ABSTRACT
If we let the knots in the Newton–Cotes formulas of Section 1.1 vary freely and compute the weights accordingly, the question raised by C.F. Gauss [1814] was the following: What is the maximum degree of polynomial exactness that can be achieved and how are the knots to be selected in order to realize this optimum? In fact, Gauss answered this question by means of the theory of continued fractions associated to hypergeometric series (see, e.g., W. Gautschi [1981] for a proof in today’s nomenclature).
