480 
[951 
951. 
N ON-EU CLIDI AN GEOMETRY. 
[From the Transactions of the Cambridge Philosophical Society, vol. xv. (1894), 
pp. 37—61. Read January 27, 1890.] 
I consider ordinary three-dimensional space, and use the words point, line, plane, 
•See., in their ordinary acceptations; only the notion of distance is altered, viz. instead 
of taking the Absolute to be the circle at infinity, I take it to be a quadric 
surface: in the analytical developments this is taken to be the imaginary surface 
(P + y 2 + z 1 + w 2 = 0, and the formulae arrived at are those belonging to the so-called 
Elliptic Space. The object of the Memoir is to set out, in a somewhat more 
systematic form than has been hitherto done, the general theory; and in particular, 
to further develop the analytical formulas in regard to the perpendiculars of two 
given lines. It is to be remarked that not only all purely descriptive theorems of 
Euclidian geometry hold good in the new theory; but that this is the case also 
(only we in nowise attend to them) with theorems relating to parallelism and 
perpendicularity, in the Euclidian sense of the words. In Euclidian geometry, infinity 
is a special plane, the plane of the circle at infinity, and we consider (for instance) 
parallel lines, that is, lines which meet in a point of this plane: in the new theory, 
infinity is a plane in nowise distinguishable from any other plane, and there is no 
occasion to consider (although they exist) lines meeting in a point of this plane, 
that is, parallel lines in the Euclidian sense. So again, given any two lines, there 
exists always, in the Euclidian sense, a single line perpendicular to each of the 
given lines, but this is not in the new sense a perpendicular line; there is nothing 
to distinguish it from any other line cutting the two given lines, and consequently 
no occasion to consider it: we do consider the lines—there are, in fact, two such 
lines—which in the new sense of the word are perpendicular to each of the given lines. 
It should be observed that the term distance is used to include inclination: we 
have, say, a linear distance between two points; an angular distance between two 
lines which meet; and a dihedral distance between two planes. But all these are