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The legacy of René Descartes should include the contributions of his student, Elisabeth of Bohemia, in the development of his ideas.
Thirteen years ago, I wrote an article for American Scientist called "A Tisket, a Tasket, an Apollonian Gasket" (January-February 2010), which explored the fascinating mathematics and history behind a certain type of fractal foam-the "Apollonian gaskets" of the title. Recently, I've begun to see that history itself is like a fractal: When you take even the smallest event and magnify it, it springs to life and turns out to have a whole story of its own.
One sentence in particular from my article nagged at me: "In 1643, in a letter to Princess Elizabeth of Bohemia, the French philosopher and mathematician René Descartes correctly stated (but incorrectly proved) a beautiful formula concerning the radii of four mutually touching circles." This offhand comment is not really wrong but also not really right, and what's worse is that I had drawn the idea from articles by other scholars without looking into it critically. In fact, this one sentence raises all sorts of questions. Why was Descartes writing to a princess, and how did they get on the subject of circles? Why did he write to her about this problem, and not to the many other mathematicians he corresponded with? And who was Princess Elizabeth, anyway?
The answers to these questions reveal the foibles and personalities of two immensely fascinating people, René Descartes and Princess Elisabeth (who signed her name with an s). The true stoThis puzzle illustrates Descartes's circle theorem, which describes the specific conditions required to have four mutually tangent circles (circles that touch at just one point). Many such quadruples can be found in this puzzle, such as the circles labeled 11,14,15, and 86. These labels refer to the circles' curvatures, which are reciprocals of their radii (Vil, Vl4, Vis, and Vs6, respectively). ry of the formula is considerably more nuanced than its conventional name, Descartes's circle theorem, implies. It introduces a mathematician who has not been widely recognized for her mathematics. And it makes one think about why certain problems or theorems become famous. Histories and textbooks of mathematics tend to shine a spotlight on certain topics, but to understand why they...