# Of what use is Newton's theory of diameters?

The theory is, as I understand it, that when there is a third degree equation, you take any line that intersects the curve thrice, and take the 'midpoint' of the ($3$) points, the infinitude of possible midpoints all lie on a common (vertical) line.

All well and good, but do any of you know of any applications of this, in the physical world or as a stepping stone to other mathematical theory? Because for all its quaintness it seems a little useless.

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This is not true of degree $2$ polynomials. And it's not exactly true of degree $3$ polynomials either. You may take any line that intersects the curve three times and the average of the three intersection points will all have the same $x$-coordinate: one-third the negative of the coefficient of $x^{3-1}$ of the polynomial, which is also the $x$-coordinate of the inflection point. –  alex.jordan Nov 24 '12 at 22:38
OK, duly edited. –  Alyosha Nov 24 '12 at 22:45
No specific application comes to my mind. But try to find yourself an application where you would solve an equation $\operatorname{cubic}(x)=\operatorname{linear}(x)$ and you expect to get three solutions. No matter how you change the parameters on the linear side, the average of the three solutions will not change. Maybe you can work that into an application. You'll need to find a situation where $y=\operatorname{cubic}(x)$ and $y=\operatorname{linear}(x)$ are relevant equations. Maybe start by finding an application where $y=\operatorname{cubic}(x)$ is a relevant equation. –  alex.jordan Nov 24 '12 at 22:51
Also, while this statement applies equally well to degree $n\geq3$ polynomials and lines that intersect them $n$ times, we've noted that for cubics the common $x$-value is the $x$-value of the inflection point. So you might research an application for an inflection point and see of you can respin that application to be about averaging the three solutions. –  alex.jordan Nov 24 '12 at 22:54
Excellent answer, you're a credit to Stackexhange (if myself as an arbiter of those sorts of things really has any weight). –  Alyosha Nov 24 '12 at 22:56