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The Project Physics Course

Reader

The Triumph of Mechanics

%}m

The Project Physics Course

Reader

UNIT

3 The Triumph of Mechanics

A Component of the
Project Physics Course

Published by

HOLT, RINEHART and WINSTON, Inc.

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This publication is one of the many
instructional materials developed for the
Project Physics Course. These materials
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many colleagues listed in the text units.

Directors of Harvard Project Physics

Gerald Holton, Department of Physics,

Harvard University
F. James Rutherford, Capuchino High School,

San Bruno, California, and Harvard University
Fletcher G. Watson, Harvard Graduate School

of Education

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(5) Portrait of Pierre Reverdy by Pablo Picasso.
Etching. Museum of Modern Art, N.Y.C.

(6) Lecture au lit by Paul Klee. Drawing. Paul Klee
Foundation, Museum of Fine Arts, Berne.

Piclure Credits

Cover picture: "Deluge." Drawing by Leonardo
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5 I

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Double-page spread on following pages:

(1) Photo by Glen J. Pearcy.

(2) Jeune fille au corsage rouge lisant by Jean
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(3)
(4)

Sources and Acknowledgments
Project Physics Reader 3

1. Silence, Please from Tales From the White Hart
by Arthur C. Clarke. Reprinted with permission of
the author and his agents Scott Meredith Literary
Agency, and David Higham Associates, Ltd.

2. The Steam Engine Comes of Age from A History
of Science and Technology by R. J. Forbes and
E. Dijksterhuis, copyright © 1963 by Penguin
Books, Ltd. Reprinted with permission.

3. The Great Conservation Principles from The
Character of Physical Law by Richard P. Feynman,
copyright © 1965 by Richard P. Feynman. Pub-
lished by the British Broadcasting Corporation
and The M.I.T. Press. Reprinted with permission.

4. The Barometer Story by Alexander Calandra
from Current Science, Teacher's Edition, Section
1, Vol. XLIX, Number 14, January 1964. Reprinted
with special permission of Current Science,
Teacher's Edition, published by American
Education Publications, copyright © 1964 by
Xerox Corp.

5. The Great Molecular Theory of Gases from
Physics for the Inquiring Mind: The Methods,
Nature, and Philosophy of Physical Science by
Eric M. Rogers, copyright © 1960 by Princeton
University Press. Reprinted with permission.

6. Entropy and the Second Law of Thermodynamics
from Basic Physics by Kenneth W. Ford, copy-
right © 1968 by Ginn and Company. Reprinted
with permission.

7. The Law of Disorder from One, Two, Three . . .
Infinity by George Gamow, copyright 1947 by
George Gamow. Reprinted with permission of
The Viking Press, Inc., and Macmillan & Co. Ltd.

8. The Law by Robert M. Coates, copyright 1947
by The New Yorker Magazine, Inc. Reprinted
with permission.

9. The Arrow of Time from Insight by Dr. J.
Bronowski, copyright © 1964 by Dr. J. Bronowski.
Reprinted with permission of Harper & Row,
Publishers, and Macdonald & Co. (Publishers)
Ltd., London.

10. James Clerk Maxwell (Part 1) by James R.
Newman from Scientific American, June 1955,
copyright © 1955 by Scientific American, Inc.
Reprinted with permission. All rights reserved.

11. Frontiers of Physics Today — Acoustics from
Physics Today by Leo L. Beranek, copyright ©
1969. Reprinted with permission.

12. Randomness and the Twentieth Century by Alfred
M. Bork. Reprinted from The Antioch Review,
volume XXVII, No. 1 with permission of the
editors.

13. Waves from Theory of Physics by Richard
Stevenson and R. B. Moore, copyright © 1967

by Richard Stevenson and R. B. IVIoore. Published
by W. B. Saunders Company. Reprinted w/ith
permission.

14. What Is a Wave? by Albert Einstein and Leopold
Infeld from The Evolution of Physics, copyright ©
1961 by Estate of Albert Einstein. Published by
Simon and Schuster. Reprinted with permission.

15. Musical Instruments and Scales from Classical
and Modern Physics by Harvey E. White, Ph.D.,
copyright 1940 by Litton Educational Publishing,
Inc. Reprinted with permission of Van Nostrand
Reinhold Company.

16. Founding a Family of Fiddles by Carleen M.
Hutchins from Physics Today, copyright © 1967
by the American Institute of Physics, New York.
Reprinted with permission.

17. The Seven Images of Science from Modern
Science and the Intellectual Tradition by Gerald
Holton from Science, Vol. 131, pp. 1187-1193,
April 22, 1960. Copyright © 1960 by the American
Association of Science. Reprinted with
permission.

18. Scientific Cranks from Fads and Fallacies in the
Name of Science by Martin Gardner, copyright ©
1957 by Martin Gardner. Published by Dover
Publications, Inc. Reprinted with permission.

19. Physics and the Vertical Jump from the American
Journal of Physics, Vol. 38, Number 7, July 1970,
by Elmer L. Offenbacher, copyright © 1970.
Reprinted with permission.

Ill

i^mii

This is not a physics textbook. Rather, it is o physics
reader, a collection of some of the best articles and
book passages on physics. A few are on historic events
in science, others contain some particularly memorable
description of what physicists do; still others deal with
philosophy of science, or with the impact of scientific
thought on the imagination of the artist.

There are old and new classics, and also some little-
known publications; many have been suggested for in-
clusion because some teacher or physicist remembered
an article with particular fondness. The majority of
articles is not drawn from scientific papers of historic
importance themselves, because material from many of
these is readily available, either as quotations in the
Project Physics text or in special collections.

This collection is meant for your browsing. If you follow
your own reading interests, chances are good that you
will find here many pages that convey the joy these
authors have in their work and the excitement of their
ideas. If you want to follow up on interesting excerpts,
the source list at the end of the reoder will guide you
for further reading.

i^0

~^^i)f</ ^

Reader 3
Table of Contents

1 Silence, Please 1

Arthur C. Clarke

2 The Steam Engine Comes of Age 12

R. J. Forbes and E. J. Dijksterhuis

3 The Great Conservation Principle 20

Richard Feynman

4 The Barometer Story 45

Alexander Calandra

5 The Great Molecular Theory of Gases 46

Eric M. Rogers

6 Entropy and the Second Law of Thermodynamics 59

Kenneth W. Ford

7 The Law of Disorder 87

George Gamow

8 The Law 125

Robert M. Goates

9 The Arrow of Time 127

Jacob Bronowski

10 James Clerk Maxwell 133

James R. Newman

1 1 Frontiers of Physics Today: Acoustics 155

Leo L. Beranek

1 2 Randomness and The Twentieth Century 167

Alfred M. Bork

13 Waves 188

Richard Stevenson and R. B. Moore

14 What is a Wave? 208

Albert Einstein and Leopold Infeld

15 Musical Instruments and Scales 213

Harvey E. White

16 Founding a Family of Fiddles 233

Carleen M. Hutchins

1 7 The Seven Images of Science 245

Gerald Holton

18 Scientific Cranks 248

Martin Gardner

19 Physics and the Vertical Jump 254

Elmer L. Offenbacher

Vil

A fictional scientist tells of an apparatus for pro-
ducing silence. Although the proposed scheme is inr
probable, the story has a charming plausibility.

1 Silence, Please

Arthur C. Clarke

An excerpt from his Tales from the White Hart, 1954.

You COME upon the "White Hart" quite unexpectedly in
one of these anonymous little lanes leading down from
Reet Street to the Embankment. It's no use telling you
where it is: very few people who have set out in a deter-
mined effort to get there have ever actually arrived. For
the first dozen visits a guide is essential: after that you'll
probably be all right if you close your eyes and rely on
instinct. Also — to be perfectly frank — we don't want any
more customers, at least on our night. The place is already
uncomfortably crowded. All that I'll say about its loca-
tion is that it shakes occasionally with the vibration of
newspaper presses, and that if you crane out of the win-
dow of the gent's room you can just see the Thames.

From the outside, is looks like any other pub — as in-
deed it is for five days of the week. The pubhc and saloon
bars are on the ground floor: there are the usual vistas of
brown oak panelling and frosted glass, the bottles behind
the bar, the handles of the beer engines . . . nothing out
of the ordinary at all. Indeed, the only concession to the
twentieth century is the juke box in the pubUc bar. It was
installed during the war in a laughable attempt to make
G.I.'s feel at home, and one of the first things we did was
to make sure there was no danger of its ever working
again.

At this point I had better explain who "we" are. That
is not as easy as I thought it was going to be when I
started, for a complete catalogue of the "White Hart's"
clients would probably be impossible and would certainly
be excruciatingly tedious. So all I'll say at this point is
that "we" fall into three main classes. First there are the
journalists, writers and editors. The journalists, of course,
gravitated here from Fleet Street. Those who couldn't
make the grade fled elsewhere: the tougher ones remained.
As for the writers, most of them heard about us from
other writers, came here f©r copy, and got trapped.

Where there are writers, of course, there are sooner or
later editors. If Drew, our landlord, got a percentage on
the literary business done in his bar, he'd be a rich man.
(We suspect he is a rich man, anyway.) One of our wits
once remarked that it was a common sight to see half a
dozen indignant authors arguing with a hard-faced editor
in one comer of the "White Hart", while in another, half
a dozen indignant editors argued with a hard-faced author.

So much for the literary side: you will have, I'd better
warn you, ample opportunities for close-ups later. Now
let us glance briefly at the scientists. How did they get in
here?

Well, Birkbeck College is only across the road, and
King's is just a few hundred yards along the Strand. That's
doubtless part of the explanation, and again personal rec-
ommendation had a lot to do with it. Also, many of our
scientists are writers, and not a few of our writers are
scientists. Confusing, but we like it that way.

The third portion of our little microcosm consists of
what may be loosely termed "interested laymen". They
were attracted to the "White Hart" by the general brou-
haha, and enjoyed the conversation and company so much

that they now come along regularly every Wednesday

which is the day when we all get together. Sometimes
they can't stand the pace and fall by the wayside, but
there's always a fresh supply.

With such potent ingredients, it is hardly surprising that
Wednesday at the "White Hart" is seldom dull. Not only
have some remarkable stories been told there, but remark-
able thmgs have happened there. For example, there was

the time when Professor , passing through on his

way to Harwell, left behind a brief-case containing well,

we'd better not go into that, even though we did so at the
time. And most interesting it was, too. . . . Any Russian
agents will find me in the comer under the dartboard. I
come high, but easy terms can be arranged.

Now that I've finally thought of the idea, it seems
astonishing to me that none of my colleagues has ever
got round to writing up these stories. Is it a question of
being so close to the wood that they can't see the trees?
Or is it lack of incentive? No, the last explanation can
hardly hold: several of them are quite as hard up as I am.

Silence, Please

and have complained with equal bitterness about Drew's
"NO CREDIT" rule. My only fear, as I type these words
on my old Remington Noiseless, is that John Christopher
or George Whitley or John Beynon are already hard at
work using up the best material. Such as, for instance, the
story of the Fenton Silencer, . . .

I don't know when it began: one Wednesday is much
like another and it's hard to tag dates on to them. Be-
sides, people may spend a couple of months lost in the
"White Hart" crowd before you first notice their exist-
ence. That had probably happened to Harry Purvis, be-
cause when I first came aware of him he already knew
the names of most of the people in our crowd. Which is
more than I do these days, now that I come to think of it.

But though I don't know when, I know exactly how it
all started. Bert Huggins was the catalyst, or, to be more
accurate, his voice was. Bert's voice would catalyse any-
thing. When he indulges in a confidential whisper, it
sounds hke a sergeant major drilhng an entire regiment.
And when he lets himself go, conversation languishes else-
where while we all wait for those cute little bones in the
inner ear to resume their accustomed places.

He had just lost his temper with John Christopher (we
all do this at some time or other) and the resulting deto-
nation had disturbed the chess game in progress at the
back of the saloon bar. As usual, the two players were
surrounded by backseat drivers, and we all looked up with
a start as Bert's blast whammed overhead. When the
echoes died away, someone said: "I wish there was a way
of shutting him up."

It was then that Harry Purvis repUed: "There is, you
know."

Not recognising the voice, I looked round. I saw a
small, neady-dressed man in the late thirties. He was
smoking one of those carved German pipes that always
makes me think of cuckoo clocks and the Black Forest.
That was the only unconventional thing about him: other-
wise he might have been a minor Treasury ofl&cial all
dressed up to go to a meeting of the Public Accounts
Committee.

"I beg your pardon?" I said.

He took no notice, but made some delicate adjust-
ments to his pipe. It was then that I noticed that it wasn't.

as I'd thought at first glance, an elaborate piece of wood
carving. It was something much more sophisticated — a
contraption of metal and plastic like a small chemical
engineering plant. There were even a couple of minute
valves. My God, it was a chemical engineering plant. . . .

I don't goggle any more easily than the next man, but I
made no attempt to hide my curiosity. He gave me a su-
perior smile.

"AU for the cause of science. It's an idea of the Bio-
physics Lab. They want to find out exactly what there is
in tobacco smoke — hence these filters. You know the old
argument — does smoking cause cancer of the tongue, and
if so, how? The trouble is that it takes an awful lot of —
er — distillate to identify some of the obscurer bye-prod-
ucts. So we have to do a lot of smoking."

"Doesn't it spoil the pleasure to have all this plumbing
in the way?"

"I don't know. You see, I'm just a volunteer. I don't
smoke."

"Oh," I said. For the moment, that seemed the only
reply. Then I remembered how the conversation had
started.

"You were saying," I continued with some feeling, for
there was still a slight tintinus in my left ear, "that there
was some way of shutting up Bert. We'd all like to hear
it — if that isn't mixing metaphors somewhat."

"I was thinking," he replied, after a couple of experi-
mental sucks and blows, "of the ill-fated Fenton Silen-
cer. A sad story — yet, I feel, one with an interesting les-
son for us all. And one day — who knows? — someone may
perfect it and earn the blessings of the world."

Suck, bubble, bubble, plop. . . .

"WeU, let's hear the story. When did it happen?"

He sighed.

"I'm almost sorry I mentioned it. Still, since you insist
— and, of course, on the understanding that it doesn't go
beyond these walls."

"Er — of course."

"Well, Rupert Fenton was one of our lab assistants. A
very bright youngster, with a good mechanical back-
ground, but, naturally, not very well up in theory. He was
always making gadgets in his spare time. Usually the idea
was good, but as he was shaky on fundamentals the things

Silence, Please

hardly ever worked. That didn't seem to discourage him:
I think he fancied himself as a latter-day Edison, and
imagined he could make his fortune from the radio tubes
and other oddments lying around the lab. As his tinkering
didn't interfere with his work, no-one objected: indeed,
the physics demonstrators did their best to encourage him,
because, after all, there is something refreshing about any
form of enthusiasm. But no-one expected he'd ever get
very far, because I don't suppose he could even integrate
e to the X."

"Is such ignorance possible?" gasped someone.

"Maybe I exaggerate. Let's say ;c e to the x. Anyway,
all his knowledge was entirely practical — rule of thumb,
you know. Give him a wiring diagram, however compli-
cated, and he could make the apparatus for you. But un-
less it was something really simple, like a television set, he
wouldn't understand how it worked. The trouble was, he
didn't realise his limitations. And that, as you'll see, was
most unfortunate.

"I think he must have got the idea whUe watching the
Honours Physics students doing some experiments in
acoustics. I take it, of course, that you all understand the
phenomenon of interference?"

"Naturally," I replied.

"Hey!" said one of the chess-players, who had given up
trying to concentrate on the game (probably because he
was losing.) "/ don't."

Purvis looked at him as though seeing something that
had no right to be around in a world that had invented
penicillin.

"In that case," he said coldly, "I suppose I had better
do some explaining." He waved aside our indignant pro-
tests. "No, I insist. It's precisely those who don't under-
stand these things who need to be told about them. If
someone had only explained the theory to poor Fenton
while there was still time. . . ."

He looked down at the now thoroughly abashed chess-
player.

"I do not know," he began, "if you have ever con-
sidered the nature of sound. Suffice to say that it consists
of a series of waves moving through the air. Not, how-
ever, waves like those on the surface of the sea — oh dear

no! Those waves are up and down movements. Sound
waves consist of alternate compressions and rarefactions."

"Rare-what?"

"Rarefactions."

"Don't you mean 'rarefications'?"

"I do not. I doubt if such a word exists, and if it does,
it shouldn't," retorted Purvis, with the aplomb of Sir Alan
Herbert dropping a particularly revolting neologism into
his killing-bottle. "Where was I? Explaining sound, of
course. When we make any sort of noise, from the faintest
whisper to that concussion that went past just now, a
series of pressure changes moves through the air. Have you
ever watched shunting engines at work on a siding? You
see a perfect example of the same kind of thing. There's a
long line of goods-wagons, all coupled together. One end
gets a bang, the first two trucks move together — and then
you can see the compression wave moving right along the
line. Behind it the reverse thing happens — the rarefaction
— I repeat, rarefaction — as the trucks separate again.

"Things are simple enough when there is only one
source of sound — only one set of waves. But suppose you
have two wave-patterns, moving in the same direction?
That's when interference arises, and there are lots of
pretty experiments in elementary physics to demonstrate
it. All we need worry about here is the fact — which I
think you will all agree is perfectly obvious — that if one
could get two sets of waves exactly out of step, the total
result would be precisely zero. The compression pulse of
one sound wave would be on top of the rarefaction of
another — net result — no change and hence no sound. To
go back to my analogy of the line of wagons, it's as if
you gave the last truck a jerk and a push simultaneously.
Nothing at aU would happen.

"Doubtless some of you will already see what I am
driving at, and will appreciate the basic principle of the
Fenton Silencer. Young Fenton, I imagine, argued in this
manner. 'This world of ours,' he said to himself, 'is too
full of noise. There would be a fortune for anyone who
could invent a really perfect silencer. Now, what would
that imply . . . ?'

"It didn't take him long to work out the answer: I told
you he was a bright lad. There was really very Uttle in
his pilot model. It consisted of a microphone, a special
amplifier, and a pair of loudspeakers. Any sound that
happened to be about was picked up by the mike, amph-

Silence, Please

fied and inverted so that it was exactly out ot phase with
the original noise. Then it was pumped out of the speak-
ers, the original wave and the new one cancelled out, and
the net result was silence,

"Of course, there was rather more to it than that. There
had to be an arrangement to make sure that the canceHing
wave was just the right intensity — otherwise you might be
worse oS than when you started. But these are technical
details that I won't bore you with. As many of you will
recognise, it's a simple appUcation of negative feed-back."

"Just a moment!" interrupted Eric Maine. Eric, I
should mention, is an electronics expert and edits some
television paper or other. He's also written a radio play
about space-flight, but that's another story. "Just a mo-
ment! There's something wrong here. You couldn't get
sUence that way. It would be impossible to arrange the
phase . . ."

Purvis jammed the pipe back in his mouth. For a mo-
ment there was an ominous bubbling and I thought of the
first act of "Macbeth". Then he fixed Eric with a glare.

"Are you suggesting," he said frigidly, "that this story
is untrue?"

"Ah — well, I won't go as far as that, but . . ." Eric's
voice trailed away as if he had been silenced himself. He
pulled an old envelope out of his pocket, together with an
assortment of resistors and condensers that seemed to have
got entangled in his handkerchief, and began to do some
figuring. That was the last we heard from him for some
time.

"As I was saying," continued Purvis calmly, "that's the
way Fenton's Silencer worked. His first model wasn't very
powerful, and it couldn't deal with very high or very low
notes. The result was rather odd. When it was switched
on, and someone tried to talk, you'd hear the two ends of
the spectrum — a faint bat's squeak, and a kind of low
rumble. But he soon got over that by using a more Unear
circuit (dammit, I can't help using some technicalities!)
and in the later model he was able to produce complete
silence over quite a large area. Not merely an ordinary
room, but a full-sized hall. Yes. . . .

"Now Fenton was not one of these secretive inventors
who won't tell anyone what they are trying to do, in case
their ideas are stolen. He was all too willing to talk. He
discussed his ideas with the staff and with the students.

whenever he could get anyone to listen. It so happened
that one of the first people to whom he demonstrated his
improved Silencer was a young Arts student called — I
think — Kendall, who was taking Physics as a subsidiary
subject. Kendall was much impressed by the Silencer, as
well he might be. But he was not thinking, as you may
have imagined, about its commercial possibilities, or the
boon it would bring to the outraged ears of suffering hu-
manity. Oh dear no! He had quite other ideas.

"Please permit me a sUght digression. At College we
have a flourishing Musical Society, which in recent years
has grown in numbers to such an extent that it can now
tackle the less monumental symphonies. In the year of
which I speak, it was embarking on a very ambitious en-
terprise. It was going to produce a new opera, a work by
a talented young composer whose name it would not be
fair to mention, since it is now well-known to you all. Let
us call him Edward England. I've forgotten the title of the
work, but it was one of these stark dramas of tragic love
which, for some reason I've never been able to under-
stand, are supposed to be less ridiculous with a musical
accompaniment than without. No doubt a good deal de-
pends on the music.

"I can still remember reading the synopsis while wait-
ing for the curtain to go up, and to this day have never
been able to decide whether the libretto was meant seri-
ously or not. Let's see — the period was the late Victorian
era, and the main characters were Sarah Stampe, the pas-
sionate postmistress, Walter Partridge, the saturnine game-
keeper, and the squire's son, whose name I forget. It's the
old story of the eternal triangle, compUcated by the vil-
lager's resentment of change — in this case, the new tele-
graph system, which the local crones predict will Do
Things to the cows' milk and cause trouble at lambing
time.

"Ignoring the frills, it's the usual drama of operatic
jealousy. The squire's son doesn't want to marry into the
Post OflBce, and the gamekeeper, maddened by his rejec-
tion, plots revenge. The tragedy rises to its dreadful cli-
max when poor Sarah, strangled with parcel tape, is found
hidden in a mail-bag in the Dead Letter Department. The
villagers hang Partridge from the nearest telegraph pole,
much to the annoyance of the linesmen. He was supposed
to sing an aria while he was being hung: that is one thing

Silence, Please

I regret missing. The squire's son takes to drink, or the
Colonies, or both: and that's that.

"I'm sure you're wondering where all this is leading:
please bear with me for a moment longer. The fact is that
while this synthetic jealousy was being rehearsed, the real
thing was going on back-stage. Fenton's friend Kendall
had been spurned by the young lady who was to play
Sarah Stampe. I don't think he was a particularly vindic-
tive person, but he saw an opportunity for a unique re-
venge. Let us be frank and admit that college life does
breed a certain irresponsibility — and in identical circum-
stances, how many of us would have rejected the same
chance?

"I see the dawning comprehension on your faces. But
we, the audience, had no suspicion when the o.'erture
started on that memorable day. It was a most distinguished
gathering: everyone was there, from the Chancellor down-
wards. Deans and professors were two a penny: I never
did discover how so many people had been bullied into
coming. Now that I come to think of it, I can't remember
what I was doing there myself.

"The overture died away amid cheers, and, I must ad-
mit, occasional cat-calls from the more boisterous mem-
bers of the audience. Perhaps I do them an injustice: they
may have been the more musical ones.

"Then the curtain went up. The scene was the village
square at Doddering Sloughleigh, circa 1860. Enter the
heroine, reading the postcards in the morning's mail. She
comes across a letter addressed to the young squire and
promptly bursts into song.

"Sarah's opening aria wasn't quite as bad as the over-
ture, but it was grim enough. Luckily, we were to hear
only the first few bars. . . .

"Precisely. We need not worry about such details as
how Kendall had talked the ingenuous Fenton into it —
if, indeed, the inventor realised the use to which his device
was being applied. All I need say is that it was a most
convincing demonstration. There was a sudden, deaden-
ing blanket of silence, and Sarah Stampe just faded out
like a TV programme when the sound is turned off. Every-
one was frozen in their seats, while the singer's lips went
on moving silently. Then she too realised what had hap-
pened. Her mouth opened in what would have been a
piercing scream in any other circumstances, and she fled
into the wings amid a shower of postcards.

"Thereafter, the chaos was unbehevable. For a few min-
utes everyone must have thought they had lost the sense
of hearing, but soon they were able to tell from the be-
haviour of their companions that they were not alone in
their deprivation. Someone in the Physics Department
must have realised the truth fairly promptly, for soon
little shps of paper were circulating among the V.LP.'s in
the front row. The Vice-Chancellor was rash enough to
try and restore order by sign-language, waving frantically
to the audience from the stage. By this time I was too sick
with laughter to appreciate such fine details.

"There was nothing for it but to get out of the hall,
which we all did as quickly as we could. I think Kendall
had fled — he was so overcome by the effect of the gadget
that he didn't stop to switch it off. He was afraid of stay-
ing around in case he was caught and lynched. As for
Fenton — alas, we shall never know his side of the story.
We can only reconstruct the subsequent events from the
evidence that was left.

"As I picture it, he must have waited until the hall was
empty, and then crept in to disconnect his apparatus. We
heard the explosion all over the college."

"The explosion?" someone gasped.

"Of course. I shudder to think what a narrow escape
we all had. Another dozen decibels, a few more phons —
and it might have happened while the theatre was still
packed. Regard it, if you Uke, as an example of the in-
scrutable workings of providence that only the inventor
was caught in the explosion. Perhaps it was as well: at
least he perished in the moment of achievement, and be-
fore the Dean could get at him."

"Stop moralising, man. What happened?"

"Well, I told you that Fenton was very weak on theory.
If he'd gone into the mathematics of the Silencer he'd
have found his mistake. The trouble is, you see, that one
can't destroy energy. Not even when you cancel out one
train of waves by another. All that happens then is that
the energy you've neutralized accumulates somewhere else.
It's rather like sweeping up all the dirt in a room — at the
cost of an unsightly pile under the carpet.

"When you look into the theory of the thing, you'll find
that Fenton's gadget wasn't a silencer so much as a col-
lector of sound. All the time it was switched on, it was
really absorbing sound energy. And at that concert, it was
certainly going flat out. You'll understand what I mean if

Silence, Please

you've ever looked at one of Edward England's scores. On
top of that, of course, there was all the noise the audi-
ence was making — or I should say was trying to make —
during the resultant panic. The total amount of energy
must have been terrific, and the poor Silencer had to keep
on sucking it up. Where did it go? Well, I don't know the
circuit details — probably into the condensers of the power
pack. By the tune Fenton started to tinker with it again,
it was like a loaded bomb. The sound of his approaching
footsteps was the last straw, and the overloaded apparatus
could stand no more. It blew up."

For a moment no-one said a word, perhaps as a token
of respect for the late Mr. Fenton. Then Eric Maine, who
for the last ten minutes had been muttering in the comer
over his calculations, pushed his way through the ring of
listeners. He held a sheet of paper thrust aggressively in
front of him.

"Hey!" he said. "I was right all the time. The thing
couldn't work. The phase and amplitude relations. . . ."

Purvis waved him away.

"That's just what I've explained," he said patiently.
"You should have been listening. Too bad that Fenton
found out the hard way."

He glanced at his watch. For some reason, he now
seemed in a hurry to leave.

"My goodness! Time's getting on. One of these days,
remind me to tell you about the extraordinary thing we
saw through the new proton microscope. That's an even
more remarkable story."

He was half way through the door before anyone else
could challenge him. Then George Whitley recovered his
breath.

"Look here," he said in a perplexed voice. "How is it
that we never heard about this business?"

Purvis paused on the threshold, his pipe now burbling
briskly as it got into its stride once more. He glanced back
over his shoulder.

"There was only one thing to do," he replied. "We
didn't want a scandal — de mortuis nil nisi bonum, you
know. Besides, in the circumstances, don't you think it
was highly appropriate to — ah — hush the whole business
up? And a very good night to you all."

The invention of the steam engine was a major factor
In the early stages of the Industrial Revolution.

The Steam Engine Comes of Age

R. J. Forbes and E. J. Dijksterhuis

A chapter from their book A History of Science and Technology, 1963.

The steam engine, coke, iron, and steel are the four principal
factors contributing to the acceleration of technology called the
Industrial Revolution, which some claim to have begun about
1750 but which did not really gain momentum until about 1830.
It started in Great Britain but the movement gradually spread to
the Continent and to North America during the nineteenth
century.

SCIENCE INSPIRES THE ENGINEER

During the Age of Projects the engineer had little help from the
scientists, who were building the mathematical-mechanical
picture of the Newtonian world and discussing the laws of nature.
However, during the eighteenth century, the Age of Reason,
when the principles of this new science had been formulated, the
scientists turned to the study of problems of detail many of which
were of direct help to the engineer. The latter was perhaps less
interested in the new ideals of 'progress' and 'citizenship of the
world' than in the new theory of heat, in applied mechanics and
the strength of materials, or in new mathematical tools for their
calculations. The older universities like Oxford and Cambridge
contributed little to this collaboration. The pace was set by the
younger ones such as the universities of Edinburgh and Glasgow,
which produced such men as Hume, Roebuck, Kerr, and Black,
who stimulated the new technology. The Royal Society, and also
new centres like the Lunar Society and the Manchester Philo-
sophical Society and the many similar societies on the Continent,
contributed much to this new technology by studying and dis-
cussing the latest scientific theories and the arts. Here noblemen,
bankers, and merchants met to hear the scientist, the inventor,
and the engineer and to help to realize many of the projects
which the latter put forward. They devoted much money to
scientific investigations, to demonstrations and stimulated in-
ventions by offering prizes for practical solutions of burning
problems. They had the capital to promote the 'progress' which
made Dr Johnson cry out: 'This age is running mad after innova-
tion. All business of the world is to be done in a new way, men
are to be hanged in a new way; Tyburn itself is not safe from the
fury of innovation!' New institutions such as the Conservatoire
des Arts et Metiers and the Royal Institution of Great Britain

The Steam Engine Comes of Age

were founded to spread the new science and technology by
lectures and demonstrations and the number of laymen attending
these lectures was overwhelming.

ENGINEERS AND SKILLED LABOUR

The new professional engineers which the ficole des Fonts et
Chaussees began to turn out were the descendants of the sappers
and military engineers. However, the new technology also needed
other types of engineers for which new schools such as the ficole
Polytechnique and the ficole des Mines were founded. In Great
Britain the State was less concerned with the education of the
new master craftsmen. They were trained in practice: such
famous workshops as that of Boulton and Watt in Soho, Birm-
ingham, or those of Dobson and Barlow, Asa Lees, and Richard
Roberts. Their success depended not only on good instruction
but also on appropriate instruments and skilled labour.

The scientists of the eighteenth century had turned out many
new instruments which were of great value to the engineer. They
were no longer made individually by the research scientist, but
by professional instrument makers in Cassel, Nuremberg, or
London, and such university towns as Leiden, Paris, and Edin-
burgh. Their instruments became more efficient and precise as
better materials became available such as good glass for lenses
and more accurate methods for working metals.

Skilled labour was more difficult to create. The older genera-
tion of Boulton and Watt had to work with craftsmen such as
smiths and carpenters, they had to re-educate them and create
a new type of craftsmen, 'skilled labour'. The design of early
machinery often reveals that it was built by the older type of
craftsmen that belonged to the last days of the guild system. The
new industrialists tried out several systems of apprenticeship
in their machine shops during the eighteenth century until they
finally solved this educational problem during the next century
and created schools and courses for workmen for the new indus-
tries, qualified to design and to make well-specified engines and
machine parts.

A factor that contributed greatly to this development was the
rise of the science of applied mechanics and the methods of
testing materials. The theories and laws which such men as
Palladio, Derand, Hooke, Bernoulli, Euler, Coulomb, and
Perronet formulated may have been imperfect but they showed

the way to estimate the strength of materials so important in
the construction of machinery, 's Gravesande and Van Muss-
chenbroek were the first to design and demonstrate various
machines for measuring tensile, breaking, and bending strengths
of various materials early in the eighteenth century. Such instru-
ments were gradually improved by Gauthey, Rondelet, and
others. The elastic behaviou'" of beams, the strength of arches,
and many other problems depended on such tests. Some scien-
tists developed tests for certain types of materials, for instance
for timber (Buffon), stone (Gauthey), or metals (Reaumur).
Such knowledge was of prime importance to the development
of the steam engine and other machinery which came from the
machine shops.

MACHINE SHOPS

The engineers who led this Industrial Revolution had to create
both the tools and the new workmen. Watt, himself a trained
instrument maker, had to invent several new tools and machines
and to train his workmen in foundries and machine shops. Hence
his notebooks are full of new ideas and machines. He invented
the copying press. His ingenious contemporaries Maudsley and
Bramah were equally productive. Joseph Bramah was respon-
sible for our modern water closet (1778) and the first successful
patent lock (1784) which no one succeeded in opening with a
skeleton key before Hobbs (1851), who spent fifty-one hours of
labour on it.

The difficulty in finding suitable labour arose from the fact that
the new machines were no longer single pieces created by one
smith, but that series of such machines were built from standard
parts which demanded much greater precision in manufacturing
such parts. The steam engine parts had to be finished accurately
to prevent the steam escaping between metal surfaces which slid
over each other, especially as steam pressures were gradually
increased to make these machines more efficient. Hence the
importance of the new tools and finishing processes, such as the
lathe and drilling, cutting and finishing machinery.

In 1797 Henry Maudsley invented the screw-cutting lathe.
Lathes originally belonged to the carpenter's shop. Even before
the eighteenth century they had been used to turn soft metals
such as tin and lead. These lathes were now moved by means
of treadles instead of a bow, though Leonardo da Vinci had

The Steam Engine Comes of Age

already designed lathes with interchangeable sets of gear wheels
to regulate the speed of the lathe. Maudsley applied similar ideas
and introduced the slide rest. Brunei, Roberts, Fox, Witworth,
and others perfected the modem lathe, which permitted moving
the object horizontally and vertically, adjustment by screws, and
automatic switching off when the operation was completed. The
older machine lathes were first moved by hand, then by a steam
engine, and finally by electric motors. Now the mass production
of screws, bolts, nuts, and other standard parts became possible
and machines were no longer separate pieces of work. They were
assembled from mass-produced parts.

The tools of the machine shop were greatly improved during
the nineteenth century, pulleys, axles, and handles being per-
fected. The new turret or capstan lathe had a round or hexagonal
block rotating about its axis and holding in a hole in each side
the cutting or planing tool needed. These tools could then at will
be brought into contact with the metal to be finished, thus per-
forming the work of six separate lathes in a much shorter time.
The turret block was made to turn automatically (1857) and
finally Hartness invented the flat turret lathe, replacing the block
by a horizontal face plate which gave the lathe greater flexi-
bility and allowed work at higher speeds. Such lathes ranged
from the small types used by the watchmaker to those for pro-
cessing large guns. This development was completed by the
introduction of high-speed tool steels by Taylor and White about
the beginning of our century, making the machine lathe a uni-
versal tool for the mass production of machine parts.

FACTORIES AND INDUSTRIAL REVOLUTION

This brought about a great change in the manufacturing process
itself. No longer were most commodities now made in the private
shops of craftsmen, but in larger workshops in which a water
wheel or a steam engine moved an axle from which smaller
machinery derived its power by means of gear wheels or belts,
each machine only partly processing the metal or material. Hence
the manufacturing process was split up into a series of opera-
tions, each of which was performed by a special piece of machin-
ery instead of being worked by hand by one craftsman who
mastered all the operations.

The modem factory arose only slowly. Even in 1 800 the word
'factory' still denoted a shop, a warehouse, or a depot; the
eighteenth century always spoke of 'mills' in many of which

the prime mover still was a horse mill or tread mill. The textile
factory law of 1844 was the first to speak of 'factories'.

It is obvious that the new factories demanded a large outlay
of capital. The incessant local wars had impoverished central
Europe and Italy and industry did not flourish there, so many
German inventors left their country to seek their fortune in
western Europe. State control of the 'manufactures' in France
had not been a success. The French government had not created
a new class of skilled labour along with the new engineers, and
Napoleon's 'self-supporting French industry' was doomed to
be a failure when overseas trade was re-established after his fall.
Neither the Low Countries nor Scandinavia had the necessary
capital and raw materials needed for the Industrial Revolution.
Only in eighteenth-century England did such a fortunate com-
bination of factors exist, a flourishing overseas trade, a well-
developed banking system, raw materials in the form of coal and
iron ores, free trade and an industry-minded middle class willing
to undertake the risks of introducing new machinery and recruit-
ing the new skilled labour from the ranks of the farmers and
immigrants from Ireland and Scotland.

Hence we find the first signs of the Industrial Revolution in
Great Britain rather than in France, which, however, soon fol-
lowed suit. Competition from Germany did not start until the
middle of the nineteenth century, and from the United States
not until the beginning of our century.

THE BEAM ENGINES

The prime mover of this new industry was the steam engine. The
primitive machine that pumped water was transformed into a
prime mover by the eff"orts of Newcomen and Watt. Thomas
Newcomen (1663-1729) and John Calley built a machine in
which steam of 100" C moved a piston in its cylinder by con-
densation (1 705). This piston was connected with the end of a beam,
the other end of which was attached to the rod of the pump or
any other machine. Most of these engines were used to drain
mines. John Smeaton (1724-92) studied the Newcomen engine
and perfected it by measurement and calculation, changing its
boiler and valves and turning it into the most popular steam
engine up to 1800.

James Watt (1736-1819), trained as an instrument maker,
heard the lectures of John Robison and Joseph Black at Edin-
burgh, where the new theory of heat was expounded and methods

The Steam Engine Comes of Age

were discussed to measure the degree and the amount of heat, as
well as the phenomena of evaporation and condensation. He
perceived that a large amount of heat was wasted in the cylinder
of the Newcomen engine, heating it by injection of steam and
cooling it by injecting cold water to condense the steam. Hence
he designed an engine in which the condensatsion took place in a
separate condenser, which was connected with the cylinder by
opening a valve at the correct moment, when the steam had
forced the piston up (1763).

Watt tried to have his engine built at John Roebuck's Carron
Iron Works in Scotland but did not find the skilled workmen
there to make the parts. So he moved southwards and started
work at the works of Matthew Boulton, who built Roebuck's
share in Watt's patents (1774). At the nearby Bradley foundry of
John Wilkinson, cylinders could be bored accurately and thus
Watt produced his first large-scale engine in 1781. The power
output of the Watt engine proved to be four times that of a
Newcomen engine. It was soon used extensively to pump water
in brine works, breweries, and distilleries. Boulton and Murdock
helped to advertise and apply Watt's engines.

THE DOUBLE-ACTING ROTATIVE ENGINE

However, Watt was not yet satisfied with these results. His
Patent of 1781 turned the steam engine into a universally
efficient prime mover. The rod on the other arm of the beam
was made to turn the up-and-down movement of the beam into a
rotative one, by means of the 'sun and planet movement' of a
set of gear wheels connecting the rod attached to the end of the
beam with the axle on which the driving wheels and belts were
fixed which moved the machines deriving their energy from this axle.
A further patent of 1782 made his earlier engine into a double-
acting one, that is a steam engine in which steam was admitted
alternately on each side of the piston. This succeeded only when
Boulton and Watt had mastered the difficult task of casting and
finishing larger and more accurate cylinders. Watt also had to
improve the connexion of the beam and the piston rod by means
of his extended three-bar system (1784) which he called the ' paral-
lel movement'. He was also able to introduce a regulator which
cut off the steam supply to the cylinder at the right moment and
leaving the rest of the stroke to the expansion of the steam made
better use of its energy.

In 1788 he designed his centrifugal governor which regulated
the steam supply according to the load keeping constant the
number of strokes of the piston per minute. Six years later he
added the steam gauge or indicator to his engine, a miniature
cylinder and piston, connected with the main cylinder. The small
piston of this indicator was attached to a pen which could be
made to indicate on a piece of paper the movements of the Uttle
piston and thus provide a control on the movements of the steam
engine proper. William Murdock (1754-1839), by inventing the
sliding valves and the means of preparing a paste to seal off the
seams between the cast iron surface of the machine parts, con-
tributed much to the success of these engines as proper packing
was not yet available.

By 1 800 some 500 Boulton and Watt engines were in operation,
160 of which pumped water back on to water wheels moving
machinery. The others were mostly rotative engines moving
other machinery and twenty-four produced blast air for iron
furnaces, their average strength being 15-16 h.p.

THE MODERN HIGH-PRESSURE STEAM ENGINE

The period 1800-50 saw the evolution of the steam engine to
the front rank of prime movers. This was achieved by building
steam engines which could be moved by high-pressure steam of
high temperature containing much more energy per pound than
the steam of 100° C which moved the earlier Watt engines. This
was only possible by perfecting the manufacture of the parts of
the steam engine, by better designing, and by the more accurate
finishing and fit of such parts.

Jabez Carter Hornblower built the first 'compound
engine', in which the steam released from the first cylinder was
left to expand further in a second one. These compound engines
did away with the Watt condenser, but could not yet compete
seriously until high pressure steam was applied. Richard Tre-
vithick and Oliver Evans were the pioneers of the high-pressure
engine, which meant more horse power per unit of weight of the
steam engine. This again meant lighter engines and the possi-
bility of using them for road and water traffic.

Nor were properly designed steam engines possible until the
theory of heat had been further elaborated and the science of
thermodynamics formulated, the theory of gases studied, and
more evidence produced for the strength of metals and materials
at high temperatures. Another important problem was the con-

The Steam Engine Comes of Age

struction of boilers to produce the high-prc^surc steam. The
ancient beehive-shaped boilers of Watt's generation could not
withstand such pressures. Trevithick created the Cornish boiler
(1812), a horizontal cylinder heated by an inner tube carrying the
combustion gases through the boiler into the flue and adding to
the fuel efficiency of the boilers. The Lancashire boiler, designed
by William Fairbairn (1844), had two tubes and became a serious
competitor of the Cornish boiler. Better grates for burning the
coal fuel were designed such as the 'travelling grate stoker' of
John Bodmer (1841), and more fuel was economized by heating
the cold feed water of the boiler with flue gases in Green's
economizer (1845). Then multitubular boilers were built in the
course of the nineteenth century, most of which were vertical
boilers, the best known of which was the Babcock and Wilcox
tubular boiler (1876).

Further factors helping to improve the design of high-pressure
steam engines were the invention of the direct-action steam pump
by Henry Worthington (1841), the steam hoist (1830), and James
Nasmyth's steam hammer (1839). In the meantime Cartwright
(1797) and Barton (1797) had f>erfected metallic packing which
ensure tight joints and prevented serious leakage.

Thus steam pressures rose from 3-5 atm in 1810 to about
5 or 6 atm in 1830, but these early high-pressure engines were
still of the beam type. Then came the much more efficient
rotation engines in which the piston rod was connected with the
driving wheel by means of a crank. Though even the early
American Corliss engine (1849) still clung to the beam design,
John M'Naught (1845) and E. Cowper (1857) introduced modern
rotative forms, which came to stay. Three -cylinder engines of this
type were introduced by Brotherhood (1871) and Kirk (1874)
and became very popular prime movers for steamships (1881).

Not until 1850 was the average output of the steam engines
some 40 h.p., that is significantly more than the 15 h.p. windmill
or water-wheel of the period. Again the steam engine was not
bound to sites where water or wind were constantly available,
it was a mobile prime mover which could be installed where
needed, for instance in iron works situated near coal fields and
iron ores. In 1700 Great Britain consumed some 3,000,000 tons
of coal, mostly to heat its inhabitants. This amount had doubled
by 1800 because of the introduction of the steam engine, and
by 1850 it has risen to 60,0(X),000 tons owing to the steam engine
and the use of coke in metallurgy. . .

A survey of the most fundamental principles that underlie all
of physics— and what they have in common.

3 The Great Conservation Principles

Richard Feynman

An excerpt from his book The Character of Physical Law, 1965.

When learning about the laws of physics you find that there
are a large number of comphcated and detailed laws, laws
of gravitation, of electricity and magnetism, nuclear inter-
actions, and so on, but across the variety of these detailed
laws there sweep great general principles which all the laws
seem to follow. Examples of these are the principles of con-
servation, certain quaUties of symmetry, the general form
of quantum mechanical principles, and unhappily, or
happily, as we considered last time, the fact that all the laws
are mathematical. In this lecture I want to talk about the
conservation principles.

The physicist uses ordinary words in a peculiar manner.
To him a conservation law means that there is a number
which you can calculate at one moment, then as nature
undergoes its multitude of changes, if you calculate this
quantity again at a later time it will be the same as it was
before, the number does not change. An example is the
conservation of energy. There is a quantity that you can
calculate according to a certain rule, and it comes out the
same answer always, no matter what happens.

Now you can see that such a thing is possibly useful.
Suppose that physics, or rather nature, is considered analo-
gous to a great chess game with miUions of pieces in it,
and we are trying to discover the laws by which the pieces
move. The great gods who play this chess play it very
rapidly, and it is hard to watch and difficult to see. However,
we are catching on to some of the rules, and there are some
rules which we can work out which do not require that we
watch every move. For instance, suppose there is one
bishop only, a red bishop, on the board, then since the

The Great Conservation Principles

bishop moves diagonally and therefore never changes the
colour of its square, if we look away for a moment while
the gods play and then look back again, we can expect that
there will be still a red bishop on the board, maybe in a
different place, but on the same colour square. This is in
the nature of a conservation law. We do not need to watch
the insides to know at least something about the game.

It is true that in chess this particular law is not necessarily
perfectly vahd. If we looked away long enough it could
happen that the bishop was captured, a pawn went down to
queen, and the god decided that it was better to hold a
bishop instead of a queen in the place of that pawn, which
happened to be on a black square. Unfortunately it may
well turn out that some of the laws which we see today may
not be exactly perfect, but I will tell you about them as we
see them at present.

I have said that we use ordinary words in a technical
fashion, and another word in the title of this lecture is
'great', The Great Conservation Principles'. This is not a
technical word : it was merely put in to make the title sound
more dramatic, and I could just as well have called it 'The
Conservation Laws'. There are a few conservation laws that
do not work; they are only approximately right, but are
sometimes useful, and we might call those the 'little' con-
servation laws. I will mention later one or two of those that
do not work, but the principal ones that I am going to
discuss are, as far as we can tell today, absolutely accurate.

I will start with the easiest one to understand, and that
is the conservation of electric charge. There is a number, the
total electric charge in the world, which, no matter what
happens, does not change. If you lose it in one place you
wiU find it in another. The conservation is of the total of all
electric charge. This was discovered experimentally by
Faraday.* The experiment consisted of getting inside a
great globe of metal, on the outside of which was a very
deUcate galvanometer, to look for the charge on the globe,

♦Michael Faraday, 1791-1867, English physicist.

because a small amount of charge would make a big effect.
Inside the globe Faraday built all kinds of weird electrical
equipment. He made charges by rubbing glass rods with
cat's fur, and he made big electrostatic machines so that the
inside of this globe looked like those horror movie labora-
tories. But during all these experiments no charge developed
on the surface ; there was no net charge made. Although the
glass rod may have been positive after it was charged up by
rubbing on the cat's fur, then the fur would be the same
amount negative, and the total charge was always nothing,
because if there were any charge developed on the inside
of the globe it would have appeared as an effect in the gal-
vanometer on the outside. So the total charge is conserved.

This is easy to understand, because a very simple model,
which is not mathematical at all, will explain it. Suppose the
world is made of only two kinds of particles, electrons and
protons - there was a time when it looked as if it was going
to be as easy as that - and suppose that the electrons carry
a negative charge and the protons a positive charge, so that
we can separate them. We can take a piece of matter and
put on more electrons, or take some off; but supposing that
electrons are permanent and never disintegrate or dis-
appear - that is a simple proposition, not even mathe-
matical - then the total number of protons, less the total
number of electrons, will not change. In fact in this particu-
lar model the total number of protons will not change, nor
the number of electrons. But we are concentrating now on
the charge. The contribution of the protons is positive and
that of the electrons negative, and if these objects are never
created or destroyed alone then the total charge will be
conserved. I want to list as I go on the number of properties
that conserve quantities, and I will start with charge
(fig. 14). Against the question whether charge is conserved
I write 'yes'.

This theoretical interpretation is very simple, but it was
later discovered that electrons and protons are not perma-
nent; for example, a particle called the neutron can disinte-
grate into a proton and an electron - plus something else

The Great Conservation Principles

(locally)

Yes

Y«

N£a«'ly

^ayrcL it a.
fiel2

Yei

•

/ei

NB This is the completed table which Professor Feynman
added to throughout his lecture.

Figure 14

which we will come to. But the neutron, it turns out, is
electrically neutral. So although protons are not perma-
nent, nor are electrons permanent, in the sense that they can
be created from a neutron, the charge still checks out; start-
ing before, we had zero charge, and afterwards we had plus
one and minus one which when added together become
zero charge.

An example of a similar fact is that there exists another
particle, besides the proton, which is positively charged. It
is called a positron, which is a kind of image of an electron.
It is just hke the electron in most respects, except that it has
the opposite sign of charge, and, more important, it is
called an anti-particle because when it meets with an elec-
tron the two of them can annihilate each other and
disintegrate, and nothing but hght comes out. So electrons
are not permanent even by themselves. An electron plus a
positron will just make light. Actually the 'hght' is invisible
to the eye; it is gamma rays; but this is the same thing for
a physicist, only the wavelength is different. So a particle
and its anti-particle can annihilate. The light has no electric

charge, but we remove one positive and one negative charge,
so we have not changed the total charge. The theory of
conservation of charge is therefore shghtly more comphca-
ted but still very unmathematical. You simply add together
the number of positrons you have and the number of
protons, take away the number of electrons - there are
additional particles you have to check, for example anti-
protons which contribute negatively, pi-plus mesons which
are positive, in fact each fundamental particle in nature has
a charge (possibly zero). All we have to do is add up the
total number, and whatever happens in any reaction the
total amount of charge on one side has to balance with
the amount on the other side.

That is one aspect of the conservation of charge. Now
comes an interesting question. Is it sufficient to say only
that charge is conserved, or do we have to say more? If
charge were conserved because it was a real particle which
moved around it would have a very special property. The total
amount of charge in a box might stay the same in two ways.
It may be that the charge moves from one place to another
within the box. But another possibility is that the charge in
one place disappears, and simultaneously charge arises in
another place, instantaneously related, and in such a
manner that the total charge is never changing. This second
possibility for the conservation is of a different kind from
the first, in which if a charge disappears in one place and
turns up in another something has to travel through the
space in between. The second form of charge conservation
is called local charge conservation, and is far more detailed
than the simple remark that the total charge does not
change. So you see we are improving our law, if it is true
that charge is locally conserved. In fact it is true. I have
tried to show you from time to time some of the possibiUties
of reasoning, of interconnecting one idea with another, and
I would now like to describe to you an argument, funda-
mentally due to Einstein, which indicates that if anything
is conserved - and in this case I apply it to charge - it must
be conserved locally. This argument relies on one thing,

The Great Conservation Principles

that if two fellows are passing each other in space ships,
the question of which guy is doing the moving and whicli
one standing still cannot be resolved by any experiment.
That is called the principle of relativity, that uniform motion
in a straight hne is relative, and that we can look at any
phenomenon from either point of view and cannot say
which one is standing still and which one is moving.
Suppose I have two space ships, A and B (fig. 15). I am

PosifcioM?. at time
of ei/£vtts

js/'*^

^ /IN,

r^/^--^

Positions at t\ft\t
When 6>sees eoents.

Figure 15

going to take the point of view that A is the one that is
moving past B. Remember that is just an opinion, you can
also look it at the other way and you will get the same
phenomena of nature. Now suppose that the man who is
standing still wants to argue whether or not he has seen a
charge at one end of his ship disappear and a charge at the
other end appear at the same time. In order to make sure it
is the same time he cannot sit in the front of the ship, be-
cause he will see one before he sees the other because of the
travel time of light; so let us suppose that he is very careful
and sits dead centre in the middle of the ship. We have
another man doing the same kind of observation in the
other ship. Now a lightning bolt strikes, and charge is
created at point x, and at the same instant at point y at the

other end of the ship the charge is annihilated, it disappears.
At the same instant, note, and perfectly consistent with our
idea that charge is conserved. If we lose one electron in one
place we get another elsewhere, but nothing passes in
between. Let us suppose that when the charge disappears
there is a flash, and when it is created there is a flash, so
that we can see what happens. B says they both happen at
the same time, since he knows he is in the middle of the
ship and the light from the bolt which creates x reaches him
at the same time as the light from the flash of disappearance
at y. Then B will say, 'Yes, when one disappeared the other
was created'. But what happens to our friend in the other
ship? He says, 'No, you are wrong my friend. I saw x
created before y'. This is because he is moving towards x,
so the light from x will have a shorter distance to travel
than the hght from y, since he is moving away from y. He
could say, 'No, x was created first and then y disappeared,
so for a short time after x was created and before y dis-
appeared I got some charge. That is not the conservation
of charge. It is against the law'. But the first fellow says,
'Yes, but you are moving'. Then he says, 'How do you know ?
I think you arc moving', and so on. If we are unable, by
any experiment, to see a diflerence in the physical lav/s
whether we are moving or not, then if the conservation of
charge were not local only a certain kind of man would see
it work right, namely the guy who is standing still, in an
absolute sense. But such a thing is impossible according to
Einstein's relativity principle, and therefore it is impossible
to have non-local conservation of charge. The locality of the
conservation of charge is consonant with the theory of
relativity, and it turns out that this is true of all the conser-
vation laws. You can appreciate that if anything is conserved
the same principle applies.

There is another interesting thing about charge, a very
strange thing for which we have no real explanation today.
It has nothing to do with the conservation law and is inde-
pendent of it. Charge always comes in units. When we have
a charged particle it has one charge or two charges, or minus

The Great Conservation Principles

one or minus two. Returning to our table, although this has
nothing to do with the consen'ation of charge, I must write
down that the thing that is conserved comes in units. It is
very nice that it comes in units, because that makes the
theory of conservation of charge very easy to understand.
It is just a thing we can count, which goes from place to
place. Finally it turns out technically that the total charge
of a thing is easy to determine electrically because the charge
has a ver>' important characteristic; it is the source of the
electric and magnetic field. Charge is a measure of the inter-
action of an object with electricity, with an electric field. So
another item which we should add to the hst is that charge
is the source of a field; in other words, electricit}' is related
to charge. Thus the particular quantity which is conserved
here has two other aspects which are not connected with
the conservation directly, but aie interesting anyway. One
is that it comes in units, and the other that it is the source
of a field.

There are many conservation laws, and I will give some
more examples of laws of the same type as the conseivation
of charge, in the sense that it is merely a matter of counting.
There is a conser\'ation law called the conservation of
bar>ons. A neutron can go into a proton. If we count each
of these as one unit, or bar\'on, then we do not lose the
number of bar)'ons. The neutron carries one bar>'onic
charge unit, or repiesents one bar>on, a proton represents
one bar>'on - all we are doing is counting and making big
words! - so if the reaction I am speaking of occurs, in
which a neutron decays into a proton, an electron and an
anti-neutrino, the total number of barvons does not change.
However there are other reactions in nature. A pr6ton plus
a proton can produce a great variet}' of strange objects, for
example a lambda, a proton and a K plus. Lambda and K
plus are names for pecuhar particles.

In this reaction we know we put two baryons in, but we see
only one come out, so possibly either lambda or K"*" has a
baryon. If we study the lambda later we discover that very
slowly it disintegrates into a proton and a pi, and ultimately
the pi disintegrates into electrons and what-not.

(iltiyi) ^-> PH-TT

What we have here is the baryon coming out again in the
proton, so we think the lambda has a baryon number of 1 ,
but the K+ does not, the K+ has zero.

On our chart of conservation laws (fig. 14), then, we have
charge and now we have a similar situation with baryons,
with a special rule that the baryon number is the number of
protons, plus the number of neutrons, plus the number of
lambdas, minus the number of anti-protons, minus the
number of anti-neutrons, and so on; it is just a counting
proposition. It is conserved, it comes in units, and nobody
knows but everybody wants to think, by analogy, that it is
the source of a field. The reason we make these tables is that
we are trying to guess at the laws of nuclear interaction, and
this is one of the quick ways of guessing at nature. If charge
is the source of a field, and baryon does the same things in
other respects it ought to be the source of a field too. Too
bad that so far it does not seem to be, it is possible, but we
do not know enough to be sure.

There are one or two more of these counting propositions,
for example Lepton numbers, and so on, but the idea is the
same as with baryons. There is one, however, which is
slightly different. There are in nature among these strange
particles characteristic rates of reaction, some of which are
very fast and easy, and others which are very slow and hard.
I do not mean easy and hard in a technical sense, in actually
doing the experiment. It concerns the rates at which the
reactions occur when the particles are present. There is a
clear distinction between the two kinds of reaction which I
have mentioned above, the decay of a pair of protons, and

The Great Conservation Principles

the much slower decay of the lambda. It turns out that if
you take only the fast and easy reactions there is one more
counting law, in which the lambda gets a minus 1, and the
K plus gets a plus 1, and the proton gets zero. This is called
the strangeness number, or hyperon charge, and it appears
that the rule that it is conserved is right for every easy re-
action, but wrong for the slow reactions. On our chart (fig.
14) we must therefore add the conservation law called the
conservation of strangeness, or the conservation of hyperon
number, which is nearly right. This is very pecuhar; we
see why this quantity has been called strangeness. It is
nearly true that it is conserved, and true that it comes
in units. In trying to understand the strong interactions
which are involved in nuclear forces, the fact that in strong
interactions the thing is conserved has made people propose
that for strong interactions it is also the source of a field, but
again we do not know. I bring these matters up to show you
how conservation laws can be used to guess new laws.

There are other conservation laws that have been pro-
posed from time to time, of the same nature as counting.
For example, chemists once thought that no matter what
happened the number of sodium atoms stayed the same. But
sodium atoms are not permanent. It is possible to transmute
atoms from one element to another so that the original
element has completely disappeared. Another law which was
for a while believed to be true was that the total mass of an
object stays the same. This depends on how you define mass,
and whether you get mixed up with energy. The mass con-
servation law is contained in the next one which I am going
to discuss, the law of conservation of energy. Of all the
conservation laws, that dealing with energy is the most
difficult and abstract, and yet the most useful. It is more
difiicult to understand than those I have described so far,
because in the case of charge, and the others, the mechanism
is clear, it is more or less the conservation of objects. This
is not absolutely the case, because of the problem that we
get new things from old things, but it is really a matter of
simply counting.

The conservation of energy is a little more difficult, be-
cause this time we have a number which is not changed in
time, but this number does not represent any particular
thing. I would like to make a kind of silly analogy to ex-
plain a little about it.

I want you to imagine that a mother has a child whom she
leaves alone in a room with 28 absolutely indestructible
blocks. The child plays with the blocks all day, and when
the mother comes back she discovers that there are indeed
28 blocks; she checks all the time the conservation of blocks!
This goes on for a few days, and then one day when she
comes in there are only 27 blocks. However, she finds one
block lying outside the window, the child had thrown it
out. The first thing you must appreciate about conservation
laws is that you must watch that the stuff you are trying to
check does not go out through the wall. The same thing
could happen the other way, if a boy came in to play with
the child, bringing some blocks with him. Obviously these
are matters you have to consider when you talk about con-
servation laws. Suppose one day when the mother comes to
count the blocks she finds that there are only 25 blocks, but
suspects that the child has hidden the other three blocks in
a little toy box. So she says, T am going to open the box'.
'No,' he says, 'you cannot open the box.' Being a very
clever mother she would say, 'I know that when the box is
empty it weighs 16 ounces, and each block weighs 3
ounces, so what I am going to do is to weigh the box'. So,
totalling up the number of blocks, she would get -

N«.0( W«W fc«„ + W«liiKt<»t,t:^-l6...

and that adds up to 28. This works all right for a while, and
then one day the sum does not check up properly. However,
she notices that the dirty water in the sink is changing its
level. She knows that the water is 6 inches deep when there
is no block in it, and that it would rise i inch if a block was

The Great Conservation Principles

in the water, so she adds another term, and now she has -
No. (i We<k5 seen + —5 — + 1 — _

^ 3«*. -fclM.

and once again it adds up to 28. As the boy becomes more
ingenious, and the mother continues to be equally ingenious,
more and more terms must be added, all of which represent
blocks, but from the mathematical standpoint are abstract
calculations, because the blocks are not seen.

Now I would hke to draw my analogy, and tell you what
is common between this and the conservation of energy, and
what is different. First suppose that in all of the situations
you never saw any blocks. The term 'No. of blocks seen' is
never included. Then the mother would always be calculating
a whole lot of terms Hke 'blocks in the box', 'blocks in the
water', and so on. With energy there is this difference, that
there are no blocks, so far as we can tell. Also, unlike the
case of the blocks, for energy the numbers that come out
are not integers. I suppose it might happen to the poor
mother that when she calculates one term it comes out
6 ^ blocks, and when she calculates another it comes out
i^ of a block, and the others give 21, which still totals 28.
That is how it looks with energy.

What we have discovered about energy is that we have a
scheme with a sequence of rules. From each different set
of rules we can calculate a number for each different kind of
energy. When we add all the numbers together, from all the
different forms of energy, it always gives the same total.
But as far as we know there are no real units, no little ball-
bearings. It is abstract, purely mathematical, that there is
a number such that whenever you calculate it it does not
change. I cannot interpret it any better than that.

This energy has all kinds of forms, analogous to the
blocks in the box, blocks in the water, and so on. There is
energy due to motion called kinetic energy, energy due to
gravitational interaction (gravitational potential energy, it

is called), thermal energy, electrical energy, light energy,
elastic energy in springs and so on, chemical energy, nuclear
energy - and there is also an energy that a particle has from
its mere existence, an energy that depends directly on its
mass. The last is the contribution of Einstein, as you un-
doubtedly know. E = mc'^ is the famous equation of the
law I am talking about.

Although I have mentioned a large number of energies,
I would hke to explain that we are not completely ignorant
about this, and we do understand the relationship of some of
them to others. For instance, what we call thermal energy is
to a large extent merely the kinetic energy of the motion of
the particles inside an object. Elastic energy and chemical
energy both have the same origin, namely the forces be-
tween the atoms. When the atoms rearrange themselves in
a new pattern some energy is changed, and if that quantity
changes it means that some other quantity also has to
change. For example, if you are burning something the
chemical energy changes, and you find heat where you did
not have heat before, because it all has to add up right.
Elastic energy and chemical energy are both interactions of
atoms, and we now understand these interactions to be a
combination of two things, one electrical energy and the
other kinetic energy again, only this time the formula for it
is quantum mechanical. Light energy is nothing but elec-
trical energy, because light has now been interpreted as an
electric and magnetic wave. Nuclear energy is not represen-
ted in terms of the others ; at the moment I cannot say more
than that it is the result of nuclear forces. I am not just
talking here about the energy released. In the uranium
nucleus there is a certain amount of energy, and when the
thing disintegrates the amount of energy remaining in the
nucleus changes, but the total amount of energy in the world
does not change, so a lot of heat and stuff is generated in
the process, in order to balance up.

This conservation law is very useful in many technical
ways. I will give you some very simple examples to show
how, knowing the law of conservation of energy and the

The Great Conservation Principles

formulae for calculating energy, we can understand other
laws. In other words many other laws are not independent,
but are simply secret ways of talking about the conservation
of energy. The simplest is the law of the lever (fig. 16).

o ^--'

Figure 16

We have a lever on a pivot. The length of one arm is 1 foot
and the other 4 feet. First I must give the law for gravity
energy, which is that if you have a number of weights, you
take the weight of each and multiply it by its height above
the ground, add this together for all the weights, and that
gives the total of gravity energy. Suppose I have a 2 lb
weight on the long arm, and an unknown mystic weight on
the other side - X is always the unknown, so let us call it
W to make it seem that we have advanced above the usual !
Now the question is, how much must W be so that it just
balances and swings quietly back and forth without any
trouble ? If it swings quietly back and forth, that means that
the energy is the same whether the balance is parallel to
the ground or tilted so that the 2 lb weight is, say, 1 inch
above the ground. If the energy is the same then it does not
care much which way, and it does not fall over. If the 2 lb
weight goes up 1 inch how far down does W go? From the
diagram you can see (fig. 3) that if AO is 1 foot and OB
is 4 feet, then when BB' is 1 inch AA' will be \ inch. Now
apply the law for gravity energy. Before anything happened
all the heights were zero, so the total energy was zero. After
the move has happened to get the gravity energy we multi-
ply the weight 2 lb by the height 1 inch and add it to the

unknown weight W times the height - i inch. The sum of
this must give the same energy as before - zero. So -

2.-^»0. ^ yjmsi be 8

This is one way we can understand the easy law, which you
already knew of course, the law of the lever. But it is interest-
ing that not only this but hundreds of other physical laws
can be closely related to various forms of energy. I showed
you this example only to illustrate how useful it is.

The only trouble is, of course, that in practice it does not
really work because of friction in the fulcrum. If I have
something moving, for example a ball rolling along at a
constant height, then it will stop on account of friction.
What happened to the kinetic energy of the ball ? The answer
is that the energy of the motion of the ball has gone into the
energy of the jigghng of the atoms in the floor and in the
ball. The world that we see on a large scale looks like a nice
round ball when we polish it, but it is really quite complica-
ted when looked at on a little scale; bilUons of tiny atoms,
with all kinds of irregular shapes. It is like a very rough
boulder when looked at finely enough, because it is made
out of these httle balls. The floor is the same, a bumpy busi-
ness made out of balls. When you roll this monster boulder
over the magnified floor you can see that the little atoms are
going to go snap-jiggle, snap-jiggle. After the thing has
rolled across, the ones that are left behind are still shaking
a httle from the pushing and snapping that they went
through; so there is left in the floor a jigghng motion, or
thermal energy. At first it appears as if the law of conser-
vation is false, but energy has the tendency to hide from
us and we need thermometers and other instruments to
make sure that it is still there. We find that energy is con-
served no matter how complex the process, even when we
do not know the detailed laws.

The first demonstration of the law of conservation of

The Great Conservation Principles

energy was not by a physicist but by a medical man. He
demonstrated with rats. If you burn food you can find out
how much heat is generated. If you then feed the same
amount of food to rats it is converted, with oxygen, into
carbon dioxide, in the same way as in burning. When you
measure the energy in each case you find out that hving
creatures do exactly the same as non-Hving creatures. The
law for conservation of energy is as true for fife as for
other phenomena. Incidentally, it is interesting that every
law or principle that we know for 'dead' things, and that we
can test on the great phenomenon of life, works just as well
there. There is no evidence yet that what goes on in hving
creatures is necessarily different, so far as the physical
laws are concerned, from what goes on in non-hving things,
although the living things may be much more complicated.
The amount of energy in food, which will tell you how
much heat, mechanical work, etc., it can generate, is
measured in calories. When you hear of calories you are not
eating something called calories, that is simply the measure
of the amount of heat energy that is in the food. Physicists
sometimes feel so superior and smart that other people
would hke to catch them out once on something. I will
give you something to get them on. They should be utterly
ashamed of the way they take energy and measure it in a
host of difi"erent ways, with different names. It is absurd that
energy can be measured in calories, in ergs, in electron volts,
in foot pounds, in B.T.U.s, in horsepower hours, in kilowatt
hours - all measuring exactly the same thing. It is hke having
money in dollars, pounds, and so on; but unHke the econo-
mic situation where the ratio can change, these dopey things
are in absolutely guaranteed proportion. If anything is
analogous, it is hke shillings and pounds - there are always
20 shilUngs to a pound. But one comphcation that the
physicist allows is that instead of having a number hke 20
he has irrational ratios like 1-6183178 shillings to a pound.
You would think that at least the more modern high-class
theoretical physicists would use a common unit, but you
find papers with degrees Kelvin for measuring energy, mega-

cycles, and now inverse Fermis, the latest invention. For
those who want some proof that physicists are human, the
proof is in the idiocy of all the different units which they
use for measuring energy.

There are a number of interesting phenomena in nature
which present us with curious problems concerning energy.
There has been a recent discovery of things called quasars,
which are enormously far away, and they radiate so much
energy in the form of Hght and radio waves that the question
is where does it come from ? If the conservation of energy
is right, the condition of the quasar after it has radiated this
enormous amount of energy must be different from its
condition before. The question is, is it coming from gravi-
tation energy - is the thing collapsed gravitationally, in a
different condition gravitationally? Or is this big emission
coming from nuclear energy? Nobody knows. You might
propose that perhaps the law of conservation of energy is
not right. Well, when a thing is investigated as incompletely
as the quasar - quasars are so distant that the astronomers
cannot see them too easily - then if such a thing seems to
conflict with the fundamental laws, it very rarely is that
the fundamental laws are wrong, it usually is just that the
details are unknown.

Another interesting example of the use of the law of
conservation of energy is in the reaction when a neutron
disintegrates into a proton, an electron, and an anti-neutrino.
It was first thought that a neutron turned into a proton plus
an electron. But the energy of all the particles could be
measured, and a proton and an electron together did not
add up to a neutron. Two possibilities existed. It might
have been that the law of energy conservation was not
right; in fact it was proposed by Bohr* for a while that per-
haps the conservation law worked only statistically, on the
average. But it turns out now that the other possibility is
the correct one, that the fact that the energy does not check
out is because there is something else coming out, something

•Niels Bohr, Danish physicist.

The Great Conservation Principles

which we now call an anti-neutrino. The anti-neutrino which
comes out takes up the energy. You might say that the only
reason for the anti-neutrino is to make the conservation of
energy right. But it makes a lot of other things right, Uke
the conservation of momentum and other conservation laws,
and very recently it has been directly demonstrated that
such neutrinos do indeed exist.

This example illustrates a point. How is it possible that
we can extend our laws into regions we are not sure about?
Why are we so confident that, because we have checked the
energy conservation here, when we get a new phenomenon
we can say it has to satisfy the law of conservation of energy ?
Every once in a while you read in the papei that physicists
have discovered that one of their favourite laws is wrong.
Is it then a mistake to say that a law is true in a region where
you have not yet looked? If you will never say that a law is
true in a region where you have not already looked you do
not know anything. If the only laws that you find are those
which you have just finished observing then you can never
make any predictions. Yet the only utility of science is to
go on and to try to make guesses. So what we always do is
to stick our necks out, and in the case of energy the most
likely thing is that it is conserved in other places.

Of course this means that science is uncertain; the mo-
ment that you make a proposition about a region of ex-
perience that you have not directly seen then you must be
uncertain. But we always must make statements about the
regions that we have not seen, or Lhe whole business is no
use. For instance, the mass of an object changes when it
moves, because of the conservation of energy. Because of
the relation of mass and energy the energy associated with
the motion appears as an extra mass, so things get heavier
when they move. Newton beheved that this was not the
case, and that the masses stayed constant. When it was dis-
covered that the Newtonian idea was false everyone kept
saying what a terrible thing it was that physicists had found
out that they were wrong. Why did they think they were
right? The effect is very small, and only shows when you get

near the speed of light. If you spin a top it weighs the same
as if you do not spin it, to within a very very fine fraction.
Should they then have said, 'If you do not move any faster
than so-and-so, then the mass does not change'? That
would then be certain. No, because if the experiment
happened to have been done only with tops of wood,
copper and steel, they would have had to say 'Tops made
out of copper, wood and steel, when not moving any faster
than so and so . . .'. You see, we do not know all the con-
ditions that we need for an experiment. It is not known
whether a radioactive top would have a mass that is con-
served. So we have to make guesses in order to give any
utihty at all to science. In order to avoid simply describing
experiments that have been done, we have to propose laws
beyond their observed range. There is nothing wrong with
that, despite the fact that it makes science uncertain. If you
thought before that science was certain - well, that is just
an error on your part.

To return then, to our hst of conservation laws (fig. 14),
we can add energy. It is conserved perfectly, as far as we
know. It does not come in units. Now the question is, is
it the source of a field? The answer is yes. Einstein under-
stood gravitation as being generated by energy. Energy and
mass are equivalent, and so Newton's interpretation that
the mass is what produces gravity has been modified to the
statement that the energy produces the gravity.

There are other laws similar to the conservation of energy,
in the sense that they are numbers. One of them is momen-
tum. If you take all the masses of an object, multiply them
by the velocities, and add them all together, the sum is the
momentum of the particles; and the total amount of mo-
mentum is conserved. Energy and momentum are now
understood to be very closely related, so I have put them in
the same column of our table.

Another example of a conserved quantity is angular
momentum, an item which we discussed before. The angular
momentum is the area generated per second by objects
moving about. For example, if we have a moving object.

The Great Conservation Principles

and we take any centre whatsoever, then the speed at which
the area (fig. 17) swept out by a line from centre to object,

Figure 17

increases, multiphed by the mass of the object, and added
together for all the objects, is called the angular momentum.
And that quantity does not change. So we have conservation
of angular momentum. Incidentally, at first sight, if you
know too much physics, you might think that the angular
momentum is not conserved. Like the energy it appears in
different forms. Although most people think it only appears
in motion it does appear in other forms, as I will illustrate.
If you have a wire, and move a magnet up into it, increasing
the magnetic field through the flux through the wire, there
will be an electric current - that is how electric generators
work. Imagine that instead of a wire I have a disc, on which
there are electric charges analogous to the electrons in the
wire (fig. 18). Now I bring a magnet dead centre along the

Figure 18

axis from far away, very rapidly up to the disc, so that now
there is a flux change. Then, just as in the wire, the charges
will start to go around, and if the disc were on a wheel it
would be spinning by the time I had brought the magnet
up. That does not look like conservation of angular momen-
tum, because when the magnet is away from the disc nothing
is turning, and when they are close together it is spinning.
We have got turning for nothing, and that is against the
rules. 'Oh yes,' you say, 'I know, there must be some other
kind of interaction that makes the magnet spin the opposite
way.' That is not the case. There is no electrical force on the
magnet tending to twist it the opposite way. The explana-
tion is that angular momentum appears in two forms: one
of them is angular momentum of motion, and the other is
angular momentum in electric and magnetic fields. There is
angular momentum in the field around the magnet, although
it does not appear as motion, and this has the opposite sign
to the spin. If we take the opposite case it is even clearer

(fig. 19).

>eof&

Figure 19

If we have just the particles, and the magnet, close together,
and everything is standing still, I say there is angular momen-
tum in the field, a hidden form of angular momentum which
does not appear as actual rotation. When you pull the mag-
net down and take the instrument apart, then all the fields
separate and the angular momentum now has to appear and

The Great Conservation Principles

the disc will start to spin. The law that makes it spin is the
law of induction of electricity.

Whether angular momentum comes in units is very diffi-
cult for me to answer. At first sight it appears that it is
absolutely impossible that angular momentum comes in
units, because angular momentum depends upon the direc-
tion at which you project the picture. You are looking at an
area change, and obviously this will be different depending
on whether it is looked at from an angle, or straight on. If
angular momentum came in units, and say you looked at
something and it showed 8 units, then if you looked at it
from a very slightly different angle, the number of units
would be very slightly different, perhaps a tiny bit less than
8. But 7 is not a httle bit less than 8; it is a definite amount
less than eight. So it cannot possibly come in units. However
this proof is evaded by the subtleties and peculiarities of
quantum mechanics, and if we measure the angular momen-
tum about any axis, amazingly enough it is always a
number of units. It is not the kind of unit, like an electric
charge, that you can count. The angular momentum does
come in units in the mathematical sense that the number we
get in any measurement is a definite integer times a unit. But
we cannot interpret this in the same way as with units of
electric charge, imaginable units that we can count - one,
then another, then another. In the case of angular momen-
tum we cannot imagine them as separate units, but it comes
out always as an integer . . . which is very peculiar.

There are other conservation laws. They are not as
interesting as those I have described, and do not deal exactly

Figure 20

with the conservation of numbers. Suppose we had some
kind of device with particles moving with a certain definite
symmetry, and suppose their movements were bilaterally
symmetrical (fig. 20). Then, following the laws of physics,
with all the movements and collisions, you could expect, and
rightly, that if you look at the same picture later on it will
still be bilaterally symmetrical. So there is a kind of con-
servation, the conservation of the symmetry character. This
should be in the table, but it is not like a number that you
measure, and we will discuss it in much more detail in the
next lecture. The reason this is not very interesting in classi-
cal physics is because the times when there are such nicely
symmetrical initial conditions are very rare, and it is there-
fore a not very important or practical conservation law. But
in quantum mechanics, when we deal with very simple
systems like atoms, their internal constitution often has a
kind of symmetry, like bilateral symmetry, and then the
symmetry character is maintained. This is therefore an
important law for understanding quantum phenomena.

One interesting question is whether there is a deeper
basis for these conservation laws, or whether we have to take
them as they are. I will discuss that question in the next
lecture, but there is one point I should Uke to make now. In
discussing these ideas on a popular level, there seem to be
a lot of unrelated concepts; but with a more profound
understanding of the various principles there appear deep
interconnections between the concepts, each one implying
others in some way. One example is the relation between
relativity and the necessity for local conservation. If I had
stated this without a demonstration, it might appear to be
some kind of miracle that if you cannot tell how fast you
are moving this implies that if something is conserved it
must be done not by jumping from one place to another.

At this point I would like to indicate how the conserva-
tion of angular momentum, the conservation of momentum,
and a few other things aie to some extent related. The con-
servation of angular momentum has to do with the area
swept by particles moving. If you have a lot of particles

The Great Conservation Principles

(fig. 21), and take your centre (x) very far away, then the
distances are almost the same for every object. In this case
the only thing that counts in the area sweeping, or in the
conservation of angular momentum, is the component of
motion, which in figure 21 is vertical. What we discover then

Figure 21

is that the total of the masses, each multiplied by its velocity
vertically, must be a constant, because the angular momen-
tum is a constant about any point, and if the chosen point
is far enough away only the masses and velocities are rele-
vant. In this way the conservation of angular momentum
implies the conservation of momentum. This in turn implies
something else, the conservation of another item which is
so closely connected that I did not bother to put it in the
table. This is a principle about the centre of gravity (fig. 22).

Figure 22

A mass, in a box, cannot just disappear from one position
and move over to another position all by itself. That is
nothing to do with conservation of the mass ; you still have
the mass, just moved from one place to another. Charge

could do this, but not a mass. Let me explain why. The laws
of physics are not affected by motion, so we can suppose
that this box is drifting slowly upwards. Now we take the
angular momentum from a point not far away, x. As the
box is drifting upwards, if the mass is lying quiet in the box,
at position 1, it will be producing an area at a given rate.
After the mass has moved over to position 2, the area will
be increasing at a greater rate, because although the altitude
will be the same because the box is still drifting upwards,
the distance from x to the mass has increased. By the con-
servation of angular momentum you cannot change the
rate at which the area is changing, and therefore you simply
cannot move one mass from one place to another unless
you push on something else to balance up the angular mo-
mentum. That is the reason why rockets in empty space
cannot go . , , but they do go. If you figure it out with a lot
of masses, then if you move one forward you must move
others back, so that the total motion back and forward of all
the masses is nothing. This is how a rocket works. At first
it is standing still, say, in empty space, and then it shoots
some gas out of the back, and the rocket goes forward. The
point is that of all the stuff in the world, the centre of mass,
the average of all the mass, is still right where it was before.
The interesting part has moved on, and an uninteresting
part that we do not care about has moved back. There is
no theorem that says that the interesting things in the
world are conserved - only the total of everything.

Discovering the laws of physics is like trying to put to-
gether the pieces of a jigsaw puzzle. We have all these dif-
ferent pieces, and today they are proliferating rapidly. Many
of them are lying about and cannot be fitted with the other
ones. How do we know that they belong together? How do
we know that they are really all part of one as yet incom-
plete picture? We are not sure, and it worries us to some
extent, but we get encouragement from the common charac-
teristics of several pieces. They all show blue sky, or they
are all made out of the same kind of wood. All the various
physical laws obey the same conservation principles.

A physicist and educator here tells a parable to illus-
trate the inadequacies he sees in the present system of
teaching.

4 The Barometer Story

Alexander Calandra

An article from Current Science, Teacher's Edition, 1964.

SOME time ago, I received a call
from a colleague who asked if I
would be the referee on the grading
of an examination question. It seemed
that he was about to give a student
a zero for his answer to a physics ques-
tion, while the student claimed he
should receive a perfect score and
would do so if the system were not set
up against the student. The instructor
and the student agreed to submit this
to an impartial arbiter, and I was
selected.

The Barometer Problem

I went to my colleague's office and
read the examination question, which
was, "Show how it is possible to deter-
mine the height of a tall building with
the aid of a barometer."

The student's answer was, "Take the
barometer to the top of the building,
attach a long rope to it, lower the ba-
rometer to the street, and then bring
it up, measuring the length of the rope.
The length of the rope is the height of
the building."

Now, this is a very interesting an-
swer, but should the student get credit
for it? I pointed out that the student
really had a strong case for full credit,
since he had answered the question
completely and correctly. On the other
hand, if full credit were given, it could
well contribute to a high grade for the
student in his physics course. A high
grade is supposed to certify that the
student knows some physics, but the
answer to the question did not con-
firm this. With this in mind, I suggested
that the student have another try at
answering the question. I was not sur-
prised that my colleague agreed to

this, but I was surprised that the stu-
dent did.

Acting in terms of the agreement, I
gave the student six minutes to an-
swer the question, with the warning
that the answer should show some
knowledge of physics. At the end of
five minutes, he had not written any-
thing. I asked if he wished to give up,
since I had another class to take care
of, but he said no, he was not giving
up. He had many answers to this prob-
lem; he was just thinking of the best
one. I excused myself for interrupting
him, and asked him to please go on.
In the next minute, he dashed oflF his
answer, which was:

"Take the barometer to the top of
the building and lean over the edge of
the roof. Drop the barometer, timing
its fall with a stopwatch. Then, using
the formula S = /s at-', calculate the
height of the building."

At this point, I asked my colleague
if he would give up. He conceded and
I gave the student almost full credit. In
leaving my colleague's office, I recalled
that the student had said he had other
answ'ers to the problem, so I asked
him what they were. "Oh, yes," said
the student. "There are many ways of
getting the height of a tall building
with the aid of a barometer. For ex-
ample, you could take the barometer
out on a sunny day and measure the
height of the barometer, the length of
its shadow, and the length of the shad-
ow of the building, and by the use
of simple proportion, determine the
height of the building."

"Fine," I said. "And the others?"

"Yes, " said the student. "There is a
very basic measurement method that

you will like. In this method, you take
the barometer and begin to walk up
the stairs. As you climb the stairs, you
mark oflF the length of the barometer
along the wall. You then count the
number of marks, and this will give
you the height of the building in ba-
rometer units. A very direct method.

"Of course, if you want a more
sophisticated method, you can tie the
barometer to the end of a string, swing
it as a pendulum, and determine the
value of g' at the street level and at
the top of the building. From the dif-
ference between the two values of 'g,'
the height of the building can, in prin-
ciple, be calculated."

Finally he concluded, "If you don't
limit me to physics solutions to this
problem, there are many other an-
swers, such as taking the barometer to
the basement and knocking on the
superintendent's door. When the
superintendent answers, you speak to
him as follows: 'Dear Mr. Superin-
tendent, here I have a very fine ba-
rometer. If you will tell me the height
of this building, I will give you this
barometer.' "

At this point, I asked the student if
he really didn't know the answer to
the problem. He admitted that he did,
but that he was so fed up with college
instructors trying to teach him how
to think and to use critical thinking,
instead of showing him the structure
of the subject matter, that he decided
to take off on what he regarded mostly
as a sham.

The kinetic theory of gases Is a marvelous structure of
Interconnecting assumption, prediction, and experiment.
This chapter supplements and reinforces the discussion
of kinetic theory In the text of Unit 3.

The Great Molecular Theory of Gases

Eric M. Rogers

An excerpt from his book Physics for the Inquiring Mind: The Methods,
Nature, and Philosophy of Physical Science, 1960.

Newton's theory of universal gravitation was a
world-wide success. His book, the Principia, ran
into three editions in his lifetime and popular studies
of it were the fashion in the courts of Europe.
Voltaire wrote an exposition of the Principia for
the general reader; books were even published on
"Newton's Theory expounded to Ladies." Newton's
theory impressed educated people not only as a
brilliant ordering of celestial Nature but as a model
for other grand explanations yet to come. We con-
sider Newton's theory a good one because it is
simple and productive and links together many
diflFerent phenomena, giving a general feeling of
understanding. The theory is simple because its
basic assumptions are a few clear statements. This
simplicity is not spoiled by the fact that some of
the deductions need difficult mathematics. The suc-
cess of Newton's planetary theory led to attempts
at more theories similarly based on the laws of
motion. For example, gases seem simple in behavior.
Could not some theory of gases be constructed, to
account for Boyle's Law by "predicting" it, and to
make other predictions and increase our general
understanding?

Such attempts led to a great molecular theory of
gases. As in most great inventions the essential dis-
covery is a single idea which seems simple enough
once it is thought of: the idea that gas pressure is
due to bombardment by tiny moving particles, the
"molecules" of gas. Gases have simple common
properties. They always fill their container and
exert a uniform pressure all over its top, bottom, and
sides, unlike solids and liquids. At constant tempera-
ture, PRESstmE • VOLUME remains constant, however
the gas is compressed or expanded. Heating a gas
increases its pressure or volume or both — and the
rate of increase with temperature is the same for all
gases ("Charles' Law"). Gases move easily, diffuse
among each other and seep through porous walls.

Could these properties be "explained" in terms of
some mechanical pictvire? Newton's contemporaries
revived the Greek philosophers' idea of matter being
made of "fiery atoms" in constant motion. Now, with
a good system of mechanics they could treat such a
picture realistically and ask what "atoms" would do.
The most striking general property that a theory
should explain was Boyle's Law.

Boyle's Law

In 1661 Boyle announced his discovery, "not
without deUght and satisfaction" that the pressures
and volumes of air are "in reciprocal proportions."
That was his way of saying: pressure ex 1/volume
or PRESSURE • VOLUME remains constant, when air is
compressed. It was well known that air expands
when heated, so the restriction "at constant tempera-
ture" was obviously necessary for this simple law.
This was Boyle's discovery of the "spring of the
air" — a spring of variable strength compared with
sohd Hooke's Law springs.

In laboratory you should try a "Boyle's-Law
experiment" with a sample of dry air, not to "dis-
cover" a law that you already know, but as a prob-
lem in precision, "your skill against nature." You

Fic. 25-1. Boyle's Law

The Great Molecular Theory of Gases

will be limited to a small range of pressures (say
% atmosphere to 2 atm. ) and your accuracy may
be sabotaged by the room temperature changing
or by a slight taper in the glass tube that contains
the sample.^ If you plot your measurements on a
graph showing pressube vs. volume you will find
they mark a hyperbola — but that is too difficult a
curve to recognize for sure and claim as verification
of Boyle's Law.^ Then plot pressure vs. 1/volume
and look for a straight line through the origin.

Boyle's measurements were fairly rough and ex-
tended only from a fraction of an atmosphere to
about 4 atm. If you make precise measurements
with air you will find that pV changes by only a few
tenths of 1% at most, over that range. Your graph of
p vs. 1/V will show your experimental points very
close to a straight line through the origin. Since
mass/volume is density and mass is constant, values
of 1/V represent density, and Boyle's Law says

Fig. 25-2. Boyle's Law Isothermals

pressure a density. This makes sense on many a
simple theory of gas molecules: "put twice as many
molecules in a box and you will double the pressure."

All the measurements on a Boyle's-Law graph
line are made at the same temperature: it is an
isothermal line. Of course we can draw several iso-
thermals on one diagram, as in Fig. 25-2.

If the range of pressure is increased, larger devia-
tions appear — Boyle's simple law is only an approxi-
mate account of real gas behavior. It fits well at low
pressures but not at high pressures when the sample
is crowded to high density. Fig. 25-3 shows the

1 Even modern glass tubing is slightly tapered, unless made
uniform by an expensive process; so when experiments "to
verify Boyle's Law" show deviations from pV = constant they
are usually exhibiting tube-taper rather than misbehavior of
air. If the air sample is replaced by certain other gases such
as COc, or by some organic vapor, real deviations from
Boyle's Law become obvious and interesting. See Ch. 30.

2 The only safe shapes of graphs for testing a law, or find-
ing one, are straight lines and circles.

experimental facts for larger pressures, up to 3000
atmospheres. (For graphs of COj's behavior, in-
cluding Hquefaction, see Ch. 30.)

Theory

Boyle tried to guess at a mechanism underlying
his experimental law. As a good chemist, he pic-
tured tiny atomic particles as the responsible agents.
He suggested that gas particles might be springy,
like little balls of curly wool piled together, resisting
compression. Newton placed gas particles farther
apart, and calculated a law of repulsion-force to
account for Boyle's Law. D. Bernoulli published a
bombardment theory, without special force-laws,
that predicted Boyle's Law. He pointed out that
moving particles would produce pressure by bom-
barding the container; and he suggested that heating
air must make its particles move faster. This was the
real beginning of our present theory. He made a
brave attempt, but his account was incomplete.
A centiu-y later, in the 1840's, Joule and others set
forth a successful 'Tcinetic theory of gases," on this
simple basic view:

A gas consists of small elastic particles in
rapid motion: and the pressure on the walls
is simply the effect of bombardment.

Joule showed that this would "explain" Boyle's Law,
and that it would yield important information about
the gas particles themselves. This was soon polished
by mathematicians and physicists into a large,
powerful theory, capable of enriching our under-
standing.

In modern theories, we call the moving particles
molecules, a name borrowed from chemistry, where
it means the smallest particle of a substance that
exists freely. Split a molecule and you have separate
atoms, which may have quite difiFerent properties
from the original substance. A molecule of water,
H,0, split into atoms yields two hydrogen atoms
and one oxygen atom, quite difiFerent from the par-
ticles or molecules of water. Left alone, these sepa-
rated atoms gang up in pairs, Hj, O2 — molecules of
hydrogen and oxygen gas. In kinetic theory, we deal
with the complete molecules, and assume they are
not broken up by collisions. And we assume the
molecules exert no forces on each other except
during collisions; and then, when they are very
close, they exert strong repulsive forces for a very
short time: in fact that is all a collision is.

You yourself have the necessary tools for con-
structing a molecular theory of gases. Try it. Assume

"BOYLE'S law" for AIR

3000

PRESSURE
(atm.)

10 ■

I '/2 "/o Com

Vo[u4ne I °/o [(nv

VOLUME SCALE
EXPANDED, X 10.
PRESSURE SCALE
COMPRESSED^ -^ 10.

REOiONi OF Boyle's simple test

300 ^00

Extendi to vcCumc -^
— >- is.ooo

Volume
7 'z % %ft

Volume
S % %fi

A 10

- \o

Fig. 25-3. Deviations from Boyle's Law for Air at Room Temperature

The curve shows the pressure: volume relationship for an ideal gas obeying Boyle's Law.

The points show the behavior of air, indistinguishable from the curve at low pressures.

that gas pressure is due to molecules bouncing
elastically on the containing walls. Carry out the
first stages by working through Problems 1 and 2.
They start with a bouncing ball and graduate to

many bouncing molecules, to emerge with a pre-
diction of the behavior of gases. After you have
tried the problems, return to the discussion of de-
tails.

The Great Molecular Theory of Gases

Difficulties of the Simple Theory

The relation you worked out in Problem 2 seems to
predict a steady pressure and Boyle's-Law behavior,
from molecular chaos. How can a rain of molecules
hitting a wall make a steady pressure? Only if the col-
lisions come in such rapid succession that their bumps
seem to smooth out into a constant force. For that the

FORCE

ON END
OF BOX

InciividuoC impacts
j of moQcuks

AnasfuuUdis
i- totdcf F-At •

TIME

SMEARED our
TO AVERAGE FORCE

\ Same totaiana

mmm^'^i);:'mwfi^m;:^;&mdMm.z,

TIME
Fig. 25-6. Smoothing Out Impacts

molecules of a gas must be exceedingly numerous, and
ver>' small. If they are small any solid pressure-gauge
or container wall will be enormously massive compared
with a single gas molecule, so that, as impacts bring it
momentum, it will smooth them out to the steady pres-
sure we observe. (What would you expect if the con-
tainer wall were as hght as a few molecules?)

The problem pretended that molecules travel
straight from end to end and never collide with each
other en route. They certainly do collide — though we
cannot say how often without further information. How
will that affect the prediction?

* PROBLEM 3. COLLISIONS IN SIMPLE THEORY

(a) Show that It does not matter, in the simple derivation
of Problems 1 and 2, whether molecules collide or not.
(Consider two molecules moving to and fro from end to end,
just missing each other as they cross. Then suppose they
collide head-on and rebound. Why will their contribution to
the pressure be unchanged? Explain with a diagram.)

(b) What special assumption about molecules is required
for (a)?

(c) Suppose the molecules swelled up and became very
bulky (but kept the some speed, mass, etc.), would the effect
of mutual collisions be an increase of pressure (for the same
volume etc.) or a decrease or whot? (Note: "bulky" means
large in size, not necessarily large in mass.)

(d) Give a clear reason for your answer to (c).

Molecular Chaos

Molecules hitting each other, and the walls, at ran-
dom— some head on, some obliquelv, some glancing —
cannot all keep the some speed t" . One will gain in a
collision, and another lose, so that the gas is a chaos of
molecules with random motions whose speeds (chang-
ing at every collision) cover a wide range. Yet they

must preserve some constancy, because a gas exerts a
steady pressure.

In the prediction p'V= (%)[Nmt3*], we do not
have all N moleciiles moving with the same speed, each
contributing m v'^ inside the brackets. Instead we have
molecule #1 with its speed t»j, molecule #2 with t;^, . . . ,
molecule N with speed o^- Then

p . V = (%) [m «,« -I- m t;j« -I- . . . -I- mtj^']

= (%) [m(t;,^-1-t;,='-h...-hV)]

= (%) ["» (N • AVERAGES*) ] See note 3.

The c* in oiu- prediction must therefore be an average
o*, so that we write a bar over it to show it is an average
value. Our theoretical prediction now runs:

PRESStIRE • VOLVTME = % N • m • U*.

We know that if we keep a gas in a closed bottle its
pressiure does not jump up and down as time goes on;
its pressiire and volume stay constant. Therefore in
spite of all the changes in coUisions, the molecular v"^
stays constant. Already our theory helps us to picture
some order — constant v^ — among molecular chaos.

A More Elegant Derivation

To most scientists the regimentation that leads to the
factor Vi is too artificial a trick. Here is a more elegant
method that treats the molecules' random velocities
honestly with simple statistics. Suppose molecule #1 is
moving in a slanting direction in the box, with velocity
Oj. (See Fig. 25-7.) Resolve this vector v^ into three

Fig. 25-7. Alternative Treatment of

Gas Molecule Motion

(More professional, less artificial.)

In this we keep the random velocities, avoiding

regimentation, but split each velocity v into three

components, ,t>, ,v, .u, parallel to the sides of the box.

Then we deal with xt;' in calculating the pressure and

arrive at the same result. Sketches show three molecules

with velocities split into components.

components along directions x, y, z, parallel to the edges
of the box. Then Uj is the resultant of ^v^ along x and
yUj along y and ^Uj along z; and since these are mutually
perpendicular, we have, by the three-dimensional form

3 Because average u' = ( sum of all the o* values ) / ( num-
ber of t;' values) = (th' -f o,' + . . . -f t>s*)/(N)

. ■ . ( Ih' 4- tij' -f . . . -f On' ) = N • ( AVERAGE u' ) Or N • «*

This V2 is called the "mean square velocity." To obtain it, take
the speed of each molecule, at an instant, square it, add all
the squares, and divide by the number of molecules. Or,
choose one molecule and average its v^ over a long time —
say a billion collisions.

Fig. 25-8. Velocity Components
Pythagoras : t>i' — xfi* + ,t>i' + xVi"

of Pythagoras' theorem: v^^ = ^v^' + jV^^ + ,v^^

And for molecule #2 v^^ = ^v^^ + yO^* + ^v^^

And for molecule #3 v^'^ = „v^^-+ jV^^ + ^v^^

and so on

And for molecule #N u^^ = i^n^ + yf n^ + z%^
Add all these equations:

= (x«i' + .^^ + xt^a' + ■ ■ • + x«n')
+ (y"i' + yV + y«3' + • • • + yUj,^)
+ (,V + ,V+zV + --- + zV)
Divide by the number of molecules, N, to get average

values: — — — — ^

t;-' = ^v^ 4- yU- + jj-

Appealing to symmetry, and ignoring the small bias
given by gravity, we claim that the three averages on
the right are equal — the random motions of a statisti-
cally large number of molecules should have the same
distribution of velocities in any direction.

To predict the pressure on the end of the box we pro-
ceed as in Problem 2, but we use v^ for a molecule's
velocity along the length of the box. (That is the velocity
we need, because ^v and ^u do not help the motion
from end to end and are not involved in the change of
momentum at each end.) Then the contribution of
molecule #1 to pressube • volume is m • ^v^^ and the
contribution of all N molecules is

m (^t;,2 + ^t;,2 -|- . . . + ^u^.^) or m • N • ^^;

and by the argument above this is m • N • (^/3)

.". PRESSURE • volume =(%) N • m • t;^

(If you adopt this derivation, you should carry through
the algebra of number of hits in t sees, etc., as in
Problem 2. )

Molecular Theory's Predictions

Thinking about molecular collisions and using
Newton's Laws gave the (%) N • m • u- prediction:
PRESSURE • VOLUME = ( V6 ) N • m ' t;^

This looks like a prediction of Boyle's Law. The
fraction {\^) is a constant number; N, the number of
molecules, is constant, unless they leak out or split

up; m, the mass of a molecule, is constant. Then if
the average speed remains unchanged, {Vs) N ' rri' v^
remains constant and therefore p • V should remain
constant, as Boyle found it does. But does the speed
of molecules remain fixed? At this stage, you have
no guarantee. For the moment, anticipate later dis-
cussion and assume that molecular motion is con-
nected with the heat-content of a gas, and that at
constant temperature gas molecules keep a constant
average speed, the same speed however much the
gas is compressed or rarefied.* Later you will receive
clear reasons for believing this. If you accept it now,
you have predicted that:

The product p-V is constant for a gas at
constant temperature.

You can see the prediction in simplest form by
considering changes of densfty instead of volume:
just put twice as many molecules in the same box,
and the pressure will be doubled.

A marvelous prediction of Boyle's Law? Hardly
marvelous: we had to pour in many assumptions —
wdth a careful eye on the desired result, we could
scarcely help choosing wisely. A theory that gathers
assumptions and predicts only one already-known
law — and that under a further assumption regard-
ing temperature — would not be worth keeping. But
our new theory is just beginning: it is also helpful
in "explaining" evaporation, diffusion, gas friction;
it predicts effects of sudden compression; it makes
vacuum-pumps easier to design and understand.
And it leads to measurements that give validity to
its owTi assumptions. Before discussing the develop-
ment, we ask a basic question, "Are there really any
such things as molecules?"

Are there really molecules?

"That's the worst of circumstantial evidence.
The prosecuting attorney has at his command
all the facilities of organized investigation. He
uncovers facts. He selects only those which, in
his opinion, are significant. Once he's come to
the conclusion the defendant is guilty, the only
facts he considers significant are those which
point to the guilt of the defendant. That's why
circumstantial evidence is such a liar. Facts
themselves are meaningless. It's onlv the inter-
pretation we give those facts which counts."
"Perry Mason" — Erie Stanley Gardner"

* Actually, compressing a gas warms it, but we believe that
when it cools back to its original temperature its molecules,
though still crowded close, return to the same average speed
as before compression.

• The Case of the Perjured Parrot, Copyright 1039, by
Erie Stanley Gardner.

The Great Molecular Theory of Gases

A century ago, molecules seemed useful: a help-
ful concept that made the regularities of chemical
combinations easy to understand and provided a
good start for a simple theory of gases. But did they
really exist? There was only circumstantial evidence
tliat made the idea plausible. Many scientists were
skeptical, and at least one great chemist maintained
his riglit to disbelieve in molecules and atoms even
until the beginning of this century. Yet one piece of
experimental evidence appeared quite early, about
1827; the Brownian motion.

The Brownian Motion

The Scottish botanist Robert Brown (1773-1858)
made an amazing discovery: he practically saw
molecular motion. Looking tlrrough his microscope
at small specks of soUd suspended in water, he saw
them dancing with an incessant jigging motion. The
microscopic dance made the specks look aUve, but
it never stopped day after day. Heating made the
dance more furious, but on cooling it returned to its
original scale. We now know that any solid specks
in any fluid will show such a dance, the smaller the
speck the faster the dance, a random motion with
no rhyme or reason. Brown was in fact watching
the effects of water molecules jostling the solid
specks. The specks were being pushed around like
an elephant in the midst of a football game.

Watch this "Brownian motion" for yourself. Look
at small specks of soot in water ("India ink") with
a high-magnification microscope. More easily, look
at smoke in air with a low-power microscope. Fill
a small black box with smoke from a cigarette or a
dying match, and illuminate it with strong white
light from the side. The smoke scatters bluish-white
light in all directions, some of it upward into the
microscope. The microscope shows the smoke as a
crowd of tiny specks of white ash which dance
about with an entirely irregular motion.^ (See Fig.
30-3 for an example)

Watching the ash specks, you can see why Brown
at first thought he saw Uving things moving, but you
can well imagine the motion to be due to chance
bombardment by air molecules. Novi'adays we not
only think it may be that; we are sure it is, because
we can calculate the effects of such bombardment
and check them with observation. If air molecules
were infinitely small and infinitely numerous, they

s There may also be general drifting motions — convection
currents — but these are easily distingiiished. An ash speck
in focus shows as a small sharp wisp of white, often oblong;
but when it drifts or dances away out of focus the micro-
scope shows it as a fuzzy round blob, just as camera pictures
show distant street lights out of focus.

would bombard a big speck symmetrically from all
sides and there would be no Brownian motion to
see. At the other extreme, if there were only a few
very big molecules of surrounding air, the ash
speck would make great violent jumps when it did
get hit. From what we see, we infer something be-
tween these extremes; there must be many mole-
cules in the box, hitting the ash speck from all sides,
many times a second. In a short time, many hun-
dreds of molecules hit the ash speck from every
direction; and occasionally a few hundreds more
hit one side of it than the other and drive it noticea-
bly in one direction. A big jump is rare, but several
tiny random motions in the same general direction
may pile up into a visible shift.* Detailed watching
and calculation from later knowledge tell us that
what we see under the microscope are those gross
resultant shifts; but, though the individual move-
ments are too small to see, we can still estimate their
speed by cataloguing the gross staggers and ana-
lysing them statistically.

You can see for yourself that smaller specks dance
faster. Now carry out an imaginary extrapolation to
smaller and smaller specks. Then what motion
would you expect to see with specks as small as
molecules if you could see them? But can we see
molecules?

Seeing molecules?

Could we actually see a molecule? That would indeed
be convincing — we feel sure that what we see is real,
despite many an optical illusion. All through the last
century's questioning of molecules, scientists agreed
that seeing one is hopeless — not just unlikely but im-
possible, for a sound physical reason. Seeing uses light,
which consists of waves of very short wavelength, only
a few thousand Angstrom Units^ from crest to crest. We
see by using these waves to form an image:

with the naked eye we can see the shape of a pin's

head, a millimeter across, or 10,000,000 AU
with a magnifying glass we examine a fine hair,

1,000,000 AU thick
with a low-power microscope we see a speck of smoke

ash, 100,000 AU

with a high-power microscope, we see bacteria, from

10,000 down to 1000 AU

but there the sequence stops. It must stop because the

wavelength of visible light sets a limit there. Waves

can make clear patterns of obstacles that are larger

8 Imagine an observer with poor sight tracing the motion
of an active guest at a crowded party. He might fail to see
the guest's detailed motion of small steps here and there,
and yet after a while he would notice that the guest had
wandered a considerable distance.

T I Angstrom Unit, 1 AU, is 10"" meter.

than their wavelength, or even about their wavelength
in size. For example, ocean waves sweeping past an
island show a clear "shadow" of calm beyond. But waves
treat smaller obstacles quite differently. Ocean waves
meeting a small wooden post show no calm behind.
They just lollop around the post and join up beyond it
as if there were no post there. A blind man paddling
along a stormy seashore could infer the presence of an
island nearby, but would never know about a small post
just offshore from him.* Light waves range in wave-
length from 7000 AU for red to 4000 for violet. An
excursion into the short-wave ultraviolet, with photo-
graphic film instead of an eye, is brought to a stop by
absorption before wavelength 1000 AU: lenses, speci-
men, even the air itself, are "black" for extreme ultra-
violet light. X-rays, with shorter wavelength still, can
pass through matter and show grey shadows, but they
practically cannot be focused by lenses. So, although
X-rays have the much shorter wavelength that could
pry into much finer structures, they give us only un-
magnified shadow pictures. Therefore the limit imposed
by light's wavelength seemed impassable. Bacteria down
to 1000 AU could be seen, but virus particles, ten times
smaller, must remain invisible. And molecules, ten times
smaller still, must be far beyond hope. Yet viruses, re-
sponsible for many diseases, are of intense medical
interest — we now think they may mark the borderhne
between living organisms and plain chemical molecules.
And many basic questions of chemistry might be an-
swered by seeing molecules.

The invisibility of molecules was unwelcome, but
seemed inescapable. Then, early in this century, X-rays
offered indirect information. The well-ordered atoms
and molecules of crystals can scatter X-rays into regular
patterns, just as woven cloth can "diffract" light into
regular patterns — look at a distant lamp at night
through a fine handkerchief or an umbrella. X-ray pat-
terns revealed both the arrangement of atoms in crystals
and the spacing of their layers. Such measurements
confirmed the oil-film estimates of molecular size. More
recently, these X-ray diffraction-splash pictures have
sketched the general shape of some big molecules —
really only details of crystal structure, but still a good
hint of molecular shape. Then when physicists still
cried "no hope" the electron microscope was invented.
Streams of electrons, instead of light-waves, pass through
the tiny object under examination, and are focused by
electric or magnetic fields to form a greatly magnified
image on a photographic film. Electrons are incom-
parably smaller agents than light-waves,» so small that

* Tiny obstacles do produce a small scattered ripple, but
this tells nothing about their shape. Bluish light scattered
by very fine smoke simply indicates there are very tiny
specks there, but does not say whether they are round or
sharp-pointed or oblong. The still more bluish light of the
sky is sunlight scattered by air molecules.

* Electrons speeding through the electron microscope be-
have as if they too have a wavelength, but far shorter than
the wavelength of light. So they offer new possibilities of
"vision," whether you regard them as minute bullets smaller
than atoms, or as ultra-short wave patterns. A technology of
"electron optics" has developed, with "lenses" for electron
microscopes and for television tubes (which are electron
projection-lanterns ) .

even "molecules" can be dehneated. Then we can "see"
virus particles and even big molecules in what seem to
be reliable photographs with huge magnifications. These
new glimpses of molecular structure agree well with
the speculative pictures drawn by chemists arguing
very cleverly from chemical behavior.

Recendy, still sharper methods have been developed.
At the end of this book you will see a picture of the
individual atoms of metal in a needle point. Whv not
show that now? Because, like so much in atomic physics,
the method needs a sophisticated knowledge of assump-
tions as well as techniques before you can decide in
what sense the photograph tells the truth. Going still
deeper, very-high-energy electrons are now being used
to probe the structure of atomic nuclei, yielding indirect
shadow pictures of them.

In the last 100 years, molecules have graduated from
being tiny uncounted agents in a speculative theory to
being so real that we e\ en expect to "see" their shape.
Most of the things we know about them — speed, num-
ber, mass, size — were obtained a century ago with the
help of kinetic theory. The theory promoted the meas-
urements, then the measurements gave validity to the
theory. We shall now leave dreams of seeing molecules,
and study what we can measure by simple experiments.

Measuring the Speed of Molecules
Returning to our prediction that:

PRESSURE • VOLUME = {\i) N ■ TTl 'V^

We can use this if we trust it, to estimate the actual
speed of the molecules. N is the number of molecules
and m is the mass of one molecule so Nm is the total
mass M of all the molecules in the box of gas. Then
we can rewrite our prediction:

PRESSURE • VOLUME = ( W ) • M 'X?

where M is the total mass of gas. We can weigh a
big sample of gas with measured volume at known
pressure and substitute our measurements in the
relation above to find the value of v^ and thus the
value of the average speed.

Fig. 25-9 shows the necessary measurements.
Using the ordinary air of the room, we measure its
pressure by a mercury barometer. (Barometer
height and the measured density of mercury and
the measured value of the Earth's gravitational field
strength, 9.8 newtons per kilogram, will give the
pressure in absolute units, newtons per square
meter. )^° We weigh the air which fills a flask. For
this, we weigh the flask first full of air at atmospheric
pressure and second after a vacuum pump has taken
out nearly all the air. Then we open the flask under
water and let water enter to replace the air pumped

1" Since we made our kinetic theory prediction with the
help of Newton's Law II, the predicted force must be in
absolute units, newtons; and the predicted pressure must be
in newtons per square meter.

The Great Molecular Theory of Gases

h meUra DENSITY OF MERCURY, d

.. .. . 2l

re«»ure of icmo.phere

\ '

: (barometer height) (dcnlity of mercury) (fuld •trenglh. g)

•. . . . % # ^R % .newtone.
P = hdg (meter.) I^ublc meter' * Eg *

= h'dg newtona/square meter

====3^ \W

Masf of air

^ ^ — •^~^y I pumped out.

KINETIC THEORY PREDICTS THAT:

lHlli'll(hMinMII!!l:rMilllll'lll'::i"">i:i'':":i»<i
AVERAGE (VELOCITY^),

DENSITY-
f

Fig. 25-9.
Measuring Molecule Velocities indikectly,

BUT SIMPLY, ASSUMING KlNETIC ThEORY.

out. Measuring the volume of water that enters the
flask tells us the volume of air which has a known
mass. Inserting these measurements in the predicted
relation we calculate o^ and thence its square root
V(f^) which we may call the "average speed," x>
(or more strictly the "root mean square," or R.M.S.
speed). You should see these measurements made
and calculate the velocity, as in the following
problem.

* PROBLEM 4. SPEED OF OXYGEN MOLECULES

Experiment shows that 32 kg of oxygen occupy 24 cubic
meters at atmospheric pressure, at room temperature.

(a) Calculate the density, MASS/VOLfME, of oxygen.

(b) Using the relation given by kinetic theory, calculate the
mean square velocity, v^, of the molecules.

(d) Also express this very roughly in miles/hour.
(Take 1 kilometer to be 5/8 mile)

Air molecules moving V* mile a second! Here is
theory being fruitful and validating its own assump-
tion, as theory should. We assumed that gases con-

sist of molecules that are moving, probably moving
fast; and our theory now tells us how fast, with
the help of simple gross measurements. Yet theory
cannot prove its own prediction is true — the result
can only be true to the assumptions that went in.
So we need experimental tests. If the theory passes
one or two tests, we may trust its further predictions.

Speed of Molecules: experimental evidence

We have rough hints from the speed of sound and
from the Brownian motion.

PROBLEM 5. SPEED OF SOUND

We believe that sound is carried by waves of compression
and rarefaction, with the changes of crowding and motion
handed on from molecule to molecule at collisions. If air does
consist of moving molecules far apart, what can you say
about molecular speed, given that the measured speed of
sound in air is 340 meters/sec («= 1 1 00 ft/sec)?

PROBLEM 6. BROWNIAN MOTION

Looking at smoke under a microscope you will see large
specks of ash jigging quite fast; small specks jig faster still.

(a) There may be specks too small to see. What motion
would you expect them to hove?

(b) Regarding o single air molecule as an even smaller "ash
speck," what can you state about its motion?

The two problems above merely suggest general
guesses. Here is a demonstration that shows that
gas molecules move very fast. Liquid bromine is
released at the bottom of a tall glass tube.* The

(a) Brvrrum diB^surxa incur, (f) Brvnune nUasciiik vacuum'.

To vacuwn
jmnw

~Au-

-\/aauun

lenntuion to enaSk
capsuCe's Cong, tfun,
neck to Se Sroken,

Fig. 25-10. Motion of Bromine Molecules:
Demonstration of Molecular Speed.

" The bromine is inserted as liquid bromine in a small glass
capsule with a long nose that can be broken easily.

liquid evaporates immediately to a brown vapor
or "gas," which slowly spreads throughout the tube.
The experiment is repeated in a tube from which
all air has been pumped out. Now the brown gas
moves very fast when released. ( In air, its molecules
still move fast, but their net progress is slow be-
cause of many collisions with air molecules.)

Direct Measurement

The real test must be a direct measurement.
Molecular speeds have been measured by several
experimenters. Here is a typical experiment, done
by Zartman. He let a stream of molecules shoot
through a slit in the side of a cylindrical drum that
could be spun rapidly. The molecules were of bis-
muth metal, boiled off molten liquid in a tiny oven
in a vacuum. A series of barriers with slits selected
a narrow stream to hit the drum. Then each time
the slit in the drum came around, it admitted a small
flock of moving molecules. With the drum at rest,
the molecules travelled across to the opposite wall
inside the drum and made a mark on a receiving
film opposite the slit. With the drum spinning, the
film was carried around an appreciable distance
while the molecules were travelling across to it, and
the mark on it was shifted to a new position. The
molecules' velocity could be calculated from the
shift of the mark and the drum's diameter and spin-
speed. When the recording film was taken out of
the drum it showed a sharp central mark of de-
posited metal but the mark made while it spun was
smeared out into a blur showing that the molecular
velocities had not all been the same but were spread
over a considerable range. Gas molecules have ran-
dom motion with frequent collisions and we must
expect to find a great variety of velocities at any
instant. It is the average velocity, or rather the root-
mean-square average, y/{v^), that is involved in
kinetic theory prediction. The probable distribution
of velocities, clustering round that average, can be
predicted by extending simple kinetic theory with
the help of the mathematical statistics of chance. In
Zartman's experiment, we e-xpect the beam of hot
vapor molecules to have the same chance distribu-
tion of velocities with its peak at an average value
characteristic of the temperature. Measurements of
the actual darkening of the recording film showed
just such a distribution and gave an average that

zartman's experiment

Rotaiinj drum -^l^i^h^JmsiJhvf

ih) Varwus Stupes of tdc rotacun of tfu drum

(c)

SPECIMEN FILM (u#iroi&(i')

^»\arks mack Cy moHcuCes cj vonWs speeds

-zero mark." made fu moUadcs wkm drum
is not syuuww

Fic. 25-11. Measuring Molecule Velocities Directly

(a) Sketch of Zartman's experiment.

(b) These sketclies show various stages

of the rotation of the drum.

agreed well with the value predicted by simple
theory (see sketch of graph in Fig. 25-12)."

Molecular Speeds in Other Gases. Diffusion

Weighing a bottle of hydrogen or helium at at-
mospheric pressure and room temperature shows
these gases are much less dense than air; and car-
bon dioxide is much more dense. Then our predic-

11 Zartman's method is not limited to this measurement.
One method of separating uranium 235 used spinning slits,
though the uranium atoms were electrically charged and
were given high speeds by electric fields. And mechanical
"chopper" systems are used to sort out moving neutrons.

Such choppers operate like traffic lights set for some constant
speed. The simplest prototype of Zartman's experiment is the
scheme shown in Fig. 8-8 for measuring the speed of a rifle
bullet.

The Great Molecular Theory of Gases

Distanci aCorw rturrd jrmn zero mark

Fic. 25-12. Results of Zartman's Experiment

The curve, drawn by a grayncss-measuring-machine, shows

the experimental results. The crosses show values

predicted by kinetic theory with simple statistics.

tion pV = {}'i) M v^ tells us that hydrogen and
helium molecules move faster than air molecules
(at the same temperature), and carbon dioxide
molecules slower. Here are actual values:

Gas

Measurements at Room

Temperature and
Atmospheric Pressure
Volume Mass

hydrogen
helium

carbon dioxide
oxygen
nitrogen
air ( % oxygen
% nitrogen )

24 cu. meters
24 "
24 "
24 "
24 "

24 "

2.0 kilograms

4.0 kg
44.0 kg
32.0 kg
28.0 kg

28.8 kg

* PROBLEM 7. SPEEDS

(i) If oxygen molecules move about i mile/sec at room
temperature, how fast do hydrogen molecules move?

(fi) How does the average speed of helium molecules com-
pare with that of hydrogen molecules at the same tem-
perature? (Give the ratio of "average" speeds.)

(iii) How does the speed of carbon dioxide molecules com-
pare with that of air molecules at the same tempera-
ture? (Give the ratio of "average" speeds.)

PROBLEM 8

Making a risky guess,* say whether you would expect the
speed of sound in helium to be the some as in air, or bigger
or smaller. Test your guess by blowing an organ pipe first
with air, then with helium (or with carbon dioxide). Or
breathe in helium and then talk, using your mouth and nose
cavities as miniature echoing organ pipes. A change in the
speed of sound changes the time taken by sound waves to

* It is obviously risky, since we ore not considering the mechonism
of sound transmission in detail. In foct there is an unexpected
factor, which is different for helium: the eose with which the gas
heats up as sound-compressions pass through. This momentary rise
of temperoture makes sound compressions travel faster. The effect
is more pronounced in helium than in air, making the speed of
sound 8% bigger thon simple comparison with air suggests.
Kinetic theory con predict this effect of specific heat, telling us
that helium must have o smaller heat capacity, for a good otomic-
molecular reason.

bounce up and down the pipe, and thus changes the fre-
quency at which sound pulses emerge from the mouth. And
that changes the musical note of the vowel sounds, which
rises to higher pitch at higher frequency.

PROBLEM 9

How would you expect the speed of sound in air to change
when the pressure is changed without any change of tem-
perature? (Try this question with the following data, for air
at room temperature: 28.8 kg of air occupy 24 cubic meters
at 1 atmosphere pressure; at 2 atmospheres they occupy
1 2 cubic meters.)

Diffusion

If molecules of different gases have such different
speeds, one gas should outstrip another when they
diffuse through long narrow pipes. The pipes must
be very long and very narrow so that gas seeps
through by the wandering of individual molecules
and not in a wholesale rush. The pores of unglazed
pottery make suitable "pipes" for this. See Fig. 25-
13a, b. The white jar J has fine pores that run right
through its walls. If it is filled with compressed gas
and closed with a stopper S, the gas will slowly leak
out through the pores into the atmosphere, as you
would expect. But if the pressure is the same (at-
mospheric) inside and out you would not expect
any leakage even if there are different gases inside
and outside. Yet there are changes, showing the
effects of different molecular speeds. The demon-
strations sketched start with air inside the jar and
another gas, also at atmospheric pressure, outside.
You see the effects of hydrogen molecules whizzing
into the jar faster than air can move out; or of air
moving out faster than COj molecules crawl in.
These are just qualitative demonstrations of "diffu-
sion," but they suggest a process for separating
mixed gases. Put a mixture of hydrogen and COj
inside the jar; then, whether there is air or vacuum
outside, the hydrogen will diffuse out faster than the
COj, and by repeating the process in several stages

Fig. 25-13a. Diffusion of Gases
Hydrogen diffuses in through the porous wall J faster
than air diffuses out.

CO,-

Carion (Unide . — (^. - -. . i i - .

Fic. 25-13b. Diffusion of Gases
Carbon dioxide diffuses in through the porous wall, J, slower than air diffuses out.

you could obtain almost pure hydrogen. This is a
physical method of separation depending on a
difference of molecular speeds that goes with a
difference of molecular masses (see Fig. 25-14). It
does not require a difference of chemical properties;
so it can be used to separate "isotopes," those twin-
brothers that are chemically identical but differ
slightly in atomic masses. When isotopes were first
discovered, one neon gas 10% denser than the other,
some atoms of lead heavier than the rest, they were
interesting curiosities, worth trying to separate just
to show. Diffusion of the natural neon mixture from
the atmosphere proved the possibility. But now with
two uranium isotopes hopelessly mixed as they
come from the mines, one easily fissionable, the
other not, the separation of the rare fissionable kind
is a matter of prime importance. Gas diffusion is
now used for this on an enormous scale. See Prob-
lem 11, and Figs. 25-15, 16 and 17. Also see Chs. 30
and 43.

START

I atm.f I atm^

1 i

Temperature

Heating a gas increases p or V or both. With a
rise of temperature there is always an increase of
pV, and therefore of (%) N m v^. Therefore making
a gas hotter increases v^, makes its molecules move
faster. This suggests some effects of temperature.

* PROBLEM 10

(a) Would you expect the speed of sound to be greater, less,
or the same in air at higher temperature? Explain.

(b) Would you expect diffusion of gases to proceed faster,
slower, or at the some rate, at higher temperature? Ex-
plain.

Kinetic Theory To Be Continued

We cannot give more precise answers to such
questions until we know more about heat and tem-
perature and energy. Then we can extract more
predictions concerning gas friction, heat conduc-
tion, specific heats; and we shall find a way of

LATER.
0.^ atm, 1.1 (Utn,

AIR . 1% ' COi •

) o » 1 ^ _ • •

■A

^/r.

v^.

««**^f

AIR

CO,

^POROUS '//
/^, BARRIER/^

Fic. 25-14. Unequal Diffusion of Gases

AIK ^ CO^ CO^ ^- AIR

Air and carbon dioxide, each originally at atmospheric pressure, are separated by a porous barrier.

luai 1

At the start, with equal volumes at the same pressure, the two populations have equal numbers of molecules.

On the average, air molecules stagger through the pores faster than CO. molecules.

Then the populations are no longer equal so the pressures are unequal.

The Great Molecular Theory of Gases

. • '(ir

. /?

m.*JA

Fig. 25-15. Sepahation of Uranium Isotopes by Diffusion of UFe Thhough a Porous Barrier
Gas molecules hit the barrier, and the walls of its pores, many times — net result: a few get through.

Soikto

from.
List itjje

Fig. 25-16a. Separation of Uranium Isotopes by
Diffusion of UFs Through a Porous Barrier.

measuring the mass of a single molecule, so that
we can count the myriad molecules in a sample of
gas. We shall return to kinetic theory after a study
of energy. Meanwhile, it is kinetic theory that leads
us towards energy by asking a question:

What is mv^?

The expression ( % ) N m t;'' is very important in
the study of all gases. Apart from the fraction (%)
it is

THE NUMBER OF MOLECULES • (mv' for one moleculc)
What is mt;* for a moving molecule? It is just the
mass multiplied by the square of the speed; but
what kind of thing does it measure? What are its
properties? Is it an important member of the series:
m mv mv^ ? We know m, mass, and treat

< ' f n V

-r^

^ ^

■%

^^<^

Fig. 25-16b. Multi-Stage Diffusion Separation

Mixture diffusing through in one stage is pumped to the

input of the next stage. Unused mixture from one stage is

recycled, pumped back to the input of the preceding stage.

Fig. 25-17. Separating Uranium Isotopes by Diffusion

To effect a fairly complete separation of

U*" F., thousands of stages are needed.

it as a constant thing whose total is universally con-
served. We know mv, momentum, and trust it as a
vector that is universally conserved. Is mv^ equally
useful? Its structure is mv • v or Ft • v or

FORCE • TIME * DISTANCe/tIME.

Then mv^ is of the form force • distance. Is
that product useful? To push with a force along
some distance needs an engine that uses fuel.
Fuel . . . money . . . energy. We shall find that
mv^ which appears in our theory of gases needs only
a constant factor (%) to make it an expression of
"energy."

PROBLEMS FOR CHAPTER 25

if 1. DERIVING MOLECULAR PRESSURE

Work through the question sheets of Problem 1 shown
earlier in this chapter. These lead up to the use of Newton's
mechanics in a molecular picture of gases.

if 2. KINETIC THEORY WITH ALGEBRA

Work through the question sheets of Problem 2.

Problems 3-10 are in the text of this chapter.

■:*: 11. URANIUM SEPARATION (For more professional
version, see Problem 3 in Ch. 30)

Chemical experiments and arguments show that oxygen
molecules contain two atoms so we write them Os; hydrogen
molecules have two atoms, written Hj; and the dense vapor
of uranium flouride has structure UFg.

Chemical experiments tell us that the relative masses of
single atoms of O, H, F, and U are 16, 1, 19, 238. Chemical
evidence and a brilliant guess (Avogadro's) led to the belief
that 0 standard volume of any gas at one atmosphere and
room temperature contains the same number of molecules
whatever the gas (the same for Oi, Hj, or UFe). Kinetic
theory endorses this guess strongly (see Ch. 30).

(a) Looking back to your calculations in Problem 7 you will
see that changing from O2 to H2 changes the mass of a
molecule in the proportion 32 to 2. For the some tem-
perature what change would you expect in the v" and
therefore what change in the overage velocity? (That is,
how fast are hydrogen molecules moving at room tem-
perature compared with oxygen ones? Give a ratio show-
ing the proportion of the new speed to the old. Note you
do not hove to repeat oil the arithmetic, just consider the
one factor that changes.)

(b) Repeat (a) for the change from oxygen to uranium
fluoride vapor. Do rough Arithmetic to find approximate
numerical value.

(c) Actually there ore several kinds of uranium otom. The
common one has moss 238 (relative to oxygen 16) but
a rare one (0.7% of the mixture got from rocks) which is
in fact the one that undergoes fission, has moss 235.
One of the (very slow) ways of separating this valuable
rare uranium from the common one is by converting the
mixture to fluoride and letting the fluoride vapor diffuse
through a porous wall. Because the fluoride of U-^^ has
a different molecular speed the mixture emerging after
diffusing through has different proportions,
(i) Does it become richer or poorer in U-^^?
(ii) Give reasons for your answer to (i).
(iii) Estimate the percentage difference between average
speeds of [U^a-Fg] and [U^-^sFg"! molecules.

(Note: As discussed in Ch. 1 1 , a change of x % in some
measured quantity Q makes a change of about
i x % in VQ ■)

12. Figs. 25-1 3a and 25-1 3b show two diffusion demon-
strations. Describe what happens and interpret the experi-
ments.

* 13. MOLECULAR VIEW OF COMPRESSING GAS

(a) When on elastic boll hits a massive wall head-on it
rebounds with much the same speed as its original speed.
The some happens when a boll hits a massive bat which
is held firmly. However, if the bat is moving towards the
ball, the ball rebounds with a different speed. Does it
move faster or slower?

(b) (Optional, hard: requires careful thought.) When the bat
is moving towards the boll is the time of the elastic
impact longer, shorter, or the some as when the bat is
stationary? (Hint: If elastic .... S.H.M. . . .)

(c) When a gas in a cylinder is suddenly compressed by the
pushing in of a piston, its temperature rises. Guess at an
explanation of this in terms of the kinetic theory of
gases, with the help of (o) above.

(d) Suppose a compressed gas, as in (c), is allowed to push
a piston out, and expand. What would you expect to
observe?

* 14. MOLECULAR SIZE AND TRAVEL

A closed box contains a large number of gas molecules
at fixed temperature. Suppose the molecules magically be-
came more bulky by swelling up to greater volume, without
any increase in number or speed, without any change of
mass, and without any change in the volume of the box.

(a) How would this affect the overoge distance apart of the
molecules, center to center (great increase, decrease, or
little change)?

(b) Give a reason for your onswer to (a).

(c) How would this affect the average distance travelled by
a molecule between one collision and the next (the
"mean free path")?

(d) Give a reason for your answer to (c).

Changes in the visible world are often the result of the rule of
probability at work in the submicroscopic world. A survey of
principles of probability, reasons why there are no perpet-
ual-motion machines, entropy and time's arrow— and much
else.

6 Entropy and the Second Law of Thermodynamics

Kenneth W. Ford

An excerpt from his book Basic Physics, 1968.

As profound as any principle in physics is the second law of thermodynamics.
Based on uncertainty and probability in the submicroscopic world, it accounts
for definite rules of change in the macroscopic world. We shall approach this law,
and a new concept, entropy, that goes with it, by considering some aspects of
probability. Through the idea of probability comes the deepest understanding of
spontaneous change in nature.

14.1 Probability in nature

When a spelunker starts down an unexplored cavern, he does not know how far
he will get or what he will find. When a gambler throws a pair of dice, he does
not know what number will turn up. When a prospector holds his Geiger counter
over a vein of uranium ore, he does not know how many radioactive particles
he will count in a minute, even if he counted exactly the number in a preceding
minute. These are three quite different kinds of uncertainty, and all of them
are familiar to the scientist.

The spelunker cannot predict because of total ignorance of what lies ahead.
He is in a situation that, so far as he knows, has never occurred before. He is
like a scientist exploring an entirely new avenue of research. He can make
educated guesses about what might happen, but he can neither say what will
happen, nor even assess the probability of any particular outcome of the
exploration. His is a situation of uncertain knowledge and uncertain probability.
The gambler is in a better position. He has uncertain knowledge but certain
probability. He knows all the possible outcomes of his throw and knows exactly
the chance that any particular outcome will actually occur. His ignorance of any
single result is tempered by a definite knowledge of average results.

The probability of atomic multitudes, which is the same as the probability
of the gambler, is at the heart of this chapter. It forms the basis for the explana-
tion of some of the most important aspects of the behavior of matter in bulk.
This kind of probability we can call a probability of ignorance — not the nearly

Figure 14.1 A tray of coins, a system governed by laws of probability.

total ignorance of the spelunker in a new cave or the researcher on a new frontier,
but the ignorance of certain details called initial conditions. If the gambler
knew with enough precision every mechanical detail of the throw of the dice
and the frictional properties of the surface onto which they are thrown (the
initial conditions) he could (in principle) calculate exactly the outcome of the
throw. Similarly, the physicist with enough precise information about the where-
abouts and velocities of a collection of atoms at one time could (with an even
bigger "in principle"*) calculate their exact arrangement at a later time.
Because these details are lacking, probability necessarily enters the picture.

The prospector's uncertainty is of still a different kind. He is coming up against
what is, so far as we now know, a fundamental probability of nature, a
probability not connected with ignorance of specific details, but rather connected
with the operation of the laws of nature at the most elementary level. In atomic
and nuclear events, such as radioactivity, probability plays a role, even when
every possible initial condition is known. This fundamental probability in
nature, an essential part of the theory of quantum mechanics, is pursued in
Chapter Twenty-Three. In thermodynamics — the study of the average behavior
of large numbers of molecules and of the links between the submicroscopic and
macroscopic worlds — the fundamental probability in nature is of only secondary
importance. It influences the details of individual atomic and molecular collisions,
but these details are unknown in any case. Of primary importance is the
probability of ignorance stemming from our necessarily scant knowledge of pre-
cise details of molecular motion.

The triumphs of thermodynamics are its definite laws of behavior for systems
about which we have incomplete knowledge. However, it should be no surprise
that laws of probability applied to large enough numbers can become laws of near
certainty. The owners of casinos in Nevada are consistent winners.

14.2 Probability in random events

We turn our attention now to a system that at first sight has little to do with
molecules, temperature, or heat. It is a tray of coins (Figure 14.1). For the
purposes of some specific calculations, let us suppose that the tray contains just
five coins. For this system we wish to conduct a hypothetical experiment and

* Because classical mechanics does not suffice to calculate exactly the outcome of an atomic
collision, this hypothetical forecast of future atomic positions and velocities could be extended
but a moment forward in time.

Entropy and the Second Law of Thermodynamics

make some theoretical predictions. The experiment consists of giving the tray
a sharp up-and-down motion so that all the coins flip into the air and land again
in the tray, then counting the number of heads and tails displayed, and repeating
this procedure many times. The theoretical problem is to predict how often a
particular arrangement of heads and tails will appear.

Table 14.1 Possible Arrangements of Five Coins

Coin 1

Coin 2

Coin 3

Coin 4

Coin 5

1 way to get 5 heads

H
H
H

H
H
T

H
T
H

H
H

H
H
H

5 ways to get 4 heads
and 1 tail

10 ways to get 3 heads

and 2 tails

10 ways to get 2 heads

and 3 tails

T
T
T

H
T
T

T
H
T

T
T
H

T
T
T

5 ways to get 1 head
and 4 tails

1 way to get 5 tails

The experiment you can easily carry out yourself. Be sure that the tray is
shaken vigorously enough each time so that at least some of the coins flip over.
Here let us be concerned with the theory. To begin, we enumerate all possible
ways in which the coins can land. This is done pictorially in Table 14.1. There
are 32 possible results of a tray shaking.* If all we do is count heads and tails
without identifying the coins, the number of possible results is 6 instead of 32
(Table 14.1). Ten of the ways the coins can land yield three heads and two tails.

* Since each coin can land in two ways, the total number of ways in which five coins ran h\xu\
is2x2x2x2x2 = 25 = 32. Three coins could land in 8 different ways (2^), four coins
in 16 ways (2*), and so on. In how many ways could 10 coins land?

There are also ten different ways to get three tails and two heads. Both four
heads and one tail and four tails and one head can be achieved in five ways.
Only one arrangement of coins yields five heads, and only one yields five tails.
These numbers do not yet constitute a prediction of the results of the experiment.
We need a postulate about the actual physical process, and a reasonable one
is a postulate of randomness: that every coin is equally likely to land heads up
or tails up and that every possible arrangement of the five coins is equally Jikely.
This means that after very many trials, every entry in Table 14.1 should have
resulted about -^ of the time. Note, however, that equal probability for each
arrangement of coins is not the same as equal probability for each possible
number of heads or tails. After 3,200 trials, for example, we would expect to have
seen five heads about 100 times, but three heads and two tails should have showed
up ten times more frequently, about 1,000 times. The exact number of ap-
pearances of five heads or of three heads and two tails or of any other combina-
tion cannot be predicted with certainty. What can be stated precisely (provided
the postulate of randomness is correct) are probabilities of each such combination.

Table 14.2 Probabilities for Different Numbers of Heads and Tails
When Five Coins Are Flipped

Probabilify

1/32 = 0.031
5/32 = 0.156
10/32 = 0.313
10/32 = 0.313
5/32 = 0.156
1/32 = 0.031

No. Heads

. Tails

Total probability = 1.000

Shown in Table 14.2 are the basic probabilities for all the possible numbers of
heads and tails that can appear in a single trial. It is interesting to present these
numbers graphically also, as is done in Figure 14.2. The probability of a certain

0.3 -

„ 0.2 h

^ 0.1 h ./

/
/

> ^,

Nunil)er of heads 0 12 3 4 5

Number of tailfl 5 4 3 2 10

Figure 14.2 Probabilities for various results of tray-shaking experiment with
five coins.

Entropy and the Second Law of Thermodynamics

Number of heads

Figure 14.3 Probabilities for various results of tray-shaking experiment with
ten coins.

number of heads plotted vs. the numbers of heads gives a bell-shaped curve, high
in the middle, low in the wings.

Table 14.3 Probabilities for Different Numbers of Heads and Tails
When Ten Coins Are Flipped

No. Heads

No. Tails

Probability

1/1024 = 0.0010

10/1024 = 0.0098

45/1024 = 0.0439

120/1024 = 0.1172

210/1024 = 0.2051

252/1024 = 0.2460

210/1024 = 0.2051

120/1024 = 0.1172

45/1024 = 0.0439

10/1024 = 0.0098

1/1024 = 0.0010
Total probability = 1.0000

The same kind of calculation, based on the postulate of randomness can be
carried out for any number of coins. For ten coins, the basic probabilities are
given in Table 14.3 and in Figure 14.3.* Two changes are evident. First, the
probability of all heads or all tails is greatly reduced. Second, the bell-shaped

' The reader familiar with binomial coeflScients may be interested to know that the number of
arrangements of n coins to yield m heads is the binomial coefficient

/n\ ^ n!

\m/ ml(Ti — m)\

Thus the probabilities in Table 14.3 are proportional to

Co")' CO' a)'

and so on.

probability curve has become relatively narrower. The greater the number of
coins, the less likely is it that the result of a single trial will be very different
from an equal number of heads and tails. To make this point clear, the probability
curve for a tray of 1,000 coins is shown in Figure 14.4. The chance of shaking
all heads with this many coins would be entirely negligible even after a lifetime
of trying. As Figure 14.4 shows, there is not even much chance of getting a
distribution as unequal as 450 heads and 550 tails.

The tendency of the probabilities to cluster near the midpoint of the graph,
where the number of heads and the number of tails are nearly equal, can be
characterized by a "width" of the curve. The width of the curve is defined to
be the distance between a pair of points (see Figures 14.3 and 14.4) outside
of which the probabilities are relatively small and inside of which the probabilities
are relatively large. Exactly where these points are chosen is arbitrary. One con-
venient choice is the pair of points where the probability has fallen to about one
third of its central value — more exactly to 1/e = 1/2.72 of its central value.
The reason for defining a width is this: It spans a region of highly probable results.
After the tray is shaken, the number of heads and the number of tails are most
likely to correspond to a point on the central part of the curve within its width.
The distribution of heads and tails is unlikely to be so unequal as to correspond
to a point on the curve outside of this central region. When the number of coins
is reasonably large (more than 100), there is a particularly simple formula for
the width of the probability curve. If C is the number of heads (or tails) at tht
center of the curve, the width W of the curve is given by

W = 2\/C. (14.1)

The half-width, that is, the distance from the midpoint to the 1/e point of the
curve, is equal to VC. This simple square root law is the reason for the particular
factor 1/e used to define the width. With this choice the probability for the result
of a tray-shaking to lie within the width of the curve is 84%.

In Figure 14.4 the value of C, the midpoint number of heads, is 500. The square
root of C is roughly 22. Thus the width of the curve is about 44, extending from

500
C

Number of heads

1.000

Figure 14.4 Probabilities for various results of tray-shaking experiment with
1,000 coins.

Entropy and the Second Law of Thermodynamics

500 — 22 = 478 to 500 + 22 = 522. The total chance for a result to lie within
this span is 84%; to lie outside it, 16%.

An important consequence of the square-root law is to sharpen the probability
curve as the number of coins increases. The ratio of the width to the total number
of coins N {N = 2C) is

W ^ Wc ^ J_
N 2C y/c'

(14.2)

This ratio decreases as C (or N) increases. For 100 coins, the width-to-number
ratio is about 1/10. For 1,000 coins, it is about 1/32. For 1,000,000 coins, it is
1/1,000. If the number of coins could be increased to be equal to the number of
molecules in a drop of water, about 10^2, the width-to-number ratio of the
probability curve would be 1/10^ i. Then the result of vigorous shaking of the
coins would produce a number of heads and a number of tails unlikely to differ
from equality by more than one part in one hundred billion. The probability curve
would have collapsed to a narrow spike (Figure 14.5).

Two more points of interest about these head-and-tail probabilities will bring
us closer to the connection between trays of coins and collections of molecules.
First is the relation between probability and disorder. Ten coins arranged as all
heads can be considered as perfectly orderly, as can an array of all tails. Five
heads and five tails, on the other hand, arranged for example as HHTHTTTHTH
or as TTHTHTHHHT, form a disorderly array. Evidently a high state of
order is associated with low probability, a state of disorder is associated with
high probability. This might be called the housewife's rule: Order is improbable,
disorder is probable. The reason this is so is exactly the same for the household
as for the tray of coins. There are many more different ways to achieve disorder
than to achieve order.

The second point of special interest concerns the way probabilities change in
time. If a tray of 1,000 coins is carefully arranged to show all heads, and is
then shaken repeatedly, its arrangement will almost certainly shift in the direction
of nearly equal numbers of heads and tails. The direction of spontaneous change
will be from an arrangement of low probability to an arrangement of high

0.5X10"
Number of heads

10^

Figure 14.5 For 10^2 coins, the probability curve is a spike much riarrower even
than the line on this graph.

probability, from order to disorder. The same will be true whenever the initial
arrangement is an improbable one, for instance 700 tails and 300 heads. If
instead we start with 498 heads and 502 tails, no amount of shaking will tend
to move the distribution to a highly uneven arrangement. This can be considered
an equilibrium situation. Repeated trials will then produce results not very
different from the starting point. Clearly there is a general rule here — a rule of
probability, to be sure, not an absolute rule: Under the action of random
influences, a system tends to change from less probable arrangements to more
probable arrangements, from order to disorder. The generalization of this rule
from trays of coins to collections of molecules, and indeed to complex systems of
any kind, is the second law of thermodynamics — a law, as we shall see, with
remarkably broad and important consequences.

14.3 Probability of position

Most of the large-scale properties of substances are, when examined closely
enough, probabilistic in nature. Heat and temperature are purely macroscopic
concepts that lose their meaning when applied to individual atoms and molecules,
for any particular molecule might have more or less energy than the average,
or might contribute more or less than the average to a process of energy exchange.
Temperature is proportional to an average kinetic energy ; heat is equal to a total
energy transferred by molecular colhsion. Because of our incomplete knowledge
about the behavior of any single molecule, and the consequent necessity of
describing molecular motion in probabilistic terms, neither of these thermal
concepts is useful except when applied to numbers so large that the laws of
probability become laws of near certainty. The same can be said of other con-
cepts such as pressure and internal energy.

A single molecule is characterized by position, velocity, momentum, and energy.
Of these, position is the simplest concept and therefore the one for which it is
easiest to describe the role of probability. Consider, for instance, an enclosure —
perhaps the room you are in — divided by a screen into two equal parts. What
is the relative number of molecules of air on the two sides of the screen? Not a
hard question, you will say. It is obvious that the two halves should contain equal,
or very nearly equal, numbers of molecules. But here is a harder question. Why
do the molecules divide equally? Why do they not congregate, at least some of
the time, in one corner of the room? The answer to this question is exactly the
same as the answer to the question : Why does a tray of coins after being shaken
display approximately equal numbers of heads and tails? The equal distribution
is simply the most probable distribution. Any very unequal distribution is very
improbable.

The mathematics of molecules on two sides of a room proves to be identical to
the mathematics of coins on a tray. By the assumption of randomness, every
single molecule has an equal chance to be on either side of the room, just as every
coin has an equal chance to land as heads or as tails. There are many different
ways to distribute the molecules in equal numbers on the two sides, but only one
way to concentrate them all on one side. If a room contained only five molecules,
it would not be surprising to find them sometimes all on a single side. The
probability that they be all on the left is 1/32 (see Table 14.1), and there is

Entropy and the Second Law of Thermodynamics

an equal probability that they be all on the right. The chance of a 3-2 distribu-
tion is 20/32, or nearly two thirds. Even for so small a number as five, a nearly
equal division is much more likely than a very uneven division. For lO^s
molecules, the number in a large room, the distribution is unlikely to deviate
from equality by more than one part in 10^*. The probability for all of the 10^8
molecules to congregate spontaneously in one half of the room is less than

This number is too small even to think about. Suddenly finding ourselves gasping
for breath in one part of a room while someone in another part of the room is
oversupplied with oxygen is a problem we need not be worried about.

The second law of thermodynamics is primarily a law of change. It states that
the direction of spontaneous change within an isolated system is from an arrange-
ment of lower probability to an arrangement of higher probability. Only if the
arrangement is already one of maximal probability will no spontaneous change
occur. Air molecules distributed uniformly in a room are (with respect to their
position) in such a state of maximal probability. This is an equilibrium situation,
one that has no tendency for spontaneous change. Nevertheless it is quite easy
through external actions to depart from this equilibrium to a less probable
arrangement. Air can be pumped from one side of the room to the other. In a
hypothetical vacuum-tight room with an impenetrable barrier dividing it in half,
almost all of the air can be pumped into one half. When the barrier is punctured,
the air rushes to equalize its distribution in space. This behavior can be described
as the result of higher pressure pushing air into a region of lower pressure. But
it can equally well be described as a simple consequence of the second law of
thermodynamics. Once the barrier is punctured or removed, the air is free to
change to an arrangement of higher probability, and it does so promptly.

It is worth noting that frequent molecular colhsions play the same role for
the air as tray-shaking plays for the coins. A stationary tray displaying all heads
would stay that way, even though the arrangement is improbable. If molecules
were quiescent, they would remain on one side of a room once placed there.
Only because of continual molecular agitation do the spontaneous changes pre-
dicted by the second law of thermodynamics actually occur.

14.4 Entropy and the second law of thermodynamics

There are a variety of ways in which the second law of thermodynamics can be
stated, and we have encountered two of them so far: (1) For an isolated system,
the direction of spontaneous change is from an arrangement of lesser probability
to an arrangement of greater probability; and (2) for an isolated system, the
direction of spontaneous change is from order to disorder. Like the conservation
laws, the second law of thermodynamics applies only to a system free of external
influences. For a system that is not isolated, there is no principle restricting its
direction of spontaneous change.

A third statement of the second law of thermodynamics makes use of a new
concept called entropy. Entropy is a measure of the extent of disorder in a system
or of the probability of the arrangement of the parts of a system. For greater
probability, which means greater disorder, the entropy is higher. An arrangement

of less probability (greater order) has less entropy. This means that the second
law can be stated: (3) The entropy of an isolated system increases or remains
the same.

Specifically, entropy, for which the usual symbol is S, is defined as Boltzmann's
constant multiplied by the logarithm of the probability of any particular state of
the system:

S = k\og P. (14.3)

The appearance of Boltzmann's constant fc as a constant of proportionality is a
convenience in the mathematical theory of thermodynamics, but is, from a funda-
mental point of view, entirely arbitrary. The important aspect of the definition
is the proportionality of the entropy to the logarithm of the probability P. Note
that since the logarithm of a number increases when the number increases, greater
probability means greater entropy, as stated in the preceding paragraph.

Exactly how to calculate a probability for the state of a system (a procedure
that depends on the energies as well as the positions of its molecules) is a
complicated matter that need not concern us here. Even without this knowledge,
we can approach an understanding of the reason for the definition expressed by
Equation 14.3. At first, entropy might seem to be a superfluous and useless
concept, since it provides the same information about a system as is provided
by the probability P, and S grows or shrinks as P grows or shrinks. Technically
these two concepts are redundant, so that either one of them might be considered
superfluous. Nevertheless both are very useful. (For comparison, consider the
radius and the volume of a sphere; both are useful concepts despite the fact that
they provide redundant information about the sphere.) The valuable aspect of
the entropy concept is that it is additive. For two or more systems brought
together to form a single system, the entropy of the total is equal to the sum
of the entropies of the parts. Probabilities, by contrast, are multiplicative. If
the probability for one molecule to be in the left half of a container is ^, the
probability for two to be there is I, and the probability for three to congregate
on one side is |. If two containers, each containing three molecules, are en-
compassed in a single system, the probability that the first three molecules are
all on the left side of the first container and that the second three are also on
the left side of the second container is ^ X ^ = ^. On the other hand, the entropy
of the combination is the sum of the entropies of the two parts. These properties
of addition and multiplication are reflected in the definition expressed by Equation
14.3. The logarithm of a product is the sum of the logarithm of the factors :

5totai = k log P1P2 = A; log Pi -f fc log P2 = ,Si -f- ^2. (14.4)

The additive property of entropy is more than a mathematical convenience.
It means that the statement of the second law can be generalized to include a
composite system. To restate it: (3) The total entropy of a set of interconnected
systems increases or stays the same. If the entropy of one system decreases,
the entropy of systems connected to it must increase by at least a compensating
amount, so that the sum of the individual entropies does not decrease.

Even though the second law of thermodynamics may be re-expressed in terms
of entropy or of order and disorder, probability remains the key underlying idea.
The exact nature of this probability must be understood if the second law is to be

Entropy and the Second Law of Thermodynamics

understood. Implicit in our discussion up to this point but still requiring emphasis
is the a priori nature of the probability that governs physical change. The state-
ment that physical systems change from less probable to more probable arrange-
ments might seem anything but profound if the probability is regarded as an
after-the-fact probability. If we decided that a uniform distribution of molecules
in a box must be more probable than a nonuniform distribution because gas
in a box is always observed to spread itself out evenly, the second law would be
mere tautology, saying that systems tend to do what they are observed to do.
In fact, the probability of the second law of thermodynamics is not based on
experience or experiment. It is a before-the-fact (a priori) probability, based on
coimting the number of different ways in which a particular arrangement could be
achieved. To every conceivable arrangement of a system can be assigned an
a priori probability, whether or not the system or that arrangement of it has ever
been observed. In practice there is no reason why the state of a system with the
highest a priori probability need be the most frequently observed. Consider
the case of the dedicated housewife. Almost every time an observant friend comes
to call, he finds her house to be in perfect condition, nothing out of place, no dust
in sight. He must conclude that for this house at least, the most probable state
is very orderly state, since that is what he most often observes. This is an after-
the-fact probability. As the housewife and the student of physics know, the orderly
state has a low a priori probability. Left to itself, the house will tend toward a
disorderly state of higher a priori probability. A state of particularly high a priori
probability for a house is one not often observed, a pile of rubble. Thus an ar-
rangement of high probability (from here on we shall omit the modifier, a priori)
need be neither frequently observed nor quickly achieved, but it is, according
to the second law of thermodynamics, the inevitable destination of an isolated
system.

In comparison with other fundamental laws of nature, the second law of thermo-
dynamics has two special features. First, it is not given expression by any
mathematical equation. It specifies a direction of change, but not a magnitude
of change. The nearest we can come to an equation is the mathematical statement,

S ^ 0. (14.5)

In words: The change of entropy (for an isolated system or collection of sys-
tems) is either positive or zero. Or, more simply, entropy does not spontaneously
decrease.

Every fundamental law of nature is characterized by remarkable generality,
yet the second law of thermodynamics is unique among them (its second special
feature) in that it finds direct application in a rich variety of settings, physical,
biological, and human. In mentioning trays of coins, molecules of gas, and disorder
in the house, we have touched only three of a myriad of applications. Entropy
and the second law have contributed to discussion of the behavior of organisms,
the flow of events in societies and economies, communication and information,
and the history of the universe. In much of the physics and chemistry of macro-
scopic systems, the second law has found a use. Only at the submicroscopic level
of single particles and single events is it of little importance. It is a startling and
beautiful thought that an idea as simple as the natural trend from order to disorder
should have such breadth of impact and power of application.

In most of the remainder of this chapter we shall be concerned with the appli-
cation of the second law of thermodynamics to relatively simple physical situations.
In Section 14.9 we return to some of its more general implications.

14.5 Probability of velocity: heat flow and equipartition

Since the velocities as well as the positions of individual molecules are generally
imknown, velocity too is subject to considerations of probability. This kind of
probability, like the probability of position, follows the rule of spontaneous change
from lower to higher probability. It should not be surprising to learn that for
a collection of identical molecules the most probable arrangement is one with equal
molecular speeds (and randomly oriented velocities). This means that available
energy tends to distribute itself uniformly over a set of identical molecules,
just as available space tends to be occupied uniformly by the same molecules.
In fact, the equipartition theorem and the zeroth law of thermodynamics can both
be regarded as consequences of the second law of thermodynamics. Energy divides
itself equally among the available degrees of freedom, and temperatures tend
toward equality, because the resulting homogenized state of the molecules is the
state of maximum disorder and maximum probability. The concentration of all
of the energy in a system on a few molecules is a highly ordered and improbable
situation analogous to the concentration of all of the molecules in a small portion
of the available space.

The normal course of heat flow can also be understood in terms of the second
law. Heat flow from a hotter to a cooler body is a process of energy transfer
tending to equalize temperature and thereby to increase entropy. The proof that
equipartition is the most probable distribution of energy is complicated and
beyond the scope of this book. Here we seek only to make it plausible through
analogy with the probability of spatial distributions.

Heat flow is so central to most applications of thermodynamics that the second
law is sometimes stated in this restricted form: (4) Heat never flows spontaneously
from a cooler to a hotter body. Notice that this is a statement about macroscopic
behavior, whereas the more general and fundamental statements of the second
law, which make use of the ideas of probability and order and disorder, refer
to the submicroscopic structure of matter. Historically, the first version of the
second law, advanced by Sadi Carnot in 1824, came before the submicroscopic
basis of heat and temperature was established, in fact before the first law of
thermodynamics was formulated. Despite a wrong view of heat and an incomplete
view of energy, Carnot was able to advance the important principle that no heat
engine (such as a steam engine) could operate with perfect efficiency. In modem
terminology, Carnot's version of the second law is this: (5) In a closed system,
heat flow out of one part of the system cannot be transformed wholly into mechan-
ical energy (work), but must be accompanied by heat flow into a cooler part
of the system. In brief, heat cannot be transformed completely to work.

The consistency of Carnot's form of the second law with the general principle
of entropy increase can best be appreciated by thinking in terms of order and
disorder. The complete conversion of heat to work would represent a transfor-
mation of disordered energy, a replacement of random molecular motion by orderly
bulk motion. This violates the second law of thermodynamics. As indicated

Entropy and the Second Law of Thermodynamics

Heat H2
into cold
region.
Entropy
increase.

Heat
engine

Heat Hi

out of hot

region.

Entropy

decrease.

Energy equal to

Hi — H2 transformed

to work.

Figure 14.6 Schematic diagram of partial conversion of disordered heat energy
to ordered mechanical energy. Heat flow out of a hot region decreases the entropy
there. A compensating increase of entropy in a cold region requires less heat.
Therefore, some heat can be transformed to work without violating the second
law of thermodynamics. Any device that achieves this aim is called a heat engine.

schematically in Figure 14.6, a partial conversion of heat to work is possible
because a small heat flow into a cool region may increase the entropy there by
more than the decrease of entropy produced by a larger heat flow out of a hot
region. At absolute zero, the hypothetically motionless molecules have maximum
order. Greater temperature produces greater disorder. Therefore heat flow into
a region increases its entropy, heat flow out of region decreases its entropy.
Fortunately for the feasibility of heat engines, it takes less heat at low tempera-
ture than at high temperature to produce a given entropy change. To make an
analogy, a pebble is enough to bring disorder to the smooth surface of a calm lake.
To produce an equivalent increase in the disorder of an already rough sea
requires a boulder. In Section 14.7, the quantitative link between heat flow and
entropy is discussed.

The reverse transformation, of total conversion of work to heat, is not only
possible but is commonplace. Every time a moving object is brought to rest
by friction, all of its ordered energy of bulk motion is converted to disordered
energy of molecular motion. This is an entropy-increasing process allowed by
the second law of thermodynamics. In general, the second law favors energy
dissipation, the transformation of energy from available to unavailable form.
Whenever we make a gain against the second law by increasing the order or
the available energy in one part of a total system, we can be sure we have lost
even more in another part of the system. Thanks to the constant input of energy
from the sun, the earth remains a lively place and we have nothing to fear
from the homogenizing effect of the second law.

14.6 Perpetual motion

We have given so far five different versions of the second law, and will add only
one more. Of those given, the first three, expressed in terms of probability, of

order-disorder, and of entropy, are the most fundamental. Worth noting in several
of the formulations is the recurring emphasis on the negative. Entropy does not
decrease. Heat does not flow spontaneously from a cooler to a hotter region.
Heat can not be wholly transformed to work. Our sixth version is also expressed
in the negative. (6) Perpetual-motion machines cannot be constructed. This
statement may sound more like a staff memorandum in the Patent Office than
a fundamental law of nature. It may be both. In any event, it is certainly the
latter, for from it can be derived the spontaneous increase of probability, of dis-
order, or of entropy. It is specialized only in that it assumes some friction,
however small, to be present to provide some energy dissipation. If we overlook
the nearly frictionless motion of the planets in the solar system and the frictionless
motion of single molecules in a gas, everything in between is encompassed.

A perpetual-motion machine can be defined as a closed system in which bulk
motion persists indefinitely, or as a continuously operating device whose output
work provides its own input energy. Some proposed perpetual-motion machines
violate the law of energy conservation (the first law of thermodynamics). These
are called perpetual-motion machines of the first kind. Although they can be
elaborate and subtle, they are less interesting than perpetual-motion machines of
the second kind, hypothetical devices that conserve energy but violate the prin-
ciple of entropy increase (the second law of thermodynamics) .

As operating devices, perpetual-motion machines are the province of crackpot
science and science fiction. As inoperable devices they have been of some signif-

CofTee container,

insulated on

top and sides.

Figure 14.7 A perpetual-motion machine of the second kind. The device labeled
MARK II receives heat energy from the coffee and converts this to mechanical
energy which turns a paddle wheel, agitating the coffee, returning to the coffee
the energy it lost by heat flow. It is not patentable.

Entropy and the Second Law of Thermodynamics

icance in the development of science. Carnot was probably led to the second law
of thermodynamics by his conviction that perpetual motion should be impossible.
Arguments based on the impossibility of perpetual motion can be used to support
Newton's third law of mechanics and Lenz's law of electromagnetic reaction,
which will be discussed in Chapter Sixteen. Any contemporary scientist with
a speculative idea can subject it to at least one quick test: Is it consistent with
the impossibility of perpetual motion?

Suppose that an inventor has just invented a handy portable coffee warmer
(Figure 14.7). It takes the heat which flows from the coffee container and, by
a method known only to him, converts this heat to work expended in stirring
the coffee. If the energy going back into the coffee is equal to that which leaks off
as heat, the original temperature of the coffee will be maintained. Is it patentable?
No, for it is a perpetual-motion machine of the second kind. Although it conserves
energy, it performs the impossible task of maintaining a constant entropy
in the face of dissipative forces that tend to increase entropy. Specifically it
violates Carnot's version of the second law (No. 5, page 441), for in one part
of its cycle it converts heat wholly to work. Of course it also violates directly
our sixth version of the second law.

One of the chief strengths of the second law is its power to constrain the behavior
of complex systems without reference to any details. Like a corporate director,
the second law rules the overall behavior of systems or interlocked sets of system
in terms of their total input and output and general function. Given a proposed
scheme for the operation of the automatic coffee warmer, it might be quite
a complicated matter to explain in terms of its detailed design why it cannot work.
Yet the second law reveals at once that no amount of ingenuity can make it
work.

14.7 Entropy on two levels

The mathematical roots of thermodynamics go back to the work of Pierre Laplace
and other French scientists concerned with the caloric theory of heat in the years
aroimd 1800, and even further to the brilliant but forgotten invention of the
kinetic theory of gases by Daniel Bernoulli in 1738. Not until after 1850 did these
and other strands come together to create the theory of thermodynamics in some-
thing like its modem form. No other great theory of physics has traveled such
a rocky road to success over so many decades of discovery, argumentation, buried
insights, false turns, and rediscovery, its paths diverging and finally rejoining
in the grand synthesis of statistical mechanics which welded together the macro-
scopic and submicroscopic domains in the latter part of the nineteenth century.

In the long and complex history of thermodynamics, the generalization of the
principle of energy conservation to include heat stands as probably the most
significant single landmark. Joule's careful experiments on the mechanical equiv-
alent of heat in the 1840's not only established the first law of thermodynamics,
but cleared the way for a full understanding of the second law, provided a basis
for an absolute temperature scale, and laid the groundwork for the submicro-
scopic mechanics of the kinetic theory. Progress in the half century before Joule's
work had been impeded by a pair of closely related difficulties: an incorrect view
of the nature of heat, and an incomplete understanding of the way in which

heat engines provide work. To be sure, there had been important insights in
this period, such as Carnot's statement of the second law of thermodynamics in
1824. But such progress as there was did not fit together into a single structure,
nor did it provide a base on which to build. Not until 1850, when the great sig-
nificance of the general principle of energy conservation was appreciated by
at least a few scientists, was Carnot's work incorporated into a developing
theoretical structure. The way was cleared for a decade of rapid progress. In the
1850's, the first and second laws of thermodynamics were first stated as general
unifying principles, the kinetic theory was rediscovered and refined, the concepts
of heat and temperature were given submicroscopic as well as macroscopic
definitions, and the full significance of the ideal-gas law was understood. The
great names of the period were James Joule, William Thomson (Lord Kelvin),
and James Clerk Maxwell in England, Rudolph Clausius and August Kronig
in Germany.

One way to give structure to the historical development of a major theory
is to follow the evolution of its key concepts. This is particularly instructive
for the study of thermodynamics, because its basic concepts — heat, temperature,
and entropy — exist on two levels, the macroscopic and the submicroscopic.
The refinement of these concepts led both to a theoretical structure for under-
standing a great part of nature and to a bridge between two worlds, the large and
the small. Of special interest here is the entropy concept.

Like heat and temperature, entropy was given first a macroscopic definition,
later a molecular definition. Being a much subtler concept than either heat or
temperature (in that it does not directly impinge on our senses), entropy was
defined only after its need in the developing theory of thermodynamics became
obvious. Heat and temperature were familiar ideas refined and revised for the
needs of quantitative understanding. Entropy was a wholly new idea, formally
introduced and arbitrarily named when it proved to be useful in expressing
the second law of thermodynamics in quantitative form. As a useful but unnamed
quantity, entropy entered the writings of both Kelvin and Clausius in the early
1850's. Finally in 1865, it was formally recognized and christened "entropy"
by Clausius, after a Greek word for transformation. Entropy, as he saw it,
measured the potentiality of a system for transformation.

The proportionality of entropy to the logarithm of an intrinsic probability
for the arrangement of a system, as expressed by Equation 14.3, was stated first
by Ludwig Boltzmann in 1877. This pinnacle of achievement in what had come
to be called statistical mechanics fashioned the last great thermodynamics link
between the large-scale and small-scale worlds. Although we now regard Boltz-
mann's definition based on the molecular viewpoint as the more fundamental,
we must not overlook the earlier macroscopic definition of entropy given by
Clausius (which in most applications is easier to use). Interestingly, Clausius
expressed entropy simply and directly in terms of the two already familiar basic
concepts, heat and temperature. He stated that a change of entropy of any part
of a system is equal to the increment of heat added to that part of the system
divided by its temperature at the moment the heat is added, provided the change
is from one equilibrium state to another:

A5 = ^ . (14.6)

Entropy and the Second Law of Thermodynamics

Here S denotes entropy, H denotes heat, and T denotes the absolute temperature.
For heat gain, A^ is positive and entropy increases. For heat loss, Ai/ is negative
and entropy decreases. How much entropy change is produced by adding or sub-
tracting heat depends on the temperature. Since the temperature T appears in
the denominator in Equation 14.6, a lower temperature enables a given increment
of heat to produce a greater entropy change.

There are several reasons why Clausius defined not the entropy itself, but the
change of entropy. For one reason, the absolute value of entropy is irrelevant,
much as the absolute value of potential energy is irrelevant. Only the change
of either of these quantities from one state to another matters. Another more
important reason is that there is no such thing as "total heat." Since heat is
energy transfer (by molecular collisions), it is a dynamic quantity measured
only in processes of change. An increment of heat AH can be gained or lost
by part of a system, but it is meaningless to refer to the total heat H stored
in that part. (This was the great insight about heat afforded by the discovery of
the general principle of energy conservation in the 1840's). What is stored
is internal energy, a quantity that can be increased by mechanical work as well as
by heat flow. Finally, it should be remarked that Clausius' definition refers not
merely to change, but to small change. When an otherwise inactive system gains
heat, its temperature rises. Since the symbol T in Equation 14.6 refers to the
temperature at which heat is added, the equation applies strictly only to incre-
ments so small that the temperature does not change appreciably as the heat
is added. If a large amount of heat is added, Equation 14.6 must be applied over
and over to the successive small increments, each at slightly higher temperature.

To explain how the macroscopic definition of entropy given by Clausius
(Equation 14.6) and the submicroscopic definition of entropy given by Boltzmann
(Equation 14.3) fit together is a task beyond the scope of this book. Nevertheless
we can, through an idealized example, make it reasonable that these two defi-
nitions, so different in appearance, are closely related. To give the Clausius
definition a probability interpretation we need to discuss two facts: (1) Addition
of heat to a system increases its disorder and therefore its entropy; (2) The dis-
ordering influence of heat is greater at low temperature than at high temperature.
The first of these facts is related to the apparance of the factor AH on the right
of Equation 14.6; the second is related to the inverse proportionality of entropy
change to temperature.

Not to prove these facts but to make them seem reasonable, we shall consider
an idealized simple system consisting of just three identical molecules, each one
capable of existing in any one of a number of equally spaced energy states.
The overall state of this system can be represented by the triple-ladder diagram
of Figure 14.8, in which each rung corresponds to a molecular energy state.
Dots on the three lowest rungs would indicate that the system possesses no internal
energy. The pictured dots on the second, third, and bottom rungs indicate that
the system has a total of five units of internal energy, two units possessed by
the first molecule, three by the second, and none by the third. The intrinsic
probability associated with any given total energy is proportional to the number
of different ways in which that energy can be divided. This is now a probability
of energy distribution, not a probability of spatial distribution. However, the
reasoning is much the same as in Section 14.3. There the intrinsic (a priori)

Figure 14.8 Idealized energy diagram for a system of three molecules, each with
equally spaced energy states. Each ladder depicts the possible energies of a par-
ticular molecule, and the heavy dot specifies the actual energy of that molecule.

probability for a distribution of molecules in space was taken to be proportional
to the number of different ways in which that distribution could be obtained.
Or, to give another example, the probability of throwing 7 with a pair of dice
is greater than the probability of throwing 2, because there are more different
ways to get a total of 7 than to get a total of 2.

Table 14.4 enumerates all the ways in which up to five units of energy can
be divided among our three idealized molecules. The triplets of numbers in the
second column indicate the occupied rungs of the three energy ladders. It is an
interesting and instructive problem to deduce a formula for the numbers in the

Table 14.4 Internal Energy Distribution for Idealized System of Three Molecules

Total

Number of Ways to

Energy

Distribution of

Energy

Distribute Energy

000

100

010

001

200

020

002

110

101

Oil

300

030

003

210

201

012

120

102

021

111

400

040

004

310

301

031

130

103

013

220

202

022

211

121

112

500

050

005

410

401

041

140

104

014

320

302

032

230

203

023

311

131

113

122

212

221

Entropy and the Second Law of Thermodynamics

last column, (hint: The number of ways to distribute 6 units of energy is 28.)
However, since this is a highly idealized picture of very few molecules, precise
numerical details are less important than are the qualitative features of the
overall pattern. The first evident feature is that the greater the energy, the more
different ways there are to divide the energy. Thus a higher probability is asso-
ciated with greater internal energy. This does not mean that the system, if isolated
and left alone, will spontaneously tend toward a higher probability state, for that
would violate the law of energy conservation. Nevertheless, we associate with
the higher energy state a greater probability and a greater disorder. When energy
is added from outside via heat flow, the entropy increase is made possible. This
makes reasonable the appearance of the heat increment factor, A//, in Equation
14.6.

Looking further at Table 14.4, we ask whether the addition of heat produces a
greater disordering effect at low temperature than at high temperature. For
simplicity we can assume that temperature is proportional to total internal
energy, as it is for a simple gas, so that the question can be rephrased: Does
adding a unit of heat at low energy increase the entropy of the system more
than adding the same unit of heat at higher energy? Answering this question
requires a little care, because of the logarithm that connects probability to en-
tropy. The relative probability accelerates upward in Table 14.4. In going from
1 to 2 units of energy, the number of ways to distribute the energy increases by
three, from 2 to 3 units it increases by four, from 3 to 4 units it increases by five,
and so on. However, the entropy, proportional to the logarithm of the probability,
increases more slowly at higher energy. The relevant measure for the increase
of a logarithm is the factor of growth.* From 0 to 1 unit of energy, the probabil-
ity trebles, from 1 to 2 units it doubles, from 2 to 3 units it grows by 67%,
and so on, by ever decreasing factors of increase. Therefore the entropy
grows most rapidly at low internal energy (low temperature). This makes
reasonable the appearance of the temperature factor "downstairs" on the right
of Equation 14.6.

This example focuses attention on a question that may have occurred to you
already. Why is it that energy addition by heat flow increases entropy, but energy
addition by work does not? The definition. Equation 14.6, makes reference to
only one kind of energy, heat energy. The difference lies basically in the
recoverability of the energy. When work is done on a system without any
accompanying heat flow, as when gas is compressed in a cylinder (Figure 14.9),
the energy can be fully recovered, with the system and its surroundings returning
precisely to the state they were in before the work was done. No entropy change
is involved. On the other hand, when energy in the form of heat flows from a
hotter to a cooler place, there is no mechanism that can cause the heat to flow
spontaneously back from the cooler to the hotter place. It is not recoverable.
Entropy has increased. In a realistic as opposed to an ideal cycle of compression
and expansion, there will in fact be some entropy increase because there will
be some flow of heat from the compressed gas to the walls of the container.

♦Logarithms are defined in such a way that the logarithms of 10, 100, 1,000, and 10,000 or of
5, 10, 20, 40, and 80 differ by equal steps. It is this feature which makes the multiphcation of a
pair of numbers equivalent to the addition of their logarithms.

Work I)einf5 done on system.
Energy added to gas

Motion

of piston — >-

;'^.->v^-::>;.^.;':>

:^;

(a)

Compressed gas has
more internal energy
but no more entropy

Expanding gas does work.

Energy is recovered.
Entropy remains constant

(b)

Motion of piston

Figure 14,9 Idealized cycle of compression and expansion of gas, accompanied
by no change of entropy. If any heat flow occurs in the cycle, entropy does
increase.

Another useful way to look at the difference between heat and work is in
molecular terms, merging the ideas of position probability and velocity or energy
probability. If a confined gas [Figure 14.9(b)] is allowed to expand until its
volume doubles [Figure 14.9(c)] what we learned about position probability
tells us that, so far as its spatial arrangement is concerned, it has experienced
an entropy increase, having spread out into an intrinsically more probable
arrangement. In doing so, however, it has done work on its surroundings and
has lost internal energy. This means that, with respect to its velocity and energy,
it has approached a state of greater order and lesser entropy. Its increase of
spatial disorder has in fact been precisely canceled by its decrease of energy
disorder, and it experiences no net change of entropy. Had we instead wanted
to keep its temperature constant during the expansion, it would have been neces-
sary to add heat (equal in magnitude to the work done). Then after the ex-
pansion, the unchanged internal energy would provide no contribution to entropy
change, so that a net entropy increase would be associated with the expansion-
arising from the probability of position. This would match exactly the entropy
increase bM/T predicted by the Clausius formula, for this change required a
positive addition of heat.

Although the macroscopic entropy definition of Clausius and the submicroscopic
entropy definition of Boltzmann are, in many physical situations, equivalent,
Boltzmann's definition remains the more profound and the more general. It
makes possible a single grand principle, the spontaneous trend of systems from
arrangements of lower to higher probability, that describes not only gases and
solids and chemical reactions and heat engines, but also dust and disarray, erosion
and decay, the deterioration of fact in the spread of rumor, the fate of mis-
managed corporations, and perhaps the fate of the universe.

Entropy and the Second Law of Thermodynamics

14.8 Application of the second law

That heat flows spontaneously only from a warmer to a cooler place is a fact
which can itself be regarded as a special form of the second law of thermo-
dynamics. Alternatively the direction of heat flow can be related to the general
principle of entropy increase with the help of the macroscopic definition of
entropy. If body 1 at temperature Ti loses an increment of heat AH, its entropy
change — a decrease — is

A5i = - ^ • (14.7)

If this heat is wholly transferred to body 2 at temperature T2, its entropy gain is

A-S2 = ^ • (14.8)

The total entropy change of the system (bodies 1 and 2) is the sum,

AS = ASi+ AS2 =Ah[^^-yJ- (14.9)

This entropy change must, according to the second law, be positive if the heat
transfer occurs spontaneously. It is obvious algebraically from Equation 14.9
that this requirement implies that the temperature Tj is greater than the tem-
perature T2. In short, heat flows from the warmer to the cooler body. In the
process, the cooler body gains energy equal to that lost by the warmer body but
gains entropy greater than that lost by the warmer body. When equality of
temperature is reached, heat flow in either direction would decrease the total
entropy. Therefore it does not occur.

A heat engine is, in simplest terms, a device that transforms heat to mechanical
work. Such a transformation is, by itself, impossible. It is an entropy-decreasing
process that violates the second law of thermodynamics. We need hardly conclude
that heat engines are impossible, for we see them all around us. Gasoline engines,
diesel engines, steam engines, jet engines, and rocket engines are all devices that
transform heat to work. They do so by incorporating in the same system a
mechanism of entropy increase that more than offsets the entropy decrease
associated with the production of work. The simple example of heat flow with
which this section began shows that one part of a system can easily lose entropy
if another part gains more. In almost all transformations of any complexity, and
in particular in those manipulated by man for some practical purpose, entropy
gain and entropy loss occur side by side, with the total gain inevitably exceeding
the total loss.

The normal mechanism of entropy gain in a heat engine is heat flow. Carnot's
great insight that provided the earliest version of the second law was the
realization that a heat engine must be transferring heat from a hotter to a cooler
place at the same time that it is transforming heat to work. How this is ac-
complished varies from one heat engine to another, and the process can be quite
complicated and indirect. Nevertheless, without reference to details, it is possible
to discover in a very simple way what fraction of the total energy supplied by
fuel can be transformed into usable work. This fraction is called the efficiency

of the engine. Refer to Figure 14.6, which shows schematically a process of partial
transformation of heat to work. From the hotter region, at temperature Tj, flows
an increment of heat H^. Into the cooler region, at temperature T2, flows heat //g.
The output work is W. The first and second laws of thermodynamics applied
to this idealized heat engine can be given simple mathematical expression.

1. Energy conservation : Hi = H2 -\- W. (14.10)

2. Entropy increase: S = ~ -^^ > Q- (14.11)

If this heat engine were "perfect"— free of friction and other dissipative effects—
the entropy would remain constant instead of increasing. Then the right side of
Equation 14.11 could be set equal to zero, and the ratio of output to input heat
would be

H2 T2

H[ = Vi- (14.12)

From Equation 14.10 follows another equation containing the ratio H2/H1,

W , H2

Substitution of Equation 14.12 into Equation 14.13 gives for the ratio of output
work to initial heat supply,

'' max ■, J- 2

-^^ = 1 - ^ • (14.14)

Here we have written W^^^ instead of W, since this equation gives the maximum
possible efficiency of the idealized heat engine. If the temperatures T^i and T2 are
nearly the same, the efficiency is very low. If T2 is near absolute zero, the
theoretical efficiency can be close to 1— that is, almost perfect.

The modern marvels of technology that populate our present world— auto-
mobiles, television, airplanes, radar, pocket radios— all rest ultimately on basic
prmciples of physics. Nevertheless they are usually not instructive as illustrations
of fundamental laws, for the chain of connection from their practical function to
the underlying principles is complex and sophisticated. The refrigerator is such
a device. Despite its complexity of detail, however, it is worth considering in
general terms. Because it transfers heat from a cooler to a warmer place, the
refrigerator appears at first to violate the second law of thermodynamics. 'The
fact that it must not do so allows us to draw an important conclusion about
the minimum expenditure of energy required to run it. The analysis is quite
similar to that for a heat engine. Suppose that the mechanism of the refrigerator
is required to transfer an amount of heat H^ out of the refrigerator each second.
If the interior of the refrigerator is at temperature Tj, this heat loss contributes
an entropy decrease equal to -Hi/T^. This heat is transferred to the surrounding
room at temperature T2 (higher than Tj), where it contributes an entropy in-
crease equal to H^/T 2. The sum of these two entropy changes is negative. Some
other contribution to entropy change must be occurring in order that the total
change may be positive, in consonance with the second law. This extra contribu-
tion comes from the degradation of the input energy that powers the refrigerator.

Entropy and the Second Law of Thermodynamics

The energy supplied by electricity or by the combustion of gas eventually reaches
the surrounding room as heat. If the external energy (usually electrical) supplied
in one second is called W, and the total heat added to the room in the same
time is called H2, energy conservation requires that H2 be the sum of H^ and W:

H2 = Hi + W.

(14.15)

The energy flow is shown schematically in Figure 14.10. At the same time the
total entropy change is given by

T2 Ti

(14.16)

Since A<S must be zero or greater, the ratio H2/H1 [= (heat added to room) /(heat
extracted from refrigerator)] must be at least equal to T2/T1. If the energy
conservation equation is written in the form

we can conclude that

Tr^i/:[^-l].

(14.17)

The right side of this inequality gives the minimum amount of external energy
input required in order to transfer an amount of heat H^ "uphill" from tempera-
ture Ti to temperature T2. As might be expected, the input energy requirement
increases as the temperature difference increases. If the temperature Tj is near
absolute zero, as it is in a helium liquefier, the external energy expended is much
greater than the heat transferred.

The real beauty of the result expressed by Equation 14.17 is its generality for
all refrigerators regardless of their construction and mode of operation. The input
energy W could be supplied by an electric motor, a gas flame, or a hand crank.
It is characteristic of the second law of thermodynamics, just as it is characteristic
of the fundamental conservation laws, that it has something important to say
about the overall behavior of a system without reference to details, perhaps

External
source 1 21 Energy W
of power

Temperature Ti.
Heat Hi removed

Mechanism of
refrigerator

Room temperature T2.
Heat H2 added

Figure 14.10 Energy and heat flow in the operation of a refrigerator.

without knowledge of details. In the small-scale world, our inability to observe
precise features of individual events is one reason for the special importance of
conservation laws. In the large-scale world, the elaborate complexity of many
systems is one reason for the special importance of the second law of thermo-
dynamics. Like the conservation laws, it provides an overall constraint on the
system as a whole.

In many applications of the second law, the concept of available energy is the
easiest key to understanding. In general, the trend of nature toward greater dis-
order is a trend toward less available energy. A jet plane before takeoff has a
certain store of available energy in its fuel. While it is accelerating down the
runway, a part of the energy expended is going into bulk kinetic energy (ordered
energy), a part is going into heat that is eventually dissipated into unavailable
energy. At constant cruising speed, all of the energy of the burning fuel goes to
heat the air. Thermodynamically speaking, the net result of a flight is the total
loss of the available energy originally present in the fuel. A rocket in free space
operates with greater efficiency. Being free of air friction, it continues to accelerate
as long as the fuel is burning. When its engine stops, a certain fraction (normally
a small fraction) of the original available energy in the fuel remains available
in the kinetic energy of the vehicle. This energy may be "stored" indefinitely in
the orbital motion of the space vehicle. If it re-enters the atmosphere, however,
this energy too is transformed into the disordered and unavailable form of internal
energy of the air. To get ready for the next launching, more rocket fuel must
be manufactured. The energy expended in the chemical factory that does this job
is inevitably more than the energy stored in the fuel that is produced.

In general the effect of civilization is to encourage the action of the second law
of thermodynamics. Technology greatly accelerates the rate of increase of entropy
in man's immediate environment. Fortunately the available energy arriving each
day from the sun exceeds by a very large factor the energy degraded by man's
activity in a day. Fortunately too, nature, with no help from man, stores in
usable form some of the sun's energy — for periods of months or years in the
cycle of evaporation, precipitation, and drainage; for decades or centuries in
lumber; for millennia in coal and oil. In time, as we deplete the long-term
stored supply of available energy, we shall have to rely more heavily on the
short-term stores and probably also devise new storage methods to supplement
those of nature.

14.9 The arrow of time

Familiarity breeds acceptance. So natural and normal seem the usual events of
our everyday life that it is difficult to step apart and look at them with a scientific
eye.

Men with the skill and courage to do so led the scientific revolution of the
seventeenth century. Since then, the frontiers of physics have moved far from
the world of direct sense perception, and even the study of our immediate en-
vironment more often than not makes use of sophisticated tools and controlled
experiment. Nevertheless, the ability to take a fresh look at the familiar and to
contrast it with what would be the familiar in a different universe with different
laws of nature remains a skill worth cultivating. For the student, and often for

Entropy and the Second Law of Thermodynamics

the scientist as well, useful insights come from looking at the familiar as if it
were unfamiliar.

Consider the second law of thermodynamics. We need not go to the laboratory
or to a machine or even to the kitchen to witness its impact on events. It is
unlikely that you get through any five minutes of your waking life without
seeing the second law at work. The way to appreciate this fact is by thinking
backward. Imagine a motion picture of any scene of ordinary life run backward.
You might watch a student untyping a paper, each keystroke erasing another
letter as the keys become cleaner and the ribbon fresher. Or bits of hair clippings
on a barber-shop floor rising to join the hair on a customer's head as the barber
unclips. Or a pair of mangled automobiles undergoing instantaneous repair as
they back apart. Or a dead rabbit rising to scamper backward into the woods as a
crushed bullet reforms and flies backward into a rifle while some gunpowder is
miraculously manufactured out of hot gas. Or something as simple as a cup of
coffee on a table gradually becoming warmer as it draws heat from its cooler
surroundings. All of these backward-in-time views and a myriad more that you
can quickly think of are ludicrous and impossible for one reason only — they
violate the second law of thermodynamics. In the actual sequence of events,
entropy is increasing. In the time reversed view, entropy is decreasing. We recog-
nize at once the obvious impossibility of the process in which entropy decreases,
even though we may never have thought about entropy increase in the everyday
world. In a certain sense everyone "knows" the second law of thermodynamics.
It distinguishes the possible from the impossible in ordinary affairs.

In some of the examples cited above, the action of the second law is obvious,
as in the increasing disorder produced by an automobile colUsion, or the increas-
ing entropy associated with heat flow from a cup of coffee. In others, it is less
obvious. But whether we can clearly identify the increasing entropy or not,
we can be very confident that whenever a sequence of events occurs in our world
in one order and not in the other, it is because entropy increase is associated
with the possible order, entropy decrease with the impossible order. The reason
for this confidence is quite simple. We know of no law other than the second
law of thermodynamics that assigns to processes of change in the large-scale world
a preferred direction in time. In the submicroscopic world too, time-reversal
invariance is a principle governing all or nearly all fundamental processes.* Here
we have an apparent paradox. In order to understand the paradox and its resolu-
tion, we must first understand exactly what is meant by time-reversal invariance.

The principle of time-reversal invariance can be simply stated in terms of
hypothetical moving pictures. If the filmed version of any physical process, or
sequence of events, is shown backward, the viewer sees a picture of something that
could have happened. In slightly more technical language, any sequence of
events, if executed in the opposite order, is a physically possible sequence of
events. This leads to the rather startling conclusion that it is, in fact, impossible

* For the first time in 1964, some doubt was cast on the universal validity of time-reversal in-
variance, which had previously been supposed to be an absolute law of nature. In 1968 the
doubt remains unresolved. Even if found to be imperfect, the principle will remain valid to
a high degree of approximation, since it has already been tested in many situations. In par-
ticular, since all interactions that have any effect on the large-scale world do obey the
principle of time-reversal invariance, the discussion in this section will be unaffected.

to tell by watching a moving picture of events in nature whether the film is
running backward or forward. How can this principle be reconciled with the
gross violations of common sense contained in the backward view of a barber
cutting hair, a hunter firing a gun, a child breaking a plate, or the President
signing his name? Does it mean that time-reversal invariance is not a valid law
in the macroscopic world? No. As far as we know, time-reversal invariance
governs every interaction that underlies processes of change in the large-scale
world. The key to resolving the paradox is to recognize that possibility does not
mean probability. Although the spontaneous reassembly of the fragments of an
exploded bomb into a whole, unexploded bomb is wildly, ridiculously improbable,
it is not, from the most fundamental point of view, impossible.

At every important point where the macroscopic and submicroscopic descrip-
tions of matter touch, the concept of probability is crucial. The second law of
thermodynamics is basically a probabilistic law whose approach to absolute
validity increases as the complexity of the system it describes increases. For a
system of half a dozen molecules, entropy decrease is not only possible, it is
quite likely, at least some of the time. All six molecules might cluster in one
comer of their container, or the three less energetic molecules might lose energy
via collisions to the three more energetic molecules ("uphill" heat flow). For a
system of lO^o molecules, on the other hand, entropy decrease becomes so im-
probable that it deserves to be called impossible. We could wait a billion times
the known lifetime of the universe and still never expect to see the time-reversal
view of something as simple as a piece of paper being torn in half. Nevertheless,
it is important to realize that the time-reversed process is possible in principle.

Even in the world of particles, a sequence of events may occur with much
higher probability in one direction than in the opposite direction. In the world
of human experience, the imbalance of probabilities is so enormous that it no
longer makes sense to speak of the more probable direction and the less
probable direction. Instead we speak of the possible and the impossible. The
action of molecular probabilities gives to the flow of events in the large-scale
world a unique direction. The (almost complete) violation of time-reversal in-
variance by the second law of thermodynamics attaches an arrow to time, a
one-way sign for the unfolding of events. Through this idea, thermodynamics
impinges on philosophy.

In the latter part of the nineteenth century, long before time-reversal in-
variance was appreciated as a fundamental law of submicroscopic nature,
physicists realized that the second law had something quite general to say about
our passage through time. There are two aspects of the idea of the arrow of
time: first, that the universe, like a wound-up clock, is running down, its supply
of available energy ever dwindling; second, that the spontaneous tendency of
nature toward greater entropy is what gives man a conception of the unique
one-way direction of time.

The second law of thermodynamics had not long been formulated in a general
way before men reflected on its implications for the universe at large. In 1865,
Clausius wrote, without fanfare, as grand a pair of statements about the world
as any produced by science: "We can express the fundamental laws of the uni-
verse which correspond to the two fundamental laws of the mechanical theory
of heat in the following simple form.

Entropy and the Second Law of Thermodynamics

"1. The energy of the universe is constant.

"2, The entropy of the universe tends toward a maximum."

These are the first and second laws of thermodynamics extended to encompass all
of nature. Are the extensions justifiable? If so, what are their implications? We
know in fact no more than Clausius about the constancy of energy and the
steady increase of entropy in the universe at large. We do know that energy
conservation has withstood every test since he wrote, and that entropy increase
is founded on the very solid principle of change from arrangements of lesser to
those of greater probability. Nevertheless, all that we have learned of nature in
the century since Clausius leaped boldly to the edge of existence should make us
cautious about so great a step. In 1865, the single theory of Newtonian mechanics
seemed to be valid in every extremity of nature, from the molecular to the
planetary. A century later we know instead that it fails in every extremity — in the
domain of small sizes, where quantum mechanics rules; in the domain of high
speed, where special relativity changes the rules; and in the domain of the very
large, where general relativity warps space and time.

The logical terminus of the imiverse, assuming it to be a system obeying the
same laws as the macroscopic systems accessible to experiment, is known as
the "heat death," a universal soup of uniform density and uniform temperature,
devoid of available energy, incapable of further change, a perfect and featureless
final disorder. If this is where the universe is headed, we have had no hints of
it as yet. Over a time span of ten billion years or more, the imiverse has been
a vigorously active place, with new stars still being born as old ones are dying.
It is quite possible that the long-range fate of the universe will be settled within
science and need not remain forever a topic of pure speculation. At present,
however, we have no evidence at all to confirm or contradict the applicability of
thermodynamics to the universe as a whole. Even if we choose to postulate its
applicability, we need not be led inevitably to the idea of the ultimate heat death.
The existence of a law of time-reversal invariance in the world of the small and
the essential probabilistic nature of the second law leave open the possibility that
one grand improbable reversal of probability could occur in which disorder is
restored to order. Finally, we can link this line of thought to the second aspect
of the arrow of time, the uniqueness of the direction of man's course through time,
with this challenging thought. If it is the second law that gives to man his sense
of time's direction, the very construction of the human machine forces us to see
the universe running down. In a world that we might look in upon from the
outside to see building order out of disorder, the less probable from the more
probable, we would see creatures who remembered their future and not their
past. For them the trend of events would seem to be toward disorder and greater
probability and it is we who would seem to be turned around.

In the three centuries since Newton, time has evolved from the obvious to the
mysterious. In the Principia, Newton wrote, "Absolute, true, and mathematical
time, of itself, and from its own nature flows equably without regard to anything
external, and by another name is called duration." This view of time as something
flowing constantly and inexorably forward, carrying man with it, persisted largely
intact imtil the revolution of relativity at the beginning of this century. The
nineteenth century brought only hints of a deeper insight, when it was appreciated

that the second law of thermodynamics differentiated between forward and back-
ward in time, as the laws of mechanics had failed to do. If time were run backward,
the reversed planetary orbits would be reasonable and possible, obeying the
same laws as the actual forward-in-time orbits. But the reversal of any entropy-
changing transformation would be neither reasonable nor possible. The second law
of thermodynamics points the way for Newton's equable flow.

Relativity had the most profound effect on our conception of time. The merger
of space and time made unreasonable a temporal arrow when there was no spatial
arrow. More recently, time-reversal invariance has confirmed the equal status
of both directions in time. Relativity also brought time to a stop. It is more
consistent with the viewpoint of modern physics to think of man and matter
moving through time (as they move through space) than to think of time itself
as flowing.

All of the new insights about time make clear that we must think about it in
very human terms — its definition, its measurement, its apparently unique direc-
tion stem not from "absolute, true and mathematical time" but from psychological
time. These insights also reinforce the idea that the second law of thermodynamics
must ultimately account for our sense of time.

It is a stimulating idea that the only reason man is aware of the past and
not the future is that he is a complicated and highly organized structure.
Unfortunately, simpler creatures are no better off. They equalize future and past
by remembering neither. An electron, being precisely identical with every other
electron, is totally unmarked by its past or by its future. Man is intelligent
enough to be scarred by his past. But the same complexity that gives him a
memory at all is what keeps his future a mystery.

EXERCISES

14.1. Section 14.1 describes three kinds of uncertainty, associated respectively with a
spelunker, a gambler, and a uranium prospector. Which of these kinds of uncertainty
characterizes each of the following situations? (1) A pion of known energy enters a bubble
chamber. The number of bubbles formed along its first centimeter of track is measured.
The number of bubbles along its second centimeter of track can then be predicted approxi-
mately, but not exactly. (2) Another pion is created in the chamber. How long it will live
before decaying is uncertain. (3) Still another pion, of energy higher than any previously
studied, strikes a nucleus. The result of the collision is uncertain. Which, if any, of these
examples of uncertainty is governed by thermodynamic probability (the probability of
atomic multitudes)?

14.2. Suppose that a small cylinder (see figure) could be so nearly perfectly evacuated
that only 100 molecules remained within it. (1) Using Figures 14.3 and 14.4 and Equation
14.1 as guides, sketch a curve of relative probability for

any number of these molecules to be found in region A, A B

which is half of the container. (2) If you placed a bet at rS. /"^ TN

even money that a measurement would reveal exactly \2 Li 1/

50 molecules in region A, would this be, from your point

of view, a good bet or a poor bet? (3) If you bet, also at even money, that a series of
measurements would show less than 60 molecules in region A more often than not, would
you be making a good bet or a poor bet?

Completely random motion, such as the thermal motion
of molecules, might seem to be out of the realm of law-
fulness. But on the contrary. Just because the motion
is completely disorderly, it is subject to statistical laws.

The Law of Disorder

George Gamow

A chapter from his book One, Two, Three . . . Infinity, 1947.

IF YOU pour a glass of water and look at it, you will see a clear
uniform fluid with no trace of any internal structure or motion
in it whatsoever ( provided, of course, you do not shake the glass ) .
We know, however, that the uniformity of water is only apparent
and that if the water is magnified a few million times, there will
be revealed a strongly expressed granular structure formed by a
large number of separate molecules closely packed together.

Under the same magnification it is also apparent that the water
is far from still, and that its molecules are in a state of violent
agitation moving around and pushing one another as though they
were people in a highly excited crowd. This irregular motion of
water molecules, or the molecules of any other material substance,
is known as heat (or thermal) motion, for the simple reason that
it is responsible for the phenomenon of heat. For, although
molecular motion as well as molecules themselves are not directly
discernible to the human eye, it is molecular motion that produces
a certain irritation in the nervous fibers of the human organism
and produces the sensation that we call heat. For those organisms
that are much smaller than human beings, such as, for example,
small bacteria suspended in a water drop, the effect of thermal
motion is much more pronounced, and these poor creatures are
incessandy kicked, pushed, and tossed around by the restless
molecules that attack them from all sides and give them no rest
(Figure 77). This amusing phenomenon, known as Brownian
motion, named after the English botanist Robert Brown, who first
noticed it more than a century ago in a study of tiny plant spores,
is of quite general nature and can be observed in the study of any
kind of sufficiently small particles suspended in any kind of
liquid, or of microscopic particles of smoke and dust floating
in the air.

If we heat the Hquid the wild dance of tiny particles suspended
in it becomes more violent; with cooling the intensity of the
motion noticeably subsides. This leaves no doubt that we are
actually watching here the effect of the hidden thermal motion
of matter, and that what we usually call temperature is nothing
else but a measurement of the degree of molecular agitation. By
studying the dependence of Brownian motion on temperature,
it was found that at the temperature of -273° C or -459° F,

Figure 77

Six consecutive positions of a bacterium which is being tossed around by
molecular impacts (physically correct; bacteriologically not quite so).

thermal agitation of matter completely ceases, and all its mole-
cules come to rest. This apparently is the lowest temperatiure
and it has received the name of absolute zero. It would be an
absurdity to speak about still lower temperatures since apparently
there is no motion slower than absolute rest!

Near the absolute zero temperature the molecules of any sub-
stance have so little energy that the cohesive forces acting upon
them cement them together into one solid block, and all they

The Law of Disorder

can do is only quiver slightly in their frozen state. When the
temperature rises the quivering becomes more and more intense,
and at a certain stage our molecules obtain some freedom of
motion and are able to slide by one another. The rigidity of the
frozen substance disappears, and it becomes a fluid. The tem-
perature at which the melting process takes place depends on the
strength of the cohesive forces acting upon the molecules. In
some materials such as hydrogen, or a mixture of nitrogen and
oxygen which form atmospheric air, the cohesion of molecules
is very weak, and the thermal agitation breaks up the frozen
state at comparatively low temperatures. Thus hydrogen exists in
the frozen state only at temperatures below 14° abs (i.e,, below
— 259° C), whereas soHd oxygen and nitrogen melt at 55° abs
and 64° abs, respectively (i.e. -218° C and -209° C). In other
substances the cohesion between molecules is stronger and they
remain soHd up to higher temperatures: thus pure alcohol re-
mains frozen up to —130° C, whereas frozen water (ice) melts
only at 0° C. Other substances remain solid up to much higher
temperatures; a piece of lead vdll melt only at +327° C, iron at
+ 1535° C, and the rare metal known as osmium remains sohd up
to the temperature of -1-2700° C. Although in the sohd state of
matter the molecules are strongly bound to their places, it does
not mean at all that they are not affected by thermal agitation.
Indeed, according to the fundamental law of heat motion, the
amount of energy in every molecule is the same for all sub-
stances, solid, hquid, or gaseous at a given temperature, and the
difference lies only in the fact that whereas in some cases this
energy suflBces to tear off the molecules from their fixed positions
and let them travel around, in other cases they can only quiver
on the same spot as angry dogs restricted by short chains.

This thermal quivering or vibration of molecules forming a
solid body can be easily observed in the X-ray photographs de-
scribed in the previous chapter. We have seen indeed that, since
taking a picture of molecules in a crystal lattice requires a con-
siderable time, it is essential that they should not move away
from their fixed positions during the exposure. But a constant
quivering around the fixed position is not conducive to good
photography, and results in a somewhat blurred picture. This

Absolute zero

f?OOrtTEnPERATURE

. Meltin(^ point

Figure 78

The Law of Disorder

eflFect is shown in the molecular photograph which is repro-
duced in Plate I. To obtain sharper pictures one must cool the
crystals as much as possible. This is sometimes accomplished by
dipping them in liquid air. If, on the other hand, one warms up
the crystal to be photographed, the picture becomes more and
more blurred, and, at the melting point the pattern completely
vanishes, owing to the fact that the molecules leave their places
and begin to move in an irregular way through the melted
substance.

After solid material melts, the molecules still remain together,
since the thermal agitation, though strong enough to dislocate
them from the fixed position in the crystalline lattice, is not yet
suflBcient to take them completely apart. At still higher tem-
peratures, however, the cohesive forces are not able to hold the
molecules together any more and they fly apart in all directions
unless prevented from doing so by the surrounding walls. When
this happens, of course, the result is matter in a gaseous state.
As in the melting of a solid, the evaporation of liquids takes place
at different temperatures for different materials, and the sub-
stances with a weaker internal cohesion will turn into vapor at
lower temperatures than those in which cohesive forces are
stronger. In this case the process also depends rather essentially
on the pressure under which the liquid is kept, since the outside
pressure evidently helps the cohesive forces to keep the molecules
together. Thus, as everybody knows, water in a tightly closed
kettle boils at a lower temperature than will water in an open one.
On the other hand, on the top of high mountains, where atmos-
pheric pressure is considerably less, water will boil well below
100° C. It may be mentioned here that by measuring the tem-
perature at which water will boil, one can calculate atmospheric
pressure and consequently the distance above sea level of a given
location.

But do not follow the example of Mark Twain who, according
to his story, once decided to put an aneroid barometer into a
boihng kettle of pea soup. This will not give you any idea of the
elevation, and the copper oxide will make the soup taste bad.

The higher the melting point of a substance, the higher is its
boiling point. Thus liquid hydrogen boils at —253° C, liquid

oxygen and nitrogen at —183° C and —196° C, alcohol at
+78° C, lead at +1620° C, iron at +3000° C and osmium only
above +5300° C.^

The breaking up of the beautiful crystalline structure of solid
bodies forces the molecules first to crawl around one another
like a pack of worms, and then to fly apart as though they were a
flock of frightened birds. But this latter phenomenon still does
not represent the limit of the destructive power of increasing
thermal motion. If the temperature rises still farther the very
existence of the molecules is threatened, since the ever increasing
violence of intermolecular collisions is capable of breaking them
up into separate atoms. This thermal dissociation, as it is called,
depends on tlie relative strength of the molecules subjected to it.
The molecules of some organic substances will break up into
separate atoms or atomic groups at temperatures as low as a few
hundred degrees. Other more sturdily built molecules, such as
those of water, will require a temperature of over a thousand
degrees to be destroyed. But when the temperature rises to
several thousand degrees no molecules will be left and the matter
will be a gaseous mixture of pure chemical elements.

This is the situation on the surface of our sun where the tem-
perature ranges up to 6000° C. On the other hand, in the com-
paratively cooler atmospheres of the red stars,^ some of the mole-
cules are still present, a fact that has been demonstrated by the
methods of spectral analysis.

The violence of thermal collisions at high temperatures not
only breaks up the molecules into their constituent atoms, but
also damages the atoms themselves by chipping off their outer
electrons. This thermal ionization becomes more and more pro-
nounced when the temperature rises into tens and hundreds of
thousands of degrees, and reaches completion at a few million
degrees above zero. At these tremendously hot temperatures,
which are high above everything that we can produce in our
laboratories but which are common in the interiors of stars and
in particular inside our sun, the atoms as such cease to exist.
All electronic shells are completely stripped off, and the matter

^ All values given for atmospheric pressure.
2 See Chapter XI.

The Law of Disorder

(Covrtesy of Dr. M. L. Huggins. Eastman Kodak Laboratory.)

PLATE I

Photograph of Hexamethylbenzene molecule magnified 175,000,000
times.

becomes a mixture of bare nuclei and free electrons rushing
wildly through space and colliding with one another with tre-
mendous force. However, in spite of the complete wreckage of
atomic bodies, the matter still retains its fundamental chemical

Icmi

\0'K

loS

lO^'Ky

I0'»K

\oh

ATomic "DucUt

bY«Qk up.

1 break u

shells
P

lO^'K ►

100°K
lO'K

- O^Ynium mclls

- Jro-n Tn?lts.

x:: — Wo'hr boils.

*^ fTo7cn Water TncllS.

<: Pvozcn alcohol mtlt"*.

<i — Li<}oicl Kydvogcn botls.
< — hro7ei^ lydvoaen m«l'J"s.

MolecoUs
brwk V)P.

Ati.o -L< — EvERvrwifvo. Froze w.
Figure 79
The destructive eflFect of temperature.

characteristics, inasmuch as atomic nuclei remain intact. If the
temperature drops, the nuclei will recapture their electrons and
the integrity of atoms will be reestablished.

In order to attain complete thermal dissociation of matter, that
is to break up the nuclei themselves into the separate nucleons
(protons and neutrons) the temperature must go up to at least
several billion degrees. Even inside tlie hottest stars we do not

The Law of Disorder

find such high temperatures, though it seems very likely that tem-
peratures of that magnitude did exist several bilhon years ago
when our universe was still young. We shall return to this exciting
question in the last chapter of this book.

Thus we see that the e£Fect of thermal agitation is to destroy
step by step the elaborate architecture of matter based on the law
of quantum, and to turn this magnificent building into a mess of
widely moving particles rushing around and colhding with one
another without any apparent law or regularity.

2. HOW CAN ONE DESCRIBE DISORDERLY MOTION?

It would be, however, a grave mistake to think that because of
the irregularity of thermal motion it must remain outside the
scope of any possible physical description. Indeed the fact itself
that thermal motion is completely irregular makes it subject to a
new kind of law, the Law of Disorder better known as the Law oj
Statistical Behavior. In order to understand the above statement
let us turn our attention to the famous problem of a "Drunkard's
Walk." Suppose we watch a drunkard who has been leaning
against a lamp post in the middle of a large paved city square
(nobody knows how or when he got there) and then has sud-
denly decided to go nowhere in particular. Thus off he goes,
making a few steps in one direction, then some more steps in an-
other, and so on and so on, changing his course every few steps
in an entirely unpredictable way (Figure 80). How far will be
our drunkard from the lamp post after he has executed, say, a
hundred phases of his irregular zigzag journey? One would at
first think that, because of the unpredictabiHty of each turn, there
is no way of answering this question. If, however, we consider
the problem a little more attentively we will find that, although
we really cannot tell where the drunkard will be at the end of his
walk, we can answer the question about his most probable dis-
tance from the lamp post after a given large number of turns. In
order to approach this problem in a vigorous mathematical way
let us draw on the pavement two co-ordinate axes with the origin
in the lamp post; the X-axis coming toward us and the Y-axis to
the right. Let R be the distance of the drunkard from the lamp

post after the total of N zigzags ( 14 in Figure 80 ) . If now Xu and
Yn are the projections of the N'^ leg of the track on the corre-
sponding axis, the Pythagorean theorem gives us apparently:

R2= (Xi+Xa+Xs- • • +X^)2+ (Y1+Y2+Y3+ • • •Y,,)2

where X's and Y's are positive or negative depending on whether
our drunkard was moving to or from the post in this particular

Figure 80
Drunkard's walk.

phase of his walk. Notice that since his motion is completely dis-
orderly, there will be about as many positive values of X's and
Y's as there are negative. In calculating the value of the square
of the terms in parentheses according to the elementary rules of
algebra, we have to multiply each term in the bracket by itself
and by each of all other terms.

The Law of Disorder

Thus:

(Xi+X2+X3+---Xj,)2
= (X1+X2+X3+ • • -Xn) {X1+X2+X3+ • • •X.v)
= Xi2+XiX2+XiX3+ • • •X22+X1X2+ • • -X^^

This long sum will contain the square of all X's (Xi^, X2^ • • • X^f^),
and the so-called "mixed products" like XiX2, X2X3, etc.

So far it is simple arithmetic, but now comes the statistical point
based on the disorderhness of the drunkard's walk. Since he was
moving entirely at random and would just as likely make a step
toward the post as away from it, the values of X's have a fifty-fifty
chance of being either positive or negative. Consequently in
looking through the "mixed products" you are likely to find always
the pairs that have the same numerical value but opposite signs
thus canceling each other, and the larger the total number of
turns, the more likely it is that such a compensation takes place.
What will be left are only the squares of X's, since the square is
always positive. Thus the whole thing can be written as
X12+X22+ --■'X^^ = N X^ where X is the average length of the
projection of a zigzag Hnk on the X-axis.

In the same way we find that the second bracket containing
Ts can be reduced to: NY^, Y being the average projection of the
link on the Y-axis. It must be again repeated here that what
we have just done is not stricdy an algebraic operation, but is
based on the statistical argument concerning the mutual cancel-
lation of "mixed products" because of the random nature of the
pass. For the most probable distance of our drunkard from the
lamp post we get now simply:

il2 = N (X2+Y2)
or

R = ^'^/X^+W
But the average projections of the link on both axes is simply
a 45° projection, so that y/X^+W right is ( again because of the
Pythagorean theorem ) simply equal to the average length of the
hnk. Denoting it by 1 we get:

R = l'y/N
In plain words our result means: the most probable distance of

OUT drunkard from the lamp post after a certain large number of
irregular turns is equal to the average length of each straight
track that he walks, times the square root of their number.

Thus if our drunkard goes one yard each time before he turns
(at an unpredictable angle!), he will most probably be only ten
yards from the lamp post after walking a grand total of a hundred
yards. If he had not turned, but had gone straight, he would be a
hundred yards away — which shows that it is definitely advan-
tageous to be sober when taking a walk.

' / /

Figure 81
Statistical distribution of six walking drunkards around the lamp post.

The statistical nature of the above example is revealed by the
fact that we refer here only to the most probable distance and not
to the exact distance in each individual case. In the case of an
individual drunkard it may happen, though this is not very prob-
able, that he does not make any turns at all and thus goes far
away from the lamp post along the straight hne. It may also
happen, that he turns each time by, say, 180 degrees thus re-
turning to the lamp post after every second turn. But if a large
number of drunkards all start from the same lamp post walking
in diflFerent zigzag paths and not interfering with one another

The Law of Disorder

you will find after a suflSciently long time that they are spread
over a certain area around the lamp post in such a way that their
average distance from the post may be calculated by the above
rule. An example of such spreading due to irregular motion is
given in Figure 81, where we consider six walking drunkards.
It goes without saying that the larger the number of drunkards,
and the larger the number of turns they make in their disorderly
walk, the more accurate is the rule.

Now substitute for the drunkards some microscopic bodies such
as plant spores or bacteria suspended in liquid, and you will have
exactly the picture that the botanist Brown saw in his microscope.
True the spores and bacteria are not drunk, but, as we have said
above, they are being incessantly kicked in all possible directions
by the surrounding molecules involved in thermal motion, and
are therefore forced to follow exactly the same irregular zigzag
trajectories as a person who has completely lost his sense of
direction under the influence of alcohol.

If you look through a microscope at the Brownian motion of a
large number of small particles suspended in a drop of water,
you will concentrate your attention on a certain group of them
that are at the moment concentrated in a given small region ( near
the 'lamp post"). You will notice that in the course of time they
become gradually dispersed all over the field of vision, and that
their average distance from the origin increases in proportion
to the square root of the time interval as required by the mathe-
matical law by which we calculated the distance of the drunkard's
walk.

The same law of motion pertains, of course, to each separate
molecule in our drop of water; but you cannot see separate mole-
cules, and even if you could, you wouldn't be able to distinguish
between them. To make such motion visible one must use two
different kinds of molecules distinguishable for example by their
different colors. Thus we can fiU one half of a chemical test tube
with a water solution of potassium permanganate, which will give
to the water a beautiful purple tint. If we now pour on the top
of it some clear fresh water, being careful not to mix up the two
layers, we shall notice that the color gradually penetrates the
clear water. If you wait suflBciently long you will find that all the

water from the bottom to the surface becomes uniformly colored.
This phenomenon, familiar to everybody, is known as diffusion
and is due to the irregular thermal motion of the molecules of dye
among the water molecules. We must imagine each molecule of
potassium permanganate as a little drunkard who is driven to
and fro by the incessant impacts received from other molecules.
Since in water the molecules are packed rather tightly (in con-
trast to the arrangement of those in a gas ) the average free path
of each molecule between two successive collisions is very short,
being only about one hundred millionths of an inch. Since on
the other hand the molecules at room temperature move with the
speed of about one tenth of a mile per second, it takes only one
million-millionth part of a second for a molecule to go from
one collision to another. Thus in the course of a single second

\ 7

Figure 82

each dye molecule will be engaged in about a million million
consecutive collisions and will change its direction of motion as
many times. The average distance covered during the first second
will be one hundred millionth of an inch ( the length of free path )
times the square root of a million millions. This gives the average
difiFusion speed of only one hundredth of an inch per second; a
rather slow progress considering that if it were not deflected by
collisions, the same molecule would be a tenth of a mile away!
If you wait 100 sec, the molecule will have struggled through
10 times (V 100 = 10) as great distance, and in 10,000 sec, that
is, in about 3 hr, the diffusion will have carried the coloring
100 times farther (V 10000 = 100), that is, about 1 in. away. Yes,

100

The Law of Disorder

diffusion is a rather slow process; when you put a lump of sugar
into your cup of tea you had better stir it rather than wait until
the sugar molecules have been spread throughout by their own
motion.

Just to give another example of the process of diffusion, which
is one of the most important processes in molecular physics, let
us consider the way in which heat is propagated through an iron
poker, one end of which you put into the fireplace. From your
own experience you know that it takes quite a long time until
the other end of the poker becomes uncomfortably hot, but you
probably do not know that the heat is carried along the metal
stick by the process of diffusion of electrons. Yes, an ordinary
iron poker is actually stuffed with electrons, and so is any metallic
object. The difference between a metal, and other materials, as
for example glass, is that the atoms of the former lose some of
their outer electrons, which roam all through the metalHc lattice,
being involved in irregular thermal motion, in very much the
same way as the particles of ordinary gas.

The surface forces on the outer boundaries of a piece of metal
prevent these electrons from getting out,^ but in their motion
inside the material they are almost perfectly free. If an electric
force is applied to a metal wire, the free unattached electrons
will rush headlong in the direction of the force producing the
phenomenon of electric current. The nonmetals on the other hand
are usually good insulators because all their electrons are bound
to be atoms and thus cannot move freely.

When one end of a metal bar is placed in the fire, the thermal
motion of free electrons in this part of the metal is considerably
increased, and the fast-moving electrons begin to diffuse into the
other regions carrying with them the extra energy of heat. The
process is quite similar to the diffusion of dye molecules through
water, except that instead of having two different kinds of par-
ticles (water molecules and dye molecules) we have here the
diffusion of hot electron gas into the region occupied hy cold
electron gas. The drunkard's walk law appHes here, however, just

^ When we bring a metal wire to a high temperature, the thermal motion
of electrons in its inside becomes more violent and some of them come out
through the svirface. This is the phenomenon used in electron tubes and
familiar to all radio amateurs.

101

as well and the distances through which the heat propagates
along a metal bar increase as the square roots of corresponding
times.

As our last example of diffusion we shall take an entirely dif-
ferent case of cosmic importance. As we shall learn in the fol-
lowing chapters the energy of our sun is produced deep in its
interior by the alchemic transformation of chemical elements.
This energy is liberated in the form of intensive radiation, and
the "particles of light," or the hght quanta begin their long jour-
ney through the body of the sun towards its surface. Since light
moves at a speed of 300,000 km per second, and the radius of
the sun is only 700,000 km it would take a light quantum only
slightly over two seconds to come out provided it moved without
any deviations from a straight line. However, this is far from being
the case; on their way out the hght quanta undergo innumerable
colhsions with the atoms and electrons in the material of the sun.
The free pass of a light quantum in solar matter is about a centi-
meter (much longer than a free pass of a molecule!) and since
the radius of the sun is 70,000,000,000 cm, our light quantum must
make (7' 10^°)^ or 5-10^^ drunkard's steps to reach the surface.

Since each step requires — or 3-10"^ sec, the entire time of

travel is 3 • 10-^ X 5 • lO^i = 1.5 • lO^^ sec or about 200,000 yr! Here
again we see how slow the process of diffusion is. It takes light
2000 centuries to travel from the center of the sun to its surface,
whereas after coming into empty intraplanetary space and
traveling along a straight line it covers the entire distance from
the sun to the earth in only eight minutes!

3. COUNTING PROBABILITIES

This case of diffusion represents only one simple example of
the application of the statistical law of probability to the problem
of molecular motion. Before we go farther with that discussion,
and make the attempt to understand the all-important Law of
Entropy, which rules the thermal behavior of every material
body, be it a tiny droplet of some liquid or the giant universe of
stars, we have first to learn more about the ways in which the

102

The Law of Disorder

probability of different simple or complicated events can be cal-
culated.

By far the simplest problem of probability calculus arises when
you toss a coin. Everybody knows that in this case (wdthout
cheating) there are equal chances to get heads or tails. One
usually says that there is a fifty-fifty chance for heads or tails,
but it is more customary in mathematics to say that the chances
are half and half. If you add the chances of getting heads and
getting tails you get ^ + 1 = 1. Unity in the theory of probabiUty
means a certainty; you are in fact quite certain that in tossing a

Figure 83
Four possible combinations in tossing two coins.

coin you get either heads or tails, unless it rolls under the sofa and
vanishes tracelessly.

Suppose now you drop the coin twice in succession or, what is
the same, you drop 2 coins simultaneously. It is easy to see that
you have here 4 different possibilities shown in Figure 83.

In the first case you get heads twice, in the last case tails
twice, whereas the two intermediate cases lead to the same
result since it does not matter to you in which order ( or in which
coin) heads or tails appear. Thus you say that the chances of
getting heads twice are 1 out of 4 or :^ the chances of getting
tails twice are also ^, whereas the chances of heads once and tails
once are 2 out of 4 or ^. Here again i + i + i = 1 meaning that you

103

III

are certain to get one of the 3 possible'combinations. Let us see
now what happens if we toss the coin 3 times. There are altogether
8 possibilities summarized in the following table:

First tossing

Second

Third

If you inspect this table you find that there is 1 chance out of 8
of getting heads three times, and the same of getting tails three
times. The remaining possibilities are equally divided between
heads twice and tails once, or heads once and tails twice, with
the probabihty three eighths for each event.

Our table of different possibilities is growing rather rapidly,
but let us take one more step by tossing 4 times. Now we have
the following 16 possibiUties:

First tossing

Second

Third

Fourth

I II II III II mill IV II iiiiiiiviiiiviv V

Here we have ^^ for the probability of heads four times, and
exactly the same for tails four times. The mixed cases of heads
three times and tails once or tails three times and heads once
have the probabilities of fie ^^ i each, whereas the chances of
heads and tails the same number of times are %6 or f.

If you try to continue in a similar way for larger numbers of
tosses the table becomes so long that you will soon run out of
paper; thus for example for ten tosses you have 1024 different
possibilities (i.e., 2x2x2x2x2x2x2x2x2x2). But it is not
at all necessary to construct such long tables since the simple
laws of probability can be observed in those simple examples that
we already have cited and then used directly in more compli-
cated cases.

First of all you see that the probability of getting heads twice
is equal to the product of the probabilities of getting it separately
in the first and in the second tossing; in fact i = i X |. Similarly

104

The Law of Disorder

the probability of getting heads three or four times in succession
is the product of probabihties of getting it separately in each
tossing (i = ixix|; ^^ = ^X^X^Xi). Thus if somebody asks
you what the chances are of getting heads each time in ten toss-
ings you can easily give the answer by multiplying ^ by ^ ten
times. The result will be .00098, indicating that the chances are
very low indeed: about one chance out of a thousand! Here we
have the rule of "multiphcation of probabilities," which states
that if you want several different things, you may determine the
mathematical probability of getting them by multiplying the
mathematical probabilities of getting the several individual ones.
If there are many things you want, and each of them is not par-
ticularly probable, the chances that you get them all are dis-
couragingly low!

There is also another rule, that of the "addition of probabilities,"
which states that if you want only one of several things (no matter
which one), the mathematical probability of getting it is the sum
of mathematical probabilities of getting individual items on your
list.

This can be easily illustrated in the example of getting an equal
division between heads and tails in tossing a coin twice. What
you actually want here is either "heads once, tails twice" or "tails
twice, heads once." The probabihty of each of the above com-
binations is ^, and the probability of getting either one of them
is I plus ^ or 4. Thus: If you want "that, and that, and that . . ."
you multiply the individual mathematical probabihties of dif-
ferent items. If, however, you want "that, or that, or that" you
add the probabilities.

In the first case your chances of getting ever)ihing you ask for
will decrease as the number of desired items increases. In the
second case, when you want only one out of several items your
chances of being satisfied increase as the Hst of items from which
to choose becomes longer.

The experiments with tossing coins furnish a fine example of
what is meant by saying that the laws of probabihty become
more exact when you deal with a large number of trials. This is
illustrated in Figure 84, which represents the probabihties of
getting a different relative number of heads and tails for two.

105

three, four, ten, and a hundred tossings. You see that with the
increasing number of tossings the probabihty curve becomes
sharper and sharper and the maximum at fifty-fifty ratio of heads
and tails becomes more and more pronounced.

Thus whereas for 2 or 3, or even 4 tosses, the chances to have
heads each time or tails each time are still quite appreciable, in
10 tosses even 90 per cent of heads or tails is very improbable.

0.75^ -.

0.5^0^

0.2$-

Figure 84
Relative number of tails and heads.

For a still larger number of tosses, say 100 or 1000, the probability
curve becomes as sharp as a needle, and the chances of getting
even a small deviation from fifty-fifty distribution becomes prac-
tically nil.

Let us now use the simple rules of probability calculus that we
have just learned in order to judge the relative probabihties of
various combinations of five playing cards which one encounters
in the well-known game of poker.

106

The Law of Disorder

In case you do not know, each player in this game is dealt
5 cards and the one who gets the highest combination takes the
bank. We shall omit here the additional complications arising
from the possibility of exchanging some of your cards with the
hope of getting better ones, and the psychological strategy of
bluffing your opponents into submission by making them believe
that you have much better cards than you actually have. Although
this bluffing actually is the heart of the game, and once led the
famous Danish physicist Niels Bohr to propose an entirely new
type of game in which no cards are used, and the players simply
bluff one another by talking about the imaginary combinations
they have, it lies entirely outside the domain of probabihty
calculus, being a purely psychological matter.

FiGUBE 85
A flush (of spades).

In order to get some exercise in probability calculus, let us
calculate the probabilities of some of the combinations in the
game of poker. One of these combinations is called a "flush" and
represents 5 cards all of the same suit (Figure 85).

If you want to get a flush it is immaterial what the first card

you get is, and one has only to calculate the chances that the

other four will be of the same suit. There are altogether 52 cards

in the pack, 13 cards of each suit,* so that after you get your first

card, there remain in the pack 12 cards of the same suit. Thus

the chances that your second card will be of the proper suit are

12/51. Similarly the chances that the third, fourth, and fifth cards

■* We omit here the complications arising from the presence of the "joker,"
an extra card which can be substituted for any other card according to the
desire of the player.

107

will be of the same suit are given by the fractions: 11/50, 10/49
and 9/48. Since you want all 5 cards to be of the same suit you
have to apply the rule of probability-multiplications. Doing this
you find that the probability of getting a flush is:

12 11 10 9 13068

— X — X — X — = or about 1 in 500.

51 50 49 48 5997600

But please do not think that in 500 hands you are sure to get a
flush. You may get none, or you may get two. This is only prob-
ability calculus, and it may happen that you will be dealt many
more than 500 hands without getting the desired combination, or
on the contrary that you may be dealt a flush the very first time
you have the cards in your hands. All that the theory of prob-

FlGURE 86

Full house.

ability can tell you is that you will probably be dealt 1 flush in 500
hands. You may also learn, by following the same methods of
calculation, that in playing 30,000,000 games you will probably
get 5 aces ( including the joker ) about ten times.

Another combination in poker, which is even rarer and there-
fore more valuable, is the so-called "full hand," more popularly
called "full house." A full house consists of a "pair" and "three of
a kind" ( that is, 2 cards of the same value in 2 suits, and 3 cards
of the same value in 3 suits — as, for example, the 2 fives and
3 queens shown in Figure 86).

If you want to get a full house, it is immaterial which 2 cards
you get first, but when you get them you must have 2 of the re-
maining 3 cards match one of them, and the other match the

108

The Law of Disorder

other one. Since there are 6 cards that will match the ones you
have (if you have a queen and a five, there are 3 other queens
and 3 other fives ) the chances that the third card is a right one
are 6 out of 50 or 6/50. The chances that the fourth card will be
the right one are 5/49 since there are now only 5 right cards out
of 49 cards left, and the chance that the fifth card will be right
is 4/48. Thus the total probability of a full house is:

6 5 4 120

— X — X — =-

50 49 48 117600

or about one half of the probabilit\' of the flush.

In a similar way one can calculate the probabilities of other
combinations as, for example, a "straight" (a sequence of cards),
and also take into account the changes in probabiht\' introduced
by the presence of the joker and the possibility of exchanging
the originally dealt cards.

By such calculations one finds that the sequence of seniority
used in poker does really correspond to the order of mathematical
probabilities. It is not known by the author whether such an
arrangement was proposed by some mathematician of the old
times, or was established purely empirically by miUions of
players risking their money in fashionable gambling salons and
little dark haunts all over the world. If the latter was the case,
we must admit that we have here a pretty good statistical study
of the relative probabilities of complicated events!

Another interesting example of probability calculation, an ex-
ample that leads to a quite unexpected answer, is the problem of
"Coinciding Birthdays." Try to remember whether you have ever
been invited to two different birthday parties on the same day.
You will probably say that the chances of such double invitations
are very small since you have only about 24 friends who are
likely to invite you, and there are 365 days in the year on which
their birthdays may faU. Thus, v^ith so many possible dates to
choose from, there must be very htde chance that any 2 of your
24 friends will have to cut their birthday cakes on the same day.

However, unbehevable as it may sound, your judgment here is
quite wrong. The truth is that there is a rather high probabihty
tiiat in a company of 24 people there are a pair, or even several
pairs, with coinciding birthdays. As a matter of fact, there are
more chances that there is such a coincidence than that there is not.

109

You can verify that fact by making a birthday list including
about 24 persons, or more simply, by comparing the birth dates
of 24 persons whose names appear consecutively on any pages of
some such reference book as "Who's Who in America," opened
at random. Or the probabilities can be ascertained by using the
simple rules of probability calculus with which we have become
acquainted in the problems of coin tossing and poker.

Suppose we try first to calculate the chances that in a company

of twenty-four persons everyone has a diflFerent birth date. Let

us ask the first person in the group what is his birth date; of

course this can be any of the 365 days of the year. Now, what is

the chance that the birth date of the second person we approach

is different from that of the first? Since this (second) person

could have been born on any day of the year, there is one chance

out of 365 that his birth date coincides with that of the first one,

and 364 chances out of 365 (i.e., the probability of 364/365) that

it does not. Similarly, the probability that the third person has a

birth date different from that of either the first or second is

363/365, since two days of the year have been excluded. The

probabilities that the next persons we ask have different birth

dates from the ones we have approached before are then: 362/365,

361/365, 360/365 and so on up to the last person for whom the

, , ,. (365-23) 342
probability is -^^- or — .

Since we are trying to learn what the probability is that one of
these coincidences of birth dates exists, we have to multiply all
the above fractions, thus obtaining for the probability of all the
persons having different birth dates the value:

364 363 362 342

365^365^365^ "'365

One can arrive at the product in a few minutes by using cer-
tain methods of higher mathematics, but if you don't know them
you can do it the hard way by direct multiplication,'^ which
would not take so very much time. The result is 0.46, indicating
that the probability that there will be no coinciding birthdays
is slightly less than one half. In other words there are only 46
chances in 100 that no two of your two dozen friends will have
* Use a logarithmic table or slide rule if you can!

110

The Law of Disorder

birthdays on the same day, and 54 chances in 100 that two or
more will. Thus if you have 25 or more friends, and have never
been invited to two birthday parties on the same date you may
conclude with a high degree of probability that either most of
your friends do not organize their birthday parties, or that they
do not invite you to them!

The problem of coincident birthdays represents a very fine
example of how a common-sense judgment concerning the
probabilities of complex events can be entirely wrong. The
author has put this question to a great many people, including
many prominent scientists, and in all cases except one^ was
oflFered bets ranging from 2 to 1 to 15 to 1 that no such co-
incidence will occur. If he had accepted all these bets he would
be a rich man by now!

It cannot be repeated too often that if we calculate the
probabilities of diflFerent events according to the given rules and
pick out the most probable of them, we are not at all sure that
this is exactly what is going to happen. Unless the number of
tests we are making runs into thousands, millions or still better
into billions, the predicted results are only "likely" and not at all
"certain." This slackening of the laws of probability when dealing
with a comparatively small number of tests limits, for example,
the usefulness of statistical analysis for deciphering various codes
and cryptograms which are limited only to comparatively short
notes. Let us examine, for example, the famous case described
by Edgar Allan Poe in his well-known story "The Gold Bug."
He tells us about a certain Mr. Legrand who, strolling along a
deserted beach in South Carolina, picked up a piece of parchment
half buried in the wet sand. When subjected to the warmth of
the fire burning gaily in Mr. Legrand's beach hut, the parchment
revealed some mysterious signs written in ink which was invisible
when cold, but which turned red and was quite legible when
heated. There was a picture of a skull, suggesting that the docu-
ment was written by a pirate, the head of a goat, proving beyond
any doubt that the pirate was none other than the famous Captain
Kidd, and several lines of typographical signs apparently indi-
cating the whereabouts of a hidden treasure (see Figure 87).

We take it on the authority of Edgar Allan Poe that the pirates

of the seventeenth century were acquainted with such typo-

® This exception was, of course, a Hungarian mathematician ( see the
beginning of the first chapter of this book).

Ill

graphical signs as semicolons and quotation marks, and such
others as: |, +, and j[.

Being in need of money, Mr. Legrand used all his mental
powers in an attempt to decipher the mysterious cryptogram and

53 ::t30S))t*;i/8«4>;);W4*.^^|4o)^85. ,4 ^. .^
^} ',%:S%htSt8S)H)ifS^^SStBB0b*8l (i^)9i }

Figure 87
Captain Kidd's Message.

finally did so on the basis of the relative frequency of occurrence
of different letters in the English language. His method was based
on the fact that if you count the number of different letters of
any English text, whether in a Shakespearian sonnet or an Edgar
Wallace mystery story, you will find that the letter "e" occurs
by far most frequentiy. After "e" the succession of most
frequent letters is as follows:

a, o, f, d, hy n, r, 5, t, u, y, c, /, g, I, m, w, h, k, p, q, x, z

By counting the different symbols appearing in Captain Kidd's
cryptogram, Mr. Legrand found that the symbol that occurred
most frequentiy in the message was the figure 8. "Aha," he said,
"that means that 8 most probably stands for the letter e."

Well, he was right in this case, but of course it was only very
probable and not at all certain. In fact if the secret message had
been "You will find a lot of gold and coins in an iron box in woods
two thousand yards south from an old hut on Bird Island's north
tip" it would not have contained a single "e"! But the laws of
chance were favorable to Mr. Legrand, and his guess was really
correct.

Having met with success in the first step, Mr. Legrand became
overconfident and proceeded in the same way by picking up the

112

The Law of Disorder

letters in the order of the probability ot their occurrence. In the
following table we give the symbols appearing in Captain Kidd's
message in the order of their relative frequency of use:

Of the character 8 there are 33

p <'

- '^ r'

; 26

4 19

0 A

t 16

lHj \

yf^o

( 16

dA/

J ^r

* 13

hK\

/Ct^

5 12

n e^

\^ a

6 11

f 8

s /

1 8

0 6

"i^

g 5

2 5

- \

i 4

3 4

f /'

^>ir

? 3

f 2

The first column on the right contains the letters of the alpha-
bet arranged in the order of their relative frequency in the
Enghsh language. Therefore it was logical to assume that the
signs hsted in the broad column to the left stood for the letters
listed opposite them in the first narrow column to the right. But
using this arrangement we find that the beginning of Captain
Kidd's message reads: ngiisgunddrhaoecr . . .

No sense at all!

What happened? Was the old pirate so tricky as to use special
words that do not contain letters that follow the same rules of
frequency as those in the words normally used in the English
language? Not at all; it is simply that the text of the message is

113

not long enough for good statistical sampling and the most prob-
able distribution of letters does not occur. Had Captain Kidd
hidden his treasure in such an elaborate way that the instructions
for its recovery occupied a couple of pages, or, still better an
entire volume, Mr. Legrand would have had a much better
chance to solve the riddle by applying the rules of frequency.

If you drop a coin 100 times you may be pretty sure that it will
fall with the head up about 50 times, but in only 4 drops you
may have heads three times and tails once or vice versa. To make
a rule of it, the larger the number of trials, the more accurately
the laws of probability operate.

Since the simple method of statistical analysis failed because
of an insufficient number of letters in the cryptogram, Mr. Le-
grand had to use an analysis based on the detailed structure of
different words in the English language. First of all he strength-
ened his hypothesis that the most frequent sign 8 stood for e by
noticing that the combination 88 occurred very often (5 times)
in this comparatively short message, for, as everyone knows, the
letter e is very often doubled in English words (as in: meet, fleet,
speed, seen, been, agree, etc. ) . Furthermore if 8 really stood for e
one would expect it to occur very often as a part of the word
"the." Inspecting the text of the cryptogram we find that the
combination ;48 occurs seven times in a few short lines. But if this
is true, we must conclude that ; stands for t and 4 for h.

We refer the reader to the original Poe story for the details
concerning the further steps in the deciphering of Captain Kidd's
message, the complete text of which was finally found to be:
**A good glass in the bishop's hostel in the devil's seat. Forty-one
degrees and thirteen minutes northeast by north. Main branch
seventh limb east side. Shoot from the left eye of the death's
head. A bee-line from the tree through the shot fifty feet out."

The correct meaning of the different characters as finaUy de-
ciphered by Mr. Legrand is shown in the second column of the
table on page 217, and you see that they do not correspond exactly
to the distribution that might reasonably be expected on the
basis of the laws of probability. It is, of course, because the text
is too short and therefore does not furnish an ample opportunity
for the laws of probability to operate. But even in this small
"statistical sample" we can notice the tendency for the letters
to arrange themselves in the order required by the theory of
probability, a tendency that would become almost an unbreak-

114

The Law of Disorder

Figure 88

able rule if the number ot letters in the message were much
larger.

There seems to be only one example (excepting the fact that
insurance companies do not break up) in which the predictions
of the theory of probability have actually been checked by a
very large number of trials. This is a famous problem of the
American flag and a box of kitchen matches.

To tackle this particular problem of probabihty you wdll need
an American flag, that is, the part of it consisting of red and
white stripes; if no flag is available just take a large piece of
paper and draw on it a number of parallel and equidistant lines.
Then you need a box of matches — any kind of matches, provided
they are shorter than the wddth of the stripes. Next you will need
a Greek pi, which is not something to eat, but just a letter of the
Greek alphabet equivalent to our "p." It looks like this: ir. In
addition to being a letter of the Greek alphabet, it is used to
signify the ratio of the circumference of a circle to its diameter.
You may know that numerically it equals 3.1415926535 . . .
(many more digits are known, but we shall not need them all.)

Now spread the flag on a table, toss a match in the air and
watch it fall on the flag (Figure 88). It may fall in such a way
that it all remains vsdthin one stripe, or it may fall across the
boundary between two stripes. What are the chances that one or
another will take place?

Following our procedure in ascertaining other probabilities.

115

we must first count the number of cases that correspond to one
or another possibility.

But how can you count all the possibihties when it is clear
that a match can fall on a flag in an infinite number of different
ways?

Let us examine the question a Httle more closely. The position
of the fallen match in respect to the stripe on which it falls
can be characterized by the distance of the middle of the match

Ark,
Figure 89

from the nearest boundary line, and by the angle that the match
forms with the direction of the stripes in Figure 89. We give
three typical examples of fallen matches, assuming, for the sake
of simplicity, that the length of the match equals the width of
the stripe, each being, say, two inches. If the center of the match
is rather close to the boundary line, and the angle is rather large
(as in case a) the match will intersect the line. If, on the con-
trary, the angle is small (as in case b) or the distance is large
(as in case c) the match will remain within the boundaries of
one stripe. More exactly we may say that the match will intersect
the hne if the projection of the half-of-the-match on the vertical
direction is larger than the half width of the stripe (as in case a),
and that no intersection will take place if the opposite is true

116

The Law of Disorder

(as in case b). The above statement is represented graphically
on the diagram in the lower part of the picture. We plot on the
horizontal axis ( abscissa ) the angle of the fallen match as given
by the length of the corresponding arc of radius 1. On the vertical
axis (ordinate) we plot the length of the projection of the half-
match length on the vertical direction; in trigonometry this length
is known as the sinus corresponding to the given arc. It is clear
that the sinus is zero when the arc is zero since in that case the
match occupies a horizontal position. When the arc is ^ tt, which
corresponds to a straight angle,'' the sinus is equal to unity,
since the match occupies a vertical position and thus coincides
with its projection. For intermediate values of the arc the sinus
is given by the familiar mathematical wavy curve known as
sinusoid. (In Figure 89 we have only one quarter of a complete
wave in the interval between 0 and ir/2. )

Having constructed this diagram we can use it with con-
venience for estimating the chances that the fallen match will or
will not cross the hne. In fact, as we have seen above ( look again
at the three examples in the upper part of Figure 89 ) the match
will cross the boundary line of a stripe if ^e distance of the
center of the match from the boundary hne is less than the cor-
responding projection, that is, less than the sinus of the arc.
That means that in plotting that distance and that arc in our
diagram we get a point below the sinus line. On the contrary
the match that falls entirely within the boundaries of a stripe
will give a point above the sinus line.

Thus, according to our rules for calculating probabihties, the
chances of intersection will stand in the same ratio to the
chances of nonintersection as the area below the curve does to
the area above it; or the probabilities of the two events may be
calculated by dividing the two areas by the entire area of the
rectangle. It can be proved mathematically {cf. Chapter II) that
the area of the sinusoid presented in our diagram equals exactly

TT W

1. Since the total area of the rectangle is z-Xl=- we find the
probabihty that the match will fall across the boundary (for

' The circiimference of a circle with the radius 1 is ir times its diameter
or 2 IT. Thus the length of one quadrant of a circle is 2 ir/4 or ir/2.

117

The interesting fact that -n- pops up here where it might be
least expected was first observed by the eighteenth century
scientist Count BuflFon, and so the match-and-stripes problem now
bears his name.

An actual experiment was carried out by a diligent Italian
mathematician, Lazzerini, who made 3408 match tosses and ob-
served that 2169 of them intersected the boundary line. The
exact record of this experiment, checked with the Buffon formula,

substitutes for tt a value of — - — -— or 3.1415929, differing from

2169 ^

the exact mathematical value only in the seventh decimal placel
This represents, of course, a most amusing proof of the validity
of the probability laws, but not more amusing than the deter-
mination of a number "2" by tossing a coin several thousand
times and dividing the total number of tosses by the number
of times heads come up. Sure enough you get in this case:
2.000000 . . . with just as small an error as in Lazzerini's deter-
mination of TT.

4. THE "MYSTERIOUS" ENTROPY

From the above examples of probability calculus, all of them
pertaining to ordinary Hfe, we have learned that predictions of
that sort, being often disappointing when small numbers are in-
volved, become better and better when we go to reaUy large
numbers. This makes these laws particularly applicable to the
description of the almost innumerable quantities of atoms or
molecules that form even the smallest piece of matter we can
conveniently handle. Thus, whereas the statistical law of Drunk-
ard's Walk can give us only approximate results when applied
to a half-dozen drunkards who make perhaps two dozen turns
each, its appHcation to billions of dye molecules undergoing
billions of collisions every second leads to the most rigorous
physical law of diffusion. We can also say that the dye that was
originally dissolved in only one half of the water in the test tube
tends through the process of diffusion to spread uniformly
through the entire hquid, because, such uniform distribution is
more probable than the original one.

118

The Law of Disorder

For exactly the same reason the room in which you sit reading
this book is filled uniformly by air from wall to wall and from
floor to ceihng, and it never even occurs to you that the air in the
room can unexpectedly collect itself in a far corner, leaving you to
suflFocate in your chair. However, this horrifying event is not at
all physically impossible, but only highly improbable.

To clarify the situation, let us consider a room divided into
two equal halves by an imaginary vertical plane, and ask our-
selves about the most probable distribution of air molecules be-
tween the two parts. The problem is of course identical with the
coin-tossing problem discussed in the previous chapter. If we
pick up one single molecule it has equal chances of being in the
right or in the left half of the room, in exactly the same way as
the tossed coin can fall on the table with heads or tails up.

The second, the third, and all the other molecules also have
equal chances of being in the right or in the left part of the room
regardless of where the others are.^ Thus the problem of dis-
tributing molecules between the two halves of the room is
equivalent to the problem of heads-and-tails distribution in a
large number of tosses, and as you have seen from Figure 84,
the fifty-fifty distribution is in this case by far the most probable
one. We also see from that figure that with the increasing number
of tosses (the number of air molecules in our case) the prob-
ability at 50 per cent becomes greater and greater, turning prac-
tically into a certainty when this number becomes very large.
Since in the average-size room there are about KF"^ molecules,*
tiie probability that all of them collect simultaneously in, let us
say, the right part of the room is:

i.e., 1 out of 10.3^<>^

On the other hand, since the molecules of air moving at
the speed of about 0.5 km per second require only 0.01 sec
to move from one end of the room to the other, their dis-
tribution in the room v^dll be reshuffled 100 times each second.
Consequently the waiting time for the right combination is

* In fact, owing to large distances between separate molecules of the gas,
the space is not at all crowded and the presence of a large number of
molecules in a given volume does not at ail prevent the entrance of new
molecules.

^ A room 10 ft by 15 ft, with a 9 ft ceiling has a voltime of 1350 cu ft, or
S-IO' cu cm, thus containing 5-10' g of air. Since the average mass of air
molecules is 3- 1-66x10-**= 5x10"* g, the total number of molecules is
5- 10V5- 10-^=10*". (^ means: approximately equal to.)

119

20299.999.999.909.999.999.999.999.998 gg^j ^g COmparcd with Only 1(F' SCC

representing the total age of the universel Thus you may go on
quietly reading your book without being afraid of being suf-
focated by chance.

To take another example, let us consider a glass of water
standing on the table. We know that the molecules of water,
being involved in the irregular thermal motion, are moving at
high speed in all possible directions, being, however, prevented
from flying apart by the cohesive forces between them.

Since the direction of motion of each separate molecule is
governed entirely by the law of chance, we may consider the
possibihty that at a certain moment the velocities of one half
of the molecules, namely those in the upper part of the glass,
will all be directed upward, whereas the other half, in the lower
part of the glass, will move downwards.^* In such a case, the co-
hesive forces acting along the horizontal plane dividing two
groups of molecules will not be able to oppose their "unified
desire for parting," and we shall observe the unusual physical
phenomenon of half the water from the glass being spontaneously
shot up with the speed of a bullet toward the ceiling!

Another possibility is that the total energy of thermal motion
of water molecules will be concentrated by chance in those
located in the upper part of the glass, in which case the water
near the bottom suddenly freezes, whereas its upper layers begin
to boil violently. Why have you never seen such things happen?
Not because they are absolutely impossible, but only because
they are extremely improbable. In fact, if you try to calculate
the probability that molecular velocities, originally distributed
at random in all directions, will by pure chance assume the dis-
tribution described above, you arrive at a figure that is just about
as small as the probability that the molecules of air will collect
in one comer. In a similar way, the chance that, because of
mutual collisions, some of the molecules will lose most of their
kinetic energy, while the other part gets a considerable excess
of it, is also negligibly small. Here again the distribution of
velocities that corresponds to the usually observed case is the
one that possesses the largest probabihty.

If now we start with a case that does not correspond to the

^''We must consider this half-and-half distribution, since the possibility
that aU molecules move in the same direction is ruled out by the mechanical
law of the conservation of momentum.

120

The Law of Disorder

most probable arrangement of molecular positions or velocities,
by letting out some gas in one comer of the room, or by pouring
some hot water on top of the cold, a sequence of physical
changes will take place that will bring our system from this less
probable to a most probable state. The gas will diffuse through
the room until it fills it up uniformly, and the heat from the top
of the glass will flow toward the bottom until all the water as-
sumes an equal temperature. Thus we may say that all physical
processes depending on the irregular motion of molecules go in
the direction of increasing probability, and the state of equilib-
rium, when nothing more happens, corresponds to the maximum
of probability. Since, as we have seen from the example of the
air in the room, the probabilities of various molecular distribu-
tions are often expressed by inconveniently small numbers (as
jO-3 1028 fQj. ^g gij. collecting in one half of the room), it is cus-
tomary to refer to their logarithms instead. This quantity is known
by the name of entropy, and plays a prominent role in all ques-
tions connected with the irregular thermal motion of matter. The
foregoing statement concerning the probability changes in
physical processes can be now rewritten in the form: Any spon-
taneous changes in a physical system occur in the direction of
increasing entropy, and the firud state of equilibrium corresponds
to the maximum possible value of the entropy.

This is the famous Law of Entropy, also known as the Second
Law of Thermodynamics ( the First Law being the Law of Con-
servation of Energy), and as you see there is nothing in it to
frighten you.

The Law of Entropy can also be called the Law of Increasing
Disorder since, as we have seen in all the examples given above,
the entropy reaches its maximum when the position and velocities
of molecules are distributed completely at random so that any
attempt to introduce some order in their motion would lead to
the decrease of the entropy. Still another, more practical, formula-
tion of the Law of Entropy can be obtained by reference to the
problem of turning the heat into mechanical motion. Remember-
ing that the heat is actually the disorderly mechanical motion of
molecules, it is easy to understand that the complete transforma-
tion of the heat content of a given material body into mechanical
energy of large-scale motion is equivalent to the task of forcing
all molecules of that body to move in the same direction. How-
ever, in the example of the glass of water that might spon-

121

taneously shoot one half of its contents toward the ceiling, we
have seen that such a phenomenon is sufficiently improbable to
be considered as being practically impossible. Thus, although the
energy of mechanical motion can go completely over into heat
{for example, through friction), the heat energy can never go
completely into mechanical motion. This rules out the possibility
of the so-called "perpetual motion motor of the second kind,"^^
which would extract the heat from the material bodies at normal
temperature, thus cooling them down and utilizing for doing
mechanical work the energy so obtained. For example, it is im-
possible to build a steamship in the boiler of which steam is
generated not by burning coal but by extracting the heat from the
ocean water, which is first pumped into the engine room, and
then thrown back overboard in the form of ice cubes after the
heat is extracted from it.

But how then do the ordinary steam-engines turn the heat
into motion without violating the Law of Entropy? The trick is
made possible by the fact that in the steam engine only a part of
the heat liberated by burning fuel is actually turned into energy,
another larger part being thrown out into the air in the form of
exhaust steam, or absorbed by the specially arranged steam
coolers. In this case we have two opposite changes of entropy
in our system: (1) the increase of entropy corresponding to the
transformation of a part of the heat into mechanical energy of
the pistons, and (2) the decrease of entropy resulting from the
flow of another part of the heat from the hot-water boilers into
the coolers. The Law of Entropy requires only that the total
amount of entropy of the system increase, and this can be easily
arranged by making the second factor larger than the first. The
situation can probably be understood somewhat better by con-
sidering an example of a 5 lb weight placed on a shelf 6 ft
above tiie floor. According to the Law of Conservation of Energy,
it is quite impossible that this weight will spontaneously and
without any external help rise toward the ceiling. On the other
hand it is possible to drop one part of this weight to the floor
and use the energy thus released to raise another part upward.

In a similar way we can decrease the entropy in one part of
our system if there is a compensating increase of entropy in its
other part. In other words considering a disorderly motion of

*^ Called so in contrast to the "perpetual motion motor of the first kind"
which violates the law of conservation of energy working without any energy
supply.

122

The Law of Disorder

molecules we can bring some order in one region, if we do not
mind the fact that this will make the motion in other parts stiU
more disorderly. And in many practical cases, as in all kinds of
heat engines, we do not mind it.

5. STATISTICAL FLUCTUATION

The discussion of the previous section must have made it clear
to you that the Law of Entropy and all its consequences is based
entirely on the fact that in large-scale physics we are always
dealing with an immensely large number of separate molecules,
so that any prediction based on probability considerations be-
comes almost an absolute certainty. However, this kind of predic-
tion becomes considerably less certain when we consider very
small amounts of matter.

Thus, for example, if instead of considering the air filling a

large room, as in the previous example, we take a much smaller

volume of gas, say a cube measuring one hundredth of a

micron^^ each way, the situation wiU look entirely different. In

fact, since the volume of our cube is 10'^^ cu cm it will contain

10-18-10-3
only ^g 30 molecules, and the chance that all of them

will collect in one half of the original volume is (^)^®=10-^^.

On the other hand, because of the much smaller size of the
cube, the molecules wHl be reshuffled at the rate of 5 • 10^ times
per second (velocity of 0.5 km per second and the distance of
only 10"^ cm ) so that about once every second we shall find that
one half of the cube is empty. It goes without saying that the
cases when only a certain fraction of molecules become con-
centrated at one end of our small cube occur considerably more
often. Thus for example the distribution in which 20 molecules
are at one end and 10 molecules at the other (i.e only 10 extra
molecules collected at one end) v^dll occur with the frequency
of ( i ) 10 X 5 • 10^0 = 10-3 X 5 X 1010 = 5 X 10^ that is, 50,000,000 times
per second.

Thus, on a small scale, the distribution of molecules in the air is
far from being uniform. If we could use sufficient magnification,
we should notice the small concentration of molecules being
instantaneously formed at various points of the gas, only to be
dissolved again, and be replaced by other similar concentrations

"One micron, usually denoted by Greek letter Mu (m), is 0.0001 cm.

123

appearing at other points. This effect is known as fluctuation of
density and plays an important role in many physical phenomena.
Thus, for example, when the rays of the sun pass through the
atmosphere these inhomogeneities cause the scattering of blue
rays of the spectrum, giving to the sky its familiar color and mak-
ing the sun look redder than it actually is. This effect of redden-
ing is especially pronounced during the sunset, when the sun
rays must pass through the thicker layer of air. Were these fluctua-
tions of density not present the sky would always look completely
black and the stars could be seen during the day.

Similar, though less pronounced, fluctuations of density and
pressure also take place in ordinary liquids, and another way
of describing the cause of Brownian motion is by saying that
the tiny particles suspended in the water are pushed to and fro
because of rapidly varying changes of pressure acting on their
opposite sides. When the liquid is heated until it is close to its
boiling point, the fluctuations of density become more pro-
nounced and cause a slight opalescence.

We can ask ourselves now whether the Law of Entropy applies
to such small objects as those to which the statistical fluctuations
become of primary importance. Certainly a bacterium, which
through all its life is tossed around by molecular impacts, will
sneer at the statement that heat cannot go over into mechanical
motion! But it would be more correct to say in this case that the
Law of Entropy loses its sense, rather than to say that it is
violated. In fact all that this law says is that molecular motion
cannot be transformed completely into the motion of large
objects containing immense numbers of separate molecules. For
a bacterium, which is not much larger than the molecules them-
selves, the difference between the thermal and mechanical motion
has practically disappeared, and it would consider the molecular
collisions tossing it around in the same way as we would consider
the kicks we get from our fellow citizens in an excited crowd.
If we were bacteria, we should be able to build a perpetual
motion motor of the second kind by simply tying ourselves to a
flying wheel, but then we should not have the brains to use it
to our advantage. Thus there is actually no reason for being
sorry that we are not bacteria!

124

The "law of averages" applies to all randomly moving
objects whether in kinetic theory or in city traffic.
This story from The New Yorker magazine raises in
fictional form the question of the meaning of a statisti'
cal law.

8 The Law

Robert M. Coates

An article from The New Yorker Magazine, 1947.

THE first intimation that things
were getting out of hand came
one early-fall evening in the late
nineteen-forties. What happened, sim-
ply, was that between seven and nine
o'clock on that evening the Triborough
Bridge had the heaviest concentration
of outbound traffic in its entire histor\ .

This was odd, for it was a weekday
evening (to be precise, a Wednesday),
and though the weather was agreeably
mild and clear, with a moon that was
close enough to being full to lure a cer-
tain number of motorists out of the
city, these facts alone were not enough
to explain the phenomenon. No other
bridge or main highway was affected,
and though the two preceding nights
had been equally balmy and moonlit, on
both of these the bridge traffic had run
close to normal.

The bridge personnel, at any rate,
was caught entirely unprepared. A main
artery of traffic, like the Triborough,
operates under fairly predictable condi-
tions. Motor travel, like most other
large-scale human activities, obeys the
Law of Averages — that great, ancient
rule that states that the actions of people
in the mass will alwa)s follow consistent
patterns — and on the basis of past ex-
perience it had alwa)s been possible to
foretell, almost to the last digit, the
number of cars that would cross the
bridge at any given hour of the day or
night. In this case, though, all rules
were broken.

The hours from seven till nenriy mid-
night are normally quiet ones on the
bridge. But on that night it was as if
all the motorists in the city, or at any
rate a staggering proportion of them,

-had conspired together to upset tradi-
tion. Beginning almost exacti)' at seven
o'clock, cars poured onto the bridge in
such numbers and with such rapidity
that the staff at tiie toll booths was over-
whelmed almost from the start. It was
soon apparent that this was no momen-
tary congestion, and as it became more
and more obvious that the traffic jam
promised to be one of truly monumental
proportions, added details of police were
rushed to the scene to help handle it.

Cars streamed in from all direc-
tions— from the Bronx approach and
the Manhattan one, from 125th Street
and the East River Drive. (At the peak
of the crush, about eight-fifteen, ob-
servers on the bridge reported that the
drive was a solid line of car headlights
as far south as the bend at Eighty-ninth
Street, while the congestion crosstown
in Manhattan disrupted traffic as far
west as Amsterdam .Avenue.) And per-
haps the most confusing thing about
the whole manifestation was that there
seemed to be no reason for it.

Now and then, as the harried toll-
booth attendants made change for the
seemingly endless stream of cars, they
would question the occupants, and it
soon became clear that the very partici-
pants in the monstrous tieup were as
ignorant of its cause as anyone else
was. A report made by Sergeant .Alfonse
O'Toole, who commanded the detail in
charge of the Bronx approach, is typical.
"I kept askin' them," he said, " 'Is there
night football somewhere that we don't
know about.' Is it the races )'ou're goin'
tor' But the funny thing was half the
time they'd be askin' fnf. '\Vhat's the
crowd for, Mac? ' they would say. And

I'd just look at them. There was one
guy I mind, in a Ford convertible with
a girl in the seat beside him, and when
he asked me, I said to him, 'Hell, you're
in the crowd, ain't you?' I said. 'What
brings you here? ' And the dummy just
looked at me. 'Me?' he says. 'I just
come out for a drive in the moonlight.
But if I'd known there'd be a crowd like
this . . .' he says. And then he asks me,
'Is there any place I can turn around
and get out of this?' " As the Herald
Tribune summed things up in its story
next morning, it "just looked as if every-
body in Manhattan who owned a
motorcar had decided to drive out on
Long Island that evening."

THE incident was unusual enough
to make all the front pages next
morning, and because of this, many sim-
ilar events, which might otherwise have
gone unnoticed, received attention. The
proprietor of the .Aramis Theatre, on
Eighth .Avenue, reported that on sev-
eral nights in the recent past his audi-
torium had been practically empty,
while on others it had been jammed to
suffocation. Lunchroom owners noted
that increasingly their patrons were de-
veloping a habit of making rims on spe-
cific items; one day it would be the roast
shoulder of veal with pan gravy that
was ordered almost exclusively, while
the next everyone would be taking the
Vienna loaf, and the roast veal went
begging. A man who ran a small no-
tions store in Bayside revealed that over
a period of four days two hundred and
scvent)-four successive customers had
entered his shop and asked for a spool
of pink thread.

Reprinted by permission.

125

The Law

These were news items that would
ordinarily have gone into the papers as
fillers or in the sections reserved for
oddities. Now, however, they seemed
to have a more serious significance. It
was apparent at last that something de-
cidedly strange was happening to peo-
ple's habits, and it was as unsettling
as those occasional moments on excur-
sion boats when the passengers are
moved, all at once, to rush to one side
or the other of the vessel. It was
not till one day in December when,
almost incredibly, the Twentieth Cen-
tury Limited left New York for Chi-
cago with just three passengers aboard
that business leaders discovered how
disastrous the new trend could be, too.

Until then, the New York Central,
for instance, could operate confidently
on the assumption that although there
might be several thousand men in New
York who had business relations in
Chicago, on any single day no more —
and no less — than some hundreds of
them would have occasion to go there.
The play producer could be sure that his
patronage would sort itself out and
that roughly as many persons would
want to see the performance on Thurs-*
day as there had been on Tuesday or
Wednesday. Now they couldn't be sure
of anything. The Law of Averages had
gone by the board, and if the effect on
business promised to be catastrophic, it
was also singularly unnerving for the
general customer.

The lady starting downtown for a
day of shopping, for example, could
never be sure whether she would find
Macy's department store a seething
mob of other shoppers or a wilderness
of empty, echoing aisles and unoccupied
salesgirls. And the uncertainty pro-
duced a strange sort of jitteriness in the
individual when faced with any impulse
to action. "Shall we do it or shan't
wer" people kept asking themselves,
knowing that if they did do it, it might
turn out that thousands of other indi-
viduals had decided similar!)'; knowing,
too, that if they didn't, they might miss
the one glorious chance of all chances
to have Jones Beach, say, practically to
themselves. Business languished, and a
sort of desperate uncertainty rode ev-
eryone.

AT this juncture, it was inevitable
-^ ^ that Congress should be called on
for action. In fact. Congress called on
itself, and it must be said that it rose
nobly to the occasion. A committee
was appointed, drawn from both Houses
and headed by Senator J. Wing Sloop-

er (R.), of Indiana, and though after
considerable investigation the commit-
tee was forced reluctantly to conclude
that there was no evidence of Com-
munist instigation, the unconscious sub-
versiveness of the people's present con-
duct was obvious at a glance. The
problem was what to do about it. You
can't indict a whole nation, particu-
larly on such vague grounds as these
were. But, as Senator Slooper bold-
ly pointed out, "You can control it,"
and in the end a system of reeduca-
tion and reform was decided upon, de-
signed to lead people back to — again
we quote Senator Slooper — "the basic
regularities, the homely averageness of
the American way of life."

In the course of the committee's in-
vestigations, it had been discovered, to
everyone's dismay, that the Law of
Averages had never been incorporated
into the body of federal jurisprudence,
and though the upholders of States'
Rights rebelled violently, the oversight
was at once corrected, both by Consti-
tutional amendment and by a law — the
Hills-Slooper Act — implementing it.
According to the Act, people were re-
quired to be average, and, as the simplest
way of assuring it, they were divided
alphabetically and their permissible
activities catalogued accordingly. Thus,
by the plan, a person whose name began
with "G," "N," or "U," for example,
could attend the theatre only on Tues-
days, and he could go to baseball games
only on Thursdays, whereas his visits
tq, a haberdashery were confined to the
hours between ten o'clock and noon on
Mondays.

The law, of course, had its disadvan-
tages. It had a crippling effect on thea-
tre parties, among other social functions,
and the cost of enforcing it was un-
believably heavy. In the end, too, so
many amendments had to be added to
it — such as the one permitting gentle-
men to take their fiancees (if accredit-
ed) along with them to various events
and functions no matter what letter the
said fiancees' names began with — that
the courts were frequently at a loss to
interpret it when confronted with vio-
lations.

In its way, though, the law did serve
its purpose, for it did induce — rather
mechanically, it is true, but still ade-
quately— a return to that average ex-
istence that Senator Slooper desired. All,
indeed, would have been well if a year
or so later disquieting reports had not
begun to seep in from the backwoods.
It seemed that there, in what had hith-
erto been considered to be marginal
areas, a strange wave of prosperity was

making itself felt. Tennessee moun-
taineers were buying Packard converti-
bles, and Sears, Roebuck reported that
in the Ozarks their sales of luxury items
had gone up nine hundred per cent. In
the scrub sections of Vermont, men who
formerly had barely been able to scratch
a living from their rock-strewn acres
were now sending their daughters to
Europe and ordering expensive cigars
from New York. It appeared that the
Law of Diminishing Returns was going
haywire, too. — Robert M. Coates

126

How can a viewer distinguish whether a film is being
run forward or backward? The direction of increasing
disorder helps to fix the direction of the arrow of time.

9 The Arrow of Time

Jacob Bronowski

A chapter from his book Insight, 1964.

This chapter and those that follow deal with time.
In particular, this chapter looks at the direction of
time. Why does time go one way only? Why cannot
we turn time backwards? Why are we not able to
travel in time, back and forth?

The idea of time travel has fascinated men. Even
folklore contains legends about travel in time. And
science fiction, from The Time Machine onwards, has
been pre-occupied with this theme. Plainly, men feel
themselves to be imprisoned in the single direction
of time. They would like to move about in time as
freely as they can move in space.

And time is in some way like space. Like space,
time is not a thing but is a relation between things.
The essence of space is that it describes an order
among things — higher or lower, in front or behind,
to left or to right. The essence of time also is that it
describes an order — earlier or later. Yet we cannot
move things in time as we can in space. Time must
therefore describe some fundamental process in
nature which we do not control.

It is not easy to discuss time without bringing in
some way of measuring it — a clock of one sort or

another. Yet if all the clocks in the world stopped,
and if we all lost all inner sense of time, we could
still tell earlier from later. The essential nature of
time does not depend on clocks. That is the point of
this chapter, and we will begin by illustrating it
from very simple and common experiences.

The three pairs of pictures point the way. They
help to show what it is that enables us to tell earlier
from later without a clock. In each pair, the pictures
are arranged at random, and not necessarily in the
sequence of time. Yet in all except the first pair, it
is easy to arrange the pictures; the sequence in time
is obvious. Only the first pair does not betray its time
sequence. What is the difference between the first
pair of pictures and the other two pairs?

We get a clue to the difference when we study the
arrangement of the things in each picture. In the first
pair, we cannot really distinguish one arrangement
from another; they are equally tidy and orderly. The
two pictures of the first pair show a shot at billiards.
The billiard balls are as well arranged after the shot
as before; there is no obvious difference between
the arrangements.

The situation is different in the other two pairs.
A broken egg is an entirely different arrangement

127

from a whole egg. A snooker pyramid is quite
different from a jumble of balls.

And not only are the arrangements here different.
Once they are different, it is quite clear which
arrangement comes before the other. Whole eggs
come before broken ones. The snooker pyramid
comes before the spread of the balls.

In each case, the earlier arrangement is more
ordered than the later. Time produces disorder; that
is the lesson of these pictures. And it is also the
lesson of this chapter. The arrow of time is loss
of order.

In a game of snooker, we know quite well that the
highly ordered arrangement of the balls at the be-
ginning of the game comes before the disordered
arrangement at the end of the first shot. Indeed, the
first shot is called 'breaking the pyramid'; and
breaking is a destructive action — it destroys order.
It is just not conceivable that fifteen balls would
gather themselves up into a pyramid, however skilful
the player. The universe does not suddenly creaie
order out of disorder.

These pictures show the same thing agam. When
a spot of powdered dye is put on the surface of

water, it spreads out and gradually dissolves. Dye
would never come out of solution and stream to-
gether by itself to gather in a spot on the surface.
Again time is breaking down order and making dis-
order. It disperses the dye randomly through the
water.

We know at once that the stones in the picture be-
low were shaped and erected a very long time ago.
Their rough, weathered surfaces bear the mark of
time, it is still possible to reconstruct the once orderly
arrangement of the stones of Stonehenge. But the
once orderly surface of each stone cannot be re-
covered. Atom by atom, the smooth surface has
been carried away, and is lost to chaos.

And here finally is the most interesting of all the
pictures in which time betrays itself. In these shots
from an old film the heroine has been tied to the
rails — a splendid tradition of silent films. A train is
approaching, but of course it stops just in time. The
role of the heroine would seem to call for strong
nerves as well as dramatic ability, if she has to trust
the engine driver to stop the locomotive exactly
where he is told. However, the last few yards of the
approach are in fact done bv a trick. The locomotive

The Arrow of Time

is started close to the heroine and is backed away;
and the film is then run backwards.

There is only one thing that gives this trick away.
When the film is run backwards, the smoke visibly
goes into the funnel instead of coming out of it. We
know that in reality, smoke behaves like the spread-
ing dye: it becomes more disorderly, the further it
gets from the funnel. So when we see disorder coming
before order, we realise that something is wrong.
Smoke does not of itself collect together and stream
down a funnel.

One thing remains to clear up in these examples.
We began with an example in which earlier and later
were equally well ordered. The example was a shot
at billiards. The planets in their orbits would be
another example, in which there would be nothing
to say which arrangement comes first.

Then does time stand still in billiards and planetary
motion? No, time is still doing its work of disorder.
We may not see the effects at once, but they are
there. For billiard balls and planets gradually lose
material from their surface, just like the stones of
Stonehenge. Time destroys their orderly shape too.
A billiard ball is not quite the same after a shot
as before it. A planet is not quite the same in each
successive orbit. And the changes are in the direction

of disorder. Atoms are lost from ordered structures
and return to chaos. The direction of time is from
order to disorder.

That is one reason why perpetual motion machines
are impossible. Time cannot be brought to a stand-
still. We cannot freeze the arrangement of the atoms,
even in a tiny corner of the universe. And that is what
we should have to do to make a perpetual motion
machine. The machine would have to remain the
same, atom for atom, for all time. Time would have
to stand still for it.

For example, take the first of these three machines
from a famous book of Perpetual Motion Machines.
It is meant to be kept going by balls in each sector,
which roll from the centre to the rim and back again
as the wheel turns. Of course it does not work. There
is friction in the bearing of the wheel, and more
friction between the balls and the tracks they run on.
Every movement rubs off a few atoms. The bearings
wear, the balls lose their smooth roundness. Time
does not stand still.

The second machine is more complicated and
sillier. It is designed to work like a waterwheel with
little balls instead of water. At the bottom the balls
roll out of their compartments down the chute, and
on to a moving belt which is to lift them to the top

129

130

The Arrow of Time

again. That is how the machine is meant to keep
going. In fact, when we built it, it came to a stop
every few minutes.

The pendulum arrangement in the third picture also
comes from the book of Perpetual Motion Machines.
A bail runs backwards and forwards in the trough
on top to keep it going. There are also elastic strings
at each end for good measure. This machine at least
works for short bursts. But as a perpetual motion
machine, it has the same defects as the others.
Nothing can be done to get rid of friction; and
where there is friction, there must be wear.

This last point is usually put a little differently.
Every machine has friction. It has to be supplied
with energy to overcome the friction. And this
energy cannot be recovered. In fact, this energy is
lost in heat, and in wear — that is, in moving atoms
out of their order, and in losing them. That is an-
other way of putting the same reasoning, and shows
equally (in different language) why a perpetual
motion machine cannot work.

Before we put these fanciful monsters out of mind,
it is worth seeing how beautifully a fine machine can
be made. It cannot conquer the disorder of time, it
cannot get rid of friction, but it can keep them to a
minimum. So on page 132 are two splendid clocks
which make no pretence to do the impossible, yet
which go as far as it is possible to go by means of
exact and intelligent craftsmanship.

These clocks are not intended to be p>erpetual
motion machines. Each has an outside source of
energy to keep it going. In the clock at the top, it is
ordinary clockwork which tips the platform when-
ever the ball has completed a run. The clock below
is more tricky: it has no clockwork spring, and
instead is driven by temp>erature differences in the
air. But even if there was someone to wind one clock,
and suitable air conditions for the other, they could
not run for ever. They would wear out. That is, their
ordered structure would slowly become more dis-
ordered until they stopped. The clock with no spring
would run for several hundred years, but it could
not run for ever.

To summarise: the direction of time in the uni-
verse is marked by increasing disorder. Even without
clocks and without an inner sense of time, we could
tell later and earlier. Later" is characterised by the
greater disorder, by the growing randomness of the
universe.

We ought to be clear what these descriptive
phrases mean. Order is a very special arrangement;
and disorder means the loss of what makes it special.
When we say that the universe is becoming more
disordered, more random, we mean that the special
arrangements in this place or that are being evened
out. The peaks are made lower, the holes are filled

131

in. The extremes disappear, and all parts sink more
and more towards a level average. Disorder
and randomness are not wild states; they are simply
states which have no special arrangement, and in
which everything is therefore near the average.

Even in disorder, of course, things move and
deviate round their average. But they deviate by
chance, and chance then takes them back to the
average. It is only in exceptional cases that a devia-
tion becomes fixed, and perpetuates itself. These
exceptions are fascinating and important, and we
now turn to them.

The movement towards randomness, we repeat, is
not uniform. It is statistical, a general trend. And
(as we saw in Chapter 8) the units that make up a
general trend do not all flow in the same direction.
Here and there, in the midst of the flow towards
an average of chaos, there are places where the flow
is reversed for a time. The most remarkable of these
reversals is life. Life as it were is running against
time. Life is the very opposite of randomness.

How this can come about can be shown by an
analogy. The flow of time is like an endless shuflling
of a pack of cards. A typical hand dealt after long
shuffling will be random— say four diamonds, a
couple of spades, four clubs, and three hearts. This
is the sort of hand a bridge player expects to pick up
several times in an evening. Yet every now and then
a bridge player picks up a freak hand. For example,
from time to time a player picks up all thirteen
spades. And this does not mean that the pack was
not properly shuflled. A hand of thirteen spades can
arise by chance, and does; the odds against it are
high, but they are not astronomic. Life started with
a chance accident of this kind. The odds against it
were high, but they were not astronomic.

The special thing about life is that it is self-
perpetuating. The freak hand, instead of disappear-
ing in the next shufile, reproduces itself. Once the
thirteen spades of life are dealt, they keep their
order, and they impose it on the pack from then on.
This is what distinguishes life from other freaks,
other deviations from the average.

There are other happenings in the universe that
run against the flow of time for a while. The forma-
tion of a star from the interstellar dust is such a
happening. When a star is formed, the dust that
forms it becomes less random; its order is increased,
not decreased. But stars do not reproduce themselves.
Once the star is formed, the accident is over. The
flow towards disorder starts again. The deviation
begins to ebb back towards the average.

Life IS a deviation of a special kind; it is a self-
reproducing accident. Once its highly ordered
arrangement occurs, once the thirteen spades happen
to be dealt in one hand, it repeats itself. The order
was reached by chance, but it then survives because
it is able to perpetuate itself, and to impose itself on
other matter.

It is rare to find in dead matter behaviour of this
kind which illustrates the way in which life imposes
its order. An analogy of a kind, however, is found
in the growth of crystals. When a supercooled solu-
tion is ready to form crystals, it needs something to
start it ofl". Now we introduce the outside accident,
the freak hand at bridge. That is, we introduce a tiny
crystal that we have made, and we drop it in. At
once the crystal starts to grow and to impose its
own shape round it.

In this analogy, the first crystal is a seed, like the
seed of life. Without it, the supercooled solution
would remain dead, unchanged for hours or even
days. And like the seed of life, the first crystal im-
poses its order all round it. It reproduces it.self many
times over.

Nearly five hundred years ago, Leonardo da Vinci
described time as the destroyer of all things. So we
have seen it in this chapter. It is the nature of time
to destroy things, to turn order into disorder. This
indeed gives time its single direction its arrow.

But the arrow of time is only statistical. The
general trend is towards an average chaos; yet there
are deviations which move in the opposite direction.
Life is the most important deviation of this kind. It
is able to reproduce itself, and so to perpetuate the
order whieh began by accident. Life runs against the
disorder of time.

132

The biography of this great Scottish physicist, renowned
both for kinetic theory and for his mathematical formu-
lation of the laws of electricity and magnetism, is pre-
sented in two parts. The second half of this selection is
in Reader 4.

10 James Clerk Maxwell

James R. Newman

An article from the Scientific American, 1955.

AMES CLERK MAXWELL was the greatest theo-
_ retical physicist of the nineteenth century. His
discoveries opened a new epoch of science, and much of what
distinguishes our world from his is due to his work. Because
his ideas found perfect expression in mathematical symbol-
ism, and also because his most spectacular triumph — the
prophecy of the existence of electromagnetic waves — was
the fruit of theoretical rather than experimental researches, he
is often cited as the supreme example of a scientist who builds
his systems entirely with pencil and paper. This notion is
false. He was not, it is true, primarily an experimentalist. He
had not the magical touch of Faraday, of whom Helmholtz
once observed after a visit to his laboratory that "a few wires
and some old bits of wood and iron seem to serve him for the
greatest discoveries." Nonetheless he combined a profound

133

physical intuition with a formidable mathematical capacity to
produce results "partaking of both natures." On the one hand,
Maxwell never lost sight of the phenomena to be explained,
nor permitted himself, as he said, to be drawn aside from the
subject in pursuit of "analytical subtleties"; on the other hand,
the use of mathematical methods conferred freedom on his in-
quiries and enabled him to gain physical insights without com-
mitting himself to a physical theory. This blending of the
concrete and the abstract was the characteristic of almost all
his researches.

Maxwell was born at Edinburgh on November 13, 1831,
the same year Faraday announced his famous discovery of
electromagnetic induction. He was descended of the Clerks of
Penicuick in Midlothian, an old Scots family distinguished no
less for their individuality, "verging on eccentricity," than
for their talents. His forbears included eminent lawyers,
judges, politicians, mining speculators, merchants, poets, mu-
sicians, and also the author (John Clerk) of a thick book on
naval tactics, whose naval experience appears to have been
confined entirely to sailing mimic men of war on the fishponds
at Penicuick. The name Maxwell was assumed by James's
father, John Clerk, on inheriting the small estate of Middlebie
from his grandfather Sir George Clerk Maxwell.

At Glenlair, a two-day carriage ride from Edinburgh and
"very much in the wilds," in a house built by his father shortly
after he married. Maxwell passed his infancy and early boy-
hood. It was a happy time. He was an only son (a sister, bom
earlier, died in infancy) in a close-knit, comfortably-off fam-
ily. John Clerk Maxwell had been called to the Scottish bar
but took little interest in the grubby pursuits of an advocate.
Instead the laird managed his small estates, took part in county
affairs and gave loving attention to the education of his son.
He was a warm and rather simple man with a nice sense of
humor and a penchant for doing things with what he called
"judiciosity"; his main characteristic, according to Maxwell's

134

James Clerk Maxwell

James Clerk Maxutell.
(The Bettmann Archive)

135

biographer Lewis Campbell,* was a "persistent practical in-
terest in all useful purposes." Maxwell's mother, Frances Cay,
who came of a well-known Northumbrian family, is described
as having a "sanguine, active temperament."

Jamesie, as he was called, was a nearsighted, lively, affec-
tionate little boy, as persistently inquisitive as his father and
as fascinated by mechanical contrivances. To discover of any-
thing "how it doos" was his constant aim. "What's the go of
that?" he would ask, and if the answer did not satisfy him he
would add, "But what's the particular go of that?" His first
creation was a set of figures for a "wheel of life," a scientific
toy that produced the illusion of continuous movement; he
was fond of making things with his hands, and in later life
knew how to design models embodying the most complex mo-
tions and other physical processes.

When Maxwell was nine, his mother died of cancer, the
same disease that was to kill him forty years later. Her death
drew father and son even more closely together, and many in-
timate glimpses of Maxwell in his younger years emerge from
the candid and affectionate letters he wrote to his father from
the time he entered school until he graduated from Cambridge.

Maxwell was admitted to Edinburgh Academy as a day
student when he was ten years old. His early school experi-
ences were painful. The master, a dryish Scotsman whose
reputation derived from a book titled Account of the Irregular
Greek Verbs and from the fact that he was a good disciplin-
arian, expected his students to be orderly, well-grounded in
the usual subjects and unoriginal. Maxwell was deficient in
all these departments. He created something of a sensation
because of his clothes, which had been designed by his strong-

* The standard biography (London, 1882) is by Lewis Campbell and William
Garnett. Campbell wrote the first part, which portrays Maxwell's life; Garnett
the second part, dealing with Maxwell's contributions to science. A shorter
biography, especially valuable for the scientific exposition, is by the mathema-
tician R. T. Glazebrook {James Clerk Maxwell and Modern ['hysics. London,
1901). In this essay, material in quotation marks, otherwise unattributed, is
from Campbell and Garnett.

136

James Clerk Maxwell

^^ sisinisiintnGiKSrafHiianinifafafgpK^

(^^yVedx Sir.

giye ^Lscinnf sstht IjJl sajs mKs Amexir^Yi.
dJi cixesb rn v4icli. lie -tfild ws how ^hese

Illuminated letter was written by Maxwell to his father in 1843, when the
younger Maxwell was II. The letter refers to a lecture by the American
frontier artist, George Catlin. (Scientific American)

137

minded father and included such items as "hygienic" square-
toed shoes and a lace-frilled tunic. The boys nicknamed him
"Dafty" and mussed him up, but he was a stubborn child and
in time won the respect of his classmates even if he continued
to puzzle them. There was a gradual awakening of mathe-
matical interests. He wrote his father that he had made a
"tetra hedron, a dodeca hedron, and two more hedrons that I
don't know the wright names for," that he enjoyed playing
with the "boies," that he attended a performance of some
"Virginian minstrels," that he was composing Latin verse and
making a list of the Kings of Israel and Judah. Also, he sent
him the riddle of the simpleton who "wishing to swim was
nearly drowned. As soon as he got out he swore that he would
never touch water till he had learned to swim." In his four-
teenth year he won the Academy's mathematical medal and
wrote a paper on a mechanical method, using pins and thread,
of constructing perfect oval curves. Another prodigious little
boy, Rene Descartes, had anticipated Maxwell in this field, but
Maxwell's contributions were completely independent and
original. It was a wonderful day for father and son when they
heard "Jas's" paper on ovals read before the Royal Society of
Edinburgh by Professor James Forbes: "Met," Mr. Maxwell
wrote of the event in his diary, "with very great attention and
approbation generally."

After six years at the Academy, Maxwell entered the Uni-
versity of Edinburgh. He was sixteen, a restless, enigmatic,
brilliantly talented adolescent who wrote not very good but
strangely prophetic verse about the destiny of matter and
energy :

When earth and sun are frozen clods.
When all its energy degraded
Matter to aether shall have faded

His friend and biographer Campl)ell records that James was
completely neat in his person "though with a rooted oi)jection
to the vanities of starch and gloves," and that he had a "pious

138

James Clerk Maxwell

horror of destroying anything — even a scrap of writing pa-
per." He had a quaint humor, read voraciously and passed
much time in mathematical speculations and in chemical, mag-
netic and optical experiments. "When at table he often seemed
abstracted from what was going on, being absorbed in observ-
ing the effects of refracted light in the finger glasses, or in try-
ing some experiment with his eyes — seeing around a corner,
making invisible stereoscopes, and the like. Miss Cay [his aunt]
used to call his attention by crying, 'Jamesie, you're in a
prop!' [an abbreviation for mathematical proposition]." He
was by now a regular visitor at the meetings of the Edinburgh
Royal Society, and two of his papers, on "Rolling Curves"
and on the "Equilibrium of Elastic Solids," were published
in the Transactions. The papers were read before the Society
by others "for it was not thought proper for a boy in a round
jacket to mount the rostrum there." During vacations at Glen-
lair he was tremendously active and enjoyed reporting his
multifarious doings in long letters to friends. A typical com-
munication, when Maxwell was seventeen, tells Campbell of
building an "electro-magnetic machine," taking off an hour to
read Poisson's papers on electricity and magnetism ("as I am
pleased with him today" ) , swimming and engaging in "aquatic
experiments," making a centrifugal pump, reading Herodotus,
designing regular geometric figures, working on an electric
telegraph, recording thermometer and barometer readings,
embedding a beetle in wax to see if it was a good conductor of
electricity ("not at all cruel, because I slew him in boiling
water in which he never kicked"), taking the dogs out, picking
fruit, doing "violent exercise" and solving props. Many of his
letters exhibit his metaphysical leanings, especially an intense
interest in moral philosophy. This bent of his thought, while
showing no particular originality, reflects his social sympathy,
his Christian earnestness, the not uncommon nineteenth-century
mixture of rationalism and simple faith. It was a period when
men still shared the eighteenth-century belief that questions of
wisdom, happiness and virtue could be studied as one studies
optics and mechanics.

139

In 1850 Maxwell quit the University of Edinburgh for
Cambridge. After a term at Peterhouse College he migrated
to Trinity where the opportunity seemed better of obtaining
ultimately a mathematical fellowship. In his second year he
became a private pupil of William Hopkins, considered the
ablest mathematics coach of his time. It was Hopkins's job to
prepare his pupils for the stiff competitive examinations, the
mathematical tripos, in which the attainment of high place
insured academic preferment. Hopkins was not easily im-
pressed; the brightest students begged to join his group, and
the famous physicists George Stokes and William Thomson
(later Lord Kelvin) had been among his pupils. But from the
beginning he recognized the talents of the black-haired young
Scotsman, describing him as "the most extraordinary man I
have ever met," and adding that "it appears impossible for
[him] to think incorrectly on physical subjects." Maxwell
worked hard as an undergraduate, attending the lectures of
Stokes and others and faithfully doing what he called "old
Hop's props." He joined fully in social and intellectual ac-
tivities and was made one of the Apostles, a club limited to
twelve members, which for many years included the outstand-
ing young men at Cambridge. A contemporary describes him
as "the most genial and amusing of companions, the pro-
pounder of many a strange theory, the composer of many a
poetic jeu d'esprit.''^ Not the least strange of his theories re-
lated to finding an effective economy of work and sleep. He
would sleep from 5 in the afternoon to 9:30, read very hard
from 10 to 2, exercise by running along the corridors and up
and down stairs from 2 to 2:30 a.m. and sleep again from
2:30 to 7. The occupants of the rooms along his track were
not pleased, but Maxwell persisted in his bizarre experiments.
Less disturbing were his investigations of the process i)y which
a cat lands always on her feet. He demonstrated that a cat
could right herself even when dropped upside down on a table
or bed from about two inches. A complete record of these valu-
able researches is unfortunately not available.

A severe illness, referred to as a "sort of brain fever,"

140

James Clerk Maxwell

seized Maxwell in the summer of 1853. For weeks he was
totally disabled and he felt effects of his illness long after-
ward. Despite the abundance of details about his life, it is hard
to get to the man underneath. From his letters one gleans evi-
dence of deep inner struggles and anxieties, and the attack of
"brain fever" was undoubtedly an emotional crisis; but its
causes remain obscure. All that is known is that his illness
strengthened Maxwell's religious conviction — a deep, ear-
nest piety, leaning to Scottish Calvinism yet never completely
identified with any particular system or sect. "I have no nose
for heresy," he used to say.

In January, 1854, with a rug wrapped round his feet and
legs (as his father had advised) to mitigate the perishing cold
in the Cambridge Senate House where the elders met and
examinations were given, he took the tripos. His head was
warm enough. He finished second wrangler, behind the noted
mathematician, Edward Routh. (In another competitive or-
deal, for the "Smith's Prize," where the subjects were more
advanced. Maxwell and Routh tied for first.)

After getting his degree. Maxwell stayed on for a while at
Trinity, taking private pupils, reading Berkeley's Theory of
Vision, which he greatly admired, and Mill's Logic, which he
admired less: ("I take him slowly ... I do not think him the
last of his kind"), and doing experiments on the effects pro-
duced by mixing colors. His apparatus consisted of a top,
which he had designed himself, and colored paper discs that
could be slipped one over the other and arranged round the
top's axis so that any given portion of each color could be
exposed. When the top was spun rapidly, the sectors of the
different colors became indistinguishable and the whole ap-
peared of one uniform tint. He was able to show that suitable
combinations of three primary colors — red, green and blue
— produced "to a very near degree of approximation" almost
every color of the spectrum. In each case the required combi-
nation could be quantitatively determined by measuring the
sizes of the exposed sectors of the primary-color discs. Thus,
for example, 66.6 parts of red and 33.4 parts of green gave

141

the same chromatic effect as 29.1 parts of yellow and 24.1
parts of blue. In general, color composition could be expressed
by an equation of the form

xX = aA + bB + cC

— shorthand for the statement that x parts of X can be matched
by a parts of A, b parts of B and c parts of C. This symbolism
worked out very prettily, for "if the sign of one of the quanti-
ties, a, 6, or c was negative, it simply meant that that color had
to be combined with X to match the other two."* The problem
of color perception drew Maxwell's attention on and off for
several years, and enlarged his scientific reputation. The work
was one phase of his passionate interest in optics, a subject to
which he made many contributions ranging from papers on
geometrical optics to the invention of an ophthalmoscope and
studies in the "Art of Squinting," Hermann von Helmholtz was
of course the great leader in the field of color sensation, but
Maxwell's work was independent and of high merit and in
1860 won him the Rumford Medal of the Royal Society.

These investigations, however, for all their importance,
cannot be counted the most significant activity of the two post-
graduate years at Trinity. For during this same period he was
reading with intense absorption Faraday's Experimental Re-
searches, and the effect of this great record on his mind is
scarcely to be overestimated. He had, as he wrote his father,
been "working away at Electricity again, and [I] have been
working my way into the views of heavy German writers. It
takes a long time to reduce to order all the notions one gets
from these men, but I hope to see my way through the subject,
and arrive at something intelligible in the way of a theory."
Faraday's wonderful mechanical analogies suited Maxwell
perfectly; they were what he needed to stimulate his own con-
jectures. Like Faraday, he thought more easily in images than

• Glazebrook, op. cit., pp. 101-102. See also Maxwell's paper. "Experiments on
Colour, as perceived by the Eye, with remarks on Colour-Blindness." Transac-
tions of the Royal Society of Edinburgh, vol. XXI, part II; collected in The
Scientific Papers of James Clerk Maxwell, edited by W. D. Niven, Cambridge,
1890.

142

James Clerk Maxwell

Color wheel is depicted in Max-
welFs essay "Experiments in
Colour, as perceived by the Eye,
with remarks on Colour-Blind-
ness." The wheel is shown at
the top. The apparatus for rotat-
ing it is at the bottom.
(Scientific American)

abstractions: the models came first, the mathematics later. A
Cambridge contemporary said that in their student days,
whenever the subject admitted of it, Maxwell "had recourse
to diagrams, though the rest [of the class] might solve the
question more easily by a train of analysis." It was his aim,
he wrote, to take Faraday's ideas and to show how "the con-
nexion of the very different orders of phenomena which he
had discovered may be clearly placed before the mathematical

143

mind."* Before the year 1855 was out, Maxwell had pub-
lished his first major contribution to electrical science, the
beautiful paper "On Faraday's Lines of Force," to which I
shall return when considering his over-all achievements in the
field.

Trinity elected Maxwell to a fellowship in 1855, and he
began to lecture in hydrostatics and optics. But his father's
health, unsettled for some time, now deteriorated further, and
it was partly to avoid their being separated that he became a
candidate for the chair of natural philosophy at Marischal
College, Aberdeen. In 1856 his appointment was announced;
his father, however, had died a few days before, an irrepar-
able personal loss to Maxwell. They had been as close as
father and son could be. They confided in each other, under-
stood each other and were in certain admirable traits much
alike.

The four years at Aberdeen were years of preparation as
well as achievement. Management of his estate, the design of
a new "compendious" color machine, and the reading of
metaphysics drew on his time. The teaching load was rather
light, a circumstance not unduly distressing to Maxwell. He took
his duties seriously, prepared lectures and demonstration ex-
periments very carefully, but it cannot be said he was a great
teacher. At Cambridge, where he had picked students, his
lectures were well attended, but with classes that were, in his
own words, "not bright," he found it difficult to hit a suitable
pace. He was unable himself to heed the advice he once gave
a friend whose duty it was to preach to a country congregation:

* The following quotation from the preface to Maxwell's Treatise on Electricity
and Magnetism (Cambridge, 1873) gives Maxwell's views of Faraday in his own
words: "Before I began the study of electricity I resolved to read no mathe-
matics on the subject till I had first read through Faraday's Experimental Re-
searches in Electricity. I was aware that there was supposed to be a difference
between Faraday's way of conceiving phenomena and that of the mathematicians
so that neither he nor they were satisfied with each other's language. 1 had also
the conviction that this discrepancy did not arise from either party being wrong.
... As I proceeded with the study of Faraday. I perceived that his method of
conceiving the phenomena was also a mathematical one. though not exhibited
in the conventional form of mathematical symbols. I also found that these
methods were capable of being expressed in the ordinary mathematical forms,
and these compared with those of the professed mathematicians."

144

James Clerk Maxwell

"Why don't you give it to them thinner?"* Electrical studies
occupied him both during term and in vacation at Glenlair.
"I have proved," he wrote in a semijocular vein to his friend
C. J. Monro, "that if there be nine coefficients of magnetic
induction, perpetual motion will set in, and a small crystalline
sphere will inevitably destroy the universe by increasing all
velocities till the friction brings all nature into a state of
incandescence. . . ,"

Then suddenly the work on electricity was interrupted by a
task that engrossed him for almost two years. In competition
for the Adams prize of the University of Cambridge (named
in honor of the discoverer of Neptune), Maxwell prepared a
brilliant essay on the subject set by the electors: "The Struc-
ture of Saturn's Rings."

Beginning with Galileo, the leading astronomers had ob-
served and attempted to explain the nature of the several con-
centric dark and bright rings encircling the planet Saturn.
The great Huygens had studied the problem, as had the
Frenchman, Jean Dominique Cassini, Sir William Herschel
and his son John, Laplace, and the Harvard mathematician
and astronomer Benjamin Peirce. The main question at the
time Maxwell entered the competition concerned the stability
of the ring system: Were the rings solid? Were they fluid?
Did they consist of masses of matter "not mutually coherent"?
The problem was to demonstrate which type of structure ade-
quately explained the motion and permanence of the rings.

Maxwell's sixty-eight-page essay was a mixture of common
sense, subtle mathematical reasoning and profound insight
into the principles of mechanics.* There was no point, he said
at the outset, in imagining that the motion of the rings was the
result of forces unfamiliar to us. We must assume that gravi-
tation is the regulating principle and reason accordingly. The
hypothesis that the rings are solid and uniform he quickly
demonstrated to be untenable; indeed Laplace had already

* Occasionally he enjoyed mystifying his students, but at Aberdeen, where, he
wrote Campbell. "No jokes of any kind are understood," he did not permit him-
self such innocent enjoyments.

* A summary of the work was published in the Proceedings of the Royal Soci-
ety of Edinburgh, vol. IV; this summary and the essay "On the Stability of the
Motion of Saturn's Rings" appear in the Scientific Papers (op. cit.) .

145

shown that an arrangement oi this kind would be so precarious
that even a slight displacement of the center of the ring from
the center of the planet "would originate a motion which would
never be checked, and would inevitably precipitate the ring
upon the planet. . . ."

Suppose the rings were not uniform, but loaded or thick-
ened on the circumference — a hypothesis for which there ap-
peared to be observational evidence. A mechanically stable
system along these lines was theoretically possible; yet here
too, as Maxwell proved mathematically, the delicate adjust-
ment and distribution of mass required could not survive the
most minor perturbations. What of the fluid hypothesis? To be
sure, in this circumstance the rings would not collide with the
planet. On the other hand, by the principles of fluid motion it
can be proved that waves would be set up in the moving rings.
Using methods devised by the French mathematician Joseph
Fourier for studying heat conduction, by means of which
complex wave motions can be resolved into their simple har-
monic, sine-cosine elements. Maxwell succeeded in demon-
strating that the waves of one ring will force waves in another
and that, in due time, since the wave amplitudes will increase
indefinitely, the rings will break up into drops. Thus the con-
tinuous-fluid ring is no better a solution of the problem than the
solid one.

The third possibility remained, that the rings consist of
disconnected particles, either solid or liquid, but necessarily
independent. Drawing on the mathematical theory of rings.
Maxwell proved that such an arrangement is fairly stable and
its disintegration very slow; that the particles may be disposed
in a series of narrow rings or may move through each other
irregularly. He called this solution his "dusky ring, which is
something like the state of the air supposing the siege of
Sebastopol conducted from a forest of guns 100 miles one
way, and 30,000 miles from the other, and the shot never to
stop, but go spinning away around a circle, radius 170.000
miles. . . ."

Besides the mathematical demonstration. Maxwell devised
an elegantly ingenious model to exhibit the motions of the
satellites in a disturbed ring, "for the edification of sensible

146

James Clerk Maxwell

composed of particles. (Scientific Amencan)

147

image-worshippers." His essay — which Sir George Airy, the
Astronomer Royal, described as one of the most remarkable
applications of mathematics he had ever seen — won the prize
and established him as a leader among mathematical physicists.

In 1859 Maxwell read before the British Association his
paper "Illustrations of the Dynamical Theory of Gases."*
This marked his entry into a branch of physics that he en-
riched almost as much as he did the science of electricity. Two
circumstances excited his interest in the kinetic theory of gases.
The first was the research on Saturn, when he encountered the
mathematical problem of handling the irregular motions of
the particles in the rings — irregular but resulting nonetheless
in apparent regularity and uniformity — a problem analo-
gous to that of the behavior of the particles of gas. The second
was the publication by the German physicist Rudolf Clausius
of two famous memoirs: on the heat produced by molecular
motion and on the average length of the path a gas molecule
travels before colliding with a neighbor.

Maxwell's predecessors in this field — Daniel Bernoulli,
James Joule, Clausius, among others — had been successful
in explaining many of the properties of gases, such as pres-
sure, temperature, and density, on the hypothesis that a gas is
composed of swiftly moving particles. However, in order to
simplify the mathematical analysis of the behavior of enor-
mous aggregates of particles, it was thought necessary to make
an altogether implausible auxiliary assumption, namely, that
all the particles of a gas moved at the same speed. The gifted
British physicist J. J. Waterson alone rejected this assumption,
in a manuscript communicated to the Royal Society in 1845:
he argued cogently that various collisions among the molecules
must produce different velocities and that the gas temperature
is proportional to the square of the velocities of all the mole-
cules. But his manuscript lay forgotten for half a century in
the archives of the Society.

Maxwell, without knowledge of Waterson's work, arrived at
the same conclusions. He realized that further progress in the
science of gases was not to be cheaply won. If the subject was

• Philosophical Magazine, January and July. 1860; also Maxwell's Scientific
Papers, op. cit.

148

James Clerk Maxwell

to be developed on "strict mechanical principles" — and for
him this rigorous procedure was essential — it was necessary,
he said, not only to concede what was in any case obvious, that
the particles as a result of collisions have different speeds, but
to incorporate this fact into the mathematical formulation of
the laws of motion of the particles.

Now, to describe how two spheres behave on colliding is
hard enough; Maxwell analyzed this event, but only as a prel-
ude to the examination of an enormously more complex phe-
nomenon — the behavior of an "indefinite number of small,
hard and perfectly elastic spheres acting on one another only
during impact."* The reason for this mathematical investiga-
tion was clear. For as he pointed out, if the properties of this
assemblage are found to correspond to those of molecular
assemblages of gases, "an important physical analogy will be
established, which may lead to more accurate knowledge of
the properties of matter."

The mathematical methods were to hand but had hitherto
not been applied to the problem. Since the many molecules
cannot be treated individually, Maxwell introduced the statis-
tical method for dealing with the assemblage. This marked a
great forward step in the study of gases. A fundamental Max-
wellian innovation was to regard the molecules as falling into
groups, each group moving within a certain range of velocity.
The groups lose members and gain them, but group population
is apt to remain pretty steady. Of course the groups differ in
size; the largest, as Maxwell concluded, possesses the most
probable velocity, the smaller groups the less probable. In
other words, the velocities of the molecules in a gas can be
conceived as distributed in a pattern — the famous bell-shaped
frequency curve discovered by Gauss, which applies to so
many phenomena from observational errors and distribution
of shots on a target to groupings of men based on height and
weight, and the longevity of electric light bulbs. Thus while
the velocity of an individual molecule might elude description,
the velocity of a crowd of molecules would not. Because this

' "Illustrations of the Dynamical Theory of Gases, op. cit.

149

method afforded knowledge not only of the velocity of a body
of gas as a whole, but also of the groups of differing velocities
composing it. Maxwell was now able to derive a precise formula
for gas pressure. Curiously enough this expression did not
differ from that based on the assumption that the velocity of
all the molecules is the same, but at last the right conclusions
had been won by correct reasoning. Moreover the generality
and elegance of Maxwell's mathematical methods led to the
extension of their use into almost every branch of physics.

Maxwell went on, in this same paper, to consider another
factor that needed to be determined, namely, the average
number of collisions of each molecule per unit of time, and its
mean free path (i.e., how far it travels, on the average, be-
tween collisions) . These data were essential to accurate formu-
lations of the laws of gases. He reasoned that the most direct
method of computing the path depended upon the viscosity of
the gas. This is the internal friction that occurs when (in Max-
well's words) "different strata of gas slide upon one another
with different velocities and thus act upon one another with a
tangential force tending to prevent this sliding, and similar in
its results to the friction between two solid surfaces sliding
over each other in the same way." According to Maxwell's
hypothesis, the viscosity can be explained as a statistical con-
sequence of innumerable collisions between the molecules and
the resulting exchange of momentum. A very pretty illustra-
tion by the Scotch physicist Balfour Stewart helps to an under-
standing of what is involved. Imagine two trains running with
uniform speed in opposite directions on parallel tracks close
together. Suppose the passengers start to jump across from one
train to the other. Each passenger carries with him a momen-
tum opposite to that of the train onto which he jumps; the
result is that the velocity of both trains is slowed just as if
there were friction between them. A similar process, said
Maxwell, accounts for the apparent viscosity of gases.

Having explained this phenomenon, Maxwell was now able
to show its relationship to the mean free path of the molecules.
Imagine two layers of molecules sliding past each other. If a
molecule passing from one layer to the other travels only a

150

James Clerk Maxwell

short distance before colliding with another molecule, the two
particles do not exchange much momentum, because near the
boundary or interface the friction and difference of velocity
between the two layers is small. But if the molecule penetrates
deep into the other layer before a collision, the friction and
velocity differential will be greater; hence the exchange of
momentum between the colliding particles will be greater.
This amounts to saying that in any gas with high viscosity the
molecules must have a long mean free path.

Maxwell deduced further the paradoxical and fundamental
fact that the viscosity of gas is independent of its density. The
reason is that a particle entering a dense — i.e., highly crowded
— gas will not travel far before colliding with another par-
ticle; but penetration on the average will be deeper when the
gas entered is only thinly populated, because the chance of a
collision is smaller. On the other hand, there will be more
collisions in a dense than in a less dense gas. On balance, then,
the momentum conveyed across each unit area per second re-
mains the same regardless of density, and so the coefficient of
viscosity is not altered by varying the density.

These results, coupled with others arrived at in the same
paper, made it possible for Maxwell to picture a mechanical
model of phenomena and relationships hitherto imperfectly
understood. The various properties of a gas — diffusion, vis-
cosity, heat conduction — could now be explained in precise
quantitative terms. All are shown to be connected with the
motion of crowds of particles "carrying with them their mo-
menta and their energy," traveling certain distances, colliding,
changing their motion, resuming their travels, and so on. Alto-
gether it was a scientific achievement of the first rank. The
reasoning has since been criticized on the ground, for exam-
ple, that molecules do not possess the tiny-billiard-ball prop-
erties Maxwell ascribed to them; that they are neither hard,
nor perfectly elastic; that their interaction is not confined
to the actual moment of impact. Yet despite the inadequacies
of the model and the errors of reasoning, the results that, as
Sir James Jeans has said, "ought to have been hopelessly
wrong," turned out to be exactly right, and the formula tying

151

the relationships together is in use to this day, known as Max-
well's law.*

This is perhaps a suitable place to add a few lines about
Maxwell's later work in the theory of gases. Clausius, Max
Planck tells us, was not profoundly impressed by the law of
distribution of velocities, but the German physicist Ludwig
Boltzmann at once recognized its significance. He set to work
refining and generalizing Maxwell's proof and succeeded,
among other results, in showing that "not only does the Max-
well distribution [of velocities] remain stationary, once it is
attained, but that it is the only possible equilibrium state, since
any system will eventually attain it, whatever its initial state."*
This final equilibrium state, as both men realized, is the ther-
modynamic condition of maximum entropy — the most dis-
ordered state, in which the least amount of energy is available
for useful work. But since this condition is in the long run also
the most probable, purely from the mathematical standpoint,
one of the great links had been forged in modern science be-
tween the statistical law of averages and the kinetic theory of
matter.

The concept of entropy led Maxwell to one of the celebrated
images of modern science, namely, that of the sorting demon.
Statistical laws, such as the kinetic theory of gases, are good
enough in their way, and, at any rate, are the best man can
arrive at, considering his limited powers of observations and
understanding. Increasing entropy, in other words, is the ex-
planation we are driven to — and indeed our fate in physical
reality — because we are not very bright. But a demon more
favorably endowed could sort out the slow- and fast-moving
particles of a gas, thereby changing disorder into order and

* "Maxwell, by a train of argument which seems to bear no relation at all to
molecules, or to the dynamics of their movements, or to logic, or even to ordi-
nary common sense, reached a formula which, according to all precedents and
all the rules of scientific philosophy ought to have been hopelessly wrong. In
actual fact it was subsequently shown to be exactly right. . . ." (James Jeans.
"Clerk Maxwell's Method," in James Clerk Maxwell, A Commemoration Vol-
ume, 1831-1931, New York, 1931.)

* Max Planck, "Maxwell's Influence on Theoretical Physics in Germany," in
James Jeans, ibid.

152

James Clerk Maxwell

converting unavailable into available energy. Maxwell imag-
ined one of these small, sharp fellows "in charge of a friction-
less, sliding door in a wall separating two compartments of a
vessel filled with gas. When a fast-moving molecule moves
from left to right the demon opens the door, when a slow mov-
ing molecule approaches, he (or she) closes the door. The
fast-moving molecules accumulate in the right-hand compart-
ment, and slow ones in the left. The gas in the first compart-
ment grows hot and that in the second cold." Thus the demon
would thwart the second law of thermodynamics. Living or-
ganisms, it has been suggested, achieve an analogous success;
as Erwin Schrodinger has phrased it, they suck negative en-
tropy from the environment in the food they eat and the air
they breathe.

Maxwell and Boltzmann, working independently and in a
friendly rivalry, at first made notable progress in explaining
the behavior of gases by statistical mechanics. After a time,
however, formidable difficulties arose, which neither investi-
gator was able to overcome. For example, they were unable to
write accurate theoretical formulas for the specific heats of
certain gases (the quantity of heat required to impart a unit
increase in temperature to a unit mass of the gas at constant
pressure and volume).* The existing mathematical techniques

* In order to resolve discrepancies between theory and experiment, as to the
viscosity of gases and its relationship to absolute temperature. Maxwell sug-
gested a new model of gas behavior, in which the molecules are no longer con-
sidered as elastic spheres of definite radius but as more or less undefined bodies
repelling one another inversely as the fifth power of the distance between the
centers of gravity. By this trick he hoped to explain observed properties of
gases and to bypass mathematical obstacles connected with computing the veloc-
ity of a gas not in a steady state. For, whereas in the case of hard elastic bodies
molecular collisions are a discontinuous process (each molecule retaining its
velocity until the moment of impact) and the computation of the distribution
of velocities is essential in solving ([uestions of viscosity, if the molecular inter-
action is by repulsive force, acting very weakly when the molecules are far away
from each other and strongly when they approach closely, each- collision may be
conceived as a rapid but continuous transition from the initial to the final veloc-
ity, and the computation both of relative velocities of the colliding molecules
and of the velocity distribution of the gas a-- a whole can be dispensed with. In
his famous memoir On the Dynamical Theory of Gases, which appeared in 1866,
Maxwell gave a beautiful mathematical account of the properties of such a sys-
tem. The memoir inspired Boltzmann to a Wagnerian rapture. He compared
Maxwell's theory to a musical drama: "At first are developed majestically the

153

simply did not reach — and a profound transformation of
ideas had to take place before physics could rise to — a new
level of understanding. Quantum theory — the far-reaching
system of thought revolving about Planck's universal constant,
h — was needed to deal with the phenomena broached by
Maxwell and Boltzmann.* The behavior of microscopic par-
ticles eluded description by classical methods, classical con-
cepts of mass, energy and the like; a finer mesh of imagination
alone would serve in the small world of the atom. But neither
quantum theory, nor relativity, nor the other modes of thought
constituting the twentieth-century revolution in physics would
have been possible had it not been for the brilliant labors of
these natural philosophers in applying statistical methods to
the study of gases.

Variations of the Velocities, then from one side enter the Ecjuations of State,
from the other the Equations of Motion in a Central Field; ever hifiher swoops
the chaos of Formulae; suddenly are heard the four words: 'Put n = 5". The
evil spirit V (the relative velocity of two molecules) vanishes and the dominat-
ing figure in the bass is suddenly silent; that which had seemed insuperable
being overcome as if by a magic stroke . . . result after result is given by the
pliant formula till, as unexpected climax, comes the Heat E(iuilibriuni of a
heavy gas; the curtain then drops."

Unfortunately, however, the descent of the curtain did not, as Boltzmann had
supposed, mark a happy ending. For as James Jeans points out, "Maxwell's be-
lief that the viscosity of an actual gas varied directly as the absolute tem[)era-
ture proved to have been based on faulty arithmetic, and the conclusions he
drew from his belief were vitiated by faulty algebra." [Jeans, op. rit.'\ It was,
says Jeans, "a very human failing, which many of us will welccmie as a bond of
union between ourselves and a really great mathematician" — even though the
results were disastrous.

* Explanation of the discrepancies they found had to await the development of
quantum theory, which showed that the spin and vibration of molecules were
restricted to certain values.

154

A fine example of the reach of a scientific field, from re-
search lab to industrial plant to concert hall.

11 Frontiers of Physics Today: Acoustics

Leo L.Beranek

An excerpt from his book Mr Tompkins in Paperback, 1965.

An intellectually vital and
stimulating field, acoustics is rich in
unsolved and intriguing research prob-
lems. Its areas of interest are per-
tinent to the activities of many tra-
ditional university departments:
mathematics, physics, electrical engi-
neering, mechanical engineering, land
and naval architecture, behavioral
sciences and even biology, medicine
and music.

On opening a recent issue of the
Journal of the Acoustical Society of
America, a leading Boston neurosur-
geon exclaimed: "It's like Alice's Won-
derland. You find a parade of papers
on echoencephalograms, diagnostic
uses of ultrasound in obstetrics and
gynecology, acoustical effects of vio-
lin varnish, ultrasonic cleavage of cy-
clchexanol, vibration analysis by holo-
graphic interferometry, detection of
ocean acoustic signals, sounds of mi-
grating gray whales and nesting ori-
ental hornets, and sound absorption
in concert halls. Certainly no other
discipline could possibly be more varie-
gated."

Acoustics assumed its modem aspect
as a result of at least seven factors.
They are:

• a research program begun in 1914
at the Bell Telephone Laboratories
(on the recording, transmission, and
reproduction of sound and on hearing)
that flourished because of the triode
vacuum tube^

• the development of quantum
mechanics and field theory, which un-

derlay Philip M. Morse's classic text
of 19362

• large government funding of re-
search and development during and
since World War II, resulting in many
valuable acoustics texts^"22 and,
since 1950, a five- fold increase in the
number of papers published annually
in the Journal of the Acoustical So-
ciety of America

• a growing public demand in the
last decade for quieter air and surface
transportation

• the tremendous growth of acous-
tics study in other countries^^

• the reconstruction of European
dwellings, concert halls and opera
houses destroyed during World War
II, and the postwar construction of
new music centers in the US, UK,
Israel and Japan^^^s

• development of the solid-state
digital computer.2*'

Instruction in acoustics has moved
steadily across departmental bound-
aries in the universities, beginning in
physics prior to the time of radio and
electronics and moving into electrical
engineering as the communication and
imderwater-acoustics fields developed.
Then, more recently, it has reached
into mechanical engineering and fluid
mechanics as the nonlinear aspects of
wave propagation and noise genera-
tion in gases, Kquids and solids have
become of prime interest. Also, be-
cause much of acoustics involves the
human being as a source, receiver and
processor of signals that impinge on

155

his ears and body, the subject has
attained vital importance to depart-
ments of psychology and physiology.

In spite of its variety and its im-
portance to other sciences, acoustics
remains a part of physics. It involves
all material media; it requires the
mathematics of theoretical physics;
and, as a tool, it plays a primary role
in solving the mysteries of the solid,
liquid and gaseous states of matter.

Frederick V. Hunt of Harvard sug-
gests that the field of acoustics might
be separated into the categories of
sources, receivers, paths, tools and
special topics. These are the catego-
ries I will use here. Scientists and en-
gineers are active in all these groups,
and each group promises exciting
frontiers for those entering the field.

SOURCES OF SOUND

The sources that we must consider in-
clude speech, music, signals, and a
variety of generators of noise.

Speech

One of the most challenging goals of

FABIAN BACHRACH

Leo L. Beranek, chief scientist of Bolt
Beranek and Newman Inc of Cambridge,
Mass., was its first president for 16
years and continues as director. He
also has a continuous association with
Massachusetts Institute of Technology
dating from 1946. Beranek holds de-
grees from Cornell (Iowa) and Harvard
and was Director of the Electro-Acous-
tics Laboratory at Harvard during World
War II. He has served as vice-president
(1949-50) and president (1954-55) of
the Acoustical Society of America, and is
a member of the National Academy of
Engineering.

mMi!

11 5 .♦: .♦: •: .♦: • .♦ .♦; >:>:>:>:>:>:>;>:5||i
!<^<i! •! t. • t. t.-r.-r..r.-r.-?.*iS>S

MJjji^^ij^^Mm^^M^^^^

SS8SS.vv>>>>2

■llilii

\ojjjj.*j.*jj.».

»SSSSSSSSS>S.

FAST FOURIER TR.ANSFORM permits a spectral analysis of sounds in near-real time.
This tree graph is used as an algorithm to obtain factored matrices in the computation
of Fourier transforms. For further details see E. O. Brigham, R. E. Morrow, IEEE
Spectrum, Dec. 1967, page 63. —FIG. 1

156

Frontiers of Physics Today: Acoustics

speech research is speech synthesis by
rule ("talking computers"). At the
simplest level we could assemble all
the basic sounds (phonemes) of
speech by cutting them from a tape

CORNELL UNIVERSITY

ELECTRONIC MUSIC. Robert A.
Moog is here seen (left) at the keyboard
of one of his "synthesizers" that generate
and modify musical sounds. — FIG. 2

recording and calling them up for
reproduction, thus producing con-
nected speech according to a set of
instructions. But this procedure
works very poorly, because percep-
tually discrete phonemes, when com-
bined by a person talking to produce
syllables, have a modifying influence
on each other. Thus stringing to-
gether phonemes to produce good
speech would require a very large
inventory' of recorded units.

Workers at the Haskins Labora-
tories and at Bell Labs agree on these
fundamentals, but have taken some-
what different approaches. Bell Labs
uses the digital computer to assemble
natural utterances by appropriate
modification of pitch, stress and dura-
tion of words spoken in isolation.
One of the Bell Labs methods applies

the principles of predictive coding.
However, the basic problem remains:
How does one structure the com-
puter "brain" so that it will select,
modify and present a sequence of
sounds that can carry the desired
meaning in easily interpretable form?
The geographers of speech have
received new impetus with relatively
recent, easy access to large computer
power. A potent tool for this work
is the fast Fourier transform, which
allows spectral analyses of sounds
with high resolution in near- real time
(figure 1). Accompanying this proc-
ess are new methods for three-dimen-
sional display of speech spectra with
continuously adjustable resolution in
time and frequency. Thus deeper in-
sights into the structure of speech
signals and their spectra are slowly
becoming possible. The problem is to
select the meaningful parameters of
the primary information-bearing parts
of speech and to learn how they are
superimposed on, or modulate, the
secondary parameters that are associ-
ated with accent and individual style
and voice quality.

Music

Currently, the availability of rich
avant-garde sounds is stirring creative
activity in acoustics and music.
Solid-state devices are generally re-
sponsible for this incipient revolution,
partly because they permit complex
machines in small space (figure 2),
but also because of their lower price.

The initial period of bizarre, ex-
perimental, musical sounds is passing;
music critics speak more frequently of
beauty and intellectual challenge.
Soon a new version of musical form
and sound will evolve and, as de-
creasing costs widen the availability
of new instruments, recreational com-
posing may eventually occupy the
leisure time of many individuals.
Hopefully these new sounds and com-
positions will excite educated people
to an extent not observed since the
18th century.

The on-line computer will also play

]57

its part, permitting traditional com-
posers to perfect their compositions
with an entire orchestra at their finger-
tips.26

Composers in all eras have had some
specific hall-reverberation character-
istics in mind for each of their works.
Some modern composers now can see
the exciting possibility of the ex-
pansion of artificial reverberation to
permit reverberation times that change
for different parts of a composition,
and are different at low, medium and
high frequencies.

Perhaps the greatest progress will
be made by those trained from youth
in both the musical arts and physics,
so that the novel ideas of the two
disciplines can be combined to pro-
duce results inconceivable to the clas-
sical composer. Early stages of this
type of education are under way in
universities in the Boston, New York
and San Francisco areas.

Noise

Noise sources must be understood if
they are to be controlled, but the
sbjdy of them has often been ne-
glected in the past. Many challenges
appear in the understanding and con-
trol of high-level, nonlinear vibrations,
including nonlinear distortion, har-
monic and subharmonic generation,
radiation forces and acoustic wind.

Aerodynamic noise looms large on
the research frontier. For example,
the periodic aerodynamic loads asso-
ciated with noise from helicopter
blades are not well understood, par-
ticularly as the stall point of the
blades is approached. Multiple-rotor
helicopters, in which one set of blades
cuts through the vortices produced by
the other set, offer important possi-
bilities for theoretical investigation.
For example the helicopter rotor must
operate in unsteady air flow, but this
condition produces uneven loadings.

SSOCIATEO ARCHITECTS

CLOWES MEMORIAL HALL, Butler University, Indianapolis. The acoustics of this
hall (Johansen and Woollen, architects, and Bolt Beranek and Newman Inc, consul-
tants) are acknowledged to be among the best of contemporary halls. Research is
needed to explain why this hall is superior, to the ears of musicians and music critics, to
Philharmonic Hall in New York. The same general principles were used in the design
of the two halls, which opened at about the same time. —FIG. 3

158

random stresses on the blades and
magnified vortex production. The
fuselage of the helicopter also affects
the noise produced.

Surprisingly, noise production by
jet-engine exhausts is not yet well
understood, although large sums of
money have been spent on "cut-and-
try" muffling.

Perhaps least understood of all
mechanical sources of noise is the im-
pact of one body on another. For
example even the sound of a hand-
clap has never been studied. The
noise of engine blocks and industrial
machinery is largely produced by im-
pacts. The production of noise by
hammers, punches, cutters, weaving
shuttles, typewriter keys and wheels
rolling on irregular surfaces is also
largely unexplored.

RECEIVERS OF SOUND

The most important receivers of sound
are people— those who use sound to
aid them, as in listening and com-
munication, and those who are
bothered by the annoying or harmful
aspects of noise. Much engineering
effort is constantly expended to better
the acoustic environment of people
at home and at work. In some areas
the basic understanding of noise prob-
lems is well developed, and engineer-
ing solutions are widely available. In
others, such understanding is only be-
ginning to emerge and engineering
solutions are still uncertain.

Variety and complexity

The intellectually interesting questions
related to human beings as receivers
of sound derive in large part from the
extraordinary variety in the physical
stimuli and the complexity of human
responses to them. The questions in-
clude: What are the few most im-
portant physical descriptions (dimen-
sions) that will capture the essence of
each complex psychophysical situa-
tion? How can the variety of stimuli
be catalogued in a manageable way
so that they can be related to the hu-

Frontiers of Physics Today: Acoustics

man responses of interest?

Many of the sources of sound are
so complex (a symphony orchestra,
for example) that simplified methods
must be used to describe them and to
arrive at the responses of things or
people to them. The dangers in sim-
plified approaches, such as statistical
methods for handhng room or struc-
tural responses, are that one may
make wrong assumptions in arriving
at the physical stimulus-response de-
scription, and that the description may
not be related closely enough to the
psychophysical responses. The pro-
cess of threading one's way through
these dangers is a large part of being
on the research frontier. Good exam-
ples of the perils are found in archi-
tectural acoustics (figure 3).

Concert halls

In 1900, Wallace C. Sabine gave room
acoustics its classical equation. 2728
Sabine's statistically based equation
for predicting reverberation time (that
is, the time it takes for sound to decay
60 decibels) contains a single term di-
rectly proportional to the volume of a
room and inversely proportional to
the total absorbing power of the sur-
faces and contents. A controversy
exists today as to its relevance to
many types of enclosure. Research at
Bell Labs, aided by ray-tracing
studies on a digital computer,^'' shows
that the influence of room shape is of
major importance in determining
reverberation time, a fact not recog-
nized in the Sabine equation. A
two- or three-term equation appears
to be indicated, but until it is available
there are many subtleties that con-
front the engineer in the application

of published sound-absorption data

on acoustical materials and ob-

jects.23-30

Reverberation time is only one of

the factors contributing to acoustical

quality in concert halls. A hall with

cither a short or a long reverberation

time, may sound either dead or live.^"

Of greater importance, probably, is

the detailed "signature" of the hall

159

reverljeration that is impressed on the
music during the first 200 milliseconds
after the direct sound from the or-
chestra is heard. 2'

It would be easy to simulate the
reverberation signature of a hall by
earphones or a loudspeaker, were it
not that spatial factors are of primaiy
importance to the listener's perception.
Reflections that come from overhead
surfaces are perceived differently
from those that come from surfaces in
front, and from surfaces to the right,
left and behind the listener. A new
approach suggests that with a num-
ber of loudspeakers, separated in space
about a listener and excited by signals
in precise relative phases, one can pro-
duce the direct analog of listening in
an auditorium.

Frequency is a further dimension.
To be optimum, both the 60-dB re-
verberation time and the 200-msec
signature of the hall should probably
be different at low, middle and high
frequencies.

There are many other subjective at-
tributes to musical-acoustical quality
besides liveness (reverberation time).
They include richness of bass, loud-
ness, clarity, brilliance, diffusion, or-
chestral balance, tonal blend, echo,
background noise, distortion and other
related binaural-spatial effects. ^s Com-
puter simulations may lead to the sep-
aration of a number of the variables in-
volved, but analog experiments con-
ducted in model and full-scale halls
will most likely also be necessary to im-
prove our understanding of the relative
importance of the many factors. These
studies would be very costly and
would need Federal support. The
prospect of greater certainty in de-
sign of concert halls makes this an ex-
citing frontier for research.

Psychoacoustics

Traditional advances in psycho-
acoustics have resulted from investiga-
tion of the basic aspects of hearing:
thresholds of audibility (both tempo-
rary and permanent), niasking loud-
ness, binaural localization, speech in-

telligibility, detectability of signals in
noise, and the like.*' Just as in the
case of structures, humans exhibit a
multiplicity of responses to different
noise situations. Those on the fore-
front of research are attempting to
find simplified statistical descriptions
of the various physical stimuli that
correlate well with several subjective
responses, such as annoyance.

As an example, a recent means for
rating the subjective nuisance value of
noises"^^ says that the nuisance value
is greater as the average level of the
noise is increased and is greater the
less steady it is. In other words, the
nuisance is related to the standard
deviation of the instantaneous levels
from the average; the background
noise, if appreciable, is part of the
average level. But there is no treat-
ment of the "meaning" in the noise
(the drip of a faucet would not be
rated high, although it might be very
annoying), or of special characteristics
—such as a shrill or warbling tone, or
a raucous character. Although this
formulation is probably an improve-
ment over previous attempts to relate
annoyance to the level of certain types
of noise, the whole subject of a per-
son's reaction to unwanted sounds is
still wide open for research.

Another forefront area of psycho-
acoustics is the response of the tactile
senses to physical stimuli, both when
the body is shaken without sound and
when the body and the hearing sense
are stimulated together. We know
that discomfort in transportation is a
function of both noise and body vibra-
tion. How the senses interact, and
whether or how they mask each other,
is not known. Neither do we under-
stand the mechanism by which the
hearing process takes place in humans
beyond the point where the mechanical
motions of the inner ear are trans-
lated into nerve impulses. We also
do not know whether extended ex-
posure to loud noise or to sonic booms
has detrimental physiological or psy-
chological effects, other than damage
to the middle ear. We have not ade-

160

quately analyzed the nonlinear be-
havior of the ear and its effect on
enjoyment of music or understanding
of speech.

UNDERWATER AND AIRBORNE
PATHS

Several major problem areas exist in
underwater and airborne sound prop-
agation. One is prediction of acoustic
propagation between two points in the
ocean over distances up to several
hundred times the depth of the water.
Involved are many alternate paths of
propagation, spatial distributions of
pressure and temperature, spatial and
temporal fluctuation resulting from
waves, suspended particles, bubbles,
deep-water currents and so on. Math-
ematical phy.sics and the computer
have proven that strictly deterministic
thinking about sound propagation is
frequently fruitless. The need is to
characterize statistically the transmis-
sion between two points in both ampli-
tude and phase. The ultimate value of
this research is to distinguish informa-
tion-bearing signals from all other
sounds in which they are immersed. •''^
Similar needs exist in air. In short,
this area is an important element of the
acoustical frontier.

Structural paths

When sound or vibration excites a
structure, waves are propagated
throughout it and sound is radiated
to the surrounding medium. An
understanding of the physics of these
phenomena, adequate to quantitative
prediction of the efi^ect of changes in
the structural design on them, is re-
quired for many applications. The
response of buildings to sonic booms,
including the noise generated by ob-
jects in the building set in vibration by
the boom, is one example.^s Many
other examples arise in connection with
buildings and transportation vehicles,
including underground, ground, ma-
rine, air and space vehicles.

Structures and the noise and vibra-

Frontiers of Physics Today: Acoustics

tion fields in them are generally com-
plex beyond description. Almost in-
variably, the vibrational properties of
an existing structure cannot be deter-
mined in a way consistent with setting
up the dynamical equations of motion
and arriving at solutions to them on a
computer. Furthermore, the real in-
terest is in predicting response for a
structure that has not been built.
Again the problem is, in principle,
deterministic (solvable on a computer)
but one does not ever know the para-
meters to use. Progress is now result-
ing from the invention of a new lan-
guage, a statistical mathematical ap-
proach, for describing what goes on.^'*
But the dangers, as in room acoustics,
are that the answer may be incomplete.
It is necessary to go back repeatedly to
the laboratory experiment and try to
improve the language, the vocabulary
of statistical assumptions, that is used
to describe the physical situation. The
added dimension of damping, non-
homogeneity of structures, and radia-
tion into media of widely different
properties (air and water) make this
field rich in research topics.-"-'^"'

ACOUSTIC TOOLS

Satisfaction of the needs of tool seek-
ers is a lush field for the acoustical
inventor. Here is where the acoustic
delay line is perfected for radar systems
and process-conbol computers; where
sound is used to help clean metals and
fabrics; where vibration is used to pro-
cess paints, candy, and soups; and
where ultrasonics is used to test mate-
rials nondestructively. Transducers of
all types, seismic, underwater, vibra-
tion, microphones, loudspeakers, and so
forth are constantly being improved.
The medical profession seeks help
from ultrasonics as a means of detect-
ing objects or growths imbedded in the
body, or as a means for producing
warming of body tissue. The whole
field of spectrographic analysis of body
sounds as an aid to medical diagnosis
is largely unexplored. Sp>ecial tools
such as sonic anemometers and sonic

161

Sou

Light

■e- —

DEBYE-SEARS SCATTERING. A beam of light, passed through
a fluid at an angle to the direction of a sound wave, diminishes in
amplitude, and first-order diffracted waves appear. — FIG. 4

Light
scattered

Light incident ^

Sound
wavefronts

BRILLOUIN SCATTERING by a sound wave wide compared
with the wavelength of the light, generates two new frequencies
— that of the light plus and minus the acoustic frequency. — FIG. 5

temperature- and velocity-sensing de-
vices, are just becoming available.

SPECIAL TOOLS

The Physical Review Letters attest to
a renaissance of acoustics in physics
during the past decade. High-fre-
quency sound waves are being used
in gases, liquids and solids as tools
for analyzing the molecular, defect,
domain-wall and other types of motions
in these media. High-frequency sound
waves interact in various media with
electric fields and light waves at
frequencies for which their wave-
lengths in the media become about
alike (typically lO^ to IO12 Hz).
From these basic investigations, prac-
tical devices are emerging for signal
processing, storage and amplification,
for testing, measurement, inspection,
medical diagnosis, surgery and ther-
apy, and for ultrasonic cleaning, weld-
ing, soldering and homogenizing.'''^

Plasma acoustics

Plasma acoustics is concerned witli the
dynamics of a weakly ionized gas.^
The electrons in the gas (with a tem-
perature of 10^ to 105 K, typically)
will draw energy from the electric field
that maintains the plasma. Because of
the lower temperature of the neutral
gas (500 K, typically), much of this
energy is transferred to the neutral-gas
particles through elastic collisions. If
this transfer is made to vary with time,
for example, by a varying external
electric field, a sound wave is gener-
ated in the neutral gas. Alternatively,
the electric field may be held constant
and the electron density varied by an
externally applied sound wave. When
the frequency and other parameters are
in proper relation, a coupling of the
electron cncrg\- to the acoustic wave
may create a p<)siti\e feedback that re-
sults ill sound amplification.

Current research involves exami-
nation of the acoustic instabilities that
result from tliis amplification and in
the determination of the conditions for

162

spontaneous excitation of normal
modes of vibration, such as in a tube.
Because there is coupling between the
neutral gas and the electrons, the
sound-pressure field can be deter-
mined in terms of the electron-density
field. Thus an ordinary Langmuir
probe, arranged to measure fluctua-
tions in the electron density, can be
used as a microphone in the weakly
ionized gas. This technique has
proved useful in the determination of
the speed of sound and the tempera-
ture, of the neutral-gas component in a
plasma. It also appears to be a prom-
ising tool in the study of density
fluctuations in jet and supersonic wind-
tunnel flow.

In a fully ionized gas there exists a
type of sound wave, called "plasma
oscillation," in which there is charge
separation. The speed of propagation
of this ion-acoustic longitudinal wave
is determined by the inertia of the
ions and the "elasticity" of the elec-
tions. In the presence of a magnetic
field, the plasma becomes nonisotropic;
the wave motion then becomes consid-
erably more complicated, creating an
interesting area for research.

Optical acoustics

The density fluctuations caused by a
sound wave in a gas, liquid or solid,
produce corresponding fluctuations in
the index of refraction, and this leads
to scattering and refraction of light.
Conversely, under certain conditions,
sound can be generated by light.^

To illustrate Debye-Sears scattering
(figure 4), a beam of light is passed
through a fluid at an angle with re-
spect to the direction of travel of a
narrow-beam sound wave. The sound
wave acts somewhat like an optical
transmission grating, except for its fi-
nite width and time and motion depen-
dence. If the light penetrates at a
right angle to the direction of propaga-
tion of the sound wave, the incident
light beam diminishes in amplitude,
and first-order diffracted waves appear
at angles ±6, where sin 6 equals the

Frontiers of Physics Today: Acoustics

ratio of the wavelength of the light to
that of the sound.

When the width of the sound wave
is very large compared to the wave-
length of light, the wavefronts of the
sound in the medium form a succes-
sion of infinite partially-reflecting
planes traveling at the speed of sound,
and the scattered light occurs in only
one direction. At very high fre-
quencies (109 to IQio Hz) the primary
scattered wave is backward, and the
effects of thermal motion of the
medium on scattering are easily ob-
served. Because thermal sound waves
travel in all directions and have a wide
frequency spectrum, frequency-shifted
light beams are scattered from them
at all angles. This phenomenon is
called "Brillouin scattering" (figure 5).
There are two Brillouin "lines" in the
scattered light, equal in frequency to
that of the light plus and minus the
acoustic frequency. These lines are
broadened by an amount of the order
of the inverse of the "lifetimes" of the
ordinary and transverse propagating
sound waves.

A very active area of research is the
determination of the acoustic disper-
sion relation for "hypersound" (fre-
quencies above 10" Hz) in fluids, with
lasers as the light sources and high-
resolution spectroscopic techniques
(for example heterodyne spectros-
copy) for the frequency analysis. ^

Other frontiers

Other areas of research are reported
in Physical Review Letters, as al-
ready mentioned, and in the Journal
of the Acoustical Society of America
and elsewhere.

One such frontier involves the col-
lective modes of vibration in liquid
helium. In particular, the sound at-
tenuation has been measured at tem-
peratures very close to absolute zero
with incredible accuracy, with and
without porous materials present in
the hquid.^^

An interesting geophysical problem
is the generation of seismic waves by
sonic booms''' from supersonic aircraft

163

at high altitudes. When the seismic
waves travel at the same speed as the
phase velocity of the air wave, efficient
and effective coupling of energy from
the acoustic mode to the seismic mode
takes place. One application of this
coupling effect is as a tool to de-
termine surficial earth structure.

Holographic imaging has attracted
interest because it offers the possi-
bility, first, of three-dimensional image
presentation of objects in opaque
gases or liquids, and, second, of re-
cording and utilizing more of the in-
formation contained in coherent
sound-field configurations than do the
more conventional amplitude-detecting
systems. •^■'^ Holographic imaging has
been done in an elementary way at
both sonic and ultrasonic frequencies
and in air and water. Figure 6 shows
an example.

Much recent research in physical
acoustics is concerned with ultrasonic
absoi-ption in solids, particularly crys-
tals, explained in terms of attenuation
by thermal photons. i** An intrinsic
mechanism for the attenuation of
ultrasonic sound in solids is the inter-
action of the mechanical (coherent)
sound wave with thermal (incoherent)
phonons, where thermal phonons are
described as the quantized thermal
vibrations of the atoms in the crystal
lattice of the solid. Because the rela-
tion between the applied force and the
atomic displacements is nonlinear, a
net "one-way" transfer of energy from
the ultrasonic wave to the thermal
phonons results. At very high fre-
quencies and low temperatures, such
interactions must be considered in
terms of discrete events, namely,
acoustic phonons interacting witli
thermal phonons. '^••'^ Also in this
field, light scattering has proven to be
a useful diagnostic tool in the study
of sound and crystal properties. '*°

Just as we may have interaction be-
tween sound waves and electrons in a
gaseous plasma, sound and electrons
may interact in certain semiconductors.
In a .semiconductor the tension and
compressions of the acoustic wave

create an electric field that moves
along with the traveling wave. If an
intense steady electric field is apphed
to the semiconductor, the free elec-
trons will try to go somewhat faster
than the sound wave, and the sound
wave will increase in amplitude, pro-
vided the thermal losses in the crystal
are not too great. This interaction re-
quires extremely pure crystalline mate-
rial.-"

Attempts are underway to make
ultiasonic delay lines adjustable, by
drawing upon the interaction between
acoustic waves and magnetic "spin
waves." Fermi-surface studies for
many metals can also be carried out by
measuring attenuation in the presence
of magnetic fields.

One application for surface (Ray-
leigh) waves on a crystalline solid is
in signal processing. Surface waves
are accessible along their entire wave-
length and are compatible with inte-
grated-circuit technology. Perhaps
such waves at GHz frequencies can
be used to build mixers, filters, cou-
plers, amplifiers, frequency shifters,
time compressors and expanders, and
memory elements. ■*-

In the study of high-frequency sur-
face waves, laser light again proves to
be a useful diagnostic tool. With it,
the thermally excited surface waves in
liquids have been studied by tech-
niques quite similar to the Brillouin
scattering from phonons.^'' One appli-
cation is determination of surface ten-
sion through observation of the mean
frequency and bandwidth of such
waves.

Many more examples of modem
physical acoustics could be cited, but
these examples should prove my open-
ing statement that acoustics is a vital,
growing field.

/ wish to thank Frederick V. Hunt, Man-
fred R. Schroeder, K. Uno Ingard, Preston
W. Smith, }r, Theodore }. Schultz and
Richard H. Lyon for their helpful com-
ments during the preparation of this
paper.

164

Frontiers of Physics Today: Acoustics

ACOUSTICAL HOLOGRAPHY. Acoustical wavefronts reflected from irregular sur-
faces can be recorded and reconstructed with coherent laser light. An advantage over
conventional holography is that optically opaque gases and liquids can be penetrated.
The experiment shown is in progress at the McDonnell Douglas Corp. — FIG. 6

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Problem," /. Acoust. Soc. Am. 44,
1074 (1968).

38. F. L. Thurstone, "Holographic Imag-
ing with Ultrasound," /. Acoust. Soc.
Am. 45,895 (1969).

39. C. Elbaum, "Ultrasonic Attenuation
in Crystalline Sohds— Intrinsic and
Extrinsic Mechanisms," Ultrasonics
7, 113 (April 1969).

40. C. Krisoher, PhD thesis, physics de-
partment, Massachusetts Institute of
Technology (1969).

41. A. Smith, R. W. Damon, "Beyond
Ultrasonics," Science and Technology
no. 77, 41 (May 1968).

42. A. P. Van der Heuvel, "Surface-wave
Electronics," Science and Technology
no. 85, 52 (Jan. 1969).

43. R. H. Katyl, U. Ingard, Phys. Rev.
Lett. 20, 248 (1968). Q

166

The use of random elements is common today not only
in science, but also in music, art, and literature. One
influence was the success of kinetic theory in the
nineteenth century.

12 Randomness and The Twentieth Century

Alfred M. Bork

An article from The Antioch Review, 1967.

■ As I write this I have in front of me a book that may be un-
famihar to many. It is entitled One Million Random Digits with
1,000 Normal Deviates and was produced by the Rand Corporation
in 1955. As the title suggests, each page contains digits — numbers
from I to 9 — arranged as nearly as possible in a completely random
fashion. An electronic roulette wheel generated the numbers in this
book, and afterwards the numbers were made even more random by
shuffling and other methods. There is a careful mathematical defini-
tion of randomness, and associated with it are many tests that one
can apply. These numbers were shuflfled until they satisfied the tests.
I want to use this book as a beginning theme for this paper. The
production of such a book is entirely of the twentieth century. It
could not have been produced in any other era. I do not mean to
stress that the mechanism for doing it was not available, although
that is also true. What is of more interest is that before the twentieth-
century no one would even have thought of the possibility of pro-
ducing a book like this; no one would have seen any use for it.
A rational nineteenth-century man would have thought it the height
of folly to produce a book containing only random numbers. Yet
such a book is important, even though it is not on any of the usual
lists of one hundred great books.

167

That this book is strictly of the twentieth century is in itself of
importance. I claim that it indicates a cardinal feature of our cen-
tury: randomness, a feature permeating many different and appar-
ently unrelated aspects of our culture. I do not claim that randomness
is the only feature which characterizes and separates twentieth-
century thought from earlier thought, or even that it is dominant,
but I will argue, admittedly on a speculative basis, that it is an
important aspect of the twentieth century.

Before I leave the book referred to above, you may be curious
to know why a collection of random numbers is of any use. The
Rand Corporation, a government-financed organization, is not likely
to spend its money on pursuits having no possible application. The
principal use today of a table of random numbers is in a calcula-
tional method commonly used on large digital computers. Because
of its use of random numbers, it is called the Monte Carlo method,
and it was developed primarily by Fermi, von Neumann, and Ulam
at the end of the Second World War. The basic idea of the Monte
Carlo method is to replace an exact problem which cannot be solved
with a probabilistic one which can be approximated. Another area
where a table of random numbers is of importance is in designing
experiments, particularly those involving sampling. If one wants,
for example, to investigate certain properties of wheat grown in a
field, then one wants thoroughly random samplings of wheat; if all
the samples came from one corner of the field, the properties found
might be peculiar to that corner rather than to the whole field.
Random sampling is critical in a wide variety of situations.

Actually, few computer calculations today use a table of random
numbers; rather, a procedure suggested during the early days of
computer development by John von Neumann is usually followed.
Von Neumann's idea was to have the computer generate its own
random numbers. In a sense numbers generated in this way are not
"random," but they can be made to satisfy the same exacting tests
applied to the Rand Table; randomness is a matter of degree. It is
more generally convenient to let the computer produce random
numbers than to store in the computer memory a table such as the
Rand Table. Individual computer centers often have their own
methods for generating random numbers.

I shall not give any careful definition of randomness, but shall

168

Randomness and The Twentieth Century

rely on intuitive ideas of the term. A formal careful definition would
be at odds with our purposes, since, as A. O. Lovejoy noted in The
Great Chain of Being, it is the vagueness of the terms which allows
them to have a life of their own in a number of different areas. The
careful reader will notice the shifting meanings of the word "ran-
dom," and of related words, in our material.

However, it may be useful to note some of the different ideas
connected with randomness. D. M. Mackay, for example, distin-
guishes between "(a) the notion of well-shuffledness or impartiality
of distribution; (b) the notion of irrelevance or absence of correla-
tion; (c) the notion of 7 don't care'-, and (d) the notion of chaos"^
Although this is not a complete, mutually exclusive classificadon —
the editor of the volume in which it appears objects to it — the classi-
fication indicates the range of meaning that "random" has even
in well-structured areas like information theory.

Let us, then, review the evidence of randomness in several
areas of twentieth-century work, and then speculate on why this
concept has become so pervasive, as compared with the limited use
of randomness in the nineteenth century.

I begin with the evidence for randomness in twentieth-century
physics. There is no need to search far, for the concept helps to
separate our physics from the Newtonian physics of the last few
centuries. Several events early in this century made randomness
prominent in physics. The first was the explanadon of Brownian
motion. Brownian movement, the microscopically observed motion
of small suspended particles in a liquid, had been known since the
early iSoo's. A variety' of explanadons had been proposed, all un-
satisfactory. But Albert Einstein showed, in one of his three famous
papers of 1905, that Brownian motion could be understood in
terms of kinetic theory:

... it will be shown that according to the molecular-kinetic theory of
heat, bodies of microscopically visible size, suspended in a liquid, will
perform movements of such magnitude that they can be easily observed

^Donald M. Mackay, "Theoretical Models of Space Perception — Appendix,"
in "Aspects of the Theory of Artificial Intelligence," The Proceedings of the
First International Symposium of Biosimulation, edited by C. A. Muses
(Plenium Press, New York, 1962), p. 240.

169

in a microscope on account of the molecular motions of heat. It is pos-
sible that the movements to be discussed here are identical with the
so-called "Brownian molecular motion." ... if the movement discussed
here can actually be observed . . . then classical thermodynamics can no
longer be looked on as applicable with precision to bodies even of di-
mensions distinguishable in a microscope. . . . On the other hand
[if] the prediction of this movement proves to be incorrect, weighty
argument would be provided against the molecular-kinetic theory
of heaL^

It is the randomness of the process, often described as a "random
walk," which is the characteristic feature of Brownian motion.

But an even more direct experimental situation focused atten-
tion on randomness. During the last years of the nineteenth century,
physicists suddenly found many new and strange "rays" or "radia-
tions," including those from radioactive substances. A series of ex-
perimental studies on alpha-rays from radioactive elements led
Rutherford to say in 1912 that "The agreement between theory and
experiment is excellent and indicates that the alpha particles are
emitted at random and the variations accord with the laws of
probability."^ These radiations were associated with the core of the
atom, the nucleus, so randomness was present in the heart of matter.

One of the two principal physical theories developed in the
past forty years is the theory of atomic structure, quantum mechan-
ics, developed during the period from 1926 to 1930. Wave mechanics,
the form of quantum mechanics suggested by the Austrian physicist
Erwin Schrodinger, predicted in its original form only the allowable
energy levels and hence the spectroscopic lines for an atom of some
particular element. Later, Max Born and Werner Heisenberg gave
quantum theory a more extensive interpretation, today called the
"Copenhagen Interpretation," which relinquishes the possibility of
predicting exactly the outcome of an individual measurement of an
atomic (or molecular) system. Instead, statistical predictions tell
what, on the average, will happen if the same measurement is per-
formed on a large number of identically prepared systems. Identical

^Albert Einstein, Investigations on the Theory of Broumian Movement, edited
by R. Fiirth, translated by A. A, Cowpcr (E. P. Dutton, New York).
^E. Rutherford, Radioactive Substances and their Radiations (Cambridge Uni-
versity Press, Cambridge. 1913), p. 191.

170

Randomness and The Twentieth Century

measurements on identically prepared systems, in this view, do not
always give the same result. Statistical ideas had been used in the
nineteenth-century physics, but then it was always assumed that the
basic laws were completely deterministic. Statistical calculations
were made when one lacked complete information or because of
the complexity of the system involved. In the statistical interpre-
tation of quantum mechanics I have just described, however, ran-
domness is not accepted purely for calculational purposes. It is a
fundamental aspect of the basic physical laws themselves. Although
some physicists have resisted this randomness in atomic physics, it
is very commonly maintained. A famous principle in contemporary
quantum mechanics, the "uncertainty principle," is closely related
to this statistical view of the laws governing atomic systems.

These examples illustrate randomness in physics; now we pro-
ceed to other areas. Randomness in art is particularly easy to discuss
because it has been so consistently and tenaciously used. My first
example is from graphic design. For hundreds of years books and
other publications have been "justified" in the margins in order to
have flush right margins in addition to flush left margins. This is
done by hyphenation and by adding small spaces between letters
and words. But recently there is a tendency toward books that are
not "justified"; the right margins end just where they naturally
end, with no attempt to make them even. This is a conscious design
choice. Its effect in books with two columns of print is to randomize
partially the white space between columns of print, instead of
maintaining the usual constant width white strip.

In the fine arts, the random component of assemblages, such
as those of Jean Tinguely, often lies in the use of "junk" in their
composition. The automobile junkyard has proved to be a particu-
larly fruitful source of material, and there is something of a random
selection there. Random modes of organization, such as the scrap-
metal press, have also been used.

In art, as elsewhere, one can sometimes distinguish two kinds
of randomness, one involving the creative technique and another
exploiting the aesthetic effects of randomness. We see examples of
this second type, called "accident as a compositional principle" by
Rudolf Arnheim, in three woodcuts by Jean Arp, entitled "Placed
According to the Laws of Chance." We would perhaps not have

171

understood the artist's intent if we did not have the titles. Arp,
Hke other contemporary artists, has returned repeatedly to the ex-
ploration of such random arrangements. As James Thrall Soby
says, "There can be no doubt that the occasional miracles of accident
have particular meaning for him. . . . One assumes that he considers
spontaneity a primary asset of art.'"*

An area which has been particularly responsive to the explora-
tion of randomness for aesthetic purposes is "op art." Again the titles
often identify this concept, as in "Random Field" by Wen- Yin Tsai.

Perhaps more common, however, is the former aspect, an
artistic technique by which the artist intentionally employs some
random element. The contemporary school of action painting is
an example. Jackson Pollock often would place his canvas on the
ground and walk above it allowing the paint to fall several feet
from his brush to the canvas. Soby describes it as follows: "Pol-
lock's detractors call his current painting the 'drip' or 'spatter'
school, and it is true that he often spreads large canvases on the floor
and at them flings or dribbles raw pigments of various colors."^ With
this method he did not have complete control of just where an
individual bit of paint fell — this depended in a complicated way on
the position of the brush, the velocity of the brush, and the con-
sistency of the paint. Thus this technique had explicit chance ele-
ments, and its results have been compared to Brownian motion.

Similarly, J. R, Rierce, in Symbols, Signals, and Noise, dis-
cussing random elements in art, gives some examples of computer-
generated art. He emphasizes the interplay of "both randomness
and order" in art, using the kaliedoscope as an example.

I will comment even more briefly on music. In Percy Granger's
"Random Round" each instrument has a given theme to play;
the entrances are in sequence, but each player decides for him-
self just when he will enter. Thus each performance is a unique
event, involving random choices. The most famous example of
random musical composition is the work of John Cage. One of
his best known works involves a group of radios on a stage, each

■*Iames Thrall Soby, Arp (Museum of Modern Art, New York, 1958).
•''James Thrall Soby, "Jackson Pollock," in The New Art in America (Fred-
erick Praeger, Inc., Greenwich, Conn., 1957).

172

Randomness and The Twentieth Century

with a person manipulating the controls. They work independently,
each altering things as he wishes, and the particular performance is
further heavily dependent on what programs happen to be playing
on the local radio stations at the time of the performance. There is
no question that Cage furnishes the most extreme example of ex-
ploitation of techniques with a chance component.

Most evidence for randomness in literature is not as clear as
in science, art, or music. The first example is clear, but perhaps
some will not want to call it literature at all. In 1965 two senior
students at Reed College saw some examples of computer-produced
poetry and decided that they could do as well. As their model was
symbolist poetry, they did not attempt rhyme or meter, although
their program might be extended to cover either or both. The com-
puter program is so organized that the resulting poem is based on
a series of random choices. First, the computer chooses randomly
a category — possibilities are such themes as "sea" or "rocks." The
program then selects (again using a built-in random number gen-
erator) a sentence structure from among twenty possibilities. The
sentence structure contains a series of parts of speech. The com-
puter randomly puts words into it, keeping within the previously
chosen vocabulary stored in the computer memory. Because of the
limited memory capacity of the small computer available, only
five words occur in a given thematic and grammatical category.
There are occasionally some interesting products.

Turning from a student effort to a recendy available commercial
product, consider the novel Composition I by Marc Saporta, which
comes in a box containing a large number of separate sheets. Each
page concludes with the end of a paragraph. The reader is told to
shuffle the pages before beginning to read. Almost no two readers
will see the pages in the same order, and the ordering is deter-
mined in a random process. For some readers the girl is seduced
before she is married, for other readers after she is married. A
similar process has been used by William Burroughs in The Naked
Lunch and elsewhere, except that in this case the shuffling is done
by the writer himself. Burroughs writes on many separate pieces
of paper and then orders them over and over in different ways
until he is satisfied with the arrangement. He has suggested that
his work can be read in other orders, and ends The Nal{ed Lunch
with iin "Atrophied Preface."

173

p. Mayersburg* has pointed out elements of chance construction
in several other writers' work. He says of Michel Botor: ''Mobile is
constructed around coincidence: coincidence of names, places, signs,
and sounds. . . . Coincidence implies the destruction of traditional
chronology. It replaces a pattern of cause and effect with one of
chance and accident." He sees another chance aspect in these writers:
they recognize that they cannot completely control the mind of
the reader.

But can we find examples in the work of more important
writers? The evidence is less direct. While contemporary artists
have openly mentioned their use of randomness, contemporary
writers and critics, with a few exceptions, have seldom been willing
to admit publicly that randomness plays any role in their writings.
But I will argue that randomness is nevertheless often there, al-
though I am aware of the difl&culty of establishing it firmly.

The cubist poets, perhaps because of their associations with
artists, did experiment consciously with randomness. The story is
told of how ApoUinaire removed all the punctuation from the proofs
of Alcools because of typesetting errors, and he continued to use
random organization in his "conversation poems" and in other work.

The "opposite of narration" defines the very quality ApoUinaire finally
grasped in following cubism into the experimental work of Delaunay, the
quality he named simultanism. It represents an effort to retain a moment
of experience without sacrificing its logically unrelated variety. In poetry
it also means an effort to neutralize the passage of time involved in the act
of reading. The fragments of a poem are deliberately kept in a random
order to be reassembled in a single instant of consciousness.'

It can be argued that James Joyce used random elements in
Ulysses and Finnegans Wa/^e. Several minor stories at least indicate
that Joyce was not unfriendly toward the use of random input. For
example, when Joyce was dictating to Samuel Beckett, there was a
knock at the door. Joyce said, "Come in," and Beckett wrote down,
"Come in," thinking that it was part of the book. He inmiediatcly

'P. Mayersberg, "The Writer as Spaceman," The Listener, October 17, 1963,
p. 607.

'Roger Shattuck, The Banquet Years (Harcourt, Brace, and Co., New York),
p. 238.

174

Randomness and The Twentieth Century

realized that Joyce had not intended to dictate it; but when he
started to erase it, Joyce insisted that it should stay. And it is
still there in Finnegans Wake, because of a chance occurrence. A
related comment is made by Budgin in James Joyce and the Maying
of Ulysses: ". . . he was a great believer in his luck. What he needed
would come to him."

Proceeding from such stories to Joyce's books, I believe that
there are random elements in the vocabulary itself. It is well known
that much of the vocabulary of Finnegans Wake differs from the
vocabulary of other English-language books. Some of the words are
combinations of other better-known English words, and others are
traceable to exotic sources. I do not think that Joyce constructed
every new word carefully, but rather that he consciously explored
randomly or partially randomly formed words. There is some
slight tradition for this procedure in such works as "Jabberwocky."

Another aspect of Joyce's writing, shared with other works of
contemporary literature, also has some connection with our theme,
although this connection is not generally realized. I refer to the
"stream of consciousness" organization. The Victorian novel was
ordered in a linear time sequence; there were occasional flashbacks,
but mostly the ordering of events in the novel was chronological.
The stream of consciousness novel does not follow such an order,
but instead the events are ordered as they might be in the mind of
an individual. This psychological ordering has distinctly random
elements. Finnegans Wake has been interpreted as one night in the
mental life of an individual. I would not claim that our conscious
processes are completely random, but I think it is not impossible to
see some random elements in them

We mentioned that it has not been customary to admit that
randomness is a factor in contemporary literature. Much of the
critical literature concerning Joyce exempHfies this. But at least one
study sees Joyce as using random components: R. M. Adams' Surface
and Symbol — the Consistency of James Joyce's Ulysses.^ Adams
relates the story of the "come in" in Finnegans Wake, and he tells
of Joyce's requesting "any God dam drivel you may remember" of

*R. M. Adams, Surface and Symbol — The Consistency of James Joyce's Ulysses
(Oxford University Press, New York, 1952).

175

his aunt. Adams points out that artists and musicians of the period
were also using chance components: "Bits of rope, match or news-
paper began to be attached to paintings, holes were cut in their
surfaces, toilet bowls and spark plugs appeared unadorned on ped-
estals as works of original sculpture. . . ." Adams calls Ulysses a
collage, and in his conclusion he cautions against trying to define
the symbolism of every tiny detail in Ulysses-. "The novel is, in part
at least, a gambler's act of throwing his whole personality — his
accidents, his skills, his weaknesses, his luck — against the world."

My final example of randomness is lighter. I am reliably in-
formed that several years ago a group of students at Harvard formed
a random number society for propagating interest in random num-
bers. Among other activities they chose each week a random
number of the week, and persuaded a local radio station to an-
nounce it!

Although the reader may not accept my thesis, I continue with
the assumption that our culture differs from the culture of the
previous few centuries partly because of an increased concern with
and conscious use of elements which are random in some sense of
the word. We have seen this use in seemingly unrelated areas, and
in ways previously very uncommon. Now we will enter on an even
more difficult problem: assuming that the twentieth century con-
sciously seeks out randomness, can we find any historical reasons
for its permeating different fields?

1 need hardly remind you of the difiiculty of this problem. The-
orizing in history has generally seemed unreasonable, except to the
theorist himself and to a small group of devoted followers. The
present problem is not general history but the even more difficult
area of intellectual history. Despite vigorous attempts to understand
cultural evolution, or particular aspects of it such as the development
of scientific knowledge, I believe it is fair to say that we know far
less than we would like to know about how ideas develop. It would,
therefore, be unreasonable for me to expect to give a rich theory of
how humans modify ideas. Instead I shall grope toward a small
piece of such a theory, basing my attempt on the evidence presented
on randomness as a twentieth-century theme.

The rough idea I shall bring to your attention might be crudely

176

Randomness and The Twentieth Century

called the "splash in the puddle" theory. If a stone is dropped in a
pond, waves travel out from the disturbance in all directions; a
big splash may rock small boats a good bit away from the initial
point of impact. Without claiming that this "mechanism" is com-
plete, I shall argue that culural evolution bears some analogy to the
splash in the puddle. Even though the nineteenth century rejected
the theme of fundamental randomness, cultural events then created
new waves of interest in randomness, which eventually, through
the traveling of the wave, affected areas at a distance from the
source. Probably one source is not enough; often one needs rein-
forcement from several disturbances to create a revolution. And the
sources themselves must be powerful if the ejffects are to be felt
at great distances in the cultural plane.

I shall note two nineteenth-century events which were power-
ful sources, and so may have contributed to a new interest in
randomness. Both are from science, but this may reflect my own
specialization in history of science; I am likely to find examples
from the area I know best. My two examples are of unequal
weight. The minor one certainly affected profoundly the physicist's
attitude toward randomness, but how widespread its effect was is not
clear. The second example, however, was the major intellectual
event of the century.

The first example is the development of kinetic theory and
statistical thermodynamics in the last half of the century, involving
Rudolf Clausius, James Clerk Maxwell, Ludwig Boltzmann, Wil-
lard Gibbs, and others. Because physicists believed that Newtonian
mechanics was the fundamental theory, they thought that all other
theories should "reduce" to it, in the same sense that all terms could
be defined using only the terms of mechanics, and that the funda-
mental principles of other areas could be deduced logically from the
principles of mechanics. This attitude, applied to thermodynamics,
led to kinetic theory and statistical thermodynamics.

In kinetic theory a gas (a word which may originally have
meant "chaos"^) was viewed as a very large number of separate
particles, each obeying the Newtonian laws of motion, exerting

^Pointed out to me by Steven Brush. See J. R. Partington, "Joan Baptist von
Helmont," Annals of Science, I, 359-384 (^936)-

177

forces on each other and on the walls of the container. To know
the positions and velocities of all the particles was impossible because
of the multitude of particles; ordinary quantities of gas contained
10"^*— one followed by twenty-four zeros — particles. This lack of
complete information made it necessary to use general properties
such as energy conservation in connection with probability con-
siderations. One could not predict where each particle would be,
but one could predict average behavior and relate this behavior to
observed thermodynamical quantities. Thus statistical thermody-
namics introduced statistical modes of thought to the physicist; but
the underlying laws were still considered to be deterministic.

A fundamental quantity in thermodynamics, entropy, was
found to have a simple statistical interpretation: it was the measure
of the degree of randomness in a collection of particles. Entropy
could be used as the basis of the most elegant formulation of the
second law of thermodynamics: in a closed system the entropy
always increases, or the degree of randomness tends to increase.

A special series of technical problems developed over the two
kinds of averaging used in statistical considerations: time-averaging,
inherently involved in all measurements; and averaging over many
different systems, the ensemble averaging of Gibbs used in the cal-
tulations. The "ergodic theorems" that were extensively developed
to show that these two averages were the same again forced careful
and repeated attention on probabilistic considerations.

My second example is the theory of evolution, almost universally
acknowledged as the major intellectual event of the last century.
Charles Darwin and Alfred Russell Wallace developed the theory
Independently, using clues from Malthus' essay on population. The
basic ideas are well known. Organisms vary, organisms having the
fittest variations survive, and these successful variations are passed
on to the progeny. The random element of evolution is in the "nu-
merous successive, slight favorable variations"; the offspring differ
slightly from the parents. Darwin, lacking an acceptable theory of
heredity, had little conception of how these variations come about;
he tended to believe, parallel to the developers of statistical thermo-
dynamics, that there were exact laws, but that they were unknown.

I have hitherto sometimes spoken as if the variations . . . had been due
to chance. This, of course, is a wholly incorrect expression, but it seems

178

Randomness and The Twentieth Century

to acknowledge plainly our ignorance of the cause of each particular
variation. ^'^

But Others were particularly disturbed by the chance factors ap-
parently at work in variations. This was one of the factors that led
Samuel Butler from his initial praise to a later critical view of
Darwin. Sir John Herschel was very emphatic:

We can no more accept the principle of arbitrary and casual variation
and natural selection as a sufiBcicnt account, per se, of the past and present
organic world, than we can receive the Laputian method of composing
books ... as a sufficient one of Shakespeare and the Principia}^

When a usable theory of heredity was developed during the next
half century, randomness played a major role, both in the occur-
rence of mutations in genes and in the genetic inheritence of the
offspring. So, almost in spite of Darwin, chance became increasingly
important in evolutionary theory. "... The law that makes and loses
fortunes at Monte Carlo is the same as that of Evolution."*"

The theory of evolution roused almost every thinking man in
the late nineteenth century. Frederick Pollock, writing about the
important British mathematician William Kingdon Clifford, says:

For two or three years the knot of Cambridge friends of whom Clifford
was a leading spirit were carried away by a wave of Darwinian en-
thusiasm: we seemed to ride triumphant on an ocean of new life and
boundless possibilities. Natural selection was to be the master-key of the
universe; we expected it to solve all riddles and reconcile all contra-
dictions.^*

This is only one account outside biology, but it illustrates how evo-
lution affected even those not directly concerned with it as a scientific
theory. It does not seem unreasonable, then, that at the same time
evolution contributed to the new attitude toward randomness. I

'°C. Darwin, Origin of the Species (first edition), p. 114.
"Sir Herschel, Physical Geography of the Globe (Edinburgh, 1861), quoted
in John C. Green, The Death of Adam (New American Library, New York),
p. 296.

i^M. Hopkins, Chance and Error— The Theory of Evolution (Kegan Paul,
Trench, Truber & Co., London, 1923).

^*W. K. Clifford, Lectures and Essays (Macmillan, London, 1886), Intro-
duction.

179

might also mention two other books that are particularly interesting
in showing the influence of evolution outside the sciences, offering
details we cannot reproduce here. One is Leo J. Henkin's Darwinism
in the English Novel i86o-igio; the other is Alvar EUegSrd's Dar-
win and the ^General Reader.

There were of course other things happening in the nineteenth
century, but these two developments were important and had far-
reaching implications outside of their immediate areas. Alfred
North Whitehead, in Science and the Modern Worlds claims that in
the nineteenth century "four great novel ideas were introduced into
theoretical science." Two of these ideas were energy, whose rise
in importance was related to thermodynamics, and evolution. It was
consistent with established tradition, however, to believe that the
use of chance in these areas was not essential. Other non-scientific
factors were also important; for example. Lord Kelvin's attitude
toward chance was colored by religious considerations. In S. P.
Thomson's hije we find a speech of his in the Times of 1903 arguing
that "There is nothing between absolute scientific belief in Creative
Power and the acceptance of the theory of a fortuitous concourse of
atoms."

According to our splash in the puddle theory, we should be able
to point out evidence that two nineteenth-century developments,
statistical mechanics and evolution, had very far-reaching effects in
areas quite different from their points of origin, effects reflecting
interest in randomness. This is a big task, but we will attempt to
give some minimal evidence by looking at the writings of two
important American intellectuals near the turn of the century, both
of whom were consciously influenced by statistical mechanics and
Darwinian evolution. The two are Henry Adams and Charles
Sanders Peirce.

We have Adams' account of his development in The Education
of Henry Adams. Even a casual glance shows how much of the
language of physics and biology occurs in the book, and how often
references are made to those areas. Chapter 15 is entitled "Dar-
winism," and early in the chapter he says:

The atomic theory; the correlation and conservation of energy; the
mechanical theory of the universe; the kinetic theory of gases; and
Darwin's law of natural selection were examples of what a young man
had to take on trust.

180

Randomness and The Twentieth Century

Adams had to accept these because he was not in a position to argue
against them. Somewhat later in the book Adams comments, in his
usual third person:

He was led to think that the final synthesis of science and its ultimate
triumph was the kinetic theory of gases. ... so far as he understood it,
the theory asserted that any portion of space is occupied by molecules of
gas, flying in right lines at velocities varying up to a mile a second, and
colliding with each other at intervals varying up to seventeen million
seven hundred and fifty thousand times a second. To this analysis — if
one understood it right — all matter whatever was reducible and the only
difference of opinion in science regarded the doubt whether a still deeper
analysis would reduce the atom of gas to pure motion.

And a few pages later, commenting on Karl Pearson's "Grammar
of Science":

The kinetic theory of gases is an assertion of ultimate chaos. In plain,
chaos was the law of nature; order was the dream of man.

Later, "Chaos was a primary fact even in Paris," this in reference
to Henri Poincare's position that all knowledge involves conven-
tional elements.

Of all Henry Adams' writings, "A Letter to American Teachers
of History" is most consistently saturated with thermodynamical
ideas. This 1910 paper^* begins with thermodynamics. It first men-
tions the mechanical theory of the universe, and then says:

Toward the middle of the Nineteenth Century — that is, about 1850 — a
new school of physicists appeared in Europe . . . made famous by the
names of William Thomson, Lord Kelvin, in England, and of Clausius
and Helmhokz in Germany, who announced a second law of thermo-
dynamics.

He quotes the second law of thermodynamics in both the Thomson
and the Clausius forms. It is not always clear how seriously one is
to take this thermodynamical model of history.

About fifteen pages into "A Letter," Darwin is presented as
contradicting the thermodynamical ideas of Thomson. He sees Dar-
win's contribution not in the theory of natural selection, but in that
the evolutionary mediod shows how to bring "all vital processes
under the lav/ of development." It is this that is to furnish a lesson
to the study of history. This apparent conflict is one of the major
subjects of the early part of the "Letter."

"Henry Adams, The Degradation of the Democratic Dogma (Macmillan and
Co., New York, 1920), pp. 137-366.

181

Thus, at the same moment, three contradictory ideas of energy were in
force, all equally useful to science:

1. The Law of Conservation

2. The Law of Dissipation

3. The Law of Evolution

The contrast Adams is making is between Darwin's ideas and Kel-
vin's ideas.

We find other similar references in Henry Adams, but this
should be enough to show his interest in Darwin and kinetic theory.
Other aspects of contemporary science also very much influenced
him; he often refers to the enormous change produced by the
discovery of new kinds of radiation at the turn of the century. He
seems to be a particularly rewarding individual to study for an
understanding of the intellectual currents at the beginning of the
century, as Harold G. Cassidy has pointed out:

Henry Adams was an epitome of the non-scientist faced with science
that he could not understand, and deeply disturbed by the technological
changes of the time. He was a man with leisure, with the wealth to
travel. With his enquiring mind he sensed, and with his eyes he saw
a great ferment at work in the World. He called it a force, and tried
to weigh it along with the other forces that moved mankind. The edu-
cation he had received left him inadequate from a technical point of
view to understand, much less cope with, these new forces. Yet his
insights were often remarkable ones, and instructive to us who look at
our own period from so close at hand.'*

As final evidence we consider the work of the seminal American
philosopher Charles Sanders Peircc. Peirce, although seldom hold-
ing an academic position, played an important role in American
philosophy, particularly in the development of pragmatism. He was
the leader of the informal "Metaphysical Club" in Cambridge dur-
ing the last decades of the century. The history and views of the
group, much influenced by evolutionary ideas, are discussed by
Philip Weiner in Evolution and the Founders of Pragmatism.

Peirce was familiar with the development of both statistical
thermodynamics and evolution, and both played an enormous role
in the development of his thought. Peirce was a scientist by occupa-
tion, so his active interest in science is not surprising. We find his
awareness of these theories (some of which he did not fully accept)
evidenced by many passages in his work, such as these comments in
"On the Fixation of Belief":

^"Harold G. Cassidy. "The Muse and the Axiom," American Scientist 51,
315 (1963).

182

Randomness and The Twentieth Century

Mr. Darwin has purposed to apply the statistical method to biology. The
same thing has been done in a widely different branch of science, the
theory of gases. We are unable to say what the movements of any par-
ticular molecule of gas would be on a certain hypothesis concerning
the constitution of this class of bodies. Clausius and Maxwell were yet
able, eight years before the publication of Darwin's immortal work, by
the apphcation of the doctrine of probabilities, to predict that in the
long run such and such a proportion of the molecules would under
given circumstances, acquire such and such velocities; that there would
take place, every second, such and such a relative number of collisions,
etc., and from these propositions were able to deduce certain properties of
gases especially in regard to the heat relations. In like manner, Darwin,
while unable to say what the operation of variation and natural selection
in any individual case will be, demonstrates that, in the long run, they
will, or would, adopt animals to their circumstances.^^ [5-362]

A second example in which Peirce links the two theories is in
"Evolutionary Lore":

The Origin of the Species was published toward the end of the year
1859. The preceding years since 1846 had been one of the most pro-
ductive seasons — or if extended so as to cover the book we are con-
sidering, the most productive period in the history of science from its
beginnings until now. The idea that chance begets order, which is one
of the cornerstones of modern physics . . . was at that time put into its
clearest light. [6.297]

He goes on to mention Quetelet and Buckle, and then begins a
discussion of the kinetic theory:

Meanwhile, the statistical method had, under that very name, been applied
with brilliant success to molecular physics. ... In the very summer pre-
ceding Darwin's publication, Maxwell had read before the British Asso-
ciation the first and most important of his researches on the subject. The
consequence was that the idea that fortuitous events may result in physical
law and Lurther that this is the way in which these laws which appear
to conflict with the principle of conservation of energy are to be explained
had taken a strong hold upon the minds of all who are abreast of the
leaders of thought. [6.297]

Peirce is not reflecting the historical attitude of the physicists
who developed statistical thermodynamics but is reading his own
views back into this work.

**C. S. Peirce, Collected Papers ed. C. Hartshorn and P. Weiss (Harvard Uni-
versity Press, Cambridge, Mass.). References are to section numbers.

183

So it is not surprising that chance plays a fundamental role in
Peirce's metaphysics. Peirce generalized these ideas into a general
philosophy of three categories, Firstness, Secondness, and Thirdness.
These three terms have various meanings in his work, but a fre-
quent meaning of Firstness is chance. He was one of the first to
emphasize that chance was not merely for mathematical conven-
ience but was fundamental to the universe. He used the word
"Tychism," from the Greek for "chance," the "doctrine that absolute
chance is a factor in the universe." [6.2000]

This view of the essential role of chance he opposed to the view
that universal necessity determined everything by fixed mechanical
laws, in which most philosophers of science in the late nineteenth
century still believed. In a long debate between Peirce and Carus
concerning this issue, Peirce says:

The first and most fundamental element that we have to assume is a
Freedom, or Chance, or Spontaneity, by virtue of which the general vague
nothing-in-particuiar-ness that preceded the chaos took on a thousand
definite qualities.

In "The Doctrine of Necessity" Peirce stages a small debate
between a believer in his position and a believer in necessity, to show
that the usual arguments for absolute law are weak. Everyday ex-
periences make the presence of chance in the universe almost
obvious:

The endless variety in the world has not been created by law. It is not
of the nature of uniformity to originate variation nor of law to beget
circumstance. When we gaze on the multifariousness of nature we arc
looking straight into the face of a living spontaneity. A day's ramble
in the country ought to bring this home to us. [6.553!

A man in China bought a cow and three days and five minutes
later a Greenlander sneezed. Is that abstract circumstance connected with
any regularity whatever? And are not such relations infinitely more fre-
quent than those which are regular? [5.342]

The necessity of initial conditions in solving the equations of
mechanics is another indication to Peirce of the essential part played
by chance. Modern scientists have also stressed the "randomness"
of initial conditions: E. P. Wigner writes, "There are . . . aspects of
the world concerning which we do not believe in the existence of any
accurate regularities. We call these initial conditions."

Peirce tells us we must remember that "Three elements are
active in the world: first, chance; second, law; and third, habit

184

Randomness and The Twentieth Century

taking." [1409] He imagines what a completely chance world
would be like, and comments, "Certainly nothing could be imagined
more systematic." For Peirce the universe begins as a state of com-
plete randomness. The interesting problem is to account for the
regularity in the universe; law must evolve out of chaos. This evo-
lutionary process is far from complete even now, and presents a
continuing process still:

We are brought, then, to this: Conformity to law exists only within a
limited range of events and even there is not perfect, for an element of
pure spontaneity or lawless originality mingles, or at least must be sup-
posed to mingle, with law everywhere. [1.407]

Thus Peirce's scheme starts with chaos and out of this by habit order-
liness comes, but only as a partial state.

What is of interest to us is the fundamental role of chance or
randomness in Peirce's cosmology, and the connection of that role
with statistical mechanics and Darwinism, rather than the details of
his metaphysics.

The two examples of Henry Adams and C. S. Peirce do not
establish the splash in the puddle, but they do serve at least to indi-
cate the influence of the Darwinian and kinetic theory ideas, and
they show the rising importance of chance.

Although I have concentrated on the relatively increased atten-
tion focused upon randomness in the twentieth century as compared
with the nineteenth century, randomness attracted some interest
before our century. One can find many earlier examples of the order-
randomness dichotomy, and there have been periods when, even
before the nineteenth century, random concepts acquired some
status. One example containing elements of our present dichotomy
is the continuing battle between classicism and romanticism in the
arts and in literature. But the twentieth-century interest, as we have
indicated, is more intense and of different quality. The chance com-
ponent has never been totally absent; even the most careful artist
in the last century could not be precisely sure of the result of his
meticulously controlled brush stroke. The classical painter resisted
chance — the goal of his years of training was to gain ever greater
control over the brush. By contrast the contemporary painter often
welcomes this random element and may even increase it. It is this
contrast that I intend to stress. Although I point to this one element,
the reader should not falsely conclude that I am not aware of non-

185

random elements. Even now randomness is seldom the sole faaor.
When Pollock painted, the random component was far from the
only element in his technique. He chose the colors, he chose his
hand motions, and he chose the place on the canvas where he wanted
to work. Further, he could, and often did, reject the total product
at any time and begin over. Except in the most extreme examples,
randomness is not used alone anywhere; it is almost always part of
a larger situation. This is J. R. Pierce's emphasis on order.

The persistence of chance elements in highly ordered societies
suggests a human need for these elements. Perhaps no society ever
described was more completely organized than Arthur C. Clarke's
fictional city of Diaspar, described in The City and the Stars. Diaspar,
with its past, and even to some extent its future, stored in the
memory banks of the central computer, has existed with its deter-
mined social structure for over a billion years. But the original
planners of the city realized that perfect order was too much for
man to bear:

"Stability, however, is not enough. It leads too easily to stagnation, and
thence to decadence. The designers of the city took elaborate steps to
avoid this, ... I, Khedron the Jester, am part of that plan. A very
small part, perhaps. I like to think otherwise, but I can never be sure. . . .
Let us say that I introduce calculated amounts of disorder into the city."*''

But our present situation confronts us with something more than a
simple dichotomy between order and disorder, as suggested in both
of the following passages, one from L. L. Whyte and one from
Erwin Schrodinger:

In his long pursuit of order in nature, the scientist has turned a corner.
He is now after order and disorder without prejudice, having discovered
that complexity usually involves both,*^

The judicious elimination of detail, which the statistical system has
taught us, has brought about a complete transformation of our knowledge
of the heavens. ... It is manifest on all sides that this statistical method
is a dominant feature of our epoch, an important instrument of pro-
gress in almost every sphere of public life.*®

'^A. C. Clarke, The City and the Stars (Harcourt, Brace and Co., New York,

1953). PP- 47-53-

**L. L. Whyte, "Atomism, Structure, and Form," in Structure in Art and in

Science, ed. G. Kepes (G. Braziller, New York, 1965) p. 20.

'"E. Schrodinger, Science and Human Temperament, trans. }. Murphy and

W. H. Johnston (W. W. Norton, Inc., New York), p. 128.

186

Randomness and The Twentieth Century

Although the use of random methods in physics and biology at
the end of the last century originally assumed that one was dealing
with areas that could not be treated exactly, but where exact laws
did exist, a subtle change of view has come about, so that now
random elements are seen as having a validity of their own. Both
Whytc and Schrodinger see the current situation as something more
than a choice between two possibilities. Whyte thinks both are
essential for something he calls "complexity." But I prefer Schro-
dinger's suggestion that the two are not necessarily opposed, and that
randomness can be a tool for increasing order. Perhaps we have a
situation resembling a Hegelian synthesis, combining two themes
which had been considered in direct opposition.

Finally I note an important twentieth century reaction to ran-
domness: Joy. The persistence of games of chance through the ages
shows that men have always derived some pleasure from random-
ness; they are important in Clarke's Diaspar, for example:

In a world of order and stability, which in its broad outlines had not
changed for a bilUon years, it was perhaps not surprising to find an
absorbing interest in games of chance. Humanity had always been fasci-
nated by the mystery of the falling dice, the turn of a card, the spin
of the pointer . . . however, the purely intellectual fascination of chance
remained to seduce the most sophisticated minds. Machines that behaved
in a purely random way — events whose outcome could never be predicted,
no matter how much information one had — from these philosopher and
gambler could derive equal enjoyment.

But the present joy exceeds even this. Contemporary man often
feels excitement in the presence of randomness, welcoming it in a
way that would have seemed very strange in the immediate past. In
some areas (literature, perhaps) this excitement still seems not quite
proper, so it is not expressed openly. But in other places randomness
is clearly acknowledged. We noted that the artist is particularly
willing to admit the use of randomness, so it is not surprising to
see an artist, Ben Shahn, admitting his pleasure: "I love chaos. It is
a mysterious, unknown road with unexpected turnings. It is the way
out. It is freedom, man's best hope."^'*

^''Quoted in Industrial Design 13, 16 (1966).

187

A survey of the chief properties of wave motion, using simple
mathematics in clear, step-by-step development.

13 Waves

Richard Stevenson and R. B. Moore

From their book Theory of Physics, 1967.

As we all know, energy can be localized in space and time. But the
place where energy is localized may be different from the place where
its use is desired, and thus mechanisms of transport of energy are of
the greatest interest.

The transport of energy is achieved in only two ways. The first
involves the transport of matter; as matter is moved its kinetic energy
and internal energy move with it. The second method is more com-
plicated and more interesting; it involves a wave process. The wave
carries energy and momentum, but there is no net transfer of mass.
There are many different types of waves, but the general nature of
the events by which energy is carried by a wave is always the same.
A succession of oscillatory processes is always involved. The wave
is created by an oscillation in the emitting body; the motion of the
wave through space is by means of oscillations; and the wave is ab-
sorbed by an oscillatory process in the receiving body.

Most waves are complex. In this chapter we study the most simple
types of waves, those for which the amplitude varies sinusoidally.

17.1 PULSES

Suppose that you are holding the end of a relatively long rope
or coil spring and that the other end is fixed to the wall.* If you
raise your hand suddenly and bring it back to its original position,
you will create a pulse which moves down the rope and is reflected
back. The sequence of events is indicated in Figure 17.1.

Any individual point on the rope simply moves up and down
as the pulse passes by. It is obvious that the pulse moves with a
certain velocity, and we might imagine that there is a certain
energy and momentum associated with it, even though there is
no transfer of mass. Keep in mind the observation that the pulse
is inverted after reflection from the wall.

Consider now another experiment with two rop>es, one light

* It is best, of course, to hang the rope from the ceiling or lay it on a smooth table
so that the rope does not sag under the action of gravity. We will draw the diagrams
with the rope horizontal, as if there were no gravitational force.

188

Waves

VJIAAAA/WWWWWVWVWWWW^^'''^

jJJJJJJJJJ.\MMfMMMN\r/^^^ ^ ^ *>

\J.KKK*^J.JJJJJJJJJJJJJJJ:

.....;/''''^^^

' Wyy^///JvvJ</AVA^^^^^^^^vvvvvAvvv»vy^vr«VITTt■l

FIGURE 17.1 This sequence of photographs
shows a pulse traveling to the left on a long coil
spring. The pulse is reflected by the fixed end
of the spring and the reflected pulse is inverted.
(From Physical Science Study Committee:
Physics. Boston, D. C. Heath & Co., 1960. Copy-
right, Educational Services Inc.)

vvv-'./.'-'.'v-vyvyyy///v\;^vv^\^^'^.^^^^^^^vvvAYVYvvvvyY»YrrrrI^

.J^^Jy/^/-rJ•/A'.wA^^^^^^^^^^^'^^vyvvvvYvvv»-rvrrl-rrI•l

JJJJ.^JJ^JJJJJJJJJJJJJJJJJ^

'■f^^S\^J^^^^..fttt****•'^'''''V^^

>jJjJJJJJJJJJ''JJ.IJJ.U^'ff^-'fJ'.\'.

•'^■^••^^^^^^,,^^^^,,^,^,^r<rryY^nnnrvrn.

and one heavy, attached to each other as in Figure 17.2. An in-
cident pulse is sent along the light rope, and when it arrives at the
junction or interface it is partially transmitted and partially re-

189

li^t rope

heavy rope

incident pulse

reflected pulse

transmitted pulse

FIGURE 17.2 An incident pulse is sent along the light rope toward the attached heavy
rope. The pulse is partially transmitted and partially reflected. The reflected pulse is in-
verted as in Figure 17.1.

fleeted. The transmitted pulse is upright, and the reflected pulse
is inverted.

We can vary the two-rope experiment by sending the incident
pulse along the heavy rope. Part is transmitted and part reflected,
but the reflected part is not inverted. This is different from the
case shown in Figure 17.3, and we conclude that the type of re-
flection depends on the nature of the interface at which reflection
occurs.

What happens when two pulses are sent along a rope and pass
over each other? If two equivalent pulses inverted with respect to
each other are sent from opp)osite ends of the rope, they will seem
to cancel each other when they meet, and at that instant the rop>e
appears to be at rest. A moment later the pulses have passed by
each other with no evident change in shape. Evidently one pulse
can move along the rope quite independently of another, and
when they meet the pulses are superimposed one on the other.

light rope

heavy rope

incident pulse

transmitted pulse

reflected pulse

FIGURE 1 7.3 T^^i'S is similar to Figure 17.2 but now the incident pulse is on the heavy rope.
Again the pulse is partially reflected and partially transmitted. However the reflected pulse
is not inverted. The nature of the reflected pulse will depend on the boundary which caused
the reflection.

190

Waves

FIGURE 17.4 Superposition implies that waves or pulses pass
through one another with no interaction. The diagram shows a
rope carrying two pulses. In (a) the pulses approach each other.
In (b) they begin to cross, and the resultant rope shape is found
by the addition of pulse displacements at each point along the
rope. At the instant of time shown in (c) there will be no net dis-
placement of the rope; if the pulse shapes are the same and their
amplitudes are opposite there will be an instantaneous cancella-
tion. In (cO the pulses move along with no change in shape or
diminution of amplitude, just as if the other pulse had not existed.

17.2 RUNNING WAVES

Let us supply a succession of pulses to our long rope, as in
Figure 17.5. This is easily enough done by jerking the end of the
rof)e up and down at regular intervals. If the interval is long
enough we would have a succession of separate pulses traveling
along the rope. Eventually, of course, these pulses will be reflected
and will complicate the picture, but for the moment we can assume
that no reflection has occurred.

FIGURE 17.5 We can send a succession of pulses along a long rope by jerking one end
up and down.

Now suppose that we apply the pulses to the rope so that there
is no interval between pulses. The result is shown in Figure 17.6.
This is obviously a special case, and we give it a special name. We
say that a wave is moving along the rope, and it is clear that the
wave is composed of a specially applied sequence of pulses. Such
a wave is called a running or traveling wave.

FIGURE 17,6 Instead of sending isolated pulses along the rope as in Figure 17.5, we move
our hand up and down continuously. Now there is no interval of time between individual
pulses, and we say that the rope is carrying a wave. The wave velocity is identical to the
velocity of the individual pulses which make up the wave.

Problem 2

191

A simple type of wave can be created by causing the end of
the rope to move up and down in simple harmonic motion. The
sequence of events by which the wave was established is shown in
Figure 17.7. The motion of the end of the rope causes the wave
pulse to move along the rope with velocity c. As the wave pulse
moves along, a point on the rope a distance / from the end of the

FIGURE 17.7 This sequence of drawings shows the means by which a wave is established
along a rope. The left hand end moves up and down in simple harmonic motion. This causes
the wave pulse to move along the rope with velocity c. The frequency of the wave will be
the same as the frequency of the event which started the wave.

rope will also start into simple harmonic motion, but it will start at
a time Ijc later than that of the end of the rope.

Consider Figure 17.8. Point A has just completed one cycle of
simple harmonic motion. It started at < = 0 and finished at f = 7,

FIGURE 1 7.8 This shows the wave form for one complete cycle of simple harmonic motion
of the source. The wave moves in the x-direction, and individual points on the rope move
in the ±K-directions. The wavelength x is the distance the wave travels for one complete
cycle of the source. The wave amplitude is a.

192

Waves

where T is the period. If the amplitude of motion is a, then the
displacement in the y direction of point A can be represented by

>^ = asin27r// (17-1)

where / is the frequency of the motion. Now as point A is just
finishing one cycle and starting another, point B is starting its
first cycle. If it is a distance \ away from A, it starts at time

ts = ^

Another point on the curve, such as X, had started at a time

With respect to point A , the motion of point X is delayed by a time
tx. We can see that the displacements of points B and X can be
represented by

yB = a sin 27r/( t 1

yx = a sin 27r/f t j

Let us return for a moment to Figure 17.8. The distance
AB = X, for one complete wave form, is called the wavelength. If
the wave has velocity c, the time required for the wave to travel
from A to B is k/c, and this will just equal the period of the simple
harmonic motion associated with the wave. That is,

However

Thus

kf=c (17-3)

This very important relationship between wavelength, frequency
and wave velocity holds for any type of wave.

We also have developed an equation which represents the
wave. For the wave moving in the positive x-direction, the displace-
ment of any point a distance x from the origin is given by ( 1 7-2).

y = a sin

^'-f)

We can simplify this by noting that

a) = 27r/

193

194

and

Thus we have

// = ! and ^ = f
^ T c k

y = a siTKoit 1

= a sin 27rf — — -J

Example. Two sources separated by 10 m vibrate according
to the equations yi = 0.03 sin Trt and 3)2 = 0.01 sin nt. They send out
simple waves of velocity 1.5 m/sec. What is the equation of motion
of a particle 6 m from the first source and 4 m from the second?

1 2

• • •

-6 m »+« — 4 m — H

We suppose that source 1 sends out waves in the +x-direction,

311 = fli sin 27rfAt ^ J

and that source 2 sends out waves in the — x-direction,

y2 — ch sin 27r/2( ^ + ~)

Then

Thus

fli = 0.03 m 02 = 0.01 m

Xi = 6 m X2 = —4 m

/i =/2 = V2 sec-^
c = 1.5 m/sec

3»i = 0.03 sin7r(/-4)

= 0.03 sin TTt cos iir — cos nt sin 47r
= 0.03 sin TTt

3»2 = 0.01 sin7r(/-8/3)

= 0.01 (sin TTt cos 877/3 — cos irt sin 87r/3)
= 0.01 (sin TTf (-1/2) - cos nt V3/2)
= -0.005 sin TTt - 0.00866 cos nt

The resultant wave motion is

y = yi+ y2
= 0.03 sin nt - 0.005 sin nt - 0.00866 cos nt
= 0.025 sin nt - 0.00866 cos nt

We will write this in the form

> = /4 sin {nt + </>)
= A sin TTf cos <f} + A cos tt^ sin 0

Waves

Thus

A^ = 0.0252 + 0.008662
, . 0.00866
"^"^==-0:025-
- <^=19.1*'

A = 0.0264 m
= 0.346

17.3 STANDING WAVES

Suppose that we have a long rope with one wave train of
angular frequency o> traveling in the +x-direction and another of
the same frequency traveling in the — x-direction. Both wave trains
have the same amplitude, and we can write the general displace-
ments as

3>+ = a sinwU j

)»- = a sin ft)( / + - 1

These two wave trains are superimposed, so the net displacement
is

y = )>+ + 31-

= a svnoiit 1 + a sin cdI < + - j

To simplify this we use the trigonometric relations

sin {d + <^) = sin B cos <^ + cos 6 sin </>
sin {B — 4>) = sin B cos </> — cos B sin <\>

(17-6)

Thus (17-6) is transformed to

y = (2a sin (nt) cos

(liX

(17-7)

FIGURE 17.9 This sequence shows pictures of standing waves at intervals of 1/4 T, where
7"= lit is the period. At f = 0, 1/2 T, T the displacement at all points is instantaneously zero.
At inter/als of X/2 along the wave there are points called nodes for which the displacement
is zero at all times.

195

This is called a standing wave. The amplitude is 2a sin cot, which
varies with time and is zero at f = 0, f = V2 T, and so forth. The
displacement on the rope will be zero for distances x, where

--(2n-l)2
w = 0, ±1,±2, . ,

(17-8)

Since w = 27r/, from (17-7), these points of zero displacement or
nodes are located at

x=(2n-l)^f

= (2n-l)2|^f
= (2n-l)|

:i7-9)

The distance between two nodes will be, therefore, nX/2 where
n = 1, 2, 3, and so forth.

It is easy to see how standing waves can be created on a string
which is fixed at one or at both ends. One wave train is caused by

FIGURE 17.10 A string of length /, such as a violin string, is clamped at both ends. Both
ends must be nodes if a standing wave is to be set up on the string. The maximum wave-
length of the standing wave will be x = 2 /. The next possible standing wave will have a
wavelength x = /. Vibrations with wavelengths different from those of the standing waves
will die out quickly.

the agency which causes the vibration, and the other wave train
arises from a reflection at the fixed end. Consider a string fixed
at both ends. Both ends must be nodes, so that if the length of the
string is /, then by (17-9)

/ =

(17-10)

196

Waves

The string can vibrate with wavelengths 2/, /, 2//3, and so forth.
Vibrations with other wavelengths can be set up of course, but
these die out very quickly. The string will resonate to the wave-
lengths given by (17-10).

It is very important that the distinction between a running
wave and a standing wave be kept in mind. The running wave is
illustrated in Figures 17.6 and 17.7. The wave disturbance moves
in one direction only and each particle through which the wave
passes suffers a sinusoidal variation of amplitude with time. The
standing wave, on the other hand, is a superposition of two run-
ning waves of the same frequency and amplitude, moving in
opposite directions. Certain points on the standing waves, the
nodes, have a constant zero amplitude even though the two run-
ning waves are continually passing through these points. Usually
a standing wave is made by the superposition of an incident wave
and the reflected wave trom some boundary.

Example. Standing waves are produced by the superposition
of two waves

^^i = 15 sin (Sirt — 5x)
y2 = 15 sin {Snt + 5x)

Find the amplitude of motion at x = 21.
We use the relationships

sin (a ± )3) = sin a cos /3 ± cos a sin ^
sin (a + /3) 4- sin (a — ^) = 2 sin a cos /3

Thus

With

y = )'i + )'2 = 30 sin Snt cos 5x

X = 21, 5x = 105 radians

= 38.47r radians

Now cos 38.477 = cos 0.47r = cos 72° = 0.309.
Thus the amplitude at x = 21 is

30 cos 38.477 = 30X0.309
= 9.27

17.4 THE DOPPLER EFFECT

We wish now to study what happens when waves from a point
source S, which moves with velocity u, are detected by an observer
O who moves with velocity v.

Let the situation be as in Figure 17.1 1. The velocities u and v
are in the positive x-direction. The velocity of the waves emitted
by S is c, and we can imagine two points A and B fixed in space

197

)'▲

FIGURE 17.11 Waves are emitted by a point source S moving
g with velocity u, and detected by an observer O moving with

9 velocity i^. At f = 0, points A and 8 are equidistant from S. A

spherical wave emitted from S at f = 0 will just reach A. and 8 at

time f = T.

equidistant from 5 at ^ = 0 such that a wave emitted at f = 0 will
just reach A and B at time t. Thus at < = 0

dist /iS = C7
dist 5S = CT

f=0

But by time t, 5 will have moved a distance wt, and then

t = 7

dist y45 = CT + WT
dist 55 = CT — MT

If the frequency of the source is /o, it will have emitted /o wave-
fronts between t = 0 and t = t. Since the first wavefront reaches
A and B at t = t, then/o wavefronts are contained in the distances
AS and BS. Thus the apparent wavelength in front of the source is

_ ^ _ c — u
for /o

and the wavelength behind the source is

. , _ AS^ _ c + u
for /o

:i7-ii:

(17-12)

Now the observer O moves with velocity v, and the speed of
the waves relative to him is c + v. Since he is behind the source he
experiences waves of wavelength X' at an apparent frequency/
given by

c + V
k'

C + V

c -\- u

(17-13)

The various expressions can easily be altered if the source is mov-
ing in a direction opposite to that of the detector.

Most of us will have noticed the Doppler effect in the change
in pitch of a horn or siren as it passes by. The Doppler effect is a
property of any wave motion, and is used, for example, by the
police in the radar sets that are employed to apprehend speeding
motorists.

Example. A proposed police radar is designed to work by the
Doppler effect using electromagnetic radiation of 30 cm wave-

198

Waves

radar

length. The radar beam is reflected from a moving car; the motion
causes a change in frequency, which is compared with the original
frequency to compute the speed.

A car moves toward the radar at 65 mph (31 m/sec). The wave-
length of the beam emitted by the radar is

On time t the source emits /o wavefronts, and these travel a dis-
tance CT — VT before reflection. Thus the wavelength of the re-
flected beam as seen by the car is

k' =

CT — VT C — V

for

As seen by the stationary radar set this wave X' reflected by the
moving car has wavelength X" and frequency/".

^'-t'

-fo

c-2v

The fractional change in frequency is

^f^fo-r^^ c-2v_2v_

c 3 X 108
= 2.06 X 10-7

/o /o

The frequency of 30 cm radiation is

Thus the change in frequency would be

A/= 2.06 X 10-7 X 10*
= 206 cps

17.5 SOUND WAVES

Waves on a string are called transverse waves because the
motion of the individual particles is perpendicular or transverse
to the direction of motion of the wave. Another type of wave is
the longitudinal wave, where the motion of the particle is along
the same line as the direction of motion of the wave.

Sound is a longitudinal wave which involves very small changes
in density of the medium through which it is propagated. That is,

199

a sound is a train of pressure variations in a substance. At any one
point there is an oscillatory variation in pressure or density.
In a solid the velocity of sound is given by

(17-14)

C= A -T

where d is the density and E is Young's modulus (the ratio of stress
to strain in the elastic region).*

In a perfect gas the velocity of sound is given by

c=\M-

y Cvd

(17-15)

where again d = m/v is the density of the gas. Since pV = RT, we
can see from (17-15) that c « T^l^.

Very interesting effects occur with sound waves when the
source emitting the wave is moving faster than the velocity of
sound. For example, in Figure 17.12, consider a source moving
with speed v > c. It moves from ^4 to B in time At and from 5 to C
in an equal time A^ When the body is at C the wave emitted at B
has spread out as a sphere of radius cAt. Similarly the wave emitted

* See Section 27.5.

FIGURE 17.12 A point source moving with velocity
V emits spherical waves. The diagram shows wave
fronts emitted at intervals Af. The wave fronts are
enclosed within a cone of angle 2d, where sin 0 =

civ. This cone is called the Mach cone. If the source
is emitting sound waves there will be a finite pres-
sure difference across the Mach cone, it is this
pressure difference that gives rise to the term
"shock wave."

200

Waves

at A has spread out as a sphere of radius 2cAf. All the waves emitted
at previous times are enclosed within a cone, called the Mach cone,
of angle 20, where sin 6 = cjv. The wave along this cone is called
a shock wave because there is a finite difference of pressure across
the front. The ratio vie is called the Mach number, after the
scientist who first proposed its use. The cone of shock waves is
called the Mach cone.

Example. The index of refraction n of a substance is the ratio
of the velocity of light in a vacuum to the velocity of light in the
substance, n = cjv. If a high speed charged particle is sent with
velocity u through the substance, the ratio (the Mach number, as
it were) ulv can be greater than unity. Then any radiation emitted
by the particle is enclosed within a cone of angle 26 where

. . t; c
s\n6 = — = —
u nu

The velocity of the particle can be greater than the velocity of
light in the medium (but never greater than the velocity of light
in a vacuum).

The radiation emitted by the particle is known as Cherenkov
radiation. By measuring 6, this phenomena finds useful applica-
tion in the measurement of the velocities of charged particles.

17.6 ENERGY OF WAVES

Let us return to the wave moving along a string. The dis-
placement of point X at time t is

y = asmJt-fj (17-16)

The velocity v of this point is

v = ^^ = awcosco{t-fj (17-17)

Now we suppose that the mass of a small element of the string at
this point is m; thus, the kinetic energy is

T = V2 mv^

= V2 ma^cj^ cos^ col

The time average of the kinetic energy is

T=y4nuiW (17-19)

And finally we define an energy density as being the average
kinetic energy per unit mass.

kinetic energy density = V4 aW (17-20)

201

Since the energy of the system is being transformed from kinetic
to potential and back again, and since Tmax = Vmax we can see that

potential energy density = V4 aW (17-21)

Thus

total energy density = V2 aW (17-22)

The energy density is proportional to the square of the amplitude
and the square of the frequency.

The same expression (17-22) holds for a sound wave. The
density d gives the mass per unit volume; thus, the total energy
per unit volume is

V2 rfaW (17-23)

We can now begin to perceive how energy is transported by
waves. By expending energy a source can cause a harmonic dis-
turbance in a medium. This disturbance is propagated through
the medium by the influence that an individual particle has on
other particles immediately adjacent to it. The motion of the par-
ticle means that it has a certain amount of energy, part kinetic and
part potential at any instant of time. At some point energy is re-
moved from the wave and presumably dissipated. And to sustain
the wave motion along the wave train, energy must be supplied by
the source.

17.7 DISPERSION OF WAVES

We have discussed only the simplest type of waves, sinusoidal
in form, and in the remainder of the book we will never have occa-
sion to talk about more complicated waves.

In a simple wave, at a point in space there is a simple harmonic
motion of mass or a sinusoidal variation of a field vector. This
local event may be parallel or perpendicular to the direction of the
wave, from which arises the terms "longitudinal" or "transverse"
waves. The wave has a frequency /and a wavelength X. The wave
velocity c is related to these by c =fK.

Strictly speaking, we should call this velocity the phase veloc-
ity, and give it another symbol V4,.

V4,=J\ (17-24)

This is because v^, gives the velocity at which an event of constant
phase is propagated along the wave. For later use we will define
another quantity, the wave vector modulus k.

'^ 27rX

Thus

V4, =

^ (17-25)

202

Waves

As might be expected, real waves are liable to be more com-
plicated than the simple waves, and we might suppose that the real
wave is a result of the superposition of many simple waves. If the
velocities of the simple waves vary with wavelength, what then is
the velocity of the resultant wave and how is it related to wave-
length?

For an example consider two simple waves of slightly different
wavelengths X and k' and velocities v and v' , but with the same
amplitude. For the resultant wave the displacement x at time t is

y = a sin {(ot — kx) + a sin {(o't — k'x) (17-26)

We can use the trigonometric identity

sm a + sm /3 = 2 sm — r-^ cos — r-^

Thus the displacement oi y becomes

,=2asi„[(^>-(*±^>]cos[(^>-(^>] ,17-27)

We will rewrite (17-27) as

y = 2a sin {(04,1 — A^) cos (cjgt — kgx) (17-28)

The individual waves correspond to the sine factor in (17-27) and
(17-28) and the phase velocity is

The cosine factor in (17-27) and (17-28) indicates that another
wave is present with velocity

_ if^a —

fa> — fa>

"' 1 " (17-30)

_ Afa>

~ ^k

This is called the group velocity, using the terminology that the
real wave is made up of a group of individual waves.
We know that (o = v<i,k, thus

Afa) = (fa> 4- Aa>) — fa)

Therefore

Afa)

(17-31)

The significance of this is that the group velocity is the veloc-
ity at which energy flows, and it is normally the only velocity that

203

can be observed for a wave train. Dispersion is said to occur when
the phase velocity varies with wavelength, that is when Af,<,/AA 7^ 0.
If there is no dispersion the phase velocity is identical to the group
velocity.

Example. An atom emits a photon of green light X = 5200 A
in T = 2 X 10~^° sec. Estimate the spread of wavelengths in the
photon.

We will consider the photon to be composed of a train of
waves. The length of the wave train is cr = 3 X 10* X 2 X 10-'° =
0.06 m.

To make the estimate we can suppose that the wave train is
made up of waves with slightly different frequencies and wave-
lengths,

y = a sin {oit — kx) + a sin {oi't — k'x)
= 2asinV2[(ft> + w')f- {k + k')x'] cos V2[(a) - a>')f - {k-k')x'\

The resultant wave has an overall frequency ¥2(0) — co') and an
overall wave vector Vzik — k'). Thus the length of the wave train
is approximated by

2/ =

277- y^ik-k')

Since k = x~r, k' = „ , • This length is given by

/ =

k-k'

We can write k' = k + AX, thus

1 = ^
AX

and we calculated / = 0.06 m. Therefore

., _ X' _ (0.52 X 10-«)'
^^ " / " 0.06

= 4.5 X 10-" m
= 4.5 X 10-2 A

204

Waves

A similar estimate of AX can be made using the uncertainty prin-
ciple in the following way. We use A£A/ ~ h, and E = hf; thus

h^f^t ~ h

■' A/

Now kf=c, therefore

(\ + AX)(/+A/) -kf=0

which gives, in absolute values,

AX^A^
^ f

Thus Af ~ -r- reduces to

AX = -TT- = —r-
fAt cAt

which is the same as the expression used in the previous calcula-
tion.

17.8 SPHERICAL WAVES

So far we have talked about only waves on a rope, and clearly
the rope was the medium which carried the wave. Many waves are
associated with a medium, but the existence of a medium is not
essential to the existence of a wave; all we need is something that
vibrates in simple harmonic motion.

Perhaps the most important types of waves are sound waves
and electromagnetic waves. A sound wave needs a medium to be
transmitted, and the vibration consists of small oscillations in the
density of the medium in the direction of propagation of the wave.
Thus a sound wave is classified as a longitudinal wave. On the
other hand, an electromagnetic wave needs no medium and con-

FIGURE 17.13 A point source S emits spherical waves. At time U the
wavefront is a sphere of radius W,; at time ft the wavefront is a sphere
of radius flj.

205

sists of oscillation of electric and magnetic field vectors perpendicu-
lar to the direction of propagation. It is classified, therefore, as a
transverse wave.

For a rope the medium extends only along a rope; thus, the
wave can be propagated only in that direction. But a sound wave
or an electromagnetic wave can be propagated in all directions at
once. Consider, as in Figure 17.13, a small source S of wave motion.
The wave front travels out in every direction from S, and we can
consider it to be spherical since no direction of propagation is
preferred over another.

Suppose that the energy associated with the wavefront is E.
This energy is distributed over the spherical wavefront of radius
R. Thus the intensity* or energy density at a point is

' = 4^ (17-32)

We can measure the intensity at two distances Ri and R2 from the
source. They will be

47rR\
E

h =
Thus

r D2

(17-33)

h R]

h Rl

That is, the intensity of a spherical wave varies inversely as the
square of the distance from the wave source. This inverse square
law applies only to spherical waves, but can be used in an approxi-
mate way to estimate the variation in intensity of waves which are
only approximately spherical.

17.9 HUYGENS' PRINCIPLE

In the last few sections we have talked about wavefronts with-
out defining them carefully. A small source S of frequency /can
emit waves of wavelength X and velocity c. We can suppose that
the source sends out wavefronts at time intervals 1//, and that
these wavefronts are separated by a distance X.

If we know the position of a wavefront at time t, how do we
find its position at time t + Af ? This problem is solved by Huygens'
principle, which states that every point on the wavefront at time t
can be considered to be the source of secondary spherical waves

* Keep in mind that the intensity of the wave is proportional to the square of the
amplitude; see Section 17.6.

206

Waves

FIGURE 17.14 If we know the position of a wavefront at time t, we can find its position at
time f + Af by Huygens' principle. Each point on the original wavefront is thought to emit
a secondary spherical wavelet. In time Af the wavelet will have a radius cAf, where c is the
wave velocity. The wavefront at time t + Af will be the envelope of all the secondary wave-
lets.

which have the same velocity as the original wave. The wavefront
at time t + At is the envelope of these secondary waves.

This is a geometric principle, of course, and is best illustrated
by a diagram. In Figure 17.14, AB is a wavefront at time t. If the
wave velocity is c, then in time At a secondary wave will travel a
distance cAt. The envelope of the secondary waves is AB, which
is therefore the position of the wavefront at time t + A^

207

Two masters of physics introduce the wave concept in
this section from c well-known popular book.

14 What is a Wave?

Albert Einstein and Leopold Infeld

An excerpt from their book The Evolution of Physics, 1961.

A bit of gossip starting in Washington reaches New
York very quickly, even though not a single individual
who takes part in spreading it travels between these
two cities. There are two quite different motions in-
volved, that of the rumor, Washington to New York,
and that of the persons who spread the rumor. The
wind, passing over a field of grain, sets up a wave
which spreads out across the whole field. Here again
we must distinguish between the motion of the wave
and the motion of the separate plants, which undergo
only small oscillations. We have all seen the waves that
spread in wider and wider circles when a stone is
thrown into a pool of water. The motion of the wave
is very different from that of the particles of water.
The particles merely go up and down. The observed
motion of the wave is that of a state of matter and not
of matter itself. A cork floating on the wave shows
this clearly, for it moves up and down in imitation of
the actual motion of the water, instead of being carried
along by the wave.

In order to understand better the mechanism of the
wave let us again consider an idealized experiment.
Suppose that a large space is filled quite uniformly with
water, or air, or some other "medium." Somewhere in
the center there is a sphere. At the beginning of the
experiment there is no motion at all. Suddenly the
sphere begins to "breathe" rhythmically, expanding
and contracting in volume, although retaining its spher-

208

What is a Wave?

ical shape. What will happen in the medium? Let us
begin our examination at the moment the sphere begins
to expand. The particles of the medium in the immedi-
ate vicinity of the sphere are pushed out, so that the
density of a spherical shell of water, or air, as the case
may be, is increased above its normal value. Similarly,
when the sphere contracts, the density of that part of
the medium immediately surrounding it will be de-
creased. These changes of density are propagated*
throughout the entire medium. The particles constitut-
ing the medium perform only small vibrations, but the
whole motion is that of a progressive wave. The essen-
tially new thing here is that for the first time we con-
sider the motion of something which is not matter, but
energy propagated through matter.

Using the example of the pulsating sphere, we may
introduce two general physical concepts, important for
the characterization of waves. The first is the velocity
with which the wave spreads. This will depend on the
medium, being different for water and air, for exam-
ple. The second concept is that of ivave-length. In the
case of waves on a sea or river it is the distance from
the trough of one wave to that of the next, or from the
crest of one wave to that of the next. Thus sea waves
have greater wave-length than river waves. In the
case of our waves set up by a pulsating sphere the
wave-length is the distance, at some definite time, be-
tween two neighboring spherical shells showing max-
ima or minima of density. It is evident that this dis-
tance will not depend on the medium alone. The rate
of pulsation of the sphere will certainly have a great
effect, making the wave-length shorter if the pulsation
becomes more rapid, longer if the pulsation becomes
slower.

This concept of a wave proved very successful in
physics. It is definitely a mechanical concept. The phe-

209

nomenon is reduced to the motion of particles which,
according to the kinetic theory, are constituents of
matter. Thus every theory which uses the concept of
wave can, in general, be regarded as a mechanical
theory. For example, the explanation of acoustical phe-
nomena is based essentially on this concept. Vibrating
bodies, such as vocal cords and violin strings, are
sources of sound waves which are propagated through
the air in the manner explained for the pulsating sphere.
It is thus possible to reduce all acoustical phenomena to
mechanics by means of the wave concept.

It has been emphasized that we must distinguish be-
tween the motion of the particles and that of the wave
itself, which is a state of the medium. The two are
very different but it is apparent that in our example of
the pulsating sphere both motions take place in the

same straight line. The particles of the medium oscillate
along short line segments, and the density increases
and decreases periodically in accordance with this mo-
tion. The direction in which the wave spreads and the
line on which the oscillations lie are the same. This
type of wave is called longitudinal. But is this the only
kind of wave? It is important for our further considera-

210

What is a Wave?

tions to realize the possibility of a different kind of
wave, called transverse.

Let us change our previous example. We still have
the sphere, but it is immersed in a medium of a differ-
ent kind, a sort of jelly instead of air or water. Further-
more, the sphere no longer pulsates but rotates in one
direction through a small angle and then back again,

always in the same rhythmical way and about a definite
axis. The jelly adheres to the sphere and thus the ad-
hering portions are forced to imitate the motion. These
portions force those situated a little further away to
imitate the same motion, and so on, so that a wave is
set up in the medium. If we keep in mind the distinc-
tion between the motion of the medium and the mo-
tion of the wave we see that here they do not lie on the
same line. The wave is propagated in the direction of
the radius of the sphere, while the parts of the medium
move perpendicularly to this direction. We have thus
created a transverse wave.

Waves spreading on the surface of water are trans-
verse. A floating cork only bobs up and down, but the
wave spreads along a horizontal plane. Sound waves,
on the other hand, furnish the most familiar example
of longitudinal waves.

211

One more remark: the wave produced by a pulsat-
ing or oscillating sphere in a homogeneous medium is
a spherical wave. It is called so because at any given
moment all points on any sphere surrounding the
source behave in the same way. Let us consider a por-
tion of such a sphere at a great distance from the
source. The farther away the portion is, and the
smaller we take it, the more it resembles a plane. We
can say, without trying to be too rigorous, that there
is no essential difference between a part of a plane and

a part of a sphere whose radius is sufficiently large. We
very often speak of small portions of a spherical wave
far removed from the source as plane ivaves. The far-
ther we place the shaded portion of our drawing from
the center of the spheres and the smaller the angle be-
tween the two radii, the better our representation of a
plane wave. The concept of a plane wave, like many
other physical concepts, is no more than a fiction which
can be realized with only a certain degree of accuracy.
It is, however, a useful concept which we shall need
later.

212

Many aspects of the music produced by Instruments,
such as tone, consonance, dissonance, and scales, are
closely related to physical laws.

15 Musical Instruments and Scales

Harvey E. White

A chapter from his book Classical and Modern Physics, 1940.

Musical instruments are often classified under one of the follow-
ing heads: strings, winds, rods, plates, and bells. One who is more or
less familiar with instruments will realize that most of these terms
apply to the material part of each instrument set into vibration when
the instrument is played. It is the purpose of the first half of this
chapter to consider these vibrating sources and the various factors gov-
erning the frequencies of their musical notes, and in the second part
to take up in some detail the science of the musical scale.

16.1. Stringed Instruments. Under the classification of strings
we find such instruments as the violin, cello, viola, double bass, harp,
guitar, and piano. There are two principal reasons why these instru-
ments do not sound alike as regards tone quality, first, the design of
the instrument, and second, the method by which the strings are set
into vibration. The violin and cello are bowed with long strands of
tightly stretched horsehair,
the harp and guitar are N ^

plucked with the fingers or
picks, and the piano is ham-
mered with light felt mallets.

Under very special condi-
tions a string may be made
to vibrate with nodes at either

end as shown in Fig. 16A. In this state of motion the string gives rise
to its lowest possible note, and it is said to be vibrating with its funda-
mental frequency.

Every musician knows that a thick heavy string has a lower natural
pitch than a thin one, that a short strong string has a higher pitch than
a long one, and that the tighter a string is stretched the higher is its
pitch. The G string of a violin, for example, is thicker and heavier
than the high pitched E string, and the bass strings of the piano are
longer and heavier than the strings of the treble.

Fig. 16A — Single string vibrating with its funda-
mental frequenqr.

213

Accurate measurements with vibrating strings, as well as theory,
show that the frequency n is given by the following formula:

= k^^i

(16^)

where L is the distance in centimeters between two consecutive nodes,
F is the tension on the string in dynes, and 7n the mass in grams of one
centimeter length of string. The equation gives the exact pitch of a
string or the change in pitch due to a change in length, mass, or tension.
If the length L is doubled the frequency is halved, i.e., the pitch is
lowered one octave. If m is increased n decreases, and if the tension F
is increased n increases. The formula shows that to double the fre-
quency by tightening a string the tension must be increased fourfold.

371

fundamental

1st overtone

Znd overtone,

3rd. overtone,

5th overtone

Fig. 16B — Vibration modes for strings of musical instruments.

16.2. Harmonics and Overtones. When a professional violinist
plays ''in harmonics" he touches the strings lightly at various points
and sets each one vibrating in two or more segments as shown in
Fig. 16B. If a string is touched at the center a node is formed at that
point and the vibration frequency, as shown by Eq. (16^/), becomes
just double that of the fundamental. If the string is touched lightly
at a point just one-third the distance from the end it will vibrate in
three sections and have a frequency three times that of the fundamental.
These higher vibration modes as shown in the figures, which always
have frequencies equal to whole number multiples of the fundamental
frequency ;;, are called overtones.

It is a simple matter to set a string vibrating with its fundamental

214

Musical Instruments and Scales

frequency and several overtones simultaneously. This is accomplished
by plucking or bowing the string vigorously. To illustrate this, a dia-
gram of a string vibrating with its fundamental and first overtone is
shown in Fig. 16C. As the string vibrates with a node at the center
and a frequency 2n, it also moves up and down as a whole with the
fundamental frequency n and a node at each end.

It should be pointed out that a string set into vibration with nodes
and loops is but an example of standing waves, see Figs. 14K and 14L.
Vibrations produced at one
end of a string send a con-
tinuous train of waves along
the string to be reflected back

from the other end. Th>s is ^'=' •'^7/ta^;ttLn,:l;eou'st'"^°"'
true not only for transverse

waves but for longitudinal or torsional waves as well. Standing waves
of the latter two types can be demonstrated by stroking or twisting one
end of the string of a sonometer or violin with a rosined cloth.

16.3. Wind Instruments. Musical instruments often classified
as "wind instruments" are usually divided into two subclasses, "wood-
winds" and "brasses." Under the heading of wood-winds we find
such instruments as the ^ute, piccolo, clarinet, bass clarinet, saxophone,
bassoon, and contra bassoon, and under the brasses such instruments as
the French horn, cornet, trumpet, tenor trombone, bass trombone, and
tuba (or bombardon) .

In practically all wind instruments the source of sound is a vibrating
air column, set into and maintained in a state of vibration by one of
several different principles. In instruments like the saxophone, clari-
net, and bassoon, air is blown against a thin strip of wood called a
reed, setting it into vibration. In most of the brasses the musician's
lips are made to vibrate with certain required frequencies, while in
certain wood-winds like the flute and piccolo air is blown across the
sharp edge of an opening near one end of the instrument setting the
air into vibration.

The fundamental principles involved in the vibration of an air
column are demonstrated by means of an experiment shown in Fig. 16D.
A vibrating tuning fork acting as a source of sound waves is held over
the open end of several long hollow tubes. Traveling down the tube
with the velocity of sound in air, each train of sound waves is reflected
from the bottom back toward the top. If the tube is adjusted to the

215

proper length, standing waves will be set up and the air column will
resonate to the frequency of the tuning fork. In this experiment the
proper length of the tube for the closed pipes is obtained by slowly
pouring water into the cylinder and listening for the loudest response.
Experimentally, this occurs at several points as indicated by the first
three diagrams; the first resonance occurs at a distance of one and one-
quarter wave-lengths, the second at three-quarters of a wave-length,
and the third at one-quarter of a wave-length. The reason for these

(a)

(b-)

(c)

(d)

(e)

\ 1

\ '

\ I

V
A

1 1

I /

\ /

-I

;l \

' —

1
1

\ 1 >

—

3 C

~— j

B a

L
N

B &L

open pipes

closed Di

pipes

Fig. 16D — The column of air in a pipe will resonate to sound of a given pitch if the length
of the pipe is properly adjusted.

odd fractions is that only a node can form at the closed end of a pipe
and a loop at an open end. This is true of all wind instruments.

For open pipes a loop forms at both ends with one or more nodes
in between. The first five pipes in Fig. 16D are shown responding to a
tuning fork of the same frequency. The sixth pipe, diagram (f), is
the same length as (d) but is responding to a fork of twice the fre-
quency of the others. This note is one octave higher in pitch. In
other words, a pipe of given length can be made to resonate to various
frequencies. Closed pipe (a), for example, will respond to other
forks whose waves are of the right length to form a node at the bottom,
a loop at the top and any number of nodes in between.

The existence of standing waves in a resonating air column may be
demonstrated by a long hollow tube filled with illuminating gas as
shown in Fig. 16E. Entering through an adjustable plunger at the left
the gas escapes through tiny holes spaced at regular intervals in a row

216

Musical Instruments and Scales

along the top. Sound waves from an organ pipe enter the gas column
by setting into vibration a thin rubber sheet stretched over the right-
hand end. When resonance is attained by sliding the plunger to the
correct position, the small gas flames will appear as shown. Where
the nodes occur in the vibrating gas column the air molecules are not
moving, see Fig. 14L (b) ; at these points the pressure is high and the
flames are tallest. Half way between are the loops; regions where the
molecules vibrate back and forth with large amplitudes, and the flames
are low. Bernoulli's principle is chiefly responsible for the pressure

>., organ pipe

gas flames

illuminatlnq ocls \

airblast

Fig. 16E — Standing waves in a long tube containing illuminating gas.

difl^erences, see ^tc. 10.8, for where the velocity of the molecules is
high the pressure is low, and where the velocity is low the pressure
is high.

The various notes produced by most wind instruments are brought
about by varying the length of the vibrating air column. This is illus-
trated by the organ pipes in Fig. 16F. The longer the air column the
lower the frequency or pitch of the note. In a regular concert organ
the pipes vary in length from about six inches for the highest note to
almost sixteen feet for the lowest. For the middle octave of the musical
scale the open-ended pipes vary from two feet for middlt C to one
foot for O- one octave higher. In the wood-winds like the flute the
length of the column is varied by openings in the side of the instru-
ment and in many of the brasses like the trumpet, by means of valves.
A valve is a piston which on being pressed down throws in an addi-
tional length of tube.

The frequency of a vibrating air column is given by the following
formula,

where L is the length of the air column, /C is a number representing
the compressibility of the gas, p is the pressure of the gas, and d is its

217

l-A-I-;^;-l 1- 1---1--

Ce)

Fig. 14L — Illustrating standing waves as they are produced with (a) the longitudinal
waves of a spring, (b) the longitudinal waves of sound in the air, and (d) the transverse
waves of a rope, (r) and (e) indicate the direction of vibration at the loops.

218

Musical Instruments and Scales

density. The function of each factor in this equation has been verified
by numerous experiments. The effect of the length L is illustrated in
Fig. 16F. To lower the frequency to half -value the length must be
doubled. The effect of the density of a gas on the pitch of a note may
be demonstrated by a very interesting experiment with the human

Do ^^

Fig. 16F — Organ pipes arranged In a musical scale. The longer the pipe the lower is
its fundamental frequency and pitch. The vibrating air column of the flute is terminated
at various points by openings along the tube.

voice. Voice sounds originate in the vibrations of the vocal cords in
the larynx. The pitch of this source of vibration is controlled by mus-
cular tension on the cords, while the quality is determined by the size
and shape of the throat and mouth cavities. If a gas lighter than air
is breathed into the lungs and vocal cavities, the above equation shows
that the voice should have a higher pitch. The demonstration can be
best and most safely performed by breathing helium gas, whose effect
is to raise the voice about two and one-half octaves. The experiment
must be performed to be fully appreciated.

16.4. Edge Tones. When wind or a blast of air encounters a
small obstacle, little whirlwinds are formed in the air stream behind
the obstacle. This is illustrated by the cross-section of a flue organ
pipe in Fig. 16G. Whether the obstacle is long, or a small round
object, the whirlwinds are formed alternately on the two sides as shown.
The air stream at B waves back and forth, sending a pulse of air first
up one side and then the other. Although the wind blows through
the opening A a.s a. continuous stream, the separate whirlwinds going
up each side of the obstacle become periodic shocks to the surrounding
air. Coming at perfectly regular intervals these pulses give rise to a

219

ViJ^^-V*

musical note often described as the whistling of the
wind. These notes are called "edge tones."

The number of whirlwinds formed per second,
and therefore the pitch of the edge tone, increases flue

with the wind velocity. When the wind howls
through the trees the pitch of the note rises and
falls, its frequency at any time denoting the velocity ; . . v^gvl
of the wind. For a given wind velocity smaller
objects g\s^ rise to higher pitched notes than large
objects. A fine stretched wire or rubber band when
placed in an open window or in the wind will be set
into vibration and giv^ out a musical note. Each
whirlwind shock to the air reacts on the obstacle
(the wire or rubber band) , pushing it first to one
side and then the other. These are the pushes that
cause the reed of a musical instrument to vibrate
and the rope of a flagpole to flap periodically in the
breeze, while the waving of the flag at the top of a
pole shows the whirlwinds that follow each other
along each side.

These motions are all "forced vibrations" in that '""^

they are forced by the wind. A stretched string or ^^^- i^G— ^ ""^^y

"'..•' . o stream or air blown

the air column in an organ pipe has its own natural across the lip of an
frequency of vibration which may or may not coin- ^^f>^, . pJP^ . ^"^ , "P

.,.,, ^ f . , T/-1 whirlwinds along both

cide with the frequency of the edge tone. If they do sides of the partition,
coincide, resonance will occur, the string or air
column will vibrate with a large amplitude, and a loud sound will result.
If the edge tone has a diff^erent frequency than the fundamental of the
string, or air column, vibrations will be set up but not as intensely as
before. If the frequency of the edge tone of an organ pipe, for example,
becomes double that of the fundamental, and this can be obtained by a
stronger blast of air, the pipe will resonate to double its fundamental
frequency and give out a note one octave higher.

16.5. Vibrating Rods. If a number of small sticks are dropped
upon the floor the sound that is heard is described as a noise. If one
stick alone is dropped one would also describe the sound as a noise,
unless, of course, a set of sticks of varying lengths are arranged in
order of length and each one dropped in its order. If this is done, one
notices that each stick gives rise to a rather delinite musical note and
the set of sticks to a musical scale. The use of vibrating rods in the
design of a musical instrument is to be found in the xylophone^ the
inarhnha, and the tviangle. Standing waves in a rod, like those in a

220

Musical instruments and Scales

stretched string, may be any one of three different kinds, transverse,
longitudinal, and torsional. Only the first two of these modes of vi-
bration will be treated here.

Transverse waves in a rod are usually set up by supporting the rod
at points near each end and striking it a blow at or near the center. As

Fig. 16H — The bars of the marimba or xylophone vibrate transversely with nodes near

each end.

illustrated in Fig. l6H(a) the center and ends of the rod move up and
down, forming nodes at the two supports. Like a stretched string of
a musical instrument, the shorter the rod the higher is its pitch, and
the longer and heavier the rod the lower is its frequency of vibration
and pitch.

The xylophone is a musical instrument based upon the transverse
vibrations of wooden rods of different lengths. Mounted as shown in
Fig. l6H(b) the longer rods produce the low notes and the shorter
ones the higher notes. The marimba is essentially a xylophone with
a long, straight hollow tube suspended vertically under each rod. Each
tube is cut to such a length that the enclosed air column will resonate
to the sound waves sent out by the rod directly above. Each resonator
tube, being open at both ends, forms a node at its center.

Longitudinal vibrations in a rod may be set up by clamping a rod
at one end or near the center and stroking it with a rosined cloth.
Clamped in the middle as sliown in Fig. 161 the free ends of the rod
move back and forth while the middle is held motionless, maintaining
a node at that point. Since the vibrations are too small to be seen
with the eye a small ivory ball is suspended near the end as shown.
The bouncing of this ball is indicative of the strong longitudinal vi-
brations. This type of vibra-

node jt-.

tion in a rod is not used in
musical instruments.

16.6. Vibrating Plates.
Although the drum or the
cymbals should hardly be
called musical instruments

\>mmmm

m//m^m/Jmmm//m'^Mm/mym ,■! , ■

■ M

Fig.

161 — Diagram of a rod vibrating longitu-
dinally with a node at the center.

221

they are classified as such and made use of in nearly all large orchestras
and bands. The noise given out by a vibrating drumhead or cymbal
plate is in general due to the high intensity of certain characteristic
overtones. These overtones in turn are due to the very complicated
modes of vibration of the source.

Cymbals consist of two thin metal disks with handles at the centers.
Upon being struck together their edges are set into vibration with a
clang. A drumhead, on the other hand, is a stretched membrane of

Fig. 16J — Chladni's sand figures showing the nodes and loops of (a) a vibrating drum-
head (clamped at the edge) and (b) a vibrating cymbal plate (clamped at the center).

leather held tight at the periphery and is set into vibration by being
struck a blow at or near the center.

To illustrate the complexity of the vibrations of a circular plate,
two typical sand patterns are shown in Fig. 16J. The sand pattern
method of studying the motions of plates was invented in the 18th
century by Chladni, a German physicist. A thin circular metal plate
is clamped at the center C and sand sprinkled over the top surface.
Then while touching the rim of the plate at two points Ni and N2 a
violin bow is drawn down over the edge at a point L. Nodes are
formed at the stationary points Ni and N2 and loops in the regions of
Li and L2. The grains of sand bounce away from the loops and into
the nodes, the regions of no motion. At one instant the regions marked
with a -|- sign all move up, while the regions marked with a — sign
all move down. Half a vibration later the -|- regions are moving
down and the — regions up. Such diagrams are called Chladni's sand
figures.

With cymbal plates held tightly at the center by means of handles
a node is always formed there, and loops are always formed at the
periphery. With a drumhead, on the other hand, the periphery is
always a node and the center is sometimes but not always a loop.

16.7. Bells. In some respects a bell is like a cymbal plate, for
when it is struck a blow by the clapper, the rim in particular is set

222

Musical Instruments and Scales

Fig. 16K — Experiment illustrating that the rim of a bell or glass vibrates with nodes

and loops.

vibrating with nodes and loops distributed in a symmetrical pattern
over the whole surface. The vibration of the rim is illustrated by a
diagram in Fig. l6K(a) and by an experiment in diagram (b). Small
cork balls are suspended by threads around and just touching the out-
side rim of a large glass bowl. A violin bow drawn across the edge
of the bowl will set the rim into vibration with nodes at some points
and loops at others. The nodes are always even in number just as they
are in cymbal plates and drumheads, and alternate loops move in while
the others move out.

Strictly speaking, a bell is not a very musical instrument. This is
due to the very complex vibrations of the bell surface giving rise to so
many loud overtones. Some of these overtones harmonize with the
fundamental while others are discordant.

16.8. The Musical Scale. The musical scale is based upon the
relative frequencies of different sound waves. The frequencies are so
chosen that they produce the greatest am.ount of harmony. Two notes
are said to be harmonious if they are pleasant to hear. If they are not
pleasant to hear they are discordant.

The general form of the musical scale is illustrated by the symbols,
letters, terms, and simple fractions given in Fig. 16L.

-O-
C

-Q-

-e-

-Q-

tomc second ^ M^^^ m SI S^A ^^^^^^

^ ^ ^ ^

Fig. 16L — Diagram giving the names, and fractional ratios of the frequencies, of the
different tone intervals on the diatonic musical scale.

223

The numbers indicate that whatever the frequency of the toriic C,
the frequency of the octave C^ will be twice as great, that G will be
three halves as great, F four thirds as great, etc. These fractions below
each note are proportional to their frequencies in whatever octave of
the musical scale the notes are located.

The musical pitch of an orchestral scale is usually determined by
specifying the frequency of the A string of the first violin, although
sometimes it is given by 7niddle C on the piano. In the history of
modern music the standard of pitch has varied so widely and changed
so frequently that no set pitch can universally be called standard.*
For many scientific purposes the A string of the violin is tuned to a
frequency of 440 vib/sec, while in a few cases the slightly different
scale of 256 vib/sec is used for the tonic, sometimes called middle C.

16.9. The Diatonic Scale. The middle octave of the diatonic
musical scale is given in Fig. 16M assuming as a standard of pitch
A = 440. The vocal notes usually sung in practicing music are given
in the second row. The ratio numbers are the smallest whole numbers
proportional to the scale ratios and to the actual frequencies.

The tone ratios given at the bottom of the scale indicate the ratio
between the frequencies of two consecutive notes. Major tones have
a ratio of 8 : 9, minor tones a ratio of 9 : 10, and diatonic semitones a
ratio 15 : 16. (The major and minor tones on a piano are called
whole tones and the semitones are called half tones.)

Other tone intervals of interest to the musician are the following:

Interval Frequency Ratio Examples

Octave 1:2 CO, DD', EE^

Fifth 2:3 CG, EB, GD^

Fourth 3:4 CF, EA, GC*

Major third 4:5 CE, FA, GB

Minor third 5:6 EG, AC^

Major sixth 3:5 CA, DB, GE'

Minor sixth 5:8 EC, AF'

A scientific study of musical notes and tone intervals shows that
harmony is based upon the frequency ratios between notes. The
smaller the whole numbers giving the ratio between the frequencies of

* For a brief historical discussion of normal standards of pitch the student
is referred to the book 'The Science of Musical Sounds" by D. C. Miller. For
other treatments of the science of music see "Sound" by Capstick, "Science and
Music" by James Jeans, and "Sound and Music" by J. A. Zahn.

224

Musical Instruments and Scales

scale notes
vocal notes

ratio numbers

frequencies

scale ratios
tone ratios

<a" i!

Z64

Z97

330

35Z

396

44C

495

5za

594

^/3

3 :9 9: JO J5:}6 3-9 9-/0 6:9 /5-/6 6:9

Fig. 16M — The diatonic musical scale illustrated by the middle octave with C as the
tonic and A = 440 as the standard pitch.

two notes the more harmonious, or consonant, is the resultant. Under
this definition of harmony the octave, with a frequenq^ ratio of 1 : 2,
is the most harmonious. Next in Hne comes the fifth with a ratio 2 : 3,
followed by the fourth with 3 : 4, etc. The larger the whole numbers
the more discordant, or dissonant, is the interval.

Helmholtz was the first to giwQ a physical explanation of the various
degrees of consonance and harmony of these different intervals. It is
based in part upon the beat notes produced by two notes of the interval.

As shown by Eq. (15<^) the beat frequency between two notes is
equal to their frequency difference. Consider, for example, the two
notes C and G of the middle octave in Fig. 16M. Having frequencies
of 264 and 396, the beat frequency is the difference, or 132. This is a
frequency fast enough to be heard by the ear as a separate note, and in
pitch is one octave below middle C. Thus in sounding the fifth, C and
G, three harmonious notes are heard, 132, 264, 396. They are har-
monious because they have ratios given by the smallest whole numbers

1:2:3.

Harmonious triads or chords are formed by three separate notes
each of which forms a harmonious interval with the other two, Avhile
the highest and lowest notes are less than an octave apart. Since there
are but six such triads they are shown below.

Harmonic Triads or Chords

Frequency Ratio Example

Major third followed by minor third 4

" fourth 3

Minor third " " major third 5

Minor third " " fourth 5

Fourth " " major third 4

Fourth " " minor third 3

5 :6

4 : 5
6, 4
6, 3
5, 3
4, 5

225

Consider the beat notes or di§erence tones between the various pairs
of notes in the second triad above. The notes themselves have fre-
quencies C = 264, F=352, and /4 = 440. The difference tones
F-C^88, ^-F^88, and A-C^^ll6. Being exactly one and two
octaves below C, one of the notes of the triad, they are in harmony
with each other. Grouping the first two beat frequencies as a single
note, all the frequencies heard by the ear have the frequencies 88,
176, 264, 352, and 440. The frequency ratios of these notes are
1:2:3:4:5, the first five positive whole numbers.

16.10. The Chromatic Scale. Contrary to the belief of many
people the sharp of one note and the jiat of the next higher major or
minor tone are not of the same pitch. The reason for this false im-
pression is that on the piano the black keys represent a compromise.
The piano is not tuned to the diatonic scale but to an equal tempered
scale. Experiments with eminent musicians, and particularly violinists,
have shown that they play in what is called pure intonation, that is, to
a chromatic scale and not according to equal temperament as will be
described in the next section.

On the chromatic scale of the musician the ratio between the fre-
quency of one note and the frequency of its sharp or flat is 25 : 24.
This ratio is just the difi^erence between a diatonic semitone and a minor

tone. I.e.

rio

25,

^4. The actual frequencies of the various

sharps and flats for the middle octave of the chromatic scale, based
upon A =z 440, are shown above in Fig. 16N. C* for example has

C^D''

p^^b

^ Z7S 28^.7 ^309.^316.8

3667 3B0.Z

F _L_L 6

412.5 422.4
I I,

4583 47SZ

A jL_i_5

major ton e rrimor ton e

semi-
tone

major tone

minor tone

major tone

semi-
tone

264-

297

330 352

396

440

49S 5Z8

whole ' tone

whole\tone

half

whole 'tone

whole •tone

tvhole\tone

half

Z61.6 I -293 7 I 329.6 349 2 j 39Z | 440 \ 493 9

277.2 311. 1 370 4153 466.1 S23.2

^^nb n*^ch c*/:b /c*>iA Ait' oh

C^D'

D^E'-

F'^G'

G'^A'

A^B'

Fig. 16N — Scale diagrams showing the diatonic and chromatic scale above and the equal

tempered scale below.

a frequency of 275 whereas D^ is 285.1. This is a difference of
10 vib/sec, an interval easily recognized at this pitch by most every-

226

Musical Instruments and Scales

370.1 415.5 466 A

ihlack

white
keys

Z6I.6 293.7 529.6 349.Z 391 440 493.9 5Z3.Z

Fig. 160 — The equal tempered scale of the
piano illustrating the frequencies of the middle
octave based upon A = 440 as the standard pitch.

one. (The sharps and flats of the semitone intervals are not shown.)

16.11. The Equal Tempered Scale. The white keys of the
piano are not tuned to the exact frequency ratios of the diatonic scale;

they are tuned to an equal
tempered scale. Each octave
is divided into twelve equal
ratio intervals as illustrated
below in Fig. 16N. The
whole tone and half tone in-
tervals shown represent the
white keys of the piano, as
indicated in Fig. 160, and
the sharps and flats represent
the black keys. Including
the black keys, all twelve
tone intervals in every octave
are exactly the same. The frequency of any note in the equal tempered
scale turns out to be 6 percent higher than the one preceding it. More
accurately, the frequency of any one note multiplied by the decimal
1.05946 gives the frequency of the note one-half tone higher. For
example, A = 440 multiplied by 1.05946 gives A^ or B^ as 466.1 vib/
sec. Similarly, 466.1 X 1-05946 gives 493.9.

The reason for tuning the piano to an equal tempered scale is to
enable the pianist to play in any key and yet stay within a given pitch
range. In so doing, any given composition can be played within the
range of a given person's voice. In other words, any single note can
be taken as the tonic of the musical scale.

Although the notes of the piano are not quite as harmonious as if
they were tuned to a diatonic scale, they are not far out of tune. This
can be seen by a comparison of the actual frequencies of the notes of
the two scales in Fig. 16N. The maximum diff^erences amount to about
1 percent, which for many people is not noticeable, particularly in a mod-
ern dance orchestra. To the average musician, however, the difl^erence is
too great to be tolerated, and this is the reason most symphony orchestras
do not include a piano. The orchestral instruments are usually tuned
to the A string of the first violin and played according to the chromatic
and diatonic scale.

16.12. Quality of Musical Notes. Although two musical notes
have the same pitch and intensity they may difl^er widely in tone quality.
Tone quality is determined by the number and intensity of the over-
tones present. This is illustrated by an examination either of the vi-

227

brating source or of the sound waves emerging from the source. There
are numerous experimental methods by which this is accomplished.

A relatively convenient and simple demonstration is given in
Fig. 16P, where the vibrating source of sound is a stretched piano
string. Light from an arc lamp is passed over the central section of
the string which, except for a small vertical slot, is masked by a screen.
As the string vibrates up and down the only visible image of the string
is a very short section as shown at the right, and this appears blurred.
By reflecting the light in a rotating mirror the section of wire draws
out a wave 1^ on a distant screen.

If a string is made to vibrate with its fundamental alone, its own
motion or that of the emitted sound waves have the form shown in
diagram (a) of Fig. 16Q. If it vibrates in two segments or six seg-
ments (see Fig. 16B) the wave forms will be like those in diagrams (b)
and (c) respectively. Should the string be set vibrating with its fun-
damental and first overtone simultaneously, the wave form will appear
something like diagram (d). This curve is the sum of (a) and (b)
and is obtained graphically by adding the displacement of correspond-
ing points. If in addition to the fundamental a string vibrates with

rotaiing
mirror

Fig. 16P — Diagram of an experiment demonstrating the vibratory motion of a stretched

string.

the first and fifth overtones the wave will look like diagram (e) . This
is like diagram (d) with the fifth overtone added to it.

It is difficult to make a string vibrate with its fundamental alone.
As a rule there are many overtones present. Some of these overtones
harmonize with the fundamental and some do not. Those which har-
monize are called harmonic overtones, and those which do not are
called anharmomc overtones. If middle C = 264 is sounded with its

228

Musical Instruments and Scales

fundamental

1st overtone
5 th overtone

(a)f(b)

(a)Hb)t(c')

Fig. 16Q — Illustrating the form of the sound waves resulting from the addition of over-
tones to the fundamental.

first eight overtones, they will have 2, 3, 4, 5, 6, 7, and 8 times
264 vib/sec. These on the diatonic scale will correspond to notes
Ci, Gi, C2, £2^ G^, X, and C\ All of these except X, the sixth over-
tone, belongs to some harmonic triad. This sixth overtone is anhar-
monic and should be suppressed. In a piano this is accomplished by
striking the string one-seventh of its length from one end, thus pre-
venting a node at that point.

16.13. The Ranges of Musical Instruments. The various octaves
above the middle of the musical scale are often labeled with numerical
superscripts as already illustrated, while the octaves below the middle
are labeled with numerical subscripts.

The top curve in Fig. 16Q is typical of the sound wave from a
tuning fork, whereas the lower one is more like that from a violin.
The strings of a violin are tuned to intervals of the fifth, G\ = 198,
D = 297, A = 440, and £i = 660. The various notes of the musical
scale are obtained by touching a string at various points, thus shorten-
ing the section which vibrates. The lowest note reached is with the
untouched Gi string and the highest notes by the E^ string fingered
about two-thirds of the way up toward the bridge. This gives the
violin a playing range, or compass, of 3^ octaves, from Gi = 198 to
<:3 = 2112.

The viola is slightly larger in size than the violin but has the same
shape and is played with slightly lower pitch and more sombre tone
quality. Reaching from Ci to C^, it has a range of three octaves.

The cello is a light bass violin which rests on the floor, is played
with a bow, has four strings pitched one octave lower than the viola,
^2, G2, Di, and Ai, and has a heavy rich tone quality. The double
bass is the largest of the violin family, rests on the floor and is played

229

with a bow. The strings are tuned to two octaves below the viola and
one octave below the cello. In modern dance orchestras the bow is
often discarded and the strings are plucked with the fingers.

Of the wood-wind instruments the jlute is nearest to the human
voice. It consists essentially (see Fig. 16R) of a straight narrow tube
about 2 feet long and is played by blowing air from between the lips
across a small hole near the closed end. The openings along the tube
are for the purpose of terminating the vibrating air column at various
points. See Fig. 16F. With all holes closed a loop forms at both
ends with a node in the middle. See Fig. l6D(d). As each hole is
opened one after the other, starting from the open end, the vibrating
air column with a loop at the opening grows shorter and shorter, giving
out higher and higher notes. To play the scale one octave higher, one
blows harder to increase the frequency of the edge tones and set the
air column vibrating, as in Fig. 16D(e), with three loops and two
nodes. Starting at middle C the flute can be extended in pitch for two
octaves, up to C-. The piccolo is a small flute, 1 foot long, and sings
one octave higher. The tone is shrill and piercing and the compass
iis Ci to A^.

The oboe is a melodic double-reed keyed instrument, straight and
about 2 feet long. It has a reedy yet beautiful quality, and starting at
Bi has a range of about two octaves. The clarinet, sometimes called

Fig. 16R — Musical instruments. Brasses: {a) horn, {b) bugle, {c) cornet, {d) trombone.
Wood-winds: {e) flute, (/) oboe, and {g) clarinet.

230

Musical Instruments and Scales

the violin of the mihtary band (see Fig. 16R), is a single-reed instru-
ment about 3 feet long. It has a range of over three octaves starting
at £i. The bass clarinet is larger than the clarinet, but has the same
shape and plays one octave lower in pitch.

The bassoon is a bass double-reed keyed instrument about 4 feet
long. The tone is nasal and the range is about two octaves starting
at Si's.

The horn is a coiled brass tube about 12 feet in length (see Fig. 16R)
but interchangeable according to the number of crooks used. It has
a soft mellow tone and starting at C2 has a range of three octaves.
The cornet, not usually used in symphony orchestras (see Fig. I6R), is
a coiled conical tube about 41/2 feet long with three valves. It has a
mellow tone starting at middle C and extends for two octaves. The
trumpet is a brass instrument having a similar shape as, and slightly
larger than, the cornet. Having three valves, it extends to two octaves
above middle C. The purpose of the valves is to vary the length of
the vibrating air column.

The trombone is a brass instrument played with a slide, is a conical
tube about 9 feet long when straightened (see Fig. 16R), and has a
tone range from F2 to C^. Since the length of the vibrating air column
can be varied at will it is easily played to the chromatic scale. The
tuba is the largest of the saxhorns and has a range from F3 to Fi.

Fig. 16S — Diagram of a phonodeik. An instrument for observing the form of sound waves.

The bugle (see Fig. I6R) is not capable of playing to the musical
scale but sounds only certain notes. These notes are the harmonic
overtones of a fundamental frequency of about GG vibrations per sec-
ond. With a loop at the mouthpiece, a node in the center, and a loop
at the flared end, this requires a tube 8 feet long. The second, third,
fourth, and fifth overtones have the frequencies 66 X 3 = 198, GG 'X
4= 264, GGX "> = 330, and GGX ^^^ 396 corresponding to d,

231

C, E, and G, the notes of the bugle. By making the lips vibrate to
near these frequencies the air column is set resonating with 3, 4, 5, or
6 nodes between the t^o open ends.

16.14. The Phonodeik. The phonodeik is an instrument designed
by D. C. Miller for photographing the minute details and wave forms
of all audible sounds. The instrument consists of a sensitive diaphragm
D (see Fig. 16S), against which the sound waves to be studied are
allowed to fall. As the diaphragm vibrates back and forth under the
impulses of the sound waves the thread T winds and unwinds on the
spindle S, turning the tiny mirror AI up and down. A beam of light
from an arc lamp A and lens L is reflected from this mirror onto a ro-
tating mirror RAi. As RAi spins around the light sweeps across a dis-
tant screen, tracing out the sound wave. The trace may be either pho-
tographed or observed directly on the screen. Persistence of vision
enables the whole curve to be seen for a fraction of a second.

Several sound curves photographed by Miller are redrawn in
Fig. 16T. In every graph except the one of the piano, the sound is

/lute

voice vowel 'a.

Aoyixi/w

bass voice

piano

Fig. 16T — Various types of sound waves in music as observed with a phonodeik or cathode

ray oscillograph.

maintained at the same frequency so that the form of each wave, no
matter how complex, is repeated the same number of times. The
tuning fork is the one instrument which is readily set vibrating with
its fundamental alone and none of its harmonics. Although each
different instrument may sound out with the same note, that is, the
same fundamental, the various overtones present and their relative
loudness determines the quality of the note identified with that
instrument.

232

The four members of the violin family have changed very
little In hundreds of years. Recently, a group of musi-
cians and scientists have constructed a "new" string
family.

16 Founding a Family of Fiddles

Carleen M. Hutchins

An article from Physics Today, 1967.

New measmement techniques combined with recent acoustics research enable
us to make vioUn-type instruments in all frequency ranges with the properties built
into the vioHn itself by the masters of three centuries ago. Thus for the first time
we have a whole family of instruments made according to a consistent acoustical
theory. Beyond a doubt they are musically successful

by Carleen Maley Hutchins

For three or folti centuries string
quartets as well as orchestras both
large and small, ha\e used violins,
violas, cellos and contrabasses of clas-
sical design. These wooden instru-
ments were brought to near perfec-
tion by violin makers of the 17th and
18th centuries. Only recendy, though,
has testing equipment been good
enough to find out just how they work,
and only recently have scientific meth-
ods of manufactiu-e been good enough
to produce consistently instruments
with the qualities one wants to design
into them. Now, for the first time,
we have eight instruments of the \ iolih
family constructed on principles of
proper resonance for desired tone
quality. They represent the first suc-
cessful application of a consistent
acoustical theorv- to a whole family of
musical instruments.

The idea for such a gamut of violins
is not new. It can be found in Mi-
chael Praetorius's Syntagma Musicum
published in 1619. But incomplete
understanding and technological ob-

stacles have stood in the way of practi-
cal accomplishment. That we can
now routinely make fine violins in a
variety of frequency ranges is the re-
siJt of a fortuitous combination:
violin acoustics research— showing a
resurgence after a lapse of 100 years—
and the new testing equipment capa-
ble of responding to the sensitivities of
wooden instruments.

As is shown in figure 1, oiu new in-
struments are tuned in alternate inter-
vals of a musical fourth and fifth over
the range of the piano keyboard.
Moreover each one has its two main
resonances within a semitone of the
tuning of its middle strings. The re-
sult seems beyond a doubt successful
musically. Over and over again we
hear the comment, "One must hear the
new instruments to believe such
sounds are possible from strings."

Catgut Acoustical Society
Groundwork in the scientific investiga-
tion of the violin was laid bv such men

233

as Marin Mersenne (1636), Ernst
Chladni (1802), Felix Savart (1819)
and Hemiann L. F. Helmholtz (1860).
Savart, who can rightly be considered
the grandfather of violin research,
used many ingenious devices to ex-
plore the vibrational characteristics of
the violin. But he was unable to gain
sufficient knowledge of its complicat-
ed resonances to apply his ideas suc-
cessfully to development and construc-
tion of new instruments. Recent re-
search that has led to our new fiddle
family is largely the work of Hermann
Backhaus, Herman Meinel, Gioacchino
Pasqualini, Ernst Rohloff, Werner Lot-
ternioser and Frieder Eggers in Eu-
rope and of the late Frederick A.
Saunders, John C. Schelleng, William
Harvey Fletcher and myself in the
United States.

Saunders, widely known for his
work on Russell-Saunders coupling, pi-
oneered violin research on this side of
the Atlantic. He was a former chair-
man of the physics department of Har-
vaid Uni\ersity, a fellow of the Na-
tional Academy of Sciences and presi-
dent of the Acoustical Society of
America. In his work on violin acous-
tics, Saunders gradually became as-
sociated with colleagues who were
highly competent in various scientific
and musical disciplines. These associ-
ates greatly furthered the development
of his work and contributed valuable
technical knowledge, but they had lit-
tle time for experimentation. Some
were skillful musicians living under

the pressure of heavy teaching and
concert schedules. Nevertheless some
were able to find time for the testing,
designing and craftsmanship needed
in the development of experimental in-
struments. In 1963 about 30 persons
associated with Saunders in this proj-
ect labeled themselves the "Catgut
Acoustical Society." This infonnal so-
ciety now has more than 100 members
(see box on page 26), publishes a
semiannual newsletter and holds one
or two meetings each year. Among its
members are acousticians, physicists,
chemists, engineers, instrument mak-
ers, composers, performing musicians,
musicologists, patrons and others who
believe that insufficient attention has
been paid to the inherent potentialities
of bowed string instruments. They
are making a coordinated effort to dis-
cover and develop these potentialities
and are encouraged that many mem-
bers of the violin fraternity share their
aims.

Among other accomplishments of
our Catgut Acoustical Society is a con-
cert played at Harvard last summer
during the meeting of the Acoustical
Society of America. It was dedicated
to Saunders and the instruments were
our eight new fiddles, which are the
outgrowth of research he began. I
write about the concert and about the
instruments as a member of the society
and as one who worked with Saunders
from 1948 until his death in 1963.
My activities include reconciliation of
the wisdom of experienced musicians

In addition to nur- W

turing her fiddle \

family, the author

shows interest in

children. .After Krul-

uatiiig from Come

she taiiKhl for IS

years in Now Yoik

schools, acquiring an

M.\ from New York

l'ni\crsity nican-

whilc. She also ,u- ^ ,#10^ i

(piiii'il a clicmist hits- Ij ^''jT

h.uul and two iliil- Ij ^^^M_

(Iren, all of whom 1 '^HV^

live MnnU'laiv. f^ ^^^u

I -Mm

and violin makers, coordination of
much technical information from
widely separated sources, and design,
construction and testing of experimen-
tal instruments. In 1937 Saunders re-
portedi in the Journal of the Acousti-
cal Society of America what later
proved to be basic to the development
of the new violin family, namely the
position of the main body resonance
as well as the main cavity resonance
in a series of excellent violins. (The
main body resonance is the lowest
fundamental resonance of the wood
structure; the cavity resonance is that
of the air in the instrument cavity.)
But the necessary knowledge of liow
to place these resonances with any de-
gree of predictability in instruments of
good tone (jualit)' was not evolved and
reported until 1960.2 The tonal effect
of this placement of the two main
resonances for each instrument and the
necessar>' scaling theory was not re-
ported until 1962.3

Between 1950 and 1958 Saunders
and I undertook a long series of exper-
iments to test various features of violin
construction one at a time. We deter-
mined effect of variations in length,
shape and placement of the f holes,
position of the bass bar and sound
post, significance of the inlay of pur-
fling around the edges of top and back
plates and frequency of the ca\ity res-
onance as a function of rib height and
f hole areas (see figure 2). Because
many of these experiments needed de-
finitive testing equipment not then
available, most of the results are still
unpublished in Saunders's notebooks.

One sobering conclusion we reached
was that with many alterations in such
features as size and shape of f holes,
position of the bass bar and sound
post, the best tonal qualities resulted
when conventional violin-making rules
were followed. In other words, the
early violin makers, working empirical-
ly b\' slow trial and error, had evoked
a s\stem that produced practically op-
timal relationships in \iolin construc-
tion.

In 1958. during a long series of ex-
periments to test the effect of moving
\i()lin and viola resonances up and
down scale, the composer in residence
at Bennington College, Henry Brant,
and the cellist. Sterling Hunkins, pro-
posed development of eight violin-
type instruments in a .scries of tunings

234

Founding a Family of Fiddles

D 293.7-

C 261.6

B 246.9

E 329.6

F 349.2

G 392

A
27.5

A
55

A
110

A
220

A
440

880 1760

FREQUENCY (cycles/sec)

iiiiiimmmiifniiimiiiniTff

TREBLE (6-D-AE)

lllllll1!ITimil!!!lllltHI!ll!

SOPRANO ((>G-D-A)

IIIIIIH!H!lllllt!lfllglTIIIIIT

MEZZO (6-D-A-E)

IIHHIf!ll!filirilllllfflltf

ALTO (C-G-D-A)

l!ll'll!!Hmi!l!IHIIHIiri[!

TENOR (G-D-AE)

lllll!l!H!ltl!ll!!lllll!!ll!!l!

BARITOFtf (C-G-D-A)

SMALL BASS (A-IW3-C)

mMUMBM

lllltltllfri!!ll!!lfflHII!lfim

A
3520

CONTRABASS (E-A-IVG)

NEW INSTRUMENT TUNING spans the piano range with eight fiddles that
range in size from 210-cm contrabass to a 27-cm treble. The conventional
violin is the mezzo of the new scries. Colored keys show tuning of new in-
struments and white dots that of conventional instruments. — FIG. 1

235

and sizes to cover substantially the
whole pitch range used in written
music; these instruments would start
with an oversize contrabass and go to
a tiny instrument tuned an octave
above the violin. Their request was
so closely related to our experimental
work that after half an hour's discus-

sion Saunders and I agreed that a seri-
ous attempt would be made to de-
velop the set. The main problem
would be to produce an instrument in
each of the eight frequency ranges
having the dynamics, the expressive
qualities and overall power that are
characteristic of the violin itself, in

contrast to the conventional viola, cello
and string bass.

Research and new fiddles
The problem of applying basic re-
search results to actual design and
construction of new instruments now
faced us. From the previous ten

Who's Who in Catgut Acoustics

Without cross fertilization of
ideas from experts in many
related disciplines our new
fiddle family could not have
evolved in the short period of
nine or ten years. No listing
of names and activities can
do justice to each one whose
thinking and skills have been
challenged and who has
given time, energy and
money. Their only reward
is sharing in the project.

The spirit of the group has
been likened to the informal
cooperation tha; flourished
among scientists in the 18th
century. In addition many
of the active experimenters
are themselves enthusiastic
string players so that a tech-
nical session is likely to end
with chamber-music playing.

In the following list I try to
include all those who have
helped along the way, listing
those who have been most
active first even though they
are not all members of CAS.
Some of the numerous
musicians are not actually
familiar with the new instru-
ments, but their comments
on earlier experimental mod-
els of conventional violins,
violas and cellos have provid-
ed musical insights and in-
formation necessary to the
new instruments.

Physicists. Basic re-
search and scaling for the
new instruments: Frederick
A. Saunders, John C. Schel-
leng and myself. Theory of

vibrations, elasticity, shear
and damping in the instru-
ments and their parts: Ar-
thur H. Benade, Frieder Eg-
gers, Roger Kerlin, Max V.
Mathews, Bernard W. Robin-
son, Robert H. Scanlan, John
C. Schelleng, Eugen J.
Skudrzyk, Thomas W. W.
Stewart, Sam Zaslavski.

Chemists. Effects of var-
nish and humidity on the in-
struments; varnish research:
Robert E. Fryxell, Morton A.
Hutchins, Louis M. Condax.

Architect. Basic design
and development of patterns
for the new violin family, and
maker of bows for them:
Maxwell Kimball.

Electronic engineers.

Norman Dooley, Francis L.
Fielding, Sterling W. Gor-
rill, A. Stuart Hegeman, Alvin
S. Hopping.

Translators. Mildred Al-
len, Edith L. R. Corliss, Don-
ald Fletcher.

Editors. Harriet M. Bart-
lett, Dennis Flanagan, Rob-
ert E. Fryxell, Mary L. Har-
bold, Martha Taylor, Alice
Torrey, Howard Van Sickle.

Photographers. Louis M.
Condax, Russell B. Kingman,
Douglas Ogawa, Peter N.
Pruyn, J. Kellum Smith.

Artist. Irving Geis.

Lawyers. Harvey W. Mor-
timer, J. Kellum Smith, Rob-
ert M. Vorsanger.

General consultants.

Alice T. Baker, Donald Engle,
Cushman Haagensen, Mary

W. Hinckley, Ellis Kellert,
Henry Allen Moe, Ethel and
William R. Scott.

Secretaries. Lorraine El
liott. Belle Magram.

Violin experts and makers
Karl A. Berger, Rene Morel
Simone F. Sacconi, Rembert
Wurlitzer, myself — and Vir
ginia Apgar, Armand Bartos
William W. Bishop, Donald L
Blatter, William Carboni
Louis M. Condax, Fred Dau
trich, Jean L. Dautrich, Louis
Dunham, Jay C. Freeman
Louis Grand, Jerry Juzek
Otto Kaplan, Gordon McDon
aid, William E. Slaby.

Violinists. Charles F. Aue
Broadus Erie, William Kroll
Sonya Monosoff, Helen Rice
Louis E. Zerbe — and Sam
uel Applebaum, Catherine
Drinker Bowen, Marjorie
Bram, Ernestine Briemeis
ter, Alan Branigan, Nicos
Cambourakis, Roy B. Cham
berlin Jr., Frank Clough
Louis M. Condax, Yoko Mat
suda Erie, Sterling Gorrill
Walter Grueninger, Ann Ha
worth, H. T. E. Hertzberg
Carol Lieberman, Max Man
del. Max V. Mathews, David
Montagu, Max Pollikoff, Ber
nard W. Robinson, Booker
Rowe, Frances Rowell, Rob
ert Rudie, Florence DuVal
Smith, Jay C. Rosenfeid.

Violists. Robert Courte,
Lilla Kalman, Maxwell Kim-
ball, David Mankovitz, Louise
Rood, Frederick A. Saunders
— and John A. Abbott, Alice

Schradieck Aue, Virginia
Apgar, Emil Bloch, Harold
Coletta, Helene Dautrich,
John D'Janni, Lillian Fuchs,
Raphael Hillyer, Henry
James, Boris Kroyt, Eugene
Lehner, Rustin Mcintosh,
John Montgomery, Elizabeth
Payne, Werner Rose, David
Schwartz, Emanuel Vardi,
Eunice Wheeler, Bernard Zas-
lav, Sam Zaslavski, myself.

Cellists. Robert Fryxell,
John C. Schelleng, India
Zerbe — and Charles F. Aue,
Joan Brockway, Roy B.
Chamberlin, Frank Church,
Elwood Culbreath, Oliver
Edel, Maurice Eisenberg,
George Finckel, Marie Gold-
man, Barbara Hendrian, Ar-
nold Kvam, Russell B. King-
man, Charles McCracken,
Stephen McGee, George
Ricci, Peter Rosenfeid, Mary
Lou Rylands, True Sackrison,
Mischa Schneider, Sanford
Schwartz, Joseph Stein, Mis-
cha Slatkin, Joseph Tekula.

Bassists. Julius Levine,
Alan Moore, Ronald Naspo,
David Walter — and Alvin
Brehm, John Castronovo,
Garry Karr, Stuart Sankey,
Charel Traeger, Howard Van
Sickle, Ellery Lewis Wilson.

Composers and conduc-
tors. Henry Brant — and
Marjorie Bram, Justin Con-
nolly, Herbert Haslam, Frank
Lewin, Marc Mostovoy, Har-
old Oliver, Quincy Porter,
Cornelia P. Rogers, Leopold
Stokcwski, Arnold M. Walter.

PHOTO BY J. KELLUM SMITH

REHEARSAL for a
concert with Henry
Brant conducting an
octet of fiddles.

236

Founding a Family of Fiddles

BRIDGE

years' experimentation, the following
four working guides were at hand:

1. location of the main body and
main cavity resonances of several
hundred conventional violins, violas
and cellos tested by Saunders and oth-
ers.i. 4-«

2. the desirable relation between
main resonances of free top and back
plates of a given instrument, devel-
oped from 400 tests on 35 violins and
violas during their construction, 2,10,11

3. knowledge of how to change
frequencies of main body and cavity
resonances within certain limits
(learned not only from many experi-
ments of altering plate thicknesses, rel-
ative plate tunings and enclosed air
volume but also from constru'^tion of
experimental instruments with varying
body lengths, plate archings and rib
heights) and of resultant resonance
placements and effects on tone quality
in the finished instruments,**-^^

4. observation that the main body

resonance of a completed violin or
viola is approximately seven semitones
above the average of the main free-
plate resonances, usually one in the
top and one in the back plate of
a given instrument.'- This obsen'a-
tion came from electronic plate test-
ing of free top and back plates of
45 violins and violas under construc-
tion. It should not be inferred that
the relation implies a shift of free-plate
resonances to those of the finished in-
strument. The change from two free
plates to a pair of plates coupled at
their edges through intricately con-
structed ribs and through an off-center
soundpost, the whole \mder varying
stresses and loading from fittings and
string tension, is far too complicated to
test directly or calculate."

What is good?

In developing the new instnmients our
main problem was finding a measura-
ble physical characteristic of the violin

INSTRUMENT PARTS, except for
scaling, have remained the same since
master makers brought the violin to
near perfection about three centuries
ago. — FIG. 2

itself that would set it apart from its
cousins, the viola, cello and contra-
bass. The search for this controlling
characteristic, unique to the violin,
led us through several hundred re-
sponse and loudness curves of violins,
violas and cellos. The picture was at
first confusing because many varia-
tions were found in the placement of
the two main resonances. However,
Saunders's tests on Jasha Heifetz's
Guamerius violin^^ showed the main-
body resonance was near the fre-
quency of the unstopped A 440-cycles-
per-second string and the main cavity
resonance at the unstopped D 294
string. Thus the two main resonances

237

of this instrument were near the fre-
quencies of its two unstopped middle
strings.

Ten violins, selected on the basis
that their two main resonances were
within a whole tone of their two open
middle strings, were found to be some
of the most musically desirable instru-
ments—Amatis, Stradivaris, Guar-
neris and several modem ones. In
marked contrast to these were all vi-
olas and cellos tested, which charac-
teristically had their main body and
cavity resonances three to four semi-
tones above the frequencies of their
two open middle strings although they
still had the same separation, approxi-
mately a musical fifth, between these
two main resonances.

We reasoned that the clue to our
problem might be this placement of
the two main resonances relative to
the tuning of the two open middle
strings. A search through many small
violins and cellos, as well as large and

small violas, showed enormous varia-
tion in the placement of these two res-
onances. We hoped to find some in-
strument in which even one of these
resonances would approximate what
we wanted for the new instruments.

In one quarter-size cello the body
resonance was right for viola tuning, D
294, but the cavity resonance was too
low at D 147. We bought this
chubby Uttle cello and reduced the
rib height nearly 4 in. (10 cm),
thereby raising the frequency of the
cavity resonance to the desired G 196.
When it was put back together, it
looked very thin and strange with ribs
only 1.5 in. (3.8 cm) high and a body
length of over 20 in. (51 cm), but
strung as a viola it had tone quality
satisfactory beyond expectations!

An experimental small viola that I
had made for Saunders proved to have
its two main resonances just a semi-
tone below the desired frequency for
violin tone range. When strung as a

1/ '^

2 3 4 S

WAVELENGTH (relativ»— log scale)

'«S

violin, this shallow, heavy-wooded in-
strument had amazing power and clar-
ity of tone throughout its range. It
sounded like a violin although the
quality on the two lower strings was
somewhat deeper and more viola-like
that the normal violin.

The next good fortune was discov-
ery and acquisition of a set of three in-
struments made by the late Fred L.
Dautrich of Torrington, Conn., during
the 1920's and '30's. He had de-
scribed them in a booklet caUed Bridg-
ing the Gaps in the Violin Family.^
His vilonia, with a body length of
20 in. (51 cm) was tuned as a viola
and played cello-fashion on a peg.
The vilon, or tenor, which looked like
a half-size cello, was tuned an octave
below the violin, G-D-A-E. His vi-
lone, or small bass, with strings tuned
two octaves below the violin, filled the
gap between the cello and the con-
trabass. These represented three of
the tone ranges we had projected for
the new violin family. Tests showed
that their resonances lay within work-
ing range of our theory. A year of
work, adjusting top and back plate
wood thicknesses for desired reso-
nance frequencies and rib heights for
proper cavity resonances in each of the
three instruments gave excellent re-
sults. The vilono proved to have ex-
actly the resonance frequencies pro-
jected for the enlarged cello, or bari-
tone. So it was moved up a notch in
the series and tuned as a cello with
extra long strings.

Dautrich's pioneering work had
saved years of cut and try. We now
had four of the new instruments in
playing condition; mezzo, alto (verti-

BODY LENGTHS for new instniments
w«re determined by plotting lengths
of known instruments against wave-
length, then extending data in a
smooth curve to include treble at one
end and contrabass at the other.
Identified points show where old and
new instruments fall. — FIG. 3

238

Founding a Family of Fiddles

cal viola), tenor and baritone. I was
able to add a fifth by making a so-
prano, using information gained from
many tests on three-quarter- and half-
size violins.

With five of the new instruments
developed experimentally and in play-
ing condition, we decided to explore
their musical possibilities and evaluate
the overaU results of our hypothesis of
resonance placement. In October
1961 the working group gathered at
the home of Helen Rice in Stock-
bridge, Mass., where Saunders and his
associates had, for some years, met fre-
quently to discuss violin acoustics and
play chamber music. Short pieces of
music were composed for the five in-

struments, and the musicians gave the
new family of fiddles its first workout.
The consensus was that our hypothesis
was working even better than we had
dared to hope! Apparently the violin-
type placement of the two main reso-
nances on the two open middle strings
of each instmment was enabling us to
project the desirable qualities of the
violin into higher and lower tone
ranges.

The next step was to explore the
resonances of various size basses to
help in developing the small bass and
the contrabass. A small three-quarter-
size bass with arched top and back
proved to have just about proper reso-
nances for the small bass. With re-

iliiliiiliiliiiliili lilijli III iiliiiliili III I ilill

STRAOIVARIUS VIOLIN 1713

HUTCHINS VIOLIN tioM

A
440

rf^r

OPEN STRINGS 6 D A E

HUTCHINS 42.5-CM
VIOLA No.34

HUTCHINS 44<:M
VIOLA No.42

c e D A

CARLOS BERGONZI III
■CELL0-l«17

HUTCHINS CEaONo.38

C G D A

p:::

TESTORE BASS
GERMAN BASS

RESONANCES

W = WOOD PRIME

C =: CAVriY

B = BODY (wood)

moval of its low E 41 string and the
addition of a high C 131 string to
bring the tuning to A-D-G-C (basses
are tuned in musical fourths for ease of
fingering) it fitted quite well into the
series as the small bass. But as yet no
prototype for the contrabass could be
located. This final addition to the se-
ries was to come later.

First musical success
By January 1962 we were ready for a
real test in which experts could hear
our six new instruments and compare
them with good conventional violins,
violas and cellos. Composers ar-
ranged special music, and professional
players had a chance to practice on
the new instruments.

Ensemble results exceeded all our
expectations. We had violin-like
quality and brilliance through the en-
tire range of tones. Our soprano pro-
duced a high clear quality that carried
well over the other instruments al-
though the high positions on its two
lower strings were weak. The mezzo
tone was powerful and clear although
somewhat viola-like on the two lower
strings. The alto (vertical viola) was
judged a fine instrument even with in-
adequate strings. The unique tone of
the tenor excited all who heard it.
The baritone produced such powerful
and clear tones throughout its range
that the cellist playing it in a Brahms
sonata commented, 'This is the first
time I have been able to hold my own
with the piano!" The small bass was
adequate but needed more work.
General comments told us that the
new instr\iments were ready to stand
on their own, musically, although
much more work was to be done on
adjustments, strings and proper bows.

End-of-scale problems

With the helpful criticisms and
suggestions that came from the first
musical test we were encouraged to

LOUDNESS CURVES are useful
evaluations of instrument character-
istics. Each is made by bowing an
instrument to maximal loudness at 14
semitones on each string and plotting
the resulting loudness ceiling against
frequency of sound. — FIG. 4

239

tackle the problems of the largest and
smallest instruments. No existing in-
struments could be adapted experi-
mentally. We had to design and build
them.

The largest bass available for testing
was a huge Abraham Prescott, with a
48-in. (122-cm) body length, made in
Concord, N.H., in the early 1800's but
even that was not big enoughl A tiny
pochette, or pocket fiddle, from the
Wurlitzer collection, with a body
length of 7 in. (18 cm) had the right
cavity resonance, but its body reso-
nance was much too low.

The body length of each of the new
instruments has been one of the con-
trolling factors in all of our experi-
ments. Thus it was decided that the
best way to arrive at the dimensions
for the largest and smallest would be
to plot a curve of body lengths of
known instruments, to check against
their resonance placement and string
tuning. This tvorking chart is shown
in figiure 3 in which linear body length
is plotted against the logarithm of
wavelength. The curve for the new
instnmients was extended in a smooth
arc to include the contrabass fre-

quency at the low end and the treble
frequency at the upper end, an octave
above the normal violin. This proce-
dure gave a projected body length of
51 in. ( 130 cm) for the contrabass and
10.5 in. (26.5 cm) for the treble. Of
course rib height and enclosed air vol-
ume were separately determined by
other considerations.

Current design practice
From all of this experience we have
developed what we might call a "de-
sign philosophy." It depends mainly
on resonance placement and loudness
curves.

Our resonance principle, according
to which each member of the new
violin family has been made, can be
stated as follows: The main body res-
onance of each of the instruments
tuned in fifths is placed at the fre-
quency of the open third string, and
the main cavity resonance at the fre-
quency of the open second string.
Another way of stating the principle,
and one that includes the instruments
tuned in fourths as well as those timed
in fifths, is this: Wood prime is
placed two semitones above the lowest

2 1.5

\^THEOnmeAL LENQTH

'\ ^mw wwTnuMcms

CONVENTIONAL

1.17

•MAN

HORIZONTAL INSTRU

VIOUN

ae7

CONTRABASS MRrTONE

TENOR VIOU

VKMJN

0.75
TREBLE

SOfRANO

0.1C7

OJS aS3 0.5 0.67 1 1.33

FREQUENCY (relative— 4oc seal*)

tone, and the cavity resonance is a
fourth above that. (Wood prime is
the strengthened frequency one oc-
tave below the main body— "wood"
—resonance.) These conditions are
exemplified in Heifetz's Guamerius
violin and many other good ones, but
they are not found in all good violins.

The loudness curve developed by
Saunders is one of our most useful
measures for evaluating overall instru-
ment characteristics. We make such a
curve by bowing an instrument as
loudly as possible at 14 semitones on
each string and plotting maximal loud-
ness against frequency. Despite una-
voidable variations in any test that re-
quires a musician to bow an instru-
ment, the loudness curve is significant
because there is a fairly definite limit
to the momentary volume an experi-
enced player can produce with a short
rapid bow stroke.

As you will see in figure 4, the loud-
ness ceiling varies for each semitone
on a given instniment. The curves of
this figure were made by bowing each
instrviment without vibrato at a con-
stant distance from a sound meter.
From them you can see the placement
of main body and cavity resonances in
eight conventional instnmients— two
violins, two violas, two cellos and two
basses. You can see that in the vio-
lins the wood prime adds power to the
low range of the G string. In the vi-
olas, cellos and basses the two main
resonances, which are higher in fre-
quency relative to string timing, create

SCALING FACTORS for old and new
instruments are a useful reference
guide for designers. — FIG. 5

240

Founding a Family of Fiddles

a condition of somewhat weaker re-
sponse on the lowest four or five semi-
tones.

Fitting fiddles to players
After you decide what kind of acous-
tics you want, you still have another
problem: You have to make fiddles
that people can play. For years we
worked toward design of an acousti-
cally good instrument with genuine
viola tone. Meanwhile we had to
keep in mind such conflicting require-
ments as large radiating areas in the
plates and adequate bow clearance in
the C bouts (figure 2). Relation of
string length to other dimensions that
define tone spacing on the fingerboard
—the viohn maker's "mensure"— is an-
other consideration important to the
player. With our acoustic pattern as a
model we undertook enlarging, scahng
and redesigning all our new instru-
ments, always keeping violin place-
ment of resonances in each tone range.

From our set of experimentally
adapted instruments, which represent
a variety of styles and designs in violin
making, we had learned many things.
The vertical viola was about right in
body dimensions, but its strings were
too long for viola fingering and too
short for cello fingering. The tenor
was too small, and the cellists were
asking for it to have strings as long as
possible. The baritone was right for
body size, but it had much too long
strings. The bass players were asking
for a long neck on the small bass and a
short one on the large bass with string
lengths as close as possible to conven-
tional.

From such comments we realized
that there were two basic designs for
ease of playing in relation to string
lengths and overall mensure of each
instrument. ControlHng factor in the
instrument mensure is placement of
the notches of the f holes because a
line drawn between these two points
dictates the position of the bridge and
the highest part of the arch of the top
plate. Mensure for the tenor and
small bass would need to be as great
as possible and for the vertical viola
and baritone it would need to be as
small as possible. Since the relative
areas of the upper and lower bouts are
critical factors in plate tuning, adjust-
ment of these mensures posed quite a
set of problems.

We developed a series of scaling
factors^ based on relative body length,
relative resonance placement and rela-
tive string tuning that could be used as
a reference guide in actual instrument
construction. Figure 5 shows the set
which has proved most useful in mak-
ing the eight new instruments as well
as those of conventional instruments.

We had a problem in measuring re-
sponses of plates of many sizes— all the
way from the 10.5-in. (26-cm) one of
the treble violin to the 51-in. (130-
cm ) one of the contrabass. We solved
it by redesigning our transducer from
a magnet-armature to a moving-coil
type. Then the wooden fiddle plate,
suspended at its comers by elastic
bands, was made to vibrate as the
cone of a loudspeaker (figure 6).

Using the know-how developed in
making and testing several hundred
violin, viola and cello plates, I could
tune the plates of new instruments so
that not only did each pair of top and
back plates have the desired frequency
relation, 2 but it also had its wood
thicknesses adjusted to give a reason-
able approach to what would be an
optimal response. ^^

As a starting guide in adjusting plate
frequencies I used the finding that a
seven-semitone interval should sepa-
rate the main body resonance of the
finished violin from the average of the
two frequencies of the free plates. It
was soon obvious, however, that this
relationship was not going to hold as
the instnmients increased in size. As
the instrument gets larger the interval
becomes smaller, but we do not have
enough data yet to make a precise
statement about it.

We used scaling theory and the
three basic acoustical tools of scientific
violin making: (a) frequency rela-
tionship between free top and back
plates, (b) optimal response in each
plate and (c) interval between body
resonance and average of free-plate
frequencies. We are able not only to
create new instruments of the violin
family but also to improve the present
members. But we have to combine
the acoustical tools with the highest
art of violin making.

Traits of family members

Any family has its resemblances and
its differences. So it is with our vio-
lins. They make a family (figure 7)

with basic traits in common. But they
also have their own personalities.

Treble ( G-D-A-E ) . The main prob-
lem with our treble has been to get the
frequencies of body and cavity reso-
nances high enough and still keep the
mensure long enough for a player to
finger consecutive semitones without
having to slide his fingers around.
We projected a theoretical body
length of 10.5 in. (26.7 cm) and a
string length of 10 in. (25.4 cm), but
to have the proper cavity resonance in
this size body, the ribs would be only
3 mm high-a potentially dangerous
structural condition! Besides we
knew of no string material that could
be tuned to E 1320 at a length of 25.4
cm without breaking. At one point
we thought we might have to resort to
a three-stringed instrument in this
range as was indicated by Michael
Praetorius in 1619.1^

The cavity-resonance problem was
solved by making six appropriately
sized holes in the ribs to raise its fre-
quency to the desired D 587. A string
material of requisite tensile strength to
reach the high E 1320 was finally
found in carbon rocket wire, made by
National Standard Company. This
proved suitable not only for the high E
string but for a number of others on
the new instruments. As a temporary
measure the ribs were made of soft
aluminum to prevent the holes from
unduly weakening the structure. Re-
design should eliminate the nasal
quality found on the lower strings and
improve the upper ones. Despite this
nasal quaHty many musicians are
pleased with the degree in which the
upper strings surpass the normal violin
in the same high range.

Plans are to redesign this instrument
in several different ways in an effort to
discover the best method of achieving
desired tone quality throughout its en-
tire range.

Soprano (C-G-D-A). The soprano
was designed to have as large a plate
area as possible, with resulting shallow
ribs and fairly large f holes to raise the
cavity resonance to the desired G 392.
The overall tone has been judged good
and is most satisfactory on the three
upper strings. The instrument needs
redesign, however, for a better quality
on the lower strings. The mensure is
as long as possible for playing con-
venience. J. S. Bach wrote for an in-

241

TESTING FIDDLES. New tech
niques enable today's makers to
achieve results their predecessors
could not produce. Redesigned trans-
ducer measures response of plate that
is made to vibrate like a loudspeaker
cone in operation. — FIG. 6

strument in this tuning, which Sir
George Grove describes in Grove's dic-
tionary:^'^ "The violino piccolo is a
small violin, with strings of a length
suitable to be tuned a fourth above the
ordinary violin. It existed in its own
right for playing notes in a high
compass. . . .It survives as the 'three-
quarter violin' for children. Tuned
like a violin, it sounds wretched, but
in its proper pitch it has a pure tone
color of its own, for which the high
positions on the ordinary violin gave
no substitute."

Mezzo (G-D-A-E). The present
mezzo with a body length of 16 in.
(40.5 cm) was added to the new
violin family when musicians found

that even an excellent concert violin
did not have the power of the other
members of the group. According
to scaling theory^^ this instrument,
which is 1.14 times as long as the
violin, has somewhat more power than
necessary to match that of the others.
So a second instrument has been de-
veloped that is 1.07 times as long as
the violin. It has violin placement of
resonances yet is adjusted to have con-
ventional violin mensure for the play-
er.'^ It has more power than the nor-
mal violin and seems most satisfactory.
In fact several musicians have indicat-
ed that it may be the violin of the fu-
ture.

Alto (vertical viola) (C-G-D-A).
The greatest difficulty with the alto
is that it puts the trained viola player
at a distinct disadvantage by taking
the viola from under his chin and set-
ting it on a peg, cello fashion on the

PHOTOS BY PETER PRUYN

floor. Even with an unusual body
length of 20 in., its mensure has been
adjusted to that of a normal 17.5-in.
(44.5-cm) viola, and some violists
with large enough physique have been
able to play it under the chin. Cello
teachers have been impressed by its
usefulness in starting young children
on an instrument that they can handle
readily as well as one they can con-
tinue to follow for a career. The
greatest advantage is the increase in
power and overall tone quality.^o
Leopold Stokowski said when he
heard this instrument in concert, "That
is the sound I have always wanted
from the violas in my orchestra. No
viola has ever sounded like that be-
fore. It fills the whole hall."

Tenor ( G-D-A-E ) . The body length
of the tenor was redeveloped from the
Dautrich vilon which had a length
ratio of 1.72 to the violin. The pres-

242

Founding a Family of Fiddles

PHOTOS BY J. KELLUM SMITH

THE WHOLE FAMILY poses
for pictures with performers
trying them out. — FIG. 7

MAX POLLIKOFF
treble

ERNESTINE BREIMEISTER
soprano

LILLA KALMAN
mezzo

ent tenor has a ratio of 1.82 with
other factors adjusted accordingly, and
the strings as long as possible for con-
venience in cello fingering. Many
musicians have been impressed with
its potential in ensemble as well as solo
work. They are amazed to find that it
is not a small cello, musically, but a
large octave violin.

The main problem for this instru-
ment is that there is little or no music
for it as yet. Early polyphonic music,
where the tenor's counterpart in the
viol family had a voice, has been rear-
ranged for either cello or viola. It has
no part in classical string or orches-
tral literature, and only a few con-
temporary compositions include it.
Grove'" has this to say: "The gradual
suppression of the tenor instrument in
the 18th century was a disaster;
neither the lower register of the viola
nor the upper register of the violon-
cello can give its effect. It is as
though all vocal part music were sung
without any tenors, whose parts were
distributed between the basses and
contraltos! It is essential for 17th
century concerted music for violins
and also for some works by Handel
and Bach and even later part-writing.
In Purcell's Fantasy on One Note the
true tenor holds the sustained C. . .
The need for a real tenor voice in the
19th century is evidenced by the many
abortive attempts to create a substi-
tute."

Baritone ( C-G-D-A ) . The body res-

onance of our baritone is nearly three
semitones lower than projected, and
this departure probably accounts for
the somewhat bass-like quality on the
low C 65.4 string. Its strings are 0.73
in. (1.8 cm) longer than those of the
average cello. One concert cellist said
after playing it for half an hour, "You
have solved all the problems of the
cello at once. But I would like a con-
ventional cello string length." Thus a
redesign of this instrument is desirable
by shortening the body length a little.
This redesign would raise the fre-
quency of the body resonance and at
the same time make possible a shorter
string.

Small bass (A-D-G-C). Our first
newly constructed instrument in the
bass range is shaped like a bass viol
with sloping shoulders, but has both
top and back plates arched and other
features comparable to viobn construc-
tion. This form was adopted partly to
discover the effect of the sloping
shoulders of the viol and partly be-
cause a set of half-finished bass plates
was available. The next small bass is
being made on violin shape with other
features as nearly like the first one as
possible. Bass players have found the
present instrument has a most desira-
ble singing quality and extreme play-
ing ease. They particularly like the
bass-viol shape. It has proved most
satisfactory in both concert and re-
cording sessions.

Contrabass (E-A-D-G). Our con-

trabass^i is 7 ft (210 cm) high overall;
yet it has been possible to get the
string length well within conventional
bass mensure at 43 in. (110 cm) so
that a player of moderate height has
no trouble playing it except when he
reaches into the higher positions near
the bridge. For sheer size and weight
it is hard to hold through a 10-hr re-
cording session as one bassist did.
When it was first strung up, the player
felt that only part of its potential was
being realized. The one construction-
al feature that had not gone according
to plan was rib thickness. Ribs were 3
mm thick, whereas violin making indi-
cated they needed to be only 2 mm
thick. So the big fiddle was opened;
the lining stripes cut out, and the ribs
planed down on the inside to an even
2 mm all over— a job that took 10 days.
But when the contrabass was put to-
gether and strung up, its ease of play-
ing and depth of tone delighted all
who played or heard it. Henry Brant
commented, "I have waited all my life
to hear such sounds from a bass."

How good are they really?
All who have worked on the new in-
struments are aware of the present
lack of objective tests on them— aside
from musician and audience com-
ments. In the near future we plan to
compare comments with adequate
tonal analyses and response curves of
these present instruments as well as
new ones when they are made. The

243

STERLING HUNKINS
alto

PETER ROSENFELD
tenor

JOSEPH TEKULA
baritone

DAVID WALTER
small bass

STUART SANKEY
contrabass

only objective evaluation so far comes
from A. H. Benade at Case Institute:
"I used my 100-W amplifier to run a
tape recorder alternately at 60 and 90
cps while recording a good violin with
the machine's gearshift set at the three
nominal 1-, 3.5- and 7.5-in/sec speeds.
This was done in such a way as to
make a tape which, when played back
at 3.5 in/sec, would give forth sounds
at the pitches of the six smaller instru-
ments in the new violin family (small
bass and contrabass excluded). There
were some interesting problems about
the subjective speed of low- compared

References

1. F. A. Saunders, "The mechanical ac-
tion of violins," J. Acoust. Soc. Am. 9,
81 (1937).

2. C. M. Hutchins, A. S. Hopping, F. A.
Saunders, "Subharmonics and plate
tap tones in violin acoustics," J.
Acoust. Soc. Am. 32, 1443 (1960).

3. J. C. Schelleng, "The violin as a cir-
cuit," J. Acoust. Soc. Am. 35, 326
(1963).

4. F. A. Saunders, "Recent work on vio-
lins," J. Acoust. Soc. Am. 25, 491
(1953).

5. F. A. Saunders, "The mechanical ac-
tion of instruments of the violin
family," J. Acoust. Soc. Am. 17, 169
(1946).

6. F. A. Saunders, unpublished note-
books.

7. H. Meinel, "Regarding the sound
quality of violins and a scientific basis
for violin construction," J. Acoust.
Soc. Am. 29, 817 (1957).

8. F. Eggers, "Untersuchung von Corp-
us-Schwingungen am Violoncello,"

with high-pitch playing, but the mu-
sician was up to it and we managed to
guess reasonably well. The playing
was done without vibrato. It is a
tribute to everyone involved in the de-
sign of those fiddles that they really do
sound like their scientifically trans-
posed cousin violin."

But as yet we know only part of
why this theory of resonance place-
ment is working so well. Probing
deeper into this "why" is one of the
challenges that lie ahead. Still un-
solved are the problems of the intri-
cate vibrational patterns within each

Acustica 9, 453 (1959).
9. W. Lottermoser, W. Linhart, "Beit-
rag zur akustichen Prufung von Gei-
gen und Bratschen," Acustica 7, 281
(1957).

10. C. M. Hutchins, A. S. Hopping, F. A.
Saunders, "A study of tap tones," The
Strand, August, September ( 1958).

11. C. M. Hutchins, "The physics of vio-
lins," Scientific American 207, no. 5,
78 (1962).

12. R. H. Scanlan, "Vibration modes of
coupled plates," J. Acoust. Soc. Am.
35,1291 (1963).

13. F. A. Saunders, C. M. Hutchins, "On
improving violins," Violins and Vio-
linists 13, nos. 7, 8 (1952).

14. F. L. Dautrich, H. Dautrich, "A chap-
ter in the history of the violin family,"
The Catgut Acoustical Society News-
letter No. 4 (1 Nov. 1965).

15. C. M. Hutchins, The Catgut Acousti-
cal Society Newsletter No. 5 ( 1 May
1966) and No. 6 ( 1 Nov. 1966).

16. M. Praelorius, Syntagma Musicum
II: de Organographia (1619); re-

free plate as compared to those in the
assembled instrument; the reasons for
the effect of moisture and various fin-
ishes on the tone of a vioUn and the
possibility of some day being able to
write adequate specifications for a
fabricated material that will equal the
tone qualities of wood!

e o •

This work has received support from the •
John Simon Guggenheim Memorial Foun-
dation, the Martha Baird Rockefeller
Fund for Music, the Alice M. Ditson
Fund of Columbia University, the Catgut
Acoustical Society and private contribu-
tions.

printed 1964 by Internationale Gesell-
schafl fiir Musikwissenschaft, Baren-
reiter Kassel, Basel, London, New
York, page 26.

17. G. Grove, Grove's Dictionary of Musks
and Musicians, 5th ed., St. Martins
Press, New York (1954). vol. 8,
page 809.

18. J. C. Schelleng, "Power relations in
the violin family," paper presented at
71st meeting. Acoustical Society of
America, Boston (3 June 1966).

19. C. M. Hutchins, J. C. Schelleng, "A
new concert violin," paper presented
to the Audio Engineering Society, 12
Oct. 1966 ( to be published ).

20. C. M. Hutchins, "Comparison of the
acoustical and constructional para-
meters of the con\entional 16 to
17-in. viola and the new 20-in. verti-
cal viola," J. Acoust. Soc. Am. 36,
1025 (1964) (abstract only).

21. C. M. Hutchins, "The new contrabass
violin," .•\merican String Teacher,
Spring 1966.

244

Some nonscientlsts hold odd views of the nature of
science. This article catalogs and analyses the most
common fallacies.

17 The Seven Images of Science

Gerald Helton

An article from Science, 1960.

Pure Thought and Practical Power

Each person's image of the role of
science may differ in detail from that
of the next, but all public images are
in the main based on one or more of
seven positions. The first of these goes
back to Plato and portrays science as
an activity with double benefits: Science
as pure thought helps the mind find
truth, and science as power provides
tools for effective action. In book 7 of
the Republic. Socrates tells Glaucon
why the young rulers in the Ideal State
should study mathematics: "This, then,
is knowledge of the kind we are seek-
ing, having a double use, military and
philosophical; for the man of war must
learn the art of number, or he will not
know how to array his troops; and the
philosopher also, because he has to rise
out of the sea of change and lay hold
of true being. . . . This will be the eas-
iest way for the soul to pass from be-
coming to truth and being."

The main flaw in this image is that
it omits a third vital aspect. Science
has always had also a mythopoeic func-
tion— that is, it generates an impor-
tant part of our symbolic vocabulary
and provides some of the metaphysical
bases and philosophical orientations of
our ideology. As a consequence the
methods of argument of science, its
conceptions and its models, have per-
meated first the intellectual life of the
time, then the tenets and usages of
everyday life. All philosophies share
with science the need to work with
concepts such as space, time, quantity,
matter, order, law, causality, verifica-
tion, reality. Our language of ideas,
for example, owes a great debt to
statics, hydraulics, and the model of
the solar system. These have furnished
jjowerful analogies in many fields of
study. Guiding ideas — such as condi-
tions of equilibrium, centrifugal and
centripetal forces, conservation laws,
feedback, invariance, complementarity
— enrich the general arsenal of imagina-
tive tools of thought.

A sound image of science must em-

brace each of the three functions.
However, usually only one of the three
is recognized. For example, folklore
often depicts the life of the scientist
either as isolated from life and from
beneficent action or, at the other
extreme, as dedicated to technological
improvements.

Iconoclasm

A second image of long standing is
that of the scientist as iconoclast. In-
deed, almost every major scientific ad-
vance has been interpreted — either tri-
umphantly or with apprehension — as
a blow against religion. To some ex-
tent science was pushed into this posi-
tion by the ancient tendency to prove
the existence of God by pointing to
problems which science could not solve
at the time. Newton thought that the
regularities and stability of the solar
system proved it "could only proceed
from the counsel and dominion of an
intelligent and powerful Being." and
the same attitude governed thought
concerning the earth's formation before
the theory of geological evolution, con-
cerning the descent of man before the
theory of biological evolution, and con-
cerning the origin of our galaxy before
modern cosmology. The advance of
knowledge therefore made inevitable
an apparent conflict between science
and religion. It is now clear how large
a price had to be paid for a misunder-
standing of both science and religion:
to base religious beliefs on an estimate
of what science cannot do is as fool-
hardy as it is blasphemous.

The iconoclastic image of science
has, however, other components not as-
cribable to a misconception of its func-
tions. For example, Arnold Toynbee
charges science and technology with
usurping the place of Christianity as
the main source of our new symbols.
Neo-orthodox theologians call science
the "self-estrangement" of man be-
cause it carries him with idolatrous

zeal along a dimension where no ulti-
mate— that is, religious — concerns pre-
vail. It is evident that these views fail
to recognize the multitude of divergent
influences that shape a culture, or a
person. And on the other hand there
is, of course, a group of scientists,
though not a large one, which really
does regard science as largely an icono-
clastic activity. Ideologically they are, of
course, descendants of Lucretius, who
wrote on the first pages of De renim
naiiira, "The terror and darkness of
mind must be dispelled not by the rays
of the sun and glittering shafts of day.
but by the aspect and the law of na-
ture: whose first principle we shall be-
gin by thus stating, nothing is ever got-
ten out of nothing by divine power."
In our day this ancient trend h;is as-
sumed political significance owing to
the fact that in Soviet literature scien-
tific teaching and atheistic propaganda
are sometimes equated.

Ethical Perversion

The third image of science is that
of a force which can invade, possess,
pervert, and destroy man. The current
stereotype of the soulless, evil scientist
is the psychopathic investigator of
science fiction or the nuclear destroyer
— immoral if he develops the weap-
ons he is asked to produce, traitorous
if he refuses. According to this view,
scientific morality is inherently nega-
tive. It causes the arts to languish, it
blights culture, and when applied to hu-
man affairs, it leads to regimentation
and to the impoverishment of life.
Science is the serpent seducing us into
eating the fruits of the tree of knowl-
edge— thereby dooming us.

The fear behind this attitude is genu-
ine but not confined to science: it is
directed against all thinkers and inno-
vators. Society has always found it
hard to deal with creativity, innovation,
and new knowledge. And since science
assures a particularly rapid, and there-

245

fore particularly disturbing, turnover of
ideas, it remains a prime target of sus-
picion.

Factors peculiar to our time intensify
this suspicion. The discoveries of
"pure" science often lend themselves
readily to widespread exploitation
through technology. The products of
technology— whether they are better
vaccines or better weapons — have the
characteristics of frequently being very
effective, easily made in large quanti-
ties, easily distributed, and very ap-
pealing. Thus we are in an inescapable
dilemma — irresistibly tempted to reach
for the fruits of science, yet, deep in-
side, aware that our metabolism may
not be able to cope with this ever-in-
creasing appetite.

Probably the dilemma can no longer
be resolved, and this increases the
anxiety and confusion concerning
science. A current symptom is the pop-
ular identification of science with the
technology of superweapons. The bomb

is taking the place of the micro>copc.
Wernher von Bruun. the place of Ein-
stein, as svnibols for modern science
and scientists. The efforts to convince
people that science itself can give man
only knowledge about himself and his
environment, and occasionally a choice
of action, have been largely unavail-
ing. The scientist as scieniisi can take
little credit or responsibility either for
facts he discovers — for he did not
create them — or for the uses others
make of his discoveries, for he gen-
erally is neither permitted nor specially
fittcil to make these decisions. They
are controlled by considerations of
ethics, economics, or politics and
therefore arc shaped by the values and
historical circumstances of the whole
society.

There are other evidences of the
widespread notion that science itself
cannot contribute positively to culture.
Toynbce, for example, gives a list of
"creative individuals," from Xenophon
to Hindenburg and from Dante to
Lenin, but docs not include a single
scientist. I cannot forego the remark
that there is a significant equivalent on
the level of casual conversation. For
when the man in the street — or many
an intellectual — hears that you are a
physicist or mathematician, he will
usually remark with a frank smile, "Oh.
I never could understand that subjec:"";
while intending this as a curious com-
pliment, he betravs his intellectual dis-
sociation from scientific fields. It is not

fashionable to confess to a lack of ac-
quaintance with the latest ephemera in
literature or the arts, but one may even
exhibit a touch of pride in professing
ignorance of the structure of the uni-
verse or one's own body, of the be-
havior of matter or one's own mind.

The Sorcerer's Apprentice

The last two views held that man is
inherently good and science evil. The
next image is based on the opposite as-
sumption— that man cannot be trusted
with scientific and technical knowledge.
He has survived only because he lacked
sufficiently destructive weapons: now
he can immolate his world. Science, in-
directly responsible for this new power,
is here considered ethically neutral.
But man, like the sorcerer's apprentice,
can neither understand this tool nor
control it. Unavoidably he will bring
upon himself catastrophe, partly
through his natural sinfulness, and
partly through his lust for power, of
which the pursuit of knowledge is a
manifestation. It was in this mood that
Pliny deplored the development of pro-
jectiles of iron for purposes of war:
"This last I regard as the most criminal
artifice that has been devised by the hu-
man mind; for. as if to bring death
upon man with still greater rapidity,
we have given wings to iron and taught
it to fiy. Let us, therefore, acquit Na-
ture of a charge that belongs to man
himself."

When science is viewed in this plane
— as a temptation for the mischievous
savage — it becomes easy to suggest a
moratorium on science, a period of
abstinence during which humanity
somehow will develop adequate spirit-
ual or social resources for coping with
the possibilities of inhuman uses of
modern technical results. Here I need
point out only the two main misun-
derstandings implied in this recurrent
call for a moratorium.

First, science of course is not an oc-
cupation, such as working in a store or
on an assembly line, that one may pur-
sue or abandon at will. For a creative
scientist, it is not a matter of free
choice what he shall do. Indeed it is
erroneous to think of him as advancing
toward knowledge; it is, rather, knowl-
edge which advances towards him,
grasps him, and overwhelms him Even
the most superficial glance at the life
and work of a Kepler, a Dalton. or a

Pasteur would clarify this point. It
would be well if in his education each
person were shown by example that
the driving power of creativity is as
strong and as sacred for the scientist
as for the artist.

The second point can be put equally
briefly. In order to survive and to pro-
gress, mankind surely cannot ever know
too much. Salvation can hardly be
ihought of as the reward for ignorance.
Man has been given his mind in order
that he may find out where he is. what
he is. who he is, and how he may as-
sume the responsibility for himself
which is the only obligation incurred in
gaining knowledge.

Indeed, it may well turn out that the
technological advances in warfare have
brought us to the point where society
is at last compelled to curb the aggres-
sions that in the past were condoned
and even glorified. Organized warfare
and genocide have been practiced
throughout recorded history, but never
until now have even the war lords
openly expressed fear of war. In the
search for the causes and prevention
of aggression among nations, we shall,
I am convinced, find scientific investi-
gations to be a main source of under-
standing. I

I
Ecological Disaster

A change in the average temperature !
of a pond or in the salinity of an ocean
may shift the ecological balance and
cause the death of a large number of
plants and animals. The fifth prevalent
image of science similarly holds that
while neither science nor man may be
inherently evil, the rise of science hap-
pened, as if by accident, to initiate an
ecological change that now corrodes
the only conceivable basis for a stable
society, .n the words o Jacques Mari-
tain, the "deadly disease" science set off
in society is "the denial of eternal truth
and absolute values."

The mam events leading to this state
are usually presented as follows. The
abandonment of geocentric astronomy
implied the abandonment of the con-
ception of the earth as the center of
creation and of man as its ultimate pur-
pose. Then purposive creation gave
way to blind evolution. Space, time,
and certainty were shown to have no
absolute meaning. All a priori axioms
were discovered to be merely arbitrary
conveniences. Modern psychology and

246

The Seven Images of Science

anthropology led to cultural relativism.
Truth itself has been dissolved into
probabilistic and indcterniinistic state-
ments. Drawing upon analogy with the
sciences, liberal philosophers have be-
come increasingly relativistic, denying
either the necessity or the possibility of
postulating immutable verities, and so
have imdcrmined the old foundations
of moral and social authority on which
a stable society must be built.

It should be noted in passing that
many applications of recent scientific
concepts outside science merely reveal
ignorance about science. For example,
relativism in nonscientific fields is gen-
erally based on farfetched analogies.
Relativity theory, of course, does not
find that truth depends on the point of
view of the observer but, on the con-
trary, reformulates the laws of physics
so that they hold good for every ob-
server, no matter how he moves or
where he stands. Its central meaning
is that the most valued truths in science
are wholly independent of the point of
view. Ignorance of science is also the
only excuse for adopting rapid changes
within science as models for antitradi-
tional attitudes outside science. In real-
ity, no field of thought is more conserv-
ative than science. Each change neces-
sarily encompasses previous knowledge.
Science grows like a tree, ring by ring.
Einstein did not prove the work of
Newton wrong; he provided a larger
setting within which some contradic-
tions and asymmetries in the earlier
physics disappeared.

But the image of science as an eco-
logical disaster can be subjected to a
more severe critique. Regardless of
science's part in the corrosion of ab-
solute values, have those values really
given us always a safe anchor? A priori
absolutes abound all over the globe in
completely contradictory varieties. Most
of the horrors of history have been
carried out under the banner of some
absolutistic philosophy, from the Aztec
mass sacrifices to the auto-da-fe of the
Spanish Inquisition, from the massacre
of the Huguenots to the Nazi gas cham-
bers. It is far from clear that any so-
ciety of the past did provide a mean-
ingful and dignified life for more than
a small fraction of its members. If,
therefore, some of the new philoso-
phies, inspired rightly or wrongly by
science, point out that absolutes have a
habit of changing in time and of con-
tradicting one another, if they invite

a re-e\amination of the bases of social
authority and reject them when those
bases prove false (as did the Colonists
in this country), then one must not
blame a relativistic philosophy for
bringing out these faults. They were
there all the time.

In the search for a new and sounder
basis on which to build a stable world,
science will be indispensable. We can
hope to match the resources and struc-
ture of society to the needs and poten-
tialities of people only if vc know
more about man. Already science has
much to say that is valuable and im-
portant about human relationships and
problems. From psychiatry to dietetics,
from immunology to meteorology, from
city planning to agricultural research,
by far the largest part of our total sci-
entific and technical cfTort today is con-
cerned, indirectly or directly, with man
— his needs, relationships, health, and
comforts. Insofar as absolutes are to
help guide mankind safely on the long
and dangerous journey ahead, they
surely should be at least strong enough
to stand scrutiny against the back-
ground of developing factual knowl-
edge.

Scientism

While the last four images implied
a revulsion from science, scientism may
be described as an addiction to science.
Among the signs of scientism are the
habit of dividing all thought into two
categories, up-to-date scientific knowl-
edge and nonsense: the view that the
mathematical sciences and the large
nuclear laboratory offer the only per-
missible models for successfully employ-
ing the mind or organizing efTort; and
the identification of science with tech-
nology, to which reference was made
above.

One main source for this attitude is
evidently the persuasive success of re-
cent technical work. Another resides in
the fact that we are passing through a
period of revolutionary change in the
nature of scientific activity — a change
triggered by the perfecting and dissem-
inating of the methods of basic research
by teams of specialists with widely dif-
ferent training and interests. Twenty
years ago the typical scientist worked
alone or with a few students and col-
leagues. Today he usually belongs to a
sizable group working under a contract
with a substantial annual budget. In the

research institute of one university
more than 1500 scientists and techni-
cians are grouped around a set of mul-
timillion-dollar machines: the funds
come from government agencies whose
ultmiate aim is national defense.

Everywhere the overlapping interests
of basic research, industry, and the mil-
itary establishment have been merged
in a way that satisfies all three. Science
has thereby become a large-scale oper-
ation with a potential for immediate
and world-wide effects. The results are
a splendid increase in knowledge, and
also side effects that are analogous
to those of sudden and rapid urbaniza-
tion— a strain on communication facil-
ities, the rise of an administrative bu-
reaucracy, the depersonalization of
some human relationships.

To a large degree, all this is unavoid-
able. The new scientific revolution will
justify itself by the flow of new knowl-
edge and of material benefits that
will no doubt follow. The danger —
and this is the point where scientism
enters — ^is that the fascination with the
Diechanisii) of this successful enterprise
may change the scientist himself and
society around him. For example, the
unorthodox, often withdrawn individ-
ual, on whom most great scientific ad-
vances have depended in the past, does
not fit well into the new system. And
society will be increasingly faced with
the seductive urging of scientism to
adopt generally what is regarded — of-
ten erroneously — as the pattern of or-
ganization of the new science. The
crash program, the breakthrough pur-
suit, the megaton effect are becoming
ruling ideas in complex fields such as
education, where they may not be ap-
plicable.

Magic

Few nonscientists would suspect a
hoax if it were suddenly announced
that a stable chemical element lighter
than hydrogen had been synthesized,
or that a manned observation platform
had been established at the surface of
the sun. To most people it appears that
science knows no inherent limitations.
Thus, the seventh image depicts science
as magic, and the scientist as wizard,
dens ex nwcliino, or oracle. The atti-
tude toward the scientist on this plane
ranges from terror to sentimental sub-
servience, depending on what motives
one ascribes to him.

247

Science's greatest men met with opposition, isolation,
and even condemnation for their novel or '"heretic"
ideas. But we should distinguish between the heretical
innovator and the naive crank.

18 Scientific Cranks

Martin Gardner

An excerpt from his book Fads and Fallacies in the Name of Science, 1957.

Cranks vary widely in both knowledge and intelligence. Some are
stupid, ignorant, almost illiterate men who confine their activities to
sending "crank letters" to prominent scientists. Some produce crudely
written pamphlets, usually published by the author himself, with long
titles, and pictures of the author on the cover. Still others are brilliant
and well-educated, often with an excellent understanding of the branch
of science in which they are speculating. Their books can be highly
deceptive imitations of the genuine article — well-written and impres-
sively learned. In spite of these wide variations, however, most pseudo-
scientists have a number of characteristics in common.

First and most important of these traits is that cranks work in
almost total isolation from their colleagues. Not isolation in the geo-
graphical sense, but in the sense of having no fruitful contacts with
fellow researchers. In the Renaissance, this isolation was not neces-
sarily a sign of the crank. Science was poorly organized. There were
no journals or societies. Communication among workers in a field was
often very difficult. Moreover, there frequently were enormous social
pressures operating against such communication. In the classic case
of Galileo, the Inquisition forced him into isolation because the
Church felt his views were undermining religious faith. Even as late
as Darwin's time, the pressure of religious conservatism was so great
that Darwin and a handful of admirers stood almost alone against the
opinions of more respectable biologists.

Today, these social conditions no longer obtain. The battle of
science to free itself from religious control has been almost completely
won. Church groups still oppose certain doctrines in biology and
psychology, but even this opposition no longer dominates scientific
bodies or journals. Efficient networks of communication within each
science have been established. A vast cooperative process of testing
new theories is constantly going on — a process amazingly free (except,
of course, in totalitarian nations) from control by a higher "ortho-
doxy." In this modern framework, in which scientific progress has
become dependent on the constant give and take of data, it is impos-
sible for a working scientist to be isolated.

248

Scientific Cranks

The modern crank insists that his isolation is not desired on his
part. It is due, he claims, to the prejudice of established scientific
groups against new ideas. Nothing could be further from the truth.
Scientific journals today are filled with bizarre theories. Often the
quickest road to fame is to overturn a firmly-held belief. Einstein's
work on relativity is the outstanding example. Although it met with
considerable opposition at first, it was on the whole an intelligent
opposition. With few exceptions, none of Einstein's reputable oppo-
nents dismissed him as a crackpot. They could not so dismiss him
because for years he contributed brilliant articles to the journals and
had won wide recognition as a theoretical physicist. In a surprisingly
short time, his relativity theories won almost universal acceptance,
and one of the greatest revolutions in the history of science quietly
took place.

It would be foolish, of course, to deny that history contains many
sad examples of novel scientific views which did not receive an un-
biased hearing, and which later proved to be true. The pseudo-
scientist never tires reminding his readers of these cases. The opposi-
tion of traditional psychology to the study of hypnotic phenomena
(accentuated by the fact that Mesmer was both a crank and a charla-
tan) is an outstanding instance. In the field of medicine, the germ
theory of Pasteur, the use of anesthetics, and Dr. Semmelweiss' in-
sistence that doctors sterilize their hands before attending childbirth
are other well known examples of theories which met with strong
professional prejudice.

Probably the most notorious instance of scientific stubbornness
was the refusal of eighteenth century astronomers to believe that
stones actually fell from the sky. Reaction against medieval supersti-
tions and old wives' tales was still so strong that whenever a meteor
fell, astronomers insisted it had either been picked up somewhere and
carried by the wind, or that the persons who claimed to see it fall
were lying. Even the great French Academie des Sciences ridiculed
this folk belief, in spite of a number of early studies of meteoric
phenomena. Not until April 26, 1803, when several thousand small
meteors fell on the town of L'Aigle, France, did the astronomers de-
cide to take falling rocks seriously.

Many other examples of scientific traditionalism might be cited,
as well as cases of important contributions made by persons of a
crank variety. The discovery of the law of conservation of energy by
Robert Mayer, a psychotic German physician, is a classic instance.
Occasionally a layman, completely outside of science, will make an
astonishingly prophetic guess — like Swift's prediction about the moons
of Mars (to be discussed later), or Samuel Johnson's belief (ex-
pressed in a letter, in 1781, more than eighty years before the dis-
covery of germs) that microbes were the cause of dysentery.

249

One must be extremely cautious, however, before comparing the
work of some contemporary eccentric with any of these earlier ex-
amples, so frequently cited in crank writings. In medicine, we must
remember, it is only in the last fifty years or so that the art of healing
has become anything resembling a rigorous scientific discipline. One
can go back to periods in which medicine was in its infancy, hope-
lessly mixed with superstition, and find endless cases of scientists with
unpopular views that later proved correct. The same holds true of
other sciences. But the picture today is vastly different. The prevail-
ing spirit among scientists, outside of totalitarian countries, is one of
eagerness for fresh ideas. In the great search for a cancer cure now
going on, not the slightest stone, however curious its shape, is being
left unturned. If anything, scientific journals err on the side of per-
mitting questionable theses to be published, so they may be discussed
and checked in the hope of finding something of value. A few years
ago a student at the Institute for Advanced Studies in Princeton was
asked how his seminar had been that day. He was quoted in a news
magazine as exclaiming, "Wonderful! Everything we knew about
physics last week isn't true!"

Here and there, of course — especially among older scientists who,
like everyone else, have a natural tendency to become set in their
opinions — one may occasionally meet with irrational prejudice against
a new point of view. You cannot blame a scientist for unconsciously
resisting a theory which may, in some cases, render his entire life's
work obsolete. Even the great Galileo refused to accept Kepler's
theory, long after the evidence was quite strong, that planets move
in ellipses. Fortunately there are always, in the words of Alfred Noyes,
"The young, swift-footed, waiting for the fire," who can form the
vanguard of scientific revolutions.

It must also be admitted that in certain areas of science, where
empirical data are still hazy, a point of view may acquire a kind
of cult following and harden into rigid dogma. Modifications of Ein-
stein's theory, for example, sometimes meet a resistance similar to
that which met the original theory. And no doubt the reader will have
at least one acquaintance for whom a particular brand of psycho-
analysis has become virtually a religion, and who waxes highly indig-
nant if its postulates are questioned by adherents of a rival brand.

Actually, a certain degree of dogma — of pig-headed orthodoxy —
is both necessary and desirable for the health of science. It forces
the scientist with a novel view to mass considerable evidence before
his theory can be seriously entertained. If this situation did not exist,
science would be reduced to shambles by having to examine every
new-fangled notion that came along. Clearly, working scientists have
more important tasks. If someone announces that the moon is made
of green cheese, the professional astronomer cannot be expected

250

Scientific Cranks

to climb down from his telescope and write a detailed refutation.
"A fairly complete textbook of physics would be only part of the
answer to Velikovsky," writes Prof. Laurence J. Lafleur, in his excel-
lent article on "Cranks and Scientists" {Scientific Monthly, Nov.,
1951), "and it is therefore not surprising that the scientist does not
find the undertaking worth while."

The modern pseudo-scientist — to return to the point from which
we have digressed — stands entirely outside the closely integrated
channels through which new ideas are introduced and evaluated. He
works in isolation. He does not send his findings to the recognized
journals, or if he does, they are rejected for reasons which in the vast
majority of cases are excellent. In most cases the crank is not well
enough informed to write a paper with even a surface resemblance to
a significant study. As a consequence, he finds himself excluded from
the journals and societies, and almost universally ignored by the
competent workers in his field. In fact, the reputable scientist does
not even know of the crank's existence unless his work is given wide-
spread publicity through non-academic channels, or unless the scien-
tist makes a hobby of collecting crank literature. The eccentric is
forced, therefore, to tread a lonely way. He speaks before organizations
he himself has founded, contributes to journals he himself may edit,
and — until recently — publishes books only when he or his followers
can raise sufficient funds to have them printed privately.

A second characteristic of the pseudo-scientist, which greatly
strengthens his isolation, is a tendency toward paranoia. This is a
mental condition (to quote a recent textbook) "marked by chronic,
systematized, gradually developing delusions, without hallucinations,
and with little tendency toward deterioration, remission, or recovery."
There is wide disagreement among psychiatrists about the causes of
paranoia. Even if this were not so, it obviously is not within the scope
of this book to discuss the possible origins of paranoid traits in indi-
vidual cases. It is easy to understand, however, that a strong sense of
personal greatness must be involved whenever a crank stands in
solitary, bitter opposition to every recognized authority in his field.

If the self-styled scientist is rationalizing strong religious convic-
tions, as often is the case, his paranoid drives may be reduced to a
minimum. The desire to bolster religious beliefs with science can be
a powerful motive. For example, in our examination of George
McCready Price, the greatest of modern opponents of evolution, we
shall see that his devout faith in Seventh Day Adventism is a sufficient
explanation for his curious geological views. But even in such cases,
an element of paranoia is nearly always present. Otherwise the pseudo-
scientist would lack the stamina to fight a vigorous, single-handed
battle against such overwhelming odds. If the crank is insincere —

251

interested only in making money, playing a hoax, or both — then
obviously paranoia need not enter his make-up. However, very few
cases of this sort will be considered.

There are five ways in which the sincere pseudo-scientist's paranoid
tendencies are likely to be exhibited.

( 1 ) He considers himself a genius.

(2) He regards his colleagues, without exception, as ignorant
blockheads. Everyone is out of step except himself. Frequently he
insults his opponents by accusing them of stupidity, dishonesty, or
other base motives. If they ignore him, he takes this to mean his
arguments are unanswerable. If they retaliate in kind, this strengthens
his delusion that he is battling scoundrels.

Consider the following quotation: "To me truth is precious. ... I
should rather be right and stand alone than to run with the multitude
and be wrong. . . . The holding of the views herein set forth has
already won for me the scorn and contempt and ridicule of some of
my fellowmen. I am looked upon as being odd, strange, peculiar. . . .
But truth is truth and though all the world reject it and turn against
me, I will cling to truth still."

These sentences are from the preface of a booklet, published in
1931, by Charles Silvester de Ford, of Fairfield, Washington, in
which he proves the earth is flat. Sooner or later, almost every pseudo-
scientist expresses similar sentiments.

(3) He believes himself unjustly persecuted and discriminated
against. The recognized societies refuse to let him lecture. The jour-
nals reject his papers and either ignore his books or assign them to
"enemies" for review. It is all part of a dastardly plot. It never occurs
to the crank that this opposition may be due to error in his work.
It springs solely, he is convinced, from blind prejudice on the part
of the established hierarchy — the high priests of science who fear to
have their orthodoxy overthrown.

Vicious slanders and unprovoked attacks, he usually insists, are
constantly being made against him. He likens himself to Bruno,
GaHleo, Copernicus, Pasteur, and other great men who were unjustly
persecuted for their heresies. If he has had no formal training in the
field in which he works, he will attribute this persecution to a scientific
masonry, unwilling to admit into its inner sanctums anyone who has
not gone through the proper initiation rituals. He repeatedly calls
your attention to important scientific discoveries made by laymen.

(4) He has strong compulsions to focus his attacks on the great-
est scientists and the best-established theories. When Newton was the
outstanding name in physics, eccentric works in that science were
violently anti-Newton. Today, with Einstein the father-symbol of
authority, a crank theory of physics is likely to attack Einstein in the
name of Newton. This same defiance can be seen in a tendency to

252

Scientific Cranks

assert the diametrical opposite of well-established beliefs. Mathema-
ticians prove the angle cannot be trisected. So the crank trisects it.
A perpetual motion machine cannot be built. He builds one. There
are many eccentric theories in which the "pull" of gravity is replaced
by a "push." Germs do not cause disease, some modern cranks insist.
Disease produces the germs. Glasses do not help the eyes, said Dr.
Bates. They make them worse. In our next chapter we shall learn
how Cyrus Teed literally turned the entire cosmos inside-out, com-
pressing it within the confines of a hollow earth, inhabited only on
the inside.

(5) He often has a tendency to write in a complex jargon, in
many cases making use of terms and phrases he himself has coined.
Schizophrenics sometimes talk in what psychiatrists call "neologisms"
— words which have meaning to the patient, but sound like Jabber-
wocky to everyone else. Many of the classics of crackpot science
exhibit a neologistic tendency.

When the crank's I.Q. is low, as in the case of the late Wilbur
Glenn Voliva who thought the earth shaped like a pancake, he rarely
achieves much of a following. But if he is a brilliant thinker, he is
capable of developing incredibly complex theories. He will be able
to defend them in books of vast erudition, with profound observations,
and often liberal portions of sound science. His rhetoric may be enor-
mously persuasive. All the parts of his world usually fit together
beautifully, like a jig-saw puzzle. It is impossible to get the best of
him in any type of argument. He has anticipated all your objections.
He counters them with unexpected answers of great ingenuity. Even
on the subject of the shape of the earth, a layman may find himself
powerless in a debate with a flat-earther. George Bernard Shaw, in
Everybody's Political What's What?, gives an hilarious description of
a meeting at which a flat-earth speaker completely silenced all op-
ponents who raised objections from the floor. "Opposition such as
no atheist could have provoked assailed him"; writes Shaw, "and he,
having heard their arguments hundreds of times, played skittles with
them, lashing the meeting into a spluttering fury as he answered
easily what it considered unanswerable."

In the chapters to follow, we shall take a close look at the leading
pseudo-scientists of recent years, with special attention to native
specimens. Some British books will be discussed, and a few Conti-
nental eccentric theories, but the bulk of crank literature in foreign
tongues will not be touched upon. Very little of it has been trans-
lated into English, and it is extremely difficult to get access to the
original works. In addition, it is usually so unrelated to the American
scene that it loses interest in comparison with the work of cranks
closer home.

253

The laws of mechanics apply, of course, equally to all mat-
ter, and therefore to the athlete, to the grasshopper, and to
the physics professor too.

19 Physics and the Vertical Jump

Elmer L. Offenbacher

An article from the American Journal of Physics, 1970.

The physics of vertical jumping is described as an interesting and "relevant" illustration for
motivating students in a general physics course to master the kinematics and dynamics of one
dimensional motion. The equation for the height of the jump is derived (1) from the kine-
matic equations and Newton's laws of motion and (2) from the conservation of energy
principle applied to the potential and kinetic energies at two positions of the jump. The
temporal behavior of the reaction force and the center of gravity position during a typical
jump are discussed. Mastery of the physical principles of the jump may promote under-
standing of certain biological phenomena, aspects of physical education, and even of docu-
ments on ancient history.

I. INTRODUCTION

When the New York Mets recently won the
1969 World Series in basebalV the New York
Times carried a front page picture of one of the
players jumping for joy into the arms of another.
Jumping for joy might occur even in a physics
class if a student should suddenly realize that he
understands something new. The something new
can be on quite an old subject. This paper ^\^ll
present some aspects of the ancient subject of
jumping, the broad jump, and the high jump.^

The physics of the vertical jump, in particular, is
sufficiently simple in its basic elements that it can
be mastered by most students in an introductory
physics course. At the same time, it has the
appealing feature for our hippie-like alienated
college student of being relevant to so many
modern experiences. Neil Armstrong's ability to
jump up high on the moon,' or Bob Beamon's
record breaking broad jump in Mexico,* or just
plain off-the-record jumping on a dance floor or
basketball court are more exciting illustrations of
the pull of gravity to the average student — and
perhaps to some professors too — than are Galileo's
bronze balls rolling down inclined planes.* (No
slight to Galileo is intended!)

Should these examples not produce enough class
participation (or even if they do), the instructor

can liven things up by on the spot jumping
experiments. For example, he can suspend from the
ceiling some valuable coin (a "copper sandwich"
quarter will do too) which is just an inch or two
above the jumping reach of a six footer (about nine
feet from the floor). The instructor then might
announce that the coin will be given to whoever
can jump up and reach it. To the surprise of most
of the class, the six footer can't quite make it.
For the participation of the shorter members of
the class, one can suspend other coins at lower
levels and allow students in certain height ranges
to jump for specific coins.*

The problem of class involvement in a recitation
section of an introductory noncalculus physics
course was the stimulus which lead the author to
research the "science" of jumping; his findings
may perhaps provide other teachers with a
stimulant (legal and harmless) for their class
discussions.^

A description of the physics of vertical jumping
can be directed towards one or more of several
goals such as (a) appUcation of one dimensional
kinematic equations of motion, (b) illustration of
Newton's third law on reaction forces,' (c) study
of nonuniformly accelerated one dimensional
motion, (d) motivation for learning the derivation

254

or kinematical equations, and (e) application of
physical principles in other disciplines such as
zoology, physical education, and physiology.

For simpUcity, the presentation which follows
will be restricted primarily to the standing vertical
jump. However, some features can easily be
extended to other kinds of jumping such as the
standing broad jump or the swimmer's dive. The
latter examples illustrate the appUcation of the
kinematic equations in two dimensions.

n. THE PHYSICS OF JUMPING:
CLASS PRESENTATION

One can initiate the class discussion on jumping
by such questions as: How high can you jump?
Could you do better if you were in a high flying
airplane, if your legs were longer, if you were in

Fig. 1. Positions of the standing vertical jump: (a) lowest
crouched position, (b) position before losing contact with
the ground, (c) highest vertical position. F = foot, S =shin,
T = thigh, and B=back. 0 marks the position of the
center of gravity.

Philadelphia or in Mexico City, if you wore
sneakers or jumped with barefeet? How good is
man as a jumper compared to a kangaroo or a
grasshopper? In what way does an individual's
physical condition affect the maximum height of
his jump?

The order and nature of the presentation can be
varied according to the instructor's imagination.

Physics and the Vertical Jump

If the class is "willing" to be taught a derivation,
then one can start by deriving

d = Vot-\-^af.

(1)

It is useful to impress upon the students the fact
that this kinematical equation for uniform
acceleration can be derived purelj^ from the
definitions of average velocity and acceleration.^
Such a derivation can be easily mastered by
virtually every student.

Derivation of the Height Equation from the
Kinematic Equations and Newton's Third Law

The definition of uniform acceleration can be
written as

v/ = Vo-\-at (2)

where V/ is the final velocity, ^o is the original
velocity, a is the uniform acceleration, and t is the
interval over which the velocity change occurred.
Eliminating the time between Eqs. (1) and (2),
one obtains the well-known relation between V/, Vo,
and the distance d over which uniform acceleration
takes place:

v/ = VQ^-\-2ad

(3)

The total distance over which the displacement of
the center of gravity takes place during the jump
may be divided into two segments (see Fig. 1),
the stretching segment S and the free flight path
H. One can now apply Eq. (3) over the two differ-
ent segments as follows. For the stretching part
{which extends from the beginning of the crouched
position to the erect position before contact ^\ith
the ground is lost [Figs. 1(a) and 1(b)]} the
acceleration, a, is given by the average net upward
force on the jumper, F„, divided by the mass of
the jumper, m. As fo = 0, .substitution in Eq. (3)
gives

vjo'={2Fn/m)S, (4a)

where

F„ = Fr-mg,

Fr is the average reaction force of the ground on
the jumper during the upward displacement S.
We have labeled the final velocity at the end of
segments as Vjq, the jumping off velocity.

For the free flight path, however, the final

255

Fig. 2. Grasshopper's jump. Just before a takeoff all the joints of the hindlimbs of a grasshopper are tightly folded up at
the sides of the body. As soon as the jump begins these joints extend. The limbs extend to their maximum extent in
about 1/30 sec. (From J. Gray, Ref. 10.)

velocity at the highest position is zero, the ac- When solved for H this gives an equation ecjuiva-
celeration is —g, and Vq = Vjq. One then obtains H, lent to (5)
the displacement of the center of gravity from the

erect position to the highest point, [^Figs. (lb) and
1 (c)] from the equation

0 = v,o'-2gH. (4b)

Or, combining (4a) and (4b) one finally obtains

H = FnS/mg. (5)

Derivation Based on Conservation of Energy

An alternative way of deriving this result using
the conservation of energy principle is as follows :
Take the crouched position as the zero reference
potential. The total amount of work done on the
jumper by the floor during the push off period is
equal to the potential energy change, mgS, plus
the kinetic energy imparted at position S which is

FndS.

At the top of the jump the Idnetic energy is again
zero and the potential energy is nuj{H-\-S).
Therefore, from the conservation of energy
principle

m(i{H+S)=mgS+ ( F„(IS.

= /■ F.IS/

mg.

(6)

Note that FnS in Eq. (5) is replaced by the
integral

/ FndS.

This makes Eq. (6) valid for nonuniform accelera-
tion, whereas use of Eq. (3) in the previous
derivation involves the assumption of uniform
acceleration.

In a typical jump a man weighing 140 lb
producing an average reaction force during take
off of about 300 lb and able to stretch over a
distance /8 of 1 .4 ft \\\\\ lift his center of gravity
1.6 ft:

\ 140 /

X 1.4 = 1.6 ft.

With the help of Ya\. (5) the student should now
be able to answer many of the questions posed
earlier such as the effect of gravity or of the length
of one's legs. Consideration of Eq. (5) might also
give the student some clue about the remarkable

256

jumping ability of the grasshopper. A series of
positions during a grasshopper's jump are shown
in Fig. 2.^° The pictures were taken at intervals of
1/120 sec. Particularly noteworthy are (1) the
rapid take off for his jump (about 1/30 sec com-
pared to the human take off time of more than
half a second) and (2) the long stretch, ,S,
permitted because of his long hind legs and their
particular construction. (See Sec. IV for further
details). Indeed, these pictures might instill the
student high jump athlete (the one who may have
succeeded in getting the valuable coin) with a
bit of humility. Whereas for a superior athlete a
jump up to ^ his height is a creditable perform-
ance, an average grasshopper can jump well over
ten times his own height and even a 5-ft kangaroo
can jump up to 8ft above the ground!

m. GERRISH'S STANDING VERTICAL JUMP

In Fig. 3(a) is shown the time record of the
center of gravity position and the ground's
reaction force Fr in a typical vertical jump of Paul
H. Gerrish, the author of a 1934 Ph.D thesis
on the subject of jumping." It is interesting to
note that during the first 0.42 sec of the jump the
reaction force is less than the static weight, ^^ and
that the downward velocity reaches a maximum
of 3.8 ft/sec. After 0.61 sec, the velocity is zero
and the drop of the center of gravity is about
1.2 ft. The upward acceleration has a duration of
0.24 sec; during that period the reaction force
varies from about twice his weight to about 2.4
times his w^eight. The ''gravity controlled" part of
his jump (from lift off to the highest point) takes
about 0.3 sec while the total time for the entire
jump is less than 2 sec.

Apparatus

Gerrish designed his own force meter." This
was a device which minimized the vibration and
inertia forces and which transmitted the force of
the jump on a platform via hydrostatic pressure to
an Ashton single spring Bourbon-type pressure
gauge. He used a calibrated 16-mm movie camera
\N-ith a speed of 53.1 frames per second for timing
the sequence of positions. He obtained the
appropriate height in each frame by aligning, with
the aid of 22 (or 44) fold magnification, a refer-

Physics and the Vertical Jump

ence mark close to the center of gravity of his body
(over the anterior superior spine of the right
ilium) with the divisions of a surveyor's measuring
rod.

Statistical Results for Other Jumpers

In Gerrish's analysis of 270 jumps of 45
Columbia University men he found that the
tallest or heaviest jumpers did not always demon-
strate greater maximum forces, velocities, powers,
or height displacements for the jump than the
shortest or Ughtest jumpers, respectively." He also
noted that maximum height displacement varied
within the rather narrow range of 1 and 2 ft and
that the subjects demonstrated a range oi minimum
forces from 15%-74% of their static weight, and a
range of rnaximum forces from 210%-375% of this
weight.

Analysis of Jump

An interesting aspect of the jump is the energy
and power requirement for jumping. From the raw
data of Fig. 3(a) one can compute the velocity
curve and then construct a power curve by
multiplying the latter with the appropriate
ordinates of the force curve. These curves are
shown in Fig. 3(b). The reader may find it
interesting to analyze these curves in detail. It
should be noted that the force and height curves
are consistent with each other, thus providing
evidence for the validity of the measurements in
Fig. 3(a). This consistency was checked in two
ways. (1) The impulse imparted to the jumper at
the end of the stretch P,o can be calculated from
the integral

Fndt.

This value agrees to within 0.4% with the momen-
tum obtained from the free flight deceleration to
maximum height, PjQ = mvjo = mg{Ti — Ti). (2)
The force curve can be integrated twice with
respect to time and the resultant curve turns out
to agree well with the height curve.

The achievement in the vertical jump is
directly related to the power developed during the
jump. This in turn depends on the steepness of the
velocity curve and the ability to maintain close

257

350-

-6 50

■

cnorc . _

►

_."

200 -

(

-450

HEIGHTS

-425

'.\

> i

> .<

7 i.

' 1

0 1

III

1 12 1.3 14 1.5

TIME IN SECONDS

(a)

0 .1 .2 .3 .4 .5 .6 .7 8 .9 1.0 I.I 1.2 1.3 1.4 1.5
TIME IN SECONDS

(b)

Fig. 3. (a) The temporal sequence of the floor's reaction
(FORCE) and the center of gravity position (HEIGHT)
during a typical vertical jump beginning with the standing
position, (b) The velocity of the center of gravity
(VELOCITY) and the applied (POWER) [the product
of FORCE of Fig. 3(a) and VELOCITY of Fig. 3(b)].

to the maximum torce durmg the 0.2 sec of the
stretching segment^^ [see Fig. 3(b)].

IV. JUMPING AND OTHER DISCIPLINES

The student's motivation for learning a new
subject is usually enhanced if he is made to realize
its connection wth other studies he is undertaking
simultaneously or in which he has some innate
interest. To cite a few examples of how an under-
standing of the physics of jumping can be helpful

in other fields of study, let us turn to an application >
in biology (the jump of the grasshopper) and to i
two applications in physical education (Olympic '
records and the Sargent jump).

Biological Application

The outstanding animal jumpers include, in
addition to the kangaroo and grasshopper, the j
frog and the flea. Although these animals perform ■
much better than man when their jumps are
measured in terms of their body length,^® the ratio
of their broad jump to high jump lengths is be- ,
tween 3 and 4 which is similar to man's perform- :
ance. Actually, the fact that animals can jump ,
higher in terms of their own dimensions is not
surprising as can be shown by the following simple
scaling argument. If it is assumed that the strength
of an animal to exert a force F is proportional to i
the cross sectional area of his muscular tissue A
then F is proportional to U, where L specifies its
linear dimension. However, the mass m for con-
stant density is proportional to U. Therefore, the
acceleration, being equal to F/m, is proportional
to L~K As the stretch distance S is propor-
tional to L, one sees from Eq. (4a) that Vjo is
unaffected by a down scaling. Therefore, the
relatively large jumping achievements of the
smaller insects are not really too surprising be 'ause
if Vjo is unaffected, so is H [see Eq. (4b)].

In a 1958 article in Scientific American, a
grasshopper's physiology, responsible for its skill •
in jumping, is described. One of its secrets lies in
the construction of its hind legs. These legs differ
from those of most other insects in that the angle
between the femur (thigh) and the tibia (shin) is
not obtuse but acute. This permits a bigger value
for S and a longer period of possible acceleration as
was mentioned before. Another feature is that its
extensor muscle (which straightens the leg) is
larger than its flexor muscle (which bends it). It
can lift off from the ground a weight ten times its
own and develops, during this feat, a tension
equal to 250 times its own weight.

Jumping and Athletics

While jumping is a component of many sports
(for example, basketball, diving, skiing) , it has for
many centuries captured men's imagination in its

258

own right. Indeed, Beamon's record-breaking
broad jump of 29 ft 2f in. bettering the previous
world record by almost two feet (1 ft 9| in.) was
the sensation of the 1968 Mexico Olympics.^

Olympic Records and the Acceleration of Gravity

In comparing record performances, one should,
in all fairness, take account of variations in g that
exist between tw^o localities. ^^ From Eq. (5) one
can culculate the difference in height, AH, which
results from the difference in the g values between
two localities. Namely, the fractional change in H
is equal to the negative of the fractional change in
g: dkH/H = — Ag/g. As the maximum variation of g
on the surface of the earth is about ^%, in a seven
foot jump AH expressed in inches may be as much
as 0.42 in. (7X12X0.005). As high jumps and
broad jumps are customarily recorded to -j-th of an
inch or even Ys-th. of an inch, a fair comparison

Fig. 4. Comparison of the takeoff position in the standing
high jump, A, and the standing broad jump of a skilled
college woman. (From J. M. Cooper and R. B. Glassow,
Ref. 22.)

may very well reverse the standings of the record
holders. An examination of the broad iump winners
at the 1948 London Olympics (Latitude 5r30'N)
and the 1956 ^Melbourne Olympics (Latitude
37°52'S) shows indeed that W. Steele's (U.S.)
performance in London of 25 ft Sre in. was better
than G. Bell's (U.S.) jump of 25 ft 8i in. in
Melbourne when their jumps are compared at the
same g. The increase in g of 1.09 cm/sec^ from

Physics and the Vertical Jump

Melbourne to London accounts for a decrease in
jump length of 0.34 in.,^^ whereas their actual
recorded difference is only 0.19 in. This "injustice"
in reality didn't matter too much because both
records were below the 26 ft 5^^ in. record
established by Jesse Owens in the 1936 Olympics
in Berlin. This wasn't broken until the 1960
Olympics, which in turn was bettered phenom-
enally two years ago by Beamon.

The Sargent Jump
(or jumping as a test of athletic ability)

The standing vertical jump has been used for
close to half a century by persons interested in
tests and measurements in physical education. In
searching for a physical ability test which would
correlate with an individual's performance in
track and field events, it was found that a particu-
lar type of vertical jump, known as the Sargent
jump,2o produced a high correlation with such
events as the six second run, running high jump,
standing broad jump, and the shot put. The
instructions for the adminstration of the Sargent
jump read as follows:

the jumper is to swing his arms downward
and backward inclinging the body slightly
forward and bending the knees to about 90°
and raising the heels. He is to pause a
"moment" in this position and then to jump
vertically upward as high as possible,
swinging his arms vigorously forward and
upward to a vertical position. Just before the
highest point of the jump he is to swing his
arms forward and downward to his side. The
end of the downward swing should be timed so
as to coincide with the reaching of the highest
point of the jump. The legs should be
stretched downward and the head should be
stretched upward without tilting the chin.^^

Gerrish's jump was similar to the Sargent jump
except for the arm movements. In his thesis, he
makes the interesting observation that the loca-
tion of the center of gravity did not vary by more
than 5 in. for various body positions (with legs
bent, trunk inclined forward, etc.) assumed
during his jump. Apparently, the purpose of the
downward swing of the arms in the Sargent j ump
is to displace the center of gravity downward with

259

respect to the body so as to increase the height
reached by the top of the head.

Broad jumping

Broad jumping is closely related to the high
jump except that the initial velocity is at an angle
less than 90° with the horizontal. Ideally, if the
body were to be considered as a free projectile, for
maximum range the take off velocity should be
directed at 45° with the horizontal (see equation
in Ref. 19) . The actual take off directions of broad
jumpers turn out to be around 30°.

The take off position in the standing high jump
and the standing broad jump of a skilled college
woman is shown in Fig. 4.^2 Note that the differ-
ence in inclination of body segments is mainly due
to the difference in foot incHnation. In the broad
jump additional distance can be gained by shifting
the center of gravity backwards through the
motion of the arms, especially if the broad jumper
grasps weights in each hand. Recent experiments
have shown that the length of the jump could be
increased by 15-20 cm if the jumper holds 5-lb
weights in each hand.^^ These experiments were
conducted in a historical-philological study to
understand some "legends" of broad jumping
feats of Phajdlos of Kroton and Chionis of Sparta.
This study arrives at the conclusion that if the
pentathlon events of these two athletes consisted of
five partial jumps (i.e., five standing broad
jumps with a pause in between each jump), then
the record distances of 55 and 52 ft, respectively
attributed to the above athletic heroes are believ-
able; yet they represent superior performances
worthy of legendary transmission.

V. SUMMARY AND SUGGESTIONS FOR
JUMPING FARTHER

It is suggested in this paper that jumping
"exercises" could provide lively, student-involv-
ing, real physical situations to teach some of the
beginning fundamentals of mechanics. Realistic
discussion of the factors affecting the "altitude"
of a jump can be conducted with the help of Fig. 3
which describes the temporal behavior of the
reaction forces and center of gravity positions in a
typical jump. Although this provides only sample
curves for a specific type of jump, most likely the
general features are similar in many other kinds
of jumps such as the broad jump or the chalk and

Sargent jumps. The phase lag of the force curve
behind the position curve and the initial dip in the
force curve are common features of all of these
jumps.

The application of the laws of mechanics to
biology and athletics will motivate some students
to make the special effort required to understand
such concepts as Newton's third law, the relation-

Fig. ."). J. Robert Oppenheimer's jump as recorded in
Halsmau's Jump Book (Simon and Schuster, New York,

19.')*)).

260

Physics and the Vertical Jump

ship of velocity to position in nonuniform accelera-
tion, the physical concept of power and momen-
tum, the effect of gravity and its variation with
position, even the meaning of conservation of
energy when biological systems are involved.
Furthermore, to the interested and capable
student, it might provide the stimulus for con-
ducting some fairly simple experiments which will
provide useful information to the coach or athletic
director as well as possibly to the psychologist and
physiologist. These experiments may include
different kinds of jumps (i.e., jumping off with one
leg, the broad jump, or the Sargent jump) or
varying the footwear used, the jumping surface or
the rest period between jumps. It should also be
possible with today's advances in photographic
techniques and data analysis to improve and
enlarge on Gerrish's work to the point where the
jumping process could be analyzed reliably in
varied situations.

In searching for references on the physics of
jumping I came across an amusing book entitled
The Jump Book (Simon and Schuster, New York,
1959) by the prize winning Life photographer
Philippe Halsman, self-styled founder of the
"science" of Jumpology. In describing and
categorizing the jumps of the famous people who
his s5Tichromzed camera caught up in the air, he
jovally suggests that jumping could be used as a
psychological test (a la Rorschach), and its analysis
could constitute a new field of psychological in-
vestigation which he named Jumpology.

Although the persons in these photographs were
evidently not instructed to jump for height, some
of them did seem to reach for that goal. One of
them was J. Robert Oppenheimer (Fig. 5). Who
knows whether having students jump up in the
physics laboratory might instantaneously identify
a potentially great scientist. One might just com-
pare the student's jump with the one by J.
Robert Oppenheimer.

1 The New York Mets won the 1969 World Series against
the Baltimore Orioles after a phenomenal rise from last
place to take the Eastern Division title and the National
League pennant.

2 A certain kind of broad jump was one of the five track
and field events of the annual Tailteann games held at
Tailtu, County Meath, Ireland as early as 1829 B.C. It
was also one of the features of the Pentathlon in the
ancient Olympic games (Encyclopedia Brittanica, 1967
Edition, Vol. 13, p. 132).

» New York Times, July 21, p. 1, Col. 3 (1969).

* New York Times, Oct. 27, Sect. 5, p. 3, Col. 6 (1968) ;
World Almanac, p. 878 (1969).

' Galileo's Two New Sciences (1638).

* The average vertical jumping height with ordinary
shoes and in shirt sleeves is about 19 inches. This figure
is based on a study made by Franklin Henry [Res. Quart.
13, 16 (1942) ] of 61 male students aged 19-24 years at the
Berkely campus of the University of California.

' H. R. Crane suggested in a recent article in this
Journal [36, 1137 (1968)] that relevant examples and
exercises be incorporated into noncalculus physics course.
An exercise involving the physics of jumping can be found

in, S. Borowitz and A. Beiser, Essentials of Physics
[(Addison- Wesley Publ. Corp., Reading, Mass., 1966),
Chap. 5, Problem 7.] Even though this is a calculus level
text, students usually are not able to solve this problem
without the use of the energy conservation principle,
which is described only later on in the text in Chap. 7.
Also see F. W. Sears and M. W. Zemansky, University
Physics (1964) Problem 6-16.

* The third law also explains the physics of walking.
For a note on the physics of walking see R. M. Sutton,
Amer. J. Phys. 23, 490 (1955).

» J. G. Potter, Amer. J. Phys. 35, 676 (1967).

1" J. Gray, How Animals Move (Cambridge University
Press, London, 1953), opposite p. 70.

" P. H. Gerrish, A Dynamical Analysis of the Standing
Vertical Jump, Ph.D. thesis Teachers College, Columbia
University, 1934.

" In all 270 tests of 45 other jumpers, he found this time
always to be less than 0.5 sec.

" Reference 11, p. 7.

" In the 1960 Rome Olympics, the much shorter Russian
Shav Lakadze won the high jump gold medal whereas the
tall world record holder at that time, John Thomas,
barely got the bronze medal. [^Olympic Games 1960,
H. Lechenperg, Ed. (A. S. Barnes & Co., 1960), p. 198.]
Also see H. Krakower, Res. Quart. 12, 218 (1941).

»^ See article by R. M. Sutton [Amer. J. Phys. 23, 490
(1955) ] for a discussion of the forces developed in the foot.

i« Reference 10, p. 69.

1' G. Hogle, Scientific American 198, 30 (1958).

1* This was pointed out by P. Kirkpatrick, Scientific
American 11, 226 (1937) ; Amer. J. Phys. 12, 7 (1944).

"The range formula applicable to broad jumping, i.e.,
R = Vit^ sin2a/g, where a is the angle of the jumping off
direction with the horizontal also gives AR/R= —Ag/g.
Therefore, AR=2o.7X12 (in.) X1.09/978=0.34 in.

» D. A. Sargent, Amer. Phys. Ed. Rev. 26, 188 (1921).

"D. Van Dalen, Res. Quart. 11, 112 (1940). Also see
footnote cited in Ref. 6.

^ J. M. Cooper and R. B. Glassow, Kinesiology (C. V.
Mosby Co., St. Louis, Mis.souri, 1963).

^Joachim Ebert: Zum Pentathlon Der Antike Unter-
suchungen uber das System der Siegerermittlung und die
Ausfiihrung des Halterensprunges, Abhandlung der
Sachsichen Akademie der Wissenschaft zur Leipzig —
Philologisch-Historische Klasse Band 56 Heft 1, Akademie
Verlag, Berlin (1963).

261

Authors and Artists

LEO L. BERANEK

Leo L. Beranek is director of Bolt Beranek and
Newman Inc., a consulting company in communi-
cations physics in Cambridge, Massachusetts.
He has been associated with MIT since 1946, and
was the director of the Electro-Acoustics Labora-
tory at Harvard during World War II. He is presi-
dent of Boston Broadcasters, Inc. He has done
work in architectural acoustics (such as designing
auditoriums), acoustic measurements, and noise
control.

JACOB BRONOWSKI

Jacob Bronowski, who received his Ph.D. from
Cambridge University in 1933, is now a Fellow of
the Solk Institute of Biological Studies in Califor-
nia. He has served as Director of General Pro-
cess Development for the National Coal Board of
England, as the Science Deputy to the British
Chiefs of Staff, and as head of the Projects
Division of UNESCO. In 1953 he was Carnegie
Visiting Professor at the Massachusetts Institute
of Technology.

ALEXANDER CALANDRA
Alexander Calandra, Associate Professor of
Physics at Washington University, St. Louis,
since 1950, was born in New York in 1911. He
received his B.S. from Brooklyn College and his
Ph.D. in statistics from New York University. He
has been a consultant to the American Council for
Education and for the St. Louis Public Schools,
has taught on television, and has. been the re-
gional counselor of the American Institute of
Physics for Missouri.

ARTHUR C. CLARKE

Arthur C. Clarke, British scientist and writer, is a
Fellow of the Royal Astronomical Society. During
World War II he served as technical officer in charge
of the first aircraft ground-controlled approach
project. He has won the Kalingo Prize, given by
UNESCO for the popularization of science. The
feasibility of many of the current space develop-
ments was perceived and outlined by Clarke in the
1930's. His science fiction novels include
Childhoods End and The City and the Stars.

ROBERT MYRON COATES

Robert Myron Coates, author of many books and
articles, was born in New Haven, Connecticut, in
1897 and attended Yale University. He is a mem-
ber of the National Institute of Arts and Letters
and has been an art critic for The New Yorker
magazine. His books include The Eater of Dark-

ness, The Outlaw Years, The Bitter Season, and
The View From Here.

E. J. DIJKSTERHUIS

E. J. Dijksterhuis was born at Tilburg, Holland, in
1892, and later became a professor at the University
of Leyden. Although he majored in mathematics ond
physics, his school examinations forced him to
take Latin and Greek, which awakened his inter-
est in the early classics of science. He published
important studies on the history of mechanics, on
Euclid, on Simon Steven and on Archimedes.
Dijksterhuis died in 1965.

ALBERT EINSTEIN

Albert Einstein, considered to be the most creative
physical scientist since Newton, was nevertheless
a humble and sometimes rather shy man. He was
born in Ulm, Germany, in 1879. He seemed to learn
so slowly that his parents feared that he might be
retarded. After graduating from the Polytechnic
Institute in Zurich, he became a junior official at
the Patent Office at Berne, At the age of twenty-six,
and quite unknown, he published three revolutionary
papers in theoretical physics in 1905. The first
paper extended Max Planck's ideas of quantization
of energy, and established the quantum theory of
radiation. For this work he received the Nobel
Prize for 1921. The second paper gave a mathe-
matical theory of Brownian motion, yielding a cal-
culation of the size of a molecule. His third paper
founded the special theory of relativity. Einstein's
later work centered on the general theory of rela-
tivity. His work has a profound influence not only
on physics, but also on philosophy. An eloquent
and widely beloved man, Einstein took an active
part in liberal and anti-war movements. Fleeing
Nazi Germany, he settled in the United States in
1933 at the Institute for Advanced Study in
Princeton. He died in 1955.

RICHARD PHILLIPS FEYNMAN

Richard Feynmon was born in New York in 1918,
and graduated from the Massachusetts Institute of
Technology in 1939. He received his doctorate in
theoretical physics from Princeton in 1942, and
worked at Los Alamos during the Second World
War. From 1945 to 1951 he taught at Cornell, and
since 1951 hos been Tolman Professor of Physics
at the California Institute of Technology. Pro-
fessor Feynmon received the Albert Einstein
Award in 1954, and in 1965 was named o Foreign
Member of the Royal Society. In 1966 he wos
awarded the Nobel Prize in Physics, which he
shared with Shinichero Tomonogo and Julian
Schwinger, for work in quantum field theory.

R. J. FORBES

R.J. Forbes, professor at the University of

Amsterdam, was born in Breda, Holland, in 1900.

262

After studying chemical engineering, he worked
for the Royal Dutch Shell Group in their labora-
tories and in refineries in the East indies.
Interested in archaeology and museum collections,
he has published works on the history of such
fields OS metallurgy, alchemy, petroleum, road-
building, technology, and distillation.

KENNETH W. FORD

Kenneth W. Ford was born in 1917 ot West Palm
Beach, Florida. He did his undergraduate work at
Harvard College. His graduate work at Princeton
University was interrupted by two years at Los
Alamos and at Project Manhattan in Princeton.
He worked on a theory of heavy elementary par-
ticles at the Imperial College in London, and at
the Max Planck Institute in Gbttingen, Germany.
Before joining the faculty at the University of
California, Irvine, as chairman of the Department
of Physics, Mr. Ford was Professor of Physics
at Brandeis University.

GEORGE GAMOW

George Gamow, a theoretical physicist from Russia,
received his Ph.D. in physics at the University of
Leningrad. At Leningrad he became professor after
being a Carlsberg fellow and a university fellow at
the University of Copenhagen and a Rockefeller
fellow at Cambridge Uni versi ty . He come to the
United States in 1933 to teach at the George
Washington University and later at the University
of Colorado. His popularization of physics are
much admired.

MARTIN GARDNER

Martin Gardner, well-known editor of the 'Mathe-
matical Games" department of the Scientific
American, was born in Tulsa, Oklahoma, in 1914.
He received a B.A. in philosophy from the Univer-
sity of Chicago in 1939, worked as a publicity
writer for the University, and then wrote for the
Tulsa Tribune. During World Wor II he served in
the Navy. Martin Gardner has written humorous
short stories as well as serious articles for such
journals as Scripta Mathematica and Philosophy
of Science, and is the best-selling author of The
Annotated Alice, Relativity for the Million, Math,

Magic, and Mystery, as well as two volumes of the
Scientific American Book of Mathemotical Puzzles

ind D

[versions.

GERALD HOLTON

Gerald Holton received his early education in
Vienna, at Oxford, and at Wesieyan University,
Connecticut. He has been at Harvard University
since receiving his Ph.D. degree in physics there
in 1948; he is Professor of Physics, teaching
courses in physics as well as in the history of

Authors and Artists

science. He was the founding editor of the
quarterly Daedalus. Professor Holton's experi-
mental research is on the properties of matter
under high pressure. He is co-director of
Harvard Project Physics.

CARLEEN MALEY HUTCHINS

Carleen Hutchins was born in Springfield, Massa-
chusetts, in 1911. She received her A.B. from
Cornell University and her M.A. from New York
University. She has been designing and construc-
ting stringed instruments for years. Her first step
was in 1942 when "I bought an inexpensive weak-
toned viola because my musical friends complained
that the trumpet I had played was too loud in cham-
ber music, as well as out of tune with the strings —
and besides they needed a viola." In 1947, while on
0 leave of absence from the Brearley School in New
York, she started making her first viola — it took
two years. She has made over fifty, selling some
to finance more research. In 1949 she retired from
teaching and then collaborated with Frederick A.
Saunders at Harvard in the study of the acoustics
of the instruments of the violin family. She has
had two Guggenheim fellowships to pursue this
study.

LEOPOLD INFELD

Leopold Infeld, a co-worker with Albert Einstein in
general relativity theory, was born in 1898 in
Poland. After studying at the Cracow and Berlin
Universities, he became a Rockefeller Fellow at
Cambridge where he worked with Max Born in
electromagnetic theory, and then a member of the
Institute for Advanced Study at Princeton. For
eleven years he was Professor of Applied Mathe-
matics at the University of Toronto. He then re-
turned to Poland and became Professor of
Physics at the University of Warsaw and until his
death on 16 January 1968 he was director of the
Theoretical Physics Institute at the University.
A member of the presidium of the Polish Academy
of Science, Infeld conducted research in theoretical
physics, especially relativity and quantum theories.
Infeld was the author of The New Field Theory,
The World in Modern Science, Quest, Albert Einstein,

and with Einstein The Evolution of Physics.

JAMES CLERK MAXWELL

See J. R. Newman's articles in Readers 3 and 4.

ROBERT B. MOORE

Robert B. Moore was born in Windsor, Newfound-
land in 1935. He attended McGill University in
Canada as an undergraduate, continued for his
Ph.D. in physics, and remained there as a pro-
fessor. He is a nuclear physicist, specializing
in nuclear spectroscopy.

263

Authors and Artists

JAMES ROY NEWMAN

James R. Newman, lawyer and mathematician,
was born in New York City in 1907. He received
his A.B. from the College of the City of New York
and LL.B. from Columbia. Admitted to the New York
bar in 1929, he practiced there for twelve years.
During World War II he served as chief intelligence
officer, U.S. Embassy, London, and in 1945 as
special assistant to the Senate Committee on
Atomic Energy. From 1956-57 he was senior
editor of The New Republic, and since 1948 had
been a member of the board of editors for
Scientific American where he was responsible for
the book review section. At the same time he was
a visiting lecturer at the Yale Law School. J.R.
Newman is the author of What is Science?, Science
and Sensibility, and editor of Common Sense of the
Exact Sciences. The World of Mathematics, and

the Harper Encyclopedia of Science. He died in
19661

ELMER L. OFFENBACHER

Elmer L. Offenbacher, born in Germany in 1923,
was educated at Brooklyn College and University
of Pennsylvania, and is professor of physics at
Temple University in Philadelphia. His primary
research field is solid state physics.

ERIC MALCOLM ROGERS

Eric Malcolm Rogers, Professor of Physics at
Princeton University, was born in Bickley,
England, in 1902. He received his education at
Cambridge and later was a demonstrator at the
Cavendish Laboratory. Since 1963 he has been the
organizer in physics for the Nuffield Foundation
Science Teaching Project. He is the author of the
textbook. Physics for the Inquiring Mind.

RICHARD STEVENSON

Richard Stevenson was born in Windsor, Ontario in
1931. He obtained a degree in mechanical engineer-
ing from MIT in 1957, and is now associate pro-
fessor of physics at McGill University in Canada.
He does research on the magnetic properties of
solids and high pressure physics.

PETER GUTHRIE TAIT

Peter Guthrie Tait, collaborator of William Thomson
(Lord Kelvin) in thermodynamics, was born at Dal-
keith, Scotland, in 1831. He was educated at the
Academy of Edinburgh (where James Clerk Maxwell
was also a student), and at Peterhouse, Cambridge.
He remained at Cambridge as o lecturer before
becoming Professor of Mathematics at Queen's
College, Belfast. There he did research on the
density of ozone and the action of the electric dis-
charge of oxygen and other gases. From 1860 until
his death in 1901 he served as Professor of Natur*
al Philosophy at Edinburgh. In 1864 he published

his first important paper on thermodynamics and
thermoelectricity ond thermal conductivity. With
Lord Kelvin he published the textbook Elements
of Natural Philosophy in 1867.

BARON KELVIN, WILLIAM THOMSON

Baron Kelvin, William Thomson, British scientist
and inventor, was born in Belfast, Ireland, in 1824.
At the age of eleven he entered the University of
Glasgow where his father was professor of mathe-
matics. In 1841 he went to Peterhouse, at Cam-
bridge University. In 1848 Thomson proposed a
temperature scale independent of the properties
of any particular substance, and in 1851 he
presented to the Royal Society of Edinburgh a
paper reconciling the work on heat of Sadi
Carnot with the conclusions of Count von
Rumford, Sir Humphrey Davy, J.R. von Mayer
and J. P. Joule. In it he stated the Second Law
of Thermodynamics. Lord Kelvin worked on such
practical applications as the theory of submarine
cable telegraphy and invented the mirror galvano-
meter. In 1866 he was knighted, 1892 raised to
peerage, and in 1890 elected president of the
Royal Society. He died in 1907.

LEONARDO DA VINCI

Leonardo da Vinci, the exemplor of 'I'uomo univer-
sale," the Renaissance ideal, was born in 1452 near
Vinci in Tuscany, Italy. Without a humanistic edu-
cation, he was apprenticed at an early age to the
painter-sculptor Andrea del Verrocchio. The first
10 years of Leonardo's career were devoted
to painting, culminating in the 'Adoration of the
Magi." Defensive to criticisms on his being "un-
lettered," Leonardo emphasized his ability as in-
ventor and engineer, becoming a fortification ex-
pert for the militarist Cesare Borgia. By 1503 he
was working as on artist in almost every field.
"Mono Lisa" and "The Last Supper" are among
the world's most famous paintings. Besides his
engineering feats such as portable bridges, ma-
chine guns, tanks, and steam cannons, Leonardo
contrived highly imaginative blueprints such as
the protoheliocop ter and a flying machine. His
prolific life terminated in the Castle of Cloux
near Amboise on May 2, 1519.

HARVEY ELLIOTT WHITE

Harvey Elliott White, Professor of Physics at the
University of California, Berkeley, wos born in
Parkersburg, West Virginia in 1902. He attended
Occidental College and Cornell University where
he received his Ph.D. in 1929. In 1929-30 he was
an Institute Research Fellow at the Physics and
Technology Institute in Germany. His special
interests are atomic spectra and ultraviolet and
infrared optics.

264

Holt/Rinehart/Winston

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