506 The British Association . [October. 
ments may be chosen ; and upon the degree of complexity 
of the subject of our choice will depend the internal struc- 
ture or manifoldness of space. Thus, beginning with the 
simplest case, a point may have any singly infinite multitude 
of positions in aline, which gives a onefold system of points 
in a line. The line may revolve in a plane about any one 
of its points, giving a twofold system of points in a plane ; 
and the plane may revolve about any one of the lines, giving 
a threefold system of points in space. Suppose, however, 
that we take a straight line as our element, and conceive 
space as filled with such lines. This will be the case if we 
take two planes, — e.g., two parallel planes, — and join every 
point in one with every point in the other. Now the points 
in a plane form a twofold system, and it therefore follows 
that the system of lines is fourfold ; in other words, space 
regarded as a plenum of lines is fourfold. The same result 
follows from the consideration that the lines in a plane, and 
the planes through a point, are each twofold. Again, if we 
take a sphere as our element we can through any point as 
a centre draw a singly infinite number of spheres, but the 
number of such centres is triply infinite ; hence space as a 
plenum of spheres is fourfold. And, generally, space as a 
plenum of surfaces has a manifoldness equal to the number 
of constants required to determine the surface. Although 
it would be beyond our present purpose to attempt to pursue 
the subject further, it should not pass unnoticed that the 
identity in the fourfold character of space, as derived on the 
one hand from a system of straight lines, and on the other 
from a system of spheres, is intimately connected with the 
principles established by Sophus Lie in his researches on 
the correlation of these figures. If we take a circle as our 
element we can around any point in a plane as a centre 
draw a singly infinite number of circles ; but the number of 
such centres in a plane is doubly infinite ; hence the circles 
in a plane form a threefold system, and as the planes in 
space form a threefold system it follows that space as a 
plenum of circles is sixfold. Again, if we take a circle as 
our element, we may regard it as a section either of a sphere 
or of a right cone (given except in position) by a plane per- 
pendicular to the axis. In the former case the position of 
the centre is threefold ; the directions of the plane, like that 
of a pencil of lines perpendicular thereto, twofold : and the 
radius of the sphere onefold ; sixfold in all. In the latter 
case the position of the vertex is threefold ; the direction of 
the axis twofold ; and the distance of the plane of section 
onefold ; sixfold in all, as before. Hence space as a plenum 
