A History of Probability and Statistics and Their Applications before 1750
ANDERS HALD Formerly Professor of Statistics University of Copenhagen Copenhagen, Denmark
@ Z E L E N C E A JOHN WILEY & SONS, INC., PUBLICATION
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A History of Probability and Statistics and Their Applications before 1750
This Page Intentionally Left Blank
A History of Probability and Statistics and Their Applications before 1750
ANDERS HALD Formerly Professor of Statistics University of Copenhagen Copenhagen, Denmark
@ Z E L E N C E A JOHN WILEY & SONS, INC., PUBLICATION
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Preface
Until recently a book on the history of statistics in the 19th century was badly needed. When I retired six years ago, I decided to write such a book, feeling that I had a good background in my statistical education in the 1930s, when the curriculum in statistics was influenced mainly by the writings of Laplace, Gauss, and Karl Pearson. Studying the original works of these authors I found no difficulty in understanding Gauss and Pearson, but I soon encountered difficulties with Laplace. The reason is of course that Gauss and Pearson are truly 19th century figures, whereas Laplace has his roots in the 18th century.
I then turned to the classical authors and worked my way back to Cardano through de Moivre, Montmort, Nicholas and James Bernoulli, Huygens, Fermat, and Pascal. Comparing my notes with Todhunter’s History, I found to my surprise that his exposition of the topics in probability theory that I found most important was incomplete, and I therefore decided to write my own account.
The present book, covering the period before 1750, is an introduction to the one I had in mind. It describes the contemporaneous development and interaction of three topics: probability theory and games of chance; statistics in astronomy and demography; and life insurance mathematics.
Besides the story of the life and works of the great natural philosophers who contributed to the development of probability theory and statistics, I have told the story of important problems and methods, in this way exhibiting the gradual advance of solving these problems. I hope to have achieved a better balance than had been achieved before in evaluating the contributions of the various authors; in particular, I have stressed the importance of the works of John Graunt, Montmort, and Nicholas Bernoulli.
The contents of the book depend heavily on research carried out by many authors during the past 40 years. I have drawn freely on these sources and
V
vi PREFACE
acknowledged my debt in the references. The manuscript was written during the years 1985-1987, so works published in 1986 and 1987 are not fully integrated in the text. Some important books and papers from 1988 are briefly mentioned.
With hesitation, I have also included some background material on the history of mathematics and the natural and social sciences because I have always felt that my students needed such knowledge. I realize of course that my qualifications for doing so are rather poor since I am no historian of science. These sections and also the biographies are based on secondary sources.
The plan of the book is described in Section 1.2. I am grateful to Richard Gill for advice on my English in Chapters 2 and
3, to Steffen L. Lauritzen for translating some Russian papers, and to Olaf Schmidt for a discussion of Chapter 10. In particular, I want to thank SBren Johansen for discussions on the problem of the duration of play.
I am grateful to two anonymous reviewers from the publisher for valuable comments on the manuscript and for advice resulting in considerable reduction of the background material. I thank the copy editor for improving my English and transforming it into American.
I thank the Institute of Mathematical Statistics, University of Copenhagen, for placing working facilities at my disposal.
I thank the Almqvist & Wiksell Periodical Company for permission to use material in my paper published in Scandinavian Actuarial Journal, 1987; the International Statistical Institute for permission to use material from three papers of mine published in International Statistical Review, 1983, 1984, and 1986; and Springer-Verlag for permission to use material from my paper published in Archive for History of Exact Sciences, 1988.
I am grateful to the Department of Statistics, Harvard University, for permission to quote from Bing Sung’s Translations from James Bernoulli, Technical Report No. 2, 1966, and to Thomas Drucker for permission to quote from his (unpublished) translation of Nicholas Bernoulli’s De Usu Artis Conjectandi in Jure.
My first book on statistics, written fifty years ago, was dedicated to G. K., so is this one.
ANDERS HALD
September I988
Contents
1 The Book and Its Relation to Other Works
1.1 Principles of Exposition, 1 1.2 Plan of the Book, 4 1.3 A Comparison with Todhunter’s Book, 8 1.4 Works of Reference, 1 I
1
2 A Sketch of the Background in Mathematics and Natural Philosophy 13
2.1 Introduction, 13 2.2 On Mathematics before 1650, 14 2.3 On Natural Philosophy before 1650, 19
3 Early Concepts of Probability and Chance
3.1 3.2 3.3
Two Concepts of Probability, 28 Probability in Antiquity and the Middle Ages, 29 Probability from the Renaissance to the Mid-17th Century, 30
4 Cardano and Liber de Ludo Aleae, c. 1565
4.1 On Games of Chance, 33 4.2 Early Attempts to Solve the Problem of Points, 35 4.3 Cardano and Liber de Ludo Aleae, 36 4.4 Galileo and the Distribution ofthe Sum of Points ofThree
Dice, c. 1620, 41
28
33
v i i
viii CONTENTS
5 The Foundation of Probability Theory by Pascal and Fermat in 1654 42
5.1 Pascal and Fermat, 42 5.2 5.3
5.4 Pascal’s Wager, 63
Pascal’s Arithmetic Triangle and Some of Its Uses, 45 The Correspondence of Pascal and Fermat and Pascal’s Treatise on the Problem of Points, 54
6 Huygens and De Ratiociniis in Ludo Aleae, 1657
6.1 Huygens and the Genesis of His Treatise, 65 6.2 De Ratiociniis in Ludo Aleae, 68 6.3 Huygens’ Five Problems and His Solutions, 74 6.4 Other Contributions by Huygens, 78 6.5 Problems, 78
65
7 John Graunt and the Observations Made upon the Bills of Mortality, 1662 81
7.1 On the Origin of the Word “Statistics”, 81 7.2 Graunt’s Discussion of the Plague Mortality, 82 7.3 John Graunt and His Obseruations Made upon the Bilk
of Mortality, 85 7.4 Graunt’s Appraisal of the Data, 89 7.5 Proportional Mortality by Cause of Death, 91 7.6 The Stability of Statistical Ratios, 92 7.7 A Test of the Hypothesis “That the More Sickly the
Year Is, the Less Fertile of Births”, 95 7.8 On the Number of Inhabitants, 96 7.9 Graunt’s Life Table, 100 7.10 Concluding Remarks about Graunt’s Obseruations, 103 7.1 1 William Petty and Political Arithmetic, 104
8 The Probabilistic Interpretation of Graunt’s Life Table 106
The Correspondence ofthe Brothers Huygens, 1669, 106 8.1 8.2 Nicholas Bernoulli’s Thesis, 1709, 110
CONTENTS
9 The Early History of Life Insurance Mathematics
9.1 The Background, 116 9.2 Jan de Witt and His Report on the Value of Life
Annuities, 1671, 122 9.3 Halley and His Life Table with Its Seven Uses, 1694, 131 9.4 Problems, 141
ix
116
10 Mathematical Models and Statistical Methods in Astronomy from Hipparchus to Kepler and Galileo
10.1 Observational Errors and Methods of Estimation in Antiquity and the Middle Ages, 144
10.2 Planning of Observations and Data Analysis by Tycho Brahe, 146
10.3 Galileo’s Statistical Analysis of Astronomical Data, 1632, 149
10.4 Mathematical Models in Astronomy from Ptolemy to Kepler, 160
10.5 Problems, 168
144
11 The Newtonian Revolution in Mathematics and Science 170
11.1 Introduction, 170 11.2 The Newtonian Revolution, 172 11.3 Newton’s Interpolation Formula, 176
12 Miscellaneous Contributions between 1657 and 1708
12.1 Publication of Works from before 1657, 183 12.2 New Contributions Published between 1657 and
1708, 184 12.3 Contributions during the Period Published after
1708, 189 12.4 A Note on Data Analysis, 190
13 The Great Leap Forward, 1708-1718: A Survey
13.1 A List of Publications, 191 13.2 Methods and Results, 192
183
191
X
14 New Solutions to Old Problems, 1708-1718
CONTENTS
196
14.1 The Problem of Points, 196 14.2 Solutions of Huygens’ Five Problems, 198 14.3 To Find the Number of Chances of Throwing s Points
with n Dice, Each Having f Faces, 204 14.4 To Find the Number of Trials Giving an Even Chance
of Getting at Least c Successes. The Poisson Approximation, 21 3
14.5 Problems, 218
15 James Bernoulli and A m Conjecfandi, 1713
15.1 15.2 15.3 15.4
15.5 15.6 15.7
15.8
James, John, and Nicholas Bernoulli, 220 Ars Conjectandi, 223 Bernoulli’s Commentary on Huygens’ Treatise, 226 Bernoulli’s Combinatorial Analysis and His Formula for the Sums of Powers of Integers, 228 Bernoulli on Games of Chance, 235 Bernoulli’s Letter on the Game of Tennis, 241 Bernoulli’s Concept of Probability and His Program for Applied Probability, 245 Problems from Ars Conjectandi and Bernoulli’s Letter on Tennis, 254
16 Bernoulli’s Theorem
16.1 Bernoulli’s Formulation of the Problem, 257 16.2 Bernoulli’s Theorem, 1713, 259 16.3 Nicholas Bernoulli’s Theorem, 171 3, 264 16.4 Some Comments by Markov, Uspensky, and
K. Pearson, 267 16.5 A Sharpening of Bernoulli’s Theorem, 270
220
257
17 Tests of Significance Based on the Sex Ratio at Birth and the Binomial Distribution, 1712-1713 275
17.1
17.2
Arbuthnott’s Statistical Argument for Divine Providence, 275 ’sGravesande’s Test of Significance, 279
xi CONTENTS
17.3 Nicholas Bernoulli’s Comparison of the Observed Distribution with the Binomial, 280
17.4 A Note on Theology and Political Arithmetic, 285
18 Montmort and the Essay d’Analyse sur les Jeux de Hazard, 1708 and 1713 286
18.1 18.2
18.3 18.4
18.5
18.6
18.7
Montmort and the Background for His Essay, 286 Montmort’s Combinatorial Analysis and the Occupancy Distribution, 292 Montmort on Games of Chance, 297 The Correspondence of Montmort with John and Nicholas Bernoulli, 3 10 Montmort and Nicholas Bernoulli on the Game of Tennis, 312 The Discussion of the Strategic Game Her and the Minimax Solution, 314 Problems from Montmort’s Essay, 322
19 The Problem of Coincidences and the Compound Probability Theorem 326
19.1 Introduction, 326 19.2 Montmort’s Formula for the Probability of at Least One
Coincidence, 1708, 328 19.3 The Results of Montmort and Nicholas Bernoulli,
19.4 De Moivre’s Derivation of the Probability of Compound Events, 1718, 336
19.5 De Moivre’s Solution of the Problem of Coincidences, 338 19.6 Some Notes on Later Developments, 340 19.7 Problems, 345
1710-1713, 330
20 The Problem of the Duration of Play, 1708-1718
20.1 Formulation of the Problem, 347 20.2 Montmort’s Discussion of the Duration of Play in
1708, 349 20.3 Nicholas Bernoulli’s Formula for the Ruin
Probability, 1713, 350
347
xii CONTENTS
20.4 De Moivre’s Results in De Mensura Sortis, 1712, 356 20.5 De Moivre’s Results in the Doctrineof Chances, 1718, 360 20.6 Problems, 373
21 Nicholas Bernoulli 375
21.1 21.2
21.3 21.4
22 De Moivre and the Doctrine of Chances, 1718, 1738, and 1756
De Usu Artis Conjectandi in Jure, 1709, 375 Solutions of Waldegrave’s Problem by Nicholas Bernoulli, Montmort, and de Moivre, 378 A Survey of Nicholas Bernoulli’s Contributions, 392 A Note on Nicolaas Struyck, 394
397
22.1 22.2 22.3 22.4
22.5 22.6 22.7
The Life of de Moivre, 397 De Merisura Sortis, 1712, 401 The Prefaces of the Doctrine of Chances, 404 A Survey of the Probability Problems Treated in the Doctrine of Chances, 408 The Occupancy Problem, 414 The Theory of Runs, 417 Problems from de Moivre’s De Mensura Sortis and the Doctrine of Chances, 422
23 The Problem of the Duration of Play and the Method of Difference Equations 425
23.1 De Moivre’s Theory of Recurring Series, 425 23.2 De Moivre’s Trigonometric Formula for the Continuation
Probability, 433 23.3 Methods of Solution of Difference Equations by Lagrange
and Laplace, 1759- 1782, 437 23.4 Solutions of the Problem of the Duration of Play by
Laplace and Lagrange, 452 23.5 Problems, 464
24 De Moivre’s Normal Approximation to the Binomial Distribution, 1733
24.1 Introduction, 468 24.2 The Mean Deviation of the Binomial Distribution. 470
468
CONTENTS xiii
24.3 De Moivre’s Approximations to the Symmetric Binomial in Miscellanea Analytica, 1730, 472
24.4 Stirling’s Formula and de Moivre’s Series for the Terms of the Symmetric Binomial, 1730, 480
24.5 De Moivre’s Normal Approximation to the Binomial Distribution, 1733, 485
24.6 Laplace’s Extension of de Moivre’s Theorem, 18 I 2, 495 24.7 The Edgeworth Expansion, 1905, 497 24.8 Daniel Bernoulli’s Derivation of the Normal Density
Function, I 770- I77 1, 500
25 The insurance Mathematics of de Moivre and Simpson, 1725- 1 756 508
25. I Introduction, 508 25.2 The Life of Thomas Simpson, 514 25.3 De Moivre’s Linear and Piecewise Linear Approximation
to Halley’s Life Table, 5 15 25.4 Simpson’s Life Table for the Population of London, 518 25.5 Single-life Annuities, 519 25.6 Joint-life Annuities, 528 25.7 Reversionary Annuities, 534 25.8 Life Assurances, Reversions, and Successive Lives, 535 25.9 Survivorship Probabilities and Expectations of Life, 539 25.10 Survivorship Insurances, 543 25.1 1 The Scottish Ministers’ Widows’ Fund of 1744, 547 25.12 Problems, 547
References
Index
549
57 1
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