## Theory Of Probability.A Historical Essay

Figures from the History of Probability and Statistics

** John Aldrich, University of Southampton, Southampton, UK.**(

__)__

__home__** 2005. Latest changes October 2012**

### Notes on the work of

A further **200+** individuals are mentioned below. Use **Search** on your browser to find the person you are interested in. It is also worth searching for the ‘principals’ for they can pop up anywhere.

The entries are arranged chronologically, so the document can be read as a story.These are the date markers

with people placed according to date of their first impact. Do not take the placings too seriously and remember that a career may last more than 50 years! At each marker there are notes on developments in the following period. There is more about and about economics than there should be but I know more about them.

For *further** on-line information* there are *links* to

· *Earliest Uses* (__ Words__ and

__) for details (particularly detailed references) on the topics to which the individuals contributed. (The__

__Symbols__*Words*site is organised by letter of the alphabet. See

__for a list of entries)__

__here__· __ MacTutor__ for fuller biographical information on the ‘principals’ (all but three)

*and*on a

*very*large ‘supporting’ cast. The MacTutor biographies also cover the work the individuals did

*outside*probability and statistics. The

**MacTutor**and

**References**links are to these pages. There is an

__to the Statistics articles on the site.__

__index__· ASA __ Statisticians in History__ for biographies of mainly recent, mainly US statisticians.

· __ Life and Work of Statisticians__ (part of the

__site) for further links, particularly to original sources.__

__Materials for the History of Statistics__· Oscar Sheynin’s *Theory of Probability: A Historical Essay*An account of developments to the beginning of the twentieth century, particularly useful for its coverage of Continental work on statistics.

· __ Isaac Todhunter’s__ classic from 1865

*for detailed commentaries on the contributions from 1650-1800. The coverage is extraordinary and the entries are still interesting—even their humourlessness has a certain charm.*

__A History of the Mathematical Theory of Probability : from the Time of Pascal to that of Laplace__· The *Mathematics Genealogy Project,***abbreviated***MGP*, which is useful for tracking modern scholars. The PhD degree is a relatively recent development and in the a

*very*recent one. See my

__.____The____Mathematics PhD in the UK__· __ Wikipedia__ for additional biographies. This is an uneven site but it has some useful articles.

The entries contain references to the following histories and books of lives. See below for more literature.

· Ian Hacking *The Emergence of Probability*, , Press 1975. (__ contents__)

· Stephen M Stigler *The History of Statistics: The Measurement of Uncertainty before 1900*, , : Press 1986. (__ contents + bibliography__)

· Anders Hald *A History of Probability and Statistics and their applications before 1750*, : Wiley 1990. (__ contents__)

· Anders Hald *A History of Mathematical Statistics from 1750 to 1930*, : Wiley 1998. (__ contents + bibliography__)

· Jan von Plato *Creating Modern Probability,*: Cambridge University Press, 1994. (__ contents__)

· *Leading Personalities in Statistical Sciences from the Seventeenth Century to the Present*, (ed. N. L. Johnson and S. Kotz) 1997. : Wiley. Contains around 110 biographies and based on entries in *Encyclopedia of Statistical Science *(ed. N. L. Johnson and S. Kotz.) **Abbreviated**** LP**.

· *Statisticians of the Centuries* (ed. C. C. Heyde and E. Seneta) 2001. : Springer. Contains 105 biographies. The coverage is restricted to individuals born before 1900. **Abbreviated**** SC**.

· *Encyclopedia of Social Measurement *(ed. K. Kempf-Leonard) 2004. : Elsevier (__ description__). Contains numerous biographies.

**Abbreviated**

**.**

*ESM*On the web (see also online biblios and texts below)

· __ Stochastikon Encyclopedia.__ Articles in English and German.

· __ Portraits of Statisticians__ on the

__site.__

__Materials for the History of Statistics__· __ Sources in the History of Probability and Statistics__ by Richard J. Pulskamp.

· *Tales of Statisticians.* Vignettes by E. Bruce Brooks.

· __ History of Statistics and Probability__ 18 short biographies from .

· __Glimpses of the Prehistory of Econometrics__*. *Montage by Jan Kiviet.

· __ Probability and Statistics Ideas in the Classroom: Lesson from History.__ Comments on the uses of history by D. Bellhouse.

· __ The History of Statistics in the Classroom.__Thumbnail sketches of Gauss, Laplace and Fisher by H. A. David.

· __ Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization__. Encyclopaedic coverage by M. Friendly & D.J. Denis.

· __ Actuarial History__. A very comprehensive collection of links, created by Henk Wolthuis,not only to actuarial science and demography, but to statistics as well.

To help place individuals I have used modern terms for occupation (e.g. physicist or statistician). For the earlier figures these terms are anachronistic but, I hope, not too misleading. I have **not** given nationality as people move and states come and go. MacTutor has plenty of geographical information.

Top

1650-1700The origins of __probability__ and __statistics__ are usually found in this period, in the mathematical treatment of games of chance and in the systematic study of mortality data. This was the age of the *Scientific Revolution* and the biggest names, __ Galileo__ (

__and__

__Materials____(4-6).) and__

__Todhunter ch.I____(__

__Newton__**) gave some thought to probability without apparently influencing its development. For an introduction to the**

*LP**Scientific Revolution*, see Westfall’s

__(1986).__

__Scientific Revolution__· There *were* earlier contributions to probability, e.g. __ Cardano__ (1501-76) gave some ‘probabilities’ associated with dice throwing, but a critical mass of researchers (and results) was only achieved following discussions between Pascal and

__and the publication of the first book by Huygens.__

__Fermat__**Hacking**

__discusses thinking__

__Chapters 1-5__*before*Pascal. James Franklin’s

*The Science of Conjecture: Evidence and Probability Before Pascal*(2001) examines this earlier work in depth. A recent issue of the

*is devoted to*

__JEHPS__*.*

__Medieval____probabilities__· Statisticsin the form of population statistics was created by Graunt. Graunt’s friend __ William Petty__ gave the name

*Political Arithmetic*to the quantitative study of demography and economics.

__was an important figure in the next generation. However the economic line fizzled out. Adam Smith, the most influential C18 British economist, wrote,“I have no great faith in political arithmetic...”__

__Gregory King__*Wealth of Nations*(1776)

__.__

__B.IV, Ch.5, Of Bounties__· A form of life insurance mathematics was developed from Graunt’s work on the __ life table__ by the mathematicians

__,__

__Halley____and__

__Hudde____. Many later ‘probabilists’ wrote on actuarial matters, including de Moivre,__

__de Witt____,__

__Simpson____,__

__Price____,__

__De Morgan__,__Gram____,__

__Thiele____, Cramér and de Finetti. In the C20 the__

__Cantelli__*Skandinavisk aktuarietidskrift*and the

*Giornale dell'Istituto Italiano degli Attuari*were important journals for theoretical statistics and probability. Actuarial questions and friendship with the actuary

__stimulated the Edinburgh mathematicians,__

__G. J. Lidstone____and__

__E. T. Whittaker____(__

__A. C. Aitken__*), to contribute to statistics and*

__MGPP____. The C17 work is discussed by__

__numerical analysis__**Hacking**(1975):

__,__

__Chapter 13__*Annuities*. See also Chris Lewin’s

__and the other historical links on the__

__The Creation of Actuarial Science____page. There are historical articles in the__

__International Actuarial Links__*. Classics are reprinted in*

__Encyclopedia of Actuarial Science__*. There is a nice review of the early literature in the*

__History of Actuarial Science____of the Equitable Life Archive.__

__catalogue__· New institutions, rather than the traditional universities, underpinned these developments. In and private discussion groups, like that of __ Mersenne,__ were forerunners of the

__and the__

__Académie des Sciences____(__

__Royal Society of London____). The latter’s__

__archives__*Philosophical Transactions*(

__) published many important contributions to probability and statistics, including papers by__

__Gallica____, de Moivre, Bayes, Pearson, Fisher, Jeffreys and Neyman. The__

__Halley____and__

__Berlin____academies were formed a bit later. The Royal Society was a forerunner of the modern scientific society, while the continental academies were more like research institutes.__

__St. Petersburg____ Life & Work__ has links to the writings of many of these people. For the period generally see

__(pp. 1-55) and__

__Todhunter ch. I-VI__**Hald**(1990, ch. 1-12).

Top

Blaise Pascal are usually found in the probabilitybetween Pascal and Fermat where they treated several problems associated with games of chance. The letters were not published but their contents were known in Parisian scientific circles. Pascal’s only probability publication was the posthumously published correspondence (1654, published in 1665 and so after Huygens’s work); this treated Traité du triangle arithmétique with probability applications. Pascal introduced the concept of Pascal’s triangle and discussed the problem of expectation. gambler’s ruin, is now often read as a pioneering analysis of decision-making under uncertainty although it appeared, not in his mathematical writings, but in the Pascal’s wagerPensées, his reflections on religion. The last chapter of the Port-Royal Logic by Pascal’s friends pp. 365ff and Nicole has a brief treatment of the use of probability in decision making, with an allusion to the wager. See Ben Rogers Arnauld,Pascal's Life & TimesLife & Work, and A.W.F. Edwards on the triangle (pp. 7-21). See also Todhunter ch.IIHald 1990, chapter 5, The Foundations of Probability Theory by Pascal and Fermat in 1654 and Hacking 1975, chapter 7, The (1654) andchapter 8, The Great Decision (1658?).Top |

Christiaan Huygens Académie des Sciences. Huygens wrote the first book on probability, a pamphlet really, Van Rekeningh in Spelen van Geluck, translated into Latin by his teacher van Schooten as De Ratiociniis in Ludo Aleae (1657) and then into English as . Huygens drew on the ideas of Pascal and Fermat, which he had encountered when he visited . Much of the book is devoted to calculating the value or, as it would be called now the The Value of all Chances in Games of Fortune etcexpectation, of a game of chance. The problems contained in the book include the gambler’s ruin and Huygens treated the hypergeometric distribution. His book was widely read and the first part ofJakob Bernoulli’s Ars Conjectandi is a commentary on it. See Life & Work and Todhunter ch.III (pp. 22-5). See also Hald (1990): Chapter 6, Huygens and De Ratiociniis in Ludo Aleae, 1657 and Hacking (1975), Chapter 11, Expectation. C.J. (Kees) Verduin is constructing an impressive site. See Peter Doyle Christiaan Huygens.Hedging HuygensTop |

No authentic portrait of Graunt is known | John Graunt Graunt is unique among the figures described here in not having had a university education. He published only one work, SC, LP, ESM. (1662). However, through this work and his friendship with Observations Made upon the Bills of Mortality he became a fellow of the Royal Society of London and his work became known to savants like William Petty,. The weekly bills of mortality, which had been collected since 1603, were designed to detect the outbreak of plague. Graunt put the data into tables and produced a commentary on them; he does basic calculations. He discusses the reliability of the data. He compares the number of male and female births and deaths. In the course of Chapter XI on estimating the population of London Graunt produces a primitive Halley see the life table articles by Glass, Renn, Benjamin and Seal. The life table became one of the main tools of demography and insurance mathematics. Halley produced a life table using data from JIA (Caspar Neumann) in . See SC for writings by Graunt, Petty and Halley. See alsoLife & Work (pp. 37-43),Todhunter ch. VHald (1990):Chapter 7, John Graunt and the Observations upon the Bills of Mortality, 1662 and Hacking (1975): Chapter 12, Political Arithmetic 1662. Top |

1700-50The **great leap forward** is Hald’s (1990) name for the decade 1708-1718: there were so many important contributions to such a greatly expanded subject. The roots of probability and statistics were quite distinct but by the early C18 it was understood that the subjects were closely related.

· Jakob Bernoulli’s *Ars Conjectandi*, like __Arnauld’s__*Logique* (1682) __ pp. 365ff__, suggested a conception of probability broader than that associated with games of chance.Bernoulli’s

__provided a theory to link between probability and data. See__

__law of large numbers__**Hacking**(1975):

__Chapters 16-7__.· __ Montmort’s__(

**)**

*SC, LP**(1708) and de Moivre’s*

__Essay d'analyse sur les jeux de hazard__*Doctrine of Chances*(1718) produced many new results on games of chance, greatly extending the work of Pascal and Huygens.

· __ Arbuthnot__’s (

**) 1710 paper**

*SC, LP*__used a__

__An Argument for Divine Providence, taken from the constant Regularity observed in the Births of both Sexes____test (__

__significance____) to establish that the probability of a male birth is not ½. The calculations were refined by__

__sign test____(__

__'sGravesande__**) and**

*LP*__(__

__Nicholas Bernoulli__**).Apart from being an early application of probability to social statistics, Arbuthnot’s paper illustrates the close connection between theology and probability in the literature of the time. The work of**

*LP*__provides another example.__

__John Craig__· Consideration of the valuation of a risky prospect, dramatised by the __ St. Petersburg paradox__(formulatedby

__in1713 and discussed by__

__Nicholas Bernoulli____)led to Daniel Bernoulli’s (1737) theory of__

__Gabriel Cramer____(or expected__

__moral expectation____).__

__utility__See __ Life & Work__ for writings by Montmort, Euler, Lagrange, etc. For the period see

__(pp. 56- 212)__

__Todhunter ch. VII-X__**Hald**(1990, ch. 1-12),

**Hacking**

__.__

__Chapter 16-9__Top

Jakob (James) Bernoulli ) and several wrote on probability. The most important contributors were Jakob, Daniel and family tree. Jakob and his younger brother Niklaus were the first of the mathematicians. Jakob studied philosophy at the but learnt mathematics on his own. Eventually he became professor of mathematics at . The posthumously publishedJohannArs Conjectandi() (1713) was his only probability publication but it was extremely influential.The first part is a commentary on Huygens’stitle pageDe Ratiociniis. The work was an important contribution to : the term combinatoricsoriginated here. (The combinatorial side of probability remained important and in late C19 Britain probability and combinatorial analysis were taught together in algebra courses.) Bernoulliused the terms permutationa priori and a posteriori to distinguish two ways of deriving probabilities (see : deduction posterior probability)a priori (without experience) is possible when there are specially constructed devices, like dice but otherwise it is possible to make a deduction from many observed outcomes of similar events. theorem, or theBernoulli’swas the work’s most spectacular contribution but see also the entries forlaw of large numbersandmorally certainThe eponymous binomial distributionBernoullitrials, numbers and random variable all refer to this Bernoulli and his Ars Conjectandi. See Sheynin ch. 3 (which also has links for the contributions of Niklaus and Jean III) and Life & Work (pp. 56-77). See Todhunter ch.VIIStigler (1986): Chapter 2, Probabilists and the Measurement of Uncertainty. See also Hald (1990): Chapter 15, James Bernoulli and Ars Conjectandi, 1713;Chapter 16, Bernoulli’s Theorem. Sheynin has recently translated of the Part IVArs Conjectandi. A new translation of the whole work has appeared: The Art of Conjecturing translated by Edith Dudley Scylla . The proceedings of a conference on the AmazonArs Conjectandi are online at the : JEHPSandpart 1part 2.Top |

Abraham de Moivre The Doctrine of Chances: or, a Method of Calculating the Probability of Events in Play (). He published other pieces on probability, putting the results into new editions of the title pageDoctrine; the third appeared in 1733. The book began with an influential definition of. De Moivre obtained the probabilityapproximation to the binomial distribution (a forerunner of the normal) and almost found thecentral limit theorem. His technical innovations included the use ofPoisson distribution, which he used to find the distribution of the sum ofprobability generating functionsvariables. De Moivre also wrote about life insurance mathematics when analysing annuities Seeuniform. Todhunter ranked de Moivre’s contributions very highly, “it will not be doubted that the Theory of Probability owes more to him than to any other mathematician, with the sole exception of survival function |

## Title: Theory of Probability. A Historical Essay

Authors:Oscar Sheynin

(Submitted on 27 Feb 2018)

Abstract: This book covers the history of probability up to Kolmogorov with essential additional coverage of statistics up to Fisher. Based on my work of ca. 50 years, it is the only suchlike book. Gorrochurn (2016) is similar but his study of events preceding Laplace is absolutely unsatisfactory. Hald (1990; 1998) are worthy indeed but the Continental direction of statistics (Russian and German statisticians) is omitted, it is impossible to find out what was contained in any particular memoir of Laplace and the explanation does not always explain the path from, say, Poisson to a modern interpretation of his results. Finally, the reader ought to master modern math. statistics. I included many barely known facts and conclusions, e. g., Gauss' justification of least squares (yes!), the merits of Bayes (again, yes!), the unforgivable mistake of Laplace, the work of Chebyshev and his students (merits and failures) etc., etc. The book covers an extremely wide field, and is targeted at the same readers as any other book on history of science. Mathematical treatment is not as difficult as it is for readers of Hald.

## Submission history

From: Oscar Sheynin [view email]**[v1]**Tue, 27 Feb 2018 15:29:25 GMT (1638kb)

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