Journey through Mathematics: Creative Episodes in Its History

479 Pages·2011·5.08 MB·English

by  Enrique A. González-Velasco (auth.)

Journey through Mathematics
Enrique A. González-Velasco
Journey through Mathematics
Creative Episodes in Its History
Enrique A. González-Velasco
Department of Mathematical Sciences
University of Massachusetts at Lowell
Lowell, MA 01854
USA
To my wife, Donna,
who solved quite a number of riddles for me.
With thanks for this, and for everything else.
TABLE OF CONTENTS
Preface ix
1 TRIGONOMETRY 1
1.1 The Hellenic Period 1
1.2 Ptolemy’s Table of Chords 10
1.3 The Indian Contribution 25
1.4 Trigonometry in the Islamic World 34
1.5 Trigonometry in Europe 55
1.6 FromViète to Pitiscus 65
2 LOGARITHMS 78
2.1 Napier’s First Three Tables 78
2.2 Napier’s Logarithms 88
2.3 Briggs’ Logarithms 101
2.4 Hyperbolic Logarithms 117
2.5 Newton’s Binomial Series 122
2.6 The LogarithmAccording to Euler 136
3 COMPLEX NUMBERS 148
3.1 The Depressed Cubic 148
3.2 Cardano’s Contribution 150
3.3 The Birth of Complex Numbers 160
3.4 Higher-Order Roots of Complex Numbers 173
3.5 The Logarithms of Complex Numbers 181
3.6 Caspar Wessel’s Breakthrough 185
3.7 Gauss and Hamilton Have the Final Word 190
4 INFINITE SERIES 195
4.1 The Origins 195
4.2 The Summation of Series 203
4.3 The Expansion of Functions 212
4.4 The Taylor and Maclaurin Series 220
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5 THE CALCULUS 230
5.1 The Origins 230
5.2 Fermat’s Method of Maxima and Minima 234
5.3 Fermat’s Treatise on Quadratures 248
5.4 Gregory’s Contributions 258
5.5 Barrow’s Geometric Calculus 275
5.6 From Tangents to Quadratures 283
5.7 Newton’s Method of Infinit Series 289
5.8 Newton’s Method of Fluxions 294
5.9 Was Newton’s Tangent Method Original? 302
5.10 Newton’s First and Last Ratios 306
5.11 Newton’s Last Version of the Calculus 312
5.12 Leibniz’ Calculus: 1673–1675 318
5.13 Leibniz’ Calculus: 1676–1680 329
5.14 The Arithmetical Quadrature 340
5.15 Leibniz’ Publications 349
5.16 The Aftermath 358
6 CONVERGENCE 368
6.1 To the Limit 368
6.2 The Vibrating String Makes Waves 369
6.3 Fourier Puts on the Heat 373
6.4 The Convergence of Series 380
6.5 The Difference Quotient 394
6.6 The Derivative 401
6.7 Cauchy’s Integral Calculus 405
6.8 Uniform Convergence 407
BIBLIOGRAPHY 412
Index 451
PREFACE
In the fall of 2000, I was assigned to teach history of mathematics on the
retirement of the person who usually did it. And this with no more reason
than the historical snippets that I had included in my previous book, Fourier
Analysis and Boundary Value Problems. I was clearly fond of history.
Initially, I was unhappy with this assignment because there were two ob-
vious difficultie from the start: (i) how to condense about 6000 years of
mathematical activity into a three-month semester? and (ii) how to quickly
learn all the mathematics created during those 6000 years? These seemed
clearly impossible tasks, until I remembered that Joseph LaSalle (chairman
of the Division of Applied Mathematics at Brown during my last years as a
doctoral student there) once said that the object of a course is not to cover
the material but to uncover part of it. Then the solution to both problems was
clear to me: select a few topics in the history of mathematics and uncover them
sufficientl to make them meaningful and interesting. In the end, I loved this
job and I am sorry it has come to an end.
The selection of the topics was based on three criteria. First, there are
always students in this course who are or are going to be high-school teachers,
so my selection should be useful and interesting to them. Through the years,
my original selection has varied, but eventually I applied a second criterion:
that there should be a connection, a thread running through the various topics
through the semester, one thing leading to another, as it were. This would give
the course a cohesiveness that tomewas aesthetically necessary. Finally, there
is such a thing as personal taste, and I have felt free to let my own interests
help in the selection.
This approach solved problem (i) andminimized problem (ii), but I still had
to learn what happened in the past. This brought to the surface another large
set of problems. The firs time I taught the course, I started with secondary
sources, either full histories of mathematics or histories of specifi topics.
This proved to be largely unsatisfactory. For one thing, coverage was not
extensive enough so that I could really learn the history of my chosen topics.
There is also the fact that, frequently, historian A follows historian B, who
in turn follows historian C, and so on. For example, I have at least four
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