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August 15th, 2016, 08:59 AM   #1
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Fourth Postulate

Can Euclid's fourth postulate be proven? In definition I-10, he defines right angles as being equal to their supplement.

Suppose I start with a line and a ray which are perpendicular as in def. I-10, and elsewhere in space, I have another line and ray, also placed as in def I-10.

Assume that one of the angles in the first figure is less than (or greater than) the corresponding angle in the other figure. If I proceed (without "moving" figures around, the way Euclid sometimes did), would I eventually reach a contradiction, or could I actually create a new geometry?
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August 15th, 2016, 10:10 AM   #2
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Quote:
Originally Posted by bsaucer View Post
Can Euclid's fourth postulate be proven?
A postulate is a statement that is assumed to be true without a proof.


Source:
http://mathbitsnotebook.com/Geometry.../BTproofs.html
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August 15th, 2016, 11:38 AM   #3
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Quote:
Originally Posted by bsaucer View Post
Can Euclid's fourth postulate be proven? In definition I-10, he defines right angles as being equal to their supplement.

Suppose I start with a line and a ray which are perpendicular as in def. I-10, and elsewhere in space, I have another line and ray, also placed as in def I-10.

Assume that one of the angles in the first figure is less than (or greater than) the corresponding angle in the other figure. If I proceed (without "moving" figures around, the way Euclid sometimes did), would I eventually reach a contradiction, or could I actually create a new geometry?
I am doing this mostly from memory, but I think you are slightly mixing up two thoughts.

First. As you obviously know, it has long been recognized that Euclid used some unstated axioms. Hilbert for example proposed a complete set of axioms to redo Euclid according to modern standards of rigor.

https://en.wikipedia.org/wiki/Hilbert%27s_axioms

Second. As you also obviously know, a great deal of work was done on geometries that use axioms different from those explicitly stated or implicitly assumed by Euclid.

https://en.wikipedia.org/wiki/Foundations_of_geometry

https://en.wikipedia.org/wiki/Absolute_geometry

Based on a quick overview of these articles and what I remember from long ago, a geometry abandoning the fourth postulate has already been developed.

EDIT: In other words, the fourth postulate is NOT a theorem. Euclid was quite right to label it as an axiom or postulate.

Last edited by JeffM1; August 15th, 2016 at 11:41 AM.
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August 15th, 2016, 01:56 PM   #4
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This question it presumably motivated by the many attempts in history to prove the fifth posulate. As has been pointed out, a postulate is an axiom, but many mathematicians felt that the 5th potulate was relatively ugly and not suited to being an axiom. They thought that it ought to be possible to prove it using the other four. We now know that this is not possible because the 5th posulate is indeed an axiom.

To the best of my knowledge nobody has seriously questioned whether the 4th postulate is an axiom or not, although I would be surprised if nobody has proved that it is independent of the other four. That is, that it cannot be proved from them.
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August 15th, 2016, 05:50 PM   #5
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Originally Posted by v8archie View Post
This question it presumably motivated by the many attempts in history to prove the fifth posulate. As has been pointed out, a postulate is an axiom, but many mathematicians felt that the 5th potulate was relatively ugly and not suited to being an axiom. They thought that it ought to be possible to prove it using the other four. We now know that this is not possible because the 5th posulate is indeed an axiom.

To the best of my knowledge nobody has seriously questioned whether the 4th postulate is an axiom or not, although I would be surprised if nobody has proved that it is independent of the other four. That is, that it cannot be proved from them.
From wiki

Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms. (3: "To describe a circle with any centre and distance radius.", 4: "That all right angles are equal to one another." ) Ordered geometry is a common foundation of both absolute and affine geometry
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August 15th, 2016, 06:00 PM   #6
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Yes, the fact that you can create a different geometry by denying the 4th postulate would suggest that it is an axiom.
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August 17th, 2016, 06:48 AM   #7
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I'd like to know more about this geometry denying the fourth postulate.
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August 17th, 2016, 07:38 PM   #8
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The three articles that I cited from wikipedia have some at least initial references to books on various kinds of non-Euclidean geometries.
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August 18th, 2016, 09:55 AM   #9
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I know there are geometries where one or more of the axioms are omitted, such as affine geometry, or absolute geometry. But I'm looking for a geometry that negates the fourth postulate. That means there would at least two right angles (each congruent to their supplement) that are not congruent to each other (one measuring less than the other). An example might be that right angles in one plane are not congruent to right angles in another plane.
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August 18th, 2016, 10:27 AM   #10
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Negating the fourth postulate is logically equivalent to omitting it. The Geometry is no longer bound by that constraint.
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