On September 30, 1744, the Swiss mathematician Gabriel Cramer (1704-1752) wrote a

Chapter 3, Problem 58

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On September 30, 1744, the Swiss mathematician Gabriel Cramer (1704-1752) wrote a remarkable letter to his countryman Leonhard Euler, concerning the issue of fitting a cubic to given points in the plane. He states two facts" about cubics: (1) Any 9 distinct points determine a unique cubic. (2) Two cubics can intersect in 9 points. Cramer points out that these two statements are incompatible. If we consider two specific cubics that intersect in 9 points (such as x* - x = 0 and y' v = 0), then there is more than one cubic through these 9 points, contradicting the first fact. Something is terribly wrong here, and Cramer asks Euler, the greatest mathematician of that time, to resolve this apparent contradiction. (This issue is now known as the Cramer-Euler Paradox.)Euler worked on the problem for a while and put his answer into an article he submitted in 1747, Sur one contradiction apparente dans la doctrine des lignes courbes [Me mo ires de I'Academie des Sciences de Berlin,4 (1750): 219-233]. Using Examples 44 through 57 as a guide, explain which of the so-called facts stated by Cramer is wrong, thus resolving the paradox.