Shifting the Earth E-Book

Arthur Mazer

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 582

Veröffentlichungsjahr: 2011

Beschreibung

Discover how mathematics and science have propelled history

From Ancient Greece to the Enlightenment and then on to modern times, Shifting the Earth: The Mathematical Quest to Understand the Motion of the Universe takes readers on a journey motivated by the desire to understand the universe and the motion of the heavens. The author presents a thought-provoking depiction of the sociopolitical environment in which some of the most prominent scientists in history lived and then provides a mathematical account of their contributions.

From Eudoxus to Einstein, this fascinating book describes how, beginning in ancient times, pioneers in the sciences and mathematics have dramatically changed our vision of who we are as well as our place in the universe. Readers will discover how Ptolemy's geocentric model evolved into Kepler's heliocentric model, with Copernicus as the critical intermediary. The author explains how one scientific breakthrough set the stage for the next one, and he also places the scientists and their discoveries within the context of history, including:

• Archimedes, Apollonius, and the Punic Wars

• Ptolemy and the rise of Christianity

• Copernicus and the Renaissance

• Kepler and the Counter-Reformation

• Newton and the Enlightenment

• Einstein and the detonation of the atom bomb

Each chapter presents the work of a single scientist or mathematician, building on the previous chapters to demonstrate the evolutionary process of discovery. Chapters begin with a narrative section and conclude with a mathematical presentation of one of the scientist's original works. Most of these mathematical presentations, including the section on Einstein's special relativity, are accessible using only basic mathematics; however, readers can skip the mathematical sections and still follow the evolution of science and mathematics.

Shifting the Earth is an excellent book for anyone interested in the history of mathematics and how the quest to understand the motion of the heavens has influenced the broader history of humankind.

Leseprobe

Table of Contents

Title Page

Copyright

Preface

Acknowledgments

Chapter 1: Perfection

Chapter 2: Perfectionists

A Tour of Athens

Plato's Challenge

Eudoxus' Universe

The Math

Chapter 3: The Iconoclast

Insanity

A Disturbing Insight

ARISTARCHUS AND On the Sizes and Distances of the Sun and Moon

Chapter 4: Instigators

A Heavy Hand

Patriarchs

The Irony

Fun

Apollonius' Osculating Circle and the Ellipse

Fun's Harvest

Chapter 5: Retrograde

The Imperial Theocracy

Ptolemy's Universe

Ptolemy and the Almagest

Inspiration

Chapter 6: Revolutionary

Possibilities

Copernicus' On Revolutions, the Pursuit of Elegance

The Model

An Elegant Result

Chapter 7: Renegades

Countering the Reformation

The Trio: Tycho, Kepler, and Rudolf

Meanwhile in Italy

New Astronomy, IN KEPLER'S OWN LIKENESS

A Path Strewn with Casualties

The Physicist's Law

Tycho's Gift

Mars' Bed

Configuring the Ellipse: The Second Law

The Mentor

Chapter 8: The Authority

Louis XIV, the Sun King

Newton, the Math King

The Influential Principia

Without Fluxions

Newton's Laws, Kepler's Laws, and Calculus

Newton's Agenda

Chapter 9: Rule Breakers

Beginnings

Debunking the Ether

The Established and the Unknown

Trouble

Exiting the Quantum Universe

Bending the Light

An Ergodic Life

Victory

The Experiment

The Famous and the Infamous

Exodus

The End

Relativity

Special Relativity

Epilogue

Bibliography

Index

Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Mazer, Arthur, 1958-

Shifting the Earth : the mathematical quest to understand the motion of the universe / Arthur Mazer.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-02427-0 (pbk.)

1. Kepler's laws. 2. Motion. 3. Celestial mechanics. I. Title.

QB355.3.M39 2012

521'.3–dc23

2011013592

Preface

There is a comic of a skinny little kid who, while standing on a stool so that he can look at himself in a mirror, views the image of a ripping, six-packed, Shwartznegger-esque champ. It's all too true to life and, like the best of comics, begs a question. Suppose that the same skinny kid looks at the mirror 35 years later when he is a pudgy, balding man. What is the image that he will see? Will he continue to see the champ, or will he have come to accept who he is, but in doing so see more possibilities for himself than the little boy could have imagined? This question is the topic of many novels and movies. It is intriguing because there is no predestined path for the boy to follow and there is no definitive outcome.

Shifting the Earth relates the story of how humanity collectively dispossessed itself of its geocentric fabrication and accepted a less prestigious position in the universe. It was not preordained that humanity would toss out its self-image, pack up what remained, and move to a new heliocentric earth. Psychological barriers resisted the move. Vested interests battled to bound humanity on a geocentric earth, and while they did not prevail, they put up a good fight. One can even imagine a scenario where they might have prevailed. The mathematicians and astronomers who contributed to the accomplishment did not live in a vacuum and were very much affected by their cultural environment. Shifting the Earth describes of the interplay between the forces tugging at the contributors in different directions and how the contributors ultimately created a prevailing force of their own.

The question of a heliocentric versus geocentric structure was indisputably resolved in the seventeenth century, a century steeped in the controversies of the Counter-Reformation. The accomplishment was the conduit to the Enlightenment and the subsequent scientific–industrial revolution that has transformed our lives. The accomplishment has received short shrift within the historical literature where philosophy lies at the center of historical evolution. Shifting the Earth holds the perspective that since Newton proposed his laws of motion with the explicit intent of determining a planet's pathway, the scientific revolution has been the main driver of history. Philosophy has become a backseat passenger that must either adapt to the discoveries of the scientists, or be forcibly removed by another philosophy that wants to get on board. Starting with the ancient Greeks, Shifting the Earth examines this transition as it unfolds.

Since the investigation into the universal order began, mathematics has guided the investigators and has also been the language through which they convey their results. This is true for geocentric theories as well as the development of the heliocentric theory. Ptolemy's universe rests firmly on Euclid's mathematical foundation, which Ptolemy skillfully exploits to describe heaven's trajectories. Fifteen centuries later, Isaac Newton's universe rests firmly on Newtonian calculus and axioms of motion. In the interim, mathematics wielded in the hands of mathematicians and astronomers was the prodding rod shepherding humanity from a geocentric to a heliocentric world. Accordingly, mathematics is central to the story and occupies a central role in Shifting the Earth. Each chapter includes a mathematical treatment of significant contributions to both the geocentric and heliocentric theories.

This book encompasses two areas, history and mathematics, which our culture normally segregates. There is no delineated boundary between these areas. History subsumes mathematics, and mathematics influences history, so the segregation is as unnatural as separating water from broth. Both constitute the chicken soup, and with the same sentiment, I uphold that history and mathematics both constitute this story.

As a preview, let me provide an offering that highlights the interplay between the history and the mathematics. The preeminent European scientist of his day, Johannes Kepler, using methods that Archimedes developed two millennia before Kepler, asserts conservation of angular momentum in integral form, and uses this assertion to hunt down the ellipse. With his discovery of the ellipse as the planetary pathway, Kepler becomes Europe's most formidable advocate of a heliocentric system. Kepler is a celebrity throughout Europe, and the Jesuit community in Vienna woos Kepler, an excommunicated Lutheran, with the hope of converting him to Catholicism. During this same time, the Jesuits in Rome were leading an Inquisition against Galileo, charging him with heresy on the basis of his arguments in favor of a heliocentric universe.

The situation is pregnant with both mathematical and historical questions. What mathematics did Archimedes develop two millennia before Kepler that assisted Kepler with his discovery? If the mathematics had been in place for so long, why did it take two from the time of Archimedes to discover the true heliocentric system? Why did the Lutherans excommunicate Kepler? Why did the Jesuits woo Kepler, while at the same time persecute Galileo? Answers to these questions require a historical as well as mathematical response. Our story must address all of these questions, or the story will be incomplete.

Concerning the mathematics, I had a few choices to make. A first decision was to determine which material to include. The combined works of Copernicus, Kepler, Galileo, and Newton are well over 2000 pages. Add to that the works of Eudoxus, Aristarchus, Archimedes, Apollonius, Hipparchus, and Ptolemy, and one can conceive of over 3000 pages of material. Clearly, this had to be pared down. I applied subjective judgment weighing two objectives, historical significance, and direct significance, to the completion of a heliocentric theory. The result is that I've sheared the mathematical hedge into a shape that pleases me, recognizing that others would do so differently.

My subjective judgment is guided by the manner in which the discovery of the ellipse as the planetary pathway unfolds. Looking backward through time, Kepler discovered the pathway for the planet Mars using Tycho Brahe's observations. Both Copernicus and Ptolemy also present their models of Mars' motion, and there is a chain that links all three works. The presentation focuses on the planet Mars so that the links in the chain are made visible.

Just as there is no delineation between mathematics and history, there is no distinct historical line that marks the beginning or the end of this story. I chose to begin with Eudoxus, a contemporary of Plato and Aristotle who gives the first known systematic, mathematical model of the motion of the heavens in the ancient Western world. One could begin at an earlier time, looking back to the Babylonians, or a later time, with Ptolemy. The former was excluded because of a blur in our knowledge of the influence of pre-Eudoxian astronomy on Eudoxus. On the mathematical front, while Eudoxus' model is not a visible link in the Ptolemaic–Copernican–Keplerian chain, it initiates Western mathematical analysis of the universe's motions and as such it is that chain's invisible anchor. On the historical front, it was the Athenian culture that fostered not only Eudoxus but also Ptolemy. To delve into Ptolemy without giving some historical context is akin to a read of Abraham Lincoln's signature Gettysburg Address without an awareness of the issues causing the Civil War. I begin with Eudoxus and Athens because they leave their signature on all that follows.

As for the story's endpoint, the decision was more difficult. Had I written this book before the twentieth century, the distinct historical line marking the end would have been unmistakable. Newtonian reasoning was the final assault that deflated the geocentric myth and launched Europe into the Enlightenment. So successful were Newton's laws and so powerful were the mathematical tools he bequeathed that another myth displaced the geocentric myth; all would be understood by Newtonian reasoning. The twentieth century revealed the existence of a distinct historical Newtonian line but also that the universe is more than the physical manifestation of Newtonian reasoning. We now know that the universe behaves in ways that completely challenge our notions of what is reasonable. This realization is inherent in our story where our balding man looks at himself in the mirror and then outwardly toward the universe and sees previously unimagined possibilities in both. Einstein was the first to expose a bizarre universe and convince a mystified world of its reality. Einstein is where I choose to end the story.

While I believe that mathematics is central to the story, popular sentiment may disagree. Showing a flexible disposition, I accommodate popular sentiment by writing in a style that allows readers to engage the mathematics to their level of comfort. If you do not find the mathematics engaging, don't fret, skip over the mathematics and enjoy the narrative.

Let me finish this preface by addressing the reader who is interested in following some or all of the mathematical content. A decision point in presenting mathematical material concerns the presentation of works that predate modern mathematical concepts and notation. I had to decide whether to derive results using only the tools available to the original authors or make full use of modern mathematics. Clarity of exposition guided my decision. So that their bosses in the State Department can understand their reports, a China specialist writes in English. Similarly, I convey results using modern concepts and notation that are familiar to the present-day reader.

Another decision that is inherently bound with the choice of material, addresses the mathematical prerequisites. My efforts are aimed at reaching as broad an audience as possible, and that means keeping the prerequisites at a minimum. In all chapters except Chapters 2 and 4, I endeavor to present material at its simplest level. This means that whenever a conflict between presenting elementary mathematics or more technically advanced mathematics arises, I choose the path of an elementary exposition even if it is the less succinct. For example, I give a mathematical presentation of Newton's laws without the use of calculus. Indeed, with the exception of Chapters 2 and 4, the mathematical material is accessible to a high school student. Okay, I confess, I couldn't resist giving a calculus-based proof of the ellipse. With the exceptions of Chapter 2, Chapter 4, and the calculus-based section of Newtonian mechanics in Chapter 8, all of the material is accessible to a high school student.

Concerning Chapter 2, this material is somewhat tangential to the remainder of the book. Chapter 2 presents Eudoxus' universe. Eudoxus' work is all the more remarkable in that he accomplished his task prior to the development of foundational mathematics that I needed to describe it. With college-level mathematics, the model becomes tractable. Without college-level mathematics, I couldn't imagine how one gives a mathematical description, let alone how one solves the problem. I had two choices, not include the work or use higher-level mathematics, and for reasons described above, I chose the latter.

Chapter 4 presents work from Apollonius' On Conics. As with Eudoxus, this work is tangential to the remaining material and far ahead of its time. While this material could be presented at an elementary level without the use of calculus, in this one instance I chose to be succinct. Aside from succinctness, the use of calculus allows for an exploration into concepts that Apollonius hints at, but are now fully developed. The mathematical material of Chapters 2 and 4 is independent of the material in the rest the book. The reader without the prerequisites may skip these sections and follow everything else.

On a final note, I was quite pleased by the accessibility of special relativity. One can give a rather complete account of the mechanical side of special relativity and remain fully bounded within a standard high school mathematics curriculum. In fact, key findings of Einstein, those that assault our common sense, are more readily accessible than the fully intuitive works of Newton. It is my hope that Shifting the Earth is an agent of the assault.

Acknowledgments

By and large, this book is not the result of my individual effort, but the circumstances in which I find myself; indulge my wish to give details. There is a nasty side of human character that takes pleasure in causing jealousy. While I like to think myself above it, I'm not. My family feeds my nasty side, for anyone who knows my wife, Lijuan, and children, Julius and Amelia, would most certainly be jealous of me. How could I write a book and hold down a job without my family behind me all the way? Lijuan gives me a pass on much of the honey-do list; our bushes perpetually look like they've had a bad-hair day. Julius and Amelia root me on and give motivation. Despite their full awareness that the question “Dad, how's your book going?” will likely lead to an enthusiastic response detailing the adversity that Kepler overcame in discovering the ellipse, a story they've been abused by more than any human deserves, they ask anyway. Then when thoughts of avoiding my writing and spending the evening watching The Simpsons™ surface, the mood to write is restored.

During the writing of this book, I had a chance to catch up with an elementary school friend, Joel Sher. I'll spare the embarrassment of the number of decades that have passed since we last met, but despite the time lapse, Joel welcomed me into his home. He gave me exclusive use of the second floor of his house for a week. It came at a critical point when I really needed the time and solitude to get the job done. Joel is as I remembered him, a good friend who is always willing to help out.

It is difficult to find Kepler's New Astronomy. I offer my thanks to Beena Morar, who not only located a copy, but also made sure the copy was exclusively available to me throughout the writing of this book.

Also, I would like to thank Professor Roger Cooke from the Mathematics Department at The University of Vermont. The mathematics editor at Wiley, Susanne Steitz-Filler, forwarded my proposal to Professor Cooke for him to review. Professor Cooke's careful line-by-line audit of the proposal was not what one would want from an IRS auditor, but exactly what one hopes for in a reviewer. His spotting of errors allowed me to pull my pants up from my ankles before going out in public. I became quite awed by Professor Cooke's knowledge. Whereas I read translations of the original Greek and Latin writings, Professor Cooke reads the original works. Not only did he point out a gaffe in a mathematical analysis; he also noted my poor theological interpretation of Ptolemy that resulted from a poor translation. His own translation and explanation of the original work influenced me to rewrite my own work. There were some disagreements between myself and Professor Cooke. If my pants are not fully buckled, I have to take responsibility, for Professor Cooke gave warning.

Finally, I would like to thank the people at Wiley and Susanne Steitz-Filler in particular. I have found my interactions with Wiley very professional and a pleasure. Susanne Steitz-Filler's support for this project is most gratifying. Without Susanne, this book would not be.

Chapter 1

Perfection

I am the circle. Within a plane, I am the set of points equidistant from a designated center. I am the perfect shape, and I boast a list of properties that no other shape can lay claim to.

Every point of my composition is equal. Segments containing midpoints and corners form the triangle, square, and every other polygon. An ellipse has two points closest to its center and another two points farthest from its center. A crescent has two vertices as well as midpoints to each of its arcs. Indeed, every shape other than mine contains points of special character, altering their status among other points within the assembly. Only I am composed of points that are all truly equal; the equality defines beauty along with perfection.

As all my points are equal, I am the only planar shape that remains unchanged by any rotation of any angle. Rotate a square by a multiple of other than 90 degrees, and it is obvious that you have altered its orientation. But rotate me through any angle, and you will perceive no change in my perfect and beautiful configuration.

I form the optimal boundary of an enclosure because the area enclosed by me is the largest area that can be enclosed within a boundary of fixed length. A triangle, square, ellipse, crescent, or any other shape with a perimeter that is the same length as my circumference, encloses a smaller area than mine. Perfection is optimal as well as beautiful.

Every line through my center forms a line of reflective symmetry about my center. Place a mirror perpendicular to any line about my center, and the semicircle in front of the mirror is self-complementary so that its image with itself forms a perfect circle. Attempt this with an equilateral triangle, and unless the mirror is perfectly placed on one of only three lines of symmetry, the component in front of the mirror is not self-complementary. An ellipse has but two lines of symmetry through its center; an isosceles triangle and crescent have only one; and a scalene triangle, like most arbitrary shapes, has none. This infinite set of symmetries expresses itself only through my perfect shape in which every point is equal.

Inscribe an equilateral triangle within me, and every line of symmetry of the triangle is also a line of symmetry of mine. This is true for a square, an octagon, a dodecagon, a heptagon, a myriagon, or any other regular polygon, one with sides of equal length. Perfection dominates the imperfect as I dominate the polygons.

My dominance of the polygons also expresses itself in the attempt of a set of regular polygons to reach perfection. Order the regular polygons by the number of sides. Then, as one indefinitely climbs up the hierarchy, the shape of the polygons approaches my perfection. Indeed, by going far enough along the hierarchy to a target polygon, the points of all the subsequent polygons can be made as close as any arbitrarily small distance to my points. While they come close to me, only an insignificant fraction of the points settle on my perfect frame. I dominate the polygons. As they strive to reach me, they strive for perfection, but can never attain it.

My perfection inspires humans in all their endeavors. For three millennia engineers have used me to transport materials across the land whether by horse-powered cart, human-powered bicycles, steam-powered locomotive, or diesel-powered 18-wheelers. Circular gears drive mechanical devices; any other shape would cause uneven wear on the equipment. Electric power generators rotate through a circle yielding controllable voltage, current, and power output.

My symmetry inspires scientists to search for symmetry in nature. They have discovered the cyclic nature of time and stamped its daily rhythms on a circular clock. In the Northern Hemisphere, the North Star is a fixed center about which all other stars rotate along a circular pathway. Toss a stone into a lake, and a scientist confirms that the waves ripple across the lake's surface in concentric circles. I provide the pattern for the eye of a hurricane and the rainbow.

Just as mathematicians have discovered an impressive list of properties that I possess, artists and architects pay homage to my beauty. The artist adorns figures of religious admiration with a halo. Light enters a house of warship through a circular stained-glass window.

As I set the standard of equality, I am the foundation of many religious and political philosophies. Monotheism places all humankind equally about God; men are the points of a circle with God as their center. Both communism and democracy strive for the equality of citizens, one through an equal distribution of goods, the other through representative government via universal plebiscite in which every citizen has an equal vote in an electoral process.

I am a universal ideal pursued by engineers, scientists, mathematicians, artists, and philosophers. They pursue me because of my perfection. But like the polygons, humankind's pursuit is in vain for perfection is impossible to attain and so easy to destroy. A bicycle wheel is never in perfect true. Lay it on its side, and it doesn't evenly rest on its surface. Even with the naked eye, one can perceive a blip as it rotates. By adjusting the tension in the spokes, one can reduce a blip, but never eliminate it, and during the course of adjustment, a new blip always arises. Once close to true, a small bump in the road perturbs the wheel yet farther from perfection.

Human philosophical and social efforts toward a perfect circle of equality meet with similar road bumps. A system of privilege blanketed the democratic ideals of ancient Athens as citizenship was limited to a small select group. Others had limited or no rights, while some of the others were no more than the property of their slave-possessing owners. Over 2000 years after Athens' zenith, an assembly of men in the city of Philadelphia founded an independent nation vested on the circle of equality with God at its center as evidenced by the words of America's Declaration of Independence:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

Echoing Athens, many of the very signers of this esteemed document left Philadelphia and returned to their estates where slaves who enriched America's founders were not among the equally created men, but were property.

It is not surprising that my perfect ideal cannot be achieved by human beings who are endowed with such imperfect character; human beings' character imperfections leach into their engineering, arts, philosophies, and institutions. So these very humans, with their imperfect character, have looked to nature to find me. Their first impressions of me as a true physical entity reflect their naivety and the immaturity of their science. Neither the stars' apparent path, the concentric waves, nor the rainbow's arc achieve my perfection. Just as a bump on the road perturbs the wheel away from true, nature introduces a wobble on the earth's axis perturbing the apparent path of the stars, and ripples on a lake perturb the concentric waves, and nonuniformity of the atmosphere perturbs the rainbow's arc. Not only am I too perfect for humankind; I am also too perfect for nature. I exist not as a fact, but as an ideal.

It is through the investigation into planetary motion that human beings revealed my true status as Utopian and not material. Humans unveiled the ellipse and replaced my perfection with its form. The scientists and mathematicians who contributed to the unveiling of the ellipse did so not in a vacuum, but with preconceived notions seeded in their instinct as well as those that society stamped on them. It was a monumental achievement to overcome humanity's obsession with perfection and find truth.

While scientists and mathematicians led the investigation, they were not the only participants. Because of the ramifications of this discovery on philosophy, religion, religious institutions, and the relation between the state and religious institutions, the highest levels of the ecclesiastic community weighed in on the debate attempting to influence its outcome. In doing so, they exposed themselves, and all except the most fervent noted that not only was the pathway of the heavens imperfect, but the institutions guiding humanity's moral precepts were themselves following an imperfect path. With the discovery of the ellipse, humans evicted me from their view of nature and their perception of their own self and institutions.

I am an ideal worth pursuing, but in pursuing me humanity has discovered a far more diverse universe with a far more interesting set of possibilities than I have to offer. Humanity has also established institutions that, however imperfect, are far more dynamic than those that claimed perfection. Shifting the Earth tells the story of how this unfolded.