The Ellipse

A Historical and Mathematical Journey

Arthur Mazer

• Publisher: Wiley
• Year: 2011
• ISBN:
  9781118211434
• Edition: 1
• Topic: Matematica
• Subtopic: Giochi in matematica

Explores the development of the ellipse and presents mathematical concepts within a rich, historical context

The Ellipse features a unique, narrative approach when presenting the development of this mathematical fixture, revealing its parallels to mankind's advancement from the Counter-Reformation to the Enlightenment. Incorporating illuminating historical background and examples, the author brings together basic concepts from geometry, algebra, trigonometry, and calculus to uncover the ellipse as the shape of a planet's orbit around the sun.

The book begins with a discussion that tells the story of man's pursuit of the ellipse, from Aristarchus to Newton's successful unveiling nearly two millenniums later. The narrative draws insightful similarities between mathematical developments and the advancement of the Greeks, Romans, Medieval Europe, and Renaissance Europe. The author begins each chapter by setting the historical backdrop that is pertinent to the mathematical material that is discussed, equipping readers with the knowledge to fully grasp the presented examples and derive the ellipse as the planetary pathway. All topics are presented in both historical and mathematical contexts, and additional mathematical excursions are clearly marked so that readers have a guidepost for the materials' relevance to the development of the ellipse.

The Ellipse is an excellent book for courses on the history of mathematics at the undergraduate level. It is also a fascinating reference for mathematicians, engineers, or anyone with a general interest in historical mathematics.

CHAPTER 1

INTRODUCTION

My first teaching job did not start out too smoothly. I would feverishly spend my evenings preparing material that I thought would excite the students. Then the next day I would watch the expression on my students’ faces as they sat through my lecture. Their expressions were similar to that on my Uncle Moe’s face when he once recalled an experience on the Bataan Death March. How could the lectures that I painstakingly prepared with the hope of instilling excitement have been as tortuous as the Bataan Death March? To find an answer to this question, I went to the source. I asked the students what was going wrong. After 16 years, with the exception of one suggestion, I have vague recollection of the students' feedback. After 16 years, with the exception of one individual, I cannot remember the faces behind any of the suggestions. Concerning the one individual, not only do I have clarity concerning her face and suggestion, but I also have perfect recollection of my response.

The individual suggested that I deliver the lectures in storylike fashion and have a story behind the mathematics that was being taught. My response that I kept to myself was “you have got to be joking.” My feeling was that mathematics was the story; the story cannot be changed to something else to accommodate someone’s lack of appreciation for the subject. This was one suggestion that I did not oblige. And while for the most part the other students responded positively to the changes that I did make, this student sat through the entire semester with her tortured expression intact.

It is difficult to recall the specifics of something that was said over 16 years ago, the contents of a normal conversation remain in the past while we move on. Despite my reaction, there must have been some meaning that resonated and continued doing so, otherwise I would have long ago forgotten the conversation. Now I see the student’s suggestion as brilliant and right on target. By not taking her suggestion, I blew the chance to get more students excited by mathematics through compelling and human stories that are at the heart of mathematics. At the time, I just did not have the vision to see what she was getting at. After 16 years, I have once more given it some thought and this book is the resulting vision. This is a mathematical story and a true one at that.

The story follows man’s pursuit of the ellipse. The ellipse is the shape of a planet’s path as it orbits the sun. The ellipse is special because it is a demonstration of man’s successful efforts to describe his natural environment using mathematics and this mathematical revelation paved the pathway from the Counter Reformation to the Enlightenment. Man pursued the ellipse in a dogged manner as if a mission to find it had been seeded into his genetic code. Through wars, enlightened times, book burnings, religious persecution, imprisonment, vanquished empires, centuries of ignorance, more wars, plagues, fear of being ridiculed, the Renaissance, the Reformation, the Counter Reformation, excommunication, witchcraft trials, the Inquisition, more wars, and more plagues, man leaf by leaf nurtured a mathematical beanstalk toward the ellipse. This book examines the development and fabric of the beanstalk. It describes the creation of geometry, algebra, trigonometry, and finally calculus, all targeted toward the ellipse.

What are the ingredients that make up a good story? Heroes: They are in this story as the book presents a glimpse of the lives of several mathematicians from Aristarchus to Leibniz who made significant contributions to the beanstalk. Villains: The story of men threatened by progress and doing their best to thwart—being central to the story of the ellipse. Struggles: The problem of planetary motion is sufficiently vexing to assure some mathematical difficulty, and as the previous paragraph indicates, additional struggles result from a tormented history. Dedication: The dedicated effort of the contributors is at once admirable and inspiring. Uncertainty: While the book reconstructs mathematical history with the certainty that man arrives at the ellipse, many contributors had absolutely no premonition of where their contributions would lead. This uncertainty is germane to our story. Character flaws: Our heroes were not perfect and their mistakes are part of the story. Tragedy: Getting speared in the back while contemplating geometry, a victim of one’s own insecurity, a burning at the stake as a victim of the Inquisition—these are a small sampling of personal tragedies that unfold as we follow the ellipse. Triumph: After a tortuous path, this story triumphantly ends at the ellipse. What else is in a good story? I dare not get explicit, but it is in there.

With such a great story, one would think that someone had told it before. Indeed, the story has been told; the most comprehensive historical presentation is Arthur Koestler’s distinguished book, The Sleepwalkers. In addition, there are history books and excellent biographies of the main contributors, mathematical history books, and books covering the various mathematical topics that are contained in this book. So what is different about this book? Simply put, the history books only address the history, the math books only address the math, and the mathematical history books only address the mathematical history. This book is a math book covering the topics of geometry, algebra, trigonometry, and calculus which contains a historical narrative that sets the context for the mathematical developments. Following my belief that separating the disciplines of the history of mathematics and science from general history is an unnatural amputation, the narrative weaves the mathematical history into the broader history of the times while focusing along the main thread of uncovering the ellipse.

There is a final category of book that readers of this book may be interested in, popular books that explain mathematical and scientific theory—books explaining general relativity, quantum mechanics, chaos theory, and string theory abound for those without the requisite mathematical background. Of necessity, the core is missing in these books, the mathematics. Just as love binds two humans in true intimacy,mathematics binds the theorist with evidence. It is difficult to have a true appreciation of the theory without the mathematics, which is unfortunate because it keeps the general public at a distance from theory. This book takes the reader through all the mathematical developments needed to uncover the ellipse, and the reader will become truly intimate with the theory. The book delves into the subjects of geometry, algebra, trigonometry, and calculus, and once the mathematical machinery is finally assembled, we stock the ellipse.

Mathematicians are explorers. They follow their imagination into new territory and map out their findings. Then their discoveries become gateways for other mathematicians who can push the path into further unexplored territories. Unlike the great sea-going explorers of the fifteenth and sixteenth centuries who were exploring the surface of a finite earth, the domain of the mathematician is infinite. The subject will never be exhausted, mathematical knowledge will continue to expand, and the beanstalk will keep growing. However, like the great explorers of the fifteenth and sixteenth centuries, mathematical journeys may target a specific objective (akin to Magellan’s circumnavigation of the world) or the consequences of mathematical journeys may be fully unrelated to their intentions (akin to Columbus’ accidental discovery of a new continent). We can even go one step further; it is possible that some mathematical journeys have no intent whatsoever other than to amuse the journeying mathematician.

This book presents mathematics as a journey. There is the intended pathway toward the ellipse and there are sojourns along bifurcating branches of the beanstalk that are unrelated to the ellipse. The journey passes through the normal high school curriculum and calculus. By placing all the subject matter together, it is possible to demonstrate relations between what are normally taught as separate disciplines. For example, the area of an ellipse, a geometric concept, is finally arrived at only after developing concepts in linear algebra and trigonometry; the approach highlights the interplay of all the disciplines toward an applied problem. In addition, setting the objective of uncovering the ellipse motivates the mathematics. For example, studies of motion motivate the presentation of calculus and the fundamental theorem of calculus is presented as a statement of the relation between displacement and velocity. The sojourns with no apparent relation to the ellipse are undertaken solely because they are irresistible.

The book allows you as a reader to plot your own course in accordance with your own purpose. Readers with excellent proficiency in calculus will certainly plot their way through the book differently from those who may be a little out of touch with their high school mathematics and calculus. And those entirely unfamiliar with one or more of the subjects will plot another course altogether. The first section of Chapter 2 hosts the main narrative and tells the story of man’s pursuit of the ellipse beginning with Aristarchus, the first known heliocentrist, and ending with Newton’s successful unveiling nearly two millennia after Aristarchus. Each subsequent chapter begins with a narrative that is pertinent to the mathematical material in the chapter. By and large the mathematical material is included for one or more of the following reasons. The material is necessary to understand the topic of the chapter and will be used in subsequent chapters, or the material presents concrete examples of relevant concepts, or I have just indulged my own fancy and included material that I find fun. Sections containing material that falls solely within the last category are clearly marked as excursions and may be skipped without compromising your understanding of the remaining material. As for the remaining material, plot your course in accordance with your own purpose. You may grasp the high-level concepts and move on, or for those who want to go through the nitty-gritty, it is in there. Enjoy your journey.

CHAPTER 2

TEE TRAIL: STARTING OUT

2.1 A STICKY MATTER

CLASSMATE: Be careful. Take such stands in the classroom only. If you speak like that in public, you could be called a heretic.

KEPLER: My beliefs are my beliefs. I will make no secret of them.

Kepler and Galileo lived during a time of transition. The church had lost much of its authority during the Reformation and answered with the Counter Reformation in an attempt to recover its former position. There were several factors contributing to the Reformation: nationalism, taxation, and a wayward clergy. The central method of the Counter Reformation was that that the church had honed over its 1000-year reign of power, fear.

For centuries the church could afford its excesses. Its position as the sole interpreter of scriptures allowed it to control human activity with the threat of eternal damnation. The message was simple and not subtle—follow the church’s dogma toward eternal salvation or suffer unimaginable consequences, not only for the short period of your life on earth, but for eternity. And the church proffered vivid descriptions of what the consequences would be so that the unimaginable became images that were seared into the minds of medieval Europe. Demons thrusting pitch forks into screaming victims, deformed beasts pursuing their victims without mercy, and rings of fire forever scorching its victims—these images of hell had been painted in medieval churches across Europe. Through fear, the church stifled intellectual development throughout the Dark Ages.

The church maintained its monopoly as the sole interpreter of scriptures through two methods. First, the predominant avenue to an education was through church seminaries or church-sponsored universities; there were few independent secular educational institutions. Second, Latin, which was only taught in the seminaries and universities, was the language of the Bible. There were no translations into local languages, so the majority of Europeans could only rely upon the church’s interpretation. In 1439, Johannes Gutenberg invented a simple device that would challenge the church’s monopoly on intellectual activity, the printing press. Soon the Bible would be printed and distributed in local languages and the masses would be free to read and interpret scriptures for themselves. The Reformation was born, and after recovering its footing, the church responded; it launched the Counter Reformation and unleashed the Inquisition.

The result was chaos as Protestants responded to the Counter Reformation with war. The Germanic states that comprised the Holy Roman Empire launched a revolt against the church-supported Habsburg dynasty. This spawned the Thirty Years’ War between Christians. Each s ide required discipline from their followers. In Italy, the church accepted no challenges to its authority and enforced its dogma with the Inquisition. Protestants followed suit, enforcing discipline the only way that they knew how, with fear. Those who did not agree with the dogma of the leading Protestant clergymen were excommunicated. It was in this environment that the Lutheran Kepler and the Catholic Galileo initiated modern science and mathematics, and it was in this environment that both were punished for their remarkable accomplishments.

Is the sun fixed, with the earth and its sister planets revolving about the sun, or is the earth fixed with all that is in heaven revolving about the earth? This seems to be an innocent question, certainly not a question that would lead to censorship, excommunication, imprisonment, torture, and execution on the pyre, with all of these indignities sponsored by an institution claiming to show humanity the way to salvation. And yet, the quest to answer this seemingly innocent question catalyzed all of these responses within the church. In those times, the church was far more politically consumed than the present-day church and political motives engendered these ugly responses. On the scientific s ide, the quest to determine the path of the planets catalyzed the development of calculus and brought science and mathematics into the modern era. This chapter follows the history of the quest in a narrative that addresses both political and scientific dimensions. The mathematics presented later in the book follows the narrative.

While there are many potential points to begin this story, we choose to begin with Aristarchus (310–230 B.C.), a Greek astronomer and mathematician from Samos. Aristarchus is the first individual known to have proposed heliocentricity based upon geometric analysis. The analysis contained two components: a method for calculating the relative size of the sun and a proposition that distance explains the fixed path of the stars from the perspective of a moving earth. This latter proposition explicitly addresses what is known as the parallax problem. Detractors of heliocentrism state that the stars would not daily appear in the same position as the earth revolves around the sun if the earth were to do so. In short order, their argument goes, the stars do appear in the same position, so the earth must be stationary. Aristarchus retorts that even though the earth moves, the stars appear fixed because the distance between the earth and stars is orders of magnitude greater than the comparatively small distances that the earth moves. With this argument, Aristarchus confronts man with the scale of the universe and how little we are within it, not a very popular notion.

Aristarchus makes another scaling argument, this one a bit more quantitative with his estimate of the relative size of the sun. This estimate demonstrates Aristarchus’ grasp of geometry while at the same time illustrating the limitations of the instruments used to take astronomical observations. The geometric argument is flawless, providing a correct equation, but the measurement of an angle required by the equation is far off base. Placing his poor measurement into the formula, Aristarchus calculated that the diameter of the sun was about 20 times that of the earth, whereas the sun’s actual diameter is on the order of 300 times that of the earth’s. Nevertheless, Aristarchus is the first to propose that the sun is the significantly larger body, likewise not a popular notion. Aristarchus’ work in which he proposes the heliocentric model did not survive. We can only conjecture that he found it more reasonable that the smaller body should orbit the larger body, not vice versa.

The prefix geo, which finds great use in the English language, has its origins in the Greek word ge, meaning “earth.” A slight permutation of geo yields ego, which is the Latin word for I. The heliocentric universe that Aristarchus proposed was much less ego friendly than the geocentric universe that had been accepted since Aristotle. The earth lies precisely in the m iddle of Aristotle’s universe and it is dominant everything else, much lighter than the earth, revolves around the earth in perfect circles. Given man’s collective ego, not even the finest snake oil salesman in history could have sold Aristarchus’ view in Aristarchus’ time. The church’s response to the heliocentric view nearly 1800 years later echoes a response by a contemporary of Aristarchus, Cleanthes. Cleanthes was so affronted by Aristarchus that he wrote a treatise entitled Against Aristarchus in which he states that “it was the duty of Greeks to indict Aristarchus of Samos on the charge of impropriety.” The charge of impropriety is eerily similar to the charge of heresy that the church would accuse adherents of heliocentrism of at a later time.

Despite Cleanthes’ appeals, there is no evidence of court action against Aristarchus. In fact, Aristarchus’ proposal was firmly rooted in Greek tradition, one that respects not only knowledge but also the quest for knowledge. Chaos often begets intellectual activity and the percolating cauldron that was Greece fits this pattern. Prior to Alexander, there was not an empire or even a monolithic civilization known as Greece. On the contrary, Greece was a constellation of city states, each with its own distinct culture. Some were ruled by tyrants and others by assembly. Some stressed military values, while others stressed arts and learning. What held them together as distinctively Greek was a common polytheistic religion, a common language, and geographic proximity. Another element binding the Greeks was the common threat of Persia, which would place them in temporary alliance. More often than not, when external threats diminished, the Greek states would war with one another.

It was not until Philip of Macedonia united the Greeks that a national entity emerged. When Philip’s son, Alexander, assumed power and established his empire, it was the Athenian culture of arts and knowledge that he exported and transplanted. This culture had an unusual tolerance for individual expression, one that resonated well with the indigenous inhabitants of the lands that Alexander conquered. The Athenian theater tradition demonstrates the high esteem that Athenians held for the right of self-expression, provided of course t...