It is contended that Quaternions (as a method) are more comprehensive and less artificial than–and, in fact, in every way far superior to–Coordinates. Thus Professor Tait, in the Preface to his Elementary Treatise on Quaternions (1867), reproduced in the second and third editions (1873 and 1890), writes–“It must always be remembered that Cartesian methods are mere particular cases of quaternions where most of the distinctive features have disappeared; and that when, in the treatment of any particular question, scalars have to be adopted, the quaternion solution becomes identical with the Cartesian one. Nothing, therefore, is ever lost, though much is generally gained, by employing quaternions in place of ordinary methods. In fact, even when quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite independent of any limitations as to choice of particular coordinate axes.” And he goes on to speak of “such elegant trifles as trilinear coordinates.” and would, I presume, think as lightly of quadriplanar coordinates. It is right to notice that the claims of quaternions are chiefly insisted upon in regard to their applications to the physical sciences; and I would here refer to his paper, “On the Importance of Quaternions in Physics” (Phil. Mag., Jan. 1890), being an abstract of an address to the Physical Society of the University of Edinburgh, Nov. 1889; but these claims certainly extend to and include the science of geometry.