Descartes’ Method for Constructing Roots of Polynomials with ‘Simple’ Curves - Conclusion, References, and About the Author

Conclusion

As noted at the start of my derivation of Descartes' method for solving sextic equations, Descartes did not explain his derivations, writing in his 'Geometry' (Smith 192):

I have also omitted here the demonstration of most of my statements, because they seem to me so easy that if you take the trouble to examine them systematically the demonstrations will present themselves to you and it will be of much more value to you to learn them in that way than by reading them.

And later, as the final sentence of his book, he wrote (Smith 240):

I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery.

Attempting to reconstruct Descartes’ thought process was not as ‘pleasurable’ for me as he might have intended, but it was certainly satisfying.

References

Boyer, Carl B. History of Analytic Geometry. Mineola, NY: Dover Publications, 2004. Originally published in Scripta Mathematica Studies series (New York), 1956. Print.

Descartes, René. 'La Géométrie,' in The Geometry of René Descartes (David Eugene Smith and Marcia L. Latham, translators). New York, NY: Dover Publications, 1996, 1954. Originally published by Open Court Publishing Co., 1925. Print.

Katz, Victor and Karen Parshall. Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press, 2014. Print.

Knorr, Wilbur Richard. The Ancient Tradition of Geometric Problems. New York: Dover, 1993. Originally published by Birkhäuser (Boston), 1986. Print.

Rickey, V. Frederick, ‘Isaac Newton: Man, Myth, and Mathematics,’ College Mathematics Journal 18 (1987), pp. 362-389.

Smith, David Eugene and Marcia L. Latham (translators). The Geometry of René Descartes. New York, NY: Dover Publications, 1996, 1954. Originally published by Open Court Publishing Co., 1925. Print. Same work as in entry for "Descartes, René," above. Since all references to this work in the present article are to the English translation, citations are to Smith rather than to Descartes.

Additional Reading

For a list of easy-to-access publications of Descartes' 'Geometry' in English, French, and Latin, see the Convergence article, "The Geometry of René Descartes."

For more about the history of solving quadratic equations, see the Convergence article, "Geometric Algebra in the Classroom" (reprint of "Geometric Approaches to Quadratic Equations from Other Times and Places").

To see how Omar Khayyam solved a cubic equation, see the Convergence article, "A GeoGebra Rendition of One of Omar Khayyam's Solutions for a Cubic Equation."

About the Author

Gary Rubinstein is a mathematics teacher at Stuyvesant High School in the Borough of Manhattan, New York, New York. One of nine public high schools in New York City with specialized and accelerated curricula, Stuyvesant is a college preparatory high school specializing in STEM subjects. Rubinstein was awarded Math for America Master Teacher Fellowships in 2006, 2010, and 2014.