1892-93.] Prof. Knott on Innovations in Vector Theory. 215 
(i, combined witli certain other simple notions. But the angle com- 
bined with these auxiliary notions is incomparably more amenable 
to analytical transformation than the simple angle of trigonometry, 
and so on — proving just as much and just as little as the great 
original itself. Heaviside, evidently with the above passage in his 
mind, says that “ The justification for the treatment of scalar and 
vector products as fundamental ideas in vector algebra is the distri- 
butive property they possess.” So be it ; and is not the quaternion 
product as grandly distributive as any ? 
(3.) To appreciate the real character of the broad geometrical 
argument advanced by Gibbs, we must consider the meaning and 
purpose of a vector analysis. We are all agreed that vectors are of 
real importance in physics. Having then formed the conception of 
a vector, we have next to find what relations exist between any two 
vectors. We have to compare one with another, and this we may 
do by taking either their difference or their ratio. The geometry 
of displacements and velocities suggests the well-known addition 
theorem a -f 3 = ft recognised by Gibbs as essentially fundamental. 
But this method, about which there is no dispute, does not seem 
to me to be more fundamental geometrically than the other method 
which gives ns the quaternion. When we wish to compare fully two 
lengths a and 5, we do not take their difference, but divide the one by 
the other. We form the quotient afb, and this quotient is defined 
as the factor which changes h into a. Kow a vector is a directed 
length. By an obvious generalisation, therefore, we compare two 
vectors by taking their quotient (a//l), and by defining this quotient 
as the factor which changes the vector /3 into the vector a. This 
is the germ out of which the whole of vector analysis naturally 
grows. A more fundamental conception it is impossible to make. 
Yet Gibbs says “it certainly does not” take rank as a fundamental 
conception in geometry with the conceptions of a vector-bounded 
area and of a vector-bounded volume, whose bounding vectors may 
have an infinity of values. 
Again, a vector is an embodiment of direction; and to know 
how to change a direction is surely demanded of a vector analyst 
from the very beginning. But a change of direction is an angular 
displacement, that is, a versor or quaternion with unit tensor. Or, 
take the case of a body strained homogeneously. The relative 
