3 On the Work of Adrain in the Theory of Errors

of dissertation. Moscow, 1967. Published as a manuscript. Inst. Istorii Estestvoznania i Tekhniki The rise of the theory of errors as a discipline belonging to experimental quantitative science is connected with the spectacular successes of instrumental astronomical observations of the 17 century and with the beginning of the epoch of meridian arc measurements (17 – 18 centuries). The advances in, and subsequent new problems of astronomy put onto the agenda various issues concerning the treatment and evaluation of the precision of instrumental observations; and the calculation of arcs of triangulation and the determination of the figure of the Earth demanded, furthermore, the ability to treat redundant systems of linear algebraic equations. The theory of errors thus naturally included a number of problems pertaining to the treatment and estimation of the precision of direct and indirect observations. The treatment of direct observations led to the justification, of qualitative, and then of a quantitative, stochastic nature, of the long prevalent arithmetic mean. The treatment of indirect measurements demanded the development of a number of algorithms which were independent of probabilistic considerations; later, however, Gauss derived the method of least squares by issuing from the principle of maximum likelihood. In the process of my work, I became acquainted with a large number of sources in mathematics, astronomy and geodesy. A large part of this literature remained little known or completely ignored. This very situation can explain to a certain degree why I succeeded in getting unexpected important results and among them 1. The description of the work of the forgotten American mathematician, Robert Adrain, who, in particular, published two derivations of the normal distribution in the theory of errors a year before Gauss [or at least not later than he]. 2. The establishment of the priority of Ernst Abbe in considering the chi-squared distribution. 3. A critique of the derivation of the normal distribution as given by Gauss in 1809. The literature on the theory of errors and the method of least squares paid great attention to the justification of the normal law. That problem occupies only a secondary place in my work. Indeed, this is perhaps a separate topic in which, first of all, it is necessary to examine the theory of elementary errors and the central limit theorem. And a large number of other derivations turned out to be dead ends and today represent only mathematical exercises. My work consists of three chapters, an appendix, and an addendum. Chapt. 1 investigates the early application of the arithmetic mean in approximate calculations, games of chance, in astronomy and theory of probability proper. That it was used in antiquity has been known for a long time. However, the connection of this fact with the stochastic nature of the mean was not noted, – and precisely this fact is stressed throughout almost the entire chapter. Furthermore, a large amount of factual material is collected there. The science of mean values, including the arithmetic mean, already existed in the Pythagorean school, and the latter had been widely applied in antiquity in approximate calculations of the areas of figures and volumes of bodies. The formula for the calculation of the area of a quadrangle as the product of half the sums of opposite sides was used in ancient Babylonia either for not exact rectangles or when the opposite sides were unequal because of the ruggedness of the terrain (A.A. Vaiman). This means that the arithmetic mean was invoked to compensate both for the lack of strict applicability of the formulas (models) and for the systematic (not random, as nowadays) errors of measurement. Commercial practice aided the spread of the idea of the arithmetic mean, and claims were even made that in this sense the sphere of economics was primary. According to Leibniz, the principle of equal allowance of equally tenable assumptions was the fundamental hypothesis in the contemporaneous theory of probability. I believe that Leibniz thought about the origin of the first stochastic concepts, probability and expectation, and that his idea provided the possibility of the later subjective understanding of probability. The history of games of chance contains evidence of the widespread knowledge of the idea of the arithmetic mean which served as an intuitive statistical indicator of the totality of possible outcomes and perhaps of the appropriate expectation. During the epoch of meridian arc measurements the arithmetic mean began to be used as an universal estimator in all stages of their treatment. In the same period and even earlier the first qualitative statements about the benefits of applying it had appeared (Picard, Condamine). In 1809 Gauss postulated the principle of the arithmetic mean and essentially used it in his derivation of the normal distribution. This attracted attention to his postulate and a number of authors tried to reduce it to a “more obvious” premise. These attempts were, however, of a purely deterministic nature and I do not dwell on them. In the first half of the 18 century Cotes applied an analogy from mechanics (the center of gravity) for justifying the arithmetic mean, and Lambert tried to substantiate it on a stochastic basis. In Chapt. 2, in connection with the history of the treatment of direct observations, I studied the work of Galileo and Lambert, then dwelt on the appropriate memoirs of Simpson and Lagrange. Also there I investi-