 So, if you lived in ancient Mesopotamia, canals were something of a fact of life. This clay tablet shows a map of part of the irrigation system near the Euphrates River, and it dates back to around 1700 BC. And not surprisingly, when Mesopotamian mathematicians came up with practical problems for their students, a lot of the problems involved canals. And if you're digging a canal there's a number of quantities of interest, and in these canal problems there are three of particular significance. The sum of the length and width, the difference of length and width, and the area. And admittedly, calling this a practical problem is something of a stretch. The actual scribal problems involved units and required some unit conversions on the fly will simplify the problem, and a typical canal problem looks something like this. A canal 14 the sum of length and width, 45 the area, find the length and width. The scribal solution would be something along the following lines. Take half the sum of the length and width, 7, multiply this by itself, 49, subtract the area, 4, find the square root, 2. So we'll add to get the length, 7 plus 2 gives us 9, and subtract to find the width, 7 minus 2 equals 5. Now the scribes just gave these steps without any sort of explanation, so let's think about why this works. So again we have three quantities of importance. The sum of the length and width, the difference of the length and width, and the area. And there's a useful relationship between these three quantities, which you should verify algebraically. Even better, if you want to get some real insight into how the ancient world looked at mathematics, you should verify this geometrically. But the important thing here is that if you know any two of these three quantities, the sum, the difference, or the area, then you can find the third. Another useful feature here is that if you add half the sum to half the difference, you get the length. And if you subtract half the difference from half the sum, you get the width. For example, the difference of length and width is 10, the area is 75, find the length and width. So we find half the difference of length and width, that's 5. We square it, that's 25. We'll add the area to get 100, find the square root, 10. So we'll add this result to half the length and width to get 15, the length. And we'll subtract, to find 5, the width. And again it may be helpful to look at this from our modern algebraic perspective. So it's useful to remember this relationship. Half the difference squared plus the area is half the sum squared. We have the difference of length and width, so half of that gets us 5. Squaring gets 25. Add the area, now that sum is going to be 100, but it's also the square of half the difference plus the area, which our identity tells us is the square of half the sum. And when we find the square root, we have half the difference and half the sum. So if we add them, we get the length, and if we subtract them, we find the width. For a similar problem, we have the sum of length and width, and we have the area. So again, half the sum squared, this time we'll subtract the area, find the square root, add to find the length, subtract to find the width. And again from our modern algebraic perspective, we have the sum, so half of that gets us 6. Squaring, and since we have the sum of length and width, we want to subtract the area, which will give us the square of half the difference. And so we can find half the difference directly, and if we add, we get the length, and if we subtract, we get the width. Now the canal problems give rise to an important problem in interpretation. We could interpret a canal problem as a quadratic equation. So we have the sum of the length and width, and we have the area, solve for length, substitute, expand, and that gives us a quadratic equation. However, it's important to keep in mind that the Mesopotamians did not view the problem this way. They did not see this as a quadratic equation, but they saw it as a canal problem.