Centres a distance

so called Watt's Curve is a tricircular plane algebraic curve of degree six. It is generated by two equal circles (radius , centres a distance apart). A line segment (length ) attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as the circles rotate (for more on Watt's Curve see [1]).

Figure 2.2: Watt's Curves for different values of and .
The Watt's Curve inspired Chebyshev to deal with the following: determine the parameters of the mechanism so that the maximal error of the approximation of the curve by the tangent on the whole interval is minimized.
In 1853 , Chebyshev published his first solutions in his "Théorie des mécanismes, connus sous le nom de parallélogrammes". He tried to give mathematical foundations to the theory of mechanisms, because practical mechanics did not succeed in finding the mechanism with the smallest deviation from the ideal run. Other techniques did not work either. Poncelet's approach did work, but only for specific cases.
Chebyshev wanted to solve general problems. He formulated the problem as follows (translated word-by-word from French):
To determine the deviations which one has to add to get an approximated value for a function , given by its expansion in powers of , if one wants to minimize the maximum of these errors between and , being an arbitrarily small quantity.
The formulation of this problem is the start of approximation in the uniform norm.