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

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Non-Euclidean Geometry Dr. Connie S. Schrock MA 722
Euclid’s Fifth Postulate The Elements – Basic facts, Theorems and Proof No 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. Definitions Axioms – relevant to all of mathematics Postulates -- particular to the mathematical subject Theorems The Elements -- Thirteen Books Books I-VI -- Plane geometry Books VII-IX -- Theory of Numbers Book X -- Incommensurables Book 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 1 To draw a straight line from any point to any point. Postulate 2 To produce a finite straight line continuously in a straight line. Postulate 3 To describe a circle with any center and radius . Postulate 4 That all right angles equal one another. Postulate 5 That, 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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