FROM NUMBER TO QUATERNION. 87 



Second, of a rotation of an angle 10 A = a 



and we can denote by IOA and represent 

 geometrically (Fig. 12) by the figure IOA the 

 operation of that multiplication. This operator 



applied to 01 i.e., to the absolute unit gives as 



OA 



result the vector whose tensor is 01 . -— = OA 



and which makes an angle a with 01 that is to 

 say gives OA. 



Therefore the operator IOA can be considered as representing 

 the complex quantity OA.* 



But with that meaning, any operator BOP such that triangle 

 BOP (Fig. 10) is directly similar to triangle IOA is equal to 

 IOA for, the two operations 



First multiplication by — - = — - 



Second rotation of a 

 are identical respectively to the preceding ones. 



Moreover, BOP operating on OB gives a vector of length 



OP 

 OB . — — = OP and making an angle a with OB that is to say 



gives OP ; therefore the operator BOP can also be considered as 



transforming by multiplication OB into OP (as operator IOA 

 transforms 01 into OA). 



Extending now these notions to space we will give to a quantity 

 (or operator) as above defined the name of quaternion. 



Two quaternions are equal when the two triangles they determine 

 are, 



First, on same plane or on parallel planes (co-planar); 

 Second, directly similar. 



Wherefore a quaternion may be transported in any way in its 

 plane and may have one of its vectors of any given length. 



* For, if multiplying the unit by a certain number we find as result 

 15, the number in question is 15. 



