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

5, 6]

SACCHERI

13

Saccheri's little book —Euclides ab omni ncevo vindicatus —
is now easily accessible.* It was published in 1733, the last
year of his life. Much of it has been incorporated in the
elementary treatment of the Non-Euclidean Geometries. A
great deal more would be found therein were it not for the fact
that he makes very frequent use of the Principle of Continuity.
It must not be forgotten that Saccheri was convinced of the
truth of the Euclidean Hypothesis. He discussed the contradictory assumptions with a definite purpose-not, like
Bolyai and Lobatschewsky, to establish their logical possibility-but in order that he might detect
the contradiction which he was persuaded C D
must follow from them. In other words,
he was employing the reductio ad absurdum
argument.
The fundamental figure of Saccheri is
the two right-angled isosceles quadrilateral A
ABDC, in which the angles at A and B are A B
right angles, and the sides AC and BD equal.
It is easy to show by congruence theorems that the angles
at C and D are equal. [Cf. ~ 28.]
On the Euclidean Hypothesis they are both right angles.
Thus, if it is assumed that they are both obtuse, or both acute,
the Parallel Postulate is implicitly denied.
Saccheri discussed these three hypotheses under the names:
The Hypothesis of the Right Angle... z C = L D = a right angle.
The Hypothesis of the Obtuse Angle... L C =L D = an obtuse
angle.
The Hypothesis of the Acute Angle... LC=L D = an acute
angle.
He showed that
According as the Hypothesis of the Right Angle, the Obtuse
Angle, or the Acute Angle is found to be true, the sum of the
angles of any triangle will be respectively equal to, greater than,
or less than two right angles.
Also that
If the sum of the angles of a single triangle is equal to, greater
than, or less than two right angles, then this sum will be equal
*Cf. Engel u. Stackel, Die Theorie der Parallellinien von Euclid bis
auf Gauss, pp. 31-136 (Leipzig, 1895).