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Overview

(2) Carlyle's and Lill's Geometric Solution of Quadratic Equations






The Scottish historian and writer Thomas Carlyle (1795-1881) devised an elegant geometrical solution to quadratic equations, based on the "Carlyle circle".

x2 + px + q = 0

The circle with the segment joining the points (0|1) and (p|q) as diameter is intersecting the p-axis, and the abscissae of these ponts of intersection are the required roots of the quadratic equation.

In 1867 by the Austrian captain of engineering Eduard Lill published a visual method of finding the real roots of polynomials of any degree.

raster box
Checking the box will mark certain points (p|q):
- p and q are multiples of the raster size, and
- the roots x1 and x2 are multiples of the raster size.

quadratic
                equation Carlyle Lill
Select the raster size, or a continuous mode ("Raster off").
A table of p, q, x1, x2 is available by "Data Window".


Web Links

Lill's method (Wikipedia)

Geometric Construction of Roots of Quadratic Equation (Cut The Knot)

Eduard Lill, radici immaginarie di un polinomio

Der Kreis von Lill, in: R. Kaendes, R. Schmidt (Hrsg.): Mit GeoGebra mehr Mathematik verstehen (Google Books)

Carlyle Circle (Wolfram MathWorld)

Carlyle Circle (Wolfram Demonstrations Project)

Applet showing Lill's method applied to quadratic equations

D. Tournès: Constructions d'équation algébriques et différentielles

T. C. Hull: Solving Cubics With Creases: The Work of Beloch and Lill (PDF)

M. E. Lill: Résolution Graphique des équations numériques de tous les degrées à une seule inconnue, et description d'un instrument inventé dans ce but, Nouvelles Annales de Mathematiques, Series 2, Vol. 6, 1867 ( PDF)

D. W. DeTemple: Carlyle Circles and the Lemoine Simplicity of Polygon Constructions (PDF)

Thomas Carlyle (MacTutor)

Print

R. Kaendes, R. Schmidt (Hrsg.): Mit GeoGebra mehr Mathematik verstehen, Vieweg+Teubner, 2011, ISBN 978-3-8348-1757-0.

A. Baeger: Eine geometrische Lösung der quadratischen Gleichung x2 + px + q = 0, in: CASIO Forum 1/2012, CASIO Europe.

E. J. Barbeau: Polynomials, Springer New York Heidelberg Berlin 2003, ISBN 0-387-40627-1, 978-0387-406275.

E. John Hornsby: Geometrical and Graphical Solutions of Quadratic Equations, The College Mathematics Journal, 1990, Volume 21, Number 5, p. 362-369
.





Updated: 2012, Apr 10