ssrs barcode font The Postulates of Euclidean Geometry in Objective-C

Create QR in Objective-C The Postulates of Euclidean Geometry

19.2 The Postulates of Euclidean Geometry
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Euclid s Elements began with only five postulates. We shall go through them with an eye for alternatives.
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POSTULATE 1:
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One and only one straight line can be drawn through any two points. ( 2, Postulate 11)
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We further postulate that there exist at least two points, A and B, so there exists a straight line.
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POSTULATE 2:
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A line segment can be extended in either direction indefinitely. ( 1, description of a line)
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Traditionally, we suppose that the line through A and B looks like Fig. 19-1(a), but our postulates do not rule out the possibility of Fig. 19-1(b). In Fig. 19-1(b), the line reaches no end in either direction, and thus in some sense can be extended indefinitely. Remember that the original meaning of geometry comes from earth and measure. The straightest line that could be drawn on the earth would wrap around to form a great circle, as discussed in 17.
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Fig. 19-1
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CHAPTER 19 Non-Euclidean Geometry
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A circle can be drawn with any center and radius. ( 2, Postulate 14)
POSTULATE 3:
When we believe this postulate, we believe that we have a plane with no end in any direction on which we could find circles with ever-larger radii. It could be that this leads to circles like ripples on the surface of a still pond, as shown in Fig. 19-2(a). However, if our plane were more like the surface of the Earth, our circles might look more like the circles of latitude, shown in Fig. 19-2(b).
Fig. 19-2
POSTULATE 4:
All right angles have the same measure.
This postulate enables us to measure angles in degrees. A plane would have to be somehow lumpy or uneven if some right angles could be bigger than others.
POSTULATE 5:
Through a given point not on a given line, one and only one line can be drawn parallel to a given line. ( 4, Parallel-Line Postulate)
This postulate is the one which establishes that our plane cannot look like a giant sphere.4 a flat plane, On 4 4 given point P not on line AB, only a point C which makes /APC > /PAB will make PC i AB , as shown in Fig. 19-3(a). On a giant sphere, straight lines are great circles which divide the sphere into two equal-sized pieces. Any two such lines will always meet at two spots at opposite points of the sphere (called antipodal points), such as X and Y, as shown in Fig. 19-3(b). Because any two straight lines meet, it is impossible for there to be parallel lines on a sphere.
Fig. 19-3
19.3 The Fifth Postulate Problem
For about 2000 years, certain mathematicians tried to use the first four postulates to prove the fifth. This challenge was called the fifth postulate problem. The first four seem to come straight from the basic tools of geometry: the straight edge (connecting points and extending straight lines), the compass (drawing circles),
CHAPTER 19 Non-Euclidean Geometry
and the square (measuring right angles). The fifth, it seemed, was something that ought to be provable. However, no one was able to succeed in proving the fifth postulate without introducing new, equivalent postulates that required belief without proof. (Actually, the fifth postulate stated previously is a simpler alternative to the one Euclid actually used.) If a postulate about parallel lines must be accepted without proof, why must it be the traditional fifth postulate of euclidean geometry
19.4 Different Geometries
In the 19th century, mathematicians finally established that there are actually three basic types of planar geometry, not one. The traditional one is called euclidean geometry, which is based on the fifth postulate (or one of the many equivalent postulates). However, we could instead believe one of the two alternate fifth postulates:
POSTULATE 5a: POSTULATE 5b:
There are no parallel lines. Through a given point not on a given line, many different lines can be drawn parallel to a given line.
If we believe Postulate 5a (along with Postulates 1 through 4), then we will end up with what is called elliptic geometry. This is the sort of geometry where a plane is actually shaped like the surface of a sphere and lines are great circles. If we believe Postulate 5b, we will end up with an even stranger geometry called hyperbolic geometry.
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