 The kingdom of Ashnuna was a part of ancient Mesopotamia. Around 1750 BC, a scribe in the kingdom of Ashnuna posed and solved the following problem. The length of a rectangle is 1, and its width is 45. Find its area and diagonal. So we have a rectangle, and we have the length, and width, and we want to know the area and the diagonal. So the area computation is pretty straightforward. We'll multiply width by length. What's interesting is the scribe's computation of the diagonal. To find the diagonal, the scribe multiplied the length by itself, which gives us 1, multiplied the width by itself to obtain 3345, added these together to obtain 1, 3345, and then found the square root to obtain 115. This is the earliest documented use of the Pythagorean theorem. And it's worth noting that, like all early uses of the Pythagorean theorem, the problem is stated in terms of the diagonal of a rectangle, and not in terms of the sides of a right triangle. Now, in this particular case, it turns out that the three lengths, the length, width, and diagonal, happen to be rational numbers. But that's not always the case. This is an artifact known as YBC7289. That YBC means Yale Babylonian Collection, and this is item number 7289 in that collection. Notice the scale here. This scale is in millimeters, and what this means is that this tablet is about 2 inches across. Now, this tablet appears to show a square cut by its diagonals, and there's numbers written along the diagonal. And if you read this number, it appears to read 1, 24, 51, 10. And if we place the sex a decimal point after the 1, we get a very good approximation to the square root of 2, which is exactly the relationship that the diagonal has to the side of the square. So not only did the Mesopotamians know the relationship between the side and diagonal of a rectangle, they also had a method of finding very good approximations to square roots that weren't rational numbers. So we know the Mesopotamians understood the relationship between the sides and diagonal of a rectangle, and they were able to find very good approximations to square roots. An even more remarkable set of values appears in Plympton 322. The numbers appear to relate to what we call Pythagorean triplets. Rational or integers a, b, and c were a squared plus b squared equals c squared. What's important to understand here is that these triplets exist independently of geometry. They are a purely number-theoretic result. And that means we have no idea why they made this tablet. In particular, there's no practical value to the Pythagorean triplets. Once you have the relationship between the side and diagonal of a rectangle, and once you have a method of finding good approximations to square roots, the existence of rational solutions, the existence of the Pythagorean triplets is interesting, but it's not practical. And that suggests that at least some people in ancient Mesopotamia were interested in purely theoretical mathematics that didn't have a practical application. The Mesopotamians also had an important result regarding the areas of circles. This was also written around 1800 BC, and this appears in a tablet, catalog as BM, British Museum 85194. And so the scribe computed the area of a circle given its diameter in the following way. First, triple the diameter to find the circumference, square the circumference, and take one-twelfth to find the area. So if we compare the Mesopotamian procedure to our own, the area of a circle in our own formula, pi r squared, the scribe's procedure, the square of three times the diameter, times a twelfth, and we can simplify and solve this for pi, and we get a Mesopotamian value of pi equal to three. Now we did this for Egyptian geometry, and the Egyptians had a slightly more accurate value for pi, except they didn't recognize the ratio of the circumference to the diameter as something of importance. In contrast, the scribe here clearly recognizes there is this ratio between the diameter and the circumference, and he explicitly states that that ratio is three. And so we can say that the Mesopotamians were the first to recognize the existence of the number pi.