276 Proceedings of Boy al Society of Edinhurgli. 
On the Intrinsic Nature of the Quaternion Method. 
By Prof. Tait. 
(Read July 2, 1894.) 
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, 2 .” 
On this I would remark as follows : — 
1. I have always maintained that it is not only not a reproach 
to, but one of the most valuable characteristics of. Quaternions 
that they are uniquely adapted to tridimensional space. In my 
Address to Section A, at the British Association Meeting in 1871, 
I said : — 
“ It is true that, in the eyes of the pure mathematician, Quatern- 
ions have one grand and fatal defect. They cannot be applied to 
space of n dimensions, they are contented to deal with those poor 
three dimensions in which mere mortals are doomed to dwell, but 
which cannot bound the limitless aspirations of a Cayley or a 
Sylvester. Prom the physical point of view this, instead of a 
defect, is to be regarded as the greatest possible recommendation. 
It shows, in fact. Quaternions to be a special instrument so con- 
structed for application to the Actual as to have thrown overboard 
everything which is not absolutely necessary, without the slightest 
consideration whether or no it was thereby being rendered useless 
for applications to the Inconceivable.'’^ 
2. Whether Quaternions are to be regarded as artificial, or the 
reverse, will obviously depend wholly upon what is to be under- 
stood by the term Quaternions. This forms the main object of the 
present paper. 
