278 » Proceedings of Royal Society of EdMiirgli. [sess. 
to which they are applied, without the necessity of appealing to co- 
ordinates at all. Co-ordinates may, however, easily he read into 
them : — when anything (such as metrical or numerical detail) is to 
be gained thereby. Quaternions, in a word, exist in space, and w'e 
lave only to recognize them : — but we have to invent or imagine co- 
>rdinates of all kinds. The grandest characteristic of Quaternions is 
heir transparent intelligibility. They give the spirit, as it were, leav- 
-,ag the mere letter aside, until or unless, it seems necessary to attend. 
;0 that also. In this respect they give a representation analogous to 
die real image of a planet in the focus of an object-glass or mirror: 
— all that is obtainable is ihere^ and you may apply your micro- 
scopes and micrometers to it if you please. But, theoretically at 
least, you may dispense with them and have recourse to your eyes 
and your yard-stick alone, if you increase the focal length, and along 
Sv^ith it the aperture, of your object-glass sufficiently. Of course 
Newton’s “ most serene and quiet air ” would be indispensable. For 
the development of this feature of my subject, and for illustrative 
examples, I refer to the B. A. Address above cited; and to the 
Address to the Edinburgh University Physical Society, alluded to 
by Prof. Cayley. 
To those who have read Poe’s celebrated tale. The Purloined 
Letter, it will be obvious that the contrast between these two 
views of Quaternions is even greater than that between the 
Parisian Police and M. Dupin himself, though of very much the 
same kind. 
There was a time, in their early history, when Professor Cayley’s 
view of Quaternions was not merely a correct one, it was the only 
possible one. But, though the name has not been altered, the 
thing signified has undergone a vital change. To such an extent, 
in fact, that we may almost look upon the quaternion of the 
latter half of this century as having, from at least one point of 
view, but little relation to that of the seven last years of the 
earlier half. 
Hamilton’s extraordinary Preface to his first great book shows 
how from Double Algebras, through Triplets, Triads, and Sets, he 
finally reached Quaternions. This was the genesis of the Quatern- 
ion of the forties, and the creature then produced is still essentially 
the Quaternion of Prof. Cayley. It is a magnificent analytical 
