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

26, 27, 28] SACCHERI'S QUADRILATERAL

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We shall see later (~ 41) that to any given segment we can
find the angle of parallelism, and that to any given acute angle
(~ 45) we can find the corresponding distance.
Thus, we can say that
If = P2, then Ifn()= 1(p2).
If p1 >P2, then II (l) < I(132).
If p1 <p32 then II(P1) > H(p2).
Also I(O)=,
IT (c)=0.
It is convenient to use the notation
x = (a), 3 = (b), etc.
Again, if the segment a is given, we can find the angle oa
[cf. ~ 41], and thus - - o. And to - - there corresponds a
2 2
distance of parallelism [cf. ~ 45]. It is convenient to denote
this complementary segment by a'.
Thus we have 7 - ()
nI (a) - 11 ().
Further, in the words of Lobatschewsky,* " we are wholly
at liberty to choose what angle we will denote by the symbol
II(p), when the line p is expressed by a negative number, so
we shall assume HI(p) + I( -p)=."
~ 28. Saccheri's Quadrilateral.
The quadrilateral in which the angles at A and B are right
angles, and the sides AC, BD equal, we shall call Saccheri's
Quadrilateral. We have seen that Saccheri
made frequent use of it in his discussion of
the Theory of Parallels.
In Saccheri's Quadrilateral, when the right
angles are adjacent to the base, the vertical
angles are equal acute angles, and the line
which bisects the base at right angles also A E B
bisects the opposite side at right angles. Fro. 27.
Let AC and BD be the equal sides, and the angles at A and
B right angles.
* Geometriscihe Untersuclhungen zur Theorie der Parallellinien, ~ 23.