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(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.

 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. 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) 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) 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