4 
versely, that the fourth proportional to any three rectangular 
vectors isa quantity distinct from every vector, and of the kind 
called real in this theory, as contrasted with the kind called 
imaginary. 
Now, in fact, what originally led the author of the present 
communication to conceive (in 1843) his theory of quaternions 
(though he had, at a date earlier by several years, speculated 
on triplets and sets* of numbers, as an extension of the theory 
of couples, or of the ordinary imaginaries of algebra, and also 
as an additional illustration of his views respecting the Sci- 
ence of Pure Time), was a desire to form to himself a distinct 
conception, and to find a manageable algebraical expression, 
of a fourth proportional to three rectangular lines, when the 
DIRECTIONS of those lines were taken into account; as Mr. 
Warrent and Mr. Peacockt had shewn how to conceive and 
express the fourth proportional to any three lines having di- 
rection, but situated 7m one common plane. And it has since 
appeared to Sir William Hamilton that the subject of quater- 
nions may be illustrated by considering more closely, though 
briefly, this question of the determination of a fourth propor- 
tional to three rectangular directions in space, rather in a geo- 
metrical than in an algebraical point of view. 
Adopting the known results above referred to, for propor- 
tions between lines having direction in a single plane (though 
varying a little the known manner of speaking on the sub- 
ject), it may be said that, in the horizontal plane, ‘* West is 
to South as South is to East,” and generally as any direction 
is to one less advanced than itself in azimuth by ninety de- 
grees. Let it be now assumed, as an extension of this view, 
that in some analogous sense there exists a fourth proportional 
* See Transactions of the Royal Irish Academy, vol. xvii. p. 422. Dublin, 
1835. 
+ Treatise on the Geometrical Representations of the Square Roots of Ne- 
gative Quantities, by the Rey. John Warren. Cambridge, 1828. 
} Treatise on Algebra, by the Rev. George Peacock. Cambridge, 1830. 
