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

14

NON-EUCLIDEAN GEOMETRY

[Cn. I.

to, greater than, or less than two right angles in every other
triangle.
Again, he showed that
The Parallel Postulate follows from the Hypothesis of the Right
Angle, and from the Hypothesis of the Obtuse Angle.
He was thus able to rule out the Hypothesis of the Obtuse
Angle; since, if the Parallel Postulate is adopted, the sum of
the angles of a triangle is two right angles, and the Hypothesis
of the Obtuse Angle is contradicted. It should be remarked
that he assumes in this argument that the straight line is
infinite. When that assumption is dropped, the Hypothesis
of the Obtuse Angle remains possible.
As we have already mentioned, Saccheri's aim was to show
that both the Hypothesis of the Acute Angle and that of the
Obtuse Angle must be false. He hoped to establish this by
deducing from these hypotheses some result, which itself
would contradict that from which it was derived, or be inconsistent with a previous proposition. So, having demolished
the Hypothesis of the Obtuse Angle, he turned to that of the
Acute Angle. In the system built upon this Hypothesis,
after a series of propositions, which are really propositions in
the Geometry of Lobatschewsky and Bolyai, he believed that
he had found one which was inconsistent with those preceding
it. He concluded from this that the Hypothesis of the Acute
Angle was also impossible; so that the Hypothesis of the
Right Angle alone remained, and the Parallel Postulate must
be true.
In his belief that he had discovered a contradiction in
the sequence of theorems derived from the Hypothesis of the
Acute Angle, Saccheri was wrong. He was led astray by the
prejudice of his time in favour of the Euclidean Geometry as
the only possible geometrical system. How near he came to
the discovery of the Geometry of Lobatschewsky and Bolyai
will be clear from the following description of the argument
contained in his Theorems 30 to 32:
He is dealing with the pencil of rays proceeding from a
point A on the same side of the perpendicular from A to a given
line b, and in the same plane as that perpendicular and the line.
He considers the rays starting from the perpendicular AB
and ending with the ray AX at right angles to AB.
In addition to the last ray AX, he shows that, on the hypo