not published until 1950, and that it derives the value of $$\pi$$ from the circumscribed circle of a hexagon, giving the value in sexagesimal fractions as

$$\frac{3}{\pi} = \frac{57}{60} + \frac{36}{(60)^{2}}$$

which yields $$\pi={3{\frac{1}{8}}}$$. Alfred S. Posamentier and Ingmar Lehmann, "$$\pi$$: A Biography of the World's Most Mysterious Number", New York: Prometheus Books 2004, likewise states (p. 44):

We now take a big leap in time to the Babylonians, which spans from 2000 BCE to about 600 BCE. In 1936 some mathematical tablets were unearthed at Susa (not far from Babylon). One of these compares the perimeter of a regular hexagon to the circumference of its circumscribed circle. The way they did this led today's mathematicians to deduce that the Babylonians used $${3{\frac{1}{8}}} = 3.125$$ as their approximation for $$\pi$$.

This seems to be a pretty good indication that no new information on Babylonian approximations to $$\pi$$ has come to light in recent years. Kazuo Muroi, "The oldest example of $$\pi \approx {3\frac{1}{8}}$$ in Sumer: Calculation of the area of a circular plot", arXiv preprint, 2016, has specific examples from Babylonian sources for the implicit use of both approximations to $$\pi$$ in the context of computing the area of a circle $$A$$ from its circumference $$c$$: $$A = \frac{c^{2}}{4\pi} \approx 0;5 \ c^{2}$$, implying $$\pi \approx 3$$; $$A = \frac{c^{2}}{4\pi} \approx 0;4,48 \ c^{2}$$, implying $$\pi \approx {3{\frac{1}{8}}}$$. He remarks:

In mathematical problems the first formula frequently occurs but the second has only been found once so far, in a problem which concerns the volume of a cylindrical log.

Muroi speculates that the Babylonians may have known more accurate approximations for $$\pi$$, which they did not use because they could not be conveniently represented as sexagesimal fractions.