1899-1900.] Dr Knott on Klein's View of Quaternions. 
33 
ticular ( ziemlich partikular ), and is inferior in simplicity to our 
original introduction of the conception. On the other hand, we 
must not conceal from ourselves that it has a great advantage 
over ours. In fact, it puts immediately in evidence the half angle 
of rotation (co/2) required for the unambiguous description of the 
quaternion, while our view of Drehstreckungen deals primarily with 
the whole angle of rotation (co), and has then to he brought into 
relation with the half angle of rotation through the somewhat 
arbitrary rules of p. 36.” 
The “it” ( sie ) of the second sentence refers presumably to 
Hamilton 1 s definition of a quaternion, although grammatically it 
refers to their own “ somewhat particular ” definition immediately 
preceding. This definition, however, is not Hamilton’s in any 
strict mathematical sense. What follows in the paragraph just 
quoted, if taken in conjunction with foregoing statements, con- 
stitutes a remarkable confession. Hamilton’s definition is first 
criticised as being “scarcely adapted to the end aimed at,” but 
now it is admitted to have “ a great advantage ” over their view 
of a Drehstreckung, which, we are nevertheless assured, “leads to 
a complete, clear, and comprehensive conception of the quaternion 
calculus ” ; and one stated reason for this great advantage is that 
their “ complete, clear, and comprehensive conception ” has to be 
eked out by means of certain “ arbitrary rules ” regarding whole 
angles and half angles of rotation. 
But, strictly and therefore mathematically speaking, their defini- 
tion has to do, not with a quaternion and two vectors , but with a 
Drehstreckung and two Wendestreckungen, whose axes are subject 
to a particular limitation. A so-called quaternion Q' is represented 
as the quotient of two vectors v' and V ; but with Q' Klein and 
Sommerfeld associate an angle of rotation double the magnitude of 
that which Hamilton would have called the angle of the quaternion 
v/V. 
In short they use Q', Q, v and V, each and all, in a double signi- 
ficance. When the exigencies of analysis demand it they simply 
follow Hamilton and Tait — that is, their analytical work is purely 
quaternionic. But when there is no direct question of establishing 
fundamental relations among the scalar quantities involved, they 
endow their so-called quaternion with powers that belong, as 
YOL. XXIII. C 
