Saccheri quadrilateral

A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book "Euclid vindicatus" (1733), an attempt to prove the parallel postulate. Since the Saccheri quadrilateral was first considered by Omar Khayyam in the late 11th century, the quadrilateral has alternatively been named the Khayyam-Saccheri quadrilateral.

For a Saccheri quadrilateral ABCD, the sides AD and BC (also called legs) are equal in length and perpendicular to the base AB. The top CD is called the summit or upper base and the angles at C and D are called the summit angles.

The advantage of using Saccheri quardrilaterals when considering the parallel postulate is that they place the mutually exclusive options in very clear terms:

:Are the summit angles right angles, obtuse angles, or acute angles?

History

Saccheri quadrilaterals were first considered by Omar Khayyam in the late 11th century in Book I of "Explanations of the Difficulties in the Postulates of Euclid".Boris Abramovich Rozenfelʹd (1988), "A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space", p. 65. Springer, ISBN 0387964584.] Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):

:Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge. [Boris A Rosenfeld and Adolf P Youschkevitch (1996), "Geometry", p.467 in Roshdi Rashed, Régis Morelon (1996), "Encyclopedia of the history of Arabic science", Routledge, ISBN 0415124115.]

Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.

It wasn't until 600 years later that Giordano Vitale made an advance on Khayyam in his book "Euclide restituo" (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.

Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.

Notes

References

*George E. Martin, "The Foundations of Geometry and the Non-Euclidean Plane", Springer-Verlag, 1975