656 
Proceedings of the Royal Society 
They are given by Newcomb in an extremely interesting article to 
which reference was made above. I arrange them in the order which 
best suits what has gone before. 
All the perpendiculars in a given plane to a given straight line 
intersect in a single point , whose distance from the straight line is \L. 
Conversely , the locus of all the points at a distance \L on straight 
lines passing through a given point in a given plane is a straight line 
perpendicular to all the radiating lines. 
The fixed point is called th q pole, and the straight locus its polar. 
If we cause the given plane to rotate about the polar the pole 
describes a straight line which may be called the conjugate of the 
given polar. 
The relation of these two lines is mutual , every point on one being 
at a distance \L from every point on the other. 
Without dwelling farther upon propositions of this kind, I proceed 
at once to establish the fundamental proposition concerning the sum 
of the angles of a plane triangle. I might follow a course like that 
adopted for hyperbolic space, but a much simpler method suggests 
itself at once as applicable to finite space. 
In the first place, since a complete plane is generated by the 
revolution of a complete straight line through 360°, it follows that 
the area of a biangle whose angle is A° is P . 
ooU 
In figure 15 let ABC be any triangle. Produce the sides to 
form biangles. Each of the biangles departs from the vertex on the 
upper side of the plane and returns to the vertex on the lower side. 
To make this clear areas in the neighbourhood of ABC in the figure 
are shaded with vertical lines when reckoned on the upper and with 
horizontal lines when reckoned on the lower side of the plane. A 
glance will show that if we take the three biangles they overlap the 
triangle ABC thrice, and that the rest of the plane is covered every 
where once on one side or the other, but nowhere on both sides. 
Hence, A denoting the area of the triangle, we have 
— p + JLp + 
360 + 360 
360 
P = JP-h2A 
