On the Importance of Quaternions in Physics, 85 



technical expression was given by Clerk-Maxwell in one of 

 his early papers, which is still in the highest degree interest- 

 ing, not only as the first step to his Theory of the Electro- 

 magnetic Field, but as giving by an exceedingly simple 

 analogy the physical interpretation of his equations. Next, 

 the narrative should go back to the establishment of the Wave- 

 theory of Light: — to the mathematics of Young and Fresnel, 

 and the experiments of Fizeau and Foucault. Maxwell's 

 theory had assigned the speed of electromagnetic waves in 

 terms of electrical quantities to be found by experiment. The 

 close agreement of the speed, so calculated, with that of light 

 rendered it certain that light is an electromagnetic pheno- 

 menon. But it was desirable to have special proof that there 

 can be electromagnetic waves ; and to measure the speed of 

 propagation of such as we can produce. Here experiment 

 was again required, and you all know how effectively it has 

 just been carried out by Hertz. It is particularly to be noticed 

 that the more important experimental steps were, almost 

 invariably, suggested by theory — that is, by mathematical 

 reasoning of some kind, whether technically expressed or not. 

 Without such guidance experiment can never rise above a 

 mere groping in the dark. 



I have to deal, at present, solely with the mathematical 

 aspect of physics ; but I have led up to it by showing its 

 inseparable connexion with the experimental side, and the 

 consequent necessity that every formula we employ should as 

 openly as possible proclaim its physical meaning. In presence 

 of this necessity we must be prepared to forego, if required, 

 all lesser considerations, not excluding even such exceedingly 

 desirable qualities as compactness and elegance. But if we 

 can find a language which secures these to an unparalleled 

 extent, and at the same time is transcendently expressive — 

 bearing its full meaning on its face — it is surely foolish at least 

 not to make habitual use of it. Such a language is that of 

 Quaternions ; and it is particularly noteworthy that it was 

 invented by one of the most brilliant Analysts the world has 

 yet seen, a man who had for years revelled in floods of sym- 

 bols rivalling the most formidable combinations of Lagrange, 

 Abel, or Jacobi. For him the most complex trains of formula -1 , 

 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 threw over all other 

 methods in their favour, devoting almost exclusively to their 

 development the last twenty years of an exceedingly active 

 life. 



Everyone has heard the somewhat peculiar, and more than 



