the_history_of_mathematics_an_introduction__6th_ed.pdf

Mathematics

The History of Mathematics: An Introduction, 6th Editi

Burton

McGraw-Hill

McGraw−Hill Primis

ISBN: 0−390−63234−1

Text:

The History of Mathematics: An

Introduction, Sixth Edition

Burton

This book was printed on recycled paper.

Mathematics

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111

MATHGEN

ISBN: 0−390−63234−1

Mathematics

Contents

Burton • The History of Mathematics: An Introduction, Sixth Edition

Front Matter

Preface

1. Early Number Systems and Symbols

Text

2. Mathematics in Early Civilizations

Text

3. The Beginnings of Greek Mathematics

Text

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. Nineteenth−Century Contributions: Lobachevsky to Hilbert

559

Text

559

iii

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

Burton: The History of

Mathematics: An

Introduction, Sixth Edition

Front Matter

Preface

Companies, 2007

P r e f a c e

Since many excellent treatises on the history of mathemat-

ics are available, there may seem little reason for writing

still another. But most current works are severely techni-

cal, written by mathematicians for other mathematicians

or for historians of science. Despite the admirable schol-

arship 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 of mathematics 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 for more 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 mathe-

matics. Thus, assorted problems of varying degrees of difﬁculty have been interspersed

throughout. Usually these problems typify a particular historical period, requiring the pro-

cedures 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 exer-

cised, both in choosing mathematicians and in treating their contributions. Nor was material

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

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