A History of Probability and
Statistics and Their
Applications before 1750

ANDERS HALD
Formerly Professor of Statistics
University of Copenhagen
Copenhagen, Denmark

@ZELENCE
A JOHN WILEY & SONS, INC., PUBLICATION

This Page Intentionally Left Blank

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

@ZELENCE
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.

Copyright 02003 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as
permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to
the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax
(978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should
be addressed to the Permissions Department, John Wiley & Sons, Inc., 1 1 1 River Street, Hoboken, NJ
07030, (201) 748-601 1, fax (201) 748-6008, e-mail: permreq@wiley.com.

Limit of LiabiIityiDisclaimer of Warranty: While the publisher and author have used their best efforts in
preparing this book, they make no representation or warranties with respect to the accuracy or
completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
representatives or written sales materials. The advice and strategies contained herein may not be
suitable for your situation. You should consult with a professional where appropriate. Neither the
publisher nor author shall be liable for any loss of profit or any other commercial damages, including
but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services please contact our Customer Care
Department within the U.S. at 877-762-2974, outside the U.S. at 3 17-572-3993 or fax 317-572-4002
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print,
however, may not be available in electronic format.

Library of Congress Cataloging-in-Publication is available.
ISBN 0-47 1-47129-I
Printed in the United States of America.
1 0 9 8 7 6 5 4 3 2 1

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.

ANDERSHALD
September I988

Contents

1 The Book and Its Relation to Other Works

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
1.1

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

28

3.1 Two Concepts of Probability, 28
3.2 Probability in Antiquity and the Middle Ages, 29
3.3 Probability from the Renaissance to the Mid-17th
Century, 30
4

Cardano and Liber de Ludo Aleae, c. 1565

4.1
4.2
4.3
4.4

33

On Games of Chance, 33
Early Attempts to Solve the Problem of Points, 35
Cardano and Liber de Ludo Aleae, 36
Galileo and the Distribution ofthe Sum of Points ofThree
Dice, c. 1620, 41
vii

viii

CONTENTS

5 The Foundation of Probability Theory by Pascal and Fermat
in 1654

42

5.1 Pascal and Fermat, 42
5.2 Pascal's Arithmetic Triangle and Some of Its Uses, 45
5.3 The Correspondence of Pascal and Fermat and Pascal's
Treatise on the Problem of Points, 54
5.4 Pascal's Wager, 63
6

Huygens and De Ratiociniis in Ludo Aleae, 1657

6.1
6.2
6.3
6.4
6.5

65

Huygens and the Genesis of His Treatise, 65
De Ratiociniis in Ludo Aleae, 68
Huygens' Five Problems and His Solutions, 74
Other Contributions by Huygens, 78
Problems, 78

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

8.1 The Correspondence ofthe Brothers Huygens, 1669, 106
8.2 Nicholas Bernoulli's Thesis, 1709, 110

106

CONTENTS

9 The Early History of Life Insurance Mathematics

ix
116

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

10 Mathematical Models and Statistical Methods in Astronomy
from Hipparchus to Kepler and Galileo

144

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
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

183

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

191

CONTENTS

X

14 New Solutions to Old Problems, 1708-1718

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, 213
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

220

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

257

16.1
16.2
16.3
16.4

Bernoulli's Formulation of the Problem, 257
Bernoulli's Theorem, 1713, 259
Nicholas Bernoulli's Theorem, 1713, 264
Some Comments by Markov, Uspensky, and
K. Pearson, 267
16.5 A Sharpening of Bernoulli's Theorem, 270

17 Tests of Significance Based on the Sex Ratio at Birth and the

Binomial Distribution, 1712-1713

17.1 Arbuthnott's Statistical Argument for Divine
Providence, 275
17.2 'sGravesande's Test of Significance, 279

275

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
18.1
18.2
18.3
18.4
18.5
18.6
18.7

286

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,
1710-1713, 330
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

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 De Usu Artis Conjectandi in Jure, 1709, 375
21.2 Solutions of Waldegrave's Problem by Nicholas
Bernoulli, Montmort, and de Moivre, 378
21.3 A Survey of Nicholas Bernoulli's Contributions, 392
21.4 A Note on Nicolaas Struyck, 394

22 De Moivre and the Doctrine of Chances, 1718, 1738, and 1756

397

22.1
22.2
22.3
22.4

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
22.5 The Occupancy Problem, 414
22.6 The Theory of Runs, 417
22.7 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

549

Index

57 1

This Page Intentionally Left Blank

A History of Probability and

Statistics and Their
Applications before 1750

This Page Intentionally Left Blank

CHAPTER 1

The Book and Its Relation to
Other Works

1.1 PRINCIPLES OF EXPOSITION
This book contains an exposition of the history of probability theory and
statistics and their applications before 1750 together with some background
material. A history should of course give an account of the time and place
of important events and their interpretations. However, opinions differ greatly
on where to put the main emphasis of interpretation.
We have attempted to cover three aspects of the history: problems,
methods, and persons. We describe probabilistic and statistical problems
and their social and scientific background; we discuss the mathematical
methods of solution and the statistical methods of analysis; and we include
the background and general scientific contributions of the persons involved,
not only their contributions to probability and statistics.
Since history consists of facts and their interpretation, history continually
changes because new facts are found in letters, archives, and books, and new
interpretations are offered in the light of deeper understanding based, in this
case, on the latest developments in probability theory, statistics, and the
history of science.
In the 17th and 18th centuries many problems were formulated as
challenge problems, and answers were given without proofs. Some books on
probability were written for the educated public and therefore contained
statements without proofs. In such cases we have tried to follow the author's
hints and construct a proof which we believe represents the author's
intentions.
The material has been ordered more according to problems and methods
1

2

THE BOOK A N D ITS RELATION TO OTHER WORKS

than according to persons in an attempt to treat the achievements of the
various authors as contributions to a general framework.
A leading principle of the exposition of probability theory and life
insurance mathematics has been to rewrite the classics in uniform modern
terminology and notation. It is clear that this principle may be criticized for
distorting the facts. Many authors prefer to recount the old proofs with the
original notation to convey the flavor of the past to the reader. There are
two essential steps in modernization that we have made here. The first is to
use a single letter, p, say, to denote a probability instead of the ratio of the
number of favorable cases to the total number of cases, a/(a + b), say, where
a and b are positive integers. This change of notation conceals the fact that
nearly all the probabilities discussed were constrained to rational fractions.
The advantage of this notation was noted by de Moivre (1738, p. 29) who
writes, “Before I make an end of this Introduction, it will not be improper
to shew how some operations may often be contracted by barely introducing
one single Letter, instead of two or three, to denote the Probability of the
happening of one Event” and, further (on p. 30), that “innumerable cases of
the same nature, belonging to any number of Events, may be solved without
any manner of trouble to the imagination, by the mere force of a proper
Notation.” However, de Moivre did not rewrite the Doctrine of Chances with
the new notation; he used it only in his Annuities upon Lives (1725 and later
editions). We have followed the advice of de Moivre and rewritten the proofs
in the new notation, feeling confident that the reader will keep in mind that
most probabilities were defined as proper rational fractions, a fact which is
nearly always obvious from the context.
The second great simplification of the proofs is obtained by the
introduction of subscripts. In analyzing some complicated games of chance,
for example, Waldegrave's problem, Nicholas Bernoulli and de Moivre had
to use the whole alphabet divided into several sections to denote probabilities
and expectations of the players corresponding to various states of the game.
De Moivre achieved some simplification by using superscripts in a few cases.
In many problems they gave the solution for two, three, and four players
only and concluded that “the continuation of this rule is manifest,” in this
way avoiding a general proof which would have been rather unintelligible.
Using modern notation with subscripts, it is easy to rewrite such proofs in
much shorter form without invalidating the idea of the proof; in fact, we
believe that our readers will get a clear idea of the proof because they are
accustomed to this symbolism, just as readers in the past understood the
original form of the proof because they were educated in that notational
tradition.
Comparison of proofs and results in a uniform notation makes evaluating
the contributions of various authors easier and minimizes the danger of

1.1 PRINCIPLES OF EXPOSITION

3

attributing too much to an individual author. Furthermore, the importance
of the results to the following period and to today becomes evident.
The same principle of exposition cannot be used for statistics, because
statistics before 1750 was nonmathematical. We shall therefore illustrate the
development of statistical methods by typical examples, giving both the
original data and their analysis at the time and adding some comments from
a modern point of view.
The book is written in textbook style, since our main purpose is to give
an account of the most important results in the classical literature. Like most
histories of mathematics and science, our exposition concentrates on results
which have proved to be of lasting importance.
The persons who laid the foundation of probability theory and statistics
were natural philosophers having a broader background and outlook than
scientists today. The word “scientist” was coined about the middle of the
19th century, reflecting an ongoing specialization and professionalization.
Nevertheless, we Shall often use the words “mathematician” and “scientist”
to stress certain characteristics of the persons involved.
To convey the flavor of classical works, we shall present quotations of
programs from the prefaces of books, the formulation of important problems,
and some heated disputes of priority.
We shall point out priorities, but the reader should be aware of the
uncertainty involved by taking note of Stigler's Law of Eponymy, (Stigler,
1980), which in its simplest form states that, “No scientific discovery is named
after its original inventor.”
The driving force behind the development of probability theory and
statistics was pressure from society to obtain solutions to important
problems for practical use, as well as competition among mathematicians.
When a problem is first formulated and its solution indicated, perhaps
only by a numerical example, the problem begins a life of its own
within the mathematical community; this leads to improved proofs and
generalizations of the problem, and we shall see many examples of this
phenomenon.
Finally, it should be noted that any history is necessarily subjective, since
the weight and interpretation of the events selected depend on the author's
interests.
For the serious student of the history of probability theory and statistics,
we can only recommend that he or she follow the advice given by de Moivre
(1738, p. 235), discussing the works of James and Nicholas Bernoulli on the
binomial distribution: “Now the Method which they have followed has been
briefly described in my Miscellanea Analytica, which the Reader may consult
if he pleases, unless they rather chuse, which perhaps would be the best, to
consult what they themselves have writ upon that Subject.”

4

THE BOOK A N D ITS RELATION TO OTHER WORKS

1.2 PLAN OF THE BOOK
A fuller title of the book would be A history ofprobability theory and statistics
and their applications to games of chance, astronomy, demography, and life
insurance before 1750, with some comments on later developments. The topics
treated may be grouped into five categories:

Background in mathematics, natural philosophy, and social conditions
Biographies
Probability theory and games of chance
Statistics in astronomy and demography
Life insurance mathematics
Probability theory before 1750 was inspired mainly by games of chance.
Dicing, card games, and lotteries, public and private, were important social
and economic activities then as today. It is no wonder that intellectual
curiosity and economic interests led to mathematical investigations of
these activities at a time when the mathematization of science was going
on. We shall distinguish three periods.
The period of the foundation of probability theory from 1654 to 1665
begins with the correspondence of Pascal and Fermat on the problem of
points, continues with Huygens' treatise on Reckoning at Games of Chance,
and ends with Pascal's treatise on the Arithmetical Triangle and its
applications. The correspondence was not published until much later. In
his treatise, Pascal solves the problem of points by recursion and finds a
division rule, depending on the tail probability of the symmetric binomial.
In their correspondence, he and Fermat had solved the same problem
also by combinatorial methods. Huygens uses recursion to solve the
problem numerically. He also considers an example with a possibly infinite
number of games, which he solves by means of two linear equations
between the conditional expectations of the two players. All three of them
solved the problem of the Gambler's Ruin without publishing their method
of solution.
After a period of stagnation of nearly 50 years, there followed a decade
with astounding activity and progress from 1708 to 1718 in which the
elementary and fragmentary results of Pascal, Fermat, and Huygens were
developed into a coherent theory of probability. The period begins with
Montmort's Essay d'Analyse sur les Jeux de Hazard, continues with de
Moivre's De Mensura Sortis, Nicholas Bernoulli's letters to Montmort,
James Bernoulli's Ars Conjectandi, Nicolaas Struyck's Reckoning of
Chances in Games, and ends with de Moivre's Doctrine of Chances. Hence,

1.2

PLAN OF THE BOOK

5

by 1718 four comprehensive textbooks were available. We shall mention
the most important results obtained. They discussed elementary rules of
probability calculus, conditional probabilities and expectations, combinatorics, algorithms and recursion formulae, the method of inclusion and
exclusion, and examples of using infinite series and limiting processes.
They derived the binomial and negative binomial distributions, the
hypergeometric distribution, the multivariate version of these distributions, the occupancy distribution, the distribution of the sum of any
number of uniformly distributed variables, the Poisson approximation to
the binomial, the law of large numbers for the binomial, and an approximation to the tail of the binomial. They solved the problem of points for a
game of bowls and for the game of tennis, Waldegrave's problem, the problem
of coincidences, and the problem of duration of play, and found the minimax
solution for the strategic game Her.
The third period, from 1718 to 1738, was a period of consolidation and
steady progress in which de Moivre derived the normal approximation
to the binomial distribution, developed a theory of recurring series,
improved his solution of the problem of the duration of play, and wrote
the second edition of the Doctrine of Chances, which became the most
important textbook before the publication of Laplace's ThPorie Analyrique
des Probabilitks in 1812.
We shall discuss these books in detail. We have, however, singled out the
most important problems for separate treatment to show how they were
solved by joint effort, often in competition among several authors.
Many problems were taken up by the following generation of
mathematicians and given solutions that have survived until today. We
shall comment on these later developments, usually ending with Laplace's
solutions.
The successful development of probability theory did not immediately
lead to a theory of statistics. A history of statistical methods before 1750
must therefore build on typical examples of data analysis; we have
concentrated here on examples from astronomy and demography.
Astronomers had been aware of the importance of both systematic and
random errors since antiquity and tried to minimize the influence of such
errors in their planning of observations and data analysis. We shall discuss
some data by Tycho Brahe from the end of the 16th century as an example.
The mathematization of science in the beginning of the 17th century
naturally led many scientists to determine not only the mathematical form
of natural laws but also the values of the parameters by fitting equations
to data. They inserted the best sets of observations in the equations, as
many as the number of parameters, solved for the parameters, calculated
the expected values, and studied the deviations between observed and

6

THE BOOK AND ITS RELATION To OTHER WORKS

calculated values. Prominent examples are Kepler's three laws on
planetary motion derived from his physical theories and data collected
by Copernicus and Tycho Brahe. Kepler's data were used by Newton to
check his axiomatic theory. Galileo used several sets of observations on
the new star of 1572 to compare two hypotheses on the position of the
star. We shall also see how Newton used an interpolation polynomial to
find the tangent to the orbit of a comet.
A paragon for descriptive statistical analysis of demographic data was
provided by Graunt's Natural and Political Observations made upon the
Bills qf Mortality in 1662. Graunt's critical appraisal of the rather
unreliable data, his study of mortality by cause of death, his estimation
of the same quantity by several different methods, his demonstration of
the stability of statistical ratios, and his life table set up new standards
for statistical reasoning. Graunt's work led to three different types of
investigations: political arithmetic; testing the stability of statist...