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

132

NON-EUCLIDEAN GEOMETRY

[CH. VI.

form in his mind. The Single Elliptic Plane and its importance
in the higher treatment of the Non-Euclidean Geometries were
first brought to light by Klein.
~ 78. We shall now show that this geometry corresponds to
Saccheri's Hypothesis of the Obtuse Angle, so that the sum of
the angles of a triangle is always greater than two right angles.
The following theorem enables us to put the proof concisely:
1. In any triangle ABC in which the angle C is a right angle,
the angle A is less than, equal to, or greater than a right angle,
according as the segment BC is less than, equal to, or greater than |.
Let P be the pole of the side AC.
A
C B P
FIG. 90.
Then P lies upon BC, and PC =.
Join AP.
Then L PAC=a right angle.
If CB > CP, then L BAC L PAC; i.e. z BAC > a right angle.
If CB = CP, then L BAC = PAC; i.e. L BAC = a right angle.
If C B < CP, then L BAC <L PAC; i.e. L BAC < a right angle.
The converse also holds.
Now consider any right-angled triangle ABC in which C is
the right angle.
If either of the sides AC or BC is greater than or equal to
F, the sum of the angles is greater than two right angles by
the above theorem.
If both sides are less than V, from D, the middle point of
the hypothenuse, draw DE perpendicular to the side BC.
Let P be the pole of DE.