26 Proceedings of Royal Society of Edinburgh. [sess. 
in which i, j, k are carefully described as three imaginary units ; 
their introduction being “etwas rein conventionelles.” Further, 
“ The magnitude T is called, after Hamilton, the Tensor of the Quater- 
nion. Therefore we may say : An ordinary rotation is a unit 
quaternion (i.e., a quaternion of tensor unity).” 
Already Klein and Sommerfeld have parted company with 
Hamilton; for, although, with Hamilton’s meanings of i,j, k, Q 
is a quaternion, the tensor of the quaternion Q is not T , hut is 
J T , and a quaternion can never he an “ ordinary rotation.” 
The geometrical meaning of the quantity Ai + Bj 4 • Gk + D 
we know, provided i, j , k are used in the Hamiltonian sense ; 
and, as will he seen later, Klein and Sommerfeld, in spite of 
guarded statements about their purely conventional character, do 
really use them in Hamilton’s sense whenever there is any 
analytical work to be done. Then, again, the operation called the 
Drehstreckung we also know, for it is a simple modification of 
an ordinary rotation. But to assert the identity of quaternion 
and rotation, and to symbolise the latter by means of an expres- 
sion appropriate to the former, — that surely is a misuse of the 
mathematical term identity, and a playing fast and loose with 
the recognised principles of mathematical symbolism. 
It is important from the outset to recognise this duality or 
ambiguity of significance attached to the symbol Q. For some 
purposes it is treated as a quaternion, and for others as a 
Drehstreckung. The avowed aim of the authors is to show that 
Hamilton’s quaternion is nothing else than a Drehstreckung, the 
name given by them to a conception which, as we learn from the 
last page of Part ii. of their Treatise, was first distinctly described 
by Gauss. Yet no one who really knows what a quaternion is 
could for a moment admit the identity. To find anything at 
all comparable to this attempt to identify two fundamentally 
different conceptions, we should have to go to old literatures in 
which the uncritical editor has pieced together into a kind of 
historic mosaic two traditions from quite different sources. As a 
foundation on which to build a mathematical superstructure, Klein 
and Sommerfeld’s Excurs uber die Quaterniontheorie suggests the 
iron and clay feet of Kebuchadnezzar’s image. Happily they do 
not try to advance their mathematical idol beyond the visionary 
