18 
Proceedings of Royal Society of Edinburgh. [sess. 
2 . 
But Prof. Klein, in the last published part of Klein u. Sommer- 
feld, Ueber die Theorie des Kreisels, p. 512, has repeated a statement 
made by him in the Mathematische Annalen (li. 128) to the effect 
that Gauss must, in future, he looked upon as, at least in some sense, 
the Inventor of quaternions. Here are the passages, the only hints 
as to the contents of this portion of Gauss’ Nachlass which it seems 
are to he given until the publication of his Gesammelte Werke, 
Bd. VIII. I translate freely. 
“ ... and further, that the bases ( Grundlagen ) of the Qua- 
ternion-theory are explicitly contained in the incidental notes 
( gelegentliclien Aufzeichnungen) of Gauss. In support of this sur- 
prising result we quote a few statements from a preliminary com- 
munication about the publication of Gauss’ Works ” 
“ ... And, what may appear even more startling, he had in 
1819 exhibited what he calls the Mutationen des Raumes (Turnings 
of Space round the origin of coordinates, coupled with general 
Dilatation), by means of the same four parameters which are em- 
ployed in the subsequent quaternion-theory ; he calls the group of 
them Mutationsskala , and gives explicitly the formulae for the com- 
position of two SJcalen (that is, the multiplication .of two quater- 
nions), using the symbolic form of writing 
(abcd).(ap y 8) = (ABCD); 
and expressly remarks that we are dealing with a non-com mutative 
process ! ” 
[Obviously, if these refer to quaternions at all, it is to their 
original, i.e., invented, form alone.] 
The note of exclamation is due to Prof. Klein. Its presence is 
puzzling, for certainly no one can imagine that a Gauss was 
required to discover that rotations are not, in general, commuta- 
tive; nor even that a Drehstreckung (the above combination of 
rotation and dilatation) depends upon four numbers. 
In the first part of this work of Klein and Sommerfeld there is 
a Digression on Quaternions, in which the Drehstreckung is directly 
identified with a quaternion. In fact, at p. 58 we find the follow- 
ing statements : — 
