1892 - 93 .] Prof. Knott on Innovations in Vector Theory. 217 
plane.” Hence a vector acts versorially. To wliich Mr Heaviside, 
in fierce denunciation : “ In a given equation [in quaternion vector 
analysis, that is] one vector may be a vector and another a qua- 
ternion. Or the same vector in one and the same equation may be 
a vector in one place and a quaternion (versor or turner) in another. 
This amalgamation of the vectorial and quaternionic functions is very 
j)uzzling. You never know how things may turn out.” Puzzling ? 
Then must Heaviside find his own system as puzzling as any. Por, 
when he writes the vector product ij = h, he is simply acting on j by 
i or on i by j, and turning it through a right angle. It is impossible 
to get rid of this versorial effect of a vector.* But, if to treat vectors 
vectorially means that we must treat them versorially, on what 
principle of straw-dividing are we to debar their quaternionic treat- 
ment ? 
(4.) Before passing on to consider the systems of the innovators, I 
shall give three other short quotations from Gibbs’s letter to Nature, 
as follows : — 
“ How much more deeply rooted in the nature of things are the 
functions Sa.yS and Vay8 than any which depend on the definition of 
a quaternion will appear in a strong light, if we try to extend our 
formulse to space of four or more dimensions. . . .” 
To elucidate the “ nature of things ” by an appeal to the fourth 
dimension — to solve the Irish question by a discussion of social life 
in Mars — is a grand conception, worthy of the scorner of the trivial 
and artificial quaternion of three dimensions. But is it not the 
glory of quaternions that it is so pre-eminently a tridimensional 
calculus? Again, further on we read : “Vectors exist in such a [four 
dimensional] space, and there must be a vector analysis for such a 
space ” — true ; and must there not be operators for changing one 
vector into another, geometrically analogous to the quaternion of 
three dimensions? Or, more generally, must there not be in n 
dimensional space the M-in-one corresponding to the 4-in-one of 3- 
diniensional space ? 
Again, we read that “ nothing is more simple than the definition 
of a linear vector function, while the definition of a quaternion 
O’Brien alone of vector analysts uses a non-scalar product of two vectors, 
Avliich does not involve directly this versorial characteristic ; but then he has 
to introduce his Directrix, so as to get out into space, as it were. 
