Lagrange and the pension funds for widows (Q2866539)

The title of the paper is misleading, since it contains the following: an introduction which sets the background, a comparison between Euler's and Lagrange's method to solve an actuarial problem concerning the funds for widows and orphans, Lagrange's meager interest in the calculus of probability, and Lagrange as member of the \textit{Académie des sciences}.NEWLINENEWLINE As a matter of fact, only until p.~78 the author is in topic. The introduction (pp.~39--54 [Section 1--4]) tells us how the necessity arose to build up a fund for the widows of the Lutheran pastors during the 17th and 18th century. While in England such types of funds were private, in the German countries they were public. In fact the noun \textit{statistic} derives from the German \textit{Statistik} = what concerns the state [Staat].NEWLINENEWLINE The necessity to help the widows was urgent, because a female in this period had few chances to obtain a job, and even if she was able to find one, she would be underpayed. Soon, the widows became poor, losing their social status. In order to change this, the husband subscribed to a fund to grant a pension to his wife in case of his death. In a while, many types of funds arose, but without a serious mathematical investigation on its functioning. For example, Kritter demonstrated that a famous fund, the so-called \textit{Calenberg fund}, was exposed to fail. What really happened.NEWLINENEWLINE Around 1767, Euler wrote an essay entitled \textit{Sur le rentes viagères}, aimed to fix the general conditions so that a fund works well, balancing taking and spending. The subject is treated in great detail on pp.~54--60. As is easy to guess, Euler's solution relies strongly on the calculus of probabilities.NEWLINENEWLINE In Prussia, Frederick II, the Great, established a fund for widows at the end of 1775: the \textit{Königlich Preußische Allgemeine Witwen-Verpflegungs-Anstalt}. Lagrange had the opportunity to read the regulations of that institute, being in doubt about its real efficacy. For this reason, he wrote an unpublished memoir, \textit{Solution d'un problème sur le rentes viagères}, now in the Bibliothèque de l'Institut de France, ms 916. The solution is almost identical to that of Euler, apart from the amount of the interests. The author treats the question on pages 61--78. In fact, Lagrange exhibited three solutions which are three variations of the same basic idea.NEWLINENEWLINE The author underestimates the closeness of Euler's solution to that of Lagrange, stressing that Lagrange was not very interested in the calculus of probability itself. This is right, but also in his solution(s) the calculus of probability is necessary. Euler's approach is very similar to that of Lagrange, changing only the historical and sociological context. Euler was in St. Petersburg, Lagrange was at the court of Frederick the Great.NEWLINENEWLINE At this point, the author shifts abruptly the focus: firstly she talks about Lagrange and his relation towards the calculus of probability (pp.~78--87 [Section 9]), to pass to Lagrange's stay in Paris (pp.~87--92 [Section 10]), where he investigated the necessary amount of food to feed France, based on the data of the army, of the average consumption in Paris, and of the average consumption in France.NEWLINENEWLINE On the following pages 93--97 [Section 11], we may appreciate Lagrange as a reader of Duvillard (1755--1832) and of Bicquilley (1738--1814), who was the first to envisage a mathematical theory for economical purposes (p.~97, footnote 2).NEWLINENEWLINE The paper ends (pp.~97--102 [Section 12]) with a listing of papers in economy, statistics or actuarial mathematics in Lagrange's own library.NEWLINENEWLINE What is strange is that the author in the final part does not recall the problem from which she started: how to build a fund for widows and orphans. From this point of view, the text is very rich of information about Lagrange, but not all concerns the main problem, or not always the links are clear. For example, what have the provisions for France to do with the funds for widows?