page_title: Carlyle circle
section_title: History
hierarchical_section_title: Carlyle circle / History
context_page_description: In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
context_section_description: According to Howard Eves (1911–2004) the mathematician John Leslie (1766–1832) described the geometric construction of roots of a quadratic equation with a circle in his book Elements of Geometry and noted that this idea was provided by his former student Thomas Carlyle (1795–1881). However while the description in Leslie's book contains an analogous circle construction, it was presented solely in elementary geometric terms without the notion of a Cartesian coordinate system or a quadratic function and its roots:
  To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle.
  — John Leslie, Elements of Geometry, prop. XVII, p. 176
  In 1867 the Austrian engineer Eduard Lill published a graphical method to determine the roots of a polynomial (Lill's method). If it is applied on a quadratic function, then it yields the trapezoid figure from Carlyle's solution to Leslie's problem (see graphic) with one of its sides being the diameter of the Carlyle circle. In an article from 1925 G. A. Miller pointed out that a slight modification of Lill's method applied to a normed quadratic function yields a circle that allows the geometric construction of the roots of that function and gave the explicit modern definition of what was later to be called Carlyle circle.
  Eves used the circle in the modern sense in one of the exercises of his book Introduction to the History of Mathematics (1953) and pointed out the connection to Leslie and Carlyle. Later publications started to adopt the names Carlyle circle , Carlyle method or Carlyle algorithm, though in German speaking countries the term Lill circle (Lill-Kreis) is used as well.  DeTemple used in 1989 and 1991 Carlyle circles to devise Compass-and-straightedge constructions for regular polygons, in particular the pentagon, the heptadecagon, the 257-gon and the 65537-gon. Ladislav Beran described in 1999, how the Carlyle circle can be used to construct the complex roots of a normed quadratic function.
caption: Carlyle's solution to Leslie's problem. The black line segment is divided in two segments in such a way that the two segments form a rectangle (green) being of equal area to another given rectangle (red).