Chapter 2 sections 5 and 6 with recording.pptx - Saccheri...

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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 AD AB BC AB 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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