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

90 NON-EUCLIDEAN GEOMETRY [co. II. 52, 53]
area. The area of a polygon will be said to be greater or less
than the area of another polygon according as its measure of
area is greater or less than the measure of area of the other.
~ 53. In the Euclidean Plane we say that a rectilinear figure
contains so many square inches (or sq. ft., etc.), and by considering a curvilinear figure as the limit of a rectilinear figure
we obtain a method of measuring curvilinear figures.
In the Hyperbolic Plane there is no such thing as a square
inch, or rectangle with equal sides, or any rectangle. To every
rectilinear figure there corresponds an equivalent Saccheri's
Quadrilateral. To all equivalent rectilinear figures there
corresponds one and the same Saccheri's Quadrilateral with a
definite acute angle.
This quadrilateral with a given acute angle can be constructed in this geometry immediately. The construction
follows from the correspondence established between rightangled triangles and the quadrilateral with three right angles.
If the acut at angle i 3, we obtain the corresponding segment
b{/3 =H(b)}, by the construction of ~ 45. We draw any
right-angled triangle with a side equal to b. The associated
quadrilateral has its acute angle equal to j3, and the Saccheri's
Quadrilateral is obtained by placing alongside it a congruent
quadrilateral.
All Saccheri's Quadrilaterals with the same acute angle are
equivalent.
Thus it will be seen that there is a fundamental difference
between measurement of length and area in the Euclidean
and the Hyperbolic Plane.* In the Euclidean, the measures
are relative. In the Hyperbolic, they are absolute. With every
linear segment there can be associated a definite angle, namely
the angle of parallelism for this segment. With every area,
a definite angle can be associated, namely the acute angle
of the equivalent Saccheri's Quadrilateral.
* Cf. Bonola, loc. cit. ~ 20. Also supra, p. 17.