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Overview

(1) Euclid's Geometric Solution of Quadratic Equations






ceckbox
Use the checkboxes (at the left or right) to select one of the two cases of q.
raster box

euclid
                raster
Checking the box will mark certain values of p (right axis)
- p is a multiple of the raster size, and
- the roots x1 and x2 are multiples of the raster size.

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

The Greeks created a geometric algebra: terms of equations were represented by sides of geometric objects, usually lines, and found by constructive methods.

The method in my applets is presented in the paper of
Hornsby. It is based on Euclid's Elements, Book II, Proposition 11, and Book VI, Propositions 28 and 29.

proof


Using the Right Triangle Altitude Theorem (Altitude-On-Hypotenuse Theorem):

q·1 = h2  and x1·x2 = h2

x1·x2 = q and  x1·x2 = |p|  (Vieta)




Web Links

Euclid's Elements

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)

Burton: The History of Mathematics, McGraw-Hill, 2006, (PDF).

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

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.





Updated: 2012, Feb 17