Example of the infinite point density of continua

Question

There is an interesting example that I've seen, possibly attributed to Hilbert. One draws a ray from the center of a circle to every point in the circle, and then one increases the radius of the circle to expose gaps between the rays. In general, this has to do with the cardinality of infinite sets. I am unable to locate this classical example via Google. Can tell me the name of this example, or link me to some material that discusses it. I want to be sure that it was Hilbert's demonstration, and I want to be sure that I'm getting everything from that was originally intended. Thanks!