276 Transactions.—Miscellaneous. 
lines in the given surface is the same. The corresponding proposition in 
spherical geometry is, that all great circles of a given sphere are equal. 
There are a great many other analogies between the imaginary surface 
here treated of and the surface of a sphere. Its straight lines, though they 
are like the straight lines of a plane, in the circumstance that any two 
of them have only one point of intersection, are in many other respects 
analogous to great circles. In any of its straight lines, for instance, 
each point has a corresponding point which is opposite to it and 
farther from it than any other point in the line. For, if by set- 
ting out from a point and travelling a finite distance in a particular 
direction we get back to the starting-point, there must be a point 
half-way on our journey which is farther from the starting-point than any 
other point in the line, and which may very appropriately be called its 
opposite point. It is an obvious corollary, that the distance between any 
two points will be the same as the distance between their opposite points. 
Let us now consider the case of a number of straight lines radiating 
from a centre. In each of them there will be a point which is opposite to 
that centre. And it will be a separate point for every separate straight 
line. For no two straight lines can have two points in common; and, 
since these radiating lines have a common centre of radiation, they can 
have no other point in common. Hence, if we suppose one of these lines 
to rotate about the centre, the point opposite to the centre will describe a 
continuous line, and one which finally returns into itself. It is the 
locus of all points in the surface opposite to the centre of radiation. What, 
now, is the character of this locus? In the first place, it is a line which is of 
the same shape all along, and of which all equal segments, therefore, can be 
made tocoincide. For any two positions of the rotating line which contain 
a given angle, may be placed upon any other two positions which contain an 
equal angle. Then, since the length of all straight lines in the surface is the 
same, the opposite points will coincide, and, by parity of reasoning, all 
intermediate points of the locus. But, in the second place, the locus is also 
of the same shape on both sides. For each point in it may be approached 
from the centre of radiation in two different ways, and it is at the same 
distance from that centre, whether it be approached in the one way or the 
other. Any particular segment, in fact, of the locus, has its extreme points 
joined to the centre of radiation by lines which are of equal length, and 
which include an equal angle—lines, therefore, whieh may be made to 
coincide. Since this is the case for any segment whatever, and for every 
subdivision of a segment, all the points of a segment will still remain on it 
if the segment be turned round and applied to itself. Hence the locus is of 
the same shape, whether viewed from the one side or the other. But, since 
