Introduction to Euclids Geometry - ESSENTIAL POINTS

  • Euclid defined a point, a line, a straight line, a surface, edges of a surface and a plane
  • These terms are now taken as undefined because the definitions are not accepted by mathematicians.
  • Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.
  • Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.

Some of Euclid’s axioms were:

  • 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 things are equal to one another.
  • Things which are halves of the same things are equal to one another.

Euclid’s postulates were:

  • 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 drawn with any centre and any radius.
  • Postulate 4 : All right angles are equal to one another.
  • Postulate 5 : If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Present Day Geometry:

  • A mathematical system consists of axioms, definitions and undefined terms.
  • Point, line and plane are taken as undefined terms.
  • A system of axioms is said to be consistent if there are no contradictions in the axioms and theorems that can be derived from them.
  • Given two distinct points, there is a unique line passing through them.
  • Two distinct lines can not have more than one point in common.
  • Playfair’s Axiom (An equivalent version of Euclid’s fifth postulate).

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