Shapes by the Numbers Coordinate Geometry Sketch 16

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Shapes by the NumbersCoordinate GeometrySketch
16

• Kristina and Jill

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Overview of contributions

• Ancient times
• Early mathematicians (16th century and earlier)
• Menaechmus
• Apollonius
• Francois Viete
• Major Contributors
• Descartes
• Fermat

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Ancient Times

• Uses
• Egypt
• Rectangular grid
• Rome
• Surveyors
• Greece
• Mapmakers

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Early Mathematicians

• Menaechmus
• Introductions of Conic Sections
• Apollonius
• Work related to loci
• Made a start in the development of analytic
  geometry
• Related geometric figures to ratios and words
• Francois Viete
• Took a leap on focusing algebra to geometric
  problems

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Fermat

• Study works of Apollonius and Viete
• He wrote a manuscript entitle Introduction to
  Plane and Solid Loci
• Developed a unique coordinate system
• His development of locus to equation
• Represented curves using algebra in two variables
• Parabola x2dy
• Hyperbola b2x2ay2
• Circle b2-x2y2
• Ellipse b2-x2ay2
• Straight line x2xyay2
• Only considered positive values
• His contributions were not published until after
  his death so a lot of credit was given to
  Descartes.

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Descartes

• Influenced by Viete and Islamic mathematicians
• Published Discourse on the Method of Rightly
  Conducting the Reason in the Search for Truth in
  the Sciences
• La Geometrie is the section in which his work on
  analytic geometry appears.
• Used single horizontal axis
• Introduced what we now know as x and y
• Represented unknowns using the letters at the end
  of alphabet and constants using letters at the
  beginning of the alphabet
• Demonstrated how algebraic equations are formed
  using their solutions

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Others that followed

• Frans van Schooten
• Translated Descartes work into Latin and added
  missing detail
• John Wallis
• Extended analytic geometry ideas to include
  negatives
• Jan de Witt
• Details on how to solve the locus problem for
  quadratic equations
• Isaac Newton
• Learned about analytic geometry on his while
  developing ideas of Calculus

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Compare and Contrast

• Fermat
• Described curve from an equation
• Used a single axis (horizontal)
• Never published his work

• Descartes
• Described algebraic equation from a curve
• Used a single axis (horizontal)
• Published his work but in French with omitted
  detail
• Dealt with more complex equations than Fermat did

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History of Geometry and Algebra

• Algebra grew out of simple manipulation of
  geometric shapes
• During the medieval period and Renaissance
  algebra was freed from Geometry
• Algebra and Geometry returned to one another in
  what we now know as analytic geometry

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Timeline

• Ancient Egypt used rectangular grid
• Same method used by Roman Surveyors and Greek
  mapmakers
• 350 B.C. Menaechmus introduces conic sections
• Approximately 250 B.C. Apollonius works on loci
  of curves
• Late 16th Century Francois Viete worked on using
  algebra in geometric problems

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Timeline Continued

• Early 17th Century Fermat and Descartes
  introduced work on analytic geometry
• 1649-1693 Van Schooten translated La Geometrie
  into Latin and added omitted detail
• Approximately 1659 Jan de Witt provided details
  to solve locus problems of quadratic equations
• End of the 17th Century analytic geometry was
  widely known throughout Europe

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Resources

• Berlinghoff, William P., and Fernando Q. Gouvea.
  Math Through the Ages a Gentle History for
  Teachers and Others. Farmington, Maine Oxton
  House, 2002. 135-140.
• Katz, Victor J. A History of Mathematics. New
  York Pearson, 2004. 260-270.
• "Analytic Geometry Marriage of Algebra
  Geometry." Think Quest. 2001. 14 Nov. 2006
  lthttp//library.thinkquest.org/C0110248/geometry/h
  istory5.htmgt.
• Gale, Thomas. "The Development of Analytic
  Geometry." Book Rags. 2006. Science and Its
  Times. 14 Nov. 2006 lthttp//www.bookrags.com/resea
  rch/the-development-of-analytic-geometr-scit-03123
  /gt.