212 
Proceedings of Royal Society of Edinburgh. [sess. 
Recent Innovations in Vector Theory. By Professor 
C. G. Knott, D.Sc., F.R.S.E. 
(Read December 19, 1892.) 
(1.) Of late years there has arisen a clique of vector analysts* 
who refuse to admit the quaternion to the glorious company of 
vectors. Their high-priest is Professor Willard Gihbs. His 
reasons against the quaternion are given with tolerable fulness in 
NaUire^ April 2, 1891. His own vector analysis is presented in a 
pamphlet, “Elements of Vector Analysis, Arranged for the Use of 
Students in Physics — not Published” (1881-4). Mr Oliver 
Heaviside, in a series of papers published recently in the Electrician^ 
and in an elaborate memoir in the Philosophical Transactions 
(1892), supports some of Gibbs’s contentions, and cannot say hard 
enough things about the quaternion as a quantity which no physicist 
v/ants. Professor Macfarlane of Texas University sides with 
Heaviside in taking umbrage at a most fundamental principle of 
quaternions, and develops a pseudo-quaternionic system of vector 
algebra non-associative in its products. 
Before proceeding to an examination of the positions taken by 
these writers, I wish to draw attention to some remarkable papers 
published between the years 1846-52, just at the time when 
Hamilton was developing the quaternion calculus. These papers 
were written by the Rev. M. O’Brien, Professor of Natural Philo- 
sophy and Astronomy in King’s College, London. They give a 
vector analysis independent of the conception of the quaternion. It 
may be safely said that the anti-quaternionic vector analysts of to- 
day have barely advanced beyond the stage reached by O’Brien in 
his third and last paper “On Symbolic Forms derived from the 
Conception of the Translation of a directed Magnitude ” {Phil. 
Trans., 1852). 
In this paper O’Brien defines two distributive products of two 
* This paper is written wholly from the point of view of mathematical physics, 
for which a vector algebra is generally admitted to be of supreme importance. 
The purely analytical aspect of cpiaternions is not contemplated. 
