Postulates and Proofs

5.1 Euclid's Postulates
In addition to the great practical value of Euclidean Geometry, the ancient Greeks also found great esthetic value in the study of geometry. Much as children assemble a few kinds blocks into many varied towers, mathematicians assemble a few definitions and assumptions into many varied theorems. The blocks are assembled with Hands, the postulates are assembled with Reason.

All of Euclidean Geometry (the thousands of theorems) were all put together with just 5 different kinds of blocks. These are called "Euclid's five postulates":

P-1 Every two points lie on exactly one line.
P-2 Any line segment with given endpoints may be continued in either direction.
P-3 It is possible to construct a circle with any point as its center and with a radius of any length. (This implies that there is neither an upper nor lower limit to distance. In-other-words, any distance, no mater how large can always be increased, and any distance, no mater how small can always be divided.)
P-4 If two lines cross such that a pair of adjacent angles are congruent, then each of these angles are also congruent to any other angle formed in the same way.
P-5 (Parallel Postulate): Given a line L and a point not on L, there is one and only one line L1 which contains the point, is in the same plane as L, and is parallel to L.

Euclid's goal was for these postulates to be (1) few in number, and (2) so obviously true that they could not possible be argued with. For over 2000 years many mathamtitions believed that the fifth postulate (the Parallel Postulate) was not needed. They believed that it could instead be proved as a theorem of the first four postulates. There were numerous attempts to do so. It was not until the nineteenth century that Lobachevski (1793-1856), Bolyai (1777-1855) and Gauss (1802-1860) finally put an end to this impossible search. Lobachevski developed theorems using Euclid's first four postulates and the negation of the Parallel Postulate. His expectation was to eventually develop two theorems which contradicted each other. This would prove that negating the Parallel Postulate is inconsistent with the first four postulates - thereby proving the Parallel Postulate (and making it the Parallel Theorem). To his surprise, however, he never obtained a contradiction. Instead, he developed a complete and consistent geometry - the first non-Euclidean Geometry. This proved that the Parallel Postulate could not be derived from the other four. This was of great mathematical and philosophical interest. From the time of the Greeks, it was believed that geometric theorems were such pure and perfect Truth that they did not need to be scrutinized by observations of the real world. Now, those statements are only true is some geometries. The only reason to prefer one geometry over another is by comparison to the real world. In spite of this philosophical and esthetic interest, it was believed that non-Euclidean geometry was of no practical value. Then, in the early 1900's, Einstein (1878-1955) developed The General Theory of Relativity which made extensive use of a particular non-Euclidean Geometry called Hyperbolic Geometry. This abstract, mathematical philosophical was now in the realm of Science.

Hyperbolic Geometry is a Non-Euclidean Geometry based on the first four of Euclid's postulates together with the following variation of the Euclidean Parallel Postulate.

5.2 Hyperbolic Parallel Postulate:
Given a line L and a point not on L, there are at least two lines L1 and L2 which contain the point, are in the same plane as L and are parallel to L.

5.3 SAS Postulate:
The SAS Postulate is a sixth postulate that is required in both Euclidean and Hyperbolic Geometry.

The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and an included angle of a second triangle, then the two triangles are congruent.

In the Elements, Euclid presents what he believes to be a proof for SAS:[Health-56]

Given:
DABC and DDEF, with sides AB @ DE, side AC @ DF, and Ð A @ Ð D.

Proof:

The flaw in the above argument is that it depends on the undefined term "move". Let "move" (in both Euclidean and Hyperbolic Geometry) be defined as a function that maps a set of points, P1, P2, P3, ... to P'1, P'2, P'3..., in such a way that for any two points Pn and Pm of the original set, the distance from Pn to Pm equals the distance from P'n to P'm. Then, for SAS to hold, it must be that for any two lines L and L', it is always possible to "move" line L so that it coincides with L'. This condition is commonly called the SAS postulate, and it is the sixth postulate in both Euclidean and Hyperbolic Geometry. [Moise-74]

5.4 Hyperbolic Geometry Proofs:
In addition to these postulates, both Euclidean and Hyperbolic geometry require a number of common notions of reason such as "Things which are equal to the same thing are also equal to one another" and "of any three points on a line, exactly one is between the other two." All of these common notions are exactly the same for both geometries. In fact, the only difference between the complete axiomatic formation of Euclidean Geometry and of Hyperbolic Geometry is the Parallel Postulate.

This is a powerful statement. It means that any proof in Euclidean Geometry which does not use the Parallel Postulate is also a proof in Hyperbolic Geometry!

Likewise, it means that Euclidean Geometry theorems that require the Parallel Postulate will be false in Hyperbolic. A striking example of this is the Euclidean Geometry theorem that the sum of the angles of a triangle will always total 180º. Figure 5.4a may help you recall the proof of this theorem - and see why it is false in Hyperbolic Geometry.

Figure 5.4a: Proof for mÐA + mÐ B + mÐ C = 180º

In Euclidean Geometry, for any DABC, there exists a unique parallel to BC that passes through point A. Then, Ð NAB @ Ð ABC and Ð MAC @ Ð ACB since they are opposite interior angles of a pair of parallel lines cut by a transversal. In Hyperbolic Geometry, however, there are an infinite number of lines that are parallel to BC and pass through point A, yet there does not exist any line such that both:
Ð NAB @Ð ABC and Ð MAC @ Ð ACB.

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