Daniel Bernoulli is one of the prominent members of the Bernoulli family from Basel, Switzerland, whose members excelled in various theoretical and applied scientific fields - especially in mathematics, probability theory, physics and medicine - in the second half of the seventeenth and in the eighteenth centuries.
The family originated from Antwerp, once under the domination of Catholic Spain. It emigrated in 1567 to Frankfurt, Germany, because of its Calvinist faith, and in the end settled in Basel in 1620.
Until Niklaus Bernoulli (1623-1708), the important wealth of the family came from the spice trade: Niklaus was himself a merchant and an officer of the city of Basel. But three of his sons, Jakob (16541705), Nikolaus (1662-1716) and Johann (1667-1748), took another route. Nikolaus was a painter and a member of the Municipality of Basel. Jakob studied philosophy and theology, and Johann medicine, but they both became renowned mathematicians, developing in particular differential and integral calculus and siding with Gottfried Wilhelm Leibniz in his quarrel with Isaac Newton - the phrase “integral calculus” is due to Jakob.Daniel Bernoulli, was born in Groningen on 8 February 1700, where his father had taught at the university since 1695, and died in Basel on 17 March 1782. He was Johann’s son and the cousin of Nikolaus (1687-1759) - alias Nikolaus I, just as his uncles were named Jakob I and Johann I by historians to distinguish them from the younger members of this dynasty, who had the same Christian names (see Figure 3) - a son of Nikolaus the painter. Daniel was a doctor in medicine and his cousin in jurisprudence but, like their predecessors, their main achievements were in mathematics and the sciences. Also like their predecessors, they travelled in Europe and were part of a network of the most eminent scientists of the time, and members of many scientific academies. (On the main members of the Bernoulli family, see O’Connor and Robertson 1997-98).
As regards the “moral sciences”, the works of Jakob, Nikolaus and Daniel are of outstanding interest thanks to their ground-breaking contributions to probability theory (Hacking 1975; Daston 1988; Hald 1990).
Jakob is the author of the celebrated Ars Conjectandi, mainly written between 1684 and 1689 but posthumously published in 1713. It was the first book written on the theory of probability, this “art of conjecturing”
Figure 3 The main members of the Bernoulli family
that he also proposed to call “stochastics”. (Pierre Remond de Montmort’s 1708 Essay d’analyse des jeux de hasard, the second edition of which was also published in 1713, was in fact written a long time after Jakob Bernoulli drafted his manuscript.) Jakob’s book elaborated considerably on Christiaan Huygens’s “De ratiociniis in ludo ale®”, a paper included as an appendix in Frans van Schooten’s 1657 book, Exercitationum mathematicarum. It deals, inter alia, with the specification of the concept of (mathematical) expectation proposed by Blaise Pascal and Huygens, the statement and proof of the binomial distribution and the (weak) “law of large numbers” (as Simeon-Denis Poisson called it later). Moreover, it introduced a first clear distinction between objective (frequentist or statistical) probabilities, where the frequencies of events are calculated from experiments or observations, and subjective (or epistemic) probabilities due to our imperfect knowledge and measuring the degree of our belief, or our “reason to believe” (Condorcet), in a statement or a proposition about things or events (see Hald 1990: 28-9, 245-7; and Daston 1988: ch. 4, 1994, for the evolution of these concepts in the eighteenth and nineteenth centuries). In addition, we owe Jakob Bernoulli also “the important distinction between probabilities which can be calculated a priori (deductively, from considerations of symmetry) and those which can be calculated only a posteriori (inductively, from relative frequencies)” (Hald 1990: 247). Finally, the fourth and unfinished part of Ars Conjectandi, which contains the statement and proof of the law of large numbers, is entitled “Usum & applicationem pr®cedentis doctrin® in civilibus, moralibus & ωcon- omicis” (“The use and application of the previous doctrine to civil, moral and economic affairs”).
It posed the fundamental question of the use of the mathematical developments of “expectatio”, made for the games of chance, to the more traditional field of the “probability” of judgements developed, for example, in jurisprudence - that is, in modern parlance, the application of probability theory to social and economic matters. Jakob could not bring his project to an end but considered it as “the main part” of his work (to Leibniz, 3 October 1703, in JEHPS 2006: 5). This part was to inspire Nikolaus’s approach and, some decades later, Condorcet’s research programme.Nikolaus who, contrary to the legend, was not the editor of Jakob’s book (Kohli 1975; Yushkevich 1987), continued the work of his uncle. In his 1709 thesis, Dissertatio inau- guralis mathematico-juridica de usu artis conjectandi in jure, and in “Specimina artis con- jectandis, ad quaestiones juris applicatae” - an abridged version of his thesis, published in 1711 in a supplement of the Leibnizian Acta Eruditorum - he used probability theory to deal with juridical and economic questions such as the reliability of witnesses and of suspicions, marine insurance, the probability of human life, life annuities, or the problem of the “absent” (after how many years can an absent person be considered as dead?) (see, for example, Hald 1990: ch. 21).