Studies in the History of Probability and Statistics. XVIII Thomas Young on Coincidences

1. Almost as soon as the calculus of probabilities began to take a definite shape mathematicians were concerned with the use of probabilistic ideas in reconciling discrepant observations. James Bernoulli's Ars Coniectandi was published in 1713. Within 9 years we find Roger Cotes (1722), in a work on the estimation of errors in trigonometrical mensuration, discussing what would nowadays be described as an estimation problem in a plane. Let p, q, r, s be four different determinations of a point o, with weights P, Q, R, S which are inversely proportional to distance from o (pondera reciproce proportionalia spatiis evagationum). Put weights P &tp, etc., and find their centre of gravity z. This, says Cotes, is the most probable site of o. (Dico punctum zfore locum obiecti maxime probabilem, quipro vero eius loco tutissime haberi potesl.) Cotes does not say why he thinks this is the most probable position or how he arrived at the rule. 2. According to Laplace this result of Cotes was not applied until Euler (1749) used it in some work on the irregularities in the motion of Saturn and Jupiter. Further attacks on the problem of a somewhat similar kind were employed by Mayer (1750) in a study of lunar libration and by Boscovich (1755) in measurements on the mean ellipticity of the earth. There was evidently a good deal of interest being taken in the combination of observations about the middle of the eighteenth century. The ideas, as was only natural, were often intuitive and sometimes obscurely expressed, but the fundamental questions seem to have been asked at quite an early stage. For example, Simpson (1757) refers to a current opinion that one good observation was as accurate as the arithmetic mean of a set, and although from that point onwards a series of writers argued for the arithmetic mean, Laplace (1774), in his first great memoir, was clearly aware that for some distributions of error there were better estimators such as the median. 3. Simpson (1756, 1757) was the first to introduce the concept of distribution of error and to consider continuous distributions. But like most of his contemporaries he regarded it as inevitable to impose two conditions: first, the distributions must be symmetrical; secondly, they must be finite in range. Lagrange reproduced Simpson's work without acknowledgement in a memoir published between 1770 and 1773, but Lagrange's contributions are more of analytical than of probabilistic interest. 4. Daniel Bernoulli was born in 1700 and lived to be 82. Throughout his productive life he made contributions to the theory of probability and although his mathematical methods are not now of much importance, the originality of his thinking on such matters as moral expectation entitles him to a permanent place among the founders of the subject. In particular, the memoir on maximum likelihood reproduced in the following pages is astonishingly in advance of its time. The author was 78 when it was published and it appears that he excogitated the basic ideas for himself without reference to previous writings. The memoir may, in actual fact, have been written rather earlier. Laplace's