Ch. 1 and 2 - Lectures 1 - 4

Ch. 1 and 2 - Lectures 1 - 4 - Math 3379 Chapter 1...

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Math 3379 Chapter 1 Prologue: Euclid’s Elements Time Line 4000 years ago practical applications Mesopotamia, Egypt, India, China 2500 years ago Greek influence brings abstraction 600 BC Thales of Miletus brings logic and logical argument 300 BC Euclid writes Elements – the organization is key 17 th Century Descartes: Cartesian plane; identification of point and number 19 th Century Saccheri, Bolyai, Lobachevsky, Gauss: Hyperbolic geometry Gino Fano – finite geometries…Geometry Explosion! 20 th Century Real numbers integrated into the study of geometry Looking at Elements 5 Common notions – common to mathematics CN 1 Things which equal the same thing are also equal to each other. Transitivity of equality. CN 2 If equals be added to equals, then the wholes are equal. A = B and C = D, so A + C = B + D. Arithmetic. CN 3 If equals be subtracted from equals, the remainders are equal. CN 4 Things which coincide with one another are equal to one another. Rigid motions and congruence. CN 5 The whole is greater than the part. A = B + C, so A > B and A > C NOTE: positive numbers 1
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5 Postulates – for Geometry alone Postulate 1 It is possible to draw a straight line through any two points. Postulate 2 Any straight line can be extended. (no concept of infinite lines) Postulate 3 It is possible to construct a circle with any given center and radius. Postulate 4 All right angles are congruent. (congruent is a modern notion) Postulate 5 Two lines cut by a transversal must intersect on the side where the sum of the two interior angles is less than 90°. This is a statement about interesting lines! Known as the Parallel Postulate, though. Definitions - A point is that which has no part – existence but no mass. A line has breadthless length – one dimension only. A straight line is a line that lies evenly with the points themselves. Proposition 1 is proved from the 5 CN and 5 P page 5 Constructing an equilateral triangle from the 5 Postulates Finish the construction. Note the problem of the existence of the apex point. 2
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Proposition 3 is proved from the 5 CN, 5 P, Propositions 1 and 2 and so on, building. Propositions 1 – 3 construction proofs Straight edge (not a ruler!) Compass (not a protractor!) Proposition 4 The familiar Side-Angle-Side Axiom We know now that there are geometries for which this isn’t true so it’s gained “axiom strength”. Further the “method of superposition” only works with the right kind of functions and space. It isn’t universally true across all geometries. Elements is the first book to logically organize and develop information about a geometry. It was THE textbook until the 20 th century. 3
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Ch. 1 and 2 - Lectures 1 - 4 - Math 3379 Chapter 1...

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