Saccheri and Lambert Chapter 2 Hyperbolic Geometry

Giovanni Girolamo Saccheri • 1667 – 1733 Italian • Jesuit Priest • Attempted to develop Euclid’s geometry without the fifth • Used reductio ad absurdum proof by assuming alternative to Euclid’s Fifth was not true

Saccheri Quadrilateral AD = BC ADAB BCAB Saccheri proved m< C = m< D. C and D are summit angles. Right Obtuse Acute

Assume that the angles are Right • Produced an equivalent to the fifth 1.AB =CD 2.The sum of the angles of an triangle = 180° 3.An angle inscribed in a semicircle is always Right. Our same geometry.

Assume that the angles are Obtuse • Eventually Produced a contradiction after proving 1. AB>CD 2.The sum of the angles of an triangle > 180° 3.An angle inscribed in a semicircle is always obtuse. These are not contradictions – these are results based on the assumption.

Assume that the angles are Acute • He thought he produced a contradiction after proving the following but he was incorrect.

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