1899 - 1900 .] Prof. Tait on a Claim made for Gauss. 23 
sufficient to state that since we now know that a Drehstreckung 
is symbolically expressed in quaternions by 
er( )r -1 , 
the resultant of two successive operations of this kind is necessarily 
ee 1 qr{ )r~ 1 q ~ ] , 
or 
ee x {qr)( )(gr)-‘; 
i.e ., it involves qr in the same extremely novel and peculiar manner 
as do the separate operators involve q and r respectively. Thus 
the multiplication of quaternions can he identified with the 
superposition of two Drehstreckungen in the same (erroneous) sense 
only as that in which a quaternion itself is identified with a 
Drelistreckung. 
It is most specially to he observed that Prof. Klein does not 
claim for Gauss any knowledge of how to add quaternions, simple 
and direct as the process is. How could Gauss have missed such 
an obvious matter if his Drelistreckung had been really a quater- 
nion ? In fact, the sum of two Drehstreckungen is not, in general, a 
Drehstreckung ; though it is, of course, a linear and vector operator. 
To add two Drehstreckungen they must first be embodied, separately, 
in any common vector, and the resulting vectors geometrically com- 
pounded. Then the Drehstreckung (if there he such) which pro- 
duces the resultant from the original vector must he found. Take 
a very simple case. Obviously we have 
- eipi - ejpj = (e 1 - efiiSip -jSjp) + (% + e)kSkp , 
The terms on the left are Drehstreckungen , applied to a common 
vector p. The right is not an embodied Drehstreckung hut a linear 
and vector function of p, which, in the particular case of e l = e, 
reduces space to an infinite straight line ! 
To add two Quaternions is a mere algebraical operation, for they 
do not require embodiment. 
Euler and Gauss, of course, easily anticipated Rodrigues in the 
mere expression of the conical rotation from one set of rectangular 
axes to another. But between that and the recognition of the 
quaternion (even as invented only) “there is a great gulf fixed”; 
and the passage across it was due entirely to Hamilton. 
