Studies in the History of Probability and Statistics. XXIII

SUMMARY This paper discusses one of D. Bernoulli's memoirs (1770-1) in which he deduced the 'De Moivre-Laplace' limit theorems, nevertheless credited to De Moivre. The memoir is described in ? 2 while ? 1 attempts to sum up Bernoulli's contributions more generally. 1. GENERAL Between 1738 and 1778 D. Bernoulli (1700-82) published seven probabilistic memoirs. The essence of these memoirs, except the memoir to be described in ?2, is given by Todhunter (1865). The memoirs contain solutions of important problems in demographic statistics (political arithmetic) and astronomy obtained with the help of probabilistic ideas and methods. As to probability and mathematical statistics proper, Bernoulli was the first to use systematically differential equations for deducing a number of formulae, one of the first to raise the problem of testing statistical hypotheses and the first to introduce ' moral expectation' (due to Cramer) and to study random processes. He is also to be credited, after Lambert, for the second introduction of the maximum likelihood principle (Bernoulli, 1961). In summary, it may be argued that D. Bernoulli's influence upon Laplace, especially concerning applications of probability, was comparable to that of De Moivre. The account of Bernoulli's memoirs given by Todhunter could well be modernized but the present paper is restricted to the description of the 1770-1 memoir, the second part of which remained unnoticed by Todhunter. For this and other reasons, Todhunter's account of the memoir is unsatisfactory and until now no one has remarked on the appearance in this memoir of the 'De Moivre-Laplace' limit theorems and of the first published small table of the normal distribution. Had these limit theorems been noticed in Bernoulli before, they possibly would not now be called only after De Moivre and Laplace.