2.6.2 Saccheri Quadrilaterals
The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than the Euclidean.
—Georg Friedrich Bernhard Riemann (1826–1866)
Girolamo Saccheri (1667–1733), an Italian Jesuit priest and mathematician, attempted to prove
Saccheri considered a certain type of quadrilateral, called a Saccheri quadrilateral, as a basis for the beginning of his work in attempting to demonstrate the Fifth Postulate. We consider Saccheri quadrilaterals here to show just how close to the Euclidean concept of parallel lines one can arrive without assuming the Fifth Postulate. Since the existence of a rectangle is equivalent to
Definition. A Saccheri quadrilateral is a quadrilateral ABCD where and are right angles and . Segment is called the base, and segment is called the summit.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
A rectangle is a quadrilateral with four right angles.
Theorem 2.16. The diagonals of a Saccheri quadrilateral are congruent.
Theorem 2.17. The summit angles of a Saccheri quadrilateral are congruent.
Proof. Given Saccheri quadrilateral ABCD with right angles and , and By Theorem 2.16, . Since , , and , we have Hence . Therefore, the summit angles of a Saccheri quadrilateral are congruent.//
Theorem 2.18. The segment joining the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both the base and summit.
Proof. Given Saccheri quadrilateral ABCD with right angles and , and Further, assume M and N are the midpoints of segment AB and segment CD, respectively. Thus A-M-B, C-N-D, , and Since the summit angles are congruent, Since , , and , we have Hence Since , , and , we have Hence Since A-M-B, and are a linear pair. Since a linear pair of congruent angles are right angles, and are right angles. Hence, by the definition of perpendicular lines, line AB is perpendicular to line MN. A similar procedure may be used to prove line CD is perpendicular to line MN. Therefore, segment MN is perpendicular to both segment AB and segment CD.//
Theorem 2.19. The summit and base of a Saccheri quadrilateral are parallel.
Theorem 2.20. A Saccheri quadrilateral is a parallelogram.
It can be proven that the existence of a rectangle is equivalent to
Prove Theorem 2.16.
Exercise 2.63. Prove Theorem 2.19.
Exercise 2.64. Prove Theorem 2.20.
© Copyright 2005, 2006, 2013 - Timothy Peil