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When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Construction of Conics

Author(s): 
Robert E. Bradley (Adelphi University) and Lee Stemkoski (Adelphi University)

There are various constructions of particular conic sections, such as Euclid's Proposition IV.5, but a geometric construction of an arbitrary conic given five points was first published by William Braikenridge [1733], although Maclaurin disputed his priority in "a rather disagreeable controversy" [Coxeter 1961a, p. 91]. Coxeter gave the construction in both [1961a, p. 91] and [1961b, p. 254]. He suggested that it was based on Pascal's celebrated theorem about the points of intersection of the sides of a hexagon inscribed in a conic section. However, it is not clear that either Maclaurin or Braikenridge knew Pascal's Theorem; see [Mills 1984]. The applet in Figure 6 illustrates that, in general, there exists a conic section passing through any five points.

Figure 6.  A conic section passing through five points. Move points \(A,B,C,D,\) and \(E\) to explore the possibilities. (Interactive applet created using GeoGebra.)
 

Robert E. Bradley (Adelphi University) and Lee Stemkoski (Adelphi University), "When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Construction of Conics," Loci (February 2014)

Dummy View - NOT TO BE DELETED

When Nine Points Are Worth But Eight: Euler's Resolution of Cramer's Paradox