The Saga of Mathematics

The Saga of Mathematics:
A Brief History

Martin Lewinter
William Widulski

Prentice Hall
ISBN: 0-13-034079-0

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This book was written to accomplish several things. Firstly, it is an inexpensive text for courses in the history of math. Secondly, it is a vehicle of the authors' passion (hopefully contagious) for mathematics. Thirdly, the book brings out the relation of mathematics to music, art, science, technology, and philosophy — in short, this is an interdisciplinary adventure for readers of all ages.

Finally, it is suitable for a "general education" course in mathematics. Teaching mathematics to nonmajors poses a difficult problem across the country. This book is, we believe, a good solution. It makes mathematics relevant to many other disciplines and emphasizes the cleverness and beauty inherent in this subject. It is our hope that the reader will dine on the banquet of ideas in this text with gusto.

The material is very accessible and, we think, well motivated. It features geometry, number theory, algebra, probability, graph theory, and ancient and modern counting systems, including binary arithmetic for the computer age. The first author has taught "History of Mathematics" for a decade and, after wandering from text to text, decided to write one that addresses the needs of the reluctant, ill-prepared (perhaps even math-phobic?) college student. With a light peppering of good-natured humor, the book touches on science, navigation, commerce, the calendar, music, art, philosophy, and, of course, history. In short, it has something for everyone.

There are many worked out examples in the text. The plentiful collection of homework exercises range from easy to hard, giving the instructor some flexibility in the level of difficulty of the course. Each chapter ends with a list of suggested reading.

Chapter 1, "Oh So Mysterious Egyptian Mathematics!," begins with primitive counting by grouping. Egyptian arithmetic, geometry, and algebra are then presented in a way that will interest history and anthropology majors. The concept of unit-based measurement is introduced. The births of astronomy, religion, and the calendar are linked.

In Chapter 2, "Mesopotamia Here We Come," the Babylonian position system and its similarity to (and differences from) the present day number system are presented in a historical context. Their algorithms for division and for calculating square roots are included. Babylon cosmology is shown to have influenced the European world-view up to the seventeenth century.

Chapter 3, "Those Incredible Greeks!," is the first chapter to feature mathematicians, their lives, and their contributions. Starting with Thales, the father of "proof," the text continues with the Pythagorean contributions to number theory (perfect, deficient, and abundant numbers, for example), tiling of the plane, and Greek "music theory" including several concrete mathematical arguments such as a proof that the angle sum of a triangle is 180 degrees and a proof of the irrationality of (the square root of 2). The chapter closes with the amazing feat of Hippocrates of Chios—the squaring of the lune.

In Chapter 4, "Greeks Bearing Gifts," we investigate the relevance of mathematics to Greek philosophy, especially to the Plato versus Aristotle dispute on the nature of universals, or "forms." Euclid is one of the heroes of Hellenistic mathematics in Alexandria and his contributions to the logical foundation of geometry and to number theory are discussed at length. His proof of the infinitude of the primes and his algorithm for the GCD of two numbers are presented. The great Archimedes' many contributions come next, and the chapter closes with Ptolemy's seed of trigonometry and his map of the ancient world.

Chapter 5 is entitled "Must All Good Things Come to an End?" The Classical world gives way to the rise of Christianity and Islam. We examine the contributions of the Islamic mathematicians and the development of Hindu-Arabic arithmetic—all in a historical perspective.

In Chapter 6, "Europe Smells the Coffee," the saga continues. Fibonacci, a.k.a. Leonardo of Pisa, writes several important books, a few universities dot the map of High Middle Ages Europe, Aquinas defends reason, and Nicole Oresme studies infinite series.

The theme of Chapter 7, "Mathematics Marches On," is the influence of mathematics on music and art. This is achieved without getting too technical. Visual perspective is described. The mathematical aspects of music such as pitch, meter, duration, volume, and intervals are presented.

In Chapter 8, "A Few Good Men," we focus on Copernicus, Brahe, Kepler, Galileo, Francis Bacon, and the revolution they ushered in. Galileo's analysis of the motion of falling objects involves an ancient Greek procedure for summing the first n odd numbers. With the invention of the printing press and the coming of the Reformation, the world was ripe for the great mathematics of the seventeenth century.

The next chapter, "A Most Amazing Century of Mathematical Marvels!," is organized as follows:

  1. Fermat and number theory
  2. Pascal and probability
  3. Descartes and analytic geometry, and
  4. Newton and the calculus.

The material is presented in some detail. The relevance to science, technology, and commerce is emphasized.

In Chapter 10, "The Age of Euler," we launch an excursion into graph theory, number theory, and the three-dimensional space R3.

Chapter 11, "A Century of Surprises," deals with some of the themes of the nineteenth century—vectors, non-Euclidean geometry, and field theory. There is a nice discussion of the sine function and its relevance to AM and FM radio.

The story ends with Chapter 12, "Ones and Zeros." The twentieth century saw the advent of the computer. Its language, binary arithmetic, is connected to the "doubling" employed in ancient Egyptian multiplication and to the position system of ancient Mesopotamia. The analysis in the text reinforces the student's grasp of the position system of decimal arithmetic.

We added an optional chapter, "Some More Math Before You Go," which contains a host of topics that may be called upon to bolster the mathematics content of a general education course. We include solving quadratic equations by factoring and by using the quadratic formula, graphing parabolas and circles, solving simultaneous equations, and reviewing operations with fractions.

We would like to thank the following people for their assistance and/or encouragement: Timothy Bocchi (a great teacher and a great friend); Dr. Jerome Huyler (author of Locke in America); Professor Frank Harary (an inspiring, world-class mathematician!); Kathy Lavelle, Bert Liberi, Joyce McQuade, and Mike Shub (for supplying some material); and the illustrator Amy Herrmann (whose entirely original artwork elevates the quality of the book enormously).

Marty Lewinter

William Widulski

May 1, 2001

© 2002 Prentice Hall Inc.

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© 2003 by Professor William F. Widulski Last Updated May 2003