Euclid's Parallel PostulateDate: 14 Feb 1995 11:36:20 -0500 From: Cerreta, Pat Subject: parallel postulate I am studying Euclid's Parallel Postulate. Why did mathematicians disagree with him? What other geometries resulted from this disagreement? What postulate replaced the Parallel Postulate? Thank You, Byram Hills Freshman Geometry Class Armonk,New York Date: 14 Feb 1995 14:59:02 -0500 From: Dr. Ethan Subject: Re: parallel postulate Howdy folks, Well, to fully explain you need to make sure you know the difference between a theorem and a postulate. A postulate is a statement about a geometrical world that is accepted without justification and makes the frame work for the geometrical world. A theorem is a conclusion that follows logically from the postulates using rational argument and does require justification. It is a result of the framework of the geometrical world. So now to Euclid's fourth and most troublesome postulate: For every line, l, and point, P, which is not on that line, there exists a unique line, m, through P that is parallel to l. (Remember parallel means that they do not intersect. It does not necessarily mean that they are always the same distance apart.) This is a postulate, so when you say that you disagree with it, that doesn't mean it is wrong it just means that you don't want to assume that it is true in your geometrical world. The disagreement came when mathematicians tried to figure out whether or not the parallel postulate had to be taken as a postulate, or whether it followed from the other four. This was hotly debated but eventually it was determined that no, it did not follow from the other three. It had to be assumed in order to get Euclidean geometry (Which for a long time was the only geometry that anybody thought mattered). From our perspective, the most important of these attempts was that made by Girolamo Saccheri. He hoped to assume that the postulate wasn't true, and show that that led to something that made no sense; thereby proving the postulate true. He was unable to find a contradiction but did unknowingly discover the first non-Euclidean geometry (i.e. one in which the fifth postulate doesn't hold) Well I guess I'm going to skip a lot of history here and skip to the results. It was eventually decided that the fifth postulate was indeed a postulate so we didn't have to assume it. The two assumptions that have been used instead are these: 1. Given a line, l, and a point, P, not on that line there are infinitely many lines passing through P that are parallel to l. (This produces hyperbolic geometry) 2. Given a line, l, and a point not on that line, P, there are no lines passing through P parallel to l. (This produces projective geometry) If this is unclear or you would like more information, please write back and I will tell you more. Ethan Doctor On Call Date: 14 Feb 1995 21:53:39 -0500 From: Dr. Ken Subject: Re: parallel postulate Hello there! I just thought I'd add to the wonderful answer of my colleague and buddy Ethan. The two kinds of Non-Euclidean geometry are what we get by assuming the _negation_ of the parallel postulate. Since the parallel postulate says, "for any line, l, and any point, p, not on that line, there is exactly one line through p which is parallel to l," the negation of it is, "there exists a line, l, and a point, p, not on that line such that there is either (a) no line through p which is parallel to l, or (b) more than one line through p which is parallel to l." It turns out that if there is one place in our geometry where we're in case (a), we can prove that we're in case (a) _everywhere_, i.e. that there are no parallel lines whatsoever in our whole system of geometry. We call this geometry (as Ethan said) projective geometry or spherical geometry. The definition is that "there exists a line, l, and a point, p, not on that line such that there is no line through p which is parallel to l," but we can prove that "given any line, l, and any point, p, there is no line through p which is parallel to l." Also, if there's any place where we're in case (b), then we're in case (b) everywhere. This is known as Hyperbolic geometry. The definition is "there exists a line, l, and a point, p, not on that line such that there is more than one line through p parallel to l," but we can actually prove that the following holds: "given any line, l, and any point, p, not on that line, there are infinitely many lines through p which are parallel to l." For what that's worth. -Ken "Dr." Math |
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