In Artin's Algebra he presents a method (that I am sure I am butchering) for classifying ideals of a given lattice $\mathbb{Z}[\sqrt{-d}]$ by taking any ideal $I$, choosing an element of minimum norm $\alpha$, drawing a rectangle with points $0,\ \alpha, \ \alpha\sqrt{-d}, \ \alpha + \alpha \sqrt{-d}$, drawing circles of radius $|\alpha|/n$ about the half-lattice points and radius $\alpha$ on the vertices of the rectangle (lattice points), and then applying a theorem that if the circles cover the rectangle, any other element not in the ideal $(\alpha)$ has to be in the half-lattice points.
What is the motivation for this argument, and what is the general method for choosing the right $n$ to cover the rectangle? (Is it bad if $n$ is too big and I use more circles than necessary?)
Edit: And why is the other element not in the $1/n$th lattice points?