528 CHAPTER 8 Geometry the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.” Euclid’s fifth postulate may be better understood by observing Fig. 8.98. The sum of angles A and B is less than the sum of two right angles (180°). Therefore, the two lines will meet if extended. In 1795, John Playfair (1748–1819), a Scottish physicist and mathematician, gave a logically equivalent interpretation of Euclid’s fifth postulate. This version is often referred to as Playfair’s postulate or the Euclidean parallel postulate. The Euclidean Parallel Postulate Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line (see Fig. 8.99). The Euclidean parallel postulate may be better understood by looking at Fig. 8.99. MATHEMATICS TODAY Black Holes On April 10, 2019, scientists revealed the first image of a black hole, shown above. The image shows a fiery ring of gravitytwisted light swirling around the edge of the black hole. It is around the edge of the black hole that the massive force of gravity causes light to bend and not travel in a straight line. The black hole is at the center of the Messier 87 galaxy and is 53 million light-years from Earth. Super massive black holes are at the center of most galaxies, including our own. They are so dense that nothing, not even light, can escape their gravitational pull. Albert Einstein predicted black holes and even predicted their shape (see Did You Know ? on page 530). This image helps confirm Einstein’s general theory of relativity. This black hole is about 6 billion times the mass of our sun. Data for this image were gathered using eight radio-telescopes around the world including the Maxwell telescope in Hawaii. Using these data to complete the image was an enormous task involving a team of more than 200 international scientists, supercomputers, and hundreds of terabytes of data. Why This Is Important Black holes provide just one example of how mathematics and specifically geometry can be used to describe our universe. We may not always see the immediate applications of many of the topics we study. However, studying mathematics may help us better understand the physical world and the universe we live in. The Fifth Axiom of Geometry Euclidean Elliptical Hyperbolic Given a line and a point not on the line, one and only one line can be drawn parallel to the given line through the given point. Given a line and a point not on the line, no line can be drawn through the given point parallel to the given line. Given a line and a point not on the line, two or more lines can be drawn through the given point parallel to the given line. A B Figure 8.98 Given line Line parallel to the given line through the given point Given point Figure 8.99 Many mathematicians after Euclid believed that this postulate was not as self-evident as the other nine postulates given by Euclid. Others believed that this postulate could be proved from the other nine postulates and therefore was not needed at all. Over the years, many mathematicians worked on the frustrating activity of trying to prove the parallel postulate used in Euclidean geometry. Eventually, Carl Friedrich Gauss (see Profile in Mathematics page 276) and Bernhard Riemann (1826–1866), and other mathematicians, realized that if the postulate is not applied, then two different types of geometry, collectively referred to as non-Euclidean geometry , can be developed. The two basic types of non-Euclidean geometries are elliptical geometry and hyperbolic geometry . Although a formal discussion of non-Euclidean geometry is beyond the scope of this book, we will provide some of the fundamental concepts. The major differences among the three geometries lie in the fifth axiom, which we summarize here. Bennymarty/123RF
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