The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

78, 79] SACCHERI'S QUADRILATERAL 135
Then we know that L AED =LEDB.
But LEDB is acute, so that both must be acute, which is
impossible.
Therefore AC must be less than BD.
Again, starting with AB and CD, which are both perpendicular
to BD, we find that CD is less than AB, so our theorem is proved.
We shall not proceed further with the formal development
of this geometry. There is no Theory of Parallels, for
parallel lines do not exist in it. There is only one kind of
circle, the locus of corresponding points upon a pencil of
straight lines. The measurement of areas follows on the same
lines as in the Hyperbolic Geometry.
Two triangles which have the same excess have equal areas,
and conversely.
The area of a triangle is proportional to its excess.