 Phaevic ritual involves the construction of altars of specific shapes and sizes. Typically the only tool used was an unmarked cord. Directions for constructing such altars appear in books called Sulbasotras, the cord rules in Sanskrit. One of the oldest of the Sulbasotras was written by Baudhiana around 800 BC. So one of the problems solved by Baudhiana was to construct a square altar. Baudhiana gave two construction methods. We'll focus on the second. Baudhiana's procedure was the following. First, take a cord twice the length of the desired square and use it to mark the east-west line. Now make a mark, a nyankana, one fourth of the distance from the middle to the end of the cord. Then tie the end points of the cord to the end points of the east-west line. And holding the cord at the nyankana, stretch it south until tight. And this actually produces a right angle, which could be used to set a second side of the square. Now these cords are typically unmarked. So you might wonder, well, how do we mark a nyankana? How do we find one fourth of the distance from the middle to the end of the cord? And if you think about that, we can actually do that by successively folding the cord in half. So if the cord is twice the length of the desired square, we'll fold it in half, then mark the end points of one side. Now one fourth of the distance from the middle to the end of the cord will fold the cord in half a couple of times. So here the cord has been marked into quarters. And now into eighths, so we can find the nyankana. We'll tie the ends to the stakes and pull until top. So why does this work? If the length of the cord is eight units, then the side of the square will be half of that four units. The nyankana will split the cord into sections of length three and five. And so the stretched cord will produce the leg and hypotenuse of a triangle with side lengths four, three, and five, which is a right triangle. And a little later on, Baviana gives the first explicit statement of the Pythagorean theorem. The squares on the side of a rectangle have a total area equal to the square on the diagonal. And again, note that this formulation of the relationship is in terms of the sides and diagonal of a rectangle and not in relationship to a right triangle. Now there's a legend that Pythagoras, who lived around 500 BC, visited India. And so this leads to some speculation. Might he have learned the Pythagorean theorem from the Indians? And it seems that the most likely answer to this question is probably not. And there's two reasons for this. First, the Pythagorean theorem, as stated by Pythagoras, had nothing to do with triangles, rectangles, or any geometric figures. It was purely a relationship among whole numbers. And second, even if Pythagoras visited India, which itself is unlikely, all of Mesopotamia is between Greece and India. And remember, the Mesopotamians knew the Pythagorean theorem as early as 1800 BC. So Pythagoras didn't have to go all the way to India to learn about the relationship. He could have learned it in Mesopotamia. However, there are some intriguing features about early Greek mathematics that suggest there may have been an Indian influence. And one problem that appears very early in Greek geometry is the transformation of areas. The general problem of this format is the following. Given a figure with a specified area, find another figure with the same area. For example, we might talk about squaring a figure. Given a figure with a specified area, find a square with the same area. And in fact, when people talk about the problem of squaring the circle, this is what they mean. Given a circle, find a square with the same area. What's interesting here is that this type of problem appears in Greek geometry, but it has no precursors in Mesopotamian or Egyptian mathematics. But, Vedic ritual often includes problems like construct a square with an area equal to the sum or difference of the areas of two given squares. So it's possible that these transformation of area problems that we see in Greek geometry may have been inspired by Vedic rituals. Let's consider one of those problems. Badiata considers the problem of constructing a square with an area equal to two given squares. So we have our two squares. And Badiata's construction, since the square on the diagonal of a rectangle has area equal to the sum of the squares on the two sides, juxtapose our two squares, and the diagonal will have the required length.