Fluent Essays

Ive attached questions that need to be answered, using the book and the discussion.follow the instructions of the question and do not use outside resources discussion.jpg questions.jpg five_equations_that_changed_the_world.pdf Unformatted Attachment Preview ~ie Equaioot ,La [LaogeJ ,Le WorlJ Bridges to Infinity: The Human Side of Mathematics ~ie ~quafiont fLaf (LangeJ fLe WorlJ The Power and Poetry of Mathematics ~ Mi~~ael Guillen, P~.D. Ii!HYPE RIO New York NI Copyright ti::l1995, Dr. Michael Guillen All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without the written permission of the Publisher. Printed in the Umted States of America. For information address: Hyperion, 114 Fifth Avenue, New York, New York 10011. Library of Congress Cataloging-In-Publication Data Guillen, Michael. Five equations that changed the world: the power and poetry of mathematics I by Michael Guillen.-lst ed. p. cm. Includes index. ISBN 0-7868-8187-9 1. Physics-Popular works. 2. Equations. I. Title. QC24.5.G85 1995 530.15~c20 95-15199 CIP Designed by Chris Welch FI1IST PAPERBACK EDITION 7 9 10 8 6 To L.urel, .Lo ~..~ DI~ .orlJ lor .La I.eHer Acknowledgments For their exceptional talent and tenacity, I wish to thank my researchers, Noe Hinojosa, Jr., Laurel Lucas, Miriam Marcus, and Monya Baker. For his extraordinary patience, friendship, and wisdom, I thank my literary agent, Nat Sobel. Also, for their enthusiasm, constructive comments, and support, special credit goes to my publisher, Bob Miller, and editor, Brian DeFiore. For their invaluable assistance, advice. and encouragement, I am indebted to: Barbara Aragon. Thomas Bahr, Randall Barone. Phil Beuth. Graeme Bird. Paul Cornish (British Information Services). Stefania Dragojlovic. Ulla Fringeli (Universitat Basel), Owen Gingerich. Ann Godoff, Heather Heiman. Gerald Holton, Carl Huss, Victor Iosilevich. Nancy Kay, Allen Jon Kinnamon (Cabot Science Library, Harvard University). Gene Krantz, Richard Leibner, - vii viii Martha Lepore, Barry Lippman, Stacie Marinelli, Martin Mattmiiller (Universitatsbibliothek Basel), Robert Millis, Ron Newburgh, Neil Pelletier (American Horticultural Society), Robert Reichblum, Jack Reilly, Diane Reverand, Hans Richner (Swiss Federal Institute of Technology), William Rosen, Janice Shultz (Naval Research Laboratory), John Stachel (Boston University), Rabbi Leonard Troupp, David Vale (Grantham Museum), Spencer Weart (American Institute of Physics), Richard Westfall, L. Pearce Williams, Ken Yanni (Hoover Dam), and Allen Zelon. If, despite the aid and comfort of these gracious people, I have made any errors, they are entirely my fault, and I thank the vigilant readers who will surely set me straight. [onlenl• .. MafLemaHeal Poefry 1 Introduction Applet anJ Orangel 9 Isaac Newton and the Universal Law oj Gravity F = G xM x m -7- d 2 Defween a RoeL anJ a DarJ lile 65 Daniel Bernoulli and the Law oj Hydrodynamic Pressure P + P x \% v 2 = CONSTANT [Iall Aef 119 Michael Faraday and the Law oj Electromagnetic Induction V x E = -aBlat An UnprohfaLle Ixperienee 165 Rudolf Clausius and the Second Law oj Thermodynamics as universe> 0 [oriotify IblleJ fLe ligLft 215 Albert Einstein and the Theory oj SpedaZ Relativity E=m xc2 InJex 267 Introduction Mll.Lemllti~lll Poeh-y • Poetry is simply the most beautifUl, impressive, and widely effective mode of saying things. MATTHEW ARNOLD M ath~matics is. a lan~age w~~se importance I c~ best ex- plam by startmg WIth a familiar story from the Blble. There· was a time, according to the Old Testament, when all the people of the earth spoke in a single tongue. This unified them and facilitated cooperation to such a degree that they undertook a collective project to do the seemingly impossible: They would build a tower in the city of Babel that was so high, they could simply climb their way into heaven. It was an unpardonable act of hubris , and God was quick to visit his wrath on the blithe sinners. He spared their lives, but not their language: fu described in Genesis 11:7, in order to scuttle the blasphemers enterprise, all God needed to do was confound their language, that they may not understand one anothers speech. Thousands of years later, we are still babbling. According to lin- 1 ~i.e ~quatioot .La D.aofjeJ .Le WorlJ 2 guists, there are about 1,500 different languages spoken in the world today. And while no one would suggest that this multiplicity of tongues is the only reason for there being so litde unity in the world, it certainly interferes with there being more cooperation. Nothing reminds us of that inconvenient reality more so than the United Nations. Back in the early 1940s, when it was first being organized, officials proposed that all diplomats be required to speak a single language, a restriction that would both facilitate negotiations and symbolize global harmony. But member nations objected-each loath to surrender its linguistic identity--so a compromise was struck; United Nations ambassadors are now allowed to speak anyone offive languages: Mandarin Chinese, English, Russian, Spanish, or French. Over the years, there have been no fewer than 300 attempts to invent and promulgate a global language, the most famous being made in 1887 by the Polish oculist L. L. Zamenhof The artificial language he created is called Esperanto, and today it is spoken by more than 100,000 people in twenty-two countries. However, as measured by the millions of those who speak it fluendy and by the historic consequences of their unified efforts, mathematics is arguably the most successful global language ever spoken. Though it has not enabled us to build a Tower of Babel, it has made possible achievements that once seemed no less impossible: electricity, airplanes, the nuclear bomb, landing a man on the moon, and understanding the nature of life and death. The discovery of the equations that led ultimately to these earthshaking accomplishments are the subject of this book. In the language of mathematics, equations are like poetry: They state truths with a unique precision, convey volumes of information in rather brief terms, and often are difficult for the uninitiated to comprehend. And just as conventional poetry helps us to see deep within ourselves, mathematical poetry helps us to see far beyond ourselves-if not all the way up to heaven, then at least out to the brink of the visible universe. MatLemalieal Poefry 3 In attempting to distinguish between prose and poetry, Robert Frost once suggested that a poem, by definition, is a pithy fonn of expression that can never be accurately translated. The same can be said about mathematics: It is impossible to understand the true meaning of an equation, or to appreciate its beauty, unless it is read in the delightfully quirky language in which it was penned. That is precisely why I have written this book. This is not so much an offspring of my last book, Bridges to Infinity: The Human Side ifMathematics, as it is its evolutionary descendant. I wrote Bridges with the intention of giving readers a sense of how mathematicians think and what they think about. I also attempted to describe the language-the numbers, symbols, and logic-that mathematicians use to express themselves. And I did it all without subjecting the reader to a single equation. It was like sweet-tasting medicine offered to all those who are afflicted with math anxiety, individuals who nonnally would not have the courage or the curiosity to buy a book on a subject that has consistently frightened them away. In short, Bridges to Infinity was a dose of mathematical literacy designed to go down easily. Now, emboldened by having written a successful book that contains no equations, I have dared to go that one step further. In this book I describe the mathematical origins of certain landmark achievements, equations whose aftereffects have pennanently altered our everyday lives. One might say I am offering the public a stronger dose of numeracy, an opportunity to become comfortably acquainted with five remarkable fonnulas in their original, undisguised forms. Readers will be able to comprehend for themselves the meaning of the equations, and not just settle for an inevitably imperfect nonmathematical translation of them. Readers of this book also will discover the way in which each equation was derived. Why is that so important? Because, to paraphrase Robert Louis Stevenson: When traveling to some exotic destination, getting there is half the fun. Ii~e Iquafioo, tLat (LaotjeJ tLe WorlJ 4 I hope that the innumerate browser will not be scared offby the zealousness of my effort. Rest assured, though these five equations look abstract, most certainly their consequences are not-and neither are the people associated with them: a sickly, love-starved loner; an emotionally abused prodigy from a dysfunctional family; a religious, poverty-stricken illiterate; a soft-spoken widower living in perilous times; and a smart-alecky, high school dropout. Each story is told in five parts. The Prologue recounts some dramatic incident in the main characters life that helps set the tone for what is to follow. Then come three acts, which I refer to as Veni, Vidi, Vici. These are Latin words for I came, I saw, I conquered, a statement Caesar reportedly made after vanquishing the Asian king Pharnaces. Veni is where I explain how the main character-the scientist-comes to his mysterious subject; Vidi explains historically how that subject came to appear so enigmatic; Vici explains how the scientist manages to conquer the mystery, resulting in a historic equation. Finally, the Epilogue describes how that equation goes on to reshape our lives forever. In preparing to write this book, I selected five equations from among dozens of serious contenders, solely for the degree to which they ultimately changed our world. Now, however, I see that the stories attached to them combine fortuitously to give the reader a rather seamless chronicle of science and society from the seventeenth century to the present. As it turns out, that is a crucial period in history. Scientifically, it ranges. from the beginning of the so-called Scientific Revolution, through the Ages ofReason, Enlightenment, Ideology, and Analysis, during which science demystified each one of the five ancient elements: Earth, Water, Fire, Air, and Ether. In that critical period of time, furthermore, we see: God being forever banished from science, science replacing astrology as our principal way of predicting the future, science becoming a paying profession, and science grappling with the ultra- MafLemafieal Poetry 5 mysterious issues of life and death and of space and time. In these five stories, from the time when an introspective young Isaac Newton sits serenely beneath a fruit tree to when an inquisitive young Albert Einstein nearly kills himself scaling the Swiss Alps, we see science wending its way from the famous apple to the infamous A-bomb. Which is to say, we see science going from being a source of light and hope to its also becoming a source of darkness and dread. Writers before me have chronicled the lives of some of these five scientists-all too often in frightfully long biographies. And writers before me have reconstructed the pedigree of some of these intellectual innovations back to the beginning of recorded history. But they have never focused their roving attentions on the small number of mathematical equations that have influenced our existence in such profound and intimate ways. The exception is Albert Einsteins famous energy equation E = m x Cl, which many people already know is somehow responsible for the nuclear bomb. But for all its notoriety, even this nefarious little equation remains in the minds of most people scarcely more than a mysterious icon, as familiar yet inexplicable as Procter & Gambles corporate logo. What exactly do the letters E, m, and c stand for? Why is the c squared? And what does it mean for the E to be equated with the m x Cl? The reader will learn the surprising answers in Curiosity Killed the Lights. The other chapters deal with scientists less well known than Einstein but who are no less important to the history of our civilization. Between a Rock and a Hard Life, for example, concerns the Swiss physicist Daniel Bernoulli and his hydrodynamic equation P + P X V2 t? = CONSTANT, which led ultimately to the modem airplane. Class Act is about the British chemist Michael Faraday and his electromagnetic equation V x E = -aBlat, which ultimately led to electricity. ~he tquafiont fLaf (LangeJ fLe WorlJ 6 Apples and Oranges tells the story of the British natural philosopher Isaac Newton and his gravitational equation F = G X M x m -7- d2-which led not to any specific invention but to an epic event: landing a man on the moon. Finally, An Unprofitable Experience is about the German mathematical physicist Rudolf Julius Emmanuel Clausius and his thermodynamic equation (or more accurately, his thermodynamic inequality) aSuniverse > O. It led neither to a historic invention or event but to a startling realization: Contrary to popular belief, being alive is unnatural; in fact, all life exists in defiance of, not in conformity with, the most fundamental law of the universe. In my last book, Bridges to lyifinity, I suggested that the human imagination was actually a sixth sense used to comprehend truths that have always existed. Like stars in the firmament, these verities are out there somewhere just waiting for our extrasensory imagination to spot them. Furthermore, I proposed that the mathematical imagination was especially prescient at discerning these incorporeal truths, and I cited numerous examples as evidence. In this book, too, readers will see dramatic corroboration for the theory that mathematics is an exceptionally super-sensitive seeingeye dog. Otherwise, how can we begin to account for the unerring prowess and tenacity with which these five mathematicians are able to pick up the scent, as it were, and zero in on their respective equations? While the equations represent the discernment of eternal and universal truths, however, the manner in which they are written is stricciy, provincially human. That is what makes them so much like poems, wonderfully artful attempts to make infmite realities comprehensible to fmite beings. The scientists in this book, therefore, are not merely intellectual explorers; they are extraordinary artists who have mastered the ex- tensive vocabulary and complex grammar of the mathematical language. They are the Whitmans, Shakespeares, and Shelleys of the quantitative world. And their legacy is five of the greatest poems ever inspired by the human imagination. F == G x M x m -:- d2 Applet anJ Oranget Isaac Newton and the Universal Law of Gravity I sometimes wish that God were back In this dark world and wide; For though some virtues he might lack, He had his pleasant side. -GAMAliEL BRADFORD ~ or the last several months, thirteen-year-old Isaac Newton had been watching with curiosity while workmen built a windmill just outside the town of Grantham. The construction project was very exciting, because although they had been invented centuries ago, windmills were still a novelty in this rural part of England. Each day after school, young Newton would run to the river and seat himself, documenting in extraordinary detail the shape, location, and function of every single piece of that windmill. He then would rush to his room at Mr. Clarkes house to construct miniature replicas of the parts he had just watched being assembled. As Granthams huge, multiarmed contraption had taken shape, therefore, so had Newtons wonderfully precise imitation of it. All that remained now was for the curious young man to come up 9 ~he tquatioot tLat (LaoljeJ tLe WorlJ 10 with something, or someone, to play the role of miller. Last night an idea had come to him that he considered brilliant: His pet mouse would be perfect for the part. But how would he train it to do the job, to engage and disengage the miniature mill wheel on command? That was what he had to puzzle out this morning on his way to school. As he walked along slowly, his brain raced toward a solution. Suddenly, however, he felt a sharp pain in his gut; his thoughts came to a screeching halt. As his minds eye refocused, young Newton came out of his daydream and beheld his worst nightmare: Arthur Storer, the sneering, taunting school bully, had just kicked him in the stomach. Storer, one of Mr. Clarkes stepsons, loved to pick on Newton, teasing him mercilessly for his unusual behavior and for fraternizing with Storers sister, Katherine. Newton was a quiet and selfabsorbed youngster, generally preferring the company of his thoughts to that of people. But whenever he did socialize, it was with girls; they were tickled by the doll furniture and ~ther toys he made for them using his customized kit of miniature saws, hatchets, and hammers. While it was common for Storer to call Newton a sissy, on this particular morning, he was insulting him for being so stupid. Unfortunately, it was true that Newton was the next-to-lowest ranking student in the whole of Granthams Free Grammar School of King Edward VI, seeded well below Storer. But the idea of this big bully thinking of himself as intellectually superior made the reclusive young mans thoughts turn from windmills to revenge. As he sat at the back of the class, Newton usually found it easy to ignore what his teacher, Mr. Stokes, was saying. This time, however, he listened with interest. The universe was divided into two realms, each obeying a different set of scientific laws, Stokes instructed. The imperfect, earthly region behaved one way, and the perfect, heavenly region behaved another; both domains, he Applet anJ Oranfjet - 11 added, had been successfully studied and their respective ordinances deduced long, long ago by the Greek philosopher Aristotle. For young Newton, suffering at the hands of an earthly imperfection such as Storer was proof enough of what Mr. Stokes was talking about. Newton hated Storer and his classmates for not liking him. Above all, he hated himself for being so unlikable that even his own mother had abandoned him. God was the only friend he had, the pious young man thought, and the only friend he needed. Newton was a much smaller person than Storer, but with Gods help, he certainly would be able to vanquish the offensive tormentor. No sooner had Mr. Stokes dismissed class that day than Newton was out the door, waiting in the nearby churchyard for the bully. Within minutes, a boisterous crowd of students gathered-round. Stokess son selected himself referee, slapping Newton on the back as if to encourage him, while winking at Storer as if to say this was going to be as entertaining as watching Daniel being fed to the lions. At first, no one cheered for young Newton. Instead, each time Storer landed a punch, the rowdy students whooped it up, egging on the ruffian to hit even harder the next time. When it seemed as if Newton had been beaten into submission, Storer straightened up and relaxed, grinning boastfully at his young peers. As he turned to walk away, however, Newton struggled to his feet: He was not about to let Storer win the right to lord over him for the rest of his life. Alerted by shouts of warning, Storer wheeled around and was greeted with a kick to his stomach and a punch to the nose; Newton had drawn blood, and that reinvigorated him. For the next several minutes, the two traded blows and wrestled one another to the ground. Time and again, Storer staggered away, thinking he had defeated Newton, only to be confronted anew. When it was all over, the crowd was stunned into silence. As the young referee stepped in to congratulate the bloodied and ex- ~he ~qualioD tLat (J.angeJ tLe WorlJ 12 hausted Newton, however, the dumbstruck students stirred and began to cheer: Daniel had become David, they declared jubilandy, as they danced around the fallen Goliath. Newton was more than satisfied with what he h ... Purchase answer to see full attachment