MODULE 6
                                   Non-Euclidean Geometry
                              Revolutions are not made; they come. – Wendell Phillips
                             I have created a new universe out of nothing. – J´onos Bolyai
                   Recall our definition of parallel lines: two lines ℓ and m are parallel if they do not intersect,
                   that is, no point P lies on both ℓ and m.
                      Threedistinct branchesofgeometryarisefromthewayweanswerthefollowingquestion:
                         Given a line ℓ and a point P not on ℓ, how many lines through P are
                         parallel to ℓ?
                   There are three possible answers:
                      Choice 1. There is one and only one parallel through P. This statement is the parallel
                   postulate (Theorem ??) and is easily shown to be equivalent to Euclid’s 5th Axiom. The
                   resulting geometry is the standard Euclidean geometry, studied by school children and
                   mathematicians for the past two thousand years, and the main focus of this text.
                      Choice 2. No parallel lines exist. This branch leads to spherical geometry. Our model
                   of spherical geometry will be the surface of the earth, discussed in the next two sections.
                      Choice 3. There is more than one parallel through P. This branch leads to hyperbolic
                   geometry. Our model of hyperbolic geometry was first proposed by the mathematician
                   Poincar´e and was later made famous by the artist Maurice Escher.
                      For thousands of years, mathematicians tried to prove the parallel postulate from the
                   other axioms of Euclid. Their minds were already made up that the only possible kind of
                   geometry is the Euclidean variety—the intellectual equivalent of believing that the earth is
                   flat. In truth, the two types of non-Euclidean geometries, spherical and hyperbolic, are just
                   as consistent as their Euclidean counterpart. The theorems in these branches look strange
                                               147
                        148                       6. NON-EUCLIDEAN GEOMETRY
                        to us because we are so used to the theorems of the geometry we were taught since grade
                        school. When you’re traveling in a foreign country, their currency never looks as real as your
                        own.
                                         1. Planet Earth and the Longitude Problem
                            We start our study of spherical geometry by looking at the surface of the earth and
                        considering the practical problem of navigation, especially from the point of view of a
                        seafaring nation like Great Britain in the 1700’s. In the next section we examine the basic
                        geometric concepts: distance, lines, segments, rays, angles, triangles, angle sum, rectangles,
                        similarity, circles, and area in this new and strange setting of spherical geometry. The last
                        section of this module takes a close look at the equally strange hyperbolic world.
                            We consider the earth to be a sphere. It’s actually slightly flatter at the North and
                        South Poles. The equatorial radius of the earth is 6378.137 km or 3963.19 miles, while the
                        polar radius is 6357 km or 3950 miles. Many books use 4000 miles as a rough approximation
                        for the radius of the earth.
                            Lines on a sphere are great circles, that is, circles whose center is the center of the
                        sphere. Longitudes, extending from the North to the South Pole, are great circles. East–
                        west latitudes are not—except for the equator. Great circles can “tilt” at any angle. The
                        great circle connecting Los Angeles and Chicago, for example, is certainly not a longitude.
                            Nowweseewhytherearenoparallel lines on a sphere. Suppose one line is the equator.
                        Then no matter what great circle you use for the second line, it must cross the equator at
                        two points. No great circle can be drawn entirely in the northern or southern hemisphere.
                        Connection to Navigation
                            Aship’s position at sea is specified by giving its longitude and latitude. Latitude tells
                        how far north (or south) you are and longitude gives your east–west position. Latitude can
                        be determined by the length of day or by the height above the horizon of the sun in daytime
                        or known guide stars at night. Unfortunately, there is no such natural way to determine
                        longitude!
                        The Longitude Prize of 1714
                            In an attempt to spur scientists, artisans, and great thinkers to find a solution to the
                        longitude problem, the Parliament of England established the Longitude Prize of 1714.
                                                 1. PLANET EARTH AND THE LONGITUDE PROBLEM                         149
                             Parliament promised a prize of 20,000 pounds—an enormous sum for the time—to anyone
                             who could accurately determine longitude at sea. They required a test voyage to the West
                             Indies—a 40 day sailing, and the solution needed to be accurate to within 1 a degree.
                                                                                                      2
                             Using time to determine longitude.
                                 Dividing the 360 degrees of the earth’s rotation into 24 time zones means that the earth
                             rotates 15 degrees every hour. Thus, the change in longitude of a seagoing vessel is directly
                             related to the difference of the time at its home port and the local time at its current location
                             at sea. To convert this time difference into geographical significance, use the formula:
                                                    change in longitude = time difference×15◦.
                             If the ship is traveling along a fixed latitude, then the distance traveled is
                                           distance = time difference ×circumference of latitude circle.
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                             Hands-on Problem. You are on the equator (r = 3963 miles).
                                 (a) How many miles comprise 90◦ of longitude?
                                 (b) How many miles comprise a single time zone?
                                 (c) How many miles have you traveled and how many degrees of longitude are you from
                             the longitude of your home port if the times are
                                            home time local time    change in longitude  miles traveled
                                             8:00 am     9:00 am
                                             2:45 am     4:30 am
                                             5:40 am     1:20 pm
                                             9:30 pm     1:55 am
                                 From this discussion we see that to determine longitude, all you need to do is compare
                             the local time at your current location with the time at your home port. Computing local
                             time was well known when the Longitude Prize was offered. Local noon occurs when the sun
                             is directly overhead and could be ascertained by the navigational tools of 1700. Computing
                             port time was the problem!
                        150                      6. NON-EUCLIDEAN GEOMETRY
                            You’re probably wondering, “Why couldn’t the ship’s navigator just take a clock on
                        board at the start of the voyage, set to the port time?” In 1700 most accurate clocks were
                        pendulumclocks. Thoughdependableonland, theyrequiredfavorableweatheratseadueto
                        the swaying of ship with the waves—especially in storms. Early clocks using a spiral balance
                        spring were not sufficiently accurate, for many reasons. Spring powered mechanisms varied
                        according to temperature and humidity. Metal parts expanded with heat or contracted with
                        cold while the viscosity of the lubrication changed with temperature. In 1700 the best spring
                        mechanized time pieces lost up to a minute a day.
                        Hands-on ProblemJust how accurate did a timepiece need to be? Suppose a clock loses
                        3 seconds per day. On a 40 day vogage along the equator, how much error in miles does
                        this amount to?
                        The Astronomy Solution.
                            The list of leading astronomers and scientists who contributed to the longitude problem
                        reads like a Who’s Who in Science of the era. Galileo advocated using the eclipses of the
                        moons of Jupiter, which occur thousands of times in a year. This idea requires accurate
                        calculations in order to predict when the moons will disappear and reappear relative to a
                        fixed viewing point on earth. He developed a special headgear with a telescope attached to
                        oneeyepieceandanemptyeyeholeintheother,tolocatethesteadylightfromJupiter. Later
                        the astronomer Ole Roemer discovered that the eclipses of the moons of Jupiter occurred
                        ahead of schedule when Jupiter was closest to the earth and behind schedule when it was
                        farthest away. This discrepancy was resolved when it was discovered that it takes light a
                        different time to to travel from Jupiter to Earth, depending on the position of the planets
                        in their respective orbits. After a while astronomers abandoned the use of the moons of
                        Jupiter as a navigation tool and began to concentrate on our own moon. Sir Issac Newton
                        solved problems about the moon’s orbit with his theory of gravitation, paving the way for
                        Nevil Maskelyne, 4th Astronomer Royal of England, who advanced the lunar method for
                        navigation and his own extensive almanac of the exact times of the rise and fall of the moon.
                        Maskelyne was the nemesis of the hero of our story, the self taught genius destined to settle
                        the Longitude problem: John Harrison.
                        John Harrison and his chronometers.
                            When he was twenty years old, John Harrison built a wooden pendulum clock, using
                        woodworking techniques he learned from his father. Nine years later he built a tower clock
                        in Brocklesbe Park which has been running continuously for 280 years, except for brief