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In 1637, the French philosopher and mathematician René Descartes (1596–1650) published his *Discours de la methodé *(see the title page)* *in which he explained his rationalist approach to the interpretation of nature. *La methodé* contained three appendices: *La dioptrique,* *Les météories,* and *La géométrie*. The last of these, *The Geometry,* was Descartes’ only published mathematical work.

**Figure 1.** Frontispiece of 1659 Latin edition of Descartes’ *Geometry* showing a portrait of the author, René Descartes

Approximately 100 pages in length, *The Geometry* was not a large work, but it promised a new approach in mathematical thinking. Descartes boasted in his introduction that “Any problem in geometry can easily be reduced to such terms that a knowledge of the length of certain straight lines is sufficient for construction." He then proceeded to show how arithmetic, algebra, and geometry could be combined to solve problems. After defining a unit length, Descartes demonstrated procedures for adding, subtracting, multiplying, and dividing line segments and for graphically determining roots of equations. But Descartes’ *La géométrie* was difficult to understand and follow. It was written in French, not the language of scholarly communication at the time, and Descartes' writing style was often obscure in its meaning.

**Figure 2.** Title page of van Schooten’s 1659 Latin edition of Descartes’ *Geometry*

In 1649, Frans van Schooten (1615–1660), a Dutch mathematician, published a Latin translation of Descartes’ *Geometry,* adding his own clarifying explanations and commentaries. Descartes’ friend and colleague, Florimond de Beaune (1601–1652), contributed an introduction to this edition. Van Schooten published a second edition in 1659, providing even more explanation and commentary. Now the message of Descartes’ *Geometry* was available to a large reading audience, and it became an influential work, spurring on the development of analytic geometry.

**Figure 3.** On Page 2 of the Latin edition of Descartes’ *Geometry,* the author demonstrated, in the uppermost illustration, the procedure for obtaining the product of two given line segments, BD and BC. A unit length AB is designated. Then when DE is constructed parallel to AC, it is found that BD x BC = AB x BE. However, AB is the unit length; therefore BD x BC = BE.

**Descartes’ Geometry in English:**

*The Geometry of Rene Descartes* (translated by David Eugene Smith and Marcia Latham), first published by Open Court in 1925 and by Dover in 1954

*Discourse on the Method, …, The Geometry, *in* Great Books of the Western World *(2^{nd} edition),* *vol. 28, Encyclopaedia Britannica, Chicago, 1990-1991

*Discourse on Method, Optics, Geometry, and Meteorology* (translated by Paul Olscamp), Bobbs-Merill, Indianapolis, 1965; revised edition, Hackett Publishing, Indianapolis, 2001

**Descartes’ Geometry in French:**

The complete text of Rene Descartes’ *La géométrie* in French is available courtesy of Project Gutenberg.

An 1886 edition of *La géométrie de Rene Descartes* published in Paris can be viewed courtesy of the Cornell University Library.

**Descartes’ Geometry in Latin:**

First, second, and third editions of Frans van Schooten’s Latin *Geometria a Renato Des Cartes* from 1649, 1659 and 1683, respectively, can be viewed at Google Books by searching for “Rene Descartes Geometria” and “Renato Descartes Geometria”.

**About Descartes and his mathematics:**

Victor J. Katz, *A History of Mathematics* (3rd ed.), Addison-Wesley, Boston, 2009

**About Descartes and his science:**

William R. Shea, *The Magic of Numbers and Motion: The Scientific Career of Rene Descartes,* Science History Publications, Watson Publishing International, Canton, Massachusetts, 1991

Frank J. Swetz (Pennsylvania State University), "The Geometry of Rene Descartes," *Loci* (March 2010), DOI:10.4169/loci003444