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.)