# Lill’s Method

On this page I will include some useful links to websites and papers that deal with Lill’s method. You can also check my papers page, to see some of my papers on the topic. I also wrote a few posts about Lill’s method. The math page not related to Lill’s method and the general resources page can also be useful.

Résolution graphique (french) The original paper by Eduard Lill. It was published in 1867 in  Nouvelles Annales de Mathématiques.

Résolution graphique des équationsalgébriques qui ont des racines imaginaires This is an article published in1868 in Nouvelles Annales de Mathématiques. The article shows that the method can deal with polynomials with complex roots (imaginary roots).

Extending Lill’s Method of 1867 This is the archived version since the original website doesn’t exist anymore. The author is Phillips Verner Bradford and he covers many topics related to Lill’s method, including angle trisection and complex root solutions.

Geometric Solutions of Algebraic Equations
This paper by Riaz offers a nice introduction to Lill’s method.

Graphical Statics – This is a book by Luigi Cremona. Chapter 6 is dedicated to Lill’s method. The mechanics of the method are described in great detail .

NOTE ON LILL’S METHOD OF SOLUTION OF NUMERICAL
EQUATIONS
– This paper by B. Meulenbeld is very important. Among many things, he shows how to derive the derivatives of polynomial equations using Lill’s method. It shows that Lill’s method can be quite a powerful mathematical tool. The original link doesn’t work anymore so I uploaded the pdf article to my website.

NOTE ON THE REPRESENTATION OF THE VALUES OF POLYNOMIALS WITH REAL COEFFICIENTS FOR COMPLEX VALUES OF THE VARIABLE
This is another paper by B. Meulenbeld. The paper is shorter than the one above but it is still very rigorous. The original link doesn’t work anymore so I uploaded the pdf article to my website.

Lill’s method and the Philo Line for Right Angles
This article shows that solving a class of third degree polynomials using Lill’s method is equivalent to finding the Philo line (also known as Philo’s line, Philon line or Philon’s line). I wrote this article before I made this website.

Wikipedia article The page has many useful links.

Mathologer Youtube video. This is a very high quality Youtube video that presents Lill’s method.

Solving Cubics With Creases: The Work of Beloch and Lill This paper by Thomas C. Hull presents the work of Margharita P. Beloch. Beloch showed that paper folding (origami) can be used to perform Lill’s method in the cubic case and thus solve general polynomials of degree three.

Origami video presentation by Marcus Herbert. The page has a Youtube video that shows how to apply Beloch’s origami technique.

Elements of Geometry and Plane Trigonometry by John Leslie This book is very similar to Euclid’s Elements. The most interesting thing about this book is that it describes the Carlyle circle method for finding the roots of quadratic equations (only real roots). The method is described starting with the page 176. The Carlyle circle is related to Lill’s method or Lill’s circle. On page 340 it mentions that Thomas Carlyle was indeed the person who discovered the method and it also presents a method used by Pappus.

Carlyle Circles and the Lemoine Simplicity of Polygon Constructions – by Duane W. DeTemple. Paper about Carlyle circles and polygon construction.

LILOV METOD ODREĐIVANJA REALNIH KORENA POLINOMA SA REALNIM KOEFICIJENTIMA (Serbian language) . This is a masters thesis by Tijana Cvetković. The paper seems to be promoting the idea that Lill’s method can be a very useful educational tool (for high school math). I found this paper because it cites two of my papers.

Lill’s Method and Horner’s Scheme . This is a blog post from the Mathematical Whetstone blog.

Geometric Solutions of Quadratic Equations – the page has links to geometric solutions for quadratic equations applets. The methods listed are Euclid’s Method, Carlyle’s Method, Von Staudt’s Method and Kumar’s Method. Kumar’s Method and Carlyle’s Method can be connected to Lill’s method (which is a generalization of both methods.

Solving Quadratic Equation | Using Set Square Only (Bikal Baral 13) This is another Youtube video that shows how to solve quadratics with a square. It’s shows the Kumar method from the previous link.