## Considérations sur la théorie mathématique du jeu.

Lyon: Chez les Frères Perisse … et se trouve à Paris chez la veuve Perise … et chez Duprat … An II, 1802.

First edition, very rare, of Ampère’s first published work, a treatise on the theory of games “noted for its elegant and polished, though simple, application of the calculus to probabilities” (Catholic Encyclopedia). “It was on the strength of this paper that he was appointed to the professorship at Lyons and later [in 1809] to a post at the École Polytechnique in Paris” (Hutchinson’s DSB). In modern terms, Ampère discusses in this work the ‘gambler’s ruin problem.’ A gambler repeatedly plays a game which he has a certain probability of winning, and each time he wins or loses his fortune increases or decreases by a fixed amount; the problem is to determine the probability that the gambler eventually loses his initial fortune. By enumerating the sequences of games, Ampère determined the probability that a gambler with initial fortune *m *is ruined after *p *wins and *m *+ *p *losses, and summed the result for all values of *p*, using d’Alembert’s ratio test to prove convergence of the resulting series. “His first mathematical paper, ‘Considerations sur la théorie mathématique du jeu’ (1802), proved that a single player inevitably would lose in a game of chance if he were opposed by a group whose financial resources were infinitely larger than his own” (DSB). “Although in 1802 Ampère was not as yet concerned with the conceptual basis of the calculus as such, in his first memoir he conclusively demonstrated his thorough familiarity with power series and his skill in the manipulation of their coefficients” (Hofmann, p. 62). This work “is so rare that that it remained unknown to H. M. Walker, *Studies in the History of Statistical Methods *and it is not even mentioned in Smith’s *History of Mathematics*” (Ernst Weil, Cat. 33, no. 43). No copy on ABPC/RBH in the past 25 years. OCLC lists 11 copies in Europe (including both issues) but none in North America.

Ampère (1775-1836) grew up in the village of Poleymieux, near Lyon. Essentially an autodidact, Ampère’s childhood came to an end in 1789 with the outbreak of the French Revolution. When Lyon fell to the troops of the Republic, his father Jean–Jacques Ampère was tried and guillotined on 23 November 1793. He married Julie Carron, from a neighbouring village, on 7 August 1799. “He was able to make a modest living as a mathematics teacher in Lyon, where on 12 August 1800 his son, Jean–Jacques, was born. In February 1802 Ampère left Lyon to become professor of physics and chemistry at the École Centrale of Bourgen-Bresse, a position that provided him with more money and, more important, with the opportunity to prepare himself for a post in the new lycée that Napoleon intended to establish at Lyons. In April of that year he began work on an original paper on probability theory that, he was convinced, would make his reputation. Thus, everything concurred to make him feel the happiest of men. Then tragedy struck. Julie had been ill since the birth of their son, and on 13 July 1803 she died. Ampère was inconsolable, and began to cast about desperately for some way to leave Lyons and all its memories. On the strength of his paper on probability, he was named *répétiteur* in mathematics at the École Polytechnique in Paris” (DSB).

“Ampère’s first mathematics essay was a consideration of the probabilities associated with a gambler’s attempt to avoid elimination from a game of chance by losing all of a finite initial fortune. The problem may first have occurred to Ampère during parlor games at Poleymieux; he commented in a letter to Couppier [a friend from a nearby village] that he sometimes lost his entire allotment of tokens in the course of an evening’s games. The composition of the memoir spanned most of 1802, and Ampère refers to it in letters to Julie beginning in April of that year. At that point he asked her to send him all his mathematics reference books and notes, material he had left behind in Lyon and had not consulted seriously for five years. His renewed interest was sparked by a desire to impress the examining committee for the forthcoming mathematics position at the Lyon Lycée. His initial insight was the discovery of a ‘direct solution’ to a problem he had posed for himself seven years earlier.

“Ampère began by considering a gambler who is assumed to wager the same specific fraction of his original fortune in every game. That is, his initial resources are divided into *m* equal parts and he wagers one of these parts in every game. What is the probability that the gambler will be eliminated from further play after any specific number of games? In any particular sequence of games the gambler will be eliminated if he wins *p* games and loses (*m + p*) games, where *p* can be any possible number of victorious games. Ampère let *q* be the ratio between the probability to win a game and the probability to lose. These two probabilities can thus be written as *q*/(1 + *q*) and 1/(1 + *q*), respectively. The first and primary goal of Ampère’s essay is to calculate the probability of the gambler being eliminated as a function of *q*” (Hofmann, pp. 62-63).

Ampère finds that the probability of elimination after *p* wins and (*m + p*) losses is given by:

*q ^{p}*(1 +

*q*)

^{– (m + 2p)}

*m*(

*m*+ 2

*p*– 1)! /

*p*!(

*m*+

*p*)!

“Ampère then calculated the general probability of elimination by considering the above probability for all values of *p*. His goal was to determine the probability of elimination by calculating the value of the sum of the resulting infinite series …

“Ampère expanded each term involving (1 + *q*) using the binomial theorem. Initially he did not notice that the expansion must actually be done for two different cases, depending on whether *q* is larger or smaller than unity. Early in 1803, when Ampère finally submitted his memoir to the First Class of the Institut, Laplace noticed this oversight and mentioned it in a letter to him. The flaw caused Ampère considerable consternation; in a letter to Julie he remarked that he initially read Laplace’s comment as the *éloge* of his memoir. But he soon recovered when he noticed that only a few pages of the paper needed to be altered and learned that his brother-in-law Marsil was willing to quickly print up a corrected insert. In the altered version Ampère considered the two possible series and used an interesting recursion formula to show that in both cases the coefficient of each power of *q *vanished when the probability sums are written out for all values of *p*. This leaves for the value of the probability [of ruin] only the first term in each series. For *q* less than 1 this value is unity and for larger values of *q* the probability is *q ^{– m}*. In other words, if the odds of winning the game are less than the odds of losing (

*q*< 1), then the gambler will inevitably be ruined eventually …

“Ampère’s isolation in Bourg was one reason for his multiple revisions of the memoir and his hesitation to consider it complete. Couppier was too polite to make objections, if indeed he had any. Ampère suspected that Clerc [i.e., François Clerc, *Professor of Mathematics at the Bourg *École centrale*] *did not fully understand the paper and thus gave untrustworthy praise. These suspicions grew when Lalande read an early version of the paper following Ampère’s presentation to the Société d’Émulation de l’Ain on 26 July 1802. The Société was one of the few centers for intellectual life in Bourg and Ampère was pleased that his memoir resulted in his election as an associate member. Lalande made a guest appearance at the meeting, studied the memoir with Clerc, and gave a positive assessment that won Ampère membership. Ampère’s enthusiasm waned when Lalande suggested that he include some numerical illustrations of the solution so as to make the results more accessible. Ampère had good reason to believe that Lalande simply did not understand the derivation and wanted to see some simple examples. Ampère was slightly offended and felt that this would give his memoir the style of a schoolboy primer …

“Several aspects of Ampère’s first significant publication are worth noting. First, it is clear that at this point in his career he had a thorough knowledge of the technique of power series expansions based on the binomial theorem. He could manipulate these series at will and was clever at using recursion relations to transform them into interesting results. Second, he was aware of the pitfalls of divergent series. Laplace’s critique drove home this point conclusively. Third, Ampère demonstrated a preference for what he called ‘direct’ proof rather than proof ‘by induction’, that is, rather than show that a result held for various values of his parameter *p* and then assume that it held for all values, he went out of his way to derive the general result directly. This was a preference that was characteristic of him, and it would recur throughout his career” (Hofmann, pp. 64-66).

There are two issues of this work, of which ours is the second. In this issue, pp. 17-20 are cancels, these two leaves being corrected following the criticism of the first issue by Laplace described above.

Poggendorff I, col. 39. Hofmann, *André-Marie Ampère. Enlightenment and Electrodynamics*, 1995.

4to (270 x 212 mm), pp. [iv], 63, [1, blank], uncut and mostly unopened. Contemporary pink wrappers lined with printer’s waste on interior (very minor wear to extremities). Housed in a custom brown cloth box. A superb copy.

Item #5378

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Price:
$25,000.00
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