A Brief Guide to the Great Equations
Robert P. Crease
is a professor in and chairman of the Department of Philosophy at Stony Brook University in New York, and historian at Brookhaven National Laboratory. He writes a monthly column, ‘Critical Point’, for
Physics World
magazine. His previous books include
The Prism and the Pendulum: The Ten Most Beautiful Experiments in Science
;
Making Physics: A Biography of Brookhaven National Laboratory
;
The Play of Nature: Experimentation as Performance
;
The Second Creation: Makers of the Revolution in Twentieth-Century Physics
(with Charles C. Mann); and
Peace & War: Reminiscences of a Life on the Frontiers of Science
(with Robert Serber). Crease’s translations include
American Philosophy of Technology: The Empirical Turn
and
What Things Do: Philosophical Reflections on Technology, Agency, and Design
. He lectures widely, and his articles and reviews have appeared in
The Atlantic Monthly
,
The New York Times Magazine
,
The Wall Street Journal
,
Science
,
New Scientist
,
American Scientist
,
Smithsonian
, and elsewhere. He lives in New York City.
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A BRIEF GUIDE TO
THE GREAT
EQUATIONS
The hunt for cosmic beauty in numbers
Robert P. Crease
Constable & Robinson Ltd
55–56 Russell Square
London WC1B 4HP
www.constablerobinson.com
First published in the USA 2008 as
The Great Equations:
Breakthroughs in Science from Pythagoras to Heisenberg
by W.W Norton & Company, Inc.,
500 Fifth Avenue, New York, NY 10110
This edition published by Robinson, an imprint of Constable & Robinson Ltd, 2009
Copyright © Robert P. Crease 2008, 2009
The right of Robert P. Crease to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act, 1988.
All rights reserved. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out or otherwise circulated in any form of binding or cover other than that in which it is published and without a similar condition being imposed on the subsequent purchaser.
A copy of the British Library Cataloguing in Publication Data is available from the British Library.
ISBN: 978-1-84529-281-2
eISBN: 978-1-47210-017-7
1 3 5 7 9 8 6 4 2
Printed and bound in the EU
beyond category
CONTENTS
1 ‘The Basis of Civilization’: The Pythagorean Theorem
Interlude: Rules, Proofs, and the Magic of Mathematics
2 ‘The Soul of Classical Mechanics’: Newton’s Second Law of Motion
3 ‘The High Point of the Scientific Revolution’: Newton’s Law of Universal Gravitation
4 ‘The Gold Standard for Mathematical Beauty’: Euler’s Equation
5 The Scientific Equivalent of Shakespeare: The Second Law of Thermodynamics
Interlude: The Science of Impossibility
6 ‘The Most Significant Event of the Nineteenth Century’: Maxwell’s Equations
Interlude: Overcoming Anosognosia; or Restoring the Vitality of the Humanities
8 The Golden Egg: Einstein’s Equation for General Relativity
9 ‘The Basic Equation of Quantum Theory’: Schrödinger’s Equation
Interlude: The Double Consciousness of Scientists
10 Living with Uncertainty: The Heisenberg Uncertainty Principle
Interlude: The Yogi and the Quantum
Conclusion: Bringing the Strange Home
Excerpts from the column, ‘Critical Point’, by Robert P. Crease, are reprinted from
Physics World
with the kind permission of the publisher.
Page
24
: Columbia University.
Pages
25
,
28
,
30
,
39
,
73
,
99
,
101
,
104
,
201
: John McAusland.
Page
43
: Robert P. Crease.
Page
58
: AIP Galilei Galileo A9.
Page
59
: AIP Newton Isaac A6.
Page
83
: AIP Newton Isaac H5.
Page
94
: AIP Euler A1.
Page
112
: Boltzmann Ludwig A4; AIP Clausius Rudolf A3; AIP Helmoltz Hermann A2; AIP Carnot Sadi A1; AIP Joule James A3; AIP Maxwell James Cleak A5; AIP Rumford Benjamin H2; AIP Emilio Segre Visual Archives, E. Scott Barr Collection; AIP Kelvin William Thomson A16; AIP Planck Max A14; Wien, Wilhelm A1.
Page
134
: AIP Maxwell James Clerk A5.
Page
140
: Maxwell, James Clerk.
A Treatise on Electricity and Magnetism, vol.
1
. Oxford: 1873.
Page
157
: Sidney Harris.
Page
167
:
Physics Today
, January 2006.
Page
179
:
Time
magazine.
Page
221
: AIP Schrodinger, Erwin A10.
Page
237
: Heisenberg Werner A15.
The first equation that most of us learn is a synonym for simplicity:
So elementary, yet so powerful! It imparts the very definition of addition: one unit plus one unit equals two units. It is powerful, too, because it exhibits the format for every other equation: in arithmetic, mathematics as a whole, physics, and other branches of science. It shows an arrangement of terms that asserts a particular kind of relationship among them. This little but fundamental equation opens so many doors that it seems like a magic wand. It is virtually
the
entrée into knowledge itself – the first little step, the basis for each of thousands of steps to follow. Richard Harrison, a poet and English teacher at Mount Royal College in Calgary, Canada, once wrote to me of this profound expression:
1 + 1 = 2 is the fairy tale of mathematics, the first equation I taught my son, the first expression of the miraculous power of the mind to change the real world. I remember my son holding up the index finger – the ‘one finger’– of each hand as he learned the expression, and the moment of wonder, perhaps his first of true philosophical wonder, when he saw that the two fingers, separated by his whole body, could be joined in
a single concept in his mind... [W]hen I saw my son’s mind open in understanding that ‘1 + 1’ was more than ‘1 + 1’ I saw that small equation as my child’s key not to what was wonderful in the outside world, but what was wonderful in him and all of us.
Harrison’s description reminds us that learning an equation, at least of the kind as fundamental as 1 + 1, is in effect a kind of journey. It is a journey that takes place in three stages. We begin naively without knowing the equation. We are led by schooling or accident or curiosity or intent to comprehend it, often accompanied by dissatisfaction and frustration. Finally, the experience of learning it transforms the way we experience the world, which fills us – naturally, if sometimes only momentarily – with wonder.
This book is about those journeys.
The first human beings lived without equations, and had no need for them. There were no equations in the Garden of Eden, not even on the Tree of Knowledge. None were present in the Sumerian paradise Dilmun, nor in the cosmic egg in which some Chinese believe that P’an Ku hatched the world, nor in any of the other places where various divine creation myths say that the first humans dwelt. Human beings did not even have the
idea
of equations. That idea is a human invention, the result of our efforts to make sense of the world. Even so, human beings did not wake up one day and suddenly decide to invent equations. They acquired the need to over time, and the idea of an equation in the scientific-technical sense first appeared late in human history.
The Latin word
aequare
means to make level or even. Many modern English words spring from this root, including adequate, equanimity, equality, equilibrium, egalitarian, equivalence, and equivocation. The word ‘equation’ at first simply meant a partitioning into equal groupings. The ‘equator’, for instance, is the
imaginary line drawn by geographers to separate the earth in two roughly equal halves. Medieval astrologers used the word ‘equation’ to refer to their practice of arbitrarily dividing up the path followed by the sun and planets into equal areas, each allegedly governed by a different constellation.
1
Meanwhile, numbers and counting were becoming important in human life. Businessmen used them in bookkeeping, finance, and budgets; religious authorities used them for record-keeping of years, seasons, and occasions such as births, deaths, and marriages; and in government officials used them in census, and for surveying and taxes.
2
This generated the need to develop symbols to stand for numbers and quantities.
3
In the third century
BC
, the Greek mathematician Diophantus took another step, using symbols to stand for
unknown
quantities, and providing some rules for operating on such quantities, including subtraction and addition. He showed not only how to use symbols to describe an unknown number so that it could be determined from known numbers (what is called a determinate equation) but also how symbols could describe something with an infinite set of solutions (a Diophantine or indeterminate equation). It was still a long route to the modern notion of equations. Even Galileo and Newton express their important results – Galileo’s law of falling bodies, and Newton’s laws of motion – in the form of ratios expressed in words, not in the familiar equation form known to science students. Not until the eighteenth century did natural scientists routinely express their conclusions in the form of equations as we know them today.