"The Greeks would have said... we know a much better way of taking a square root. ...the ancient Greeks thought entirely geometrically, not arithmetically. And they would... do the following. If you want to solve x^2 = N, you should first... think of whether N is bigger than or equal to one. Suppose that case 1) N < 1. ...Draw a [horizontal] line segment of length one and then [within and from the end of that segment]... make a segment of size N. And then with the center of the [length one] segment you draw a circle so this is a [unit length] diameter. And you... [draw a vertical line from the end of the N segment inside the circle] up here [to intersect the circle] and then... look at this quantity x... this [top angle of the largest triangle circumscribed by the circle] is a right angle by Thales theorem, so we have some similar triangles. So [side x, side 1 from the large circumscribed triangle] \frac{x}{1} = \frac{N}{x} [side N, side x from the small left triangle] by similar \triangle's . And so x^2 = N. So Geometrically finding a square root is... a relatively simple... rule or construction, but arithmetically much more difficult. What happens if N is bigger than one? Well then you just interchange the roles of the N and the one. Case 2) N \ge 1. So you start by having a diameter of size N and then you make [a line segment of length] 1 here [from the end of the segment of length N to within that segment] and then otherwise do exactly the same thing [as in the above, case 1]. ...x will be square root, x^2 = N, by the same argument."