164 ROCHESTER ACADEMY OF SCIENCE. [Jan. I4, 



Applying the six operations to this new quantity, the quaternion, 



we get 



The sum of two quaternions is a quaternion ; \ 



The difference of two quaternions is a quaternion ; J 

 The product of two quaternions is a quaternion ; 

 The quotient of two quaternions is a quaternion ; 

 The reversion of quaternion is a quaternion ; 

 The mean reversion of a quaternion is a quaternion ; 



Out of the weights of the points, then, we get two fundamental 

 quantities : scalars, vectors (and their combination, quaternions) and 

 no others. The introduction of such terms as versors, turning factors, 

 quadrantal versors, etc., etc., is merely cumulative, and unnecessary 

 except as a convenient mode of designating some particular phase of 

 the two fundamental quantities, the scalar and the vector. As we 

 have seen, a quaternion is either the product of two vectors, or the 

 sum of a scalar and a vector, and as a symbol of operation designates 

 the conversion of one vector into another. This avoids such artificial 

 and mystifying definitions of a quaternion as, a quasi mechanical 

 operator .... composed of a magnitude and a turning factor, etc. 



A vector, considered as a symbol of operation, is a mean reverser. 

 This is the inevitable result of our definition of multiplication (loc. 

 cit. ). When the vector occurs in combination with a scalar (the 

 quaternion) it does not necessarily need a new name (turning factor, 

 etc. ). It simplifies not only the theory but also the use of quaternions 

 to consider them as simply a combination of a scalar and a vector, to 

 be operated with and upon by the rules applicable to scalars and 

 vectors. This method of looking at scalars and vectors (quaternions) 

 binds the whole subject of Reals, Imaginaries and Quaternions into a 

 united and homogeneous whole, the quaternion being the inevitable 

 logical expansion of the elementary algebraic conceptions. It avoids 

 all assumptions as to " the definition of the multiplication of i into j ", 

 all arbitrary retention or rejection ' ' of the old laws of multiplication ' ' , 

 all introduction of arbitrary " laws", all attaching to multiplication of 

 " any signification we please when we speak of vectors ", etc. 



IV. 



The weights of the points have by their characteristic of difference 

 from unity (their only characteristic) indicated the operation of multi- 

 plication, and its inverse, and nothing else. 



