# 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," Convergence (February 2014)