Sách Mathematics - McGraw Hill

Mathematics - McGraw
-Hill


Contents
Burton ã The History of Mathematics: An Introduction, Sixth Edition
Front Matter 1
Preface 1
1. Early Number Systems and Symbols 4
Text 4
2. Mathematics in Early Civilizations 36
Text 36
3. The Beginnings of Greek Mathematics 87
Text 87
4. The Alexandrian School: Euclid 144
Text 144
5. The Twilight of Greek Mathematics: Diophantus 216
Text 216
6. The First Awakening: Fibonacci 272
Text 272
7. The Renaissance of Mathematics: Cardan and Tartaglia 303
Text 303
8. The Mechanical World: Descartes and Newton 338
Text 338
9. The Development of Probability Theory: Pascal, Bernoulli, and Laplace 438
Text 438
10. The Revival of Number Theory: Fermat, Euler, and Gauss 495
Text 495
11. NineteenthưCentury Contributions: Lobachevsky to Hilbert 559
Text 559
12. Transition to the Twentieth Century: Cantor and Kronecker 651
Text 651
13. Extensions and Generalizations: Hardy, Hausdorff, and Noether 711
Text 711
Back Matter 741
General Bibliography 741
Additional Reading 744
The Greek Alphabet 745
Solutions to Selected Problems 746
Index 761
Some Important Historical Names, Dates and Events 787


P r e f a c e
Sincemany excellent treatises on the history ofmathematics
are available, there may seem little reason for writing
still another. But most current works are severely technical,
written by mathematicians for other mathematicians
or for historians of science. Despite the admirable scholarship
and often clear presentation of these works, they are not especially well adapted
to the undergraduate classroom. (Perhaps the most notable exception is Howard Eves’s
popular account, An Introduction to the History of Mathematics.) There seems to be room
at this time for a textbook of tolerable length and balance addressed to the undergraduate
student, which at the same time is accessible to the general reader interested in the history
of mathematics.
In the following pages, I have tried to give a reasonably full account of how
mathematics has developed over the past 5000 years. Because mathematics is one of the
oldest intellectual instruments, it has a long story, interwoven with striking personalities
and outstanding achievements. This narrative is basically chronological, beginning with the
origin ofmathematics in the great civilizations of antiquity and progressing through the later
decades of the twentieth century. The presentation necessarily becomes less complete for
modern times, when the pace of discovery has been rapid and the subject matter more
technical.
Considerable prominence has been assigned to the lives of the people responsible
for progress in the mathematical enterprise. In emphasizing the biographical element, I can
say only that there is no sphere in which individuals count formore than the intellectual life,
and that most of the mathematicians cited here really did tower over their contemporaries.
So that they will stand out as living figures and representatives of their day, it is necessary
to pause from time to time to consider the social and cultural framework that animated
their labors. I have especially tried to define why mathematical activity waxed and waned
in different periods and in different countries.
Writers on the history of mathematics tend to be trapped between the desire to
interject some genuine mathematics into a work and the desire to make the reading as
painless and pleasant as possible. Believing that any mathematics textbook should concern
itself primarily with teaching mathematical content, I have favored stressing the mathematics.
Thus, assorted problems of varying degrees of difficulty have been interspersed
throughout. Usually these problems typify a particular historical period, requiring the procedures
of that time. They are an integral part of the text, and you will, in working them,
learn some interesting mathematics as well as history. The level of maturity needed for this
work is approximately the mathematical background of a college junior or senior. Readers
with more extensive training in the subject must forgive certain explanations that seem
unnecessary.
The title indicates that this book is in no way an encyclopedic enterprise. Neither
does it pretend to present all the important mathematical ideas that arose during the vast
sweep of time it covers. The inevitable limitations of space necessitate illuminating some
outstanding landmarks instead of casting light of equal brilliance over the whole landscape.
In keeping with this outlook, a certain amount of judgment and self-denial has to be exercised,
both in choosingmathematicians and in treating their contributions. Nor wasmaterial
selected exclusively on objective factors; some personal tastes and prejudices held sway.
It stands to reason that not everyone will be satisfied with the choices. Some readers will
raise an eyebrow at the omission of some household names of mathematics that have been
either passed over in complete silence or shown no great hospitality; others will regard the
scant treatment of their favorite topic as an unpardonable omission. Nevertheless, the path
that I have pieced together should provide an adequate explanation of how mathematics
came to occupy its position as a primary cultural force inWestern civilization. The book is
published in the modest hope that it may stimulate the reader to pursue the more elaborate
works on the subject.
Anyone who ranges over such a well-cultivated field as the history of mathematics
becomes so much in debt to the scholarship of others as to be virtually pauperized. The
chapter bibliographies represent a partial listing of works, recent and not so recent, that in
one way or another have helpedmy command of the facts. To the writers and tomany others
of whom no record was kept, I am enormously grateful.
New to This Edition
Readers familiar with previous editions of The History of Mathematics will find
that this editionmaintains the same overall organization and content.Nevertheless,
the preparation of a sixth edition has provided the occasion for a variety of small
improvements as well as several more significant ones.
Themost pronounced difference is a considerably expanded discussion of Chinese
and Islamicmathematics in Section 5.5.Asignificant change also occurs in Section 12.2with
an enhanced treatment of Henri Poincar´e’s career. An enlarged Section 10.3 now focuses
more closely on the role of the number theorists P. G. Lejeune Dirichlet and Carl Gustav
Jacobi. The presentation of the rise of American mathematics (Section 12.1) is carried
further into the early decades of the twentieth century by considering the achievements of
George D. Birkhoff and Norbert Wiener.
Another noteworthy difference is the increased attention paid to several individuals
touched upon too lightly in previous editions. For instance, material has been added
regarding the mathematical contributions of Apollonius of Perga, Regiomontanus, Robert
Recorde, Simeon-Denis Poisson, Gaspard Monge and Stefan Banach.
Beyond these textualmodifications, there are a number of relativelyminor changes.
Abroadened table of contentsmore effectively conveys thematerial in each chapter,making
it easier to locate a particular period, topic, or great master. Further exercises have been introduced,
bibliographies brought up to date, and certain numerical information kept current.
Needless to say, an attempt has been made to correct errors, typographical and historical,
which crept into the earlier versions.
Acknowledgments
Many friends, colleagues, and readers—too numerous to mention individually—
have been kind enough to forward corrections or to offer suggestions for the book’s
enrichment. I hope that they will accept a general statement of thanks for their
collective contributions. Although not every recommendation was incorporated, all
were gratefully received and seriously considered when deciding upon alterations.
In particular, the advice of the following reviewers was especially helpful in the
creation of the sixth edition:
Rebecca Berg, Bowie State University
Henry Gould, West Virginia University
Andrzej Gutek, Tennessee Technological University
Mike Hall, Arkansas State University
Ho Kuen Ng, San Jose State University
Daniel Otero, Xavier University
Sanford Segal, University of Rochester
Chia-Chi Tung, Minnesota State University—Mankato
William Wade, University of Tennessee
A special debt of thanks is owed my wife, Martha Beck Burton, for providing
assistance throughout the preparation of this edition; her thoughtful comments significantly
improved the exposition. Last, I would like to express my appreciation to the staff members
of McGraw-Hill for their unfailing cooperation during the course of production.
Any errors that have survived all this generous assistance must be laid at my door.
D.M.B.