Great Moments in Mathematics

by Howard Eves

Overview

What a splendid addition this is to the Dolciani Mathematical Exposition series! This second set of lectures on great moments in mathematics (after 1650) is a fascinating collection of pivotal points in the historical development of mathematics...The four lectures devoted to the liberation of geometry and algebra are of particular interest. The lectures should be required reading for all teachers of mathematics. --Herbert Fremont, The Mathematics Teacher Eves is never less than tantalizing and usually inspiring...each 'great moment' has detailed exercises following it, as these have been carefully chosen to illustrate the depth of the ideas in question. --C. W. Kilmister, The London Times, Higher Education Supplement As is usual with Eves' work, the books are well written and entertaining. They give an historical background to many of the best known mathematical results, and, in addition, provide interesting pieces of information about the mathematicians involved. Eves includes relevant exercises at the end of each chapter. These are a good source of different, interesting problems, and when combined with the material in the chapter, could form the basis for a mathematical project...Eves' book provides an interesting, well-written, and enjoyable account. You won't be disappointed. --David Parrott, The Australian Mathematics Teacher

Product Details

ISBN-13:	9780883853115
Publisher:	American Mathematical Society
Publication date:	12/31/1983
Series:	Dolciani Mathematical Expositions Series , #7
Edition description:	New Edition
Pages:	263
Product dimensions:	5.50(w) x 8.40(h) x 0.60(d)

Read an Excerpt

Preface

Many who were growing up in America during the period from 1928 to 1942 will recall the extreme pleasure they experienced when they tuned in their radios to the NBC "Music Appreciation Hour." During those fourteen years an estimated five million school children and a large unknown number of just plain devotees listened each week to the beautifully modulated and hypnotic voice of Walter Damrosch, eminent musicologist and master popularizer the "great moments in music," and ever after, their lives were enriched by knowing something of the noble accomplishments of the world's famous composers.

Some years later, while serving in the Mathematics Department of Oregon State College, it occurred to me that what Walter Damrosch so magnificently did for music could perhaps also be done for mathematics. Why not, I thought, develop a set of lectures devoted to the enthralling Great Moments in Mathematics? I reacted to the idea with enthusiasm. I would aim the lectures, I decided at a more specific audience than that chosen by Walter Damrosch—I had in mind, of course, a college and college-community audience. My major hope was to reach, without any great mathematical demands, anyone interested in learning something about the outstanding achievements in mathematics over the ages. The hole thing was to be an intellectual adventure, with no truly prohibiting or frightening prerequisites. And yet, at the same time I also wanted to give something that would challenge a good mathematics student and perhaps by of use to teachers of mathematics.

The somewhat conflicting aims of the lecture series were finally met as follows. A lecture sequence of some sixty chronologically ordered Great Moments in Mathematics was designed, along with ample problem material, sometimes in the form of "junior" research, bearing on the subject matter of the lectures. The lectures, each fifty minutes long, and enough of them to carry through two semesters, were to be offered at two meetings each week, and the associated problem materials was to be discussed at a third weekly meeting. The two-meting sequence was to constitute an appreciation course, open to auditors and to seekers of a total four elective college credits; for the first semester (which is covered in the present volume) an acquaintance with high school mathematics was the only prerequisite. The three-meeting sequence was to constitute a mathematics course, open to qualified students and teachers seeking a total of six college credits in mathematics; some what stiffer mathematics demands—say, mathematics through beginning calculus—were made of those registering for the extended sequence.

Upon being invited by the Publication Committee for the Dolciani Mathematical Expositions to write up the Great Moments in Mathematics lecture series, I decided, because of space problems, to attempt a curtailed version. Only forty of the lectures were selected, the first twenty from the period before 1650, and the remaining twenty from the period after 1650. Herewith are the twenty.

Each selected lecture has been mercilessly pared down, inasmuch as a transcript of a complete fifty-minute lecture would run far too many pages. Thus, almost all of the humor and anecdotal material so fitting in an oral presentation, as well as many of the cultural ramifications and side trails, and, of course all the visual props in the form of models, display, maps, portraits, and over-head-projector material, are omitted.

Consider for example, Lecture 9 of the curtailed series, devoted to Archimedes and his method of equilibrium. Recently, at an oral presentation of this lecture, I had a t the lecture desk a reproduction of an ancient Greek sand tray, a specimen of a palimpsest, an attractively boxed loculus Archimedius, a small demonstration model of an Archimedean screw, a large calibrated circular cylinder with a heavy removable inscribed sphere, working models of the three classes of levers for comparing their mechanical advantages, and a compound pulley attached to a heavy weight, which was almost effortlessly moved every now and then during the lecture. I showed, on an overhead projector, transparencies of the three questionable medallion portraits of Archimedes, a picture of the interesting mosaic portraying Archimedes' last moments now residing in the Municipal Art Institute at Frankfurt am Main, a portrait of Heiberg, a picture of a sculpted bust of Marcellus, and a map of ancient Syracuse. To lighten the oral presentation I introduced bits of humor—bits which might appear somewhat ridiculous if reproduced here in print. But, as I learned years ago at Harvard from my mentor Julian Lowell Coolidge, a touch of clownery can have a place in an oral presentation. The mathematical demonstration in the lecture, which appears so terse and stark in the written version, was carefully, slowly, and meticulously performed at a blackboard, so that the audience could almost see the balance bar. And, along with all the shortcuts and omissions, the lyrical and poetical flights of an oral presentation are also missing in the written version. What has been said of Lecture 9 can also be said—sometime, it is true, not fully—of each of the other lectures in the series.

So, here, with sincere apologies, are cruelly condensed versions of some of the lectures in the Great Moments in Mathematics. It could be that the only proper way to preserve the lectures would be on videotape, or, better, on educational TV, delivered by a gifted lecturer and all the props and marvels possible with such a presentation.

A few closing words are perhaps in order. The selection of the Great Moments is, of course, my own, and could well differ from a selection made by someone else. Some of the Great Moments can be precisely pinpointed in the time strip—others only vaguely. It must also be remembered that a moment in history is sometimes an inspired flash and sometimes an evolution extending over a long period of time. Much of the subject matter and many of the problems of the lectures subsequently found a place in my Introduction to the History of Mathematics and in An Introduction to the Foundations and Fundamental Concepts of Mathematics, which I wrote with Carroll V. Newsom, and in my four Mathematical Circles books. Finally, in a few spots, for the sake of brevity and to avoid complexities beyond the scope of the lectures, certain minor simplifications have been introduced that are hoped to be essentially unimportant so far as the purpose and the honesty of the lectures are concerned.

Howard Eves

Fox Hollow, Lubec, Maine

Winter 1977-78

Table of Contents

Contents
Preface
Lecture One. Scratches and grunts
Keeping count by a one-to-one correspondence (many millennia ago)
Lecture Two. The greatest Egyptian pyramid
Introducing the volume of a truncated square pyramid (ca. 1850 B.C.)
Lecture Three. From the laboratory into the study
Introduction of deductive procedures into mathematics (ca. 600 B.C.)
Lecture Four. The first great theorem
The Pythagorean theorem (ca 540 B.C.)
Lecture Five. Precipitation of the fires crisis
The discovery of irrational magnitudes (ca. 540 B.C.)
Lecture Six. Resolution of the first crisis
6. The Eudoxian theory of proportion (ca. 370 B.C.)
Lecture Seven. First steps in organizing mathematics
7. Material axiomatics (ca. 350 B.C.)
Lecture Eight. The mathematicians' bible
8. Euclid's Elements (ca. 300 B.C.)
Lecture Nine. The thinker and the thug
9. Archimedes on the sphere (ca. 240 B.C.)
Lecture Ten. A boost from astronomy
10. Ptolemy's construction of a table of chords (ca. 130)
Lecture Eleven. The first great number theorist
11. Diophantus and his Arthimetica (ca. 250)
Lecture Twelve. The syncopation of algebra
12. The first steps toward algebraic symbolism (ca. 250)
Lecture Thirteen. Two Early computing inventions
13. The abacus (uncertain, but early)
The Hindu-Arabic numeral system (before 800)
Lecture Fourteen. The poet-mathematician of Khorasan
Omar Khayyam's geometrical solution of cubic equations (ca. 1090)
Lecture Fifteen. The blockhead
Fibonacci and his Liber abaci (1202)
Lecture Sixteen. An extraordinary and bizarre story
The algebraic solution of cubic equations (1554)
The algebraic solution of quartic equations (1554)
Lecture Seventeen. Doubling the life of the astronomer
Napler's invention of logarithms (1614)
Lecture Eighteen. The stimulation of science
Galileo and the science of dynamics (1589 ff)
Kepler's laws of planetary motion (1619)
Lecture Nineteen. Slicing it thin
Cavalieri's method of indivisibles (1635)
Lecture Twenty. The transform-solve-invert technique
The invention of analytic geometry (1637)
Hints for the solution of some of the exercises
Index