88 G. FLEURI. 



Thus we see that a quaternion BOP is a natural generalization 

 of a complex quantity and can be at the same time considered 

 either as the symbol of a multiplication transforming its initial 

 vector OB into its terminal one OP ;* or as a translation com- 

 bined by multiplication with a rotation;! or as a magnitude. 



But whilst two numbers were sufficient to determine an ordinary 

 or plane complex quantity, four numbers are now necessary for 

 the determination of a space complex quantity or quaternion since 

 we have also to determine its plane. 



But, obviously in our notion of a quaternion we must include 

 the notion of space vector — a quantity depending only on three 

 numbers. We are therefore naturally conduced to examine the 

 particular cases when a quaternion depends on three numbers 

 only. The simplest one is for the angle of the quaternion equal 

 one right angle. In that case we can suppose that the quaternion 

 represents a vector perpendicular to its plane (in a direction such 

 that the rotation of the quaternion is from right to left for an 

 observer standing up along the vector with his feet on the plane 

 determined by the quaternion) and of length equal to its tensor. 

 However arbitrary this definition may appear, it will be sufficiently 

 justified if we obtain in the case of coplanar quadrantal quaternions 

 the same results as before for complex quantities — and this is 

 easily verified. { 



Fundamental operations on quaternions. 

 First of all, we will notice that we may easily bring any two 

 quaternions to be collinear, that is to say to have their origins 

 coinciding. For that purpose, we have simply to transport the 



* The consideration of a quantity as a magnitude or as a symbol of 

 operation is so familiar to us in mathematics that we are sometimes 

 hardly able to distinguish between those ideas. A number is very often 

 considered as a symbol of addition or of multiplication and it is according 

 to those considerations that we may write 7 = \~, for 7 and \*- are 

 magnitudes of different kinds. 



f Corresponding to a complex quantity a a put under form a x l a# 

 X We may notice that a space vector becomes thus a generalization of 

 a purely imaginary quantity. 



