## On the Ancient Babylonian Value for Pi

I have written about the ancient Egyptian value for $\pi$ before, concluding that while the Egyptians had a procedure for finding the area of a circle, they didn’t have any real understanding of the ratio.

Conversely, the Babylonians found $\pi$ as a ratio (3.125) but, oddly enough, didn’t handle circles as well as the Egyptians.

Consider this tablet from the Yale Babylonian Collection known as YBC 7302:

Since the photo is hard to interpret, here’s a version with the numbers made clear:

That’s cuneiform, which fortunately in numbers isn’t too hard to read.

Based on positional context, 3 seems to be the circumference of the circle. Using the formulas $C=2 \pi r$ and $A= \pi r^2$,

$3=2 \pi r$
$\frac{3}{2 \pi}=r$
$A=\pi(\frac{3}{2 \pi})^2$
$A=\frac{9}{4 \pi}$

While we’re used to a base 10 system ($d_1 + \frac{d_2}{10} + \frac{d_3}{100} + \frac{d_4}{1000} + ...$) the Babylonians used a base 60 system ($d_1 + \frac{d_2}{60} + \frac{d_3}{60^2} + \frac{d_4}{60^3} + ...$). Also at the time the Babylonians had no place value, so if there’s the number “45” only context can tell if they mean 45, 45/60, or 45/60*60.

If we take the 45 from the tablet to mean 45/60:

$\frac{45}{60} = \frac{9}{4 \pi}$

and the Babylonian value of $\pi$ turns out to be simply 3.

Now, if you remember my Egyptian post, their value of $\pi$ was also pulled out of a circle area procedure in the exact same manner. Arguably it is unfair to go any further; there are good reasons to call the Babylonian value of $\pi$ 3 and stop there.

However, there was some tablets found in 1936 which throw the case for a loop.

The tablets were found in Susa, in ancient times capital city of the Elamite Empire. (It also happens to be one of the oldest continuously inhabited cities in the world, at more than 7000 years.)

There are pictures in the Textes mathématiques de Suse, but I cannot reproduce them for copyright reasons.

(For this part I’m referring to The Exact Sciences in Antiquity by Otto Neugebauer.)

One of the tablets (this one, I think) contains a list of geometrical constants. For example, it gives the number $\frac{5}{3}$ in relation to a regular pentagon, apparently meaning that:

The area of a pentagon = $\frac{5}{3}$ * The side of the pentagon2

The actual number here should be $\frac{ \sqrt{25+10\sqrt5}}{4}$, or approximately 1.72, so the Babylonians were off by only 0.06. The tablet also gives constants used in the same way for the hexagon (2.625, about 0.027 off the real value) and for the heptagon (3.683, about 0.049 off the real value).

One of the other constants is 24/25. On the tablet it matches with a circle with a hexagon inscribed inside.

Suppose we take 24/25 to be the ratio of the perimeter of the hexagon p to the circumference of the circle C.

$\frac{24}{25}=\frac{p}{C}$

If the radius of the circle is r, adding some equilateral triangles reveals the perimeter of the hexagon is 6r. So:

$\frac{24}{25}=\frac{6r}{2\pi r}$
$\frac{24}{25}=\frac{6}{2\pi}$
$\frac{24}{25} \cdot \frac{2}{6}=\frac{1}{\pi}$
$\frac{48}{150} = \frac{1}{\pi}$
$\frac{150}{48} = \pi$
$\frac{25}{8} = \pi$

or alternately 3.125.

Because this is given as an actual fixed ratio (rather than being extrapolated from a circle area procedure) it’s arguably the first discovered value for $\pi$. It’s also intriguing in that relating the inscribed polygon to the circle is how Archimedes gets the first truly rigorous calculation of $\pi$ — he just adds more sides to get a closer estimate. However, I can’t give the Babylonians full laurels because this is the only place this value of $\pi$ appears. They never reapplied it back to any problem requiring the area or circumference of a circle.

### 12 Responses

1. when did the babylonians do this exactly?

2. The Babylonians value of Pi may be proved geometrically. Please refer to the following link for a paper published recently.

http://www.ijoart.org/docs/Geometric-Estimation-of-Value-of-Pi.pdf

3. awsome

4. Is there any research that shows that Pi was used? if so HOW???

5. sir, namasthe! The exact pi value is 14_root2/4. Equal to 3.14644660942…. If you are interested your postal address please. Regards yours faithfully rsj reddy india.

6. the day I was born

7. Thanks . helped for my Pi Day project.

8. […] Pi has been recognized for just about 4000 years and our earliest known determination of it was the Ancient Babylonians on a tablet, who recognized that the circumference of a circle was roughly 3 x the diameter, π = 3. Just about […]

9. […] Pi has been recognized for just about 4000 years and our earliest known determination of it was the Ancient Babylonians on a tablet, who recognized that the circumference of a circle was roughly 3 x the diameter, π = 3. Just about […]

10. […] the Babylonians did discover a more accurate approximation for pi (namely, ) by circumscribing a regular hexagon into a circle, but it seems that the babylonians […]

11. […] It’s defined to be the ratio between the circumference of a circle and the diameter of that circle. This ratio is the same for any size circle, so it’s intrinsically attached to the idea of circularity. The circle is a fundamental shape, so it’s natural to wonder about this fundamental ratio. People have been doing so going back at least to the ancient Babylonians. […]

12. […] It’s defined to be the ratio between the circumference of a circle and the diameter of that circle. This ratio is the same for any size circle, so it’s intrinsically attached to the idea of circularity. The circle is a fundamental shape, so it’s natural to wonder about this fundamental ratio. People have been doing so going back at least to the ancient Babylonians. […]