A Brief Guide to the Euclidean Postulates Found in Euclid’s Elements

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Tall Triangles and Parallel Lines image by Mike DeHaan

These tall triangles cross parallel lines: Image by Mike DeHaan

Now, look at the diagram with four triangles.

The sum of the the interior angles of any triangle is 180 degrees.

As the sum of the angles at the base increases, the angle at the apex opposite them must decrease.

Once the sum of the green angles is 180 degrees, “zero” remains for the apex’s angle.

One might picture that top point moving farther and farther “up,” finally disappearing when the distance is infinite.

That’s not an accurate mathematical statement, but it is a way to imagine how the angles at the base relate to the lines becoming parallel.

It is possible to violate the parallel postulate, but only for geometry other than a flat “plane” surface.

For example, use the earth’s equator as the base line. Each line of longitude meets the equator at a 90 degree angle.

Therefore, the sum of the interior angles between two lines of longitude is 180 degrees. Thus, longitude lines are parallel.

However, all longitude lines meet at the North Pole, and also at the South Pole. Therefore, Euclidean geometry’s parallel postulate does not apply to geometry on a sphere.

Euclid’s Elements

Along with many definitions, Euclid’s axioms and postulates allowed him to demonstrate hundreds of propositions in his Elements, as he laid the foundations of geometry.

References:

Weisstein, E. W. Euclid’s Postulates. (2012). MathWorld-A Wolfram Web Resource. Accessed July 4, 2012.

Douglass, C. Euclid. (2007). Math Open Reference. Accessed July 4, 2012.

Fitzpatrick, R. Euclid’s Elements of Geometry. (1885). University of Texas. Accessed July 4, 2012.

Swartz, N. Axioms and Postulates of Euclid. Simon Fraser University. Accessed July4, 2012.

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