Carlyle circle

However there are practical problems for the implementation of the procedure; for example, it requires the construction of the Carlyle circle for the solution of the quadratic equation x2 + x − 214 = 0.

[3] 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).

[4] 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:[5] To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle.In 1867 the Austrian engineer Eduard Lill published a graphical method to determine the roots of a polynomial (Lill's method).

[7] 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.

Ladislav Beran described in 1999 how the Carlyle circle can be used to construct the complex roots of a normed quadratic function.