![]() ![]() There is more than one line parallel to the given line Euclid already proved that parallel lines exist using 2nd postulate Weak results, convinced almost no oneĩ Prove by contradiction by denying 5th postulate Use negation, find contradiction, show it can be proved from other 4 postulates w/o direct proofĨ 2 Part Negation There are no lines parallel to the given line If it can be, replace with it’s negation, put contradiction into system 4. Axioms contain no contradictions because of real-world models 2. Italian school teacher & scholar He approached the Parallel Postulate with these 4 statements: 1. Proclus already knew this! Most current geometry books use this instead of the 5th postulate Through a point not on a given line, exactly one line can be drawn in the plane parallel to the given line. There is no upper bound to the area of a triangle. Straight lines parallel to a third line are parallel to each other. The sum of the interior angles in a triangle is two right angles. ![]() For any three non-collinear points, there exists a circle passing through them. There exists a pair of similar non-congruent triangles. ![]() That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles Doesn’t say || lines exist!Ĥ Isn’t it a Theorem? Most convinced it was Euclid not clever enough?ĥth century, Proclus stated that Ptolemy (2nd century) gave a false proof, but then went on to give a false proof himself! Arab scholars in 8th & 9th centuries translated Greek works and tried to prove postulate 5 for centuries That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right anglesģ Parallel Postulate 5. To draw a straight line from any point to any other Any straight line segment can be extended indefinitely in a straight line To describe a circle with any center and distance That all right angles are equal to each other 5. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |