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: geodesics converge positive curvature geodesics retain constant spacing zero curvature flat (Euclidean) geodesics diverge negative curvature