1899-1900.] Dr Knott on Klein's View of Quaternions. 
31 
complete, clear and comprehensive conception of the quaternion 
calculus. It has, in addition, the advantage of indicating clearly 
the sphere of applicability (Anivendungsgebiet) of quaternions. . . . 
Quaternions will be in place when we wish to have a convenient 
algorithm for the combination of rotations and dilatations.” If 
that were all, the quaternion might as well have never existed ; 
for a Drehstreckung is not a very practical dynamic conception, 
although the rotation is of fundamental importance. It has, of 
course, been long recognised by workers in quaternions that the 
quaternion method lends itself powerfully to the treatment of all 
kinds of strains ; but because it is peculiarly fitted to attack 
general problems in the rotation of a rigid body, it does not neces- 
sarily follow, as Klein and Sommerfeld seem to suggest, that its 
value in other directions is insignificant. 
Regarding Hamilton’s definition of a quaternion as the quotient 
of two vectors, Klein and Sommerfeld remark : — “ As the basis of 
a theory this definition is scarcely adapted to the end aimed at ; 
for the expression ‘ quotient of two vectors ’ requires first an 
explanation of itself, and, unless that he given,* diverts our 
attention wholly to a vague ( unklar ) analogy with the rules of 
ordinary algebra. The definition may, of course, be theoretically 
justified, and has indeed certain advantages, to be mentioned 
immediately ; hut it does not seem appropriate to begin with it.” 
To this expression of an opinion — and it is little else — the 
natural reply is, Why not ? Is Hamilton’s “ Quotient of two 
Vectors” the only expression in mathematics that requires to 
he explained? Hamilton, indeed, carefully guarded his readers 
against reading into the meaning of the word “ quotient ” more 
than is essentially involved in it, namely, the operator ajb , which 
changes b into a. The laws of its operation depend on the kind 
of quantities represented by b and a. If b and a are ordinary 
numbers, the quotient is the ordinary fraction ; if b and a are 
vectors, the quotient is a quaternion. What can he simpler in 
conception or more complete in statement ? On the other hand, 
it is very questionable indeed if the profoundest meditation on 
* The introduction of this phrase might easily suggest to the reader 
that Hamilton had erred in not sufficiently explaining his meaning. On the 
contrary, Hamilton’s explanations are always full — almost prolix at times. 
