HPS 0410  Einstein for Everyone 
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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 5^{NONE} 
Euclidean Geometry Flat Euclid's Postulate 5 
Hyperbolic Geometry Negative Curvature Postulate 5^{MORE} 

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.