Geometry pp 158-184 | Cite as

The Theory of Parallels

  • Richard S. Millman
  • George D. Parker
Part of the Undergraduate Texts in Mathematics book series (UTM)


The concept of parallel lines has led to both the most fruitful and the most frustrating developments in plane geometry. Euclid (c. 330–275 B.C.) defined two segments to be parallel if no matter how far they are extended in both directions, they never meet. Note that he was interested in segments rather than lines. This follows the general preference of the time for finite objects. The idea of never meeting is, however, infinite in nature. How then does one determine if two lines are parallel?


Parallel Line Critical Function Hyperbolic Plane Hyperbolic Geometry Unique Line 
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Copyright information

© Springer-Verlag Inc. 1981

Authors and Affiliations

  • Richard S. Millman
    • 1
  • George D. Parker
    • 2
  1. 1.Department of Mathematical and Computer ScienceMichigan Technological UniversityHoughtonUSA
  2. 2.Department of MathematicsSouthern Illinois UniversityCarbondaleUSA

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