Robert Adrain and the Method of Least Squares

TLDR: Adrain's derivations and applications of the method of least squares in modern terminology were discussed in this article, and the question of the originality of Adrain's work was treated in Section 7.

Abstract: The method of least squares is a very important numerical technique of applied mathematics where it is used for the adjustment of observations, statistical estimation, curve fitting, etc. Publications on the method by A. M. Legendre, Robert Adrain and C. F. Gauss originally appeared in the first decade of the nineteenth century. The rival claims of Legendre and Gauss for priority of discovery generated considerable controversy in the years following. For a long time the relatively unavailable publications of Robert Adrain on the method remained comparatively unknown, but in 1980 they were reprinted in Stigler [1 ; Vol. 1]. The primary purpose of this paper is to present Adrain's derivations and applications of the method of least squares in modern terminology. A sketch of Adrain's mathematical career is given in Section 2. A brief history of the adjustment of observations in the eighteenth century and of the method of least squares is given in Section 3. A surveying problem which was the stimulus for R. Adrain's work on least squares is given in Section 4. Adrain's derivations of the normal law and of the method of least squares are discussed in Section 5 and his applications of the method in Section 6. Finally the question of the originality of Adrain's work is treated in Section 7.