1 The Greek Method of Exhaustion

65 Pages·2012·4.36 MB·English

by  Desouza, Chelsea

The Greek Method of Exhaustion:
Leading the Way to Modern Integration

M.S. Thesis

Presented in Partial Fulfillment of the Requirements for the Degree M.S. Mathematics for
Educators in the Graduate School of The Ohio State University

By
Chelsea E. DeSouza
Graduate Program in Mathematics

The Ohio State University
2012

Mathematics. Committee:
Dr. Rodica Costin, Advisor. Dr. Herb Clemens, Advisor
1
Copyright by
Chelsea E. DeSouza
2012
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Abstract

This paper is a critical look at the Method of Exhaustion and how it is related to
some of the more modern techniques of integration. This text moves through the
development of the Method of Exhaustion to some applications. These applications
include estimating the area of a circle using the Method of Exhaustion, and finding the
exact area of a parabolic region by Archimedes quadrature of the parabola. These ideas
are extensively developed and all constructions and proofs are given in the text and
appendices. Similar techniques are then suggested to the reader to extend these results to
find certain volumes in the appendix. Finally, a comparison is drawn between the Method
of Exhaustion and integration techniques taught in modern classroom settings.

ii
Dedication

This document is dedicated to my family and my professors who have all helped
me a great deal in my mathematical career, which would not have started without all of
your guidance.

iii
Acknowledgments

Thank you to my two advisors, Dr. Rodica Costin and Dr. Herb Clemens who
were both instrumental in the development of this thesis. A big thanks also to the entire
staff of the Ohio State Mathematics Department, who pushed me to be better with my
mathematics.

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Vita
2007-2010 ......................................................Undergraduate TA, Mills College
2008-2010 ......................................................Undergraduate Grader, Mills College
2009................................................................CMOP, Undergraduate Research Associate
2010................................................................B.S. Mathematics, Mills College
2010-2012 ......................................................Graduate Teaching Associate, Department
of Mathematics, The Ohio State University
2012................................................................M.S. Mathematics for Educators, the Ohio
Fields of Study
M.S. Mathematics for Educators

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Table of Contents
Abstract ............................................................................................................................... ii

Dedication .......................................................................................................................... iii

Acknowledgments .............................................................................................................. iv

Vita ...................................................................................................................................... v

List of Tables ................................................................................................................... viii

List of Figures .................................................................................................................... ix

Chapter 1: Introduction ...................................................................................................... 1

Chapter 2: Estimating the Area of the Circle ...................................................................... 3

Chapter 3: Archimedes Quadrature of the Parabola ........................................................... 5

Chapter 4: A Modern Integration Technique: The Trapezoidal Method .......................... 19

Chapter 5: The Method Exhaustion: Leading the Way to Modern Integration ................ 22

Bibliography ..................................................................................................................... 23

Appendix A: Some Ruler and Compass Constructions and Proofs .................................. 25

Appendix
 B: Every isosceles can be bisected into two congruent right triangles ............ 41

Appendix C: Area of any Isosceles Triangle .................................................................... 43

Appendix D: π ................................................................................................................... 44

Appendix E: Estimating Volumes using the Method of Exhaustion ............................... 45

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Appendix F: Calculus Based Explanations of Property 1, 2, and 3 .................................. 46

Appendix G: Examples of Using the Quadrature of the Parabola to find the Area of the
Bounded Region ................................................................................................................ 50

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List of Tables

Table 1: Areas of n-gons Inscribed in a Circle ................................................................... 4

viii
List of Figures

Figure 1: A Series of Regular Polygons Inscribed within a Circle ..................................... 2

Figure 2: Isosceles Triangle Inscribed in a Circle .............................................................. 3

Figure 3: Arbitrary Isosceles Triangle ................................................................................ 4

Figure 4: Special Triangle in a Parabolic Segment ............................................................. 5

Figure 5: The Parabolic Segment ........................................................................................ 6

Figure 6: Properties of P ..................................................................................................... 6

Figure 7: Property 2 ............................................................................................................ 7

Figure 8: Adding Green Triangles ...................................................................................... 8

Figure 9: Triangles of the Quadrature ................................................................................. 8

Figure 10: Points M and C .................................................................................................. 9

Figure 11: Point D ............................................................................................................... 9

Figure 12: Triangle Congruence ....................................................................................... 10

Figure 13: Triangular Height ............................................................................................ 11

Figure 14: Heights of the Two Triangles .......................................................................... 11

Figure 15: Triangle P2ZM2 and M2OC ........................................................................... 12

Figure 16: External and Internal Angles of Triangles from Figure 16 ............................. 12

Figure 17: Rotation of the Triangle .................................................................................. 13

Figure 18: Relation of Triangle PMB and PCB ................................................................ 14

Figure 19: Height of PMB and PCB ................................................................................. 14

Figure 20: Unit Square ...................................................................................................... 16

Figure 21: Dividing S up into Four Equal Parts ............................................................... 16

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