My title is purposely ambiguous, because it has to represent two things—I intend to treat not only of what a quaternion really is, but also of its self-containedness, or independence.

Professor Cayley has just stated that “while co-ordinates are applicable to the whole science of geometry, and are the natural and appropriate basis and method in the science, Quaternions seem to me a particular and very artificial method for treating such parts of the science of three-dimensional Geometry as are most naturally discussed by means of the rectangular co-ordinates x, y, z.”