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.
© Copyright 2012 Mike DeHaan, All rights Reserved. Written For: Decoded Science
I think we can and should remove the word “infinite” from explanations in plane Geometry.. For example, instead of saying “they will never meet even if extended infinitely far” we should say “they will never meet no matter how far they are extended”. The concept of infinitely is not needed for plane geometry and its definition (if it exists) is beyond the scope of plane geometry.
i think 5 postulate is not easy to understand
i think that greg is absoulutely correct , no doubt!
I don’t think what you’re saying is correct regarding Postulate 3. If Euclid allowed such an operation then he would not have to prove Proposition 2 in such a complicated manner. Prop 2 states “To place a straight line equal to a given straight line with one end at a given point.” If he could just measure the length of the line and draw a circle of that radius around the point then this would be trivial, but he doesn’t do it that way, he in fact invokes a quite complicated argument to prove this Proposition.
Therefore I think the right way to interpret Postulate 3 is that given any two points, one can draw a circle with center equal to one of the points and passing through the other point. Prior to proving a length can be duplicated, it is exactly this way that he draws circles – there are always two points.
yes, greg is right!!!!!