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### Keywords:

• Art of conjecturing;
• probability without chance;
• Huygens;
• Jacob Bernoulli, De Moivre, Bernoulli's fundamental theorem or weak law of large numbers

### Summary

The Tercentenary of the publication of Jacob Bernoulli's Ars Conjectandi (The Art of Conjecturing) provides an opportunity to look at the origins of mathematical probability from Jacob Bernoulli's point of view. Bernoulli gave a mathematically rigorous proof of what has come to be called the weak law of large numbers, relevant to discovering ratios of unknown factors through sampling. The Art of Conjecturing was a bridge between the mathematics of expectation in games of chance as found in Huygens's On Reckoning in Games of Chance and mathematical probability as found in Abraham De Moivre's The Doctrine of Chances. This paper looks at the conceptual context as well as the mathematics of Bernoulli's book.

### Introduction

The Tercentenary of Jacob Bernoulli's Ars Conjectandi (The Art of Conjecturing) in 2013 provides an appropriate moment to focus on Jacob Bernoulli (1654–1705) and on his book. In this paper, I explain why the publication of Ars Conjectandi in 1713 is appropriately considered the start of mathematical probability. The most obvious reason for beginning mathematical probability with The Art of Conjecturing is that at the end of the book, Bernoulli gave a rigorous proof of an early fundamental theorem of mathematical probability which has become known as the weak law of large numbers. Another reason, which is not obvious to many of today's probabilists and statisticians, is that before Bernoulli's work, there existed a mathematics of games of chance but that mathematics did not involve probability—not the Latin word probabilis, not relative frequencies and not degrees of certainty. As William Feller writes, ‘the modern student has no appreciation of the modes of thinking, the prejudices, and other difficulties against which the theory of probability had to struggle when it was new' (Feller, 1957, 2). Or, third, it could be said that mathematical probability began with Ars Conjectandi in the sense that the work inspired a flurry of activity among mathematicians after little attention had been paid to the mathematics of games of chance between Christiaan Huygens' De ratiociniis in ludo aleae (On reckoning in games of chance) published in 1657 and Pierre Rémond de Montmort's Essay d'Analyse sur les jeux de hazard, published in 1708, more than 50 years later—recognising that it was reports of the contents of the manuscript of Ars Conjectandi at the time of Bernoulli's death in 1705 that led Montmort to write his Essay d'analyse and then led to the work of Abraham De Moivre. Bernoulli died before he was able to finish his book, but the proof of his fundamental theorem raised the hope that mathematical probability could be applied to a much broader array of phenomena than games of chance, provided that ratios of alternative cases in real world situations, much like tokens in an urn, could be determined a posteriori or by experience.

### The Contents of the Volume Containing The Art of Conjecturing

When Bernoulli died in 1705, he left the manuscript of The Art of Conjecturing incomplete, but what he had completed was published 8 years later in 1713 from the carefully copied manuscript that had been held in the meanwhile by his immediate family. The work was not, as is frequently alleged, edited by his nephew Nicolaus Bernoulli (Yushkevich, 1987, 301). As published in 1713, Ars Conjectandi had four parts. Part I reproduced Huygens' On Reckoning in Games of Chance along with Bernoulli's notes. Part II covered combinations and permutations and Part III the application of combinations and permutations to games of chance. Part IV applied the mathematics of the preceding parts by analogy to civil, moral and economic matters. In Chapter 5 of Part IV, Bernoulli demonstrated his theorem showing how ratios of cases for and against a given outcome might be learned a posteriori or from experience.

After Ars Conjectandi itself (239 pages), the publisher included in the same bound volume a five-part Treatise on Infinite Series (66 pages), the parts of which had been defended by Bernoulli's students in their M.A. oral exams: Part I had been defended by Jacob Fritz in 1689; Part II was defended by Jerome Beck in 1692; Part III was defended by Jacob Hermann in 1696; Part IV was defended by Nicolaus Harscher in 1698; and Part V was defended by Nicolaus Bernoulli (Jacob Bernoulli's nephew) in 1704. Finally, the volume included Lettre à un Amy sur les Parties du Jeu de Paume (Letter to a Friend on Sets [or Parts] in Court Tennis, 35 pages), which was found unpublished in Bernoulli's papers after his death. This letter, which serves in part to compensate for the lack of concrete examples of using ratios derived from experience at the end of Part IV of Ars Conjectandi, is related to theses on tennis in Latin which Bernoulli had proposed for disputation on 12 February 1686 and to an entry in Bernoulli's research notebook Meditationes, where Bernoulli had noted that in using a posteriori ratios to estimate the ratio of cases for and against each player there is less danger of erring the more times one has observed the outcomes. The title Ars Conjectandi indicates that what Bernoulli hoped to produce was not a science, but a practical art (in the sense of the Greek word techne, representing a humanly created instrument), one which might enable a person to make wise decisions, while admitting that certainty is impossible.

### The Historical Background to The Art of Conjecturing

As a discipline, probability is now part of mathematics. It was first axiomatized by A. N. Kolmogorov in 1933. Like Euclidean geometry, mathematical probability can prove theorems that follow from its axioms and other principles. In the late 17th and early 18th centuries, the discipline of mathematical probability began to be assembled from parts that previously had little to do with each other. One part was the mathematics of expectation in games of chance, represented most prominently by Christiaan Huygens' On Reckoning in Games of Chance (1657). Another part was the mathematics of combinations and permutations. Resources also came from algebra, the mathematics of infinite series, the use of logarithms, the drawing of curves and so forth.

Some of these mathematical resources had previously been contained in books of commercial arithmetic, which, in turn, had roots in mathematics written in Arabic. This connection is now most familiar through the 13th-century work of Leonardo of Pisa, also known as Fibonacci, who reported in his Liber Abaci that he had learned much of what was contained in his book in northern Africa, where his father had been employed. Although not so well known, there were many other occasions of interaction between Islamic, Jewish and Christian mathematics in late mediaeval and early modern times, not least in the context of commerce. Some of the cultural context of the mathematical resources was imported into the European Christian context along with the mathematics itself.

Huygens' On Reckoning in Games of Chance had been appended to Frans van Schooten's Exercitationes mathematicae, which included other short works such as a hundred propositions of arithmetic and geometry, Apollonius of Perga on plane loci and a work on conic sections. Both Huygens and Bernoulli had studied commercial mathematics, on the one hand, and works of classical Greek mathematics, such as those by Euclid and Apollonius, on the other. Jans Jansz Stampioen de Jonge, Huygens' tutor, listed among the books he was prepared to use in his teaching works by Apollonius, Witelo, Kepler, Descartes and Vieta, but also the Arithmetica of Antoni Smiters and Die Coss of Christoff Rudolff (in the edition of Michael Stifel). Jacob Bernoulli used the same version of Christoff Rudolff's Die Coss in teaching Paul Euler, the father of Leonhard Euler (Heefer, 2007). Huygens' On Reckoning in Games of Chance and Bernoulli's The Art of Conjecturing straddled the borderline between pure or abstract, and applied, or what Bernoulli himself called concrete mathematics. In The Art of Conjecturing, Bernoulli derived and demonstrated his results, as would be the case in pure mathematics, but ultimately he hoped that the work would be of use in the real world.

In theses he proposed for a disputation to take place on 12 February 1686, Bernoulli's Miscellaneous Theses XI and XII stated:

• The concrete mathematical disciplines such as physics, medicine, astronomy, optics, statics, ballistics (and if you wish astrology), etc., add to abstract mathematics only certain principles, as foundations. [These principles] are partly proved elsewhere and partly drawn from experience alone. On these principles one may reason further with no less geometrical rigour than one reasons in abstract mathematics on the basis of common notions or innate axioms. Thus physics presupposes the laws of motion; medicine supposes the fabric of the human body; astronomy the fabric or system of the world; astrology the influx of the stars on sublunar things and that the fate of men, of cities, and regions, depends on the configuration of the heaven that obtains when they emerge or take on their original parts; catoptrics assumes that the angles of incidence and reflection are equal; dioptrics that the sines of the angles of incidence and refraction are proportional; statics that the moments increase with distance from the fulcrum; ballistics that the spaces traversed by a falling weight are as the squares of the times.

• Whence it is clear that the certitude of these sciences depends uniquely on the certitude of these principles and not on the mode of forming conclusions, all of which should be deduced from the principles by the most evident reasoning. This is the reason why abstract mathematics is of invincible certainty, why astrology is vain and futile, while the others are of a middle certainty between these two. This is because such are the principles on which they are erected (Bernoulli, 1744: 233–234; my translation).

These theses from Bernoulli's 12 February 1686 disputation are important for understanding the conception of the art of conjecturing Bernoulli proposed. In his essentially Aristotelian/Euclidean conception of autonomous scientific disciplines in deductive or demonstrative format, the principles are not all thought to be self-evident or necessary, but rather encompass many principles that are empirical in origin and that may be uncertain. Most of the proof of Bernoulli's fundamental theorem at the end of Part IV of The Art of Conjecturing consists of purely mathematical lemmas, which, after the proof, he then interpreted as representing possible outcomes of repeated experiments. Thus, The Art of Conjecturing included parts that were abstract and other parts that were concrete, adding concrete principles to abstract mathematics, but, because of the incomplete state of the work, it lacked nearly all of the empirical principles relating to civil, moral and economic problems that rightly should have been included.

Much has been written about the transition from qualitative logico–verbal Aristotelian sciences in the high and late Middle Ages to a quantitative and mathematical approach to physical sciences in the 17th century. Bernoulli's proof of his fundamental theorem provides an alternative way in which mathematics might assist natural sciences, namely by providing a complex mathematical model—with which the mathematician is able to draw inferences—to compare to real world structures. Thus, the scientist may investigate whether real world distributions are isomorphic to distributions of terms in expansions of binomials taken to high powers.

Bernoulli defined the art of conjecturing in terms of probabilities. In Chapter I of Part IV of The Art of Conjecturing, Bernoulli wrote:

Probability, indeed, is degree of certainty, and differs from the latter as a part differs from the whole. Truly, if complete and absolute certainty, which we represent by the letter a or by 1, is supposed, for the sake of argument, to be composed of five parts or probabilities, of which three argue for the existence or future existence of some outcome and the others argue against it, then that outcome will be said to have 3a / 5 or 3/5 of certainty. One thing therefore is called more probable than another if it has a larger part of certainty, even though in ordinary speech a thing is called probable only if its probability notably exceeds one-half of certainty (Bernoulli, 2006, 315-6; Bernoulli, 1713, 211).

Then in Chapter II, Bernoulli defined ‘conjecture':

To conjecture about something is to measure its probability. Therefore we define the art of conjecture, or stochastics, as the art of measuring the probabilities of things as exactly as possible, to the end that, in our judgements and actions, we may always choose or follow that which has been found to be better, more satisfactory, safer, or more carefully considered. On this alone turns all the wisdom of the philosopher and all the practical judgement of the statesman.

Probabilities are assessed according to the number together with the weight of the arguments that in any way prove or indicate that something is, will be, or has been. By weight I mean probative force (Bernoulli, 2006: 317–318; Bernoulli, 1713: 213).

The change in the meaning of the word ‘probability' from what it means for Bernoulli to meaning relative chances of happening is first apparent in Abraham De Moivre's The Doctrine of Chances (1718, 1738, 1756), where De Moivre opens the work with the definition:

• The Probability of an event is greater or less, according to the number of Chances by which it may happen, compared with the whole number of [1718 edit.: the number of all the] Chances by which it may either happen or fail.

• Wherefore, if we constitute a Fraction whereof the Numerator be the number of Chances whereby an Event may happen, and the Denominator the number of all the Chances whereby it may either happen or fail, that Fraction will be a proper designation of the Probability of happening. (De Moivre, 1756, 1–2. Point 1, with the differences noted, is already in the 1718 edition, whereas point 2 is not.)

Thus, the quantification of probability as degree of certainty enters Part IV of Bernoulli's The Art of Conjecturing, but the quantification of probability as relative chances does not appear until De Moivre's The Doctrine of Chances. Mathematics was to be the preeminent tool of Bernoulli's art of conjecturing because it enabled one to reason infallibly from premises or principles to conclusions, but this did not require or guarantee that the premises in themselves be certain. For Bernoulli, the probability at issue was a property of opinions or propositions rather than simply of events (cf. Gillies, 1994, 1408). For instance, to take examples used by Bernoulli, in the proposition ‘It is probable that Maevius is guilty of the murder of Titius', the word ‘probable' is being used in an epistemic sense and signifies a relatively high degree of certainty. By contrast, in the proposition ‘The probability of being killed by lightning within the next year is less than 1 in 1,000,000', the word ‘probable' is being used in a frequentist sense and could be replaced by ‘chances'.

In the conception ultimately deriving from Aristotle still present in Bernoulli's work, ‘science' (Greek: episteme, Latin: scientia) is defined as knowledge of conclusions demonstrated on the basis of true principles. As in Euclidean geometry, these principles might be axioms (principles accepted universally as self-evident) or they might be postulates (principles accepted by practitioners of the given discipline, such as the parallel postulate for Euclidean geometry), or they might simply be definitions. In empirical sciences like astronomy, the majority of the principles would be based on sense, memory and experience, as can be understood from Bernoulli's miscellaneous theses about concrete mathematics quoted earlier. An empirical principle in astronomy might be that the Moon circles through the stars once a month on a path intersecting the ecliptic at an angle of about five degrees.

In contrast to sciences were other practical disciplines called ‘arts'. Such disciplines, for instance ethics, cannot demonstrate their conclusions but at best make probable arguments as Aristotle argued at the beginning of his Nicomachean Ethics (1094b12–27; cf. Sylla, 1990). When he conceived of The Art of Conjecturing, Bernoulli's goal was to apply mathematics, considered by many the most certain of the sciences, to subject matters like ethics, politics and economics, in which exact science was considered beyond reach, in order to increase the degree of probability of practical conclusions. The result would be an art enabling better decisions than could be made without mathematical assistance, but not a certain science.

If probability enters Part IV of The Art of Conjecturing in only one of the senses it has today, its role is even more restricted in Parts I–III. Some earlier thinkers had assumed that there were factors affecting humans called by such names as fortune, fate, chance or lot. One might have good or bad fortune or a good or bad lot in life. Perhaps surprisingly from our point of view, Bernoulli, like others in the 17th century, assumed that the laws of nature are deterministic and gave little or no role to chance (Daston, 1992). Rather, starting from Huygens' On Reckoning in Games of Chance, Bernoulli developed a set of mathematical tools to be applied to calculating expectations in games of chance. The mathematics was not new but was an extension of mathematical tools already included in commercial arithmetic or reckoning books, supplemented in places by classical Greek mathematics. Thus, the main formula for calculating expectation in a game of chance where the player might receive one or another prize under various conditions had the same form as the rule for the unit value of a mixture—the so-called rule of mixture or regula alligationis (Bernoulli/Sylla, 2006: 56, 138). The same formula could be used for calculating the centre of gravity. For the art of conjecture, the formula might be used with a different interpretation of the significance of the variables. At the same time, as Bernoulli himself noted in the theses he proposed for disputation in competition for the open chair in mathematics at Basel in 1687, algebra can be used to provide as many new rules as one might wish (Bernoulli, 1744: 296).

In his 1709 law dissertation which drew on the unpublished manuscript of The Art of Conjecturing which he had doubtless seen as one of Bernoulli's last students, Jacob Bernoulli's nephew Nicolaus Bernoulli summarised Huygens' first three propositions along with Bernoulli's notes, as well as the application of the same formulas to probability in Part IV as saying:

Multiply what happens in individual cases by the number of cases in which that result happens, sum the products and divide by the sum of all the cases. The quotient will show what probably will happen, whether that denotes the value of the expectation or the degree of probability sought (Sylla, 2006: 55–56).

When this transfer of the same formula from games to the probability of opinions is examined, it can be seen that degree of probability here holds the same mathematical position in the formula as expectation in games—probability here is not simply frequency but also somehow involves the range of possible costs or benefits in different cases.

In the Port Royal logic, now thought to have been written by Antoine Arnald and Pierre Nicole but first published under a pseudonym in 1662, a point was made that people sometimes pay attention only to the amount to be gained or lost, whereas they should be paying attention both to the amount gained or lost and to the relative likelihood of the various outcomes:

But with respect to events which concern men and which they may to some extent affect, either avoiding the event or exposing themselves to it, many people fall into an illusion which is the more deceptive the more reasonable it appears. This is the illusion of regarding only the greatness and importance of the advantage hoped for or the disadvantage feared rather than considering the probability (l'apparence & la probabilité) of the advantageous or disadvantageous event's occurring or not occurring....

The defect of the earlier reasonings is that they fail to take into account in addition to the good or evil of the event itself the probability (la probabilité) that the event will or will not occur. The good or evil of an event should be considered in view of the event's likelihood of occurrence (& regarder geometriquement la proportion que toutes ces choses ont ensemble) (English: Arnauld, 1964, 354–55; French: Arnauld & Nicole, 1981, 352–53).

Here Arnauld and Nicole seem to be using the French ‘probabilité' in a way related to De Moivre's later use of ‘probability' in The Doctrine of Chances, although they alternate it with ‘regarder geometriquement la proportion que toutes ces choses ont ensemble' which is not unlike the sorts of expressions that Bernoulli used, talking of ratios of cases, not calling them probabilities. And the same text in the Port Royal logic (Part IV, ch. 16) goes on to say:

In some cases the likelihood of success is so slight that no matter how great the advantage or how small the expense, good sense advises against risking a wager....These reflections may appear trifling, and indeed they are if they stop here.... Their principal use is to make us more reasonable in our hopes and fears. Many people, for example, are extremely frightened when they hear thunder. If the thunder makes them think of God and death, so much the better; we cannot think on these things too much. But if it is simply the danger of being killed by the thunder which causes this excessive apprehension, such dread can easily be shown to be unreasonable. For of two million persons killed, only one is killed by thunder; and we can indeed say that there is scarcely any violent death less common. Since the fear of an evil ought to be proportionate not only to the magnitude of the evil but also the probability (probabilité) of occurrence and since there is scarcely any kind of death rarer than death by thunder, there is hardly anything that ought to occasion less fear–especially since that fear does nothing to avoid such a death (Arnauld, 1964, 357; Arnauld & Nicole, 1981, 354).

Jacob Bernoulli mentions a similar argument in Article 65 of his research notebook, the Meditationes, just after an article on the fourth problem that Huygens had left for the reader's solution in On Reckoning in Games of Chance (Sylla, 2006, 25).

Bernoulli referred to the Port Royal logic in the introduction to his proof of his fundamental theorem:

This empirical way of determining the number of cases by experiments is neither new nor uncommon. The author of The Art of Thinking, a man of great acuteness and talent, made a similar recommendation in Chapter 12 and following of the last part [Part IV], and everyone consistently does the same thing in daily practice (Bernoulli, 2006, 328; Bernoulli, 1713, 225).

Here, in talking of an empirical way of determining the number of cases by experiments, Bernoulli seems to be reading more into The Art of Thinking than was really there. Even so, this was not enough in Bernoulli's view. It was also necessary to consider how much experience would enable a person to have what degree of confidence in the results achieved. By means of his fundamental theorem, Bernoulli claimed to show that there was no limit to the degree of probability or confidence that could be achieved that an accurate answer had been found.

Nevertheless, in Jacob Bernoulli's conception, the art of conjecturing would lead to reasonable but not certain guides to decision making. Before describing the mathematics of the art of conjecturing, Bernoulli stated nine ‘general rules or axioms, which simple reason commonly suggests to a person of sound mind, and which the more prudent constantly observe in civil life' (Bernoulli, 2006: 318; Bernoulli, 1713: 214). Some of these are:

• There is no place for conjectures in matters in which one may reach complete certainty.

• It is not sufficient to weigh one or another argument. Instead we must bring together all arguments that we can come to know and that seem in any way to work toward a proof of the thing.

• We should pay attention not only to those arguments that serve to prove a thing, but also to all those that can be adduced for the contrary, so that, when both groups have been properly weighed, it may be established which arguments preponderate.

• Remote and universal arguments are sufficient for making judgements about universals, but when we make conjectures about individuals, we also need, if they are at all available, arguments that are closer and more particular to those individuals.

• In matters that are uncertain and open to doubt, we should suspend our actions until we learn more. But if the occasion for action brooks no delay, then between two actions we should always choose the one that seems more appropriate, safer, more carefully considered, or more probable, even if neither action is such in a positive sense.

• In our judgements we should be careful not to attribute more weight to things than they have. Nor should we consider something that is more probable than its alternatives to be absolutely certain, or force it on others.

• Because, however, it is rarely possible to obtain certainty that is complete in every respect, necessity and use ordain that what is only morally certain be taken as absolutely certain (Bernoulli, 2006: 318–321; Bernoulli, 1713: 214–217).

Here, axioms (or, perhaps better, maxims) 8 and 9 indicate that Bernoulli did not want to exaggerate the certainty that could be attained by the art of conjecturing, but neither did he want to forestall all action until absolute certainty could be achieved.

Bernoulli had defined ‘morally certain' by saying:

Something is morally certain if its probability comes so close to complete certainty that the difference cannot be perceived. ... if we take something that possesses 999/1000 of certainty to be morally certain, then something that has only 1/1000 of certainty will be morally impossible (Bernoulli, 2006, 316; Bernoulli, 1713, 211–212).

One would, however, act on the basis of merely moral certainty only if ‘the occasion for action brooks no delay' (axiom 7). In choosing numbers for an example of the results of his fundamental theorem Bernoulli, chose a probability of 1 000 to 1 that the observed ratio falls between the ratios 29:50 and 31:50, but he did not there call a probability of 1 000 to 1 morally certain. Moreover, in line with the label ‘moral', moral certainty was to be chosen institutionally. In commenting on his ninth rule or axiom, Bernoulli wrote:

It would be useful, accordingly, if definite limits for moral certainty were established by the authority of the magistracy. For instance, it might be determined whether 90/100 of certainty suffices or whether 999/1000 is required. Then a judge would not be able to favour one side, but would have a reference point to keep constantly in mind in pronouncing a judgement (Bernoulli, 2006: 321; Bernoulli, 1713: 217).

When Bernoulli gave an example in his letter on tennis of determining a ratio of underlying cases leading to observed outcomes a posteriori, he suggested making perhaps 200 or 300 observations. Only rarely, if ever, would he have demanded a probability as high as 1 000 to 1.

When Bernoulli distinguished between necessity and contingency in Part IV of The Art of Conjecturing—where ‘probable' had traditionally meant that the chances of occurrence were (well) over half—he distinguished three senses of ‘necessary':

Something is necessary if it cannot not exist, now, in the future, or in the past. This necessity may be physical, hypothetical, or contractual. It is physically necessary that fire burn, that a triangle have three angles equal to two right angles, and that a full moon occurring when the moon is at a node be eclipsed. It is hypothetically necessary that something, while it exists or has existed, or while it is assumed to exist or have existed, cannot not exist or not have existed. It is necessary in this sense that Peter, whom I know and posit to be writing, is writing. Finally, there is the contractual or institutional necessity by which a gambler who has thrown a six is said to win necessarily if the players have agreed beforehand that a throw of six wins (Bernoulli, 2006, 316; Bernoulli, 1713, 212).

When Huygens and Bernoulli calculate expectation, it is contractual or institutional necessity (i.e. the players' agreement to abide by the rules of the game) that determines the necessity. Then when Bernoulli turns to the probability of conjectures, it is still more often contractual than physical necessity that determines the probability or certainty. This, for instance, would be the case in deciding when a person who has been absent for a long time without communication may be declared dead.

### Was the 1654 Correspondence of Pascal and Fermat Important for Bernoulli?

When Bernoulli set out to write The Art of Conjecturing, he took as the two main resources for his mathematics Huygens's short work on reckoning in games of chance (which, with Bernoulli's notes, formed Part I of The Art of Conjecturing) and the existing mathematics of combinations and permutations (which provided the kernel of Part II of The Art of Conjecturing). In 1654, a number of letters had been exchanged between Pascal and Fermat on the mathematics of games of chance. Fermat advocated using the mathematics of combinations and permutations to calculate how the prize money in a multiple stage game of chance should be divided if the game is broken off before its intended end. Whereas most earlier mathematicians who had addressed such a problem paid attention to what had happened in the game up to that point, Fermat instead looked to the ways in which the game might end if it were completed.

In the next year, Christiaan Huygens travelled from the Netherlands to Paris and, while there, heard about the Pascal–Fermat correspondence but did not learn its content. When he returned to the Netherlands, he wrote On Reckoning in Games of Chance, in which he said:

Lest anyone give me undeserved glory of first discovering this matter, it should be known that this calculus was already discussed some time ago by some of the most outstanding mathematicians of all of France. These scholars are accustomed to challenging each other with very difficult questions, while they keep their methods secret, so that I have had to develop the whole subject matter from its first elements. Even now, I do not know whether they used the same principle as I have used, but I have again and again experienced that their solutions to problems and mine agree beautifully (Bernoulli, 2006, 132; Bernoulli, 1713, 2).

Instead of founding the mathematics of games of chance on probability in the sense of relative chances, Huygens founded it on what a player should justly pay to play a game of chance with prizes. Huygens opens On Reckoning in Games of Chance by saying:

Although the outcomes of games that are governed purely by lot (sors) are uncertain, the extent to which a person is closer to winning than to losing always has a determination. Thus, if a person undertakes to get a six on the first toss of a die, it is indeed uncertain whether he will succeed, but how much more likely he is to fail than to succeed is definite and can be calculated. Similarly, if I were to contend with someone on the understanding that three games are needed to win, and I had already won one game, it would still be uncertain which of us would win three games first. Yet we can calculate with the greatest certainty how great my expectation (expectatio) and my opponent's expectation should be appraised to be. From this we can also determine how much greater a share (portio) of the stakes I should get than my opponent if we agree to quit with the game unfinished, or how much should be paid by someone who wanted to continue the game in my place and with my lot (Bernoulli, 2006: 132; Bernoulli, 1713: 3).

He then states the fundamental principle that lies behind his calculations as follows:

I use the fundamental principle that a person's lot (sors) or expectation to obtain something in a game of chance should be judged to be worth as much as an amount such that, if he had it, he could arrive again at a like lot or expectation contending under fair conditions (Bernoulli, 2006: 133; Bernoulli, 1713: 3–4).

In his note to this passage, Bernoulli tries to explain Huygens' principle in more popular terms:

I will try to demonstrate it by reasoning that is more popular than the previous and more adapted to common comprehension. I posit only this as an axiom or definition: Anyone may expect, or should be said to expect, just as much as he will acquire without fail.... It can be seen from what we have said that we are not using the word expectation in its ordinary sense, according to which we are commonly said to expect or to hope for what is best of all, though worse things can happen to us. Here account is taken of the extent to which our hope of getting the best is tempered and diminished by fear of getting something worse. So by its ‘value' we always mean something intermediate between the best we hope for and the worst we fear (Bernoulli, 2006: 134; Bernoulli, 1713: 5).

The Latin word underlying this approach was sors, while expectatio was a neologism that van Schooten introduced in the translation of Huygens' work from Dutch, alternating expectatio with sors. Because of the requirement or presumption of equity, Huygens assumes that what one pays in to play a game should equal one's expectation.

In his historical introduction to Die Werke von Jakob Bernoulli, Bd. III, B. L. van der Waerden writes that, before Jacob Bernoulli, probability calculations only taught about chances in games of luck (Chancen in Glücksspiel). The concept of ‘probability' (Wahrscheinlichkeit) was occasionally used, but the central concept around which everything revolved was the concept ‘value of a game', or in other words the expectation of winnings. Jacob Bernoulli was the first to recognise the importance of the concept of probability for the whole of human life (van der Waerden, 1975, 2). How Bernoulli came to the complex of questions that he researched and how he came to prove the law of large numbers can be seen in his research notebook, Meditationes, parts of which are published for the first time in Bd. III of Die Werke von Jakob Bernoulli. Through his proof of the law of large numbers, van der Waerden writes, Bernoulli became a founder of mathematical statistics.

After reviewing the writings of Cardano, Fermat, Pascal and Huygens, van der Waerden provides an enlightening discussion of the slow introduction of technical terminology into what would become mathematical probability. In particular, although the Dutch and French languages included words that could be understood as meaning relative frequency (chance and kans), the Latin translation of Huygens' On Reckoning in Games of Chance, with its introduction of expectatio in contexts in which sors had previously appeared, had no such technical term (van der Waerden, 1975, 10). In Huygens' work, Bernoulli found the technical term ‘valor expectationis' meaning ‘Erwartungswert' and the word ‘Chance' did not appear (van der Waerden 1975, 12).

By the time Bernoulli wrote his commentary on Huygens' work, he had seen the Varia Opera Mathematica of Fermat, which contained the 29 July, 24 August and 27 October, 1654 letters of Pascal to Fermat, but none of the letters of Fermat to Pascal on this subject (Hald, 1990, 45). The latter first appeared in the Oeuvres de Blaise Pascal, not published until 1779. Although in a letter sent to Bernoulli in April 1705 Leibniz had mentioned Pascal's Arithmetic Triangle, published in 1665, which would have been more significant than the letters for Bernoulli's work, Bernoulli apparently had not seen it (as argued in Edwards, 2002, 123, 131). In their correspondence, Bernoulli asked Leibniz to loan him Jan de Witt's pamphlet on annuities, but Leibniz could not find the copy he had among his papers, and furthermore, Leibniz told Bernoulli, Jan de Witt proposed only to find the arithmetic mean between what was equally uncertain, and the same approach could be found in Huygens' work on games of chance and in Pascal's Arithmetic Triangle, so in Leibniz's opinion it was not urgent for Bernoulli to have in hand Jan de Witt's work (Bernoulli, 1993, 143).

Thus, to sum up, the correspondence of Pascal and Fermat stimulated Huygens to write his short work on calculations in games of chance, without Huygens learning much about their methods. Bernoulli's work relied on that of Huygens. While Bernoulli could read the July and August letters of Pascal to Fermat in Fermat's Varia Opera Mathematica, he seems not to have read the letters carefully: in describing what Pascal wrote, Bernoulli used the Latin assequi (follow) where Pascal had used the word demonstrare (demonstrate), thus seeming not to catch Pascal's distinction between getting an answer and demonstrating that it is correct (Bernoulli, 1713, 107; David, 1962, 233; Fermat, 1679, 180).

Because of Bernoulli's scant and possibly inaccurate knowledge of Pascal's contributions to the background of mathematical probability, then, from the perspective of Bernoulli, the contribution of the correspondence of Pascal and Fermat is slight, despite the fact that since the time of the second edition of Pierre Rémond de Montmort's Essay d'analyse sur les jeux de hazard (1713) the correspondence of Pascal and Fermat has often been taken as the beginning of mathematical probability.

### Bernoulli's Work as a Link Between Huygens and De Moivre

At Jacob's death, his student, Jacob Hermann, was asked to go through his papers and to provide information to those who intended to prepare éloges. Inspired by reports of the unfinished Art of Conjecturing, Pierre Rémond de Montmort set to work. In his Essay d'analyse sur les jeux de hazard (1708), Montmort wrote that mathematicians of the preceding 50 years had made great achievements in applying mathematics to physics, but it would be even more glorious if mathematics could serve to rule judgements and conduct in practical life. This is what Bernoulli had proposed to do in his book to be named De arte conjectandi, l'art de deviner if his premature death had not prevented it. Fontenelle and Saurin had each given a short description of the proposed book in their éloges published in the Histoire de l'Academie (1705) and in the Journaux des Sçavans de France (1706), respectively. Although he did not know for which games Bernoulli had determined the relative shares (partis) of the players, Montmort proposed to help fill the gap left because of Bernoulli's early death by the publication of his own work. He himself, however, found it too difficult to deal with practical decision making, and so concentrated on games and on the mathematics of combinations and permutations (Montmort, 1708).

Not long after Montmort's book, Abraham De Moivre, a Huguenot who had left France for England after the Revocation of the Edict of Nantes, published in Latin, appearing in the Philosophical Transactions of the Royal Society of London for 1712, a shorter work with the title De mensura sortis (On the measure of lot). In a letter to Francis Robartes prefaced to De mensura sortis, De Moivre gave faint praise to the Essay d‘analyse (Hald, 1990: 291; Hald is translating from Montmort's quotation of this passage). Montmort resented this slighting of his contribution and responded to it in an advertissement to the second, 1713, edition of his Essay d'analyse. In his response, Montmort describes in detail both the correspondence of Pascal and Fermat and Pascal's Arithmetic Triangle along with the short treatises appended to it, including the one on application of the arithmetic triangle to games of chance (Montmort, 1713: xxvii–xxxi). Thus, starting the history of mathematical probability with Pascal and Fermat rather than with Huygens was part of Montmort's effort to distinguish himself from Huygens and to assert the originality and value of his own work. If Montmort's original contribution to the mathematics of games of chance did not put him among the most important contributors, his contribution to the writing of the history of mathematical probability had the effect of giving more credit to the French, including himself and the correspondence of Pascal and Fermat.

The real breakthrough in thinking about games broken off prematurely was to develop a way to enumerate systematically in a way to avoid miscounts all the equally likely alternative outcomes and then to examine them to see which players would have won what in various scenarios. This is the approach that Fermat used in the 1654 Pascal–Fermat correspondence and it is also used by Huygens and Bernoulli in some cases. In Pascal's letters that Bernoulli could read, Pascal's short cut solution to the division problem was, like that of Cardan, Huygens and even Bernoulli in the early parts of his book, based on sors (i.e. the French sort). Pascal writes that ‘Let us say that the first man had won twice and the other once; now they play another game, in which the conditions are that [le sort est tel que], if the first wins, he takes all the stakes ... if the other wins it, then they have each won two games, and therefore, if they wish to stop playing, they should each take back their own stake.' The translator of this passage into English (Maxine Merrington in David, 1962) who chose to translate ‘le sort' by ‘conditions' may have been puzzled by the appearance of le sort here, but, as indicated earlier, it is the word that shortly thereafter was displaced by expectatio in van Schooten's Latin translation of Huygens' On Reckoning in Games of Chance.

Thus, before Bernoulli's The Art of Conjecturing, mathematical probability as such did not exist. Yes, there was a mathematics of games of chance which we now include in teaching elementary mathematical probability, but the early modern mathematicians were not thinking of probabilities in the sense of relative chances in developing it, and neither were they thinking of probability in the sense of degrees of certainty. It is only in The Art of Conjecturing that an extended attempt is made to develop a mathematics of epistemic probability in the sense of degrees of certainty and it is really only in Abraham De Moivre's Doctrine of Chances that probability is defined in terms of relative chances in the external world. For Jacob Bernoulli, the point is that the epistemic probability of a proposition, such as the proposition ‘a white token will be drawn out of the urn', should be assigned by a prudent person taking into account the relative numbers of white and non-white tokens in the urn as estimated by repeatedly drawing and replacing tokens from the urn.

Thus, there were two main tracks along which mathematical probability developed, which tracks sometimes ran parallel or diverged and sometimes crossed. One was the track of developing pure and applied mathematics such as commercial arithmetic, algebra, logarithms, infinite series, the mathematics of combinations and permutations and so forth as so on. The second track did not involve mathematics for the most part, but rather ethics, law and probable opinions on subject matters where scientific certainty was not possible (Franklin, 2001). In B. L. van der Waerden's view (van der Waerden, 1975: IX), Christiaan Huygens and Jacob Bernoulli created the theory of probability nearly ex nihilo (fast aus dem Nichts geschaffen). Girolamo Cardano, Pierre Fermat and Blaise Pascal had calculated probabilities and expectations for a few games of chance, van der Waerden wrote, but no one before Huygens had tried to present an organised theory. Then in The Art of Conjecturing, the mathematical track merged with the track on which people had been trying to develop a non-mathematical art of good judgement, such as found in the Port Royal logic or Ars Cogitandi. In the art of conjecturing, frequencies would play a role, along with costs and benefits and other such factors, in support of rational decision making, or, in other words, forming probable opinions. Then, Bernoulli tied the parts of the art of conjecture together and opened new territory with his proof of his fundamental theorem.

### How Bernoulli's Fundamental Theorem Fit into The Art of Conjecturing

With this background, it is much easier to understand why Bernoulli was so gratified at having proved his fundamental theorem. In roughly 1689, Bernoulli first entered into his research journal a proof of the theorem that was to be the last item in his The Art of Conjecturing. After the proof, he wrote:

Nota Bene. I esteem this discovery more than if I had given the quadrature of the circle itself, which even if it were found very great, would be of little use. (Bernoulli, 1975: 88.)

Why was Bernoulli so thrilled, if that is not too strong a word, with what he had shown? He was thrilled because his theorem indicated a way around a road block that stood in the way of his development of an art of conjecturing, in which mathematics could be used to make better decisions in conditions of uncertainty. He apparently had decided to try to develop a mathematical art of conjecturing applied to civil, moral and economic issues before he had the proof of this fundamental theorem. In the opening chapters of Book IV of The Art of Conjecturing, Bernoulli tried to show how the mathematics of games of chance, such as found in Huygens' On Reckoning in Games of Chance, and the mathematics of combinations and permutations could be applied to civil, moral and economic problems. He took the algebraic formula that had been used to calculate the unit price of a mixture, or the expectation of a player in a game of chance, and tried to show how it could be adapted, for instance, to weigh the arguments and evidence that might indicate that a given person had or had not committed a crime. It was an uphill battle, however, because there were no uniform scales of measure on which, say, an eyewitness report of a crime could be weighed together with or against the discovery of a bloody weapon or the presence or absence of a motive. Bernoulli wrote:

I cannot conceal here that I foresee many problems in particular applications of these rules that could cause frequent delusions unless one proceeds cautiously in discerning arguments. For sometimes arguments can seem distinct that in fact are one and the same. Or, vice versa, those that are distinct can seem to be one. Sometimes what is posited in one argument plainly overturns a contrary argument (Bernoulli, 2006, 325; Bernoulli, 1713, 221–222).

In the next chapter of Part IV, he went on:

It was shown in the preceding chapter how, from the numbers of cases in which arguments for things can exist or not exist, indicate or not indicate, or also indicate the contrary, and from the forces of proving proportionate to them, the probabilities of things can be reduced to calculation and evaluated. From this it resulted that the only thing needed for correctly forming conjectures on any matter is to determine the numbers of these cases accurately and then to determine how much more easily some can happen than others. But here we come to a halt, for this can hardly ever be done. Indeed, it can hardly be done anywhere except in games of chance (Bernoulli, 2006, 326; Bernoulli, 1713, 223).

Bernoulli then mentioned some objections to his approach that had been raised in letters that Leibniz had sent him in the early 1700s and added others:

But what mortal, I ask, may determine, for example, the number of diseases, as if they were just as many cases, which may invade at any age the innumerable parts of the human body and which imply our death? And who can determine how much more easily one disease may kill than another—the plague compared to dropsy, dropsy compared to fever? Who then can form conjectures on the future state of life and death on this basis? Likewise who will count the innumerable cases of the changes to which the air is subject every day and on this basis conjecture its future constitution after a month, not to say after a year? Again, who has a sufficient perspective on the nature of the human mind or on the wonderful structure of the body so that they would dare to determine the cases in which this or that player may win or lose in games that depend in whole or in part on the shrewdness or the agility of the players? In these and similar situations, since they may depend on causes that are entirely hidden and that would forever mock our diligence by an innumerable variety of combinations, it would clearly be mad to want to learn anything in this way (Bernoulli, 2006, 327; Bernoulli, 1713, 224).

Such objections are certainly daunting, but Jacob Bernoulli saw a way forward, namely to learn the ratios of possible cases, not a priori but a posteriori, or from experience:

Nevertheless, another way is open to us by which we may obtain what is sought. What cannot be ascertained a priori, may at least be found out a posteriori from the results many times observed in similar situations, since it should be presumed that something can happen or not happen in the future in as many cases as it was observed to happen or not to happen in similar circumstances in the past (Bernoulli, 2006, 327; Bernoulli, 1713, 224).

After applying this reasoning to the problems of diseases and life expectancy, Bernoulli applied it to weather forecasting and to players in a game:

Likewise if someone for several years past should have observed the weather and noted how many times it was clear or rainy or if someone should have very frequently watched two players at a game and should have seen how many times this or that player won, just by doing so one would have discovered the ratio that probably exists between the numbers of cases in which the same outcomes can happen or not happen in the future in circumstances similar to the previous ones (Bernoulli, 2006, 327; Bernoulli, 1713, 225).

When pushed, Bernoulli refined such suggestions. When Leibniz in his correspondence argued that new diseases may arise and thereby change the situation, Bernoulli agreed that, indeed, one would have to make new observations (Sylla, 1998). And in his Lettre à un Amy sur les Parties du Jeu de Paume, Bernoulli argued that to determine the ratio of cases for winning or losing at tennis, one should not count the games won or lost, but the strokes won or lost, because a person may win many strokes and still lose the game (Sylla, 2013).

It is well known that if something has happened frequently in the past, Bernoulli wrote, it may also happen in that way in the future, and the more experience one has, the less one is likely to err in assuming something similar in the future, but he wants to show more than this:

Something else remains to think about, which perhaps no one has considered up to this point. It remains, namely, to ask whether, as the number of observations increases, so the probability increases of obtaining the true ratio between the numbers of cases in which some event can happen and not happen, such that this probability may eventually exceed any given degree of certainty. Or whether, instead, the problem has an asymptote, so to speak; whether, that is, there is some degree of certainty that may never be exceeded no matter how far the number of observations is multiplied, so that, for example, we may never be certain that we have discovered the true ratio of cases with more than a half or two-thirds or three-fourths parts of certainty (Bernoulli, 2006, 328; Bernoulli, 1713, 225).

Of course, one will not be able to find the ratio of cases exactly but only within some range of values:

Lest, however, these things be misunderstood, it must be carefully noted that we do not wish the ratio between the numbers of cases that we have undertaken to determine by experiments to be taken precisely or as an indivisible (for, if it were, then the opposite would occur, and it would become less probable that the true ratio had been found as more observations were taken). Rather the ratio should be defined within some range, that is, contained within two limits, which can be made as narrow as anyone might want (Bernoulli, 2006, 329; Bernoulli, 1713, 226).

Bernoulli concludes:

This, therefore, is the problem that I have proposed to publish in this place, after I have already concealed it for twenty years. Both its novelty and its great utility combined with its equally great difficulty can add to the weight and value of all the other chapters of this theory (Bernoulli, 2006, 329; Bernoulli, 1713, 227).

After quoting and responding to the objections that Leibniz had made in correspondence (without naming him), Bernoulli turns, in the next and last chapter of his book, to his proof.

Thus, what Bernoulli's theorem was intended to show was that if there are definite underlying factors (which Bernoulli said was assured by God's omniscience and omnipotence), then the underlying ratios of possible cases in apparently irregular processes will manifest themselves if enough outcomes in similar cases are observed. His resulting theorem can be used as a formula to indicate how many observations would be needed to achieve how great a probability that the ratio of cases will fall within chosen limits. The assumption is that one is looking for a ratio between integers, such as how many cases one tennis player has to win a stroke or a game or a set against another, although the numbers might be in the thousands or more. Furthermore, it is assumed that what one actually observes in practice may result from innumerable factors combining together. Despite the use of the letter p in modern rewritings of Bernoulli's key formula, the ratio of cases expected or observed is not called a ‘probability'.

The structure of Bernoulli's proof is entirely mathematical. He shows that if higher and higher powers of a binomial are taken, more and more of the value of the terms of the binomial expansion will fall within limits as narrow as you please around the largest term. Once the mathematical result is achieved, Bernoulli points out that the first term of the expansion represents the relative number of outcomes in which all the results will be of a first type, the second term represents the number of outcomes in which all but one of the outcomes are of the first type and so forth through the terms. Let the first type of outcome be called fertile and the second type be called sterile. Bernoulli says:

Let the number of fertile cases and the number of sterile cases have exactly or approximately the ratio r / s, and let the number of fertile cases to all the cases be in the ratio r / (r + s) or r / t, which ratio is bounded by the limits (r + 1)/t and (r–1)/t. It is to be shown that so many experiments can be taken that it becomes any given number of times (say c times) more likely that the number of fertile observations will fall between these bounds than outside them, that is, that the ratio of the number of fertile to the number of all the observations will have a ratio that is neither more than (r + 1)/t nor less than (r–1)/t. (Bernoulli, 2006, 337; Bernoulli, 1713, 236).

As formulated, the theorem enables the user to choose values for r, s and c, and then to calculate how many observations or experiments will be required (what power of the binomial—for ease of calculation taking powers some number n times t, the sum of r + s) to make it c times more likely that the observed ratio of results will fall inside the limits than outside. To make the limits narrower, Bernoulli proposes taking, for example, 30/50 or 300/500 rather than 3/5. If c is taken as 1 000 and if the ratio is taken as 30/50, then it would take 25 550 observations to assure that it is 1 000 times more likely that the observed ratio of outcomes will fall within 31/50 and 29/50 than outside, where logarithms are used to facilitate adding the respective ranges of terms of the binomial expansion, using the lemmas previously proved. If c were set at 10 000, then 31 258 experiments would be required, and if c were set at 100 000, then 36 966 experiments would be required, and so forth to infinity continually adding 5 708 experiments when the desired probability is multiplied by 10.

Interestingly, when Bernoulli proved his theorem around 1689 in his Meditationes, he used 3 and 2 as the numbers of fertile and sterile cases and 5 as their sum, which meant that the limits of the numbers he was looking for were 10 plus or minus 1, or 9 and 11, rather than 29/50 and 31/50 as in The Art of Conjecturing, so that the number of observations needed for a probability of 1000 to 1 of falling within the limits required only 5 875 or 5 713 observations rather than 25 550 (Bernoulli, 1975, 87–88; there are some changes in the manuscript Meditationes not noted in the transcription). That he chose to test for narrower limits in the final manuscript of The Art of Conjecturing, thus requiring more observations, is evidence of his wish to show that there is no limit to the probability that can be achieved and of his corresponding lack of concern for the number of observations needed, rather than being evidence that he was aiming for moral certainty.

### Abraham De Moivre and Bernoulli's Theorem

Abraham De Moivre made two, seemingly inconsistent, comments about this proof and the related proof provided by Nicolaus Bernoulli in a letter to Montmort published in the 1713 edition of Essay d'Analyse. In a letter to Nicolaus Bernoulli in March 1714, De Moivre said ‘the problem of experiences is of infinite beauty', by which he refers to Bernoulli's proof of his fundamental theorem (Sylla, 2006, xvi). Many others later admired the rigour and beauty of Bernoulli's proof, and I think it is natural that De Moivre would have appreciated it as well.

In his own later work, De Moivre used related binomial expansions and summations of terms, but in introducing his own thinking in the third (1756) edition of The Doctrine of Chances, De Moivre wrote:

But suppose it should be said that notwithstanding the reasonableness of building Conjectures upon Observation, still considering the great Power of Chance, Events might at long run fall out in a different proportion from the real Bent which they have to happen one way or the other; and that supposing for Instance that an Event might as easily happen as not happen, whether after three thousand Experiments it may not be possible it should have happened two thousand times and failed a thousand; and that therefore the Odds against so great a variation from Equality should be assigned, whereby the Mind would be the better disposed in the Conclusions derived from the Experiments.

In answer to this, I'll take the liberty to say, that this is the hardest Problem that can be proposed on the Subject of Chance, for which reason I have reserved it to the last.... in order thereto, I shall here translate a Paper of mine which was printed November 12, 1733, and communicated to some Friends, but never yet made public, reserving to myself the right of enlarging my own Thoughts, as occasion shall require (De Moivre, 1756, 242).

Then, the translation of the 1733 paper begins:

Although the Solution of Problems of Chance often requires that several Terms of the Binomial (a + b)n be added together, nevertheless in very high Powers the thing appears so laborious, and of so great difficulty, that few people have undertaken that Task; for besides James and Nicolas Bernoulli, two great Mathematicians, I know of no body that has attempted it; in which , tho' they have shewn very great skill, and have the praise which is due to their Industry, yet some things were farther required; for what they have done is not so much an Approximation as the determining very wide limits, within which they demonstrated that the Sum of the Terms was contained (De Moivre, 1756, 243).

De Moivre goes on to refer to his own Miscellanea Analytica, where he had described the Bernoullis' methods, and to suggest that the reader might rather want to consult the Bernoullis' own works. Here, if I understand correctly, De Moivre is thinking not of the structure of Jacob Bernoulli's proof, which he had thought of infinite beauty, but rather of the task of calculating the required sums of terms. Taking B to be the number of which a certain series is the Hyperbolic Logarithm, De Moivre reports:

When I first began that inquiry, I contented myself to determine at large the Value of B, which was done by the addition of some Terms of the earlier written series; but as I perceived that it converged but slowly, and seeing at the same time that what I had done answered my purpose tolerably well, I desisted from proceeding farther till my worthy and learned Friend Mr. James Stirling, who had applied himself after me to that inquiry, found that the Quantity B did denote the square root of the Circumference of a Circle whose Radius is Unity.... I own with pleasure that this discovery, besides that it has saved trouble, has spread a singular Elegancy on the Solution (De Moivre, 1756, 244).

Thus Bernoulli's proof of his fundamental theorem could be infinitely beautiful in De Moivre's eyes and yet not provide a practical and reasonably short method of calculating the sums of terms sought. In corollaries, then, De Moivre goes on to calculate, for instance, that if an infinite number of Experiments were taken (note that De Moivre does not shy away even from an infinite number in the course of his proof!), the probability that an event which has an equal number of chances to happen and to fail will fall within the limits and will be expressed by the decimal 0.682688 (De Moivre, 1756, 246). Thus, like Jacob Bernoulli, De Moivre is interested in advancing mathematical probability towards both practical and theoretical ends.

Given the way that The Art of Conjecturing breaks off just after the proof of Bernoulli's fundamental theorem, he is left with a theorem that applies to binomial expansions, but with nothing more than references to God to justify the idea that expansions of powers of binomials may represent real world situations (Bernoulli, 2006, 315; Bernoulli, 1713, 210–211). The certainty of what happens in the world is used to justify Bernoulli's proof of his fundamental proposition, but it is an arm's length justification, because the proof itself is abstract mathematics. With his theological background, Bernoulli might well have believed that the natural world is such that his fundamental theorem would apply. I do not know of any evidence that he worried about the applicability of his fundamental proposition, but in effect, his proposition was conditional: if there are processes in the world that can be represented by a binomial expansion, then it will be possible to detect underlying ratios with a probability of being correct that increases as the number of observed outcomes is increased. Repeated independent trials where there are only two possible outcomes for each trial and their probabilities remain the same throughout the trials are now named ‘Bernoulli trials' (e.g. Feller, 1957, 135). As Feller writes (1957, 136), ‘the Bernoulli scheme of trials is a theoretical model, and only experience can show whether it is suitable for the description of specified experiments'.

De Moivre, like Bernoulli a Protestant (of the Reformed, rather than Lutheran branch), made remarks consistent with Bernoulli's views:

Remark I. From what has been said, it follows, that Chance very little disturbs the Events which in their natural Institution were designed to happen or fail, according to some determinate Law..... But we have seen in our LXXII Problem that altho' Chance may produce an inequality of appearance, and still a greater inequality according to the length of time in which it may exert itself, yet the appearances, either one way or the other, will perpetually tend to a proportion of Equality.....And thus in all Cases it will be found, that altho' Chance produces Irregularities, still the Odds will be infinitely great, that in the process of Time, those Irregularities will bear no proportion to the recurrency of that Order which naturally results from ORIGINAL DESIGN (De Moivre, 1756, 250, 251).

While Bernoulli's proof of his theorem was purely mathematical, De Moivre appears to be talking about the world. De Moivre's second remark confirms this impression:

Remark II. As, upon the Supposition of a certain determinate Law according to which any Event is to happen, we demonstrate that the Ratio of Happenings will continually approach to that Law, as the Experiments or Observations are multiplied: so, conversely, if from numberless Observations we find the Ratio of the Events to converge to a determinate quantity, as to the Ratio of P to Q; then we conclude that this Ratio expresses the determinate Law according to which the Event is to happen...

Again, as it is thus demonstrable that there are, in the constitution of things, certain Laws according to which Events happen, it is no less evident from Observation, that those Laws serve to wise, useful and beneficent purposes; to preserve the steadfast Order of the Universe, to propagate the several Species of Beings, and furnish to the sentient Kind such degrees of happiness as are suited to their State.

..hence, if we blind not ourselves with metaphysical dust, we shall be led, by a short and obvious way, the acknowledgment of the great MAKER and GOVERNOUR of all; Himself all-wise, all-powerful and good (De Moivre, 1756, 251–252).

Chance, as we understand it, supposes the Existence of things, and their general known Properties : that a number of Dice, for instance, being thrown, each of them shall settle upon one or other of its Bases. After which, the Probability of an assigned Chance, that is of some particular disposition of the Dice, becomes as proper a subject of Investigation as any other quantity or Ratio can be (De Moivre, 1756, 253).

I shall only add, That this method of Reasoning may be usefully applied in some other very interesting Enquiries, if not to force the Assent of others by a strict Demonstration, at least to the satisfaction of the Enquirer himself: and shall conclude this Remark with a passage from the Ars Conjectandi of Mr. James Bernoulli, Part IV, Cap. 4. where that acute and judicious Writer thus introduceth his Solution of the Problem for Assigning the Limits within which, by the repetition of Experiments, the Probability of an Event may approach indefinitely to a Probability given, ‘Hoc igitur est illud Problema &c.' This, says he, is the Problem which I am now to impart to the Publick, after having kept it by me for twenty years: new it is, and difficult: but of such excellent use, that it gives a high value and dignity to every other Branch of this Doctrine. Yet there are Writers, of a Class indeed very different from that of James Bernoulli, who insinuate as if the Doctrine of Probabilities could have no place in any serious Enquiry; and that Studies of this kind, trivial and easy as they be, rather disqualify a man for reasoning on every other subject. Let the Reader chuse (De Moivre, 1756, 254).

In 1836, Siméon Denis Poisson made a just estimation of the value of Bernoulli's fundamental theorem at the same time as he was proposing the law of large numbers:

In the preamble that I read some months ago at the Academy of the work on The Probability of Criminal Judgments on which I am currently working, I considered the law of large numbers as a fact that we observe in things of all kinds.... One should not confuse this general law with the beautiful theorem of Jacob Bernoulli, on the demonstration of which he meditated, as is known, for twenty years. According to this theorem, outcomes occur very nearly, in a long series of trials, in proportion to their respective probabilities. But one should not lose sight of the fact that he supposes that the chances remain constant, while, to the contrary, the chances of physical phenomena and of moral matters almost always vary continually, without any regularity and often to a great extent. Nevertheless, constant observation shows us that for each kind of outcomes the ratio of the number of times that they occur to the total number of trials is without noticeable variation when the numbers are very great (Sylla, 2006, 41n103).

As Poisson says, Bernoulli's theorem assumes that the chances remain constant. Given a binomial distribution, Bernoulli elegantly shows what will result. Whereas Bernoulli's proof is purely mathematical, Poisson claims that his law of large numbers is empirical, that despite the variations of chances in the external world, ‘for each kind of outcomes the ratio of the number of times that they occur to the total number of trials is without noticeable variation when the numbers are very great'. Bernoulli's fundamental proposition is not a law of nature, but a mathematical theorem applying to binomial expansions. In a sense, then, it provides a mathematical model that could be paired with empirical results to see whether the natural world contains distributions like those of powers of binomials.

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