Pascal's Arithmetical Triangle: The Story of a Mathematical Idea
224
Pascal's Arithmetical Triangle: The Story of a Mathematical Idea
224eBook
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| ISBN-13: | 9780486840765 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 06/12/2019 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 224 |
| File size: | 16 MB |
| Note: | This product may take a few minutes to download. |
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PREFACE
The origins of this book lie in my undergraduate training in genetics in 1956, and since they involve one of the greatest scientists of the twentieth century, R. A. Fisher, about whom any anecdote is worth recording, I shall risk a charge of self-indulgence and describe several.
In 1956 Fisher, aged 66, was in his last year as the Professor of Genetics at Cambridge, and I was the sole student accepted to read genetics in this, my last (third) undergraduate year. I knew very little statistics, so at the end of one of Fisher's early lectures I asked him how I should learn some. With a twinkle i n his eye he remarked that he had written one or two books on statistics, and perhaps I might start by reading them. I bought Statistical Methods for Research Workers (12th ed.) and The Design of Experiments (6th ed.) and began with the former. I encountered the binomial distribution in Chapter III, "Distributions," and was fascinated by Fisher 's description of fitting it to some extensive data on the distribution of the sexes in families of eight children. He points out the excess variance in the data, and, after rejecting as a possible explanation the inclusion of multiple births, especially twins, he leaves the reader to think of his own explanation.
It occurred to me that a binomial distribution in which the sole parameter (in this case the probability of a male birth) varied from sample to sample would display this excess variance. By a stroke of good fortune, I had just learned from some lectures I had recently attended that a distribution called the "β" was appropriate for describing such variation. I applied it to the parameter of the binomial distribution, performed the necessary integration, and was delighted to find that it "came out." Of course, I later learned that the resulting "β-binomial distribution" was not new, but the thrill of discovery cast a warm glow over everything to do with binomial coefficients. There was a calculator (electro-mechanical) in the departmental library and I used it to fit the new distribution to the data; when Fisher came in and saw what I was doing he reprimanded me for not concentrating on my forthcoming examinations, but h is tone belied his words, for he was glad to see a student taking an initiative. Later, as a graduate student, I gave my first seminar-on the "Binomial Distribution." Fisher sat quietly through it, and I do not recall any comment s.
As the years went by I collected many results involving binomial coefficients, and during my Christmas vacation of 1978 I settled down to write a little book about them; the first half to treat their purely mathematical properties, and the second some of their applications. When nearly complete I decided to add a brief account of their history, only to find that the books I had consulted in the hi story of mathematics were often in error. Early on in my attempt to rectify this I called for one of the two copies of Pascal's Traité du triangle arithmetique from Cambridge University Library and discovered to my astonishment that Pascal had, already in 1654, written the book I had just finished in exactly the format I had chosen-pure mathematics first, applications second.
It is not often that an author finds himself forestalled by 324 years. My reaction was to write Pascal's Arithmetical Triangle to help remedy the ignorance surrounding the history of the Arithmetical Triangle and Pascal's role in it. The book was practically complete by 1981 but, after several rejection s, was not published until 1987. Even then it was hardly noticed because the publisher forgot to send out any review copies, and when they were finally dispatched the journals said the book was too old to review. Nevertheless, perhaps faute de mieux, it has slowly established itself as the standard work on the subject.
The welcome decision of the Johns Hopkins University Press to bring out a paperback edition affords an opportunity to add an epilogue referring the reader to works published after the original edition, or not noticed in that edition. The main text, however, is unchanged except for the correction of a number of misprints, scholarship in the intervening years having not necessitated any retraction or major revision. One misprint, pointed out by Frank Ruskey of the University of Victoria, British Columbia, has not been corrected: in Fig. 9 Stifel's figurate triangle has 156 instead of 165.
A third appendix, previously unpublished, has been added giving an extended account of the part of Bernoulli's Ars conjectandi that corresponds to the first part of Pascal 's Traité. It describes it in the same manner in which the latter is described and analyzed in Chapter VI. Originally written in 1985 as part of a commentary intended to accompany a new translation of Ars conjectandi, it is included here with the permission of Glenn Shafer and Edith Sylla, whose grant from the National Endowment for the Humanities supported its writing. In the present context it may conveniently be regarded as an extended footnote to pages 123-5 of Chapter X.
A fourth appendix gives a table of the binomial coefficients below 30,000 in ascending order of magnitude, omitting of course the first two rows and first two columns of Pascal's triangle in which every positive integer occurs. Such a table serves as an 'index' to the initial part of Pascal's triangle and in particular to the tables edited by Miller (1954).
All the figures reproduced from books (except Fig. 19) are from copies in Cambridge University Library and are reproduced by permission of the Syndics of the Library. In amongst the difficulties associated with the original publication of Pascal's Arithmetical Triangle this acknowledgment was regrettably omitted, and it gives me pleasure to include it now, all the more so because in the intervening years I have been able to repay some of my debt to that great library by serving as a Syndic myself and as Chairman of the Syndicate for a period of five years. One of the Library's copies of Pascal' s Traite is available at www.lib.cam.ac.uk/RareBooks/PascalTraite.
A. W. F. EDWARDS
Gonville and Caius College, Cambridge
Preface to the original edition
The Arithmetical Triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, and (barely concealed) the Fibonacci numbers from medieval Italy. It reveals patterns which delight the eye, raises questions which tax the number-theorists, and amongst the coefficients "There are so many relations present that when someone finds a new identity, there aren't many people who get excited about it any more, except the discoverer!" [1].
It was first written down long before 1654, the year in which Blaise Pascal wrote his Traité du triangle arithmetique, but it was this work that brought together all the different aspects of the numbers for the first time. In it Pascal developed the properties of the numbers as a piece of pure mathematics (often using mathematical induction in his proofs) and then, in a series of appendices, showed how these properties were relevant to the study of the figurate numbers, to the theory of combinations, to the expansion of binomial expressions, and to the solution of an important problem in the theory of probability . That the Arithmetical Triangle should bear Pascal's name cannot be disputed.
The subsequent flowering of the theory of probability gave added importance to the binomial coefficients through the central role played by the binomial distribution, and in the development of analysis in the seventeenth century they cropped up at every turn, directly influencing the discoveries of Wallis, Newton and Leibniz. More recently the growing importance of combinatorial theory has fuelled fresh interest in the coefficients.
In the history of these numbers it will be important to differentiate between the three principal ways in which they arise. We shall therefore distinguish three types of numbers. By the figurate numbers we shall mean the numbers which occur in a certain family of progressions in which the lth number of the kth progression, denoted by flk, is the sum of the first l numbers in the previous, or (k - 1)th, progression, the first such progression being the series of integers, 1, 2, 3, 4, ...; by the combinatorial numbers we shall mean the numbers which give the number of combinations of n different things taken r at a time, denoted by nCr; and by the binomial numbers we shall mean the coefficients which occur in the expansion of the nth power of the binomial expression (a + b), the coefficient of arbn-r being denoted by [MATHEMATICAL EXPRESSION OMITTED]. Analogously, an array of numbers will be called a Figurate Triangle, a Combinatorial Triangle, or a Binomial Triangle, according to the context in which it has been given. Such arrays will not necessarily be triangular — indeed, since they are capable of indefinite extension they can hardly be thought of as having any particular shape — but it is convenient, and consistent, to use Pascal's word. Of course, all these Triangles turn out to be the same thing, and once we have established their identity we will be able to relax; this artificial terminology and refer to Pascal's Arithmetical Triangle, and to its numbers as simply binomial coefficients, the dominant name for them.
It was Montmort [2] who, in 1708, first attached the name of Pascal to the Combinatorial Triangle ("Table de M. Pascal pour les combinaisons"), though his array was not actually in Pascalian format (see Frontispiece) but a staggered version of it (Fig. 1).
Then, in his M iscellanea Anal yticaof 1730, De Moivre christened Pascal's form "Triangulum Arithmeticum PASCALIANUM" [3], reserving the plain "TRIANGULUM ARITHMETICUM" [4] for Montmort's format. It is to De Moivre, therefore, that I owe the title of this book.
There appears to be no previous full-length history of the Arithmetical Triangle, and in English not even a brief one. For German readers there are Tropfke [5] (1924) and Knobloch [6] (1973) and for French readers the somewhat inaccessible account of Bosmans [7] published in 1906. Of the many extant tables of the Arithmetical Triangle we may note especially the beautiful Table of Binomial Coefficients edited by Miller [8] just before computers rendered such compilations obsolete.
Although the present work does not set out to provide a comprehensive account of Pascal's contribution to the theory of probability, it covers the greater part of it. To render it complete in this respect two of the author's papers, describing the Problem of Points and the Gambler's Ruin Problem, are reprinted as appendices.
During the writing I have been deeply indebted to Dr D. T. Whiteside, who, when he might so justifiably have concluded from my first page of notes that my cause was hopeless, instead embarked on a long correspondence which has directed, corrected, encouraged, and reprimanded me in just the right proportions. It is small wonder that earlier writers in this field made so many mistakes, for they did not have his fund of knowledge to call upon. But he did not, alas, write this book, whose mistakes are all my own.
Next it is a pleasure to thank the Master and Fellows of Gonville and Caius College, not only in their corporate capacity for possessing original editions of Bernoulli, De Moivre, Herigone, Montmort, Oughtred, Vieta and Wallis and for providing excellent working conditions in College, but also individually for answering a torrent of questions, mathematical, logical, linguistic, and historical. The following colleagues have borne the brunt of these : Dr W. J. Macpherson, Dr J. Needham, Professor C. 0. Brink, Reverend J. V. M. Sturdy, Dr E. F. Timms, Dr J. E. J. Altham, Dr J. H. Conway, Dr K. J. Martin, Dr P. J. Bayley, Dr R. H. S. Carpenter, Dr T. J. Pedley and Dr A. D. Barbour. Miss Sarah Law, also of Caius College, kindly read the whole text and made many helpful suggestions, whilst Miss Ann Edwards, of Trinity Hall, similarly helped with the references.
Finally, I am grateful to many correspondents for their help, which I acknowledge in the notes accompanying the chapters.
A. W. F. EDWARDS
Gonville and Caius College, Cambridge
(Continues…)
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Table of Contents
ContentsPreface to the Johns Hopkins edition
Preface to the original edition
1. The figurate numbers
2. Three combinatorial rule.
3. The combinatorial numbers in India
4. The combinatorial numbers in the West
5. The binomial numbers
6. Pascal's Treatise on the Arithmetical Triangle, Part 1
7. Pascal's Treatise, Part II, and associated tracts
8. The arithmetical triangle in analysis
9. The binomial and multinomial distributions
10. Bernoulli's ars conjectandi
Epilogue
Appendix I. Pascal and the Problem of Points
Appendix II. Pascal's Problem: The: "Gambler's Ruin"
Appendix III: Commentary on the Ars conjectandi
Appendix IV: Table of binomial coefficients
References
General index
Index of names