MATHEMATICS AND ASTRONOMY. 
41 
there is an assumption that the axes of the two angles are coincident ; 
and that Argand’s meaning is incorrect. 
The ideas of Warren in his Treatise on the geometrical representation of 
the square roots of negative quantities, 1828, are essentially the same as 
those of Français, but they receive a more complete development. 
It is curious to find, considering the intensely geometrical character of 
quaternions, that Hamilton was led by the Kantian ideas of space and time 
to start out with the theory that algebra is the science of time, as geometry 
is the science of space, and that he strove hard to find on that basis a 
meaning for the square root of minus one. But having observed the suc 
cess, so far as the plane is concerned, of the geometrical theory of Argand, 
Français and Warren, he adopted a geometrical basis and took up the 
problem of extending their method to space. What he sought for was 
the product of two directed lines in space, in the sense of a fourth pro 
portional to two given lines and an initial line. He perceived that one 
root of the difficulty which had been experienced lay in regarding the 
initial line as real, and the two perpendiculars as expressed by imagina 
rles ; and, looking at the symmetry of space, adopted the view that each of 
the three axes should be treated as an imaginary. He was thus led to the 
principle that if i, j, k denote three mutually rectangular axes, then 
i 2 = —1, j* = —1, k 2 = —1, 
and if JJa denote any vector of unit length ( Ua) 2 — —1. Hence follows 
the paradoxical conclusion that the square of a directed magnitude is 
negative, which is contrary to the principles of analysis. An after devel 
opment of Hamilton’s was to give to i, j, k a double meaning, namely : to 
signify not only unit vectors, but to signify the axes of quadrantal ver 
sons. But in the quaternion we have for the first time the clear distinc 
tion between a line and a geometric ratio. In a paper read before this 
Association last year 1 have given reasons for believing that the identifi 
cation of a directed line with a quadrantal quaternion is the principal 
cause of the obscurity in the method, and of its want of perfect harmony 
with the other methods of analysis. 
The imaginary symbol, notwithstanding its apparent banishment from 
space, reappears in Hamilton’s works as the coefficient of an unreal qua 
ternion. He appears to hold that there is a scalar \/—1 distinct from 
that vector -[/—1 which can be replaced by i, j, k. In the recent edition 
of Tait’s Treatise on Quaternions, Prof. Cayley contributes an analytical 
theory of quaternions, in which the components w, x, y, z of a quaternion 
are considered in the most general case to have the form a + &j/—I 
where j/—1 is the imaginary of ordinary algebra. Thus it appears as 
if we were landed in an analytic theory of quaternions instead of a qua- 
ternionic theory of analysis. 
In a work recently published on quaternions ( Theorie der Quaternionen, 
by Dr. Molenbroek), the principal novelty is the introduction of the sym 
bol Ÿ— 1 with the meaning attached to it by Bueé, namely: to denote