 So, apologies in advance for my horrific pronunciation of Chinese names and words. Diehui, who lived in the 3rd century AD, described a method of finding square roots in his commentary on the Ge Xiang Xuanshu. Well, go through the method, but it's important to keep in mind that the method is actually optimized for use on a counting board, so translating it into a written form makes it much more complicated than it actually is. So let's try to find the square root of 5329. So we'll set down the radicand, which is referred to as the he, and a placeholder, which is referred to as the geyi. Now first we'll determine the magnitude, the number of digits in the root. And we'll do that in the following way. Since this is a square root that we're trying to find, we'll move the placeholder two spaces at a time until our next move would take us past the leading digit of the root. And in this case, we can only move the placeholder one place, so we know that our root is a two-digit number. Now it's convenient to ignore all the digits to the right of the placeholder, so we'll throw down a piece of paper, also a Chinese invention, and cover up those extra digits. And to start off with, we'll guess the first digit of the root. Now this is the only tricky part of the process, and the best we can say is that it will be the largest digit that works when we apply our next procedure. And unfortunately we can't get too much more specific. But here's what we might try, since the square root of 53, remember we're ignoring the other digits, is a little bit more than 7, we'll guess that the first digit is going to be 7. So we'll set that down. Now we make what it will be convenient to call the first pass. And just a quick note, since this is a square root, there is no second pass, but the terminology will be useful for later. So here goes. In our first pass, we'll multiply the guess 7 by the placeholder 1, and record the result, which is known as the yinfa, above the placeholder. Next, we'll multiply the guess 7 by the result, also 7, and we'll set down the product. And now we're going to subtract from the digits of our radicand, so 53 minus 49. Now we've completed our first pass, there is no second pass, so now we'll double the partial root, 7, and record the result as a new yinfa. And we'll line the digits up with the placeholder. Now don't forget we had a bunch of digits underneath this piece of paper, so let's lift that piece of paper up, and now we're going to shift. The placeholder moves two places. If you remember that we moved the placeholder over two spaces to get it to where it is, we're just moving it back. Now the yinfa is going to move one place, and now we're set up for the next step. Again, it's convenient to ignore all the digits to the right of the placeholder, but there aren't any. And now we want to guess the largest digit that works when we perform the first pass. So we'll guess 3, and we'll see in a moment what we mean by 3 is the largest digit that works. We'll guess 3, we'll multiply our guess by the placeholder, and add it to the yinfa. Next, we'll multiply our guess by the yinfa, and subtract from the radicand. And so here's the thing to notice, if we guessed anything larger than 3, we wouldn't be able to do the subtraction. And if we guessed anything smaller than 3, we could have guessed a larger number. So that's what we mean by guessing the largest number that works. Now since there's nothing left of the root, we're actually done, and we've found the square root. So let's find the square root of 85,849. So we'll set down the root and the placeholder, and we'll determine the number of digits of the root. So we'll move the placeholder two spaces at a time, once, twice. And since we've had to move the placeholder twice, that tells us that the root is actually going to be a three digit number. So again, we guess the first digit, which is the largest digit that allows the first pass to work. So we guess 2, multiply guess by placeholder to get the yinfa, multiply guess by yinfa, subtract the result from the radicand, double the root, and record the result as a new dingfa, and shift. The placeholder moves two places, and the dingfa moves one place. And on to the next. We'll guess the next digit, which again is going to be the largest that will make the first pass work. Now it's worth noticing here that we'll be multiplying our guess by 40-something, and the product has to be less than 458. So let's guess 9. The guess 9 times the placeholder is added to the dingfa. The guess 9 times the dingfa, 49, is subtracted from the radicand. So we'll write down that product for 41 and subtract. And now our root so far begins to 9. We double it and record that as a new dingfa, and shift. The dingfa moves one place, and the ge moves two places. And again, we'll guess the next digit of the root as the largest that will make the first pass work. And again, whatever our guess is, this will be multiplied by 580-something, and it will be subtracted from the radicand, which means it's got to be less than 1749. So let's guess about 3. And we'll guess times placeholder, and add to dingfa, guess times dingfa, and subtract. And since there's nothing left of the radicand, our root is going to be 293.