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

134

NON-EUCLIDEAN GEOMETRY

aCH. VI.

Thus we have proved that
3. The sum of the angles of any triangle is greater than two
right angles.
The amount by which the sum of the angles of a triangle
exceeds two right angles is called its Excess.
~ 79. Saccheri's Quadrilateral, and the Quadrilateral
with three Right Angles and one Obtuse Angle.
Let AC and BD be equal perpendiculars to the segment AB.
The quadrilateral ABDC we have called Saccheri's Quadrilateral.
Let E, F be the middle points of AB and CD.
We know that E F is perpendicular to both AB and CD; and
that the angles ACD and BDC are equal.
But the sum of the angles of a quadrilateral must be greater
than four right angles, since it is made up of two triangles.
It follows that the angles at C and D are obtuse.
C F D C
E D
A E B A B
FIG. 93. FIG. 94.
Thus the Elliptic Geometry corresponds to Saccheri's Hypothesis
of the Obtuse Angle.
Now let ABDC (Fig. 94) be a quadrilateral in which the angles
at A, B, and D are right angles.
The angle at C must be obtuse by ~ 78.
Each of the two sides containing the obtuse angle in a quadrilateral with three right angles is less than the side opposite to it.
To prove this, we proceed as follows:
If AC is not less than BD, it must be either greater than it
or equal to it.
But we know that if AC=BD, L ACD=LBDC, which is
impossible, as one is obtuse and the other a right angle.
If AC>BD, cut off AE=BD, and join ED.