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Euler's Resolution Vindicated

After describing his beautiful example, Euler wrote these words:

"And from this one will easily understand that every time that two curves of the third order intersect at 9 points, these points will be such that they do not completely determine a curve of the third order, and that in the general equation, after one had applied it to these nine points, a coefficient will remain undetermined. In these cases, therefore, there will be not only two curves of the third order, but an infinity of curves of this order, which can all be described by these nine points." [Euler 1750a, p. 232]

A modern reader might reasonably object that Euler has not proved the general claim he made in this passage. Rather, having given an illustrative example, he simply asks the reader to understand -- or perhaps simply to accept -- that the general case is similar. However, Charlotte Scott [1898, p. 263] points to this very passage as an indication that Euler's explanation of Cramer's Paradox goes "just a little bit further" than the one that Cramer himself gave in [Cramer 1750]. Further reading on Euler's contribution to the resolution of Cramer's Paradox can be found in [Sandifer 2007].

An entirely satisfactory resolution of Cramer's Paradox would have to wait until the 19th century, when the concepts of linear algebra were more well formed. In fact, nineteenth century mathematicians continued to work on generalizations and elaborations of Cramer's Paradox. For more on the further development of this problem, see the historical introduction to [Eisenbud 1996]. In particular, Chasles fully vindicated Euler's resolution of Cramer's Paradox for cubic curves; see [Eisenbud 1996, p. 301]. Cayley worked on the higher degrees and proved the following, as quoted in [Scott 1898]:

Theorem (Cayley 1843, parenthesized hypothesis added 1887):
Suppose \( n \ge l \), \( n \ge m \), \( n < l+m \), and let \[ \delta = \frac{1}{2}(l+m-n-1)(l+m-n-2). \] If a curve of order \( n \) passes through \( lm-\delta \) of the points of intersection of two curves of order \( l \) and \( m \), then it necessarily passes through the remaining \( \delta \) points (unless those \( \delta \) points lie on a curve of order \( l+m-n-3 \) ).

Euler's example is indeed generic: in the case \( l=m=n=3 \), Cayley's theorem says that a curve of order 3 that passes through any 8 of the 9 points of intersection of two curves of order 3 must necessarily pass through all 9. Thus, the rank of the linear system determined by these 9 points can't possibly be more than 8. In the more general case of Cramer's Paradox, if \( l=m=n \ge 3\), then Cayley's theorem says that a curve of order \( n \) passing through any \( \frac{n^2+3n}{2}-1 \) of the \( n^2 \) points of intersection of two curves of order \( n \) necessarily passes through all \( n^2 \) points, unless those remaining points all lie on a curve of order \( n-3 \).