We present a new method of initial orbit determination for angles-only data. The technique is applicable with only three data sets but the formalism can incorporate arbitrarily large amounts of data. The algorithm rests on the fact that the orbital plane is usually very well determined, as a consequence of the central nature of the gravitational force, and a theorem of W. R. Hamilton's from 1846. His result states that the velocity vector describes a circle in the orbital plane in a particular manner. Tests on a variety of deep-space artificial satellites show that the method has promise. Unfortunately it does not work very well on the stressing cases of current practical application-high eccentricity orbits. As no other method does either, the opportunities still exist for a significant advance in this area of celestial mechanics.