# When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - An Infinite Family of Curves

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Consider the cubic equation for Euler's Elegant Example: $t(y^3-y) = s(x^3-x).$

As long as $t \ne 0,$ we can divide through by $t.$  If we then let $p={s}/{t},$ we have the parameterized cubic equation $y^3-y = p(x^3-x).$

Now for any real number that we assign to the parameter $p,$ we have an equation whose graph passes through the nine points of the $3 \times 3$ grid. You can explore these curves using the applet in Figure 8. You can set particular values of $p$ using the slider control, or put the applet into play mode and watch as $p$ cycles through values between $-4$ and $+4.$

Figure 8.  Euler's Elegant Example. Set values of $p$ using the slider control, or click the arrow at lower left and watch as $p$ cycles through values between $-4$ and $+4.$ (Interactive applet created using GeoGebra.)

For the particular values $p=0,\, p=1,$ and $p=−1,$ we can factor the parameterized cubic equation $y^3-y = p(x^3-x)$ into factors of lower order.

 Value of $p$ Factorization $−1$ $(y+x)(y^2−xy+x^2−1)=0$ $0$ $y(y−1)(y+1)=0$ $1$ $(y−x)(y^2+xy+x^2−1)=0$
In the case $p=0,$ the graph consists of three horizontal lines: $y=0,$ $y=1$ and $y=-1.$  If $p=1,$ the graph consists of the line $y=x$ and a conic section, namely a skewed ellipse with its major axis on the line $y=-x$ and its minor axis on the line $y=x.$  The case $p=-1$ is similar, with the roles of the lines $y=x$ and $y=-x$ reversed.  For all other values of $p,$ the equation can't be factored over the real numbers and the curve consists of a single continuous line.

There is one more important special case, when $t=0$ in the original equation $t(y^3-y) = s(x^3-x)$ for Euler's Elegant Example. In this case, we have $x(x-1)(x+1) = 0,$ so that the graph consists of three vertical lines: $x=0,$ $x=1$ and $x=-1.$ This is the asymptotic case to which the parameterized cubic equation $y^3-y = p(x^3-x)$ tends when either $p \rightarrow +\infty$ or $p \rightarrow -\infty.$

Robert E. Bradley (Adelphi University) and Lee Stemkoski (Adelphi University), "When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - An Infinite Family of Curves," Convergence (February 2014)