Shapes by the Numbers Coordinate Geometry Sketch 16 Kristina and Jill

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

Ancient Times Uses Egypt Egypt Rectangular grid Rome RomeSurveyors Greece GreeceMapmakers

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

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 His development of locus to equation Represented curves using algebra in two variables Parabola: x 2 =dy Parabola: x 2 =dy Hyperbola: b 2 +x 2 =ay 2 Hyperbola: b 2 +x 2 =ay 2 Circle: b 2 -x 2 =y 2 Circle: b 2 -x 2 =y 2 Ellipse: b 2 -x 2 =ay 2 Ellipse: b 2 -x 2 =ay 2 Straight line: x 2 ±xy=ay 2 Straight line: x 2 ±xy=ay 2 Only considered positive values His contributions were not published until after his death so a lot of credit was given to Descartes.

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. 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 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

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

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

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

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 16 th Century Francois Viete worked on using algebra in geometric problems

Timeline Continued Early 17 th Century Fermat and Descartes introduced work on analytic geometry 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 17 th Century analytic geometry was widely known throughout Europe

Resources Berlinghoff, William P., and Fernando Q. Gouvea. Math Through the Ages: a Gentle History for Teachers and Others. Farmington, Maine: Oxton House, Katz, Victor J. A History of Mathematics. New York: Pearson, "Analytic Geometry: Marriage of Algebra & Geometry." Think Quest Nov Gale, Thomas. "The Development of Analytic Geometry." Book Rags Science and Its Times. 14 Nov