What is the Newton's general theory of diameters? I was reading a book on Mathematics, which contained this topic. I was not able to grasp the concept. There was not much info on internet also. It was as follows:

Let an $n$th order curve be given, a curve which is represented by an $n$th degree algebraic equation in two unknowns; then an arbitrary straight line intersecting it has in general $n$ common points with it. Let $M$ be the point of the secant that is the "center of gravity" of these points of its intersection with the given $n$th order curve, i.e., the center of gravity of a set of $n$ equal point masses situated at these points. It turns out that if we take all possible sets of mutually parallel secants and for each of them consider these centers of mass $M$, then for any given set of parallel secants all the points $M$ lie on a straight line. Newton called this line the "diameter" of the $n$th order curve corresponding to the given direction of the secants.

Has someone heard about it. If yes, please explain it.
 A: In the quoted text it is assumed that an algebraic curve
$$C:=\{(x,y)\in{\mathbb R}^2\>|\> p(x,y)=0\}$$
is given. Here $p$ is a polynomial of degree $n$ in the real variables $x$, $y$. This means that
$$p(x,y)=\sum_{j\geq0, \>k\geq0} a_{jk}\> x^j y^k$$
with real $a_{jk}$, and that $a_{jk}=0$ when $j+k>n$, while at least one $a_{jk}$ with $j+k=n$ is nonzero. The text then talks about a family $(\ell_c)_{c\in I}$ of parallel secants of $C$, whereby it is assumed that each $\ell_c$ intersects $C$ in exactly $n$ points. (This last assumption is necessary when applying Vieta's theorem later on.)
For simplicity assume that we can  write the equation of the curve $C$ in the form
$$x^n +p_1(y) x^{n-1}+p_2(y) x^{n-2}+\ldots +p_{n-1}(y) x+p_n(y)=0$$
with polynomials  $y\mapsto p_j(y)$  $(0\leq j\leq n)$ of degree $\leq j$. Here we have arranged the terms appearing in $p$ according to descending powers of $x$.  As family of parallel secants we may take a family $(\ell_c)_{c\in I}$ of horizontals $\ell_c: \>y=c$, without loss of generality. By Vieta's theorem the centroid $M_c$ of $\ell_c\cap C$ is given by
$$M_c=\left({-p_1(c)\over n},c\right)\ .$$
As $y\mapsto p_1(y)$ is of degree $\leq1$ the $M_c$ $(c\in I)$ are lying on a line.
An example: The equation
$$5x^2-2xy+3y^2 -8x+2y=0$$
describes an ellipse $E$. Each complete family of parallel lines contains a subfamily whose members actually intersect $E$ in two points, and to this subfamily the above can be applied.
