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541 
962. 
COORDINATES VERSUS QUATERNIONS. 
[From the Proceedings of the Royal Society of Edinburgh, voi. xx. (1895), pp. 271—275. 
Read July 2, 1894.] 
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. 
I wish to examine into these claims on behalf of quaternions. My own view is 
that quaternions are merely a particular method, or say a theory, in coordinates. I 
have the highest admiration for the notion of a quaternion ; but (I am not sure 
whether I did or did not use the illustration many years ago in conversation with 
Professor Tait), as I consider the full moon far more beautiful than any moonlit