660 Proceedings of the Royal Society 
before proving that when two straight lines cut one another the 
vertically opposite angles are equal ! 
Appendix on the Trigonometry of Elliptic and Hyperbolic Space. 
The following appears to me to be the simplest, and at the same time the 
most instructive way of establishing the Trigonometry of Elliptic and Hyper- 
bolic Space. 
The method might, indeed, by assuming proper axioms, be made to take the 
place of the preceding synthesis. As it is, I shall base it upon the results of 
that synthesis. What I shall want are mainly the propositions concerning 
the excess or defect of plane triangles, and the conclusion founded on them 
that homaloidal trigonometry may be applied to figures, all of whose dimen- 
sions are infinitely small compared with the linear constant of space. 
Let KA and LB (fig. 17) be two straight lines in the same plane at an infinitely 
small distance apart. They may be either non-intersectors, whose minimum 
distance d is infinitely small, or intersectors which make a very small angle a 
with each other at their point of intersection. 
Let KL, AB, CD be lines making equal angles with KA and LB ; and let 
KA= LB=r, AC = BD = efoyAB = D, CD = D+c£D, where dr is infinitely small 
compared with r, dD infinitely small compared with D ; D of course is in- 
finitely small compared with p, the linear constant of space. 
Further, let [_ LBA = L KAB^J-0, and L LDC = L KCD = £-e-dd. 
A A 
Since all the dimensions of ABDC are infinitely small compared with p, 
we may apply Euclidean trigonometry. Draw B m parallel to AC. Then 
i ABm — ^-6, AB = Cm, and Dm - dD. 
2 sin|DBm=:2 sin ( - Q)—^ ; 
( 1 ) 
Now the excess of ABDC = 
2 (f + # y 
its area = D dr. Hence 
which gives 
D dr=p 2 e 
2 de + 
Z Tr + 
Whence by (1) 
<PD 
dr* 
- do'j 
2 7r — 2 dd ; and 
+ ? = °- 
( 2 ) 
