MATH325: Lecture 20

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Conclusion

Remember that in straightedge and compass construction we were able to do draw a line through two existing points, construct a circle with any existing center and radius, and construct points of intersection of circles, lines, etc.

With these tools we were able to construct regular polygons, perpendicular and parallel lines. However, simply constructing these figures was not enough to prove that they had the properties we desired.

Euclid may have been the first mathematician to try to systematically prove that these constructions were accurate. His definitions, common notions and five postulates form the foundation for his 13 books of proofs of geometric concepts and for much of what is taught in high school geometry classes today.

However, Euclid left some important questions unanswered, including:

• Can two circles intersect in more that two points?
• If a line intersects an edge of a triangle, must it intersect the boundary of the triangle at some other point?
• Can the fifth postulate be proven based on the first four postulates?

Riemann, Lobachevski and others discovered hyperbolic and spherical geometry while trying to derive a contradiction to the fifth postulate. Mathematicians such as Jordan and Pasch worked hard to answer the other questions.

In addition to reinforcing Euclid's work, mathematicians have extended it in new directions. Arthur Baragar, the author of our textbook, chose to add axioms dealing with isometries to Euclid's five postulates. René Descartes introduced coordinate geometry. Other mathematicians have tried to prove results similar to Euclid's using axioms based on paper folding or isometries rather than straightedge and compass construction.

We have studied the foundations of geometry; much has been built on those foundations, as I hope you someday have the opportunity to discover.