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

99, 100, 101] THE UNIT NOMINAL SEGMENT

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Thus the point P divides the diameter in the ratio e: 1.
The unit segment is thus fixed for any position in the domain
of the nominal points, since the segment OP can be " moved"
Fio. 113.
FIG. 113.
so that one of its ends coincides with any given nominal
point.
A different expression for the nominal length, viz.,,, 1 AV/ BV\
k log( )U/ )u
would simply mean an alteration in this unit, and taking
logarithms to the base a instead of e would have the same
effect.
~ 101. We are now able to establish some further theorems
of Hyperbolic Geometry, using the metrical properties of this
Nominal Geometry.
In the first place we can say that Similar Triangles are
impossible. For if there were two nominal triangles with the
same angles and not congruent, we could " move " the second
so that its vertex would coincide with the corresponding angular
point of the first, and its sides would lie along the same nominal
lines as the sides of the first. We would thus obtain a "quadrilateral" whose angles would be together equal to four right
angles; and this is impossible, since we have seen that the
sum of the angles in these nominal triangles is always less
than two right angles.
We also see that parallel lines are asymptotic; that is, they
continually approach each other. This follows from the
figure for nominal parallels and the definition of nominal
length.