1893-94.] Prof. Cayley on Coordinates versus Quaternions. 271 
Coordinates versus Quaternions. By Professor Cayley. 
(Read July 2, 1894.) 
It is contended that Quaternions (as a method) are more compre- 
hensive and less artificial than — and, in fact, in every way far 
superior to — Coordinates. Thus Professor Tait, in the Preface to 
his Elementary Treatise of Quaternions (1867), reproduced in the 
second and third editions (1873 and 1890), writes — “It must 
always he 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 he adopted, the quaternion solution 
becomes identical with the Cartesian one. N’othing, 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 state- 
ment 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 
