216 Proceedings of Royal Society of Edinlurgli. [sess. 
vector of any two of its points is changed into its new value by 
a process which is a combination of turning and stretching. A 
simpler description cannot he imagined. It is completely symbo- 
lised by the quaternion with its tensor and versor factors. And 
this, we are taught, is trivial and artificial — as trivial, say, as the 
versor operation which every one performs when estimating the 
time that must be allowed to catch a train. 
Professor Gibbs would have us base the whole of vector analysis on 
the two geometrical ideas embodied in the formulae Va^ and - SyVa/?. 
But any thoughtful student, approaching the subject in this way, 
will almost certainly be struck with the arbitrariness of the defini- 
tions made at the outset. There is no very apparent reason, at 
first, why an area should be represented by a vector line drawn per- 
pendicular to it ; while the transition from SyYa/3 to Sa/3 can 
hardly fail to appear somewhat mysterious. This is the method 
adopted by Clifford in his Dynamic; but I believe this development 
to be logically faulty, and quite unsuited to a student otherwise 
ignorant of the properties of vector products. In quaternions, how- 
ever, the quotient and product of two vectors being clearly defined 
— for aP is the operator which changes into a — it soon appears 
that to every quaternion q there is a conjugate f, so that their sum 
is a scalar and their difference a vector. From that the geometrical 
meanings of Va/5 and are at once obtained, and the whole system 
is established firm and sure.* 
At a recent meeting of the Physical Society of London, Professor 
Henrici is reported to have said, ‘‘Vectors must be treated vecto- 
rially and Mr Heaviside echoes the strain in § 175 of his “ Electro- 
magnetic Theory,” as published in the Electrician. On the same 
sapient principle, I suppose, scalars must be treated scalarially, 
rotors rotorially, algebra algebraically, and geometry geometrically. 
That is to say, the remark is a very loose statement of a truism, or 
it is profound nonsense. Strictly speaking, to treat vectorially is 
to treat after the manner of vectors, or to treat as vectors do. 
How, what does a vector do ? Professor Gibbs, the prince of vector 
purists, says, on page 6 of his pamphlet, that “ the effect of the skew 
[or vector] multiplication by a [any unit vector] upon vectors in a 
plane perpendicular to a is simply to rotate them all 90° in that 
See article “Quaternions” in Chambers’s Encyclopccdia (New Edit., 1892). 
