Saccheri quadrilateral

A Saccheri quadrilateral is a quadrilateral in the absolute geometry having the characteristics that two adjacent internal angles are right angles, and two opposed sides, against which these angles are of equal length. Such squares were defined by the Italian mathematician Giovanni Girolamo Saccheri in the first third of the 18th century, and examines, after which they are named today. His original aim was, Euclid's fifth postulate, the parallel axiom to derive a proof by contradiction from the other axioms.

For the first time a square of this type has been studied by the Persian mathematician Omar Khayyam in the late 11th century, it is the square well (correct ) is called Khayyam - Saccheri quadrilateral. Whether Saccheri of Khayyam knew writings, is unknown.

History and characteristics

• In plane geometry course every Saccheri quadrilateral is a rectangle.
• In the flat absolute geometry of the following theorem holds:

• From this theorem it follows that the two interior angles of a Saccheri quadrilateral is predicated on the definition in the nothing, in the absolute geometry must always be equal to each other.
• Saccheri correctly showed essentially that in which he uses axioms of absolute geometry, the axiom groups I- III and V was broadly equivalent to the much later defined by David Hilbert axiom system of Euclidean geometry, the relevant angle is not dull can be. Hilbert axiom system has to be formulated so that its axioms without the axiom of parallels both Euclidean (there is a clear parallel through a point ) as (there are several parallels through a point ) also admit hyperbolic models of geometry. In order to understand the elliptical geometry, in which through a point outside a line, there is no parallel, axiomatic, Hilbert's axioms of arrangement ( group II) and the congruence ( group III) are often replaced by weaker axioms of motion in absolute geometry. A recent axiomatization of absolute geometry, which is based entirely on the concept of motion, is the absolute metric geometry.
• In contrast, Saccheri was evidence that these angles may not be sharp, wrong.

The parallel axiom expressed by Saccheri quadrilaterals

Each of the following statements is based on the axioms of absolute geometry ( according to Hilbert ) equivalent to the parallel postulate ( Axiom IV Hilbert ):

• There is a rectangle.
• In a and thus in every Saccheri quadrilateral all interior angles are right.