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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.

5^ONE. 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:

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

5^NONE. 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 5^NONE 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 5ONE	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.

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

Copyright John D. Norton. December 28, 2006. February 16, 2022.