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Saccheri quadrilaterals. 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 1733 book Euclides ab omni naevo vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum.
English: Diagram of Saccheri quadrilaterals (right, obtuse, acute) Italiano: Quadrilatero di Saccheri (retto, ottuso, acuto) Français : Quadrilatère de Saccheri (droit, obtus, aigu)
The Saccheri quadrilateral was also first considered by Omar Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid. [14] Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri ), Khayyám was not trying to prove the parallel postulate as such but to ...
The theorems of Alhacen, Khayyam and al-Tūsī on quadrilaterals, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo ...
Lambert quadrilateral fundamental domain in orbifold *p222 *3222 symmetry with 60-degree angle on one of its corners. *4222 symmetry with 45-degree angle on one of its corners. The limiting Lambert quadrilateral has three right angles, and one 0-degree angle with an ideal vertex at infinity, defining orbifold *∞222 symmetry.
The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were part of a line of research on the parallel postulate continued by later European geometers, including Vitello (c. 1230 – c. 1314), Gersonides (1288–1344), Alfonso, John ...
A Saccheri quadrilateral is a quadrilateral which has two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, and right angles if the ...
Brahmagupta (597–668) – Euclidean geometry, cyclic quadrilaterals; Vergilius of Salzburg (c.700–784) – Irish bishop of Aghaboe, Ossory and later Salzburg, Austria; antipodes, and astronomy; Al-Abbās ibn Said al-Jawharī (c. 800–c. 860) Thabit ibn Qurra (826–901) – analytic geometry, non-Euclidean geometry, conic sections