Introduction to Euclid's Geometry

• History & Background

  • Origin of the word 'Geometry' (Geo = Earth, metrein = to measure)
  • Thales: First known mathematician, gave the first logical proof
  • Euclid: Father of Geometry, compiled the treatise 'Elements' (divided into 13 books)

• Basic Terms (Undefined & Defined)

  • Point: A dot with no breadth, length, or thickness
  • Line: Breadthless length
  • Plane / Surface: Has only length and breadth
  • Line Segment: Part of a line with two end points
  • Ray: Part of a line with one end point

• Euclid's Definitions (Book 1 of Elements)

  • Total of 23 definitions given by Euclid
  • Focus on fundamental spatial objects (point, line, surface)

• Euclid's Axioms (Common Notions)

  • Applicable to all branches of mathematics
  • Things which are equal to the same thing are equal to one another
  • If equals are added to equals, the wholes are equal
  • If equals are subtracted from equals, the remainders are equal
  • Things which coincide with one another are equal to one another
  • The whole is greater than the part
  • Things which are double of the same thing are equal to one another
  • Things which are halves of the same thing are equal to one another

• Euclid's Five Postulates

  • Postulate 1: A straight line may be drawn from any one point to any other point
  • Postulate 2: A terminated line can be produced indefinitely
  • Postulate 3: A circle can be described with any center and any radius
  • Postulate 4: All right angles are equal to one another
  • Postulate 5 (Parallel Postulate): 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

• Equivalent Versions of Euclid's 5th Postulate

  • Playfair’s Axiom: Through a given point not on a line, only one parallel line can be drawn to the given line
  • Infinite number of lines can pass through a single point, but only one is parallel

• Consistency of Systems

  • A geometry system must be consistent (free from contradiction)
  • Independence of postulates (especially the 5th postulate leading to Non-Euclidean geometries)