When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Conic Exceptions

Leonhard Euler wrote a letter to Gabriel Cramer on October 20, 1744. Although all of Euler's other letters to Cramer were preserved in an archive in Geneva, Switzerland, this particular one went missing long ago and was unknown when scholars compiled a catalog of Euler's correspondence [Euler 1975]. The lost letter became known to Euler scholars at the meeting of the Euler Society in August 2003, which both authors of this article attended. At some point in the 20th century, it had found its way into the private collection of Bern Dibner (1897-1988). Dibner was an engineer, entrepreneur and philanthropist, as well as a historian of science. Over the course of his long life, he amassed an impressive private collection of rare books, manuscripts and letters. He donated about a quarter of this collection to the Smithsonian Library in 1974 and Euler's lost letter to Cramer was part of that gift. Mary Lynn Doan, professor of mathematics at Victor Valley College, had contacted the Dibner Library of the Smithsonian Institution in the summer of 2003 and learned that it has a small collection of documents by Leonhard Euler from Dibner's collection [Euler Papers]. She brought a photocopy of the letter with her to the Euler Society's meeting that summer and one of the authors (Bradley) was able to identify the addressee as Cramer. Shortly afterwards, he brought the letter to the attention of Andreas Kleinert, one of the editors of Euler's Opera Omnia [Euler]. The letter, in its original French, will appear in a forthcoming volume of Euler's correspondence.

In his letter to Cramer, Euler observed that there are exceptions to the rule that five points determine a conic section [Euler 1744b, p. 3]:

We may clarify this ... by considering lines of the second order, for the determination of which 5 points may not always be sufficient. For when all the five points are arranged on a straight line so that they give, for example, these equations

\(x=0;\) \(y=0;\)

\(x=1;\) \(y=1;\)

\(x=2;\) \(y=2;\)

\(x=3;\) \(y=3;\)

\(x=4;\) \(y=4;\)

all of the coefficients of the general equation \(\alpha yy + \beta xy + \gamma xx + \delta y + \varepsilon x + \zeta = 0\) will not be determined, for after having introduced all of the given determinations, we are brought to this equation \(\alpha yy - (\alpha + \gamma)xy + \gamma xx  + \delta y - \delta x = 0,\) so that there still remain two coefficents to be determined. If from the five given points there had been but 4 arranged in a straight line, then there would remain but one coefficient to be determined.

Euler did not use ordered pairs as we do today, but he was asking Cramer to consider the problem of trying to fit the points \((0,0),\) \((1,1),\) \((2,2),\) \((3,3)\) and \((4,4)\) to the general equation of the second degree, \[\alpha y^2 + \beta xy + \gamma x^2 + \delta y + \varepsilon x + \zeta = 0.\] (It's interesting to note that Euler wrote \(yy\) and \(xx\) where we would write \(y^2\) and \(x^2.\)) In this case, it turns out there is no unique solution to this equation.