Probability and exams: The work of Antonio Bordoni

Abstract

The Italian mathematician Antonio Bordoni is mainly known for his adherence to the Lagrangian approach to the foundations of calculus and for his role in creating an important school of mathematics. In this paper, I consider his less known work on the application of probability to design exams and analyze their outcomes. Within this framework, he obtained in 1837, as Mondésir and Poisson, the result that would lead Catalan to formulate his “new principle” of probability (Jongmans and Seneta, 1994). Moreover, in 1843, Bordoni also gave an early complete proof of the finite rule of succession.

Sommario

Il matematico italiano Antonio Bordoni è noto soprattutto per l'adesione all'approccio lagrangiano ai fondamenti dell'analisi e per il ruolo nella creazione di un'importante scuola matematica. In questo lavoro considero contributi meno noti di Bordoni sulle applicazioni del calcolo delle probabilità alla progettazione degli esami ed all'analisi dei loro esiti. In questo àmbito egli ottenne nel 1837, al pari di Mondésir e Poisson, un risultato che porterà Catalan a formulare il suo “nuovo principio” del calcolo delle probabilità (Jongmans and Seneta, 1994). Inoltre Bordoni fornì nel 1843 una delle prime dimostrazioni complete della regola finita di successione.

Introduction

The figure of Antonio Bordoni (1788-1860)—who taught Geodesy, Higher Calculus (Calcolo sublime) and Hydrometrics during his long stay at the University of Pavia—has been the subject of historical studies (Bottazzini, 1989, 2007), which have highlighted his unconditional support of the Lagrangian approach to calculus, even after he had had the opportunity to read Cauchy's papers. Bordoni also influenced, more or less directly, Brioschi, Beltrami, Casorati, and Cremona, who were among the protagonists of mathematical research in Italy during the second half of the 19th century and had a high esteem of his value as a mathematician.¹ Moreover, they insisted in recalling his main results (Rosso, 2017),² also stressing that his teaching duties had limited his penchant for pure mathematics, since he was charged to teach engineering and architecture students, focusing on mathematics that could be applied in their professional lives. His scientific production covers pure and applied mathematics: from differential geometry, finite difference calculus and rational mechanics³ to geodesy and applied hydrodynamics, but his best known work is the treatise Lezioni di calcolo sublime in two volumes (Bordoni, 1831), a tribute to Lagrangian calculus.

In this paper, I will study his less known work on probability and its applications, consisting of a monograph and two memoirs.⁴ Bordoni presented the basic elements of probability theory in his lectures on Higher Calculus together with exercises formalized through finite difference equations.⁵ In this way he ideally continued the tradition opened by Vincenzo Brunacci (1804) who, however, had never published papers on probability. Bordoni's research was concerned with a peculiar application which, to my knowledge, had not been considered before: the design and the analysis of the outcome of examinations. Essentially, he tried to answer these questions: How can probability be used to make examination procedures fair? What can we conclude about the effective preparation of a student, given the outcome of his/her exam?⁶

There are documents that testify to the use of random mechanisms to select the questions in the exams (Ferraresi, 2015, pp. 496-497)⁷ at Bologna and Pavia,⁸ and it is likely that Bordoni himself, as a student, had experienced these procedures. Part of Bordoni's results rely upon an admittedly rigid application of the urn scheme since he replaced an urn containing black and white balls with an urn containing all admissible questions: a student can answer part of them—white balls—while he ignores the answer to the remaining questions (black balls). Despite this serious limitation, whose importance will be discussed in Section 4, there are two problems tackled by Bordoni that deserve attention. First, in his monograph (Bordoni, 1837, p. 114)—which we analyze in Section 2 by commenting some propositions—he proved a symmetry property of the hypergeometric distribution that coincides with a result published in the same year by Émile Mondésir⁹ (1837) and by Siméon-Denis Poisson (1837); this very property would lead Catalan to formulate a principe de probabilités (Catalan, 1877) whose history has been studied by Jongmans and Seneta (1994) and by Sheynin (2002) (see also (Bair, 2014) for the pedagogical aspects of this principle). A second result—analyzed in Section 3 with the rest of the memoir (Bordoni, 1843a)—is one of the early proofs of the finite rule of succession, first proved by Prevost and L'Huilier (1799) and then to be rediscovered more and more times again (Zabell, 1989, 2005). In both cases, the proofs given by Bordoni depend upon elementary summation techniques, which he frequently employed in his work and are discussed in the Appendix to this paper.

For ease of the reader, we note that general references on the topics discussed in this paper are (Dale, 1999), (Hald, 1998, 2007), and (Zabell, 2005).