65 
Years ago, the writer became acquainted with the method 
in analytical geometry which regards a plane and a straight 
line — not as the pure abstractions of the mathematician, 
but — as small portions of the surface of a sphere and of the 
periphery of a circle respectively in the limits in which the 
radii of the sphere and circle become infinite ; and, whilst 
noting that these definitions were open to serious objections 
(chiefly as giving sides to a plane and a straight line of an 
unsymmetrical character) he was much struck by the extent 
to which they enlarged the field in which geometrical inter- 
pretation might be consistently applied to algebraical forms 
of expression. He became strongly impressed with the 
notion that fruitful speculation would lie in the direction 
indicated — of regarding a 
of some kind, and that 
cause them, at infinity of 
distance from the parts 
considered, to deviate 
symmetrically from the 
positions due to the 
abstract conceptions of 
a plane and a straight 
line. The rotation of the 
system of circles shown 
on the margin about the 
straight line AB as an 
axis will yield examples 
of a straight line and a 
one or two pairs of impossible values are given for the co-ordinates of 
the points of intersection, then as many pairs of real values will be 
given by elimination between the equations of the supplementary pair 
of conics which lies in the plane that contains the two centres ; but he 
has yet to apply the analytical test of the correctness of this impression. 
The nature of this test will be to find whether for each case in which the 
primitive pair of equations gives x 2 or y 2 negative, the supplementary 
pair gives x 2 or y 2 positive, and vice versa. 
plane and a straight line as limits 
kind such as would, if possible, 
