When Nine Points Are Worth But Eight: Euler's Resolution of Cramer's Paradox

Editor’s note: This article was published in April 2011 and updated in February 2014.

Overview

Gabriel Cramer and Leonhard Euler both wrote important books on the theory of equations in the mid 18th century. During the years leading up to their publications, they carried on a friendly and fruitful correspondence. One topic they discussed was a paradox that was first noticed by Maclaurin: that nine points should be sufficient to determine a curve of order three, and yet two different curves of order three could intersect in up to nine different places. Although this situation has come to be known as Cramer's Paradox, it was Euler who first suggested the resolution of this apparent contradiction, in a letter that was lost long ago but rediscovered in the Smithsonian Institution in 2003.

In this paper, we investigate the properties of algebraic curves of order two and higher and describe Cramer's Paradox and Euler's resolution, including his elegant example of an infinite family of cubic curves that all pass through the same nine points. We also provide the first English translation of Euler's long lost letter of October 20, 1744.