277 
1893 - 94 .] Prof. Tait on the Quaternion Method. 
3. Though the passage quoted above contains no statement as to 
the relative merits of Quaternions, and Co-ordinates, as instruments 
(in the region which is common to them) it is clear from other 
passages in his paper that Prof. Cayley holds that Quaternions are, 
at best, superfluous : — he allows that they enable us to effect great 
abbreviations, but he insists that, to be applied or even understood, 
they must be reconverted into the y, ^ elements of which they 
are, in his view, necessarily composed. 
But another great analyst, who certainly devoted vastly more 
time and attention to Quaternions than it can have been possible 
for Prof. Cayley to devote, took a very different view of the 
matter : — 
“ It is particularly noteworthy that [Quaternions were] invented 
by one of the most brilliant Analysts the world has yet seen, a 
man who had for years revelled in floods of symbols rivalling 
the most formidable combinations of Lagrange, Abel, or Jacobi. 
For him the most complex trains of formulse, of the most artificial 
kind, had no secrets : — he was one of the very few who could afford 
to dispense with simplifications : yet, when he had tried quaternions, 
he thrcAv over all other methods in their favour, devoting almost 
exclusively to their development the last twenty years of an exceed- 
ingly active life.” 
It will be gathered from what precedes that, in my opinion, the 
term Quaternions means one thing to Prof. Cayley and quite another 
thing to myself ; — thus 
To Prof. Cayley Quaternions are mainly a Calculus, a species of 
Analytical Geometry ; and, as such, esse7itially made up of those 
co-ordinates which he regards as “ the natural and appropriate basis 
of the science.” They artfully conceal their humble origin, by an 
admirable species of packing or folding : — but, to be of any use, 
they 
doubly dying, must go down 
To the vile dust from which they sprung ! 
To me Quaternions are primarily a Mode of Bepresentation : — im- 
mensely superior to, but of essentially the same kind of usefulness 
as, a diagram or a model. They are, virtually, the thing repre- 
sented : and are thus antecedent to, and independent of, co- 
ordinates : giving, in general, all the main relations, in the problem 
