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any moonlit view, so I regard the notion of a quaternion as far 
more beautiful than any of its applications. As another illus- 
tration which I gave him, I compare a quaternion formula to a 
pocket-map — a capital thing to put in one’s pocket, but which 
for use must be unfolded : the formula, to be understood, must 
be translated into coordinates. 
I remark that the imaginary of ordinary algebra — for distinction call 
this 0 — has no relation whatever to the quaternion symbols ^, h ; 
in fact, in the general point of view, all the quantities which present 
themselves are, or may be, complex values a-\-0h, or, in other 
words, say that a scalar quantity is in general of the form a -1- 6b. 
Thus quaternions do not properly present themselves in plane or 
two-dimensional geometry at all — although, as will presently appear, 
we may use them in plane geometry ; but they belong essentially to 
solid or three-dimensional geometry, and they are most naturally 
applicable to the class of problems which in coordinates are dealt 
with by means of the three rectangular coordinates y, 2 :. 
In plane geometry, considering an origin 0, and through it two 
rectangular axes Ox, Oy, then in coordinates we determine the 
position of a point by means of its coordinates x, y ; or, writing 
X, y, z to denote given linear functions of the original rectangular 
coordinates x, y, we may, if we please, determine it by trilinear 
coordinates, or say by the ratios x\y \ z. The advantage is, that 
we thereby deal with the line infinity as with any other line, 
whereas with the rectangular coordinates x, y the line infinity pre- 
sents itself as a line sui gpneris, and that we thereby bring the 
theory into connection with that of the homogeneous functions 
{*¥, y, 2 )" 
In quaternions, the position of a point is determined in reference 
to the fixed point 0, by its vector a, which is in fact =ix+jy, 
where i,j are the quaternion imaginaries 
ij = -ji), but the idea is to use as little as possible the foregoing 
equation a = ix-\-jy, and thus to conduct the investigations inde- 
pendently, as far as may be, of the particular positions of the axes 
Ox, Oy. 
As the most simple example, take the theorem that the lines 
joining the extremities of equal and parallel lines in a plane are 
