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

72

NON-EUCLIDEAN GEOMETRY

[CH. ITn.

The angle EDB of the quadrilateral ABDE, in which three
angles are right angles, is a right angle, or an acute angle,
according as ED is equal to or greater than AB (cf. ~ 29).
With centre A describe a circle whose radius is equal to ED.
It will intersect DB at a point C, coincident with B, or between B and D.
E D
A BN
FIG. 48.
The angle which the line AC makes with DB is the angle of
parallelism corresponding to the segment BD.
Therefore a parallel to AN can be drawn by making the angle
BDM equal to the angle ACB.
Bolyai's proof is omitted for the reasons named above; but
it should be remarked that his construction holds both for the
Euclidean and Non-Euclidean Geometries; in his language it
belongs to the Absolute Science of Space.
~ 42. The correspondence which we have established in ~ 35
between the right-angled triangle and the quadrilateral with
three right angles and one acute angle, leads at once to Bolyai's
construction.
We have seen that, to the right-angled triangle a, b, c, (X, a),
there corresponds a quadrilateral with three right angles and an
acute angle 3, the sides containing the acute angle being c and
1, and the other two, a and m'.
Therefore we can place the right-angled triangle in the
quadrilateral, so that the side a of the triangle coincides with
the side a of the quadrilateral, and the side b of the triangle
lies along the side I of the quadrilateral. Then the hypothenuse
of the triangle will be parallel to the side c of the quadrilateral,

since it makes an angle r with '
since it makes an angle -i with m'.