Summary: This paper is concerned with the historical development of a traditional procedure for determining appropriate values for the parameters defining a linear relationship. This traditional procedure is variously known as the minimax absolute residual, Chebyshev, or \(L_{\infty}\)-norm procedure. Besides being of considerable interest in its own right as one of the earliest objective methods for estimating the parameters of such relationships, this procedure is also closely related to Rousseeuw’s least median of squared residuals and to the least sum of absolute residuals or \(L_{1}\)-norm procedures. The minimax absolute residual procedure was first proposed by Laplace in 1786 and developed over the next 40 years by de Prony, Cauchy, Fourier, and Laplace himself. More recent contributions to this traditional literature include those of de la Vallée Poussin and Stiefel. Nowadays, the minimax absolute residual procedure is usually implemented as the solution of a primal or dual linear programming problem. It therefore comes as no surprise to discover that some of the more prominent features of such problems, including early variants of the simplex algorithm are to be found in these contributions. In this paper we re-examine some of the conclusions reached by Grattan–Guinness (1970), Franksen (1985) and Grattan–Guinness (1994) and suggest several amendments to their findings. In particular, we establish the nature of de Prony’s geometrical fitting procedure and trace the origins of Fourier’s prototype of the simplex algorithm.