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PERSPECTIVE ON HISTORY Sketch of Euclid Names often associated with the early development of Greek mathematics, beginning in approximately 600 B . C ., include Thales, Pythagoras, Archimedes, Appolonius, Diophantus, Eratosthenes, and Heron. However, the name most often associated with traditional geometry is that of Euclid, who lived around 300 B . C . Euclid, himself a Greek, was asked to head the mathematics department at the University of Alexandria in Egypt, which was the center of Greek learning. It is believed that Euclid told Ptolemy the local ruler that “There is no royal road to geometry,” in response to Ptolemy’s request for a quick and easy knowledge of the subject. Euclid’s best-known work is the Elements, a systematic treatment of geometry with some algebra and number theory. That work, which consists of 13 volumes, has dominated the study of geometry for more than 2000 years. Most secondary-level geometry courses, even today, are based on Euclid’s Elements and in particular on these volumes: Book I: Triangles and congruence, parallels, quadri- laterals, the Pythagorean theorem, and area relationships Book III: Circles, chords, secants, tangents, and angle measurement Book IV: Constructions and regular polygons Book VI: Similar triangles, proportions, and the Angle Bisector theorem Book XI: Lines and planes in space, and parallelepipeds One of Euclid’s theorems was a forerunner of the theorem of trigonometry known as the Law of Cosines. Although it is difficult to understand now, it will make sense to you later. As stated by Euclid, “In an obtuse- angled triangle, the square of the side opposite the obtuse angle equals the sum of the squares of the other two sides and the product of one side and the projection of the other upon it.” While it is believed that Euclid was a great teacher, he is also recognized as a great mathematician and as the first author of an elaborate textbook. In Chapter 2 of this textbook, Euclid’s Parallel Postulate has been central to our study of plane geometry. PERSPECTIVE ON APPLICATION Non-Euclidean Geometries The geometry we present in this book is often described as Euclidean geometry. A non-Euclidean geometry is a geometry characterized by the existence of at least one contradiction of a Euclidean geometry postulate. To appreciate this subject, you need to realize the importance of the word plane in the Parallel Postulate. Thus, the Parallel Postulate is now restated. The Parallel Postulate characterizes a course in plane geometry; it corresponds to the theory that “the earth is flat.” On a small scale most applications aren’t global, the theory works well and serves the needs of carpenters, designers, and most engineers. To begin the move to a different geometry, consider the surface of a sphere like the earth. See Figure 2.53. By definition, a sphere is the set of all points in space that are at a fixed distance from a given point. If a line segment on the surface of the sphere is extended to form a line, it becomes a great circle like the equator of the earth. Each line in this geometry, known as spherical geometry, is the intersection of a plane containing the center of the sphere with the sphere. m a and m are lines in spherical geometry Figure 2.53 b These circles are not lines in spherical geometry PARALLEL POSTULATE In a plane, through a point not on a line, exactly one line is parallel to the given line. Spherical geometry or elliptic geometry is actually a model of Riemannian geometry, named in honor of Georg F. B. Riemann 1826–1866, the German mathematician responsible for the next postulate. The Reimannian Postulate is not numbered in this book, because it does not characterize Euclidean geometry. To understand the Reimannian Postulate, consider a sphere Figure 2.54 containing line 艎 and point P not on 艎. Any line drawn through point P must intersect 艎 in two points. To see this develop, follow the frames in Figure 2.55, which depict an attempt to draw a line parallel to 艎 through point P. the surface for study, mathematicians use a saddle-like surface known as a hyperbolic paraboloid. See Figure 2.56. A line 艎 is the intersection of a plane with this surface. Clearly, more than one plane can intersect this surface to form a line containing P that does not intersect 艎. In fact, an infinite number of planes intersect the surface in an infinite number of lines parallel to 艎 and containing P. Table 2.5 compares the three types of geometry. a P Figure 2.54 b P Consider the natural extension to Riemannian geometry of the claim that the shortest distance between two points is a straight line. For the sake of efficiency and common sense, a person traveling from New York City to London will follow the path of a line as it is known in spherical geometry. As you might guess, this concept is used to chart international flights between cities. In Euclidean geometry, the claim suggests that a person tunnel under the earth’s surface from one city to the other. A second type of non-Euclidean geometry is attributed to the works of a German, Karl F. Gauss 1777–1855, a Russian, Nikolai Lobachevski 1793–1856, and a Hungarian, Johann Bolyai 1802–1862. The postulate for this system of non-Euclidean geometry is as follows: This form of non-Euclidean geometry is termed hyperbolic geometry. Rather than using the plane or sphere as P a Small part of surface of the sphere Figure 2.55 b Line through P “parallel” to on larger part of surface P c Line through P shown to intersect on larger portion of surface P P d All of line and the line through P shown on entire sphere P Figure 2.56 RIEMANNIAN POSTULATE Through a point not on a line, there are no lines parallel to the given line. LOBACHEVSKIAN POSTULATE Through a point not on line, there are infinitely many lines parallel to the given line.