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