# 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 $$p \rightarrow -\infty.$$