The smallpox vaccine: the dispute between Bernoulli and d’Alembert and the calculus of probabilities

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The urgent and dramatic need to introduce and promote a vaccine against smallpox, a scourge for society at the end of the 1700s, provided the occasion for a lively debate between Daniel Bernoulli and Jean Le Rond d’Alembert. This article discusses the motivations and arguments of the dispute, illustrating the probabilistic model proposed by Bernoulli to justify the greater “reasonableness” of the campaign in favour of the vaccine, and the objections raised by d’Alembert. The aim of this analysis is, beginning with a reconstruction of the contributions of the two authors, to show how the newly founded “art of conjecture” was the object of divergent interpretations, from the characterisation of its theoretical principles, such as the concept of expectation, to the question of its legitimacy in applications as a “guide to practical living”.