3 
2, Ys 25 of the imaginary trinomial ix+jy+4z, 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 «, y, 2 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 
difficulty 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 ¢, 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 isa 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- 
B2 
