1899-1900.] Prof. Tait on a Claim made for Gauss. 
19 
“ Eine Quaternion bedeutet nichts anderes als die Operation der 
Drehstreckung 
“ Eine gewohnliche Drehung ist eine Einheitsquaternion. ” 
Hence, of course, the claim made for Gauss to at least a share in 
the invention of quaternions. 
Unfortunately for such a conclusion, a Drehstreckung is not a 
Hamiltonian quaternion at all, hut a totally different kind of con- 
cept. It is obviously only a very limited form of linear and vector 
operator (kinematically a strain) depending upon four constants 
instead of the usual nine ; and might, perhaps (but on that account 
solely), have been designated by the name quaternion, had the 
name not been already more worthily bestowed. 
3. 
A quaternion, as Hamilton gave it, forms an indispensable part 
of any conceivable complete theory of vectors. It expresses the 
relation of one vector to another, or supplies the factor required to 
convert one into the other. It is completely determined by these 
tico alone , and is thus a conception as real as either. In this 
sense it was called by Hamilton a Biradial. It has a plane (or 
rather an aspect ), an angle, and the ratio of the lengths of its two 
legs; and all hiradials characterized by like conditions of these 
kinds are regarded as equivalent to one another. [Equality of 
angles implies that they are to he measured in the same sense .] A 
quaternion, therefore, when applied to any vector in or parallel to 
its oim plane , turns it through a given angle in or parallel to that 
plane, and alters its length in a given ratio. When the legs of 
the biradial are equal, and its angle a right angle, the quaternion 
(as Hamilton showed) is fully represented by the unit-vector per- 
pendicular to its plane. All these particular statements are con- 
tained in the general expression 
?=/?/ 
(cos A + e sin A), 
where /3 and a are the vector legs of the biradial, b and a their 
lengths, A its angle, and e the unit-vector perpendicular to its 
plane. Obviously, when this is applied to a vector which is not 
perpendicular to e, the result is a new Quaternion , not a vector. 
