316 316                                                                                                                                                                                                                                                                                            MATHEMATICS MATHEMATICS 
                                                                                316 316                                                                                                                                                                                                                                                                                                                                                                             MAGAZINE MAGAZINE 
                                                                                                                                                                                                                                                                                                                                                                                   MATHEMATICS MATHEMATICS MAGAZINE MAGAZINE 
                                                                                                                               are are 
                                                                               the the ideas ideas                                               to to be be found found scattered scattered                                                                                                          the the first first 25 25                                                               of of                                                                           [4], [4], while while 
                                                                                                                                                                    47 47 of of do do                                                                       through through                                                                                   pages pages                                   Klingenberg Klingenberg                                                              - - 
                                                                                                                              6 6 on on                                                                           Carmo Carmo                                             states states a a                                                                                      of of our our 
                                                                               Exercise Exercise                                                       p. p.                                                                                            [2] [2]                                                   special special case case                                                                        formula formula for for L2 L2                                                              L1. L1. 
                                                                                                                 excellent excellent  sources sources are are available available                                                                                                                                 for for                                                                                                          in in 
                                                                               Many Many                                                                                                                                                                                          [5, [5,  6] 6]                                      anyone anyone interested interested                                                                         delving delving  further further 
                                                                                into into differential differential 
                                                                                                                                                                 geometry. geometry. 
                                                                                REFERENCES REFERENCES 
                                                                                1. 1.                           A. A.                                                                                                             coordinates coordinates and and 
                                                                                           John John                                                   Plane Plane                                                                                                                                                                                                      this this MAGAZINE MAGAZINE 64 64                                                                          75-91. 75-91. 
                                                                                                                           Baker, Baker,                                      curves, curves,                                                                                                                                     numbers, numbers,                                                                                                  (1991), (1991), 
                                                                                                                                                                                                            polar polar                                                                          winding winding 
                                                                                2. 2.  Manfredo Manfredo P. P. do do                                                                                                                                                                                                    and and 
                                                                                                                                                                                         Differential Differential                                                                         Curves Curves 
                                                                                                                                                          Carmo, Carmo,                                                                 Geometry Geometry of of                                                                           Surfaces, Surfaces, Prentice-Hall, Prentice-Hall,                                                                                           Cliffs, Cliffs, NJ, NJ, 
                                                                                            1976. 1976.                                                                                                                                                                                                                                                                                                                                 Englewood Englewood 
                                                                                                                                                                                                                                          A A 
                                                                                3. 3.  William William                                                                                                                                             First First                                                                               New New                                      1995. 1995. 
                                                                                                                             Fulton, Fulton,                                                                                                                            Course, Course,                                                                           York, York, 
                                                                                                                                                          Algebraic Algebraic Topology: Topology:                                                                                                       Springer, Springer, 
                                                                               4. 4.                                                                                              A A                                     in in 
                                                                                           Wilhelm Wilhelm                                                                                  Course Course                                                                                                                                                        New New                                      1978. 1978. 
                                                                                                                                                                                                                                   Differential Differential Geometry, Geometry,                                                                                                      York, York, 
                                                                                           Barrett Barrett  Klingenberg, Klingenberg,                                                                                                                                                             2nd 2nd                   Springer, Springer, 
                                                                                5. 5.                                                                                                                                                                                                                                              Academic Academic                                                 San San                                       1997. 1997. 
                                                                                                                        O'Neill, O'Neill,                                                                                                                                                                         ed., ed.,                                                 Press, Press, 
                                                                                                                                                          Elementary Elementary                                                                        Geometry, Geometry, 
                                                                                                                                                                                                         Differential Differential                                                                                                                                                                                    Diego, Diego, 
                                                                                6. 6.  Michael Michael                                                       A A                                                                    Introduction Introduction to to                                                                                                                         3rd 3rd                         Vol. Vol.                    Publish Publish or or 
                                                                                                                              Spivak, Spivak,                         Comprehensive Comprehensive                                                                                                Differential Differential Geometry, Geometry,                                                             ed., ed.,                            1, 1,                                               Perish, Perish, 
                                                                                                                                  1999. 1999. 
                                                                                           Houston, Houston, 
                                                                                                                                                                                                                                                                                                                     Proof Proof 
                                                                                                                                                                                                                                                                                                                                                                           Without Without 
                                                                                                                                                                                                                        Lunes Lunes                                                                                                                                                                                                                         Words: Words: 
                                                                                                                                                                                                                                                                                   and and the the Regular Regular 
                                                                                                                                                                                                                                                                                                                                                                                                                                            Hexagon Hexagon 
                                                                                                                                                              If If                                                                                                                                                                       in in 
                                                                                              THEOREM. THEOREM.                                                            a a regular regular hexagon hexagon is is  inscribed inscribed                                                                                                               a a circle circle and and six six  semicircles semicircles  con- con- 
                                                                                structed structed on on its its                                                                          then then the the area area of of the the                                                                                                                                            the the 
                                                                                                                                                          sides, sides,                                                                                                                      hexagon hexagon equals equals                                                                       area area of of the the six six lunes lunes 
                                                                                                                                                                             whose whose diameter diameter                                                                                             in in                                                                                                                                                                                             plus plus 
                                                                               the the area area of of a a circle circle                                                                                                                                      is is                                                                                  to to one one of of the the 
                                                                                                                                                                                                                                                                         equal equal                               length length                                                                                      sides sides of of the the 
                                                                                                                                                   of of                                             ca. ca. 440 440                                                                                                                                                                                                                                                            hexagon. hexagon. 
                                                                                [Hippocrates [Hippocrates                                                        Chios, Chios,                                                              B.C.E] B.C.E] 
                                                                                              Proof Proof 
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                                                                                                                                                                                                                                                                                                                                                               LEWIS LEWIS  AND AND  CLARK CLARK  COLLEGE COLLEGE 
                                                                                                                                                                                                                                                                                                                                                                                                    PORTLAND,OR PORTLAND,OR                                                                 97219 97219 
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