HPS 0410 Einstein for Everyone

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Euclid's Postulates and Some Non-Euclidean Alternatives

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

The five postulates on which Euclid based his geometry are:

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any center and distance.

4. That all right angles are equal to one another.

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.


Playfair's postulate, equivalent to Euclid's fifth, was:

5ONE. Through any given point can be drawn exactly one straightline parallel to a given line.


In trying to demonstrate that the fifth postulate had to hold, geometers considered the other possible postulates that might replace 5'. The two alternatives as given by Playfair are:

5MORE. Through any given point MORE than one straight line can be drawn parallel to a given line.

5NONE. Through any given point NO straight lines can be drawn parallel to a given line.


Once you see that this is the geometry of great circles on spheres, you also see that postulate 5NONE cannot live happily with the first four postulates after all. They need some minor adjustment:

1'. Two distinct points determine at least one straight line.

2'. A straight line is boundless (i.e. has no end).


Each of the three alternative forms of the fifth postulate are associated with a distinct geometry:

Spherical Geometry
Positive curvature
Postulate 5NONE
Euclidean Geometry
Flat
Euclid's Postulate 5
Hyperbolic Geometry
Negative Curvature
Postulate 5MORE
Straight lines Finite length; connect
back onto themselves
Infinite length Infinite length
Sum of angles of a triangle More than 2 right angles 2 right angles Less than 2 right angles
Circumference of a circle Less than 2π times radius 2π times radius More than 2π times radius
Area of a circle Less than π(radius)2 π(radius)2 More than π(radius)2
Surface area of a sphere Less than 4π(radius)2 4π(radius)2 More than 4π(radius)2
Volume of a sphere Less than 4π/3(radius)3 4π/3(radius)3 More than 4π/3(radius)3

In very small regions of space, the three geometries are indistinguishable. For small triangles, the sum of the angles is very close to 2 right angles in both spherical and hyperbolic geometries.


For convenience of reference, here is the summary of geodesic deviation, developed in the chapter "Spaces of Variable Curvature"

The effect of geodesic deviations enables us to determine the curvature of space by experiments done locally within the space and without need to think about a higher dimensioned space into which our space may (or may not) curve.

Positive geodesics converge positive curvature
Flat geodesics retain
constant spacing
zero curvature
flat (Euclidean)
Negative geodesics diverge negative curvature

Copyright John D. Norton. December 28, 2006.