X, y, z, of the imaginary trinomial ix +jy + hz, and to denote 

 that trinomial by some single letter (taken often from the 

 Greek alphabet). And on account of the facility with which 

 this so called imaginary expression, or square root of a nega- 

 tive quantity, is constructed by a right line having direction in 

 space, and having x, y,z for its three rectangular components, 

 or projections on three rectangular axes, he has been induced 

 to call the trinomial expression itself, as well as the line which 

 it represents, a vector. A quaternion may thus be said to 

 consist generally of a real part and a vector. The fixing a 

 special attention on this last part, or element, of a quaternion, 

 by giving it a special name, and denoting it in many calcula- 

 tions by a single and special sign, appears to the author to 

 have been an improvement in his method of dealing with the 

 subject : although the general notion of treating the consti- 

 tuents of the imaginary part as coordinates had occurred to 

 him in his first researches. 



Regarded from a geometrical point of view, this alge- 

 braically imaginary part of a quaternion has thus so natural 

 and simple a signification or representation in space, that the 

 diflaculty is transferred to the algebraically real part ; and we 

 are tempted to ask what this last can denote in geometry, or 

 what in space might have suggested it. 



By the fundamental equations of definition for the squares 

 and products of the symbols i,j, k, it is easy to see that any 

 (so-called) real and positive quantity is to any vector what- 

 ever, as that vector is to a certain real and negative quantity ; 

 this being indeed only another mode of saying that, in this 

 theory, every vector has a negative square. Again, the product 

 of any two rectangular vectors is a third vector at right angles 

 to both the factors (but having one or other of two opposite 

 directions, according to the order in which those factors are 

 taken) ; a relation which may be expressed by saying, that 

 the fourth proportional to the real unit and to any two rect- 

 angular vectors is a third vector rectangular to both ; or, con- 



B 2 



