Aaron’s Academy of Beauty Power of Poetry of Mathematics Book Questions I want to solve all of these questions using the book that I have attached.The instructions and questions in “Questions” file attached.and the book is “Book.pdf” file attached ~i”e Equa’ioot
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Bridges to Infinity: The Human Side of Mathematics
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The Power and Poetry of Mathematics
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
Five equations that changed the world: the power and poetry of
mathematics I by Michael Guillen.-lst ed.
1. Physics-Popular works. 2. Equations. I. Title.
Designed by Chris Welch
FI1IST PAPERBACK EDITION
.Lo ~..~ DI~ .orlJ lor .La I.eHer
For their exceptional talent and tenacity, I wish to thank my researchers, Noe Hinojosa, Jr., Laurel Lucas, Miriam Marcus, and
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,
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.
Applet anJ Orangel
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
Michael Faraday and the Law oj Electromagnetic Induction
V x E = -aBlat
An UnprohfaLle Ixperienee
Rudolf Clausius and the Second Law oj Thermodynamics
[oriotify IblleJ fLe ligLft 215
Albert Einstein and the Theory oj SpedaZ Relativity
Poetry is simply the most beautifUl,
impressive, and widely effective mode of
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 another’s speech.”
Thousands of years later, we are still babbling. According to lin-
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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.
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
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 character’s 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-
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 Einstein’s 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 &
Gamble’s 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.
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“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
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
In this dark world and wide;
For though some virtues he might
He had his pleasant side.
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. Clarke’s house to construct
miniature replicas of the parts he had just watched being assembled.
As Grantham’s huge, multiarmed contraption had taken shape,
therefore, so had Newton’s wonderfully precise imitation of it. All
that remained now was for the curious young man to come up
~h’e tquatioot tLat (LaoljeJ tLe WorlJ
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 mind’s 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. Clarke’s stepsons, loved to pick on Newton,
teasing him mercilessly for his unusual behavior and for fraternizing with Storer’s 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 Grantham’s 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 man’s 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
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 God’s 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.
Stokes’s 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
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-
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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…
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