 Theon of Alexandria explains a method of square roots in his commentary on Ptolemy's Almagest. The method is based on a literal interpretation of the square root of n as the length of one side of a square with area, n. Or we can think about the side as being the root from which the square grows. We'll illustrate Theon's method and find the square root of 123,201. So first we note that 100 squared is 10,000 and 1,000 squared is 1,000,000, so the side of the square must be somewhere between 100 and 1,000. Now we note that 300 squared is 90,000, which is smaller, while 400 squared is 160,000, which is larger, so the length of the square is 300 plus some amount. So we start with an area of 123,201 and a side length of more than 300. And we can think about our square as being big enough to contain a square of side 300. So we'll draw that in. And remember if it's not written down, it didn't happen. Let's go ahead and indicate this square has a side of 300. So our initial square has an area of 300 squared, 90,000. So the area of the remaining nomen will be, to get the remaining area of 33,201, we'll extend our original square. Since we know the remaining length is less than 100, we'll extend it by a 10. And by trial and error, we find extending the side of our square by 50 will make the resulting nomen have an area of, you can think of this as 250 by 300 rectangles plus a square of side 50. And again if it's not written down, it didn't happen. Let's go ahead and label the area of the nomen. So remember we wanted to get the remaining area 33,201. We got 32,500, so now the remaining area is 701. So now we have a square with a side of 300 plus 50, 350, that remaining area of 701. If we extend our square of side 350 by one more unit, the resulting nomen will have an area of, which is again, two 350 by one rectangles at a square of side one. And altogether our areas are, so our square on the side of 300 plus 50 plus one has an area of 123,201. So the square with an area of 123,201 has a side of 351. So the square root is 351.