654 
Proceedings of the Royal Society 
two intersecting complete straight lines enclose a plane figure. Such 
a figure I call a biangle. 
Two biangles are congruent when their angles are equal. All 
complete straight lines are of the same length, and all the straight 
lines emanating from the same point intersect in the same second 
point. 
These propositions are all equivalent to one another, and are equally 
true for single or double elliptic space. The last of them is a mere 
truism for single elliptic space. The following demonstration, which 
holds good for single or double elliptic space, may help to render 
the matter clearer. 
Let APBQA A'P'B'Q'A' (fig. 12) be two biangles having the angles 
A and A' equal. If A'B' be placed on AB so that A lies on A', and 
AT' along AP, then A'Q' will lie along AQ, since the angles at A are 
equal ; hence by the fundamental property of a straight line APB 
and A'P'B' must wholly coincide, and AQB and A'Q'B' must wholly 
coincide ; and hence B' must fall on B. It is to be noticed that the 
biangles are multiply congruent. 
Next, suppose AKA', AK'A' (fig. 13) to be any pair of intersecting 
straight lines. Let AL bisect the angle A and cut the lines in J and J'. 
Since AJ and AJ' are equiangular biangles, they are congruent ; from 
this it follows at once that J and J' must coincide with each other, 
and therefore each with A'. Hence the bisector of the angle A passes 
through A ' ; and it and AKA' and AK'A' are all of equal length. 
We may next bisect either of the halves of A, and so on ; and we 
may double any of the angles thus obtained as often as we please. 
Hence the propositions stated above are completely proved. The 
length L of a complete straight line is therefore an absolute linear 
constant which characterises an elliptic space. 
In single elliptic space the least distance between two points can 
never be greater than \L, and the greatest distance can never be greater 
than L. 
This is obvious, since the whole length of a complete straight line 
through the two points is L. 
If we consider the plane determined by two intersecting straight 
lines AO A, BOB, and if we pass from 0 along OA through a length 
L , we return to 0, but find ourselves on the opposite side of the plane 
to that from which we started , and only arrive at the same point 0 
