NOTES
Titles of books, papers and journals abbreviated here are given more fully in the bibliography. The following abbreviations for Laplace's works are used throughout these notes:
Essai - Essai Philosophique sur les Probabilites.
T.A.P. I. (or II.) - Theone Analytique des Probabilites, Book I (or II), as reprinted in Volume 7 of the (Euvres Completes de Laplace. When "Supple• ment" is added to "T.A.P." , the reference is to the appropriately numbered Supplement appearing in the third edition of the T .A.P. as printed in the (Euvres Completes de Laplace.
o. C. - (Euvres Completes de Laplace.
Le~on - Sur les probabilites, a lecture given in the tenth session of the Le~ons de Mathematiques.
The 1932 German translation of the Essai, edited by R. von Mises, will be referenced throughout as ''von Mises."
A reference "see Note n" will always indicate Note n to the article cur• rently under comment.
My indebtedness to the writer (Bernard Bru) of the notes in the Bru/Thom edition [1986] of the Essai will be evident to anyone who has studied that work. The notes by Hilda Pollaczek-Geiringer in von Mises [1932] have also been useful; these notes should be referred to for details of the "collective" approach to probability.
125 126 Notes: Foreword
Foreword
1. This lecture, originally entitled "Sur les probabilites," was given in the tenth session of the Le~ons de Mathimatiquesj it was published in 1812. An early version of the opening part of the Essai was published in 1810j it is reprinted in full in Gillispie [1979]. 2. Writing of Laplace's work on the apparent anomalies of the mean motions of Jupiter and Saturn, Wilson explains that, in Laplace's terminology, Analyse refers primarily to the set of operational symbols and sym• bolic procedures developed by Leibniz, the Bernoullis, Eu• ler, and others, for the formulation and solution of differen• tial equations. Although the Analyse employed by Laplace consisted of algorithmic processes, their successful use was far from being merely a matter of calculation. The user needed to assess and choose among procedures with a clear idea of the goal at which he aimed, the kind of solution to be expected or hoped for. [1985, pp. 24-25] However, not too much violence will be done to this notion if we think of Analyse as "mathematics." 3. The word "chance," in French as in English, is patient of a number of interpretations, the most common being "probability" and "for• tuitous event." It is with the latter meaning that Laplace uses the word in the Essai. The use of "chances" in the sense of ''things which may befall" was not uncommon in the 18th and 19th centuries. For example, Emerson writes Chance is an event, or something that happens without the design or direction of any agentj and is directed or brought about by nothing but the laws of nature. [1776, p. 2] This is perhaps why Bayes felt it necessary to state in his Essay of 1763 "By chance I mean the same as probability" [po 376].
On probability
1. A similar sentiment had been expressed earlier by David Hume, who, in Section VIII of his Inquiry Concerning Human Understanding, wrote "It is universally allowed, that nothing exists without a cause of its existence" [Hume, 1894, p. 366]. Laplace is using "cause" in a fairly usual sense here. Keynes [1973] notes the "metaphysical diffi• culties which surround the true meaning of cause" [po 306], and goes on to say Notes: On probability 127
I have followed a practice not uncommon amongst writers on probability, who constantly use the term cause, where hypothesis might seem more appropriate. [loco cit.] De Morgan is even more forthright: he says By a cause, is to be understood simply a state of things antecedent to the happening of an event, without the in• troduction of any notion of agency, physical or moral. [1838, p. 53] Quetelet in fact distinguishes three principal classes of causes, viz. Constant causes are those which act in a continuous man• ner, with the same intensity, and in the same direction. Variable causes act in a continuous manner, with ener• gies and tendencies which change either according to de• termined laws or without any apparent law ... Accidental causes only manifest themselves fortuitously, and act indifferently in any direction. [1849, p. 107] One is tempted to say of cause what Mr. Pickwick said of politics, viz. "The word ... comprises, in itself, a difficult study of no incon• siderable magnitude." (See Chap. XV of Dickens [1837].) 2. What appears in earlier editions as un effet ("an effect") appears mistakenly in the Bru/Thom edition as en effet ("indeed"). 3. See Leibniz [1710]. The following extracts are pertinent: (a) All we have just said likewise agrees perfectly with the maxims of philosophers, who teach that a cause could not operate, without having a disposition to the action; and it is this disposition that contains a predetermina• tion, be it what the agent has received from without, or be it what he has had by virtue of his own previous constitution. [§46] (b) Therefore I acknowledge indifference only in one sense, which it gives notice of in the same way as contingency, or non-necessity. But as I have carefully explained more than once, I do not acknowledge an indifference of equi• librium, and I do not believe that one ever chooses, when one is completely indifferent. Such a choice would be a kind of pure chance, without conclusive reasoning, and as much visible as hidden. But such a chance, such an absolute and real fortuitousness, is a chimera that is never found in nature. Wise men all agree that chance is only a specious thing, like fortune; it is ignorance 128 Notes: On probability
of causes that leads to it. But if there had been such a vague indifference, or else if one had chosen with• out there having been anything to lead one to choose, the chances would be something real, similar to what is found in that small change of direction in particles, occurring for no rhyme or reason, to the feeling of Epi• curus, who introduced it to eschew necessity, of which Cicero has justifiably made so much fun. [§303]
4. Laplace had expressed this sentiment before: in his [1773] he wrote
The present state of the system of nature is clearly a sequel to what it was a moment before, and, if we conceive of an intelligence that, at a given instant, encompasses all the relationships of the beings of this universe, it would be able to determine at any time whatsoever, in the past or in the future, the respective position, the movements and generally the attachments of all these beings. [O.C. VIII, p.I44] Laplace's position was by no means unreservedly accepted. Poincare in fact wrote A very slight cause, which escapes us, determines a con• siderable effect which we can not help seeing, and then we say this effect is due to chance. If we could know exactly the laws of nature and the situation of the universe at the initial instant, we should be able to predict exactly the sit• uation of this same universe at a subsequent instant. But even when the natural laws should have no further secret for us, we could know the initial situation only approxi• mately. If that permits us to foresee the subsequent situa.• tion with the same degree of approximation, this is all we require, we say the phenomenon has been predicted, that it is ruled by laws. But this is not always the case; it may hap• pen that slight differences in the initial conditions produce very great differences in the final phenomena; a slight error in the former would make an enormous error in the latter. Prediction becomes impossible and we have the fortuitous phenomenon. [1946, pp. 397-398] 5. The word ''world'' may be interpreted throughout this translation in the broad sense of ''universe.''
6. An anonymous tract of 1716 was devoted to a discussion of the evils to be expected from the appearance of a particularly noteworthy aurora borealis. De Morgan comments on the work as follows: Notes: On probability 129
The prodigy, as described, was what we should call a very decided and unusual aurora borealis. The inference was, that men's sins were bringing on the end of the world. [1915, vol. I, p. 134) 7. The evil effects foreshadowed by the appearance of comets had long been recognized. In his essay entitled An Excellent Discourse of the Names, Genus, Species, Efficient and Final Causes of all Comets, &c., Wharton wrote It has been a received Opinion in all Ages, that Comets are certain FUnebrious Appearances, secret Fires and Torches of Death rather than of Life, and were ever look'd upon as the threatening Eyes of Divine Vengeance, and the Tongue of an Ireful Deity, portending the Death of Princes, Plague of the People, Famine, and Earthquakes, with horrible and terrible Tempests. [1683, p. 159) As holders, or at least reporters, of these opinions Wharton cited Aristotle, Cicero, Pliny, the Holy Fathers (Tertullian, St Augustine), meteorologers (Fromandus) and astronomers (Tycho Brahe, Kepler). He also noted the role played by the fixed stars near to which the comet appears - for example, The Dominion of Mercury portends great Calamity unto all those that Live by their own Industry, and such as love and favour the Muses, with the Death of some great Personage, Wars, Famine, and Pestilence; of Diseases, the Phrenzy, Lethargy, Epilepsie, and griefs of the Head. [1683, p. 173)
Further appended was a list of the years (from 13 to 1618 AD) in which comets appeared, and the names of kings, popes, princes and other luminaries who died in those years. Writing some little time after Laplace, Charles Mackay commented on this matter in his entertaining work (published under different titles in 1841, 1852 and 1980) as follows: The appearance of comets has been often thought to foretell the speedy dissolution of this world. Part of this belief still exists; but the comet is no longer looked upon as the sign, but the agent of destruction. So lately as in the year 1832 the greatest alarm spread over the continent of Europe, es• pecially in Germany, lest the comet, whose appearance was then foretold by astronomers, should destroy the earth. The danger of our globe was gravely discussed. Many persons refrained from undertaking or concluding any business dur• ing that year, in consequence solely of their apprehension 130 Notes: On probability
that this terrible comet would dash us and our world to atoms. [1980, p. 258] Mrs Piozzi recalls an anecdote of Dr Johnson that runs as follows: The famous distich too, of an Italian improvisatore, who, when the Duke of Modena ran away from the comet in the year 1742 or 1743, Se al venir vestro i principi sen' vanno Deh venga ogni di - durate un anno; which (said he [Le. Ursa Major]) would do just as well in our language thus; If at your coming princes disappear, Comets! come every day - and stay a year. [1932, p. 47] 8. But see The Bible, the Book of Joshua, chap. 10, vs 12 & 13. In his essay "Additional remarks on miracles," John Tyndall says Let us take as an illustration the miracle by which the vic• tory of Joshua over the Amorites was rendered complete. In this case the sun is reported to have stood still for 'about a whole day' upon Gibeon, and the moon in the valley of Ajalon. An Englishman of average education at the present day would naturally demand a greater amount of evidence to prove that this occurrence took place, than would have satisfied an Israelite in the age succeeding that of Joshua. For to the one, the miracle probably consisted in the stop• page of a fiery ball less than a yard in diameter, while to the other it would be the stoppage of an orb fourteen hundred thousand times the earth in size. And even accepting the interpretation that Joshua dealt with what was apparent merely, but that what really occurred was the suspension of the earth's rotation, I think the right to exercise a greater reserve in accepting the miracle, and to demand stronger evidence in support of it than that which would have sat• isfied an ancient Israelite, will still be conceded to a man of science. [1879, vol. 2, pp. 35-36] Writing of meteoric lights and the terror they sometimes induce in mankind, Pliny the Elder says My own view is that these occurrences take place at fixed dates owing to natural forces, like all other events, and not, as most people think, from the variety of causes invented by the cleverness of human intellects; it is true that they were the harbingers of enormous misfortunes, but I hold Notes: On probability 131
that those did not happen because the marvellous occur• rences took place but that these took place because the misfortunes were going to occur, only the reason for their occurrence is concealed by their rarity, and consequently is not understood as are the risings and setting of the planets described above and many other phenomena. [§97, ,XXVI] Further, in §140, ,LIV, he writes "Historical record also exists of thunderbolts being either caused by or vouchsafed in answer to cer• tain rites and prayers." In his essay "Reflections on prayer and natural law" Tyndall says
As regards direct action upon natural phenomena, man's wish and will, as expressed in prayer, are confessedly pow• erless. [1879, vol. 2, p. 12] 9. In 395 AD the Roman Empire was divided into two parts (unlike "all GaUl"): the Western Empire, ruled by Rome or Milan, and the Eastern, governed from Constantinople. While the former fell in 476 AD, the latter only reached its end with the capture of Constantinople by Mehmed II in 1453. In his Anatomy of Melancholy, which appeared in various editions from 1621 to 1676, Robert Burton wrote Now the instrumental causes of these our infirmities, are as diverse as the infirmities themselves; stars, heavens, ele• ments, &c. And all those creatures which God hath made, are armed against sinners ... The heavens threaten us with their comets, stars, planets, with their great conjunctions, eclipses, oppositions, quartiles, and such unfriendly aspects. The air with his meteors, thunder and lightning, intemper• ate heat and cold, mighty winds, tempests, unseasonable weather; from which proceed dearth, famine, plague, and all sorts of epidemical diseases, consuming infinite myriads of men. [Part 1, Sect. 1, Mem. 1, Subs. 1] [1877, pp. 83-84] Some two centuries later we find Gibbon writing Our habits of thinking so fondly connect the order of the universe with the fate of man, that this gloomy period of history [the reigns of Valerian and Gallienus in the 3rd century AD] has been decorated with inundations, earth• quakes, uncommon meteors, preternatural darkness, and a crowd of prodigies fictitious or exaggerated. [1896, vol. 1, p. 281] 132 Notes: On probability
Tyndall, in his essay "Reflections on prayer and natural law" , writes
In the fall of a cataract the savage saw the leap of a spirit, and the echoed thunderpeal was to him the hammer-clang of an exasperated god. Propitiation of these terrible pow• ers was the consequence, and sacrifice was offered to the demons of earth and air. But observation tends to chasten the emotions and to check those structural efforts of the intellect which have emotion for their base. One by one natural phenomena came to be associated with their proximate causes; the idea of direct personal volition mixing itself with the economy of nature retreating more and more. [1879, vol. 2, p. 1]
When the great comet of 1456 appeared with a tail extending over sixty degrees, it was
curved like a scymeter ... and in the superstitious spirit of that age, it was regarded as a celestial sign of the success of the Thrkish invasion of Europe, from its resemblance to a Thrkish sabre. [Lardner, 1859, vol. 2, p. 71] Lardner further embellishes Laplace's somewhat stark reporting of the comet, the Thrks and the pope as follows:
[The comet] was regarded as presaging the rapid success of the Thrks under Mohammed II., who had taken Con• stantinople, advanced to the walls of Vienna, and struck terror into the whole Christian world. Pope Calixtus II. [sic], terrified for the fate of Christianity, directed the thun• ders of the Church against the enemies of the faith ter• restrial and celestial, and in the same bull exorcised the Thrks and the comet; and in order to perpetuate this man• ifestation of the power of the Church, he ordained that the bells should be rung at noon, a custom still observed in Catholic countries. Neither the progress of the comet, nor the victorious arms of the Mohammedans, were, how• ever, arrested. The comet tranquilly proceeded in its orbit, passing through its appointed changes, regardless of the thunders of the Vatican, and the Thrks established their principal mosque in the Church of St. Sophia. [Lardner, 1859, vol. 2, pp. 79-80]
10. See Halley's Astronomical Tables . .. [1705a] and also his paper "As• tronomiae Cometicae Synopsis" published in the Philosophical Trans• actions in the same year. Notes: On probability 133
11. See Seneca's Naturales Quaestiones. The passage cited by Laplace runs as follows: The time will come when diligent research over very long periods will bring to light things which now lie hidden. A single lifetime, even though entirely devoted to the sky, would not be enough for the investigation of so vast a sulr ject. What about the fact that we do not divide our few years in an equal portion at least between study and vice? And so this knowledge will be unfolded only through long successive ages. There will come a time when our descen• dants will be amazed that we did not know things that are so plain to them. [1972, Book 7, Chapter XXV] A similar sentiment was expressed, though perhaps with more sym• pathy for human frailty, by Mackay, who wrote There are so many wondrous appearances in nature for which science and philosophy cannot even now account, that it is not surprising that, when natural laws were still less understood, men should have attributed to supernat• ural agency every appearance which they could not oth• erwise explain. The merest tyro now understands various phenomena which the wisest of old could not fathom. [1980, p.464]
12. This is discussed in Clairaut [1760]: see also his [1758]. 13. Compare the first and second principles given in the next article. Whether suppose is translated here as "implies" or "supposes" has a subtle effect on the meaning.
14. See Donkin [1851, p. 356] for a result bearing on the way in which probabilities ought to be changed.
15. In the Le~on this sentence is followed by the words "The apposite appreciation of the equally possible cases is one of the most delicate points of the analysis of chance," a sentiment that is repeated in T.A.P. II, §l. 16. The ne contiennent que ("contain only") of earlier editions appears erroneously as ne contiennent pas ("do not contain") in the edition put out by Bru and Thom. 17. Hartley comments on the effects of a narrative on the mind as follows: H it be asked, how a narration of an event supposed to be certainly true, supposed doubtful, or supposed entirely 134 Notes: On probability
fictitious, differs in its effect upon the mind, in the three circumstances here alleged, the words being the same in each, I answer, first, in having the terms true, doubtful, and fictitious, with a variety of usual associates to these, and the corresponding internal feelings of respect, anxiety, dislike, &c. connected with them respectively; whence the whole effects, exerted by each upon the mind, will differ considerably from one another. Secondly, If the event be of an interesting nature, as a great advantage accruing, the death of a near friend, the affecting related ideas will re• cur oftener, and by so recurring agitate the mind more, in proportion to the supposed truth of the event. And it con• firms this, that the frequent recurrency of an interesting event, supposed doubtful, or even fictitious, does, by de• grees, make it appear like a real one, as in reveries, reading romances, seeing plays, &c. This affection of mind may be called the practical assent to past facts; and it frequently draws after it the rational, as in the other instances above alleged. The evidence for future facts is of the same kind with that for the propositions concerning natural bodies, being like it, taken from induction and analogy. This is the cause of the rational assent. The practical depends upon the re• currency of the ideas, and the degree of agitation produced by them in the mind. Hence reflection makes the practical assent grow for a long time after the rational is arisen to its height; or if the practical arise without the rational, in any considerable degree, which is often the case, it will gener• ate the rational. Thus the sanguine are apt to believe and assert what they hope, and the timorous what they fear. [1834, pp. 208-209]
Recall too the words of Montaigne, who, in his essay "That One should interfere soberly in judging divine ordinances" (Book I, Chapter 32), wrote "nothing is so resolutely believed as that which one knows least" (see Montaigne [1958, p. 160] for an alternative reading).
18. In the sixth edition of 1672 of his Pseudodoxia Epidemica, Sir Thomas Browne cites as the third cause of common errors
the Credulity of men, that is, an easie assent to what is obtruded, or a believing at first ear, what is delivered by others. This is a weakness in the understanding, without examination assenting unto things, which from their N a• tures and Causes do carry no perswasion; whereby men often swallow falsities for truths, dubiosities for certainties, Notes: General principles 135
feasibilities for possibilities, and things impossible as pos• sibilities themselves. [Book I, chap. 5] In his essay "Science and the 'spirits'," Tyndall writes It [i.e. Science] keeps down the weed of superstition, not by logic but by slowly rendering the mental soil unfit for its cultivation. [1897, vol. 1, pp. 503-504]
19. Writing of a work by J[ames] L[aurie], de Morgan says Nothing, says the motto of this work, is so difficult to de• stroy as the errors and false facts propagated by illustri• ous men whose words have authority. I deny it altogether. There are things much more difficult to destroy: it is much more difficult to destroy the truths and real facts supported by such men. And again, it is much more difficult to prevent men of no authority from setting up false pretensions; and it is much more difficult to destroy assertions of fancy spec• ulation. Many an error of thought and learning has fallen before a gradual growth of thoughtful and learned opposi• tion. But such things as the quadrature of the circle, etc., are never put down. And why? Because thought can in• fluence thought, but thought cannot influence self-conceit: learning can annihilate learning: but learning cannot anni• hilate ignorance. [1915, vol. II, p. 6] 20. In his Religio Medici Browne writes
Where we desire to be informed, 'tis good to contest with men above our selves; but to confirm and establish our opinions, 'tis best to argue with judgments below our own. [1904, Part I, Sect. 6]
General principles of the probability calculus
1. Recall that in the previous article, "On probability," this ratio was defined as "la mesure de cette probabilite" (emphasis added). Kneale [1949] traces the indifference principle to James Bernoulli's Ars Con• jectandi [1713, Part IV, Chap. 4], where we read nullaque perspicitur ratio, cur htEc vel ilia potius e:cire debeat quam fJ'UlElibet alia, that is, "because no reason is seen why this, that or the other should occur" . The "equally likely" approach may be criticised on three scores: (1) it is a petitio principii, in as much as "equally possible" can mean noth• ing but "equally probable"; (2) how are "equally probable" cases to be determined - for example, what happens in a biased experiment? 136 Notes: General principles
(This is tied up with Laplace's second principle.); (3) one must rely on something besides evidence to decide when equiprobability is present - for example, each side of a fair die is assigned a probability of 1/6 because there is no reason to favour one side rather than another, because the die is geometrically equally favoured, etc. Further details may be found in the notes by Pollaczek-Geiringer in von Mises [1932, pp. 180 et seqq.]. Keller [1986], in a careful analysis of the coin-tossing case, shows that for a sufficiently large value of the initial velocity 11. (and possibly also of the angular velocity w), the {unique} probabil• ity of a head is approximately 1/2, no matter what the continuous prior density p( 11., w) may be. A similar example, dealing with rouge et noir, or a roulette wheel, is discussed in §4 of Chapter 11 of Poincare's Science and Hypothesis (see Poincare [1946, pp. 167-168]).
2. Burnside, in his Theory of Probability [1928], suggests that the gen• eral condition "all n results are equally likely" should be replaced by "each two of the n results are equally likely" (or, equivalently, "for each condition A the N A results which satisfy this condition are equally likely"). Such an interpretation of Laplace's principle permits its use in the definition of conditional probability. (Burnside, op. cit. p. 102, also argues for "assumed equally likely" rather than "equally likely".) See also Burnside [1925], and, for a study of equipossibility theories of probability, Hacking [1971]. The importance of classical equiprobability theory to modem sta• tistics should not be dismissed: Chuaqui [1991, pp. vii, 24, 25, 71] notes that, by using non-standard analysis, one can show that any random process (and hence any probability space) can be approxi• mated (in a precise sense) by a probability measure given in terms of equiprobability.
3. Possibly interpreted as an integral - the first edition of the En• cycloptedia Britannica [1771] in fact refers to calculus integralis or summatorius as a method of snmming up differential quantities; that is, from a differential quantity given, to find the quantity from whose differencing the given differential results. 4. Probabilites, in T.A.P. II, §1. On Laplace's distinction between ''pos• sibility" and ''probability'' see Hacking [1971].
5. The game croix 011. pile was at one time called "cross or pile" in English: see Brewer [1978] for suggested reasons for the name. 6. Laplace offers no suggestion as to how the equally possible cases should be determined. From the discussion in this paragraph, such Notes: General principles 137
determination appears to be based on permutations, each of the four outcomes H H, HT, T H, TT (where H stands for heads and T for tails) being assigned the same probability. The assignment of equal probabilities to the only possible combinations {H, H}, {H, T}, {T, T} will lead to the results of D'Alembert given in Laplace's next paragraph. Johnson, in the Appendix on Eduction to his Logic, Part III [1924], in fact uses both a combination postulate and a permuta• tion postulate in the development of his theory. Hacking [1975, p. 51] suggests that the attribution of equal probabilities to permutations rather than combinations is arguably an empirical fact.
7. See volume 4 of Diderot and D'Alembert's Encyclopedie [1754]. The article "Croix ou pile (analyse des hasards)" may be found on pages 512-513. Here D'Alembert discusses the results that conventionally obtain when a coin is tossed two or three times, and then writes Thus there are properly only three possible combinations: heads on the first toss; tails, heads on the first and second tosses; and tails, tails on the first and second tosses. Thus it is only a 2 to 1 bet. A similar argument is given for three tosses of the coin. However, in the second memoir "Reflexions sur Ie calcul des proba• bilites" of his Opuscules Mathematiques of 1761, D'Alembert changed his tune. The relevant passage is long, but since this partial recan• tation is often neglected in considerations of D'Alembert's work on probability, it seems expedient to give it here in full as follows: I would not, however, regard in all rigour the three events {results} in question here as equally possible. For (1) It could indeed happen (and I myself am inclined to believe it), that the case tails, heads is not exactly as possible as the case heads alone; but the ratio of the possibilities seems inestimable to me. (2) It could happen again that the event tails, heads is a little more possible than tails, tails, for the simple reason that in the latter case the same outcome happens twice in a row; but the ratio of the possibilities (supposing that they are unequal), is no easier to deter• mine in this second case, than in the first. Thus it could happen very often that in the case proposed, the ratio of the probabilities would be neither 3 to 1 nor 2 to 1 (as we have supposed in the Encyclopedie) but incommensu• rable or inestimable, a mean between these two numbers. I believe, however, that this incommensurable {number} is nearer to 2 than 3, because once again there are only three possible cases, and not four. I likewise believe, and for the 138 Notes: General principles
same reasons, that in the case in which one plays {a game consisting of} three tosses, the ratio of 3 to 1, given by my method, is much nearer the true value than the ratio of 7 to 1, given by the ordinary method - and which seems exorbitant to me. Better to fix the conditions of the question, let us re• strict our attention to the case in which one plays {a game consisting of} two tosses. At first sight it is certain that the probability of getting heads on the first toss, is equal to that of getting tails on the same first toss; the difficulty is reduced to knowing; (1) what is the ratio of the prob• ability of getting tails on the first toss, to the probability of getting heads on the second toss, when one has thrown tails on the first, and whether, as a result, there ought to be a second toss; (2) if the probability of getting tails on the second toss, when one has thrown tails on the first toss, is equal to or a little less than that of getting heads on the second toss, when one has thrown tails on the first; and if these probabilities are not equal, what is their ratio? (pp.21-22] Note also the comments on D'Alembert in Hacking [1971].
8. Les plus delicats ("the most delicate") in the Le~on. 9. The notion of "independence" occurs explicitly in the first edition of de Moivre's treatise The Doctrine of Chances of 1718, where we find the words it follows, that if a Fraction expresses the Probability of an Event, and another Fraction the Probability of another Event, and those two Events are independent; the Probabil• ity that both those Events will happen, will be the Product of those two Fractions. (p. 4] In the third edition of the Doctrine de Moivre provides a more precise definition: Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other. Two Events are dependent, when they are so connected together as that the Probability of either's happening is altered by the happening of the other. [1756, p. 6] 10. This principle is expressed more expansively in T.A.P. II, §1. In mod• ern notation it can be written as Notes: General principles 139
11. This word is missing in both the Le~on and T.A.P. II, §1. It appears that Laplace did not always find it necessary to distinguish between "possible cases" and "equally possible cases" - though perhaps we should follow the dictum of the economist Alfred Marshall who, ac• cording to Keynes [1972, p. 211], believed that "earlier writers of repute must be held to have meant what is right and reasonable, whatever they might have said." 12. The phrase un nombre de morceaux peu considerable is une epaisseur peu considerable ("a relatively small thickness") in the Le~on. 13. See also Bayes [1763, p. 378] and de Moivre [1756, §9]. The principle, which can be expressed in modern notation as
is given more precisely in T.A.P. II, §1 as follows: If the simple events are linked together in such a way that the supposition that the first occurs has an influence on the probability that the second occurs, the probability of the compound event will be given by determining: (1) the probability of the first event; (2) the probability that, this event having occurred, the second will occur. [O.C. VII, p.182] It is sometimes argued that attention needs to be paid to the time sequence of events (see, for example, Shafer [1982]). However the ex• ample following the above principle (loc. cit.) shows that no time factor (e.g. past versus future events) is intended here. It is only af• ter this that Laplace suggests that the example shows the influence of past events on future ones. (Notice, incidentally, that Burnside's extension of the equipossibility assumption - see Note 2 above - is needed in this example.)
14. That is, Pr[E2 lEI] = Pr[EI n E2] / Pr[EI]. 15. For instance de Beguelin [1767], and D'Alembert in the article "Prob• abilite" in Volume 13 [1775] of the Eneyclopedie, pp. 393-400.
16. This is followed in the Le~on by the (important) phrase les coups passes influent done alors sur la probabiliU des evenements futurs (''the past throws then therefore have an influence on the probability of future events").
17. Laplace was more cautious at this point in the Le~on, where the words le bonheur constant est were le bonheur est sou vent ("success is often"). 140 Notes: General principles
18. See Laplace [1774a] and [1778].
19. Laplace is not always careful to distinguish between absolute and con• ditional probabilities - a fact which occasionally makes the reading of his work rather difficult. This principle can be written as
Pr[E I Gi ] > Pr[E I Gj ] ==? Pr[Gi I E] > Pr[Gj I E] , Vi "I j
20. That is, Pr[Gi I E] = Pr[E I Gi ] / E j Pr[E I Gj ]. 21. See D'Alembert [1768].
22. In his Doutes et questions ... [1768] D'Alembert sets out three ar• rangements, viz.
Constantinopolitanensibus
or aabceiiilnnnnnooopssstttu
or nbsaeptolnoiauostnisnictn
and points out that while all three are mathematically speaking equally possible, no man of sense would regard them as equally possible phys• ically speaking. A similar example is considered in Crabbe's Natural Theology of 1840, where the finding of twelve of the twenty-four de• tached letters of the alphabet arranged in alphabetical order and in a straight line, is discussed. In this Crabbe finds ''the argument for the existence of a Sovereign Intellect, derived from chance" [po 28]. His other comments on chance are of interest; for example, he says "Where mind does not intervene, necessity exhibits the phenomena we call chance" [po 28], "Chance has a legitimate existence not as a cause, but as the effect, of undirected necessity" [po 29], "The doc• trine of chances, or, to speak more correctly, disorderly necessity, is not a chimera, but a legitimate branch of science, being under cer• tain rules" [po 29], "chance is not opposed to necessity ... Chance or disorderly necessity is opposed to design" [p.30]. In the same vein Sir Thomas Browne writes in his Religio Medici 'Tis not a ridiculous devotion to say a prayer before a game at Tables; for even in sortilegies and matters of greatest un• certainty, there is a setled and preordered course of effects. It is we that are blind, not Fortune: because our Eye is too dim to discover the mystery of her effects, we foolishly paint her blind, and hoodwink the Providence of the Almighty. [1904, Part I, Sect. 18] Notes: General principles 141
23. Here the Lefon has the phrase cet arrangement ne serait ni plus ni moins possible en lui-meme, et cependant (''this arrangement would be neither more nor less possible in itself, and yet"): the omission is perhaps unhappy. 24. Either "apprehend" or "comprehend" may be used to translate saisir: for the distinction see Fowler's Modem English Usage. 25. As one of the 12 fallacies to which he finds gamblers to be most prone, Epstein [1967] notes that one in which "The sample space of 'unusual' events is confused with that of low-probability events" [po 413]. In similar vein, Feller [1968] comments on the ill-advised description of the Poisson distribution as ''the law of small numbers or ofrare events" [§VI.7 p. 159], while Borel [1965] has much to say on probabilities that are negligible from the human, terrestrial, cosmic and supercosmic perspectives (note particularly Chapter 6). 26. That is, Pr[E2 I E1] = Li Pr[Gi I E1] Pr[E2 I Gi], where E1 and E2 denote the observed and future events respectively. It is sufficient to assume here that for each i,
Pr[E1 n E2 I Gi ] = Pr[E1 I Gi ] Pr[E2 I Gi ] This is not explicitly stated by Laplace. 27. Let H1 and H2 be the two possible hypotheses in this example, and Wi the event that a white ball is obtained on the ith draw, iE {1,2,3}. The quaesitum is seen to be Pr [Wa I W1 n W2]. In order to show, as Laplace claims, that 2 Pr[Wal W1 nW2] = E Pr[Hi I W1 nW2]Pr[Wa l Hi] , i=l it is sufficient to assume some sort of conditional independence - for instance, that Pr [W1 n W2 n Wa I Hi] = Pr [W1 n W2 I Hi] Pr [Wa I Hi] for each i E {1,2}. 28. Recall that Bayes [1763] in fact gave an argument for his choice of a uniform prior: see Dale [1982] and Stigler [1982], [1986b}. 29. This is essentially a discrete form of Bayes's Theorem, a version that today might be stated as follows: let {Ai} be a sequence of mutually exclusive and exhaustive events, each of positive probability, and let B be an event of positive probability. Then Pr[Ale I B] = Pr[Ale] Pr[B I Ale] En Pr[An] Pr[B I An} (Slight variations in the conditions of this theorem are possible.) 142 Notes: On expectation
30. That is, Pr[F I E] = l:i Pr[E I Hd Pr[F I Hd / l:i Pr[E I Hi]. 31. This is what John Venn, in his Logic of Chance [1866/1962]' named Laplace's Rule of Succession. Boldrini [1972, p. 133] refers to this rule as "traditional notation." Put not too precisely, the rule runs as follows: let an experiment, patient of only two outcomes, which we shall call "success" and "failure," be repeated n times, the constant probability of "success" in each of these independent trials being p. If r successes have been observed in these n trials, then the probability of a success on the next trial will be
1 1 pr+1{1_ pt-r dP/J pr(1- p)n-r dp = (r + 1) . J (n + 2) o 0 For a good discussion of this rule and various extensions of it see Zabell [1988] and [1989].
32. This well-known example of the rising of the sun was in fact first discussed by Richard Price in his communication of Bayes's Essay to John Canton of the Royal Society in 1763. Thus Price, rather than Laplace, may well be viewed as the originator of the rule of suc• cession. Laplace seems to take a year as 365.2426 days, to get his 1,826,213 days in 5,000 years (see Pearson [1978, p. 659]). Pearson (op. cit. pp. 659-660) notes that some justification or a reference to the Euler-MacLaurin Bridge is required for Laplace's passage from l: xp+1 / l: x P to Iol x p+1 dx / I; xP dx, when p is as large as 1,826,213. The early history of this example is discussed in Zabell [1988], while a detailed study of the rule of succession is given in Zabell [1989].
33. In all editions of the Essai this work is in fact cited as Arithmitique politique. The mistake might follow from the fact that, in Buffon's es• say entitled "Etat general des naissances, des mariages, et des morts dans la ville de Paris, depuis l'annee 1709 jusques et compris l'annee 1766 inclusivement" (an essay dealing with subjects in which Laplace is known to have been interested), there is a reference to "Le Cheva• lier Petty, dans son Arithmetique politique" . A similar statement may be found in the preface to Montmort [1713], viz. "1'Arithmetique poli• tique du Chevalier Petty" (p. xx].
On expectation
1. The word esperance is translated here sometimes as "hope" and some• times as "expectation." The former translation, bearing as it does a connotation of both desire and expectation, is perhaps more happily Notes: On expectation 143
contrasted with "fear", crainte, than the latter. Spinoza provides the following useful definitions: Hope is an inconstant pleasure, arising from the idea of something past or future, whereof we to a certain extent doubt the issue. Fear is an inconstant pain, arising from the idea of some• thing past or future, whereof we to a certain extent doubt the issue. [1919, p. 176] 2. The introduction of this term is usually attributed to Huygens [1657]: for comment on the contributions of Pascal and Fermat see Shoesmith [1983]. De Morgan applies the word "expectation" to ''that state of things for the production of which there is an even chance" [1838, p.88], the term "mathematical expectation" being defined in the RULE. Multiply each gain or loss by the probability of the event on which it depends; compare the total result of the gains with that of the losses: the balance is the average required, and is known by the name of the mathematical expectation. [po 97] 3. This problem, known today as the St Petersburg pamdox, was solved by Daniel Bernoulli [1730-1731]. The question was initially put by Nicholas Bernoulli to Montmort - see the 1713 edition (i.e. the sec• ond) of the latter's Essay d'analyse sur les jeux de hazard (the rele• vant letter does not appear in the first edition). The question was also asked of Buffon by Cramer - see the former's Essai d'arithmetique momle. According to Pollaczek-Geiringer, in her notes to von Mises [1932], the difficulty seen by Laplace vanishes when the correct col• lective is defined. See Jorland [1987] and Szekely [1986, p. 28] for discussions of the problem. Proctor [1889, pp. 148-161] discusses the role of the Russian government ("which has at all times been notably ready to take advantage of scientific discoveries" lop. cit., p. 148]) in a lottery based on the system of this paradox. The plan of the game can be changed so as to ensure fairness; indeed, Feller notes that It is perfectly possible to determine entrance fees with which the Petersburg game will have all the properties of a "fair" game in the classical sense, except that these entrance fees will depend on the number of trials instead of remaining constant. Variable entrance fees are undesirable in gam• bling halls, but there the Petersburg game is impossible anyway because of limited resources. [1968, p. 252] 4. See his "Specimen theoriae novae de mensura sortis" [1730-1731]. The rule given in §12 runs as follows: 144 Notes: On expectation
Any gain must be added to the fortune previously pos• sessed, then this sum must be raised to the power given by the number of possible ways in which the gain may be obtained; these terms should then be multiplied together. Then of this product a root must be extracted the degree of which is given by the number of all possible cases, and fi• nally the value of the initial possessions must be subtracted therefrom; what then remains indicates the value of the risky proposition in question. [1730-1731; Sommer's trans• lation, p. 28] See Daston [1988, p. 94] for a discussion of Buffon's version of moral expectation. 5. That is, if y = R(x) denotes the relative value of the sum x, then dy = dx/x, whence y = lnx, or, more generally, y = bln(x/a), where a and b are arbitrary constants and a is positive. Suppose now that a player starting with fortune a has chance P. of gaining x., where LiP. = 1, and let Y=Lbp. In(a+x.)-blna . • IT X is the physical fortune that corresponds to this moral fortune, then Y = blnX -blna , and so X = IT(a + X.)Pi . • It is then X - a that Laplace refers to as the "physical fortune that procures for an individual the same moral advantage that results to him from his expectation." Bernoulli also considers more general functions y = ",(x) in his memoir: for further details see Todhunter [1865, art. 381]. 6. Literally, ''whose value may always be supposed to be non-zero", but we shall assume that values are always non-negative. See Todhunter [1865, art. 379]. 7. A similar notion had previously been expressed by Daniel Bernoulli, who wrote There is then nobody who can be said to possess nothing at all in this sense unless he starves to death. For the great majority the most valuable portion of their possessions so defined will consist in their productive capacity, this term being taken to include even the beggar's talent. [1730-1731; Sommer's translation, p. 25] Notes: On expectation 145
8. In a footnote to her translation of Bernoulli's "Specimen theoriae novae de mensura sortis" Sommer provides a literal translation of Bernoulli's emolumentum medium as "mean utility" (p. 24]. The the• ory of expected utility received its major thrust in von Neumann & Morgenstern [1944], though earlier work had been done by F.P. Ramsey. "Utilities," writes DeGroot, "are numerical representations of [one's] tastes and preferences" [1970, p. 86]. For further details see Fishburn [1970]. Cournot was not very flattering in his opinion of the difference expressed between mathematical and moral expectation: he wrote By the distinction between mathematical and moral expec• tation people have wanted to explain the advantage that assurance policies procure for the persons assured, each of them providing a source of profit for the underwriter. But, to our mind, these explanations, despite the names of the authors who have proposed them, are vague and arbitrary, and there is no real reason to have recourse to them. [1843, §184, p. 334] 9. The argument is more clearly given in D. Bernoulli's "Specimen theo• riae novae de mensura sortis," in particular §§4,7,1O,11,12. See Chap• ter 10 of T.A.P. II for details of Laplace's treatment of this matter. 10. The argument here seems to be based on the fact that, for any pin the interval (0,1),
{I + x)P {1- X)l-p $ 2PP {1-p)l-p •
11. This same example is given in Bernoulli [1730-1731]. 12. Suppose that the player, starting off with a, has chance Pi of winning the amount Xi, i E {1,2}. Then the physical fortune expected is {a + xl)Pl{a - X2)P2. If the game is fair, so that (pdP2) = (X2/Xl), then X2 Xl Pl= P2= i Xl +X2 Xl +X2 and it follows, if we regard Xl and X2 as integers, from the geo• metric/arithmetic mean inequality that the above expression for the physical fortune is less than a. {Note that the fortune may be written
[{a + Xlrl:2 (a _ X2)Zl]l/(Zl +Z2) ,
that is, the geometric mean of X2 quantities each equal to a + Xl and Xl quantities each equal to a - X2i while the arithmetic mean is
x2{a + Xl) + xl{a - X2) Xl +X2 which reduces to a.) 146 Notes: On analytical methods
13. R.A. Proctor has devoted the whole of his Chance and Luck [1889] to an exposure of the drawbacks of, and errors inherent in, gambling, and he remarks at the end of his preface "I wish I could hope that it [his book] would serve the higher purpose of showing that all forms of gambling and speculation are essentially immoral". Proctor had clearly studied - or at least read - Laplace's E88aij the example given in footnote c-c in the present translation of the article "On expectation" is given in his book (without attribution). Further, the preface begins "The false ideas prevalent among all classes of the com• munity, cultured as well as uncultured, respecting chance and luck, illustrate the truth that common consent (in matters outside the in• fluence of authority) argues almost of necessity error" - a sentiment similar to that which is expressed in the second paragraph of the ar• ticle "On probability" in the E88ai.
On analytical methods in the probability calculus 1. Writing of this article, Todhunter says This section may be regarded as a complete waste of spacej it would not be intelligible to a reader unless he were able to master the mathematical theory delivered in its appropriate symbolical language, and in that case the section would be entirely superfluous. [1865, art. 935] It is nevertheless a discussion of what was an important part of Laplace's work, and as such merits at least an attempt at under• standing on our part.
2. That is, (1 + a)n = E:=o (:)a 8 • For a detailed study of the binomial and cognate numbers see Edwards [1987]. The "choose" symbol is due to A. von Ettingshausen - see Cajori [1929, vol. II, §439].
3. That is, the number of permutations of n things taken 8 at a time is (:) x 8! = n(n - 1) ... (n - 8 + 1). Notice that Laplace writes n(n -1) ... (n - r + l)jr! for the binomial coefficient (;) (see T.A.P. II, §3). The former, in lending itself more readily to generalization, is perhaps preferable. 4. As Pollaczek-Geiringer has pointed out (see von Mises [1932, p. 190]), there is a slight error here: what Laplace gives as n - 8 in the pre• ceding sentence should be r - 8. For what is required is the number of combinations of n - 8 (non-distinguishable) items taken r - 8 at a time (i.e. from n - 8 items, a sample of r - 8 is taken). Clearly
T-8-s _ 8 i§2n - illn . r 8 Notes: On analytical methods 147
Alternatively, and perhaps more simply, one can consider the choice of n - r items from n - 8, which gives the numerator of the left-hand side of the above expression as (:=~), which of course is equal to (;=:), as before. 5. In this lottery, tickets bearing numbers in the set {I, 2, ... ,90} were sold and draws for winning numbers were held in different French cities, five numbers being taken on each draw. He who won on a single number won 15 times his stake, while those who won on 2, 3, 4 or all 5 numbers received 270, 5,500, 75,000 and 1,000,000 times the stake. The numerical values of the probabilities of winning these amounts are 1 2 1 1 1 18' 801' 11,748' 511,038' 43,949,268 respectively. The expectations, for a stake M, are
(15x l8 -1)M = -!M
(270 x 8~1 -l)M
= _1562 M (5,500 x 11~48 - 1) M 2937 etc. In each case the expectation of the ticket holder is negative, while that of the runner of the lottery is positive (for further details see Uspensky [1937, pp. 19--20, 108-109]). Laplace was not in favour of this lottery. It was introduced into France in 1758 and abolished in 1832. For further discussion see Cournot [1843, §8]. According to Szekely [1986], The first public lottery awarding money prizes, the Lotto de Firenze, was established in Florence in 1530. [po 17] For general remarks on lotteries see Proctor [1889, pp. 135, 138-144]. 6. More correctly, the coefficient of ambn- m. 7. This Polla.czek-Geiringer (see von Mises [1932, p. 191]) sees as an example of the solution of the so-called Bernoulli problem, wherein the probability wn(m) that, from an urn containing a white and b black balls, m white balls are drawn in n draws (with replacement) is wn(m) = (:)pmqn-m ,
where p = aJ(a + b) and q = bJ(a + b). The number of cases in which this result obtains is then (a + b)nwn(m), or (;;')ambn- m. Laplace's contribution to probability, in connexion with this matter, was the 148 Notes: On analytical methods
limiting form as n -+ 00, that is, the exp(-x2 ) law. De Moivre's approximation to the binomial distribution is discussed in Hald [1990, chap. 24] and Stigler [1986b, pp. 70-88], while Laplace's extension of de Moivre's theorem is examined in Hald lop. cit. §24.6]. 8. See Laplace [1773], [1774b]. 9. This problem is a simplified version of one proposed and solved by Waldegrave ("an English Gentleman, a member of the family of milord Waldegrave, who had married a natural daughter of King James"), and detailed in a letter from Montmort to Nicolas Bernoulli of the 1st of March, 1712, the latter replying on the 2nd of June, 1712 (these letters are reprinted in Montmort [1713]). A modern discussion of the problem may be found in Hald [1990, §21.2].
10. Symbolically, Pn = ~ Pn-l. 11. The functional equation of the previous note, together with the initial condition P2 = ~, leads to Pn = (~)n-l. Hence the probability that the game ends on or before the nth round is ~; (~)i-l = 1- (~)n-l. 12. See (Euvres de Blaise Pascal, vol. 3 [1908], or (Euvres de Fermat, vol. 2 [1894]. The pertinent correspondence consists of two letters: one from Fermat to Pascal in 1654, and a reply of the 29th of July of that year. The problem was apparently put to Pascal by the Chevalier de Mere, though in Fermat's Varia Opera of 1679 the name is left blank. Pascal describes the proposer of the question as follows: il a tres bon esprit, mais il n'est pas geometre (c'est, comme vous s~avez, un grand de/aut) - that is, "he has a very good mind, but he is not a mathematician (that is, as you know, a great defect)". A summary of the whole business is given in Quetelet [1849, p. 252]. 13. This question, the problem 0/ points, is discussed in Edwards [1987], Hald [1990, §§4.2, 5.3, & 14.1] and Todhunter [1865, see the Index].
14. That is, Pm,n = ~(Pm-l,n + Pm,n-l). 15. Consider the first order partial difference equation of the preceding note, together with the initial conditions Po,n = 1 , Pm,O = °given in the text. Then, for example,
PI,1 = ~(Po,l + PI,O) = ~
PI,2 = ~(Po,2 + PI,I) = i
P2,1 = ~(Pl,l + P2,o) = ~ etc. Pollaczek-Geiringer (see von Mises [1932, pp. 192-193]) shows that if we suppose, more generally, that the initial chances of A's and Notes: On analytical methods 149
B's winning are p and q respectively, and if the players lack m and n points to complete the game, then n-l Pm,n = pm ~ (m)i qi Ii! , i=O where (x)n is Pochhammer's symbol, which is defined by (x)o = 1 and (x)n = x(x + 1) ... (x + n - 1). On putting P = q = ! in this formula, we get, for example, P2,3 = ~~. 16. See Laplace [1782]' [181Ob] and T.A.P. I. The term was also used by Condorcet [1769]. Examples of Laplace's method are discussed in Rald [1990, pp. 450-452]. For a study of the early history of generating functions see Seal [1949] and Sheynin [1973], and for a mathematical treatment see MCBride [1971]. 17. This "integrating" is to be understood as an operation inverse to that of taking finite differences. This operator is denoted by p-l (Milne• Thomson [1933]) or .6-1 (Jordan [1965], Richardson [1954]). 18. What is meant here is a system of n partial finite difference equations with n dependent indices (or variables), and an arbitrary number of independent indices. 19. Jordan writes The origin of this Calculus may be ascribed to Brook Tay• lor's Methodus Incrementorum (London, 1717), but the real founder of the theory was Jacob [sic] Stirling, who in his Methodus Differentialis (London, 1730) solved very ad• vanced questions, and gave useful methods, introducing the famous Stirling numbers. [1965, p. 1] 20. See de Moivre's Miscellanea Analytica [1730, pp. 28-33]: his work on recurring series is discussed in Rald [1990, pp. 370-372]. 21. See Lagrange [1775, pp. 151-251]. 22. This paragraph is extraordinarily difficult to translate without using mathematical symbols - Pearson in fact writes I do not believe that even a first-class mathematician would appreciate what this calculus [Le. that of generating func• tions] means by reading Laplace's verbal descriptions of it! [1978, p. 662] I have allowed myself considerable licence in the translation, and shall give, in this note, the details in a mathematically assimilable form. Consider the geometric progression a + ar + ar2 + ar3 + . .. , 150 Notes: On analytical methods
with scale of relation
Un - rUn-l ,n ~ 1 . (1)
Now replace n by n - 1, and multiply the result by the constant k: this yields kUn-l - krun-2 . Subtraction of this from the original scale of relation (1) yields
Un - (r + k)Un-l + krun_2 .
Similarly, consider the geometric progression
with scale of relation
Vn - kVn-l ,n ~ 1 . (2)
On replacement of n by n - 1 and multiplication by r in (2), we get
rVn-l - krvn-2 .
Subtraction of this from (2) results in
Vn - (r + k)Vn-l + krvn-2 . Now add the two given progressions together. This gives
the associated scale of relation being
Wn - (r + k)Wn-l + krwn_2 . (3) Now consider the recurring series (of order 2)
Co + Clt + C2t2 + C3t3 + ...
with scale of relation (4)
Setting 0:1 = r + k and 0:2 = rk we get the equation 0:2 = r(O:l - r), or (5) so that the recurring series of order 2 may be written as the sum of two geometric progressions if (5) has two distinct real roots, these Notes: On analytical methods 151
roots in fact being the common ratios of the geometric progressions. As an example of this last point, consider the recurring series
eo +Cl +C2 +C3 + ...
where eo = 1, Cl = 1 and, for n :2: 2, Cn = 6Cn-l - 8Cn-2. Then C2 = -2 , C3 = -20, C4 = -104, ... and the given series is then 1 + 1 - 2 - 20 - 104 - ... (6) In this case (5) above becomes r2 - 6r + 8 = (r - 2)(r - 4) . We thus form the two geometric progressions
On setting
2a + 4b = 1 (= eo) ; 4a + 16b = 1 (= Cl) , we get a = i ' b = - ~. Substitution of these values in (7) and the summing of the two geometric progressions yields
i(2 + 22 + 23 + ...) - ~(4 + 42 + 43 + ...)
= 1 + 1 - 2 - 20 - 104 .. · , which is exactly the recurring series given in (6). More details on this matter can be found in de Moivre [1711], [1718] and [1730], Bald [1984] and [1990, pp. 370-372, §23.1], Ball & Knight [1948] and von Mises [1932, p. 194]. 23. The notation used in the 5th edition (V, T, etc.) is different to that used in the 1at edition. Further, Laplace uses t, x, 6 etc. in the 5t h, but not in the 1at.
24. Thus V(t) = E~-oo azF is the generating function of the sequence (az ) - or of the set {az }.
25. We may write this as follows: suppose that V(t) = Eaztz. Then (1 + 2t)V(t) = Ez(az + 2az_l)tZ , and cp(l)(x) = az + 2az-l is the "derived" function of the primitive function cp(x) =a z. H we also have (1 + 2t)V(t) = E bztZ , then the sequence (bz) = (az + 2az-l). (I have purposely not indicated the range of summation - here and elsewhere; this can easily be done by the reader.) 26. I have avoided Laplace's term "derivative"; its use, here and else• where, is open to misinterpretation. 152 Notes: On analytical methods
27. In the previous example (see Note 25), oax == ax + 2ax-l.
28. If V(t) = LaxtX , then (l/t)V(t) = Lax+ltx. Notice that negative exponents may appear.
29. Thus if V(t) = LaxtX , then
Z(t)V(t) == [(l/t) - l]V(t) = L(ax+l - ax)tX ,
30. Symbolically,
vrn = V(Z + l)n
n = L ('~)VZj j=O 3
But we have already seen that vrn = Lax+ntX ; and hence, on equating coefficients of t X , we have
Using Z = T - 1 we can show similarly (see von Mises [1932, p. 197]) that
31. With T and Z as defined here,
32. The technical details of this paragraph may be expressed - not al• together rigorously - as follows: let V(t) = Laxtx. Then
V(t)[(l/t) - It = L(L\nax)tX • x Notes: On analytical methods 153
Let V denote the finite integral operator. The primitive function of E:z: Vn(ana:z:)t:Z: is found by noting that
Evn(ana:z:)t:Z: = E(anx)[(I/t) - l]n . :z: :z:
n Notice that ana:z: = E (-1); (j)a:z:+; (Le. the index changes in steps ;=0 of 1), while 6na:z: = a:z:+n (i.e. the index changes in steps of n units). 33. That is, write the difference equation in the form
f(a:z:+n,a:z:+n-b ... ,a:z:) = 0,
and substitute 1 for a:z:+n, t for a:z:+n-l, t2 for a:z:+n-2, etc. 34. About 150 years later the same sentiment was expressed by MCBride, who wrote For a given set of functions it is desirable to have as many generating functions as possible from which to choose the one best suited for a particular use. [1971, p. 5] 35. Once again it is perhaps easier to see what is going on here if we resort to symbolism. Let 6 denote the difference operator and suppose that Vet) = EYlctlc and 6yIc == aoYo + alYl + ... + anYn , an:l 0 . Setting 6yIc equal to zero, we can write Yn = bn-1Yn-l + bn-2Yn-2 + ... + boYo (1) On multiplying V in turn by bn-1t, bn_2t2, ... ,b1tn- 1 and botn, and on subtracting the sum (bn-lt + bn_2t2 + ... + bltn- 1 + botn)v from V, we find, using (1), that
V = Yo + (Yl - YObn-1)t + ... + (Yn-l - Yn-2bn-l - ... - Yobdtn- 1 1 - bn-lt - bn_2t2 - ... - botn Notice that the denominator in this ratio (the differential scale, in de Moivre's work) may be found by writing (1) in the form
Yn - bn-1Yn-l - bn- 2Yn-2 - ... - boyo = 0 and by substituting 1 for Yn, t for Yn-l, t2 for Yn-2, ... , and tn for Yo. See also Hald [1990, pp. 371 (equ!l (26» and 427]. 36. If Vet, t') = E a:z:,:z:lt:Z:t':Z:/, then
37. This interpolation is more clearly described in T.A.P. I, §§3, 4. The third section begins with the words The whole theory of the interpolation of series reduces to the determination, for every value of i, of the value of Yx+i, as a function of the terms that precede or that follow YX. Denoting by u(t) = L, YxtX the generating function of the sequence x (yx), Laplace derives the expressions
Yx+i =
i i (i2 _1) 3 3 +2 (~Yx + ~ Yx-d + 2 1.2.3 (~ Yx-l + ~ Yx-2)
and
1 ( ) [(i+!)2-ll 1 (A2 A2) Yx+i = 2 Yx + Yx-l + 1.2 2 u Yx-l + u Yx-2
[(i+!)2-l1[(i+!)2-tll(~4y +~4y )+ ... + 1.2.3.4 2 x-2 x-3
[(i+!)2-11 [(i+!)2-tl ~5 Y + ... } + 1.2.3.4.5 x-3 , formulae to be used for the interpolation between an odd number 2x + 1 and an even number 2x of equidistant quantities respectively. Laplace concludes §4 by writing All these expressions {the ones for Yx+i given above and others similarly derived} for Yx+i are identical, and are such that if One imagines a parabola having i as abscissa and Yx+i as ordinate, and with equation giving the expression for Yx+i, this curve will pass through the points with ordi• nates Yx, Yx+b Yx+2, etc.j Yx-b Yx-2, etc. Thus, by taking successive finite differences of any number whatsoever of co-ordinates, one may draw a parabola through the points having these co-ordinates. 38. See Leibniz's Opem Omnia [1768] (ed. Dutens), vol. 3. Note in partic• ular §LXXVI [pp. 421-425] entitled "Symbolismus memorabilis cal• culi algebraici & infinitesimalis, in comparatione potentiarum & dif• ferentiarumj & de Lege Homogeneorum Transcendentali". Notes: On analytical methods 155
39. See Lagrange [1772].
40. Let V = E ak,ltk sl, and suppose that T = t + s - 2, as in the text. k,l Then VT = E(ak-l,l + ak,l-l - 2ak,l) tksl . On denoting the parenthesized expression by 6ak,l, we obtain, on setting it equal to zero,
6ak,l == ak-l,l + ak,l-l - 2ak,l = 0 . On using this relation one finds that
V = [!t(s) + h(t)] /(t + s - 2) ,
a similar result holding for a general T.
41. See Laplace [1779] and T.A.P. I, pp. 77-80. The problem, that of the vibrating string, had been the subject of some dispute between D'Alembert, Euler, Daniel Bernoulli and Lagrange.
42. Leibniz Opera Omnia [1768] (ed. Dutens), vol. 3, §LXXII [pp. 406- 410], "Epistola ad v. d. Christianum Wolfium, professorem Matheseos Halensem, circa scientiam Infinite.» (Here Leibniz deals with infinite series for (1-x)-t, (1 +x)-t, (1 + x2)-1, and term-by-term integra• tion. He also considers what happens when 1 is substituted for x in such expansions.) 43. In Figure 1, I1x == E, and s denotes the subsecant. Clearly y l1y Ey -; = E' or 7 = l1y
Now y = f(x) ===* y + l1y = f(x + I1x) , and so (y+l1y)-y f(x+l1x)-f(x) ll.x = I1x 1 = ll.x [f(x) + I1x.f'(x) + ... - f(x)]
-+ f'(x) as I1x -+ 0 .
For any P point on a curve (see Figure 2), the orthogonal projections of the normal and the tangent onto the x-axis are sn and st. 156 Notes: On analytical methods
y
L1 Y I ----l I
o x x+L1x x ...... __ ---5 ---..~~...-.- L1 x --II~~
Figure 1.
Y
,
~------~~~~~.. x
Figure 2. Notes: On analytical methods 157
44. Boyer [1959, p. 164] notes that, even though Lagrange, Laplace and Fourier have all described Fermat as the true originator of the calcu• lus, "Poisson has correctly pointed out that Fermat does not deserve such a designation, inasmuch as he failed to recognize the problem of quadratures as the inverse of that of tangents." Cajori [1929, ,539, p. 181] notes - contra Laplace - that "This symbolism [e.g. dx] was immensly superior to the a chosen by Fermat as the small increment of x." 45. Edwards shows that the "binomial theorem for positive integral index can therefore be attributed to AI-Kashi (1427) at the very latest" [1987, p. 52]. It in fact seems reasonable to believe Omar Khayyam's claim to have written on the raising of the binomial to various powers (see Edwards loco cit.). 46. Leibniz Opera Omnia [1768] (ed. Dutens), vol. 3, §XIII [pp. 167-172], "Nova methodus pro maximis et minimis, itemque tangentibus, quae non fractas, nec irrationales quantitates moratur, & singulare pro illis calculi genus." See also §XIX [pp. 188-194], "De geometria recondita et analysi indivisibilium atque infinitorum." 47. See Leibniz Opera Omnia [1768] (ed. Dutens), volume 3, §XLVII [pp. 301-302], "Considerations sur la difference qu'il y a entre l'Ana• lyse ordinaire, & Ie nouveau Calcul des Transcendantes". See also his Nouveaux essais sur l'entendement par l'auteur du systeme de l'harmonie preestablie, IV, chap. 7, No.6. 48. Laplace [1778] and [1782]. 49. See Jordan [1965, §174 - in particular, p. 579]. 50. Laplace [1782] and T .A.P. II, §33. The technique was independently used by Euler: see his Opera Omnia, series 1, vol. 19. Pertinent papers from this volume are "De integrationibus maxime memorabilibus ex calculo imaginariorum oriundis" [pp. 1-44], and "Ulterior disquisitio de formulis integralibus imaginariis" [pp.268-286]. 51. La Geometrie. J. Maire, Leyden [1637]. 52. Arithmetica Infinitorum [1657]. See also the first Supplement to T.A.P., (Euvres Completes VII, pp. 473-479. 53. On Wallis's use of induction see Scott [1938, p. 30]. 54. That is, a~ = vram.
55. Thus a-m = l/am . 158 Notes: On games of chance
56. For a discussion of Wallis's product, viz.
1 1+1 2+9 2+25 2+49 :n-;---;--2 + etc. , which approaches nearer and nearer to the exact value, the terms being alternately greater than and less than ~, is discussed in Chapter 4 of the latter work. 57. This opinion is endorsed by Scott [1938, p. 35]. The relevant passage in the letter from Newton to Oldenburg of the 13th June, 1676, runs as follows: For as analysts, instead of aa, aaa, etc., are accustomed to write a2, a3 , etc., so instead of ..;0., n, ..;c: a5 , etc. I write ai, at, ai, and instead of l/a, l/aa, l/a3 , I write a-I, a-2 , a-3 • [Newton, 1960, p. 3~ 58. The calculus exponentialis is defined in the first edition of the Ency• c1op£dia Britannica as "a method of differencing exponential quan• tities, and summing up the differentials of exponential quantities" [1771, vol. 2, p. 8]. 59. Leibniz, Opera Omnia [1768] (ed. Dutens), vol. 3, [pp. 416-425]. 60. Lagrange, (Euvres [1867-1892], vol. 3, [pp. 441-476].
On games of chance 1. Among those who had contributed to this topic, up to the time at which Laplace was writing, we can mention the Bernoullis, Fermat, Montmort and Pascal. 2. In T.A.P. II, Chapter 2, §4, Laplace shows that this probability is
.,::In[s(s - 1) ... (s - r + 1)Ji [n(n - I) ... (n - r + I)]i where s is to be put equal to 0 in the result. Notes: Unknown inequalities between chances 159
3. Laplace [1774b], [1780], [1783b] and T.A.P. II, Chapter 2, §4. 4. See Laplace [181Ob] and T.A.P. II, §1O. Fieller [1930-1931], using more extensive tables than were available to Laplace, found that the latter's conclusion was wrong, as a result of the neglect of the ap• proximate nature of the result. The odds are in fact exactly the same for 23,780 and 23,781 games: Fieller gives the required probability as 0.516687.
On unknown inequalities between chances
1. Laplace [1774a], [1778] and T.A.P. II, §34. 2. Rather more generally than in the text, suppose that the schemes
(a) Pr[H] = ~ - f, Pr[T] = ~ + f ;
(b) Pr[H] = ~ + f, Pr[T] = ~ - f ,
are equally probable. Then
3. This follows from the fact that, for p E (0,1), (p - (1 - p)]2n ~ 0, with equality if and only if p = ~. (Expand the bracketed term and gather the even and odd terms together separately.) 4. This paragraph may be interpreted as follows (see Pearson [1978, p. 664] and Pollaczek-Geiringer's note in von Mises [1932, p. 200]):
First coin Second coin
H T H T
~+p ~-p ~+q ~-q
By a conditional probability argument (and recall that a T on the second coin names the tail side of the first coin a head),
Pr[H1] = Pr[Hl I H 2] Pr[H2] + Pr[Tl I T2] Pr[T2]
= (~+p)G+q)+G-p)(~-q)
~ +2pq. 160 Notes: Probability & the repetition of events
Thus (assuming independence!)
Pr[H1H1] = (~+ 2pq)2 Now this results from supposing that p and q are both positive - that is, that the bias is in favour of heads in each coin. The other cases yield ~ -2pq , if p > 0 and q < 0 Pr[HtJ = 1~ - 2pq , if P < 0 and q > 0 ~ + 2pq , if P < 0 and q < 0 Thus, on averaging over these four possibilities, and regarding them as equally probable, we get
Pr[H1H1] = ~ [(~ + 2pq)2 + (~ _ 2pq)2 + (~ _ 2pq)2 + (~ + 2pq)2]
= ~ +4~q2. Since q2 < ~, we clearly have l + 4~q2 < l + p2, the right-hand side of this inequality being the probability of getting heads twice in succession with only one biased coin (cf. Note 2 above). 5. The dependantes of the 1st edition of the Essai appears (probably by error) as independantes in the Bru/Thom edition. 6. The process described here is a Markov chain: for further reference see Sheynin [1976, p. 166].
Probability & the repetition of events 1. This article is developed in T.A.P. II, Chapter 3. 2. Consider an urn containing N balls, W of which are white, and sup• pose that w white balls are drawn in a sample of size n (we shall not worry about the (important) distinction between random variables and their values). Let p denote the probability that a white ball is drawn from the urn. Then the theorem as given in the first edition of the Essai is
('
('
3. See Bernoulli's Ars Conjectandi, Book 4. Laplace makes no mention of de Moivre's work on this result - see Hald [1990], Pearson [1924] and Stigler [1986b]. 4. According to Hacking, the "dead letter" question
was discussed by polymath Thomas Young in 1819. He as• sured his readers that it implied no 'mysterious fatality', but the example was used for decades. [1990, p. 117] 5. These principles were mentioned in the foreword to the Essai. 6. This passage appears only after the first edition (of February 1814): by the time of the issue of the second edition, in November 1814, Napoleon had been deposed. Indeed, the first edition (1812) of the T.A.P. was dedicated to Napoleon, the dedication running, in trans• lation, as follows: Sire, The benevolence with which Your Majesty has deigned to entertain my 'Ih1.ite de Mecanique Celeste has inspired me with the desire of dedicating to You this Work on the Probability Calculus. This delicate calculus encompasses the most important questions of life, which are indeed, for the most part, only problems in probability. In this respect it ought to interest Your Majesty, whose talent it is fully to appreciate and properly to encourage everything which may contribute to intellectual progress and public prosper• ity. I venture to beseech You graciously to receive this ad• ditional token of respect inspired by the keenest gratitude and by the deepest sentiments of admiration and respect, with which I am, Sire, Your Majesty's very humble and very obedient servant and faithful subject, Laplace. Writing of the suppression of this dedication in subsequent editions of the Essai, Todhunter says Laplace has been censured for suppressing this dedication after the fall of Napoleon; I do not concur in this censure. The dedication appears to me to be mere adulation; and it would have almost a satire to have repeated it when the tyrant of Europe had become the mock sovereign of Elba or the exile of St Helena: the fault was in the original publication, and not in the final suppression. [1865, art. 931] 162 Notes: Probability & the repetition of events
Pearson's views are similar: he writes However we may judge of Laplace's original rendering unto Caesar of that which is Caesar's, it is perfectly clear that no publisher in 1814 could be found, or if found would have been permitted, to reprint in Paris in the year of the Emperor's deposition that dedication! [1929, p. 209] and, in the same paper, The permission to dedicate a new and important work to the sovereign was, in that day, equivalent to the statement that the book was approved by the state and it thus formed a much desired and excellent publisher's advertisement. [1929, p. 210] In a similar, though perhaps less adulatory, vein, Quetelet began his first letter to H.R.H. the Grand Duke of Saxe Coburg and Gotha with the words
The interest which your Highness has seemed to take in the study of the Theory of Probabilities, and especially in the application that can be made of it to the moral and political sciences, encourages me to hope that the new developments which I shall have the honour of presenting to you will not be unworthy of your attention. There is scarcely in fact a branch of our knowledge whose aim can be more philosophical, or more directly useful. [1849, p. 1] 7. Laplace's possibilites des evenements simples is translated, here and elsewhere, in von Mises [1932, pp. 47-48] as Ausgangswahrschein• lichkeiten - i.e. "initial probabilities" . 8. See Laplace [1774b], [1778] and T.A.P. II, Chapters 2 & 6. Pollaczek• Geiringer (von Mises [1932, p. 202]) describes it as ''the second Law of Large Numbers"; and she regards the Bayes problem as apparently (sichtlich) an inverse of the Bernoulli problem, although she perhaps goes further than is strictly correct in her subsequent discussion, when she frames the Bayes-Laplace theorem as follows: consider an urn containing black and white balls, and sup• pose that 0: is the relative frequency of white balls in n draws. Then the probability that x, the true proportion of white balls in the urn, lies between 0: - f and 0: + f tends to 1 as n tends to infinity. This result is independent of the nature of the a priori probability. [op. cit., p. 203] 9. This is really an inverse to Bernoulli's theorem - see Dale [1988]. Notes: Probability & the repetition of events 163
10. The discussion of the last two paragraphs is perhaps more clearly expressed in Chapter 6 of T.A.P. II. The results may be summarized by saying that, if X denotes the unknown probability of the simple event, and if E denotes the observed event, then
f3n / 1 Pr[an < X < ,an] = ! Pr[E I X = X] dx ! Pr[E I X = X] dx , an 0 where an and ,an are numbers whose behaviour with increasing n is given in the paragraphs under comment. 11. This topic is discussed in T.A.P. II, Chapter 6. 12. In Black's translation of Humboldt [1811] we read
In the capital of Mexico there were born in five years, be• tween 1797 and 1802,
In the parishes of Male births Female births
the Sagrario 3, 705 3, 603
of Santa Cruz 1,275 1,167
At Panuco and Y guala, two places situated in a very warm and very unhealthy climate, there was not one register in which the excess was not on the side of the male births. [Footnote: At Panuco, the parish registers give, from 1793 to 1802, for 674 male births, 550 female births. At Y guala there were 1,738 boys for 1,635 girls.] In general, the pro• portion of male to female births appears to me, in New Spain, to be as 100 : 97; which indicates an excess of males somewhat less than in France, where for 100 boys there are born 96 girls. [Book II, chap. 7, p. 190] 13. See Note 33 on the article "General principles of the probability cal• culus" . 14. The figures given by Buffon for the period 1770 to 1774 were in fact 36 boys and 37 girls. Laplace seems to have confused these figures with the total number of births of all parishes before Carcelle-Ie-Grignon in Buffon's list (a list of parishes having more female than male births in that period, the total numbers for the 42 parishes being 1,690 male and 1,840 female births). The result is a straightforward application of Bayes's Theorem: if p denotes the probability of the birth of a girl, then, with a uniform 164 Notes: Probability & the repetition of events
prior and under the assumption that m male and n female births have been observed,
1 1 Pr[p > ~] = Jxn(l- x)mdx / Jx n(l-x)mdx. 1/2 0 With m = 983 and n = 1,026 (Laplace's figures), the (exact) value 0.8312486408 is obtained for this probability: it is 0.5462196914 for Buffon's m = 36 and n = 37 (less than 1 - (~)2, the chance of not getting two heads in two tosses). For a different interpretation, see Pearson [1978, pp. 665-666]. 15. See Laplace [1783a]. 16. 2,037,615 : (110,312 + 105,287)/3 = 28.3528448. The method used here is identified by Pearson [1978, p. 666] as that of John Graunt. For further details of the latter's work on life-tables see Hald [1990]. 17. See T.A.P. II, §31. Pearson [1978, p. 666] considers Laplace's odds to be wrong, because of the assimilation of this question to an urn problem. 18. In this translation the word "infinite" may often be understood as being synonymous with "indefinitely large" . 19. According to Hacking [1990, pp. 59-60], Charles Babbage added an• other factor - that of differential infanticide - to the reasons dis• cussed by Laplace for the proportional excess of male over female births. "Babbage," notes Hacking, obtained the results of the Prussian census of 1828 and the ratios of male and female births for the preceding decade, cross-classified as illegitimate and legitimate. Among the legitimate, males exceed females by 10.6 births to 10, as opposed to less than 10.3 to 10 for the illegitimate. [1990, p. 60]
20. Pearson [1978, p. 666] finds "a more reliable explanation" for the birth ratio observed in the Foundling Hospital to be that the bulk of the foundlings were illegitimate children, of whom a far larger number are stillborn or die at birth than of legitimate children; this great difficulty and mortality at birth would affect more seriously the male sex. In the case of stillborn children, the sex-ratio is far higher than in the case of viable children. There is also higher mortality of males than of females in the case of first-borns, who would form a large majority of the foundlings. [loco cit.] Notes: Application to natural philosophy 165
21. See, for example, Arbuthnot [1710] and's Gravesande [1774a]; details of these writings may be found in Stigler [1986b, pp. 225-226] and Hald [1990, §§17.1 & 17.2].
22. Pearson [1978, p. 666] notes that "Laplace only replaces Providence by 'regular causes' ", and he goes on to discuss sex ratios from a genetic viewpoint.
23. The odds are quoted as 2 : 1 in Laplace [1783b]: see also T.A.P. II, §33.
24. This example was first given in Daniel Bernoulli [1769]: see also Laplace [1810b] and also T.A.P. II §17, in which latter work dif• ference equations are used to solve the problem. Pollaczek-Geiringer (in von Mises [1932, pp. 203-205]), on the other hand, uses stochastic matrices in her solution.
25. An early model of the type described (for two urns) here was proposed by Daniel Bernoulli [1766-1767], [1769]. A comprehensive survey of urn models is provided in Johnson and Kotz [1977] - see, in partic• ular, §4.8 on urn transfer models.
26. See Laplace [1809] and T.A.P. II, Chapter 4. This result is now known as the centrollimit theorem.
27. Thus if X denotes the error, or deviation from the true value, based on n observations, then, for some constant k < 1,
As Pearson [1978, p. 652] has noted, the ratio Laplace is trying (some• what unsuccessfully) to express without using symbols is really
Application to natural philosophy
1. See Laplace [1816a], [1818a] & [1818c] for further development of part of the content of this section, one which, in the first edition, was headed Du calcul des probabilitfs, applique a la recherche des phenomenes et de leurs causes ("On the probability calculus, applied to the examination of phenomena and their causes"). 166 Notes: Application to natural philosophy
2. The present nice distinction between ''variables'', "parameters", "es• timators" and "estimates" was not observed in Laplace's time. It seems that by elements here Laplace usually means what we today would call "parameters", for in the second supplement to the T.A.P. he writes
the analytic expression of these laws depends on constant coefficients which I shall call parameters {elements in the original}. Using probability theory, one can determine the most probable values of these parameters, and if, on substi• tuting them into the analytic expressions, these expressions are satisfied by all the observations, within the limits of pos• sible error, one will be sure that these are laws of nature, or at least that they are not very different from such laws. [O.C. VII, p. 558]
However, I have also translated elements by "estimators" where it seemed more appropriate to do so. Of course, in connexion with plan• etary orbits, the word "elements" would be correct (the elements of an orbit are its size, shape and position, together with the time at which the planet is at some particular position - see Moulton [1922, p.248]). 3. The situation here is analogous to that in regression, where, to con• sider only the simplest case in which E{Yi) = 0: + (3Xi and Var{Yi) = (72, i E {I, 2, ... , n}, we find estimators a and ~ of 0: and (3, and where, under additional assumptions about the distribution of the Yi, something can be said about the (joint) distribution of a and ~. More particularly, and less rigorously, suppose that y = !(x), and that z is the correction to be made to the approximate value x. Then
y = !(x) = !(x + z) !(x) + zf'{x) + ... Thus, approximately, y = !(x) + zf'{x), or, in the notation used in T.A.P. II, §20,
(3 = h+ pz.
Now Laplace supposes that z is itself susceptible of an error €, so that
(3 + € = h + pz, or € = pz - 0:, (1) Notes: Application to natural philosophy 167
where a: = (3 - h. It is (1) that is the equation of condition. Sim• ilar equations are obtained for each of s observations, addition of these equations giving
8-1 8-1 8-1 ~fi = Z~Pi - ~a:i , i=O i=O i=O and on supposing that the sum of these errors is zero, we get
Alternatively, we may suppose that some linear combination ~ mifi of the errors is zero, which assumption leads to
Z = ~miai, /~miPi, . Extensions to more than one element and a comparison with least squares may be found in T.A.P. II, §§2o-24.
4. That is, the error law is proportional to e-h2z2 , where, in modern notation, h2 = 1/(2u2), with u2 denoting the variance. The positive square root h is called the precision (see Lindley [1980, p. 8]), while h2 is what Laplace calls the weight.
5. Thus if n observations Xl, X2, ... ,Xn of the ''true'' value a are made, the optimal estimate of a is then ~~ h~ Xi /~~ h~, where hi is the weight of Xi, and ~ h~ is the weight of the estimate. By the appro• priate choice of the original law, we may summarize the discussion here as follows:
{Xj}i '" N .I.D.(lLj, uJ) ::} X '" N ( ~lLj /n , ~uJ /n 2 )
6. That is, if X '" N(O, 1/2h2 ), then
f3 2 Pr[a: < X < (31 =.foh Je- h z 2 dx, a
or, more generally, if X", N(IL, u 2 ), then
7. That is, if X has mean °and variance u 2 , then X / u has mean °and variance 1; and in particular, if X", N(O, u2 ) then X/u '" N(O, 1). 168 Notes: Application to natural philosophy
8. Roughly speaking, this says that the weight h2 is proportional to [E(Xi - X}2]-1, which we recognize as the reciprocal of the more common expression that the sample variance, that is, 82 , is propor• tional to E(Xi - X}2. 9. This is the least-squares rule, among whose early proponents Legen• dre and Gauss should be mentioned (see Plackett [1972]). 10. Pearson [1978, p. 668] notes that "[Laplace] had not seen that there was a fundamental distinction between large and small samples." However, at the end of Chapter 4 of T.A.P. II we find Laplace writing when there are a small number of observations, the choice of these systems {that is, the systems used in finding the estimators} depends on the law of the errors of each obser• vation. But if one considers a large number of observations ... this choice becomes independent of that law ... [po 354] 11. A. Bouvard [1806-1808]. 12. J. Bradley [1798-1805]. 13. According to Laplace [1818a], Bouvard used the equations of condi• tion of Jupiter's movement to conclude that the mass of Saturn was 3,~12 th of that of the sun. Laplace's formulae, also applied to these equations of condition (126 in number), showed that the mass of Sat• urn lay between the limits 3 5i.t..08 ± l~ with probability ~~:g~. In the first supplement to T.A.P., these limits are given as
1 ±(1 1) 3,512.3 100 x 3, 534.08
with probability 11,327 11,328' Laplace goes on to say, in the first supplement, Newton has found, using Pound's observations on the great• est elongation of Saturn's fourth satellite, that the mass of this planet is equal to 1/3,012, which exceeds the preceding result by a sixth. One may then bet millions of milliards to 1 that Newton's result is in error, and this is not surprising when one considers the difficulty of observing the greatest elongations of Saturn's satellites. [po 520] Modem versions of the mass of Saturn vary: for example, Lang [1980, p. 526] gives the inverse mass of this planet (with the mass of the sun taken as I) as 3,498.5, while Moore et al. [1983, p. 440] give 3,499.5 (in this latter work the absolute mass of Saturn is stated to be 5.684 x 1026 kg). See Sheynin [1984a] for a history of statistical methods in astronomy. Notes: Application to natural philosophy 169
14. Laplace, in [1818a] and T.A.P. (first supplement), gives the mass of Jupiter, as determined by Newton from Pound's observations, as l,~7 th that of the sun. The quotation given in the preceding Note is followed by the words
The ease of observing [the greatest elongations] of Jupiter's satellites has, as we have seen, made the value derived by Newton from Pound's observations much more precise. [p.520] Poisson [1837, §113] notes that other posterior observations have yielded a different value for this mass, which he gives as l,JSO th of that of the sun. The work cited is by (i) Encke on the perturbations of the comet of period 1,024 days, (ii) Gauss and Nicolal, on the per• turbations of the minor planets Vesta and Juno, and (iii) G.B. Airy, on the elongations of Jupiter's satellites. Poisson (loc. cit.) attributes the error in Laplace's result to inaccuracies in some of the measure• ments entering into his equations. G.B. Airy (1801-1892) showed in 1837 that the mass of Jupiter was l,~7 th that of the sun, a figure that is endorsed by Lang [1980, p. 526] and Moore et al. [1983, p. 440].
15. See the second and third supplements to T .A.P.
16. Recall that, if {Xi} is a sequence of independent random variables each Normally distributed with precisions {hi}, then ~~ aiXi has a Normal distribution with weight
1 n 2 "" a· H2 = L..J h~ . 1 '
17. On reasons for the choice of the simplest laws in science, see Good [1950, p. 60] and Jeffreys [1983, pp. 47, 414]. Newton, in Book III of his Principia Mathematica, gives the following rule: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes. [1934, p. 398]
18. This result seems to have some similarity to one of Legendre's: see Grattan-Guinness [1990, pp. 339--342] and Legendre [1799a] & [1799b]. (Laplace held Legendre's viscera in abomination.) 170 Notes: Application to natural philosophy
19. The import of this last phrase, which appears again in the following paragraph, seems to be that the probability density function becomes more sharply peaked as the number of observations increases. 20. Recall that the arithmetic mean is that value of u which minimizes E~(Xi-U)2, while the median is the value of u minimizing E~ IXi-ul. 21. In the second Supplement to the T.A.P. Laplace discusses some ex• amples in which his methode de situation is preferable to the optimal method. He concludes that there is then an advantage in thus correcting the result given by the optimal method. When one is ignorant of the probability law of the errors in the observations this cor• rection becomes impracticable. [O.C. VII, p. 580] 22. The writer of the notes to the Bru/Thom edition of the Essai sug• gests that this passage was added to the 5th edition contra Gauss (ef. T.A.P. II, p. 353). 23. See T.A.P. II, §25 (pp. 360-362). 24. In Brahe's Astronomiae Instauratae Progymnasmatum, pars secunda, we find the words Experiments applied to many things prove that the corre• sponding motions of the Moon do not conform to the same equalisation of the natural days, which the sun produces, except in so far as it depends on that one's {Le. the sun's} real motion, in which that degree of difference, as it were, is absorbed. [1915, p. 101] 25. Mayer [1770]. 26. Possible references are Mason [1780J, Mayer & Mason [1787], and Lalande [1792J. 27. See Laplace [1813] and his Traite de Mecanique Celeste, Book XVI, Chapter 1 (O.C. V, p. 408). 28. Burg [1806]. 29. Laplace [1786]. 30. Lagrange [1783]. 31. Laplace [1785a], [1785b] and T.A.P. II, §25 (pp. 362-364). 32. Today Jupiter is known to have 13 satellites (see Lang [1980, p. 543]), the named ones being 10, Europa, Ganymede and Callisto (the others are Jupiter V to Jupiter XIII). Notes: Application to natural philosophy 171
33. The writer of the notes to the Bru/Thom edition of the Essai points out that it is not so much the probability calculus that was used here, but rather the repeated coincidence of observations with the simple law ml - 3m2 + 2m3 = 7r. See Laplace [1788} and [1789a]. 34. See Pliny's Natu.ral History, Book II; the relevant passage is a long one, but I give it in full here (in translation) for completeness: XCIX. About the nature of bodies of water a great deal has been said. But the rise and fall of the tides of the sea is ex• tremely mysterious, at all events in its irregularity; however the cause lies in the sun and moon. Between two risings of the moon there are two high and two low tides every 24 hours, the tide first swelling as the world moves upward with the moon, then falling as it slopes from the midday summit of the sky towards sunset, and again coming in as after sunset the world goes below the earth to the lowest parts of the heaven and approaches the regions opposite to the meridian, and from that point sucking back until it rises again; and never flowing back at the same time as the day before, just as if gasping for breath as the greedy star draws the seas with it at a draught and constantly rises from another point than the day before; yet returning at equal intervals and in every six hours, not of each day or night or place but equinoctial hours, so that the tidal peri• ods are not equal by the space of ordinary hours whenever the tides occupy larger measures of either diurnal or noc• turnal hours, and only equal everywhere at the equinox. a It is a vast and illuntinating proof, and one of even divine ut• terance, that those are dull of wit who deny that the same stars pass below the earth and rise up again, and that they present a similar appearance to the lands and indeed to the whole of nature in the same processes of rising and setting, the course or other operation of a star being manifest be• neath the earth in just the same way as when it is travelling past our eyes. Moreover, the lunar difference is manifold, and to begin with, its period is seven days: inasmuch as the tides, which are moderate from new moon to half-moon, therefrom rise higher and at full moon are at their maximum; after that they relax, at the seventh day being equal to what they were at first; and they increase again when the moon di-
a The Roman hour was a twelfth part of actual daytime or night-time, thus varying in length throughout the year; and only at the equinox was a diurnal hour equal to a nocturnal hour, an exact twenty-fourth of day and night. 172 Notes: Application to natural philosophy
vides on the other side, at the union of the moon with the sun being equal to what they were at full moon. When the moon is northward and retiring further from the earth the tides are gentler than when she has swerved towards the south and exerts her force at a nearer angle. At every eighth year the tides are brought back at the hundredth circuit of the moon to the beginnings of their motion and to corresponding stages of increase. They make all these increases owing to the yearly influences of the sun, swelling most at the two equinoxes and more at the autumn than the spring one, but empty at mid-winter and more so at midsummer. Nevertheless this does not occur at the exact points of time I have specified, but a few days after, just as it is not at full or new moon but afterwards, and not imme• diately when the world shows or hides the moon or slopes it in the middle quarter, but about two equinoctial hours later, the effect of all the occurrences in the sky reaching the earth more slowly than the sight of them, as is the case with lightning, thunder and thunder-bolts. But all the tides cover and lay bare greater spaces in the ocean than in the rest of the sea, whether because it is more furious when moved in its entirety than when in part, or because the open extent feels the force of the starb when it marches untrammeled with more effect, whereas narrow spaces hinder the force, which is the reason why neither lakes nor rivers have tides like the ocean (Pytheas of Mar• seilles states that north of Britain the tides rise 120 ft.) But also the more inland seas are shut in by land like the water in a harbour; yet a more untrammeled expanse is subject to the tidal sway, inasmuch as there are several instances of people making the crossing from Italy to Utica in two days in a calm sea and with no wind in the sails when a strong tide was running. But these motions are observed more round the coasts than in the deep sea, since in the body too the extremities are more sensitive to the pulse of the veins, that is of the breath. But in most estuaries owing to the different risings of the stars in each region the tides occur irregularly, varying in time though not in method, as for instance in the Syrtes. 35. A relevant passage from Kepler's Astronomia nova runs as follows (see also Donahue [1992, p. 56]):
b i.e. the moon Notes: Application to natural philosophy 173
The orbit of the attractive force, a force which is centred on the Moon, extends all the way to the Earth, and it draws the sea towards the Torrid Zone, particularly when it is overhead in one or other of its passages, imperceptibly in an enclosed sea, perceptibly there where the Ocean bed is deepest, and the sea has great freedom in returning by the same path, and when this occurs the shores of the Zones and of the bordering Regions are exposed, and anywhere even in the Torrid Zone the bays of the neighbouring Ocean are more withdrawn. Therefore, the sea having arisen in the deeper bed of the Ocean, it may happen that in its narrower bays, even in bays that are not particularly narrowly con• fined, the sea may be seen, while the Moon is present, to flee away from it; so the waters subside, as the opening takes away much of the water. When indeed the Moon passes quickly over the zenith, since the sea is unable to follow so swiftly, the Ocean flows even through the Torrid Zone towards the West, until it strikes the opposite shores, and is made to change its di• rection by them; the coming together of the waters is de• stroyed by the waning of the Moon, like an army that is on the march towards the Torrid Zone, inasmuch as it has been abandoned by a region which had summoned it to help; and having mastered the attack, as in the case of sea-going vessels, it returns and attacks its own shores and covers them; and that force produces, through the absence of the Moon, another force; until the Moon returning takes control of this force, and regulates it, and drives it around together with its own motion. Thus shores that are equally exposed are all at the same time filled; indeed they are later more drawn back; some in different ways on account of the various openings of the Sea. [1609, p. 26] According to Sheynin, Kepler ... considered himself to be the founder of scientific astrology, a science of the general influence of heaven upon earth. [1978, p. 487] The history of the use of statistics in meteorology is discussed in Sheynin [1984b]. 36. See Newton's Principia Mathematica. In Book I, Proposition LXVI, Theorem XXVI, Corollaries XVIII and XIX, Newton considered the motion of fluid bodies and the movement of water on the surface of the globe, while in Book III, Proposition XXXVI, Problem XVII he 174 Notes: Application to natural philosophy
considers how "To discover the force of the Sun for moving the Sea" , and in Proposition XXXVII, Problem XVIII he proposes "To discover the force of the Moon for moving the Sea." For general remarks on the influence of the moon on the weather see Sheynin [1984b, §2]. 37. See Laplace [1775], [1776], [1790], [1818b], [1818d], as well as items 555-567 (mainly from Laplace's Mecanique celeste) in the Tables generales, O.C. XIV, pp. 449-550. 38. See Laplace [1790], [1818b] and [1818d]. 39. Laplace [1818b] and Stigler [1975]. 40. Ramond de Carbonnieres [1808-1811], [1813-1815a] and [1813-1815b]: for details of his work see Le Breton [1813]. 41. Laplace [1825]. 42. A modern work (Lang [1980, p. 543]) identifies five satellites of Uranus and ten of Saturn. In 1986 Voyager 2 raised these numbers to 15 each - see Stewart [1989, p. 16]. 43. Laplace's "43 motions" are made up as follows:
11 + 18 + 1 + 6 + 1 + 4 + 1 + 1 , these numbers representing the 11 planets, 18 satellites, and the rota• tions of the sun, the six planets, the moon, Jupiter's satellites (Lang [1980, p. 543] gives 13 satellites about Jupiter), Saturn's rings and one of its satellites.
44. The odds are 243- 1 (i.e. 4,398,046,511,104) to 1, "for all might go round either ways!" (Pearson [1978, p. 668]).
45. See Laplace [1796]. The Exposition du Systeme du Monde, like the Essai Philosophique sur les Probabilitis, went through five editions; and also like the Essai, it served as a popularisation of views expressed in more detail elsewhere - in this case the M ecanique Celeste, the five volumes of which were published in 1799 (I & II), 1803 (III), 1805 (IV), and 1825 & 1827 (V). 46. See Herschel [1786], [1789], [1791] and [1802]. 47. This is then a sort of ergodic result. 48. Michell [1767]. 49. Bessel [1818]' [1822(?)] and [1841-1842]. Notes: Application to natural philosophy 175
50. See Laplace's Exposition du Systeme du Monde (O.C. VI), Book V, Chapter 6 (pp. 476-478 in particular), and the "Note VlIet Derniere" (pp. 498-509). A pertinent passage runs as follows: thus one may conjecture that the planets have been formed at these successive limits, by the the condensation of the belts of gases, which it must, on cooling down, have left behind in its equatorial plane. [po 500] while on p. 504 we have "Under our hypothesis, the comets are alien to the planetary system." The New Scientist of the 10th of August, 1991, carried a report of a computer simulation of the solar system by George Wetherill of the Carnegie Institute of Washington. Assuming the concentra• tion of millions of planetesimals in a ring half an astronomical unit wide about the sun, Wetherill produced four planets similar to the inner planets of the solar system. In the presence of a Jupiter-like planet, and under the assumption that the planetesimals were con• centrated in a ring between 0.45 and 3.3 astronomical units from the sun, repeated simulations yielded on the average (27 repetitions) 4.2 planets between "Jupiter" and the "sun", one of these usually being approximately of the size, and in the position, of the Earth.
51. Whether "meteors" or "meteorites" is meant here is uncertain: the original reads aerolithes. 52. Moulton notes, however, that
The most remarkable thing about the head of a comet is that it nearly always contracts as the comet approaches the sun, and expands again when the comet recedes ... John Herschel suggested that the contraction may be only appar• ent, the outer layers of the comet becoming transparent as it approaches the sun. This suggestion contradicts the ap• pearances and seems to be extremely improbable ... When a comet is far from the sun, its tail is small, or may be entirely absent; as it approaches the sun, the tail develops in dimensions and splendor, and then diminishes again on its recession from the sun. [1922, pp. 316-317]
53. In Laplace [1816b], the odds are given as 8,263 to 1. 54. Halley's comet, according to Moore et al. [1983, p. 402], has been traced back to 240 BC, and it may also be the comet which was seen in 467 BC. It returned in 83 BC and 11 BC. Returns after the date given by Laplace were in 1835, 1910 and 1986. Moore et al. (loc. cit.) point out that "Because of perturbations by Jupiter and Saturn, the 176 Notes: Probability & the moral sciences
revolution period of the Comet is not constant, and may be anything between 74 and 78 years." 55. This comet, one of Jupiter's family (because its aphelion is near Jupiter's orbit - see Moulton [1922, p. 319]), was discovered by J.L. Pons in 1818: its periodicity - 3.3 years - was established by J.F. Encke. 56. The major researcher in this field at this time was A.G.A.A. Volta. 57. A phenomenon investigated by F.A. Mesmer. 58. See Gavarret [1840].
Probability & the moral sciences 1. Writing of the French term, Hacking explains that Science momle does not denote that priggish entity that we in English call morals. It is more to be understood as a science of mreurs, of customs, of society. In the course of ef• fecting its mid-nineteenth-century reforms, Cambridge Uni• versity introduced a faculty of moral sciences to embrace economics, politics, psychology, metaphysics and ethics. [1990, pp. 37-38] 2. The importance of probability in statecraft was stressed by Arbuth• not in the preface to his translation of Huygens's De Ratiociniis in Ludo Alere, where he wrote all the Politicks in the World are nothing else but a Kind of Analysis of the Quantity of Probability in casual Events, and a good Politician signifies no more, but one who is dextrous at such Calculations; only the Principles which are made use of in the Solution of such Problems, cannot be studied in a Closet, but acquired by the Observation of Mankind. [1770, vol. 2, p. 260.]
On the means of a number of observations 1. The mean error as given in the Essai, viz. V''LJx; - zi /n.j2; , may be obtained from that given in the discussion of T.A.P. II, §20, by Todhunter [1865, arts 1007-1009] as
VC£q?K'LJl?) - (Eq; 6;)2 / V27r n Eq? Notes: On the probability of testimony 177
Figure 3.
by setting each qi = 1. This amounts to supposing that Ai + U, rather than Ai +uqi, is the corrected value of the Ai> the approximate value of the function (whose determination is required) at the ith observation. 2. Consider Figure 3, in which h denotes the (true) height of the object being measured, Ii is the "given distance", and d the distance from which h is actually measured. Let hI and h2 denote the measurements of h made from Ii and d respectively (note that h2 corresponds to the estimate given by the rule in the text). If the angles (It and (}2 are approximately the same, then
hI h2 -~-Ii d' or
3. See T.A.P. II, §20, and also the second displayed equation in Note 1. 4. That is, the least-squares rule - see Stigler [1981].
On the probability of testimony 1. The eleventh chapter of T.A.P. II is devoted to this topic, one which was not discussed in the first edition of the Theorie. The subject was a favourite one at that time, and indeed it had received attention at earlier times - see, for example, Craig [1699], and its modern in• terpretation in Stigler [1986a], Condorcet [1785], and Lambert [1764] (see Sheynin [1971a]). Augustus de Morgan's views on testimony are 178 Notes: On the probability of testimony
discussed in Hailperin [1988]. According to Sheynin, "Markov de• clared that the study of testimonies was the weakest section of the theory of probability" [1989, p. 340]. For modern comment on testi• mony and miracles see Kruskal [1988] and Sobel [1987]. Mill was not impressed by applications of the probability calculus to questions of testimony or judicial decisions. In Book III, Chapter XVIII, §3, of the second edition of 1846 of Mill [1843] we read
It is obvious, too, that even when the probabilities are de• rived from observation and experiment, a very slight im• provement in the data, by better observations, or by tak• ing into fuller consideration the special circumstances of the case, is of more use than the most elaborate application of the calculus to probabilities founded on the data in their previous state of inferiority. The neglect of this obvious re• flection has given rise to misapplications of the calculus of probabilities which have made it the real opprobrium of mathematics. It is sufficient to refer to the applications made of it to the estimation of the credibility of witnesses, and of the correctness of the verdicts of juries. In regard to the first, common sense would dictate that it is impossible to strike a general average of the veracity, and other qual• ifications for true testimony, of mankind, or of any class of them; and if it were possible, such an average would be no guide, the credibility of almost every witness being either below or above the average. And even in the case of an individual witness, persons of common sense would draw their conclusions from the degree of consistency of his statements, his conduct under cross-examination, and the relation of the case itself to his interests, his partiali• ties, and his mental capacity, instead of applying so rude a standard, (even if it were capable of being verified,) as the ratio between the number of truths and falsehoods which he may be supposed to tell in the course of his daily life. Again, on the subject of juries, or other tribunals, some mathematicians have set out from the proposition that the judgment of anyone judge, or juryman, is, at least in some small degree, more likely to be right than wrong, and have concluded that the chance of a number of persons concur• ring in a wrong verdict is diminished, the more the number is increased; so that if the judges are only made sufficiently numerous, the correctness of the judgment may almost be reduced to certainty. I say nothing of the disregard shown to the effect produced on the moral position of the judges by multiplying their numbers; the virtual destruction of Notes: On the probability of testimony 179
their individual responsibility, and weakening of the ap• plication of their minds to the subject. I remark only the fallacy of reasoning from a wide average, to cases necessar• ily differing greatly from any average. It may be true that taking all causes one with another, the opinion of anyone of the judges would be oftener right than wrong; but the argument forgets that in all the cases in which it is really of much consequence what the tribunal is, the proposition might probably be reversed; besides which, the cause of er• ror, whether arising from the intricacy of the case or from the prejudices or infirmities common to human nature, if it acted upon one judge, would be extremely likely to affect all the others in the same manner, or at least a majority, and thus render a wrong instead of a right decision more probable, the more the number was increased. 2. "Probability," wrote Joseph Butler in 1736, "is the very guide of life" [Butler 1834, p. xxii]. Laplace's view that this guide is also to be regarded as a guard is endorsed by Lucas [1970, p. 1]. Hume's thoughts, however, were slightly different: in his Inquiry concerning Human Understanding (Essay XXXIX in Hume [1894]), he asserted that "Custom ... is the great guide of human life" [po 334]. The notion was also discussed by Voltaire [1772], who wrote Almost the whole of human life revolves around probabil• ities. Everything that is not proved to our eyes, or recog• nized as true by people clearly interested in repudiating it, everything is, at best, only probable ... Since uncertainty is almost always the lot of man, you would very rarely make up your mind if you had to wait for proof. However it is necessary to take sides: and it is not necessary to do this by chance. It is then necessary to our weak nature, blind, always subject to error, to study probabilities with as much care as we learn arithmetic and geometry. [1843, vol. 5, p.609] 3. This problem is a generalization of one from Condorcet [1784]. For criticism of Laplace's method see Poisson [1837, §§36-40] and Cournot [1843, §§224-225]. 4. The figure is incorrectly given as 1/10 in the Bru/Thom edition of the Essai. 5. Laplace has here dans ("in" or "under"), but a joint probability seems to be meant, and so "and" is a more natural translation. 6. The absence of symbols, and the lack of a clear distinction between conditional and unconditional probabilities, make this paragraph a 180 Notes: On the probability of testimony
bit obscure. Here is my interpretation of the example. Let 0, E, HI and H2 be defined as follows: o 79 announced (i.e. the observed event). E 79 drawn. HI the witness tells the truth. H2 the witness lies.
Our first job is to determine Pr[OIHI ] and Pr[0IH2] "a priori", as Laplace says. Now
Pr[79 announcedlwitness tells the truth]
= Pr[79 drawn]
1 = 1,000 Thus
Pr[O n HI] = Pr[OIHI ] Pr[HI ]
1 9 9 = --x- 1,000 10 10,000 .
To determine Pr[0IH2], notice firstly that if the witness announces 79, but we know that he lies, then 79 was not drawn. The probability that any number other than 79 was drawn is 999/1,000. Further, the probability that from the 999 numbers not drawn he chooses 79 as the one to be announced is 1/999. Thus
Pr[0IH2] = Pr[79 announced Iwitness lies] 999 1 = --x- 1,000 999
1 = 1,000· Thus
Pr[0IH2] Pr[H2]
1 1 = --x- 1,000 10
1 = 10,000 . Notes: On the probability of testimony 181
Then, by the sixth principle,
Pr[OIH1) Pr[H1) Pr[H1IO) = Pr[OIH1) Pr[H1) + Pr[OIH2) Pr[H2)
Pr[O n HI) =
9 10· That is,
Pr[E) Pr[79 drawn)
Pr[witness tells the truth 179 announced)
9 10' and similarly Pr[E) = 1/10. 7. The example discussed in this paragraph may be clarified as in the discussion in the preceding note. The only point needing further am• plification is perhaps the derivation of the upper bound 10/121. My understanding of it can be given as follows: 1 10 1 --x-=-- 1,000 10 1,000
999 1 1 111 = 1,000 x 9 x 10 = 10,000 ' and hence
Pr[E) = Pr[H1IO) = 10/121.
It does, however, seem strange that we can put Pr[HI) = 10/10 while keeping Pr[H2) at 1/10. 8. Laplace has here "and" rather than "given", but a conditional prob• ability is clearly meant. 9. "White" is mistakenly given as "black" in the Essai.
10. The example discussed in this paragraph may be clarified as in the discussion in Note 6. In a more general notation, however, it runs as follows: let an urn contain n (> 1) balls, n - 1 of which are black while 1 is white, and let 0, E, HI and H2 be defined by 182 Notes: On the probability of testimony
o the white ball is announced as having been drawn (the observed event). E the white ball is drawn. Hi the witness tells the truth. H 2 the witness lies. Further, let Pr[Hd = p. Then, as before, 1 Pr[OnHd Pr[OIHd Pr[Hd = - x p n n-l Pr[OIH2] Pr[H2 ] = -- x (1 - p) . n Thus
Pr[OIHd Pr[Hd + Pr[OIH2] Pr[H2 ]
(n - 1)(1 - p)/n (p/n) + (n - 1)(1 - p)/n
(n-l)(I-p) (n-l)(I-p)+p' or, in words,
Pr[H210] = (number of black balls) x Pr[witness lies] numerator + Pr[witness tells the truth] (the reason for this formulation will become apparent in the next note). The examples discussed here are criticised by Mill [1846, Book III, Chapter XXV, §6]: a useful discussion of Mill's and Laplace's posi• tions is given in Pearson [1978, pp. 680-684]. 11. The argument in this paragraph is more fully given in T.A.P. II, Chapter XI, §45. Briefly, the discussion there runs as follows: con• sider an urn containing 1 white and n - 1 black balls. One ball is drawn, and a witness to the extraction asserts that the ball drawn is white. Let O,E,Hi,H2,H3 and H4 be defined by
o the white ball is asserted to have been drawn (the observed event). E the white ball is drawn. Hi the witness tells the truth & is not misled. H2 the witness tells the truth & is misled. H3 the witness lies & is not misled. H4 the witness lies & is misled. Notes: On the probability of testimony 183
Suppose that
Pr [the witness tells the truth] = p Pr [the witness is not misled] = r, and that telling the truth and being misled are independent events. Then (cf. Notes 6 and 10) 1 - xpr n n-l = --- xp(l-r) n n-l = --- x (l-p)r n 1 Pr[OIH4] = - x (1- p)(l- r) . n Then (note that only the first and last of these four probabilities refer to cases in which the white ball is drawn)
pr+(I-p)(I-r) = pr + (1 - p)(1 - r) + (n - 1)[P(1 - r) + r(1 - p)] , and hence (or similarly)
Pr[black ball drawnlO] = Pr[EIO]
= I-Pr[EIO]
=~ ____~(n_-_1~)~[P(~I_-_r~)+~r(~I_-~p~)] ______(n - 1)[P(1 - r) + r(1 - p)] + pr + (1- p)(1 - r) Recall that n - 1 is the number of black balls, and compare the last formula in Note 10.
12. The reference is to Diderot's article "Certitude" in the Encyclopedie. See also the anonymous article, attributed to Hooper, in the Philo• sophical Transactions of 1699 (reasons for the attribution are given in Dale [1992]). This view had previously been criticized by Hume in his Essay "Of miracles" (Hume [1894, pp. 553-568]), and by Prevost and Lhuilier [1797].
13. This example is discussed ~ in a slightly more general setting - in T .A.P. II, Chapter XI, §46. More details may be found in Pearson [1978, pp. 673-675]. 184 Notes: On the probability of testimony
14. Some assumption as to the independence of the witnesses is needed here: the matter is discussed in Pearson [1978, pp. 674-675]. 15. Racine, Abrege de l'Histoire de Port-Royal. The relevant passage, translated from Vol. 4 of the 1921 edition of his (Euvres, runs The crowd increased day by day at Port-Royal, and God Himself seemed to take pleasure in allowing the people's devotion by the quantity of new miracles that took place at that church. Not only did the whole of Paris have recourse to the sacred thorn and to the prayers of the nuns, but requests were sent from all over the kingdom for things that had touched this relic; and these things, it has been said, effected many miraculous cures. [po 491]
The incident also received attention in Hume's Essay "Of miracles" (Hume [1894, pp. 553-568]), and a more recent discussion may be found in Shiokawa [1977): a summary of the salient features follows. Marguerite Perier, the niece and goddaughter of Blaise Pascal, was born in Clermont on the 5th of April, 1646. About the end of 1652 she began to suffer from a disease of the left eye. A lachrymal fistula was diagnosed, and an operation suggested, both diagnosis and rem• edy being confirmed by doctors in Paris. However, the seriousness of the operation persuaded Marguerite's mother and uncle to try, as a last resort, the remedy recommended by one M. de Chatillon, who promised to cure the young girl in six months without cauterization and with some eye-drops. Marguerite and her sister Jacqueline were thus sent by their father, Gilberte, to Port-Royal, where their pater• nal aunt Jacqueline was a nun. No improvement in the condition having been seen in the first half• year, the treatment was continued for a similar period, with the same negative result. The process was continued, with no improvement• indeed, the impediment seemed to get worse, and a tumour the size of a hazelnut developed at the corner of the eye. The doctors were now unanimous in advocating an operation, but became more and more pessimistic about the result as the malady progressed. Marguerite's parents eventually agreed to the operation, and her father left home on the 29th of March, 1656, for Paris. On his arrival there, he was told by Pascal that the child had been miraculously cured on the 24th. A reliquary, claimed to contain a splinter of a thorn from the Sacred Crown, had been exposed for adoration in the Abbey, and the mistress of novices, Sister Flavie, had persuaded Marguerite to touch it to her eye. The cure, if not instantaneous, was certainly evident within a few hours. Doctors and surgeons were called in, and pronounced the cure to be permanent and miraculous. Notes: On the probability of testimony 185
While investigations into the cure were going on, other cures were produced by the reliquary, while hostile clerics castigated the great throng flocking to Port-Royal because of the false miracles that the Jansenists said had happened there. On the 18th of October, the vicar-general, Alexandre de Hodencq, convened a commission which officially recognized the miracle. Marguerite reached a ripe old age, living until 1733. The full details of this event were given in a letter from Mother Angelique to Louise Marie de Gonzaga, Queen of Poland. This letter, and appropriate amplification, can be found in Sainte-Beuve [1954]. In his Dictionary of Mimcles [1884] E. Cobham Brewer discusses cures effected by relics (pp. 264-268) and cures effected by relics of the crucifixion (pp. 269-274). There is, however, no mention of the Perier miracle. The case is also discussed in Mackay [1980, p. 697], where it is noted that "How [the thorn] came [to the Abbey], and by whom it was preserved, has never been explained." Pearson [1978, p. 681] notes that when considering the supernat• ural, one must take cognisance of two factors, viz. (i) the events that are reported as having happened, and (ii) the interpretation of these events. He suggests that in the case of Mlle. Perier, one may well accept that she was cured and that this cure was effected neither by remedies nor by natural processes. This gives (i) above. However, it is (ii) that gives rise to the miracle. 16. These are Locke's actual words, and not a translation from the French: see Book IV, Chapter XVI, nO. 13 of Locke's Essay. For comment on this passage see Pearson [1978, p. 677]. 17. See Craig [1699] and Stigler [1986a]. 18. Literally, "an infinity of blissful lives" . 19. Pascal's argument may be found in his Pensees, No. 233 (ed. Brun• schvicg). The relevant passage runs in Krailsheimer's translation as fol!ows: ... let us say: 'Either God is or he is not.' But to which view shall we be inclined? Reason cannot decide this question. Infinite chaos separates us. At the far end of this infinite distance a coin is being spun which will come down heads or tails. How will you wager? Reason cannot make you choose either, reason cannot prove either wrong . . . . you must wager. There is no choice, you are already committed. Which will you choose then? Let us see: since a choice must be made, let us see which offers you the least interest. You have two things to lose: the true and the good; and two things to stake: your reason and your 186 Notes: On the probability of testimony
will, your knowledge and your happiness; and your nature has two things to avoid: error and wretchedness. Since you must necessarily choose, your reason is no more affronted by choosing one rather than the other. That is one point cleared up. But your happiness? Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist. 'That is wonderful. Yes, I must wager, but perhaps I am wagering too much.' Let us see: since there is an equal chance of gain and loss, if you stood to win only two lives for one you could still wager, but supposing you stood to win three? You would have to play (since you must necessarily play) and it would be unwise of you, once you are obliged to play, not to risk your life in order to win three lives at a game in which there is an equal chance of losing and winning. But there is an eternity of life and happiness. That being so, even though there were an infinite number of chances, of which only one were in your favour, you would still be right to wager one in order to win two; and you would be acting wrongly, being obliged to play, in refusing to stake one life against three in a game where out of an infinite number of chances there is one in your favour, if there were an infinity of infinitely happy life to be won. But here there is an infinity of infinitely happy life to be won, one chance of winning against a finite number of chances of losing, and what you are staking is finite. That leaves no choice; wherever there is infinity, and where there are not infinite chances of losing against that of winning, there is no room for hesitation, you must give everything. And thus, since you are obliged to play, you must be renouncing reason if you hoard your life rather than risk it for an infinite gain, just as likely to occur as a loss amounting to nothing. For it is no good saying that it is uncertain whether you will win, that it is certain that you are taking a risk, and that the infinite distance between the certainty of what you are risking and the uncertainty of what you may gain makes the finite good you are certainly risking equal to the infinite good that you are not certain to gain. This is not the case. Every gambler takes a certain risk for an uncertain gain, and yet he is taking a certain finite risk for an uncertain finite gain without sinning against reason. Here there is no infinite distance between the certain risk and the uncertain gain: that is not true. There is, indeed, an infinite distance Notes: On elections and decisions 187
between the certainty of winning and the certainty of losing, but the proportion between the uncertainty of winning and the certainty of what is being risked is in proportion to the chances of winning or losing. And hence if there are as many chances on one side as on the other you are playing for even odds. And in that case the certainty of what you are risking is equal to the uncertainty of what you may win; it is by no means infinitely distant from it. Thus our argument carries infinite weight, when the stakes are finite in a game where there are even chances of winning and losing and an infinite prize to be won. This is conclusive and if men are capable of any truth this is it. [Pascal, 1966, pp. 150-152] For a discussion of this argument see Hacking [1975, chap. 8]. 20. See T.A.P. II, p. 466 for more detail. 21. Describing Craig as "a good mathematician", de Morgan suggests that It is likely enough that Craig took a hint, directly or indi• rectly, from Mohammedan writers, who make a reply to the argument that the Koran has not the evidence derived from miracles. They say that, as evidence of Christian miracles is daily becoming weaker, a time must at last arrive when it will fail of affording assurance that they were miracles at all: whence would arise the necessity of another prophet and other miracles. [1915, vol. I, pp. 130-131].
On elections and decisions of assemblies 1. The word translated as "elections" in the heading to this article is choix, "choices": within the article choix is not used in the sense of "elections". Further, assemblees may be translated either as "as• semblies" or "meetings"; I have chosen the former, though the latter might occasionally be more suitable (perhaps Laplace had the Leg• islative and Constituent Assemblies of the Revolution in mind: see the EncycloptEdia Britannica, 14th ed., vol. 9, pp. 771-774). A major part of this article appears in the Le~on (O.C. XIV, pp. 173-176): it is to be found on pp. 73-76 of the 1st edition of the Essai. 2. Laplace [181Oa]. 3. See also Condorcet [1785, pp. clxxvi et seqq.]. 188 Notes: On elections and decisions
4. This section is in the main a ''popularisation'' of the first supplement to the Theone. However, in that part of the supplement dealing with judgments (O.C. XIV, pp. 520-530), there is no mention of this in• finite urn model. (The only mention here of an urn model occurs at the start of this section, where Laplace says I have likened, in no. 50 of Book II, the judgment of a court that decides between two contradictory opinions to the re• sult of the testimony of several witnesses to the extraction of a number from an urn containing only two numbers. [p.520], but no amplification of this is given.)
5. Laplace [1778], T.A.P. II §15 and Gillispie [1979]. 6. After the brief discussion of the ''infinite urn" model, Laplace switches to a consideration of the case in which the voters merely rank the propositions in a probability ordering. However a problem arises in the interpretation of items (1), (2) and (3). The original reads (1) l'unite divisOO par Ie nombre des propositions; (2) la quantite precooente augmentee de l'unite divisOO par Ie nom• bre des propositions moins une; (3) cette seconde quantite augmentee de l'unite divisOO par Ie nom- bre des propositions moins deux, et ainsi du reste.
IT we denote by n the number of propositions, then (1) is clearly ~. The absence of punctuation in (2) (and likewise in (3», however, results in either of the formulations ~ + n~l or (~ + 1) / (n - 1). No punctuation, in fact, was used in any of the various editions of the Essai, though in the Le~on (2) is given as l'unite divisOO par Ie nombre des propositions, plus l'unite divisOO par ce nombre diminue d'un. [O.C. XIV, p. 176]
(with (3) similarly punctuated), and this is the punctuation used in von Mises [1932, p. 102]. Writing on this point, Karl Pearson says ''This is a good illustr~ tion of one of the evils of turning mathematical formulae into words" [1978, p. 687]. He examines the situation from ''the common-sense standpoint" (loc. cit.), and assimilates the problem to the require• ment that the voter divide a straight line of length 1 into n segments with Xl > X2 > .,. > X n , E~ Xi = 1, and Xi the probability given to the ith proposition. This process of cutting the line into n - 1 Notes: On judicial decisions 189
(ordered) segments is now repeated over and over. The mean value of the rth largest segment will then be .!. {.!. + _1_ + ... + __1_} n n n-1 n-r+1'
which supports the interpretation of (2) as ~ + n': 1 . This partitioning of a line into a number of segments is discussed in T.A.P. II, §15 and in Todhunter [1865, art. 989], Dirichlet's inte• gral theorem being used for the solution. An alternative approach, perhaps one that is slightly easier to follow (contra Pearson), is given by Crofton [1885, art. 42]. 7. Commenting on this passage, Pearson writes This appears to me to overlook the fact that diverse po• litical opinions are usually held by diverse social classes in the community, and whereas the extension of the franchise might give representation to all these opinions, the changes of opinion in a single voting class may be considerable from time to time, but will not cover the opinions of all classes. Whether it is desirable to give all classes educated or uned• ucated a vote is another matter, but I think Laplace has not drawn a proper distinction between change of opinion with time in a limited voting class and differences of opinion represented by parties in an election on a comprehensive franchise. He winds up, however, with a piece of political thinking, which seems to me to indicate that he was not without wisdom as a politician. [1978, p. 689]
On the probability of judicial decisions 1. See the first supplement to T.A.P., O.C. VII, pp. 520-530. The dis• cussion here follows Condorcet [1785]. Laplace became interested in the theory of judgments after his elevation to the peerage, a body which was endowed with superior jurisdiction. The history of some attempts to apply probability to such "moral problems" is discussed in Sheynin [1973, pp. 296-297]. 2. Tribunaux is translated here sometimes as "courts" and sometimes as "tribunals". 3. The French judicial system in the early 19th century is described in the 14th edition of 1939 of the EncycloptEdia Britannica as follows: The Constituent Assembly decided on the complete reor• ganization of the administration of justice. This was ac• complished on a very simple plan, which realized that ideal 190 Notes: On judicial decisions
of the two degrees of justice which ... was that of France under the ancien regime. In the lower degrees it created in each canton a justice of the peace (juge de paix) ... He judged, both with and without appeal, civil cases of small importance; and, in cases which did not come within his competency, it was his duty to try to reconcile the par• ties. In each district was established a civil court composed of five judges. This completed the judicial organization, ex• cept for the court of cassation [i.e. a court of appeal], which had functions peculiar to itself, never judging the facts of the case but only the application of the law. For cases com• ing under the district court, the Assembly had preserved the right of appeal in cases involving sums above a certain figure. With regard to criminal prosecutions, there was in each department a court which judged crimes with the as• sistance of a jury; it consisted of judges borrowed from dis• trict courts, and had its own president and public prosecu• tor. Correctional tribunals, composed of juges de paix, dealt with misdemeanours. The Assembly preserved the commer• cial courts, or consular jurisdictions, of the ancien regime. [vol. 9, p. 772]
4. The law was passed on the 16th of August, 1790, and was confirmed by the constitution of the year III {1794-1795} of the Republican calendar. The Appeal Courts were established by law on the 27th Ventose - the windy month - in the year VIII {i.e. the 18th of March, 1800}.
5. Extenuating circumstances were recognized only in 1832 {new Penal Code, art. 463}.
6. Let p = Pr[X has committed offence F]. Should one not consider both Pr[X will commit F] and Pr[X will again commit F I p is large]? 7. Rather more generally than is discussed here, Laplace supposes in T.A.P., Supplement 1, that, in a tribunal of size p + q, p vote for conviction and q for acquittal. Assuming too that the reliability of a judge is uniformly distributed over [1/2,1], Laplace shows that the probability that the judgement of the tribunal is correct is
1 n-1 {n + I}! = 2n +1 ~ j! {n - j + I}!
For further details see Hacking [1984] and Poisson [1837, p. 364]. Notes: On judicial decisions 191
8. This eight-man jury was a special tribunal established by Napoleon for bandits - see Hacking [1990, p. 92]. 9. Laplace's reasoning in this paragraph seems a little confusing. In an attempt to clear things up, let us consider first the case of the eight judges, and suppose that five are needed to vote for conviction. The probability of an equitable judgment is then, on our using Bayes's Theorem with a uniform prior, found to be
and hence the probability of a wrong decision is
= 10.5 (6,4)
= 0.25390625.
H, following Pearson [1978, p. 693], we suppose that the chance that a judge makes an equitable decision is 0.8 rather than 0.5, then the probability of a fair judgment is
and the probability of a wrong decision is
= 10.2 (6,4)
= 0.085641728.
As a final example, consider the case of 212 judges, with 112 voting for conviction. Using the Normal approximation to the cumulative 192 Notes: On tables of mortality
binomial, we find that the probability of a wrong decision is
10 .5 (112, 100) = L..J~ (2X11) (21)211 ",=112
PrIX > 111.5]
= Pr[Z > 0.83]
0.2033, (where Z has approximately a standard Normal distribution), or, as Laplace says, very nearly t. If, on the other hand, we have 12 jurors, all of whom are required to deliver positive votes in order that the accused be convicted, then the probability of a wrong decision is
0.0001220703125 ,
or 8 :92' The remaining numerical results in this section may be sim• ilarly verified: for more details see Pearson [1978, pp. 690-693]. The beta distribution, and the incomplete beta function, used in a probabilistic context, are due to Thomas Bayes [1763] (see Sheynin [1971b] and, for more detail, Dutka [1981]). 10. The Scottish "not proven" verdict. 11. See Condorcet [1785, pp. cxxvi et seq.]. From 1810, 7 votes out of 12 were needed for conviction. This changed to 8 out of 12 in 1831, and then back to 7 out of 12 in 1835. For a detailed discussion of the role of probability in law see Eggleston [1983] and DeGroot et al. [1986]. Bertrand [1972, chap. XIII] writes somewhat disparagingly of the work of Laplace, among others, on the decisions of jurors. Amplification of the remarks on this article can be found in Hack• ing [1990], Chapter 11.
On tables of mortality and associations in general 1. See T.A.P. II, §§35-37. Pearson is somewhat scathing of Laplace's work on this matter: he writes "I do not think from Laplace's de• scription of a life-table he could ever have actually worked one out" [1978, p. 694]. Early tables of mortality were given by de Witt [1671], Graunt [1662]' Halley [1693] (see Halley [1942]), and Petty [1690] and [1899]. Notes: On tables of mortality 193
2. Consider the years 0, 1,2, ... , n, and let B be the number of births and 8i the number of children who reach year i. The following table may be constructed:
Year Survivors Differences 0 B B-81 1 8 1 81 -82 2 82
n-l 8n - l 8n - l - 8n n 8n Laplace's first method for finding the mean life-span, M, gives 1 [ 1 3 2n - 1] M = B (B - 81)2 + (81 - 82)2 + ... + (8n - l - 8n )-2-
On our supposing that 8n = 0, i.e. that all of the initial B individuals eventually die, we have 1 M = 2B [(B + 281 + 282 + ... + 28n -l) + B - B]
1 1 = B (B + 8 1 + 82 + ... + 8n - l ) - 2 ' which is the second result given in the text. 3. In Quetelet [1849, p. 43], the average duration of life is stated to be "about thirty-two years for Belgium and France: in England it reaches thirty-three years." Nearer our time, the complete life table for the total population of the United States, 195~1961, as given in Gross & Clark [1975], shows, of 100,000 born alive,
Age number living at beginning of age interval Si 74-75 50,888 75-76 48,170 Before that, the Abridged Life Tables for the total U.S. population, 1960, (see Chiang [1968, p. 201]) gave
Age l:z; (of 100,000 at 0 -1) -=~~----~~~~70-75 60,448 75-80 47,469 194 Notes: On tables of mortality
And earlier still, Von Mises [1932] gave the mean life-span "in most European countries" as being between 55 and 58 years (p. 207], while the United States Life Tables for 1930 contain the following:
Year Of 100,000 born alive, xtox+1 the number alive at the beginning of year of age 65-66 52,964 white males in 66-67 50,917 continental U.S., 67-68 48,781 1929-1931
69-70 52,264 white females 70-71 49,932 } 1929-1931
51-52 51,325 negro males 52-53 48,875 } 1929-1931
53-54 50,323 negro females 54-55 48,705 } 1929-1931. 4. Laplace seems to have been the first to have attempted to discuss the precision: see T.A.P. II, §§35-37. 5. See Quetelet [1836]; the following passages are particularly perti• nent: Book I, Chapter 3, "De l'inHuence des causes perturbatrices sur Ie nombre des naissances. 1. Influence des professions, de la nour• riture, etc." (pp. 111-114]; Book I, Chapter 4, "De l'inHuence des causes perturbatrices sur Ie nombre des deces. 1. Influence des pro• fessions, du degre d'aissance, etc." (pp. 213-217]; Book III, Chapter 3, "Developpement du penchant au crime. §II. Influence des lumieres, des professions et du climat sur Ie penchant au crime" [pp. 186--220]. 6. Jenner [1798]. 7. Daniel Bernoulli [1760]. 8. See D'Alembert's Opuscules Mathimatiques [1761]: eleventh memoir, "Sur l'application du Calcul des Probabilites a. l'inoculation de la petit Verole" [pp. 26--95]. The following two quotations are indicative of D'Alembert's views on the matter: (a) In the first place, the hypothesis made by the illustrious Mathe• matician [i.e. Daniel Bernoulli] on the number of people of each age who contract smallpox and on the number of those who die from it, seems to be completely gratuitous [po 31], Notes: On tables of mortality 195
(b) But what will one learn from this difference in mortality? One will learn, I hope, that the mean lifetime of those who are inocu• lated - that is to say, the time that each of them may reasonably expect to live after having undergone inoculation - exceeds the mean lifetime of those of the same age who resign themselves to waiting for smallpox; one will determine, for each age, by how much the mean lifetime in the first case exceeds that in the sec• ond; & as a result one will obtain, on comparing these two risks, the time that one may expect to add to one's life by allowing oneself to be inoculated. Now this knowledge does not seem to me to be sufficient to determine, in a satisfactory manner, the advantages of inoculation (p. 32].
9. This last appeared in quite a different form in the Lefon, viz. The father of a family, whose affection for his children in• creases with them, must not hesitate to submit them to an operation that will free them from the anxiety and the dangers of so merciless a disease, and that will assure his offspring of his care and of their education.
As the writer ofthe notes in the Bru/Thom [1986] edition of the Essai puts it, Laplace seems to have discovered a paternal instinct rather late! 10. As recently as 1967, it was estimated that smallpox victims numbered 10-15 million per year (see World Health Forum 8, 1987, p. 283). The last known case of endemic smallpox occurred in a 23-year-old hospi• tal cook in the Somalian port of Merka on the 26th of October, 1977, although a laboratory-associated case of variola major was found in England on the 27th of August, 1978. For comments on the history of the disease and its eradication, see the issue of the World Health Forum cited above and Fenner et al. [1988]. 11. Duvillard de Durand [1806].
12. See Malthus [1798]. Incidentally, Keynes, in his Essays in Biography, points out that the name "Malthus" is derived from "malt house", and should be pronounced accordingly. 13. Humboldt [1811]. In Volume I, Book 2, Chapter 4 we find the words "Aux Etats-Unis, on a vu doubler la population, depuis l'annee 1774, en vingt-deux ans." In the English edition of 1811 we have "In the United States we have seen the population double, since 1784, every twenty or twenty-three years." 196 Notes: On institutions & the probability of events
On institutions & the probability of events 1. In the first edition of the Essai this article is entitled "On benefits depending on the probability of events" . Ofthis article Pearson writes "I do not find much of novelty in the section" [1978, p. 696].
2. For more detail ofthe work ofthis entire article see T.A.P. II, §§38-40.
3. This is a simple problem in compound interest.
4. The idea of a sinking-fund, for liquidating public debts, was proposed by Richard Price in 1772. 5. In symbols, V = S p (1 + i)-n. 6. In common life-table notation this may be expressed as
sex = S IX+5 (1 + i)-5 , Ix where S denotes the endowment. 7. Letting Dk = (1 +i)-k Ik we have nEx = Dx+n/ Dx, and the present value of a whole-life annuity of S units is
1 ax = Sf) L:Dx+k. x k=O
8. Maritime insurance is discussed in Bernoulli [1730-1731, §15].
9. See Daniel Bernoulli [1730-1731] and T.A.P. II, §§41-43.
10. Laplace, Lagrange, Monge, Borda and Condorcet were all members of the Academy's commission on weights and measures.
11. The passage from here to the end of this article appeared for the first time in the 5th edition - perhaps directed at Gauss [1819-1822].
12. See Laplace [1774a], [1778] and T.A.P. II, §20. 13. Let rl, r2 = rl + q, r3 = rl + q/, etc. be the observations, arranged in increasing order of magnitude. Let X denote the true value: then X - rl = x is the correction to be made to rl, X - r2 = x + q is the correction to be made to r2, and so on. Now let f denote the function from which the observed results are taken. If the difference between the observation yielding the true value al + x = X and the observation yielding al is called the error Cl in the observation - i.e. if Cl = f(al + x) - f(ad = x.j'(e) , Notes: On illusions in estimation 197
then Cl is proportional to x, and similarly for a2,a3, etc. If .,pl(cl) denotes the probability of a given error Cl in the first observation, and is accordingly a function of x, then .,pl(cd = CPl(X). Similarly .,p2(c2) = CP2(X), etc. The probability p(x) of the simultaneous exis• tence of all these errors is then the product of these error functions. The determination of X may then be effected in two ways: either (1) take X to be the most probable value, i.e. one for which the product p(x) is maximal, or (2) take X to be the abscissa for which the area under the curve p(x) to the right is equal to that to the left. Then
00 ! (x-X)p(x)dx -00
should be minimized. Notice that these two procedures coincide when the error curve is the Normal distribution: they do not do so in gen• eral.
14. See Lambert [1760) and D. Bernoulli [1777).
On illusions in the estimation of probabilities 1. For the distinction between "delusions" and "illusions" see Fowler's Modern English Usage. In Book 1, Aphorisms XXIII, XXXIX et seqq., of his NoVtLm Organum, Francis Bacon discusses various kinds of illu• sions under the headings Idols of the Tribe, Idols of the Cave, Idols of the Market-place and Idols of the Theatre: note that "idola, as used by Bacon, means ... not an object of worship, but an illusion or false appearance - the original sense of the Greek word" (Bacon [1905, p. 65)). Francis Galton writes A convenient distinction is made between hallucinations and illusions. Hallucinations are defined as appearances wholly due to fancy; illusions, as fanciful perceptions of objects actually seen. There is also a hybrid case which de• pends on fanciful visions fancifully perceived. [1907, pp. 122-123)
2. The French lottery contained 90 numbers, 5 being drawn at a time. If all 5 were winning numbers, the combination was termed a quine. Four winning numbers out of 5 yielded a quaterne, while 3, 2 and 1 winners were called ternes, ambes and extraits. The probabilities of these events are given in Note 5 on the article "On analytical methods in the probability calculus." De Morgan [1915, vol. I, p. 280) mentions a book (one I have not seen) in which "all the drawings of the French lottery (two or 198 Notes: On illusions in estimation
three, each month) from 1758 to 1830" are listed. This 1830 work, by Menut de Saint-Mesmin (see also his [1830?]), must have come, like the Pickwick, the Owl, and the Waverley pen, as a boon and a blessing to those gamblers who believed in the usefulness of such data. In this same volume of his Budget of Paradoxes, de Morgan discusses the French lottery [po 281].
3. If one knows that Pr[head] = 1/2 = Pr[tail] , then a long sequence of heads, say, provides no reason for a strong belief that tails will occur on the next toss. However, if one does not know the probability of obtaining a head, then a long sequence of heads will, by Laplace's extension of Bayes's Theorem, strengthen the expectation of a head on the following toss.
4. This was added in 1825, after the publication of birth statistics by the Bureau des Longitudes (see the earlier article "On the laws of probability resulting from the indefinite repetition of events"). 5. Writing of man's misguided view of his own place in the grand scheme of things, Mackay says An undue opinion of our own importance in the scale of creation is at the bottom of all our unwarrantable notions in this respect. How flattering to the pride of man to think that the stars in their courses watch over him, and typify, by their movements and aspects, the joys or the sorrows that await him! He, less in proportion to the universe than the all-but invisible insects that feed in myriads on a sum• mer's leaf are to this great globe itself, fondly imagines that eternal worlds were chiefly created to prognosticate his fate. How we should pity the arrogance of the worm that crawls at our feet, if we knew that it also desired to know the se• crets of futurity, and imagined that meteors shot athwart the sky to warn it that a tom-tit was hovering near to gob• ble it up; that storms and earthquakes, the revolutions of empires, or the fall of mighty monarchs, only happened to predict its birth, its progress, and its decay! [1980, pp. 281-282]
6. An example of the type discussed here had been given in the Le~on. This prompted a reply from Prevost [1811], to which the present discussion is in a sense a response. Prevost's argument runs in part as follows:
This gives the explication of a kind of paradox, noticed (without explanation) by De La Place (Beoles normales, vol. 6). When one considers the posterior probability of Notes: On illusions in estimation 199
getting an ace four times in a row, after five throws have been made, in which one has thrown it only twice, one finds that this probability is much greater than one had previ• ously considered it to be, according to the known equality between ace and non-ace. Indeed, in the first case, it is equal to 1/14. In the second case, it is 1/16. This seems shocking, since the event that results in an ace only twice in five throws, must be supposed to occur less often, and not the contrary, as the calculus seems to show. The expli• cation of the paradox is this: if one had not carried out any experiment, the nature ofthe die {which, by hypothesis, is determined only by experiment {or experience}) would be entirely unknown. As a result, the probability of getting an ace four times in a row would be equal to 1/5. But once the experiment has been performed, this probability is at most only 1/14. Thus it has indeed considerably decreased. Putting this problem slightly more generally, let us suppose that the faces of a die are marked either with an ace or with a non-ace, but that the number of each is unknown. Let us suppose too that in p + q tosses of the die exactly p aces have been obtained. Then the probability that r + s further tosses will yield exactly r aces is (p;r) (q:s) / (p;!:::s) .
On putting p = 2, q = 3, s = 0 and r = 4 we obtain ::84:::160' or 1/14. IT, however, it was known that the die had as many aces as non-aces, the required probability would be {1/2)4, or 1/16. For further details see Todhunter [1865, art. 856]. 7. These last three figures are given in the reverse order in the first edition of the Essai. 8. In arguing that
where the Hi denote the various hypotheses and B and W denote the events ''four black balls" and "one white ball" respectively, Laplace is once again tactily assUllling conditional independence. 9. Suppose first that the urn contains n white and n black balls, with Pr[B] = !. Then, assuming independence, we have Pr[4 blacklW] = Pr[4 black] = {1/2)4 . Next consider the scheme presented in the following table: 200 Notes: On illusions in estimation
W 1 3/4 2 2/4 3 1/4
Suppose too that Pr[H,] = 1/3, i E {I, 2, 3}. Then, as Laplace states, it follows (by a discrete version of Bayes's Theorem) that
Pr[H1IW] = 3/6, Pr[H2IW] = 2/6, Pr[HaIW] = 1/6 . FUrther,
Pr[NB ~ NwIW] - Pr[Hl V H2IW]
= Pr[H1IW] +Pr[H2IW]
= 5/6,
where N B and Nw denote respectively the numbers of balls that are black and that are white. Under a suitable assumption as to the independence of the draws, it follows that Pr[4 blacklW] = E, Pr[4 blacklH, A W] Pr[H,IW] = EPr[4, blackIH,] Pr[H,IW]
= (1/4)4(3/6) + (2/4)4(2/6) + (3/4)4(1/6)
= 29/384. Note that the probability in this second case is greater than that (viz. 1/16) found in the first case mentioned above. 10. De Morgan notes the old definition of a paradox as "something which is apart from general opinion, either in subject-matter, method, or conclusion" [1915, vol. I, p. 2]. The meaning has, of course, become somewhat broader by now (perhaps regrettably so); Szekely writes: It is important to distinguish paradoxes from fallacies. The first one is a true though surprising theorem while the sec• ond one is a false result obtained by reasoning that seems correct. [1986, p. xii]
11. Let N B and Nw represent the numbers of black and of white balls respectively. The first draw suggests that Nw > NB. This is not inconsistent with the possibility that, in fact, N B > Nw, a hypothesis which is inconsistent with N B = Nw . Notes: On illusions in estimation 201
12. Indeed, if the urn contains n black and white balls, not all of the same colour, and if q(k, n) is the probability of getting k black balls following the drawing of a white ball on the first draw, then we shall have q(k,n) > (1/2)11: for k E {4,5,6, ... } and n E {3,4,5, ... }. To verify this assertion, consider the following scheme:
W B Pr[WIHil Pr[BIHil
n-l 1 HI n-l 1 n n n-2 2 H2 n-2 2 n n
1 n-l 1 n-l n n
Then
n-l 1 L Pr[WIHil = -(1 + 2 + ... + (n - 1» = (n - 1)/2 . i=1 n Hence, assuming that the Hi are a priori equally probable, we have, for i E {I, 2, ... , n - I},
= (n - i) / (n - 1) = 2{n - i) n 2 n(n-l)
Thus, since q(k, n) == Pr[B1 .•• BklWl,
n-l q(k,n) = LPr[B1 ••• BkIW A HilPr[HilWl i=1
n-l = L Pr[BI ..• BklHil Pr[HilWl i=1
= n-l (i)k 2(n - i) tr n n(n-l) 202 Notes: On illusions in estimation
or n-1(.)k( .) 1 q(k,n) = 2 t; ~ 1- ~ n -1 ' (1) where Bi denotes the obtaining of a black ball on the ith draw. For n = 4, as in Laplace's example, we thus have
= 2 (3+3k +2k+1). 3 X 4.\:+1 A simple calculation now yields 5 1 q(I,4) = 12 < 2
5 1 q(2,4) = 24 < 22
23 1 q(3,4) = 192 < 23
29 1 q(4,4) = 384 > 24 . Further,
1 2 ( k k+1) 1 q(k,4) > 2k <=:=} 3 X 4k+1 3 + 3 + 2 > 2k
If k = x + 4, with x E {O, 1,2, ... }, then 1 q(k,4) > 2k <=:=} 3 + 32:+4 - 22:+6 > 0
<=:=} 3 + (34 X 32:) - (26 X 22:) > 0 .
Since 34 > 26 and 32: ~ 22:, it follows that this last inequality is true, and hence 1 q(k,4»2k ' kE{4,5, ...}. It can be shown, more generally, that 1 q(k,n) > 2k ' n E {3,4, ... } , k E {4,5, ...}. Notes: On illusions in estimation 203
Indeed, on writing (1) as
n-l n-l 1 q(k, n) = k+1 ~ _ 1) [ n L ik - L ik+1 , n n i=l i=l and on summing the series, one can show that, for all n > 2, 1 q(k,n) < 2k ,k E {I,2,3}
1 q(4,n) > 24
To prove that q(k, n) > I/2k (which we know to be true for k = 4) implies that q(k + 1, n) > I/2k +1 (for all integral n > 2), let 1 r(k, n) = q(k + 1, n) - 2 q(k, n) (2)
Then n-l r(k,n) = n:I~(i/n)k+1(I-i/n)
n-l _ ~ . 2 "'(i/n)k(I- i/n) 2 (n-I)£;:
n-l = (n _ 1\ nk+2 ~ik(n - i)(2i - n)
1 = (n _ 1) nk+2 S(k, n) (3) say. Now S(k,n) = Ik(n-I)(2-n)+2k(n-2)(4-n)
+ 3k(n - 3)(6 - n) + ... + (n - 3)k3(n - 6)
+ (n - 2)k2(n - 4) + (n - I)kI(n - 2) . (Notice that this series will contain an even (odd) number of terms if n is odd (even), and that if n is 3, say, it will contain only the first two of the above terms - neither of which will be zero - while if n = 4 it will contain only the first three terms, the central one being zero. Similar remarks of course apply to other values of n.) Consider the first and last terms in the expansion of S(k, n), viz. Ik(n -1)(2 - n) and (n -I)kI(n - 2) . 204 Notes: On illusions in estimation
Since n > 2, neither is zero, and 12 - nl = In - 21. Since (n - 1)k > 1 and k > 1 (replacement of > by 2: where necessary will cover the case k = 1), we have
(n - 1)k1(n - 2) + lk(n -1)(2 - n) > 0
(remember that (2 - n) < 0). Similarly, on considering the second and second-to-last terms of S(k, n), viz.
2k(n-2)(4-n) and (n-2)k2(n-4) ,
we note that 14 - nl = In - 41 and 2k(n - 2) < (n - 2)k2 (recall that n must be larger than 4 for these terms to appear in the series). Thus
(n - 2)k2(n - 4) + 2k(n - 2)(4 - n) > 0 .
Continuing with this "pairing" of terms, we find that S(k, n) > 0, and hence, from (2) and (3)
1 q(k + 1, n) - '2 q(k, n) > 0 ,
or 1 q(k + 1, n) > '2 q(k, n) . It then follows by induction, and using the already proved fact that q(4,n) > 1/2\ that q(k,n) > 1/2k for all n E {3,4, ... } and all k E {4,5, ... }. We may further deduce from (1) that, in the limit as n -+ 00, q(k, n) tends to
1 1(k) 2 Jxk(1 - x) dx o
2 (k+1)(k+2) .
Notice that 1(k) < 1/2k for k E {1,2,3}, while 1(4) > 1/24. Further• more, if m is such that 1( m) > 1/2m , which is equivalent to saying that 2m +1 > (m + 1)(m + 2), then
2m +2 =2x2m +1 > 2x(m+1)(m+2)=
> [(m+1)+1](2m+2)
> [(m+1)+1][(m+1)+2], Notes: On illusions in estimation 205
or 2 1 I(m + 1) = [(m + 1) + l][(m + 1) + 2] > 2m +1 Since 1(4) > 1/24, it follows by induction that I(k) > 1/2k for all k E {4,5, ... }.
13. Sainte-Beuve devotes one of his Literary Portraits - one which first appeared in the Revue des Deux M andes, 1st January, 1848 - to the chevalier Antoine Gombaud (or Gombault) de Mere (1607-1684) (see Sainte-Beuve [1882, vol. III, pp. 85-128]). He supposes the knight to have been one of those who was "frivolous in serious matters, and pedantic in trivia" [po 85]; and, describing him as a type, writes if one wishes today to study one of the most honourable characters of the 17th century, one could choose no better or no more suitable subject [than de Mere]. [pp. 85-86] 14. Maugre opinions to the contrary (see, for example, Hacking [1975]), Schneider [1980] avers that the concept of probability certainly ex• isted before the 17th century, although it was not applied to games of chance. Schneider also discusses (op. cit.) why it was not possible to connect the concepts of chance and probability until that century.
15. See Note 12 of the Article "On analytical methods in the probability calculus" .
16. According to Hald [1990, p. 55], the method of solution of de Mere's problem had earlier been given by Cardano: it amounts to finding the smallest (integral) value of n, the number of trials, for which 1 - qn 1 --n- ~ 1, or qn < - q - 2
For 1- q = p = 1/6 we get n = 4 and odds of 671 : 625; for p = 1/36, the corresponding values are 25 and 506 : 494 (the odds are 491 : 509 for n = 24). Thus the probability of no six in 4 tosses of a single die is (5/6)4, while that of at least one six is 1 - (5/6)4 ~ 0.5177. Similarly, the probability of at least one double six in 4 tosses of two dice is 1 - (35/36)4 ~ 0.4194. The argument advanced by de Mere supposes that n ex 6- k ,k E {1,2, ... }. Since, for fair dice, p = 6- k and (1- p)n ~ e-np , the solution of (1 - p)n = 1/2 is approximately
n = 6k In 2 .
For k = 1 this gives n = 4.16 and for k = 2, n = 24.95. Hald notes too (loc. cit.) that the approximate solution was first given by de Moivre in 1712. The problem was also considered by Huygens in 1657 206 Notes: On illusions in estimation
(see Hald, op. cit. p. 71). (See also Hald [1990, p. 213].) For further discussion see Todhunter [1865, pp. 8-12, 63], and, for de Moivre's approximations, see Todhunter, op. cit. pp. 144-145. 17. See Leibniz [1768], Vol. 3, §LXXII (pp. 406-410), "Epistola ad v. d. Christianum Wolfium, professorem Matheseos Halensem, circa scien• tiam Infinite." Here Leibniz deals with infinite series for (1 - x)-l, (1 +X)-l, (1 +X2)-1 and term-by-term integration. He also considers what happens when 1 is substituted for x in such expansions. 18. Daniel Bernoulli [1771], [1772a], [1772b], [1773&] and [1773b]. 19. Note the expansion (1 + t)-l = 1 - t + t2 - t3 + ... (a)
= (1 - t) + (t2 - t3) + ... (b)
= 1 - (t - t2) - (t3 - t4) + .. . (c) Substitution of t = 1 in (b) and (c) respectively yields
~ =0+0+0+···
~=1+0+0+.·. Of course, the expansion in (a) is valid only for It I < 1: the legitimacy of such expansions, or of the convergence of the infinite series, was not formerly considered (see also Note 25 below).
20. Grandi (1671-1742) was in fact not a Jesuit but a Camaldulian, i.e. a monk of the order founded by St Romuald. 21. See Leibniz [1768], Vol. 3, §LVIII (pp. 346-348), "De inventione arith• meticae binariae." §LXlX (pp. 349--350; 351-354), "Epistolae duae ad Joh. Christ. Schiilenburgium, De Arithmetica. Dyadica.." §LXVIII (pp. 390--394), "Explication de l'arithmetique binaire, qui se sert des seules caracteres 0 & 1: avec des Remarques sur son utilite, & sur ce qu'elle donne Ie sens des anciennes figures Chinoises de Fohy." See also Volume 4, Part 1, (pp. 169--210) "Lettre de M. G.G. De Leib• niz sur la philosophie Chinoise, a M. De Remond," Section Four (pp. 207-219), "Des caracteres dont Fohi fondateur de l'Empire chinois s'est servi dans ses ecrits, et de l'arithmetique binaire." Needham, [1956, vol. 2], relates that Leibniz, as the result of contact with Fr. Joachim Bouvet (one of the Jesuit missionaries in China), became aware of the Book of Changes. It was apparently Bouvet's idea to interpret the I Ching hexagrams as numbers written in the binary system, and possibly as a consequence of this notion, Leibniz, in the Notes: On illusions in estimation 207
further development of his system of binary arithmetic, "gave way to the temptation of seasoning the discussion with metaphysics" (Hardy [1991, p. 13]). 22. According to Leibniz [1768) Vol. 4, Part 1 (pp. 78-86), "Praefatio libro inscripto novissima sinica" [1697), Phil(l)ipe-Marie Grimaldi (1638- 1712) followed Pere Ferdinand Verbiest (1623-1688) in this august position. A picture of Verbiest (Nan Huai-Jen) "habited as a Chinese official, with his sextant and his celestial globe" appears as Plate LXV (facing p. 50) in Needham, vol. 3 [1959). 23. K'ang-hi (1662-1723). 24. Laplace is perhaps a little severe in his opinion of the work on diver• gent trigonometric series by Bernoulli and Lagrange. For a discussion of this work during the Revolution see Nielsen [1927, pp. 25-29). 25. (See Note 19 above.) We have
l+t 1 + t+t2 = 1- t3
(a)
= (1 - t2) + (t3 - t5) + .. . (b)
= 1 - (t2 - t3) - (t5 - t6) - .. . (c)
On putting t = 1, we get, from (b) and (c) respectively,
~ =0+0+0+···
~=1-0-0-···
Once again the expansion (a) holds only for It I < 1. Hardy [1991, p. 14) notes that, for any m < n, the sum min may be obtained for 1-1 + 1-· . '. This follows from a principle due to Euler.
26. We find, in Laplace [1789b), the words ... at the time of Hipparchus the year was longer by about 10"1 than it is today. [O.C. XI, p. 491) 27. In his Exposition du Systeme du Monde, Book 1, Chapter III, Laplace writes 208 Notes: On illusions in estimation
This motion [of the equinoxes over the ecliptic] is not ex• actly the same in all centuries, and this results in a slight inequality in the length of the year in the tropics; it is now about 13/1 shorter than in the time of Hipparchus. (p. 18]
Further, in his [1818d] he says
If one extends this result to the Earth, and if one con• siders that the duration of the day has not varied, since Hipparchus, by a hundredth of a second, as I have shown by the comparison of observations with the theory of the secular equation of the Moon, one will be of the opinion that, since that time, the variation of the temperature of the interior of the Earth has been imperceptible. [O.C. XII, p. 463]
We also find Fourier writing One may conclude, with certainty, that the falling of the temperature in a century is smaller that 57~OO of a degree centigrade. [1890, p. 286]
28. The impossible of the fourth (O.C.) edition appears as possible in the Bru/Thom edition. 29. In Chapter VI, "Considerations sur Ie Systeme du Monde et sur les progres futurs de l'Astronomie," of Book V of Exposition du Systeme du Monde, Laplace writes
But so many species of extinct animals, about whose struc• ture M. Cuvier has been able, by rare sagacity, to find out, through the numerous fossilized bones that he has de• scribed - do they not indicate a tendency for nature to change those very things that seem the most permanent? The magnitude and importance of the solar system cannot be an exception to this general law, for they are relative to our own insignificance; and that system, however vast it may seem to us, is only an imperceptible speck in the universe. (p. 480] 30. Laplace's reference to "the ancient philosopher" here is well put, for in Bacon's Novum Organum [1905], a footnote to this anecdote (Book I, Aphorism XLVI) states that "This story is told of Diagoras by Cicero, De Nat. Deor., iii, and of Diogenes the Cynic by Diogenes Laertius." The story is to be found in Montaigne's essay Des prognostications (where it is ascribed to Diagoras qui Jut surnomme l'Athie - that is "Diagoras, who was called the Atheist"), and in view of Laplace's Notes: On illusions in estimation 209
other references to Montaigne in this article, it is possible that he is using the latter's essay here. 31. See Cicero De Divinatione [1959], Book II, LXXII, nO 148-150. The translation, by W.A. Falconer, runs as follows: Then let dreams, as a means of divination, be rejected along with the rest. Speaking frankly, superstition, which is widespread among the nations, has taken advantage of human weakness to cast its spell over the mind of almost every man. This same view was stated in my treatise On the Nature of the Gods; and to prove the correctness of that view has been the chief aim of the present discussion. For I thought that I should be rendering a great service both to myself and to my countrymen if I could tear this supersti• tion up by the roots. But I want it distinctly understood that the destruction of superstition does not mean the de• struction of religion. For I consider it the part of wisdom to preserve the institutions of our forefathers by retaining their sacred rites and ceremonies. Furthermore, the celes• tial order and the beauty of the universe compel me to confess that there is some excellent and eternal Being, who deserves the respect and homage of men. Wherefore, just as it is a duty to extend the influence of true religion, which is closely associated with the knowl• edge of nature, so it is a duty to weed out every root of superstition. For superstition is ever at your heels to urge you on; it follows you at every turn. It is with you when you listen to a prophet, or an omen; when you offer sacrifices or watch the flight of birds; when you consult an astrologer or a soothsayer; when it thunders or lightens or there is a bolt from on high; or when some so-called prodigy is born or is made. And since necessarily some of these signs are nearly always being given, no one who believes in them can ever remain in a tranquil state of mind. Sleep is regarded as a refuge from every toil and care; but it is actually made the fruitful source of worry and fear. In fact dreams would be less regarded on their own account and would be viewed with greater indifference had they not been taken under the guardianship of philosophers - not philosophers of the meaner sort, but those of the keenest wit, competent to see what follows logically and what does not - men who are considered well-nigh perfect and in• fallible. Indeed, if their arrogance had not been resisted by Carneades, it is probable that by this time they would have been adjudged the only philosophers. While most of 210 Notes: On illusions in estimation
my war of words has been with these men, it is not because I hold them in especial contempt, but on the contrary, it is because they seem to me to defend their own views with the greatest acuteness and skill. Moreover, it is character• istic of the Academy to put forward no conclusions of its own, but to approve those which seem to approach near• est to the truth; to compare arguments; to draw forth all that may be said in behalf of any opinion; and, without as• serting any authority of its own, to leave the judgement of the inquirer wholly free. That same method, which by the way we inherited from Socrates, I shall, if agreeable to you, my dear Quintus, follow as often as possible in our future discussions.
32. In Burton [1621] a clear distinction between religion and superstition is drawn.
True religion and superstition are quite opposite, Longe di• versa camijicina et pietas, as Lactantius describes, the one erects, the other dejects; illorum pietas, mem impietas; the one is an easy yoke, the other an intolerable burden, an ab• solute tyranny; the one a sure anchor, a haven; the other a tempestuous ocean; the one makes, the other mars; the one is wisdom, the other is folly, madness, indiscretion; the one unfeigned, the other a counterfeit; the one a diligent observer, the other an ape; one leads to heaven, the other to hell. [Part III, Sect. IV, Mem. 1, Subs. III] [1877, p. 684]
33. Mackay notes that the pilgrims to Jerusalem in the year 1000 AD held the view that
thunder was the voice of God, announcing the day of judg• ment. Numbers expected the earth to open, and give up its dead at the sound. [1980, p. 258]
34. In his chapter on fortune-telling, Mackay discusses the "sciences, so called" [1980, p. 282] of astrology, augury (using the flight or entrails of birds), geomancy (foretelling by lines, circles and other mathe• matical figures), necromancy (the summoning of departed shades), palmistry, oneiro-criticism (dreams) and divination of many kinds - indeed, he adduces a list of 52 types of divination, ranging from Stere• omancy (divining by the elements) to Lampadomancy (divining by candles and lamps). Quetelet [1849, p. 180] also expresses his dissatis• faction with astrology, alchemy and magic as compared to astronomy, chemistry and physics. Notes: On illusions in estimation 211
35. Laplace follows Bonnet [1755] (a work described by the author on page 5 of the Avertissement as cet Ouvrage de ma jeunesse - that is, "this juvenilia"), [1760] and [1769]. 36. "The following considerations are completely independent of the site of this seat and of its nature." (Laplace's own Note.) David Hartley [1749] use "sensorium" (at least initially) in a broad sense: he writes " ... the sensorium, fancy, or mind (for these 1 consider as equivalent expressions in our entrance upon these disquisitions,)" [1834, p. 7], and later on, "The brain may therefore, in a common way of speaking, be reckoned the seat of the sensitive soul, or the sensorium, in men" [1834, p. 20].
37. "I denote here by the term principles the general relationships be• tween phenomena." (Laplace's own Note.) 38. The first two chapters of the first volume of Smith's The Theory of Moral Sentiments [1792] are devoted to a study of sympathy, a term defined there as follows:
Pity and compassion are words appropriated to signify our fellow-feeling with the sorrow of others. Sympathy, though its meaning was, perhaps, originally the same, may now, however, without much impropriety, be made use of to de• note our fellow-feeling with any passion whatever. [pp. 6-7]
See also the Article De la Sympathie ("On Sympathy") in Cabanis [1802]. Here Cabanis discusses various forms of sympathy (sympathy and instinct, moral sympathy, nervous sympathy, &c.) and expresses the following sentiment (one similar to Laplace's): AB a tendency of one living being to others of the same, or different, species, sympathy is contained in the domain of instinct; it is, in a sense, instinct itself, if one wishes to consider it from the widest point of view.
(Reprinted in Cabanis [1956, Vol. XLIV, 1, 1st Part, p. 565].) 39. "The account that Montaigne gives in his essays on the friendship that existed between him and La Bootie offers a most remarkable example of an extremely rare kind of sympathy." (Laplace's own note.) 40. This principle is fundamental to associationist psychology, and was known from the 17th century. See Locke, An Essay Concerning Hu• man Understanding, Book II, Chapter X, for a discussion of contem• plation, memory, and the fixation of ideas. For further comment see Hartley [1749] and Priestley [1790]. The writer of the notes in the Bru/Thom edition of the Essai suggests that Laplace may have read 212 Notes: On illusions in estimation
of this principle in Bonnet's Essai de Psychologie, reprinted in Vol. XV of his (Euvres [1779-1783]. Here we have When two or more movements have been carried out either at the same time or one after the other in the organ of thought, if one of these movements is repeated de novo, all the others will be repeated with it and, with them, the ideas that have been attached to them. All sciences and all arts rest on this law: what! The whole system of Mankind depends on it. [po 152]
41. In France, Condillac (1715-1780) followed the ideologists Cabanis, Volney, Garat, Destutt de Tracy, Daunou, Laromiguiere, ... 42. In Bonnet's (Euvres [1779-1783], Vol. XV, we read I have shown that since our ideas, of every kind, recall one another to mind, and that since they are all originally re• lated to the mind, it is necessary that all the sensible fibres of every kind communicate with one another either imme• diately or indirectly. Thus they may acquire a customary disposition to set one another in motion in a determined and constant order. It is always by the repetition of the same movements in the same sense that one manages to acquire this disposition. [po 26]
43. Cabanis, Vol. 1, [1802, p. 187]. 44. In his Essai de Psychologie ((Euvres, Vol. XVI) Bonnet writes The somnambulist is not an automaton. All his movements are directed by a inspiration that he sees very clearly; but his vision is turned inwards . " [po 129] 45. This anecdote may be found in Bonnet's Essai Analytique ((Euvres [1779-1783]' Vol. XIV). I repeat it, in translation, here, for the sake of supplying omissions made by Laplace. I know a respectable man, in good health, frank, sound in judgement and memory, who, while wide awake and in• dependently of any impression from the outside, sees time after time before him the figures of men, women, birds, car• riages, buildings, etc. He sees these figures making various movements; approaching, going away, Beeing; decreasing and increasing in size; appearing, disappearing, reappear• ing; he sees buildings rising up before his eyes, and expos• ing to his view all the facets of their external construction. The tapestries of these rooms seem to him to be suddenly Notes: On illusions in estimation 213
changed into richer tapestries of another kind. Another time, he sees the tapestries covered in pictures represent• ing different scenes. Another day, instead of tapestries and furniture, there are only the bare walls and only a collec• tion of rough materials are presented to him. On another occasion there is scaffolding; but if I were to go into too great detail, I would describe the phenomenon, and I wish only to mention it. All these pictures seem to him to be perfectly clear, and to affect him as vividly as the objects themselves would were they present; but, they were only pictures; for the men and the women did not speak, and no noise reached his ear. All of this seems to have its seat in that part of the brain that responds to the organ of sight. [§676] In a footnote it is pointed out that ''this respectable old man is M. Charles Lullin, my maternal grandfather, who died in 1761 in his 92nd year." In his chapter on witchcraft, Mackay says
We all know the strange pranks which imagination can play in certain diseases; that the hypochondriac can see visions and spectres; and that there have been cases in which men were perfectly persuaded that they were teapots. Science has lifted up the veil, and rolled away all the fantastic hor• rors in which our forefathers shrouded these and similar cases. [1980, p. 464] 46. Helvetius, in his De I'Esprit of 1758, writes, in Discours IV. Des differents Noms donnes a I'Esprit, ch. II. De l'Imagination & du Sen• timent, Several people have confused memory with imagination. They are not aware that the words are not exactly synony• mous; that the memory consists of the distinct recollection of objects that are presented to us; & the imagination, in the combination, the new collection of images and a rela• tionship of propriety perceived between these images and the sentiment one wishes to arouse. [po 263] Bonnet's view, however, as related in his Essai Analytique (Euvres [1779-1783], Vol. XIII), is rather different. He says But the signs of our ideas are figures or sounds. Thus they affect the eye or the ear. Thus they keep to the fibres of the eye or of the ear. These fibres proceed to lead to the seat of the soul: there there are other fibres that correspond to 214 Notes: On illusions in estimation
the former, even if they are not a simple extension of it. The conservation and the recalling of the sign or of the word thus operate by a mechanism similar to that in which the conservation and the recalling of the idea attached to this sign or to this word operate. Memory does not differ essentially from imagination; I've said it before somewhere. [art. 223, pp. 180-181] 47. Once again in the Essai Analytique we find Bonnet writing The memory, by which we retain the ideas of things, has been associated with the body, since causes that affect only the body, weaken the memory, even suppressing it or con• firming it. By how many very constant and very diverse facts has medicine not established this truth! How many illnesses or accidents have there been that have been followed by a weakening or even a loss of memory! How many other for• tuitous incidents have singularly changed this faculty, or have appeared to give it new force! It would be useless for me to insist any more on a truth so well known: the memory still is retained to old age; & it is the very procedures that one uses to improve and strengthen it {i.e. the memory}, that tend to confirm this same truth. [art. 57, pp. 44-45] 48. This psychological trait - one that is often perceived by mathemati• cians as characteristic - was well described by Poincare as follows: For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian func• tions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great num• ber of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hyperg~ ometric series; I had only to write out the results, which took but a few hours. Then I wanted to represent these functions by the quo• tient of two series; this idea was perfectly conscious and deliberate, the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and I succeeded without difficulty in forming the series I have called theta-Fuchsian. Notes: On illusions in estimation 215
Just at this time I left Caen, where I was then living, to go on a geological excursion under the auspices of the school of mines. The changes of travel made me forget my math• ematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non• Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience' sake I verified the result at my leisure. Then I turned my attention to the study of some arith• metic questions apparently without much success and with• out a suspicion of any connection with my preceding re• searches. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and im• mediate certainty, that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry. Returned to Caen, I meditated on this result and de• duced the consequences. The example of quadratic forms showed me that there were Fuchsian groups other than those corresponding to the hypergeometric series; I saw that I could apply to them the theory of theta-Fuchsian series and that consequently there existed Fuchsian func• tions other than those from the hypergeometric series, the ones I then knew. Naturally I set myself to form all these functions. I made a systematic attack upon them and car• ried all the outworks, one after another. There was one however that still held out, whose fall would involve that of the whole place. But all my efforts only served at first the better to show me the difficulty, which indeed was some• thing. All this work was perfectly conscious. Thereupon I left for Mont-Valerien, where I was to go through my military service; so I was very differently oc• cupied. One day, going along the street, the solution of the difficulty which had stopped me suddenly appeared to me. I did not try to go deep into it immediately, and only after my service did I again take up the question. I had all the elements and had only to arrange them and put them to- 216 Notes: On illusions in estimation
gether. So I wrote out my final memoir at a single stroke and without difficulty. I shall limit myself to this single example; it is useless to multiply them. [1946, pp. 387-389]
49. Ball and Coxeter note that the gift of an excellent memory, like the gift of the wicked fairy in "The Sleeping Beauty" , is not an unmixed blessing. They remark that A.C. Aitken, who besides being a calcu• lating prodigy and an excellent mathematician, was possessed of a remarkable memory, "once remarked that he had to be careful what he read for entertainment, because of the difficulty of forgetting it afterwards" [1974, p. 386]. 50. This reference - and indeed this whole paragraph - appears for the first time in the 5th edition.
51. The phrase translated here as "he hummed quietly ... them to sing" appears in Barbeu-Dubourg [1773] as "he turned his thoughts to hum• ming very quietly in another room the tune that he wished them to take up." There are other small changes between the original and Laplace's version: none is significant. 52. See Helvetius [1758], Discours III, Si l'Esprit Doit Etre Considere comme un Don de la Nature, 01.£ comme un EJJet de l'Education. Chapter IV, "De l'inegale capacite d'attention." Pertinent passsages are the following: As it is the attention, more or less great, that engraves objects more or less deeply in the memory, which in fact understands better or less well the relationships which form most of our judgements, true or false; and as it is finally to that attention that we owe almost all our ideas; it is, I say, clear that the unequal strength of men's minds depends on the unequal capacity of their attention. [po 24]
It seems that, in the coincidence of fortuitous incidents nec• essary to form a witty man, the different capacity of the at• tention which would be able to produce a more or less great force, according to temperament, is of no importance. [pp. 25--26]
53. According to the Encycloptedia Britannica (14th edition), panoramas, suggested by the German architectural painter Breisig, were first ex• ecuted by the Edinburgh artist Robert Barker in 1788. In its simplest form, the panorama consists of a painting - say of a landscape - painted on the inner surface of a cylinder, at the centre of which the observer stands, the latter then sees a picture all around him, as he Notes: On illusions in estimation 217
would in reality. Panoramas were introduced into France in 1799 by Robert Fulton. 54. The writer of the notes in the Bru/Thom edition of the Essai suggests that the panoramas Laplace might have seen were (i) l'Entrevue de TiZsitt, painted by Pierre Prevost in 1807, (ii) La Vue de Jerusalem, painted in 1819, and (iii) Vue d'Athenes, painted in 1821. 55. David Hume (1711-1776) and Condorcet. Laplace gives a physiolog• ical basis for Condorcet's theory of the reason of belief. 56. Here Laplace is using the Bossut edition of Pascal's works (La Haye 1779, vol. 2, II, III, 5 & 6, pp. 207-209), which goes back to the Port• Royal edition which, in turn, did not conform to the manuscript. The exact version given in the Brunschvicg & Boutroux edition [1908] of Pascal's (Euvres, runs (in translation) as follows: "I would soon have given up these pleasures," they say, "if I had faith." But I say to you: "You would soon have faith, if you had given up these pleasures. Now, it is up to you to begin. I would give you faith if I were able to do so; but I cannot, and therefore I cannot test the truth of what you say. But you can easily give up your pleasures and test whether what I tell you is true." [N°. 240.]
For we must not forget ourselves: we are automaton as much as mind; and it follows from this that the instrument by which we are persuaded is not only demonstration. How few things can be demonstrated! Proofs satisfy only the mind; habit gives us the strongest and the most believable proofs; it disposes the automaton, which carries away the mind with it without thinking. Who has proved that the day will dawn tomorrow, and that we shall die? And what is more widely believed? It is thus habit that convinces us of this; and it is this that makes so many Christians, it is this that makes Turks, pagans, professions {or trades}, soldiers, etc. (Christians have an advantage over pagans in receiving faith at baptism.) In a word, it is necessary to have recourse to habit once the mind has realized where the truth lies, so that we may steep and stain ourselves in that belief, which constantly escapes us; for it is too much trou• ble to have proofs always at hand. It is necessary to acquire a more facile belief, which is that of habit, which, without violence, artifice or argument, makes us believe things, and disposes all our powers to this belief, so that our soul lights naturally upon it. When one believes only because of the 218 Notes: On illusions in estimation
strength of one's convictions, and the automaton is dis• posed to believe the opposite, that is not enough. It is then necessary to make both of our two parts believe; the mind, for those reasons that it is sufficient to see only once in one's life, and the automaton, by habit, and not allowing it any disposition to the contrary. Incline my heart, 0 God [Ps. CXVIII, 36]. The reason works slowly, and with so many looks at so many principles, which must of necessity be al• ways present, that it is constantly deadened or mistaken, when not all its principles are present. Sentiment does not work like that: it operates instantaneously, and is always ready to operate. It is necessary then to put our faith in sentiment; or else it will always be shaky. [N°. 252.]
57. In the 4th edition of the Essai Laplace added the following note at this point: "Here Pascal loses sight of the opinion that he had just recommended for the acquiring of faith: namely, to begin with ex• ternal acts." The writer of the notes in the Bru/Thom edition of the Essai cites a personal communication from G. Th. Guilbaud, in which it is stated that the incoherence arises in the Port-Royal edition of Pascal's Pensees, and is not to be found in the manuscripts. 58. Augustus's superstitions are detailed in Suetonius [1957, Chapter II, §§90-92]. Particularly interesting is the last of these sections, in which we read Augustus had absolute faith in certain premonitory signs: considering it bad luck to thrust his right foot into the left shoe as he got out of bed, but good luck to start a long jour• ney or voyage during a drizzle of rain, which would ensure success and a speedy return. Prodigies made a particularly strong impression on him. Once, when a palm tree pushed its way between the paving stones in front of the Palace he had it transplanted to the inner court beside his family gods, and lavished care on it. When he visited Capri, the drooping branches of a moribund old oak suddenly regained their vigour, which so delighted him that he arranged to buy the island from the City of Naples in exchange for ls• chia. He also had a superstition against starting a journey on the day after a market-day, or undertaking any impor• tant task on the Nones of a month - although, in this case, as he explained to Tiberius in a letter, it was merely the unlucky non-sound of the word that affected him.
59. "Probabilities" is used here in the colloquial sense, without any idea of the probability calculus. Notes: On approaches to certainty 219
60. The passage, from Montaigne's essay De l'Experience, is given in the original as 0 que c'est un doux et mol chevet, et sain, que l'ignomnce et l'incuriosite, d reposer une teste bien faicte!; that is, "Oh how soft and luxurious and wholesome a pillow, on which to rest a well-made head, is ignorance and incuriosity!" (Various translations give the words une teste bien faicte as "a well-made head", "a well-schooled head", "a prudent head", and "a well-contrived head"; the phrase seems remarkably awkward.)
On various approaches to certainty
1. See the Le~on, pp. 150-153. 2. The first announcement by Fermat of his famous result was made in a letter to Frenicle [de Bessy] in August (?) 1640 (see (Euvres de Fer• mat, Vol. 2 [1894, pp. 205-206]). After giving the double progression
123 456 789 10 248 16 32 64 128 256 512 1024
11 12 13 14 15 16 2048 4096 8192 16384 32768 65536
Fermat writes: But this is what I find most remarkable: namely, that I am almost persuaded (1) that all the numbers - each in turn increased by 1 - whose exponents are numbers in the double progression {Le. all numbers of the form 22n + I} are prime numbers, like
3, 5, 17, 257, 65337, 4294967297
and the following number of 20 digits
18446744 073709551617; etc. I do not have a precise proof of this, but I have excluded so many divisors by infallible proofs, and I have so great an insight into the matter, which confirms my belief, that I would have difficulty in recanting. (The large number given above is in fact 264 + 1 - see also Note 3.) Later in his "De solutione problematum geometricorum per cur• vas simplicissimas et unicuique problematum generi proprie conven• tientes, Dissertatio Tripartita", Part 3 (pp. 127-131), (Euvres de Fermat, Volume 1 [1891]), Fermat affirms 220 Notes: On approaches to certainty
I am certain that numbers which are themselves powers of 2 squared and increased by 1, {i.e. numbers of the form 22" + I}, are always prime numbers, and for a long time Analysts have suggested that the theorem, namely that
3, 5, 17, 257, 65337
and so on to infinity are prime numbers, is true. In August 1659, in a letter to Carcavi {(Euvres de Fermat, Volume 2 [1894, pp. 431-436]) he hinted at a method of proof: Then I have considered certain questions which, although answered in the negative, are still very difficult, the method for obtaining the descent being entirely different from the preceding, as it is easy to prove. These {questions} are the following ... All the square powers of 2, increased by 1, are prime numbers. [pp. 433-434]
3. Euler [1732-1733] showed, as Laplace says, that 232 + 1 has a factor of 641 - in fact,
232 + 1 = 641 x 6700417
Further, 264 + 1 is divisible by 274,177: indeed, no further Fermat primes - beyond those given by Fermat (see Note 2) are known. Dickson [1919] states that for n > 1, F.. = 22" + 1 is prime if and only if it divides kQ + 1 {where a = (F.. - 1)/2), where k is any quadratic non-residue of F .. , as 5 or 10. See also Hardy and Wright [1960, §2.5].
4. Cournot, on the other hand, believed that induction was not attached to the theory of probabilities, but was linked up with philosophical probability.
5. This "law" was apparently discovered by Laplace: see the Ency• cloptedia Britannica, 14th edition, Vol. 13, p. 190. 6. The actual passage, given in Book II of the Novum Organum (see Bacon [1905]), runs as follows:
Again, let the nature investigated be the Spontaneous M~ tion of Rotation; and in particular, whether the Diurnal Motion, whereby to our eyes the sun and stars rise and set, be a real motion of rotation in the heavenly bodies, or a motion apparent in the heavenly bodies, and real in Notes: On approaches to certainty 221
the earth. We may here take for an Instance of the Fin• gerpost the following. If there be found in the ocean any motion from east to west, however weak and languid; if the same motion be found a little quicker in the air, especially within the tropics, where because of the larger circles it is more perceptible; if the same motion be found in the lower comets, but now lively and vigorous; if the same motion be found in planets, but so distributed and graduated, that the nearer a planet is to the earth its motion is slower, the further a planet is distant from the earth its motion is quicker, and quickest of all in the starry sphere; then indeed we should receive the diurnal motion as real in the heav• ens, and deny such motion to the earth; because it will be manifest that motion from east to west is perfectly cosmi• cal, and by consent of the universe; being most rapid in the highest parts of the heavens, and gradually falling off, and finally stopping and becoming extinct in the immoveable, - that is, the earth. [1905, Aphorism XXXVI]
(The term Instances of the Fingerpost (originally Instantias Crucis) indicates, in analogy with "Signpost", a place where roads part, and is used to indicate that several directions may be taken.) This Spontaneous Motion of Rotation receives further attention in Aphorism XLVIII, where it is considered as the seventeenth motion. 7. In his essay "On the hypothesis that animals are automata, and its history" of 1874, Huxley notes Descartes's "startling conclusion" that "brute animals are mere machines or automata, devoid not only of reason, but of any kind of consciousness" (Huxley [1904, p. 216]). A particularly pertinent passage occurs in Descartes's Reponses de l'auteur aux quatriemes objections faites par Monsieur Arnauld, doc• teur en theologie, where we read but as for the souls of beasts ... I shall say further here that it seems to me to be a most remarkable thing, that no movement can take place, either in the bodies of beasts, or even in our own, if these bodies do not in themselves have all the organs and instruments by means of which these very movements would be carried out by a machine . . . . it will be easy to see that all the actions of beasts are similar only to those that we carry out without using our minds. For this reason we shall be obliged to conclude, that we know indeed of no other principle of movement in them than the mere disposition of their organs and the contin• ual affluence of animal spirits produced by the heat of the 222 Notes: Historical note
heart, which thins and refines the blood; and at the same time we shall recognize that we have had no reason be• fore to attribute any other principle to them, except that, not having distinguished these two principles of movement, and seeing that the one, which depends only on animal spirits and organs, exists in beasts as well as in us, we have thoughtlessly concluded that the other, which depends on mind and on thought, was also possessed by them. (This passage may be found in Descartes [1953, pp. 447-449] in French, while a translation into English of a longer passage, in which the above is incorporated, may be found in Huxley [1904].) Recently Dennett has explored intentional systems, such a system being one ''whose behavior is reliably and voluminously predictable via the intentional strategy" [Dennett, 1987, p. 15]. The intentional strategy, in turn, consists, roughly speaking, of treating the object whose behavior you want to predict as a rational agent with beliefs and desires and other mental stages [Dennett, loco cit.] Thus intentional systems include not only humans but also groups of humans, animals, machines and combinations of men and machines. 8. See Laplace [1780] and [1806]. Goethe's novel Die Wahlverwandt• schaften [1806] deals with this topic.
9. De Morgan reports that in 1836 the two stars of'Y Virginis were so near together that ''they appeared to be one as much with the tele• scope as without it" [1915, vol. I, p. 317]. This sighting was apparently followed by an article in the Church 0/ England Quarterly Review, in which the notion that there could be double stars was "implied to be imposture or delusion" [De Morgan, loco cit.]. 10. Discovered by Benjamin Franklin (1706-1790) in 1749.
11. In analogy with Newtonian Mechanics, phenomena are regarded as "explained" when they have been traced back to an "effective force" . This is not necessarily unconditionally acknowledged as a leading principle of explanation today (von Mises [1932, p. 210]).
Historical note on the probability calculus 1. This article is one of the first histories of the probability calculus. Among 19th century writings on this subject we should mention Gouraud [1848], Lubbock and Drinkwater-Bethune [c. 1830], and, of course, Todhunter [1865]. More recent works include Pearson and Notes: Historical note 223
Kendall [1970], Kendall and Plackett [1977], Pearson [1978], Hald [1990], Stigler [1986b] and Dale [1991] (all of which contain the word "History" in their titles).
2. See the article translated here as "On analytical methods in the prob• ability calculus."
3. Huygen's tract was entitled De Ratiociniis in Alea; Ludo on the title page, and De Ratiociniis in Ludo Alea; on the first page - see Stigler [1988]. 4. de Witt [1671].
5. Halley [1693]. In his 1942 edition of Degrees of Mortality of Mankind, by Edmund Halley, Reed says "this first table of mortality has re• mained the pattern for all subsequent tables, as to its fundamental form of expression" (introduction, p. iv). 6. Stirling's formula, perhaps more correctly the Stirling-de Moivre for• mula, is today often given in the form
the sign""' being used to indicate that the ratio of the two sides tends to 1 as n - 00. Alternatively, we can write
log x! = log v'21r + (x + ~) log x - S ,
where
Stirling [1730] gave the formula
logx! = ~log271"+(x+~)log(x+~)-(x+~)
- 1 + 7 - ... 2 x 12(x+~) 8 x 360(x+ ~)3
with the rule for the continuation of the series. De Moivre, in the Supplement to his Miscellanea Analytica, gave the series
logx! ~ log271"+(x+ ~)logx-x
Bl 1 B2 1 n 1 Bn 1 +lx2;--3x4x3 +···+(-1) + (2n-1)2nx2n - 1 ··· 224 Notes: Historical note
(where the Bi are the Bernoulli numbers), and Stirling, again, eval• uated the constant. For further details see Tweedie [1922], particu• larly pp. 203-205. Thomas Bayes was apparently the first to note the asymptotic nature of the series: see Bayes [1763] and Dale [1991, §4.2.1]. 7. See Note 56 on the Article "On analytical methods in the probability calculus." 8. Deparcieux [1746]. 9. Kersseboom [1970]. 10. Wargentin [1766].
11. See Buffon, (EU1J1'eS, Volume 12 [1827]. Dupre de Saint-Maur collected data from three parishes in Paris and twelve in the country. The tables he produced, given in the "Suite de l'Histoire des animaux, Suite de l'histoire de l'homme, De la vieillesse et de la mort" (Buffon, loco cit., pp. 3-39 & 4(}-49), were, wrote Buffon, "the only ones from which one is able to establish the probabilities of human lives in general with some certainty" [p.39]. 12. Simpson [1742]. 13. SiiBmilch [1765]. 14. The reference seems to be to Messance, referred to in the Catalogue general des livres imprimes de la Bibliotheque Nationale as "collector of election votes in St.-Etienne." Possible works are Messance [1766] and [1788]. 15. Moheau [1778]. 16. Price [1783]. 17. Baily [1810/1813]. 18. Duvillard de Durand [1787]. 19. Lagrange [1770-1773]. 20. Laplace [1778]. 21. Cotes [1722]. 22. See the 2nd Supplement to T.A.P., O.C. VII.
23. See the 3rd Supplement to T.A.P., O.C. VII. 24. See the 3rd Supplement to T.A.P., O.C. VII, and Laplace [1830]. APPENDIX
Editions of the Essai
As has already been mentioned, five editions of the Essai were published during Laplace's lifetime, while several others appeared after his death in 1827. There have also been a number of translations into other languages: those I have managed to trace are listed below. (The abbreviation O.C. is used for (Euvres Completes de Laplace.)
1. French editions of the Essai. 1810 Notice sur les probabilites. Annuaire presente Ii S.M. l'Empereur et Roi par le Bureau des Longitudes pour l'an 1811. pp. 98-125. Paris: Courcier. (Reprinted in Gillispie [1979].) 1812 Sur les probabilites. A lecture given at the Ecole Normale in 1795. Journal de l'Ecole Polytechnique VIle £3 VIlle Cahiers. (Reprinted O.C. XN, pp. 146-177.) [These two works are essentially preliminary versions of the Essai.]
1814 (February) 18t edition of the Essai Philosophique. Paris: Mme Ve Courcier. (A facsimile was published in 1967 by Culture et Civili• sation, Brussels.)
1814 (November) 2nd edition ofthe Essai Philosophique. Paris: Ve Courcier. (This served as the introduction to the 2nd edition of 1814 of Laplace's Theorie Analytique des Probabilites.) 1816 (July) 3rd edition of the Essai Philosophique. Paris: Ve Courcier.
1819 (October) 4th edition of the Essai Philosophique. Paris: ve Courcier. (This became the introduction to the 3rd edition of 1820 of the Theorie Analytique des ProbabiliUs.) 1825 (February) 5th edition of the Essai Philosophique. Paris: Bachelier. 1840 6th edition of the Essai Philosophique. (Identical to the 5th edition.) Paris: Bachelier. 225 226 Appendix
1840 rth edition of the Essai Philosophique. (Identical to the 5th edition.) Brussels: Societe BeIge de Librairie. 1920 Essai Philosophique. (With an introduction by X. Torau-Bayle, and edited by E. Chiron.) Paris: Bibliotheque de philosophie moderne. 1921 Essai Philosophique. (With a biographical note by M. Solovine). Paris: Gauthier-Villars et Cie.
1986 Essai Philosophique. (Re-issue of the 5th edition, with a preface by Rene Thorn and a postscript by Bernard Bru.) Paris: Christian Bour• gois.
2. Translations.
2.1 English. A Philosophical Essay on Probabilities. 1902. Translation of the 6th edition by F.W. Truscott and F.L. Emory. New York: John Wiley and Sons. (2nd edition 1917. Reprinted, with an introductory note by E.T. Bell, in 1951 & 1952 by Dover Publications, Inc., New York.)
2.2 German. Philosophischer Versuch tiber Wahrscheinlichkeiten. 1819. Translation ofthe 3rd edition by F.W. T6nnies, with notes by K.C. Langsdorf. Heidelberg: K. Groos. Philosophischer Versuch tiber Wahrscheinlichkeiten. 1886. Translation of the 6th edition by N. Schwaiger. Leipzig: Duncker & Humblot. Philosophischer Versuch tiber Wahrscheinlichkeiten. 1932. Translation of the 5th edition by H. LOwy, and edited by R. von Mises, with notes by H. Pollaczek-Geiringer. Leipzig: Akademische Verlagsgesellschaft. M.B.H.
2.3 Italian. Saggio Filosofico sulle Probabilita. 1951. Translated by S. Oliva, with an introduction by F. Albergamo. Bari: Lat• erza.
2.4 Russian. Opyt filosofii teorii veroiatnostei. 1908. Moscow. BIBLIOGRAPHY
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Venn, J. 1866. The Logic of Chance. London: Macmillan. Reprinted, 1962 (reprint of 3rd edition of 1888), New York: Chelsea Publishing Com• pany. Voltaire (Franc;ois-Marie Arouet). 1843. (Euvres Completes de Voltaire. vol. 5. Paris: Firmin Didot Freres. von Mises, R. (Ed.) 1932. P.S. de Laplace (1814): Philosophischer Ver• such iiber die Wahrscheinlichkeit. Leipzig: Akademische Verlagsge• selleschaft M.B.R. von Neumann, J. & Morgenstern, O. 1944. Theory of Games and Econom• ic Behavior. Princeton: Princeton University Press.
Wargentin, P.W. 1766. Mortaliteten i Sverige. Kongl. vetenskaps acade• miens handlingar. Reprinted in 1930 as Tables of Mortality based upon the Swedish population, prepared and presented in 1766 by Per Wilhelm Wargentin. Stockholm: I. Raeggstrom. Wharton, G. 1683. The Works of That Late Most Excellent Philosopher and Astronomer, Sir George Wharton, Bar. Collected into one En• tire Volume. By John Gadbury, Student in Physick and Astrology. London: Printed by R.R. for John Leigh. Wilson, C. 1985. The great inequality of Jupiter and Saturn: from Kepler to Laplace. Archive for History of Exact Sciences 33: 15-290.
Zabell, S.L. 1988. Buffon, Price, and Laplace: scientific attribution in the 18th century. Archive for History of Exact Sciences 39: 173-181. -- 1989. The rule of succession. Erkenntnis 31: 283-321. GLOSSARY
This glossary contains a list of terms, used in the text and in the notes, whose meaning may require amplification. Also provided are (very) short biographies of persons mentioned in the text. Further details may be found in the following works: Biographie Universelle, Ancienne et Modeme. 1811-1828. 52 vols. Paris: L.G. Michaud. Dictionary of National Biography. 1949-1950. 21 vols. + supplements. London: Oxford University Press. Dictionary of Scientific Biography. 1970-1990.18 vols. New York: Charles Scribner's Sons. EncycloptEdia Britannica. 1986. 15th edition. Encyclopaedia of Mathematics. 1988-1993. Dordrecht: Kluwer Academic Publishers. Encyclopedia of Statistical Sciences. 1982-1988. New York: John Wiley & Sons. Grand Larousse Encyclopedique. 1960. Paris: Librairie Larousse. James, G. & James, R.C. (eds) 1959. Mathematics Dictionary. Princeton, New Jersey: D. van Nostrand Co., Inc. Kruskal, W.H. & Tanur, J.M. 1978. International Encyclopedia of Statis• tics. 2 vols. New York: The Free Press. Marriott, F.H.C. 1990. A Dictionary of Statistical Terms. Harlow, Essex: Longman Scientific & Technical.
Millar, D., Millar I., Millar J. & Millar, M. 1989. Chambers Concise Dic• tionary of Scientists. Edinburgh: W.R. Chambers Ltd. Mitton, J. 1991. A Concise Dictionary of Astronomy. Oxford: Oxford Uni• versity Press.
253 254 Glossary
Newcomb, S. 1960. A Compendium of Spherical Astronomy, with its ap• plications to the determination and reduction of positions of the fixed stars. New York: Dover Publications, Inc.
O'Muircheartaigh, O. & Francis, D.P. 1981. Statistics: a dictionary of terms and ideas. London: Arrow Books.
The Oxford English Dictionary. 1989. Oxford: Clarendon Press.
aIllortize: to discharge a debt, plus interest, by periodic payments. analogy: a kind of inference used in the establishing of new theorems (arguing roughly that if two things agree in some respects, they may well agree in others). analysis: the art of discovering the truth or falsehood of a proposition, or its probability or impossibility. Although used in this transla• tion roughly as synonymous with mathematics, the term refers more specifically to those branches of mathematics that use mainly meth• ods of algebra and the (differential and integral) calculus. analysis, non-standard: consider a certain mathematical structure M and a first-order logico-mathematical language C. Using model the• ory one constructs a non-standard model of M that is a proper ex• tension of M. For example, if M is the field of real numbers, then the non-standard elements of M may be regarded as "infinitesimal" (i.e. infinitely large or infinitely small, but non-zero) reals. animal magnetism: see mesmerism. annuity: a series of payments made at regular intervals. anomaly: an angle used in describing the motion of a body in an elliptical orbit. The true anomaly is the angle between the line joining the body to the focus of the ellipse, and the line joining the focus to the point on the body's orbit that is nearest to the focus. (The mean and eccentric anomalies are yet more awkward to express in words.) apogee: the point in the moon's orbit that is furthest from the Earth. aruspex (or haruspex): an ancient Roman soothsayer, who divined by inspection of the entrails of victims. Glossary 255 astrologer: practitioner of a tradition that purports to connect human traits, affairs and the course of events with the positions of the sun, moon and planets in relation to the stars. atom: the smallest stable unit of a chemical element, or the smallest par• ticle of an element that can enter into, or be expelled from, chemical combination. augur: a Roman religious official, who predicted future events and advised on the course of public business by reading omens derived from the flight, singing and feeding of birds, the appearance of the entrails of sacrificial victims, and celestial phenomena. Augustus Caesar (Gaius Octavius) (63 BC-14 AD): first Roman em• peror. Great-nephew of Julius Caesar, and introducer of an autocratic regime. Deified after death.
Bacon, Francis (1561-1626): English statesman and natural philosopher; proponent of the inductive method in science; critic of Aristotle and the deductive method. Baily, Francis (1774-1844): English astronomer, and a founder of the As• tronomical Society of London (later the Royal Astronomical Society). Vivid describer of Baily's beads (beads of light seen around the edge of the moon immediately before and after a total solar eclipse). Barbeu Dubourg, Jacques (1709--1799): French medical man. Friend of Benjamin Franklin, and the official representative in Paris from 1775 to 1777 of the American insurgents. Writer of works on botany. Bayes, Thomas (1702 (?)-1761): English Presbyterian minister. Wrote (a) a treatise on fluxions (q.v.) in defence of Newton, (b) a paper in which the divergence of the series for log x! was realized, and (c) a posthumously published work of fundamental importance in modern statistical theory (Bayesian statistics). Bernoulli, Daniel (1700--1782): nephew of James (q.v.), son of Johann (= Jean). Studied medicine and mathematics. Professor of Mathe• matics at the Imperial Academy in St Petersburg, and later Professor of Physics at Basel. Awarded prizes ten times by the French Academy of Sciences. His 1738 paper on probability introduced the idea of moml expectation, or marginal utility. Also noted for his defence of inoculation for smallpox and for a statistical test of randomness for the planetary orbits. Bernoulli, James (= Jakob, Jacques) (1654-1705): studied theology. Professor of Mathematics at Basel. His posthumous Ars Conjectandi 256 Glossary
of 1713 discussed Huygens's De Ratiociniis in Ludo AlelE, gave a sys• tematic development of permutations and combinations (with appli• cations), and - in the fourth and most important part - discussed when unknown probabilities could be determined from experience.
Bernoulli, Nicolas (1687-1759): nephew of James and Johann, son of Nicolas. Studied jurisprudence. Professor of Mathematics in Padua, and later Professor of Logic and then of Jurisprudence in Basel. Edited James Bernoulli's Ars Conjectandi and corresponded with 's Gravesande on the observed regularity in the birth ratio. Also noted for his correspondence with Montmort (see the latter's Essay of 1713).
Bessel, Friedrich Wilhelm (1784-1846): German mathematician and astronomer. Noted for his detection of the companion of Sirius, for Bessel functions, and for the analysis of perturbations in stellar and planetary motions. He was the first to measure a star's distance by the measurement of its parallax. He suggested that the irregularities in the orbit of Uranus were caused by the presence of an unknown planet, but died before Neptune was discovered. bias: an effect resulting in the systematic (but not necessarily intentional), rather than the random, distortion of a statistical result. binary star: a pair of stars, orbiting about each other and bound by their mutual gravitational attraction.
n binomial formula: (a + b)n = 2:= G)akbn- k. k=O Bonnet, Charles (1720-1793): Swiss naturalist and philosophical writer. Discoverer of parthenogenesis and developer of the catastrophe the• ory of evolution. The first to use the term "evolution" in a biological context. His Essai Analytique of 1760 anticipated physiological psy• chology.
Bouvard, Alexis (1767-1843): astronomer and director of the Paris ob• servatory. Discovered eight comets. Compiled astronomical tables of Jupiter, Saturn and Uranus. Valuable assistant to Laplace, who left to him the detailed calculations for the Mecanique Celeste.
Brahe, Tycho (1546-1601): Danish astronomer, and greatest pre-tele• scopic observer. Producer of an exceedingly accurate star catalogue of 777 stars.
Buffon, George-Louis Leclerc, comte de (1707-1788): French natur• alist and polymath. Exhibited early ideas on the evolution of species. Glossary 257
Burg, Johann Tobias (1766-1834): Austrian astronomer, who worked on the Viennese ephemerides (q.v.). Improved on Laplace's perturba• tion theory of the moon's orbit by adding more terms of the pertur• bation function (for the disturbing influence of the sun and for the oblateness of the Earth). His lunar ephemerides proved to be much more accurate than those of his predecessors. calculus, integral: the study of integration and its applications.
Calixtus III (1378-1458): Alfonso de Borgia. Pope from 1455 to 1458. In• stituted the Feast ofthe Transfiguration in 1457 to commemorate the repulsion of the Turks from Belgrade in 1456: during his pontificate Joan of Arc was proclaimed innocent. cause, final: in natural theology, the design, purpose or end of the ar• rangements of the universe. celestial equator: the projection into space of the Earth's equatorial plane. celestial sphere: the sky, when considered as a hollow sphere on which the positions and motions of astronomical bodies can be described.
Chaldean: one skilled in occult learning or astrology. chance: a kind of probability - physical, or statistical, as opposed to inductive. Applies to events (or trials) when there are a number of possible outcomes, and when one is uncertain which of these will obtain. Chance, in Laplace's theory, is only a measure of our ignorance of the outcome of a trial. chances: the ways in which things fall out; casual or fortuitous circum• stances; opportunities that come one's way.
Cicero, Marcus Tullius (106-43 BC): Roman orator and politician. Stu• dent of dialectic, rhetoric and law. His extant letters give a unique knowledge of Roman life and history. Clairaut, Alexis Claude (1713-1765): French mathematician; contribu• tor to mathematics, mechanics, celestial mechanics, optics and geo• desy. Read paper before the French Academy, on the properties of curves he had discovered, at the age of twelve. Known for his the• orem connecting the gravity at points on the surface of a rotating ellipsoid with the compression and centrifugal force at the equator. Calculated the perihelion of Halley's comet. Detected the singular solutions of certain differential equations. 258 Glossary combination: an arrangement of a set of objects in which order is not of importance. Thus, in three tosses of a coin, the arrangements HHT,HTH and THH (where H stands for heads and T for tails) are a single combination. common ratio: the ratio between any two consecutive terms in a geo• metric progression. For example, in the series a + ar + ar2 + ... , the common ratio is r.
Cotes, Roger (1682-1716): English mathematician and philosopher. Plumian professor of astronomy and natural philosophy at Cam• bridge. Newton is reputed to have said "Had Cotes lived, we might have known something" .
Craig, John (? -1731): Scottish philosopher and mathematician. A friend of Newton, engaging in controversy with James Bernoulli. Said to have been "an inoffensive, virtuous man" .
Cygnus: an elliptical galaxy (the Swan): a strong radio source.
D'Alembert, Jean Ie Rond (1717-1783): a French mathematician who worked on partial differential equations, solving the vibrating string problem and the general wave equation. He applied the calculus to celestial mechanics. declination: a co-ordinate defining position on the celestial sphere (q.v.) in the equatorial co-ordinate system: the equivalent of latitude on the Earth. deduction: the method of inferring from accepted principles. de Mere, Antoine Gombaud (Gombault), chevalier (1607-1684): a gambler and proposer of some problems to Pascal: the discussion of these questions, between the latter and Fermat, contributed greatly to the development of probability theory. de Moivre, Abraham (1667-1754): a French protestant who fled to Eng• land when he was 21 to escape the religious persecution that followed on the repeal ofthe Edict of Nantes in 1685. He lived by tutoring and was made F.R.S. in 1697. His Doctrine of Chances (1718/1738/1756) contains general laws of addition for probabilities; the binomial dis• tribution; probability generating functions; difference equations and their solution by recurring series; the limiting form of (;)px (l_p)n-x, for x E {O, 1, ... , n} and 0< p < 1, with (i) n --+ 00 and np finite, and (ii) n --+ 00 and np --+ 00. He also published work on the mathematics of life contingencies. Glossary 259
Deparcieux, Antoine {1703-1768}: French mathematician and maker of sundials. Investigated problems in hydraulics, but is especially re• membered for his mortality tables and work on annuities. Descartes, Rene {1596-1650}: French philosopher and mathematician; creator of analytical geometry. de Witt, Johan {1625-1672}: studied law; prominent Dutch statesman. He contributed to actuarial science and mathematics, and did some fundamental work on the calculation of the values of annuities using mortality tables. Raadspensionaris in 1652 and re-appointed every five years thereafter, the last appointment being in 1668. difference equation: a finite difference analogue of a differential equa• tion. Simply a relationship of the form
among {k + 1} successive terms of a sequence {ai}. diviner: one who practises divination; a soothsayer, seer, prophet or ma• gician. double star: see binary star.
Dupre de Saint-Maure, Nicolas Fran~ois {1695-1774}: French econ• omist and publisher of a mortality table. Duvillard de Durand, Emmanuel Etienne {1755-1832}: French sta• tistician and politician. Publisher of a mortality table. eccentricity: a parameter used in describing a conic section {circle, ellipse, parabola, hyperbola}, and hence an element in the description of the path of a body about the sun. ecliptic: the mean plane of the orbit of the Earth about the sun. Ecoles Normales: colleges of education.
economy, animal: the organization, internal constitution, apportionment of functions, of the animal kingdom.
elements (of planets, &lc.): a set of parameters completely defining the shape, orientation and timing of orbital motion, e.g. eccentricity, pe• riod, perihelion distance. elongation: the angular distance between the sun and a planet {or the moon} when observed from the Earth. 260 Glossary
ephemerides: astronomical almanacs in which the daily positions of the sun, the moon and the planets are tabulated. Epicurus (341-270 Be): Greek philosopher. Established a school in Athens, whose regimen was characterized by simplicity and by dic• tatorial dogmatism. The subject of scandalous accusations by the Stoics. A prolific writer, his philosophical outlook was fundamentally ethical. His teaching in physics was based on the views of Democritus as regards atoms, though this debt was not acknowledged. equation of condition: let x, Y, z, ... be varying quantities whose values may be obtained at specific moments by observation. These variables are known functions oftime, t, and of other quantities a, b, c, . .. called elements, these quantities either being constant or having variations that are known in advance. Suppose now that we need to determine a, b, c, . .. from observed values of x, Y, z, ... , t. Expressions ofthe form
xI=fI(a,b,c, ... ,td, x2=h(a,b,c, ... ,t2)
YI =gl(a,b,c, ... ,td , Y2=g2(a,b,c, ... ,t2) zl=hl(a,b,c, ... ,td, z2=h2(a,b,c, ... ,t2) etc. are called equations of condition, because they express the con• ditions that the elements a, b, c, ... must satisfy if the computational results are to agree with the observational results. equinox: either of the two points at which the ecliptic and the celestial equator intersect: the times when the sun passes through these points (approximately the 23rd of September and the 21st of March). Euler, Leonhard (1707-1783): Swiss mathematician. The most prolific mathematician the world has seen, he contributed to all areas of pure and applied mathematics. event: anyone of a set of possible outcomes. events, mutually exclusive and exhaustive: two events are mutually exclusive if they cannot both occur at the same time (cf. disjoint sets). A set of events is exhaustive if it includes all possible values. expectancy: the position of being entitled, at some future time, to some or other possession, e.g. as a reversion (q.v.). expectation, mathematical: if g( Xl, X2, ... , Xn) is (the value of) a func• tion of the values of the variates (q.v.) X I ,X2 , ... ,Xn, then the ex• pected value of 9 is given by Glossary 261
where F is the joint distribution function of the variates Xl! ... ,Xn . In the discrete univariate case, this expression becomes
E[X] = LXPr[X = xl,
- an analogue of the ordinary arithmetic mean, viz.
expectation, moral: the product of the probability of an outcome and its utility.
Fermat, Pierre de (1601-1665): French mathematician. His correspon• dence with Pascal in 1654 on the problem of points led to the birth of probability theory. finite difference: the differences of y = /(x) are the differences between values of y for two different values of x. Compare
ll./(x) = [/(x + h) - /(x)J/h h with d /(x) = lim ll./(x) . dx h.-O h fiuxion: in Newton's fluxionary calculus, a curve is considered as described by a flowing point. The infinitely short path traced out in an infinitely short time is called the moment of the flowing quantity, while the ratio of the moment to the corresponding time is called the flw:ion. The quantity generated is called a fluent.
Franklin, BeJ\iamin (1707-1790): North American statesman, printer, journalist and physicist. Published Poor Richard's Almanac at age 27. Experimenter and theorist on static electricity, and inventor of bifocal spectacles.
Galilei, Galileo (1564-1642): Italian astronomer and physicist: discov• erer of Jupiter's moons and the laws governing falling bodies. Died in the year in which Newton was born. His work was carried out in the modem style, viz.: by observation, experiment and the use of mathematics. 262 Glossary
generating function: the expansion of a function f of a variable t as
is the generating function of the sequence (all:). Grandi, Francesco Luigi Guido (1671-1742): member of the religious order of the Camaldulians (founded by St Romuald). Professor of philosophy, first at Florence and then at Pisa. He was deposed from his position of Abbot of St Michael at Pisa owing to the offence his writings on the history of the Camaldulians occasioned, and was immediately made professor of mathematics in Pisa by the grand duke. His work in mathematics was on conic sections, acoustics, the rectification of the cissoid, and on problems of the logarithmic curve. He also defended Galileo's position on the motion of the Earth. gravitation, universal: a law formulated by Newton, according to which the force of attraction between two particles of masses ml and m2 is proportional to the product of the masses and inversely proportional to the square of the distance between the particles. That is,
F = kmlm2/r2 ,
where k, the universal constant of gravitation, is experimentally found to be 6.675 x 10-8 cm3 per gram sec2 •
Halley, Edmond (1656-1742): English astronomer and physicist. Pro• duced, in St Helena, the first accurate catalogue of stars of the south• ern sky. Computed the orbits of comets, and, noting that the comets of 1531, 1607 and 1682 were similar, he deduced that they were the same, and predicted its return for 1758. He showed conclusively that comets were celestial bodies and not meteorological phenomena. He suggested that nebulae were clouds of interstellar gas in which for• mative processes were under way and also realized that the aurora borealis was magnetic in origin. The first mortality tables were of his construction. heads or tails: a game in which a coin is tossed, one being required to guess whether it will land heads uppermost or not. Hipparchus (c.17k.125 BC): Greek astronomer and geographer. Discov• erer of the precession of the equinoxes, and constructor of the first star catalogue. Determined the lengths of the sidereal and tropical years. Glossary 263
Hudde, Johan van Waveren (1628-1704): Netherlands mathematician and statesman. He worked on algebraic methods for solving equa• tions of higher degrees algebraically, and on the problem of extreme values and tangents to algebraic curves. He was interested in physics and astronomy, and produced microscopes with spherical lenses. He also worked on life annuities, and anticipated the series expansion for In{1 + x) (in 1656) and the use of space co-ordinates. Humboldt, Alexander von (1769-1859): German explorer (particularly of Central and South America) and meteorological pioneer. First pro• poser of the Panama Canal. Introduced isotherms and isobars on weather maps. Huygens, Christiaan (1629-1695): born and died in the Netherlands. Studied mathematics and law. Wrote the first book on probability, viz. Van Rekeningh in Spelen van Geluck, the Latin version of which (De Ratiociniis in Ludo Aleae) appeared first. Made major contribu• tions to physics and astronomy. Based his ideas of probability on the notion of (the value of an) expectation, taking the latter as primitive. hypothesis: an assumption serving as the starting point of an investiga• tion, its probable truth or falsehood not being considered.
independent: two events A and B are said to be (statistically or stochas• tically) independent (with respect to a probability Pr) if
Pr[AIB) = Pr[A) and Pr[BIA) = Pr[B).
To avoid difficulties occasioned by events of zero probability, it is perhaps wiser to define independence by
Pr[A n B) = Pr[A) Pr[B) .
index: an exponent, as the x in aX, or a subscript, as the k in ak. induction: reasoning from the particular to the general. inequalities (in orbits, &lc.): irregularities in the motion of orbits. inequality, geometric/arithmetic mean: for any non-negative values al,a2,···,an ,
inference (vs conclusion): the process of drawing judgements. 264 Glossary
Lagrange, Joseph Louis (173~1813): French mathematician. Develop• er of mechanics, using the calculus of variations and the calculus of 4-dimensional space. Contributor to the 3-body problem. Lambert, Johann Heinrich (1728-1777): autodidact. Writer on philos• ophy, logic, mathematics and the philosophy of knowledge. Known for his hygrometer and his cosine law in photometry. In his Photome• tria of 1760, he formulated what can now be seen as the method of maximum likelihood for a location parameter (q.v.). Also known (see his Neues Organon of 1764) for his treatment of non-additive probabilities. Laplace, Pierre-Simon (1749-1827): noted for his work not only in probability and statistics, but also in physics, pure mathematics and celestial mechanics. Minister of the Interior under Napoleon and later made a marquis by Louis XVIII. His literary style led to his election to the Academie Fran~ in 1816. Made contributions to generating and characteristic functions, Bayesian inference, the Normal distri• bution and the central limit theorem, and various applications. Legendre, Adrien Marie (1752-1833): French mathematician. During the Revolution he was engaged in geodetic operations and was in• strumental in the adoption of the metric system. His mathematical work included contributions to spherical trigonometry, the theory of numbers, elliptic integrals, the calculus of variations, least squares, and Legendre polynomials.
Leibniz, Gottfried Wilhelm (1~1716): German mathematician and polymath. Independent discoverer, with Newton (q.v.), ofthe calculus and combinatorial analysis. Invented a calculating machine (much superior to Pascal's), and conceived of a universal language for logic. libration: a real or an apparent irregularity in the moon's motion; any of the several effects that alter precisely which hemisphere of the moon's surface can be seen from the Earth. limpidity: clearness, transparence. Locke, John (1632-1704): English political and educational philosopher who laid the epistemological foundations of modern science. lottery: an arrangement for the distribution of prizes by chance among the purchasers of tickets. In Britain, stat. 5 Geo. 1. c. 9. declared lotteries to be ''public nuisances" . lottery, French: this is discussed in Note 5 to the article "On analytical methods in the probability calculus" and in Note 2 to "On illusions in the estimation of probabilities" . Glossary 265
Louis XIV (1638-1715): king of France; married to the Infanta Maria Theresa. His court was noted for its brilliance. During his reign the Edict of Nantes was revoked, which led to de Moivre's fiight to Eng• land.
Mason, Charles (1728-1786): worked on astronomy and geodesy. Re• membered in the United States of America for the Mason and Dixon line. Mayer, Johann Tobias (1723-1762): German astronomer and self• taught mathematician. Cartographer and inventor of the repeating circle (q.v.), originally designed for sea navigation but later used on land also. Held the chair of economy and mathematics in Gottingen. Studied the libration (q.v.) of the moon and compiled lunar and solar tables. mesmerism: the doctrine popularized by F.A. Mesmer (1734-1815) in which a hypnotic state can be produced by an influence (== animal magnetism) exercised by an operator over the will and nervous system of the subject; the process of inducing such a hypnotic state, or the state itself. Michell, John (1724-1793): English astronomer and divine. Discoverer of the existence of physical double stars, and giver of a realistic estimate of a stellar distance. Montmort, Pierre Ramond de (1678-1719): French mathematician. Best known for his work on matches in elementary probability the• ory. Worked with John Bernoulli on the problem of points and with Nicolas Bernoulli on the problem of the duration of play.
nebula: a cloud of interstellar dust and gas. Newton, Isaac (1642-1727): English scientist; a major contributor to algebra, astronomy, the infinitesimal calculus, numerical methods, mathematical and experimental physics. Successor to Isaac Barrow as Lucasian professor in Cambridge in 1669. Warden of the Mint (1696), Master of the Mint (1699), knighted by Queen Anne (1705). Corresponded with Samuel Pepys (1693) on a question of dicing. node: either of the points on the celestial sphere where a reference plane intersects the plane of an orbit.
oblateness: a measure of the amount by which the shape of a body devi• ates from a perfect sphere. 266 Glossary
odds: the ratio of the bet or wager of one player to that of another. offerings, ex voto: an offering (to a god) made in pursuance of avow. Oldenburg, Henry (c.1618-1677): born in Bremen, Germany. Spent 15 years as secretary to the Royal Society in England, in which position he founded a complete system of records, built up an international correspondence between scientists, and provided a monthly account of scientific developments. operator: a symbol indicating an operation or a series of such operations, and itself subject to algebraical operations. opposition: the position of one of the superior planets (Mars, Jupiter, Saturn, Uranus, Neptune or Pluto) when it is opposite the sun in the sky. parameter: an unknown quantity that may take on values in a certain set; often a quantity that enters into the distribution of a variate or a statistic (Le. a function of a variate or variates). Statistics formed from samples taken from a population are used to estimate parame• ters. parameter, location: a measure of central tendency, as a mean, median or mode. Pascal, Blaise (1623-1662): French contributor to mathematics, comput• ing machines, physics, theology and philosophy. His correspondence on probabilistic matters with Fermat (q.v.) in 1654 had as central no• tion that of mathematical expectation. Worked on the arithmetical triangle. The "double six" problem was put to him by de Mere. Perier, Marguerite (1646--1733): Pascal's niece and goddaughter. For details of her miraculous cure from a lachrymal fistula see Note 15 to the article "On the probability of testimony." perigee: the point at which the moon is closest to the Earth. perihelion: the point of closest approach of a body to the sun. permutation: an arrangement of a set of objects in which order is of importance. Thus, in three tosses of a coin, the arrangements HHT, HTH, THH are three different permutations. perturbation: a disturbance in the uniform motion of a body under a stable gravitational force. Pleiades: a cluster of stars (the Seven Sisters) in the constellation Taurus. Glossary 267
Pliny the Elder (Gaius Plinius Secundus) (c. AD 23-79): Roman poly• math. His Natural History is comprised of 37 books, dealing, among others, with the heavenly bodies, geography and ethnography, zool• ogy, botany, medical botany, medicines derived from the bodies of man and other land animals, mineralogy, statuary and painting. precession: the uniform motion of the axis of rotation of a freely rotating body that is acted upon by a torque caused by external gravitational effects. present value: a sum of money which, including interest, will be equal to a given sum at some given future time - or will be equal to several sums at different specified times, as in the present value (i.e. cost) of an annuity. Price, Richard (1723-1791): English (Welsh-born) non-conformist min• ister, moral and political philosopher. Instrumental in founding an early English assurance society. Conveyor of the posthumously pu~ lished papers of Bayes (q.v.) to the Royal Society. primitive function: an analytic or geometric form from which another is derived; a function whose derivative is being considered. By analogy with the latter use I have used primitive function in connexion with finite differences. principle of indifference (principle of insufficient reason): if there is no known reason for predicating of something any particular one of a number of alternatives, then each of these alternatives should be asserted with the same probability. The term was introduced by J.M. Keynes in his 7reatise on Probability in 1921. probability measure: a measure P on an arbitrary space 0 for which P[O] = 1. An event E then becomes a measurable set and its proba• bility is PIE]. progression, arithmetic: a series (or sequence) of numbers arranged in increasing or decreasing order and such that the difference between successive pairs of terms is constant. e.g. a + (a + d) + (a + 2d) + ... progression, geometric: a series (or sequence) of numbers arranged in increasing or decreasing order and such that successive pairs of terms have the same ratio. e.g. a + ar + ar2 + ... Ptolemaic system: a geocentric model of the solar system, expounded by Ptolemy (c.lOo-170) in his work Almagest. quadrature: the position of the moon (or a planet), as seen from the Earth, when its angular distance from the sun is goo. 268 Glossary
quartile: the aspect of two heavenly bodies that are 90° distant from each other. quine: see Note 2 to the article "On illusions in the estimation of proba• bilities" .
Racine, Jean (1639-1699): French tragic dramatist: a playwright and po• etical artificer. Ramond de Carbonnieres, Louis, baron (1755---1827): French politi• cian and naturalist. Wrote on the natural sciences, with special at• tention being paid to meteorology and the geology of the Alps and the Pyrenees. random: non-deterministic, occurring purely by chance or independently of other events. random mechanism: a device for yielding things randomly. random process: a family of variates {Xt : t E T}. For example, T may be a time range and Xt the observation taken at time t. random variable: see variate. repeating circle: an instrument used in land surveying for the measure• ment of angles. The graduated limb of this apparatus consists of an entire circle. reversion: a sum which is to be paid on the death of a person, especially as a result of life-insurance; the return of an estate to the grantor (or his heirs) on the expiration of the grant, or the estate itself; the right of succeeding to an estate. See also expectancy. scale of relation: a number of terms included between two points in a series or progression, from which any term of a recurring series may be found when a sufficient number of preceding terms are given. secular equation : an equation in the changes in the orbits or periods of revolution of the planets. Seneca, Lucius Annaeus (c.4 BC--65 AD): rhetorician and philosopher: ordered by Nero - whom he rivaled in oratory and poetry, and whom he disparaged as poet and singer - to commit suicide. Writer on practical ethics, astronomy, meteorology, and of some tragedies. sensorium: the seat of sensation in the brain of man and the animals; the percipient centre to which sense impressions are transmitted by the nerves. Glossary 269 series, recurring: a series in which any term is formed by the addition of a certain number of preceding terms, multiplied or divided by any given numbers whether positive or negative. This may be expressed symbolically as
Yx = ax-1Yx-l + a x -2Yx-2 + ... + alYl . series, recurro-recurring: this differs from a recurring series in that the general term is a function of two indices. For example,
nYx = An.nYx-l + Bn ·n Yx-2 + ... + N n +Hn-1.n-1yx + Mn-1·n-1Yx-l + ... Simpson, Thomas (1710-1761): English mathematician (by trade a weaver). Writer of a number of textbooks on the calculus, geometry, algebra, probability and statistics - in fact he gained a reputation as the author of one of the two best treatises on the fiuxionary cal• culus. Noted for his formula for the finding of the area under a curve (Simpson's Rule), though this was in fact known before his work. sinking fund: a fund accumulated by periodic investments for some spe• cial purpose: the amount of the fund is the amount of the annuity formed by the payments. solstices: points on the ecliptic where the sun reaches its maximum or minimum declination, or the times at which this occurs (approxi• mately the 21st of June and the 21st of December). stake: that which is placed at hazard, to be taken by the winner of the game. Stirling, James (1692-1770): Scottish mathematician of Jacobite sym• pathies. Proceeded from the University of Glasgow to that of Oxford, leaving the latter for Venice (hence his nickname of "The Venetian"). Here he became friendly with Nicolas Bernoulli, then in Padua. He surveyed the Clyde, with the aim of making it navigable by a series of locks. His contributions to mathematics include work on the dif• ferential method, Stirling numbers, and the logarithmic equivalent of succession, rule of: suppose that all of a sample of size n from a popula• tion are found to have a certain property. Then the probability that the next individual to be sampled from this population will have that property is (n + l)/(n + 2). Using a uniform prior, we get this result from 270 Glossary
sufficient reason, principle of: the principle according to which noth• ing occurs for which someone who has sufficient knowledge could not give a reason that is sufficient to determine why it is as it is and not otherwise. SiiBmilch, Jean-Pierre (1708-1767): German theologian and economist. The first in Germany to try to relate ethics and political economy, he attempted to show the hand of Providence in human affairs. syzygy: when the sun, Earth and moon are approximately in a straight line.
Taylor, Brook (1685-1731): English mathematician. Developer of the calculus of finite differences, and discoverer of Taylor's Theorem. Turned, in his later years, to philosophy and religion. tontine: a tontine annuity is one that is bought by a group of people, the share of each member who dies being divided among the survivors. The last survivor gets the whole annuity for the rest of his life. triangulation: the surveying and mapping of a region by the tracing and measuring of a network of triangles, specifically, by measuring the angles and one side of each triangle. variate: a quantity having a numerical value for each member of a popu• lation, these values occurring according to a frequency distribution.
Wallis, John (1616-1703): English mathematician: made Professor of Mathematics at Oxford by Oliver Cromwell. Had some success in teaching deaf-mutes to speak, and found the product for 1r, 4 3x3x5x5x7x··· 1r-2x4x4x6x6x····
Wargetin, Pehr Wilhelm (1717-1783): worked in astronomy and de• mography. Constructed tables of the orbits of Jupiter's moons. One of the modern founders of population statistics. A prime mover in the Stockholm Academy of Sciences. weight: a numerical value associated with an observation and indicative of the latter's degree of importance among a set of such observations.