1.) A unique straight line can be drawn from any point to any point.
2.) A finite straight line can be produced continuously in a straight line.
3.) A circle may be described completely by centre and radius.
4.) All right angles are equal.
5.) If a transversal falls on two lines in such a way that the interior angles on one side of the transversal are less than two right angles, then the lines meet on the side on which the angles are less than two right angles.
A consistent logical system for which one of these postulates is modified in an essential way is a non - Euclidean geometry. Although there are different types of Non-Euclidean geometry which do not use all of the postulates or make alterations of one or more of the postulates of Euclidean geometry, hyperbolic and elliptic are usually most closely associated with the term Non-Euclidean Geometry. Many models of the Universe use these two geometries.
Hyperbolic geometry is based on changing Euclid's parallel (5 ^{th }) postulate, which may also be stated : one and only one parallel to a given line goes through a given point not on the line, while elliptic geometry results from a modification of postulate 2, which allows for lines of infinite length, which are denied in Elliptic geometry, where only finite lines are assumed. Elliptic Geometry includes the geometry of the surface of a sphere. Any line drawn on a sphere meets itself, so has finite length as shown below.
]]>For a triangle drawn on the surface of a sphere, the internal angles do not sum to 180, nor do the ordinary sin rule, cosine rule or Pythagoras theorem apply. Instead we use the half side formulae:
whereare the internal angles of the spherical triangle andare the lengths of the sides,being oppositeoppositeandoppositeand S is half the sum of the anglesand
]]>That between any two points a unique straight line can be drawn.
A line can be extended without limit on length.
A circle is completely described by it's centre and radius.
All right angles are equal.
The parallel postulate: "That, if a straight line falling on two straight lines make 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." This is shown below.
Other geometries than Euclidean geometry is possible, in which the postulates are abandoned or modified. These are the 'non – Euclidean geometries.'
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