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

88

NON-EUCLIDEAN GEOMETRY

[CH. III.

Therefore the defects of any two equivalent triangles are
equal.
The theorem enunciated at the head of this section is thus
established: a necessary and sufficient condition for equivalence of triangles is equal defect.
5. A triangle is said to be equivalent to the sum of two
other triangles, when the three triangles can be broken up
into a finite number of triangles, such that the triangles in the
partition of the first are congruent in pairs with the sum of the
triangles in the partitions of the other two.
Now the defect of each triangle is equal to the sum of the
defects of the triangles into which it is divided.
It follows that if a triangle is equivalent to the sum of two
other triangles, its defect is equal to the sum of their defects.
~ 52. If we regard area as a concept associated with a rectilinear figure, just as length is with a straight line, it is obvious
that equivalent figures have equal area.* And if, further, we
regard the area of a rectilinear figure as a magnitude to which
we can ascribe the relations of sum, equality and inequality,
greater and less, we obtain at once from the theorems of ~ 51
the result that the areas of triangles are proportional to their
defects. Indeed if we start with any triangle as the triangle of
unit area, a triangle which is n times this triangle will have n
times its defect.
But closer examination of the argument shows that in this
treatment of the question of area various assumptions are
made; and the work of some mathematicians of the present
day has put the theory of area on a sounder logical basis.t
This more exact treatment of the theory of area in the Hyperbolic Plane is simple, and will now be given:
The measure of of ae of a triangle is defined as k2 multiplied
* Hilbert distinguished between equivalent polygons, as defined above,
and polygons which are equivalent by completion. Two polygons are
said to be equivalent by completion, when it is possible to annex to them
equivalent polygons, so that the two completed polygons are equivalent.
If the Postulate of Archimedes is adopted, polygons, which are equivalent by completion, are also equivalent. Hilbert was able to establish
the theory of area on the doctrine of equivalence by completion without
the aid of the Postulate of Archimedes. Loc. cit. Chapter IV.
+fCf. Art. VI. by Amaldi, in Enriques' volume referred to above.
Also Finzel, Die Lehre vom YFldcheninhalt in der allgemeinen Geometrie
(Leipzig, 1912).