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

A NOTE TO THE READER This book has been electronically reproduced t7om digital information stored at John Wiley & Sons, Inc. We are pleased that the use of this new technology will enable us to keep works of enduring scholarly value in print as long as there is a reasonable demand for them. The content of this book is identical to previous printings.

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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