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C'este donc par
seulement par elle,
que l'on peut se
faire une idee juste
et approfondie de ce
que c'est qu'une
Quoted by T. H. Huxley in Fortnightly Review, Vol. II, N.S. 5.
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When Nine Points are Worth But Eight: Euler's Resolution of Cramer's Paradox
Abstract: 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 Institute 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.
[Note: This article includes several interactive worksheets displayed using the GeoGebra applet, which requires Java version 1.4.2. One of the worksheets uses GeoGebra 4.0 Beta, which requires Java version 5.]
Bradley, Robert E. and Lee Stemkoski, "When Nine Points are Worth But Eight: Euler's Resolution of Cramer's Paradox," Loci (April 2011), DOI: 10.4169/loci003651