"Descartes' parabola" and the traditional parabola : a reconstruction of a historical method with the help of the "mathematica" program

Abstract

This study concentrates on Descartes’ geometry, especially the Descartes’ parabola and traditional parabola. Who is Descartes? René Descartes (1596-1650) was a 17th century French philosopher, mathematician and a man of science whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry. His work had a great influence on both mathematicians and philosophers. In mathematics Descartes chief contribution was in analytical geometry. Descartes made other known contributions to mathematics. He was the first to use the first letters of the alphabet to represent known quantities, and the last letters to represent unknown ones. Descartes also formulated a rule known as Descartes' rule of signs, for finding the positive and negative roots of an algebraic equation. First, this study concentrates on the Descartes’ studies of Pappus’ problem. Also I explicitly explain how Descartes’ found the traditional parabola and Descartes’ parabola, and how he used the four and five lines Pappus’ problems. Secondly, this study concentrates on the Descartes’ “construction” [that means geometrical solution] of equations by using Descartes’ parabola and the traditional parabola. I clearly explain Descartes’ construction of third and fourth degree equations by circle and traditional parabola, and the construction for fifth and sixth degree equations by using circle and Descartes’ parabola. Finally, I also explain the construction of higher degree equations. Furthermore I give three numerical examples by solving them with the mathematica program, which was designed by Stephen Wolfram.