Non-Euclidean Geometry Lecture 1a with narration.pdf -...

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Non-Euclidean GeometryDr. Connie S. SchrockMA 722
Euclid’s Fifth Postulate The ElementsBasic facts, Theorems and ProofNo examples, narrative, or explanations written about 2300 years ago no original copies exist. The most widely circulated book next to the Bible. Central to mathematics for 2000 years.DefinitionsAxioms relevant to all of mathematicsPostulates -- particular to the mathematical subject Theorems The Elements-- Thirteen Books Books I-VI -- Plane geometryBooks VII-IX -- Theory of NumbersBook X -- IncommensurablesBook XI-XIII -- Solid Geometry
Euclid Common Notions (Axioms)1.Things which equal the same thing also equal one another. 2. If equals are added to equals, then the wholes are equal. 3. If equals are subtracted from equals, then the remainders are equal. 4. Things which coincide with one another equal one another. 5. The whole is greater than the part.
Postulate 1To draw a straight line from any point to any point.Postulate 2To produce a finite straight line continuously in a straight line.Postulate 3To describe a circle with any center and radius.Postulate 4That all right angles equal one another.Postulate 5That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

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