Laguerre's theorem on power of a point w.r.t. an algebraic curve

Question

So on Wikipedia article for a power of a point there is a short section about Laguerre's theorem. The problem is, the article has no references, and whenever I'm trying to Google it the only things I get are either power diagrams or some different definitions of power of a point w.r.t. algebraic curve, but none of these seem to trace back to Laguerre's work.

So my question is, as expected

Can anyone give a reference to Laguerre's original work, or at least provide a place where properties of his definition (such as independence of the choice of the circle) are proven?

Answer 1

I am not allowed to comment so I am posting here. I found a scholarly journal from Cambridge University entitled "A New Proof of Laguerre's Theorem About The Zeros of Polynomials"

http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ33_01%2FS0004972700002951a.pdf&code=2102886a73cf51a49ea5704c1f57a0b3

Journal of Pure and Applied Mathematics - Laguerre, Sur La Théorie Des Équations Numériques

https://ia600401.us.archive.org/24/items/thoriedesquatio00lagugoog/thoriedesquatio00lagugoog.pdf

Answer 2

The source for this theorem is Laguerre, Sur les courbes planes algébriques, 1865, pg 20, Théorème 1. There the theorem is stated without proof, and then all but forgotten in the mists of history. The paper is worth a look, whatever your level of French math comprehension. If I read it correctly, Laguerre defines power of a point $(\xi,\eta)$ w.r.t. to an algebraic curve $f(x,y)=0$ as $f(\xi,\eta)$, up to some multiplicative constant. This is consistent with the power of a point w.r.t to a circle.

A good place to start the literature search is Thomas & Porta, Clifford’s Identity and Generalized Cayley-Menger Determinants (there's a download link), which fortuitously came out earlier this year.

The Laguerre theorem is briefly discussed at the end of Section 4, leading to two papers by Neville:

• Neville, Products of lengths of tangents and of normals, which gives further references in the second and final paragraph.

• Neville, The power of a point for a curve., where Laguerre's theorem is stated on the final page, after preliminaries and proof.

Another proof is given in Coolidge Algebraic Plane Curves, pg 176.