[ 725 ] 
XVII. On a New Geometry of Space. By J. Plucker, of Bonn, For. Memb. B.S. 
Received December 22, 1864, — Read February 2, 1865. 
I. On Linear Complexes of Bight Lines. 
1. Infinite space may be considered either as consisting of points or transversed by 
planes. The points, in the first conception, are determined by their coordinates, by x, 
y, z for instance, taken in the ordinary signification ; the planes, in the second conception, 
are determined in an analogous way by their coordinates, introduced by myself into 
analytical geometry, by t, u, v for instance. 
The equation 
tx-\-uy-\-vz-\- 1=0 
represents, in regarding x, y, z as variable and t, u, v as constant, a plane by means of 
its points. The three constants t, u, v are the coordinates of this plane. The same 
equation, in regarding t, u, v as variable, x, y, z as constant, represents a point by means 
of planes passing through it. The three constants are the coordinates of the point. 
A point given by its coordinates and a point determined by its equation, or geome- 
trically speaking by an infinite number of planes intersecting each other in that point, 
are quite different ideas, not to be confounded with one another. That is the case also 
with regard to a plane given by its coordinates and a plane represented by its equation, 
or considered as containing an infinite number of points. Hence is derived a double 
signification of a right line. It may be considered as the geometrical locus of points, or 
described by a point moving along it, and accordingly represented by two equations in 
x, y, z, each representing a plane containing that line. But it may likewise be con- 
sidered as the intersection of an infinite number of planes, or as enveloped by one of 
these planes, turning round it like an axis ; accordingly it is represented by two equa- 
tions in t, u, v, each representing an arbitrary point of the line. The passage from one 
of the two conceptions to the other is a discontinuous one*. 
2. The geometrical constitution of space, hitherto referred either to points or to planes, 
may as well be referred to right lines. According to the double definition of such lines, 
there occurs to us a double construction of space. 
In the first construction we imagine infinite space to be transversed by lines them- 
selves consisting of points. An infinite number of such lines pass in all directions 
through any given point ; each of these lines may be regarded as described by a moving 
* According to this discontinuity, a plane curve represented by ordinary coordinates may have a conjugate 
which disappears if the same curve he represented by means of line-coordinates. See “ System der analytischen 
Geometrie,” n. 330. 
5 H 
MDCCCLXV. 
