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Full text of "Reader 1 - Concepts of Motion, Project Physics"

The Project Physics Course 



Reader 



1 



Concepts of Motion 




The Project Physics Course 



Reader 



UNIT 



"l Concepts of Motion 



A Component of the 
Project Physics Course 




Published by 

HOLT, RINEHART and WINSTON, Inc. 

New York, Toronto 



This publication is one of the many 
instructional materials developed for the 
Project Physics Course. These materials 
include Texts, Handbooks, Teacher Resource 
Books, Readers, Programmed Instruction 
Booklets, Film Loops, Transparencies, 16mm 
films and laboratory equipment. Development 
of the course has profited from the help of 
many colleagues listed in the text units. 



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

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

p. 91 Dr. Harold E. Edgerton, Massachusetts Institute 
of Technology, Cambridge. 



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 



Copyright © 1970, Project Physics 

All Rights Reserved 

SBN 03-084558-0 

1234 039 98765432 

Project Physics is a registered trademark 



Picture Credits 

Cover photo and photo facing page 1 by Herbert 
Matter, of Alexander Calder's "Hanging Mobile, 1936.' 
Courtesy of the Museum of Modern Art, New York 
from the collection of Mrs. Meric Callery, New York. 



2 4 

5 I 

3 * 



(1) 
(2) 

(3) 
(4) 



Photo by Glen J. Pearcy. 

Jeune fille au corsage rouge lisant. Jean Baptiste 

Camille Corot. Painting. Collection BiJhrle, Zurich. 

Harvard Project Physics staff photo. 

Femme lisant. Georges Seurat, Conte crayon 

drawing. Collection C. F. Stoop, London. 



Sources and Acknowledgments 
Project Physics Reader 1 

1. The Value of Scierice, by Richard P. Feynman, 
in Frontiers in Science, edited by Edward 
Hutchings, Jr., Basic Books, Inc., Publishers, 
New York, copyright © 1958. Reprinted with 
permission. 

2. Close Reasoning, by Fred Hoyle, in The Black 
Cloud, Harper & Row, Publishers, Inc., New 
York, copyright © 1957 by Fred Hoyle. 
Reprinted with permission. 

3. On Scientific Method, by P. W. Bridgman, 
in Reflections of a Physicist. Reprinted with 
permission of the Philosophical Library, Inc., 
Publishers, New York, copyright © 1955. 

4. How To Solve It, by G. Polya, in How To Solve It. 
Reprinted with permission of Princeton University 
Press, copyright © 1957. 

5. Four Pieces of Advice to Young People, by Warren 
Weaver, a talk given in Seattle during the Arches 
of Science Award. Copyright © January 1966 

by The Tennessee Teacher, publishers. Reprinted 
with permission. 

6. On Being the Right Size, by J. B. S. Haldane, 
copyright 1928 by Harper & Brothers, copyright © 
1956 renewed by J. B. S. Haldane. Reprinted with 
permission of Harper & Row, Publishers, and 

Mrs. Helen Spurway Haldane and Chatto 
and Windus, Ltd. 

7. Motion in Words, from Motion by James B. 
Gerhart and Rudi Nussbaum, copyright © 1966, 
The University of Washington, Seattle. Reprinted 
with permission. 

8. Motion, by Richard P. Feynman, Robert B. 
Leighton, and Matthew Sands from The Feynman 
Lectures on Physics, Vol. I, copyright © 1963 
by Addison-Wesley Publishing Company, Inc. 
Reprinted with permission. 

9. Representation of Movement, by Gyorgy Kepes, 
from Language of Vision, copyright 1944 by Paul 
Theobald and Company, Chicago, III. Reprinted 
with permission. 

10. Introducing Vectors, from About Vectors, by 
Banesh Hoffmann, copyright © 1966 by Prentice- 
Hall, Inc. Reprinted with permission. 

11. Galileo's Discussion of Projectile Motion, from 
Foundations of Modern Physical Science, by 



Gerald Holton and Duane H. D. Roller, copyright 
© 1958 by Addison-Wesley Publishing Company, 
Inc. Reprinted with permission. 

12. Newton's Law of Dynamics, by Richard P. 
Feynman, Robert B. Leighton, and Matthew Sands, 
from The Feynman Lectures on Physics, Vol. I, 
copyright © 1963 by Addison-Wesley Publishing 
Company, Inc. Reprinted with permission. 

13. The Dynamics of a Golf Club, by C. L. Stong, 
copyright © 1964 by Scientific American, Inc. 
All rights reserved. Reprinted with permission. 

14. Bad Physics in Athletic Measurements, by Paul 
Kirkpatrick, from The American Journal of 
Physics, Vol. 12, copyright 1944. Reprinted 
with permission. 

15. The Scientific Revolution, by Herbert Butterfield, 
copyright © 1960 by Scientific American, Inc. 
All rights reserved. Reprinted with permission. 
Available separately at 200 each as Offprint 
No. 607 from W. H. Freeman and Company, 
660 Market Street, San Francisco, California. 

16. How the Scientific Revolution of the Seventeenth 
Century Affected Other Branches of Thought, 



by Basil Willey, from A Short History of Science, 
a symposium, published in 1951. Reprinted 
with permission. 

17. Report on Tait's Lecture on Force, at British 
Association, 1876, by James Clerk Maxwell, from 
the Life of James Clerk Maxwell. Macmillan & 
Company, London, 1884. 

18. Fun in Space, by Lee A. DuBridge in The American 
Journal of Physics. November 1960. Reprinted 
with permission. 

19. The Vision of Our Age, from Insight by J. 
Bronowski, copyright © 1964 by J. Bronowski. 
Reprinted with permission of Harper & Row, 
Publishers, and Curtis Brown, Ltd. 

20. Becoming a Physicist, from The Making of a 
Scientist, by Anne Roe. Reprinted with permission 
of Dodd, Mead and Co., and Brandt & Brandt. 

21 . Chart of the Future, by Arthur C. Clarke, from 
Profiles of the Future — An Inquiry into the Limits 
of the Possible, by Arthur C. Clarke, copyright © 
1962 by Arthur C. Clarke. Reprinted with permis- 
sion of Harper & Row, Publishers, and Victor 
Gollancz, Ltd. 



Ill 










IV 




This is not a physics textbook. Rather, it is a 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 reader will guide you 
for further reading. 



%» 









Reader 1 
Table of Contents 

1 The Value of Science 1 

Richard P. Feynman 

2 Close Reasoning 7 

Fred Hoyle 

3 On Scientific Method 1 8 

Percy W. Bridgman 

4 How to Solve It 20 

George Polya 

5 Four Pieces of Advice to Young People 21 

Warren Weaver 

6 On Being the Right Size 23 

J. B. S. Haldane 

7 Motion in Words 28 

James B. Gerhart and Rudi H. Nussbaum 

8 Motion 31 

Richard P. Feynman, Robert B. Leighton and Matthew Sands 

9 Representation of Movement 44 

Gyorgy Kepes 

1 Introducing Vectors 60 

Banesh Hoffmann 

1 1 Galileo's Discussion of Projectile Motion 72 

Gerald Holton and Duane H. D. Roller 

1 2 Newton's Laws of Dynamics 77 

Richard P. Feynman, Robert B. Leighton and Matthew Sands 

1 3 The Dynamics of a Golf Club 91 

C. L. Stong 



VI 






1 4 Bad Physics in Athletic Measurements 95 

p. Kirkpatrick 

1 5 The Scientific Revolution 1 01 

Herbert Butterfield 

1 6 How the Scientific Revolution of the Seventeenth 1 09 
Century Affected Other Branches of Thought 

Basil Willey 

1 7 Report on Tait's Lecture on Force, at British 116 
Association, 1876 

James Clerk Maxwell 

18 Fun in Space 117 

Lee A. DuBridge 

1 9 The Vision of Our Age 1 22 

J. Bronowski 

20 Becoming a Physicist 133 

Anne Roe 

21 Chart of the Future 148 

Arthur C. Clarke 



VII 




A still photo of the Calder mobile shown in motion 
on the cover. 



An outstanding contemporary theoretical physicist rem- 
inisces informally about science and its role in society. 
Feynman stresses the importance in science, and else- 
where, of admitting that one does not know all the an- 
swers. 



The Value of Science 

Richard P. Feynman 

An excerpt from Frontiers of Science, 1 958. 

From time to time, people suggest to me that scientists ought 
to give more consideration to social problems— especially that 
they should be more responsible in considering the impact of 
science upon society. This same suggestion must be made to 
many other scientists, and it seems to be generally believed that 
if the scientists would only look at these very difficult social 
problems and not spend so much time fooling with the less vital 
scientific ones, great success would come of it. 

It seems to me that we do think about these problems 
from time to time, but we don't put full-time effort into them— 
the reason being that we know we don't have any magic for- 
mula for solving problems, that social problems are very much 
harder than scientific ones, and that we usually don't get any- 
v/here when we do think about them. 

I believe that a scientist looking at nonscientific problems is 
just as dumb as the next guy— and when he talks about a non- 
scientific matter, he will sound as naive as anyone untrained in 
the matter. Since the question of the value of science is not a 
scientific subject, this discussion is dedicated to proving my 
point— by example. 

The first way in which science is of value is famihar to every- 
one. It is that scientific knowledge enables us to do all kinds 
of things and to make all kinds of things. Of course if we make 
good things, it is not only to the credit of science; it is also to 
the credit of the moral choice which led us to good work. Sci- 
entific knowledge is an enabling power to do either good or 
bad— but it does not carry instructions on how to use it. Such 
power has evident value— even though the power may be negated 
by what one does. 

I learned a way of expressing this common human problem 
on a trip to Honolulu. In a Buddhist temple there, the man in 
charge explained a little bit about the Buddhist religion for 
tourists, and then ended his talk by telling them he had some- 
thing to say to them that they would never forget— and I have 
never forgotten it. It was a proverb of the Buddhist religion: 



"To every man is given the key to the gates of heaven; the 
same key opens the gates of hell." 

What then, is the value of the key to heaven? It is true that 
if we lack clear instructions that determine which is the gate to 
heaven and which the gate to hell, the key may be a dangerous 
object to use, but it obviously has value. How can we enter 
heaven without it? 

The instructions, also, would be of no value without the key. 
So it is evident that, in spite of the fact that science could 
produce enormous horror in the world, it is of value because it 
can produce something. 

Another value of science is the fun called intellectual enjoy- 
ment which some people get from reading and learning and 
thinking about it, and which others get from working in it. This 
is a very real and important point and one which is not con- 
sidered enough by those who tell us it is our social responsi- 
bility to reflect on the impact of science on society. 

Is this mere personal enjoyment of value to society as a 
whole? No! But it is also a responsibihty to consider the value 
of society itself. Is it, in the last analysis, to arrange things so 
that people can enjoy things? If so, the enjoyment of science is 
as important as anything else. 

But I would like not to underestimate the value of the world 
view which is the result of scientific effort. We have been led 
to imagine all sorts of things infinitely more marvelous than 
the imaginings of poets and dreamers of the past. It shows that 
the imagination of nature is far, far greater than the imagination 
of man. For instance, how much more remarkable it is for us 
all to be stuck— half of us upside down— by a mysterious attrac- 
tion, to a spinning ball that has been swinging in space for bil- 
lions of years, than to be carried on the back of an elephant 
supported on a tortoise swimming in a bottomless sea. 

I have thought about these things so many times alone that 
I hope you will excuse me if I remind you of some thoughts 
that I am sure you have all had— or this type of thought— which 
no one could ever have had in the past, because people then 
didn't have the information we have about the world today. 
For instance, I stand at the seashore, alone, and start to think. 
There are the rushing waves . . . mountains of molecules, each 
stupidly minding its own business . . . trillions apart . . . yet 
forming white surf in unison. 

Ages on ages . . . before any eyes could see . . . year after 
year . . . thunderously pounding the shore as now. For whom, 
for what? ... on a dead planet, with no life to entertain. 

Never at rest . . . tortured by energy . . . wasted prodigiously 

by the sun . . . poured into space. A mite makes the sea roar. 

Deep in the sea, all molecules repeat the patterns of one 

another till complex new ones are formed. They make others 

like themselves . . . and a new dance starts. 



The Value of Science 



Growing in size and complexity . . . living things, masses 
of atoms, DNA, protein . . . dancing a pattern ever more intricate. 

Out of the cradle onto the dry land . . . here it is standing 
. . . atoms with consciousness . . . matter with curiosity. 

Stands at the sea . . . wonders at wondering ... I ... a uni- 
verse of atoms ... an atom in the universe. 

THE GRAND ADVENTURE 

The same thrill, the same awe and mystery, come again 
and again when we look at any problem deeply enough. With 
more knowledge comes deeper, more wonderful mystery, luring 
one on to penetrate deeper still. Never concerned diat the an- 
swer may prove disappointing, but with pleasure and confidence 
we turn over each new stone to find unimagined strangeness 
leading on to more wonderful questions and mysteries— certainly 
a grand adventure! 

It is true that few unscientific people have this particular 
type of religious experience. Our poets do not write about it; 
our artists do not try to portray this remarkable thing. I don't 
know why. Is nobody inspired by our present picture of the 
universe? The value of science remains unsung by singers, so 
you are reduced to hearing— not a song or a poem, but an eve- 
ning lecture about it. This is not yet a scientific age. 

Perhaps one of the reasons is that you have to know how to 
read the music. For instance, the scientific article says, perhaps, 
something hke this: "The radioactive phosphorus content of 
the cerebrum of the rat decreases to one-half in a period of 
two weeks." Now, what does that mean? 

It means that phosphorus that is in the brain of a rat (and 
also in mine, and yours) is not the same phosphorus as it was 
two weeks ago, but that all of the atoms that are in the brain 
are being replaced, and the ones that were there before have 
gone away. 

So what is this mind, what are these atoms with conscious- 
ness? Last week's potatoes! That is what now can remember 
what was going on in my mind a year ago— a mind which has 
long ago been replaced. 

That is what it means when one discovers how long it takes 
for the atoms of the brain to be replaced by other atoms, to 
note that the thing which I call my individuality is only a pat- 
tern or dance. The atoms come into my brain, dance a dance, 
then go out; always new atoms but always doing the same 
dance, remembering what the dance was yesterday. 

THE REMARKABLE ffiEA 

When we read about this in the newspaper, it says, *TTie 
scientist says that this discovery may have importance in the 
cure of cancer." The paper is only interested in the use of the 
idea, not the idea itself. Hardly anyone can understand the 



importance of an idea, it is so remarkable. Except that, possibly, 
some children catch on. And when a child catches on to an 
idea like that, we have a scientist. These ideas do filter down ( in 
spite of all the conversation about TV replacing thinking), and 
lots of kids get the spirit— and when they have the spirit you 
have a scientist. It's too late for them to get the spirit when they 
are in our universities, so we must attempt to explain these ideas 
to children. 

I would now like to turn to a third value that science has. 
It is a little more indirect, but not much. The scientist has a 
lot of experience with ignorance and doubt and uncertainty, 
and this experience is of very great importance, I think. When 
a scientist doesn't know the answer to a problem, he is ig- 
norant. When he has a hunch as to what the result is, he is 
uncertain. And when he is pretty dam sure of what the result 
is going to be, he is in some doubt. We have found it of para- 
mount importance that in order to progress we must recog- 
nize the ignorance and leave room for doubt. Scientific knowl- 
edge is a body of statements of varying degrees of certainty- 
some most unsure, some nearly sure, none absolutely certain. 

Now, we scientists are used to this, and we take it for granted 
that it is perfectly consistent to be unsure— that it is possible 
to live and not know. But I don't know whether everyone real- 
izes that this is true. Our freedom to doubt was bom of a 
struggle against authority in the early days of science. It was a 
very deep and strong struggle. Permit us to question— to doubt, 
that's all— not to be sure. And I think it is important that we 
do not forget the importance of this struggle and thus perhaps 
lose what we have gained. Here lies a responsibility to society. 

We are all sad when we think of the wondrous potentialities 
human beings seem to have, as contrasted with their small ac- 
comphshments. Again and again people have thought that we 
could do much better. They of the past saw in the nightmare 
of their times a dream for the future. We, of their future, see 
that their dreams, in certain ways surpassed, have in many ways 
remained dreams. The hopes for the future today are, in good 
share, those of yesterday. 

EDUCATION, FOR GOOD AND EVIL 

Once some thought that the possibilities people had were 
not developed because most of those people were ignorant 
With education universal, could all men be Voltaires? Bad can 
be taught at least as eflBciently as good. Education is a strong 
force, but for either good or evil. 

Communications between nations must promote understand- 
ing: so went another dream. But the machines of communication 
can be channeled or choked. What is communicated can be 
truth or lie. Communication is a strong force also, but for 
either good or bad. 



The Value of Science 



The applied sciences should free men of material problems 
at least. Medicine controls diseases. And the record here seems 
all to the good. Yet there are men patiently working to create 
great plagues and poisons. They are to be used in warfare to- 
morrow. 

Nearly everybody dislikes war. Our dream today is peace. In 
peace, man can develop best the enormous possibilities he 
seems to have. But maybe future men will find that peace, too, 
can be good and bad. Perhaps peaceful men will drink out of 
boredom. Then perhaps drink will become the great problem 
which seems to keep man from getting all he thinks he should 
out of his abilities. 

Clearly, peace is a great force, as is sobriety, as are material 
power, communication, education, honesty and the ideals of 
many dreamers. 

We have more of these forces to control than did the ancients. 
And maybe we are doing a little better than most of them 
could do. But what we ought to be able to do seems gigantic 
compared with our confused accomplishments. 

Why is this? Why can't we conquer ourselves? 

Because we find that even great forces and abilities do not 
seem to carry with them clear instructions on how to use them. 
As an example, the great accumulation of understanding as to 
how the physical world behaves only convinces one that this 
behavior seems to have a kind of meaninglessness. The sciences 
do not directly teach good and bad. 

Through all ages men have tried to fathom the meaning of 
life. They have realized that if some direction or meaning could 
be given to our actions, great human forces would be unleashed. 
So, very many answers must have been given to the question 
of the meaning of it all. But they have been of all different 
sorts, and the proponents of one answer have looked with horror 
at the actions of the believers in another. Horror, because from 
a disagreeing point of view all the great potentialities of the 
race were being channeled into a false and confining blind 
alley. In fact, it is from the history of the enormous monstrosities 
created by false belief that philosophers have realized the ap- 
parently infinite and wondrous capacities of human beings. The 
dream is to find the open channel. 

What, then, is the meaning of it all? What can we say to 
dispel the mystery of existence? 

If we take everything into account, not only what the an- 
cients knew, but all of what we know today that they didn't 
know, then I think that we must frankly admit that we do 
not know. 

But, in admitting this, we have probably found the open 
channel. 

This is not a new idea; this is the idea of the age of reason. 
This is the philosophy that guided the men who made the 



democracy that we live under. The idea that no one really knew 
how to run a government led to the idea that we should ar- 
range a system by which new ideas could be developed, tried 
out, tossed out, more new ideas brought in; a trial and error 
system. This method was a result of the fact that science was 
already showing itself to be a successful venture at the end 
of the i8th century. Even then it was clear to socially-minded 
people that the openness of the possibihties was an opportunity, 
and that doubt and discussion were essential to progress into 
the unknown. If we want to solve a problem that we nave 
never solved before, we must leave the door to the unknown 
ajar. 

OUR RESPONSEBILTTY AS SCIENTISTS 

We are at the very beginning of time for the human race. 
It is not unreasonable that we grapple with problems. There 
are tens of thousands of years in the future. Our responsibihty 
is to do what we can, leam what we can, improve the solutions 
and pass them on. It is our responsibility to leave the men of 
the future a free hand. In the impetuous youth of humanity, 
we can make grave errors that can stunt our growth for a long 
time. This we will do if we say we have the answers now, so 
young and ignorant; if we suppress all discussion, all criticism, 
saying, "This is it, boys, man is savedl" and thus doom man for 
a long time to the chains of authority, confined to the limits 
of our present imagination. It has been done so many times 
before. 

It is our responsibility as scientists, knowing the great prog- 
ress and great value of a satisfactory philosophy of ignorance, 
the great progress that is the fruit of freedom of thought, to 
proclaim the value of this freedom, to teach how doubt is not 
to be feared but welcomed and discussed, and to demand this 
freedom as our duty to all coming generations. 



This chapter from a science fiction novel by a present- 
day astronomer offers some non-fiction insight Into the 
way the scientist works. Another chapter from this 
same novel is in the Unit 2 Reader. 



2 Close Reasoning 

Fred Hoyle 

A chapter from his book The Black Cloud, 1957. 



It is curious in how great a degree human 
progress depends on the individual. Humans, numbered in 
thousands of milhons, seem organised into an ant-Hke so- 
ciety. Yet this is not so. New ideas, the impetus of all 
development, come from individual people, not from cor- 
porations or states. New ideas, fragile as spring flowers, 
easily bruised by the tread of the multitude, may yet be 
cherished by the solitary wanderer. 

Among the vast host that experienced the coming of the 
Cloud, none except Kingsley arrived at a coherent under- 
standing of its real nature, none except Kingsley gave the 
reason for the visit of the Cloud to the solar system. His first 
bald statement was greeted with outright disbelief even by 
his fellow scientists — Alexandrov excepted. 

Weichart was frank in his opinion. 

"The whole idea is quite ridiculous," he said 

Marlowe shook his head. 

"This comes of reading science fiction." 

"No bloody fiction about Cloud coming straight for 
dam' Sun. No bloody fiction about Cloud stopping. No 
bloody fiction about ionisation," growled Alexandrov. 

McNeil, the physician, was intrigued. The new develop- 
ment was more in his line than transmitters and aerials. 

"I'd like to know, Chris, what you mean in this context 
by the word 'alive.' " 

"Well, John, you know better than I do that the distinc- 
tion between animate and inanimate is more a matter of 
verbal convenience than anything else. By and large, inani- 
mate matter has a simple structure and comparatively 
simple properties. Animate or living matter on the other 
hand has a highly complicated structure and is capable of 
very involved behaviour. When I said the Cloud may be 
alive I meant that the material inside it may be organised 
in an intricate fashion, so that its behaviour and conse- 
quently the behaviour of the whole Cloud is far more 
complex than we previously supposed." 



"Isn't there an element of tautology there?" — from 
Weichart. 

"I said that words such as 'animate' and 'inanimate' 
are only verbal conveniences. If they're pushed too far they 
do appear tautological. In more scientific terms I expect the 
chemistry of the interior of the Cloud to be extremely 
complicated — complicated molecules, complicated structures 
built out of molecules, complicated nervous activity. In 
short I think the Cloud has a brain." 

"A dam' straightforward conclusion," nodded Alexan- 
drov. 

When the laugh had subsided, Marlowe turned to Kings- 
ley. 

"Well, Chris, we know what you mean,* at any rate we 
know near enough. Now let's have your argument. Take 
your time. Let's have it point by point, and it'd better be 
good." 

"Very well then, here goes. Point number one, the tem- 
perature inside the Cloud is suited to the formation of 
highly complicated molecules." 

"Rightl First point to you. In fact, the temperature is 
perhaps a little more favourable than it is here on the 
Earth." 

"Second point, conditions are favourable to the forma- 
tion of extensive structures built out of complicated mole- 
cules." 

"Why should that be so?" asked Yvette Hedelfort. 

"Adhesion on the surface of solid particles. The density 
inside the Cloud is so high that quite large lumps of solid 
material — probably mostly ordinary ice — are almost certainly 
to be found inside it. I suggest that the complicated mole- 
cules get together when they happen to stick to the surfaces 
of these lumps." 

"A very good point, Chris," agreed Marlowe. 

"Sorry, I don't pass this round." McNeil was shaking 
his head. "You talk of complicated molecules being built 
up by sticking together on the surface of solid bodies. Well, 
it won't do. The molecules out of which living material is 
made contain large stores of internal energy. Indeed, the 
processes of life depend on this internal energy. The trou- 
ble with your sticking together is that you don't get energy 
into the molecules that way." 

Kingsley seemed unperturbed. 

"And from what source do the molecules of living crea- 
tures here on the Earth get their internal supplies of en- 
ergy?" he asked McNeil. 

"Plants get it from sunlight, and animals get it from 
plants, or from other animals of course. So in the last 
analysis the energy always comes from the Sun." 

"And where is the Cloud getting energy from now?" 

The tables were turned. And as neither McNeil nor any- 
one else seemed disposed to argue, Kingsley went on: 



Close Reasoning 



"Let's accept John's argument. Let's suppose that my 
beast in the Cloud is built out of the same sort of molecules 
that we are. Then the light from some star is required in 
order that the molecules be formed. Well, of course star- 
light is available far out in the space between the stars, but 
it's very feeble. So to get a really strong supply of light the 
beast would need to approach close to some star. And 
that's just what the beast has donel" 

Marlowe became excited. 

"My God, that ties three things together, straight away. 
The need for sunlight, number one. The Cloud making a 
bee-line for the Sun, number two. The Cloud stopping 
when it reached the Sun, number three. Very good, 
Chris." 

"It is a very good beginning, yes, but it leaves some 
things obscure," Yvette Hedelfort remarked. "I do not 
see," she went on, "how it was that the Cloud came to be 
out in space. If it has need of sunlight or starlight, surely it 
would stay always around one star. Do you suppose that 
this beast of yours has just been born somewhere out in 
space and has now come to attach itself to our Sun?" 

"And while you're about it, Chris, will you explain how 
your friend the beast controls its supplies of energy? How 
did it manage to fire off those blobs of gas with such 
fantastic speed when it was slowing down?" asked Leices- 
ter. 

"One question at a timel I'll take Harry's first, because 
it's probably easier. We tried to explain the expulsion of 
those blobs of gas in terms of magnetic fields, and the expla- 
nation simply didn't work. The trouble -was that the re- 
quired fields would be so intense that they'd simply burst 
the whole Cloud apart. Stated somewhat differently, we 
couldn't find any way in which large quantities of energy 
could be localised through a magnetic agency in compara- 
tively small regions. But let's now look at the problem 
from this new point of view. Let's begin by asking what 
method we ourselves would use to produce intense local 
concentrations of energy." 

"Explosions!" gasped Barnett. 

"That's right, explosions, either by nuclear fission, or 
more probably by nuclear fusion. There's no shortage of 
hydrogen in this Cloud." 

"Are you being serious, Chris?" 

"Of course I'm being serious. If I'm right in supp>osing 
that some beast inhabits the Cloud, then why shouldn't he 
be at least as intelligent as we are?" 

"There's the slight difficulty of radioactive products. 
Wouldn't these be extremely deleterious to living ma- 
terial?" asked McNeil. 

"If they could get at the living material, certainly they 
would. But although it isn't possible to produce explosions 
with magnetic fields, it is possible to prevent two samples of 



material mixing with each other. I imagine that the beast 
orders the material of the Cloud magnetically, that by 
means of magnetic fields he can move samples of material 
wherever he wants inside the Cloud. I imagine that he takes 
very good care to keep the radioactive gas well separated 
from the living material — remember I'm using the term 
'living' for verbal convenience. I'm not going to be drawn 
into a philosophical argument about it." 

"You know, Kingsley," said Weichart, "this is going 
far better than I thought it would. What I suppose you 
would say is that whereas basically we assemble materials 
with our hands, or with the aid of machines that we have 
made with our hands, the beast assembles materials with 
the aid of magnetic energy." 

"That's the general idea. And I must add tliat the beast 
seems to me to have far the better of it. For one thing he's 
got vastly more energy to play with than we have." 

"My God, I should think so, billions of times more, at 
the very least," said Marlowe. "It's beginning to look, 
Chris, as if you're winning this argument. But we objectors 
over here in this corner are pinning our faith to Yvette's 
question. It seems to me a very good one. What can you 
offer in answer to it?" 

"It is a very good question, Geoff, and I don't know 
that I can give a really convincing answer. The sort of idea 
I've got is that perhaps the beast can't stay for very long 
in the close proximity of a star. Perhaps he comes in pe- 
riodically to some star or other, builds his molecules, which 
form his food supply as it were, and then pushes off again. 
Perhaps he does this time and time again." 

"But why shouldn't the beast be able to stay perma- 
nently near a star?" 

"Well, an ordinary common or garden cloud, a beastless 
cloud, if it were permanently near a star, would gradually 
condense into a compact body, or into a number of com- 
pact bodies. Indeed, as we all know, our Earth probably 
condensed at one time from just such a cloud. Obviously 
our friend the beast would find it extremely embarrassing to 
have his protective Cloud condense into a planet. So 
equally obviously he'll decide to push off before there's 
any danger of that happening. And when he pushes off 
he'll take his Cloud with him." 

"Have you any idea of how long that will be?" asked 
Parkinson. 

"None at all. I suggest that the beast will push off 
when he's finished recharging his food supply. That might 
be a matter of weeks, months, years, millennia for all I 
know." 

"Don't I detect a slight smell of rat in all this?" 
Barnett remarked. 

"Possibly. I don't know how keen your sense of smell is, 
Bill. What's your trouble?" 



10 



Close Reasoning 



"I've got lots of troubles. I should have thought that 
your remarks about condensing into a planet apply only to 
an inanimate cloud. If we grant that the Cloud is able to 
control the distribution of material within itself, then it 
could easily prevent condensation from taking place. After 
all, condensation must be a sort of stability process and I 
would have thought that quite a moderate degree of con- 
trol on the part of your beast could prevent any condensa- 
tion at all." 

"There are two replies to that. One is that I believe the 
beast will lose his control if he stays too long near the Sun. 
If he stays too long, the magnetic field of the Sun will 
penetrate into the Cloud. Then the rotation of the Cloud 
round the Sun will twist up the magnetic field to blazes. All 
control would then be lost." 

"My God, that's an excellent point." 

"It is, isn't it? And here's another one. However dif- 
ferent our beast is to life here on Earth, one point he 
must have in common with us. We must both obey the 
simple biological rules of selection and development. By 
that I mean that we can't suppose that the Cloud started 
oflE by containing a fully-fledged beast. It must have started 
with small beginnings, just as life here on Earth started 
with small beginnings. So to start with there would be no 
intricate control over the distribution of material in the 
Cloud. Hence if the Cloud had originally been situated 
close to a star, it could not have prevented condensation 
into a planet or into a number of planets." 

"Then how do you visualise the early beginnings?" 

"As something that happened far out in interstellar 
space. To begin with, life in the Cloud must have depended 
on the general radiation field of the stars. Even that would 
give it more radiation for molecule-building purposes than 
life on the Earth gets. Then I imagine that as intelligence 
developed it would be discovered that food supplies — i.e. 
molecule-building — could be enormously increased by mov- 
ing in close to a star for a comparatively brief i)eriod- 
As I see it, the beast must be essentially a denizen of 
interstellar space. Now, Bill, have you any more troubles?" 

"Well, yes, I've got another problem. Why can't the 
Cloud manufacture its own radiation? Why bother to 
come in close to a star? If it understands nuclear fusion to 
the point of producing gigantic explosions, why not use 
nuclear fusion for producing its supply of radiation?" 

"To produce radiation in a controlled fashion requires a 
slow reactor, and of course that's just what a star is. The 
Sun is just a gigantic slow nuclear fusion reactor. To pro- 
duce radiation on any real scale comparable with the Sun, 
the Cloud would have to make itself into a star. Then the 
beast would get roasted. It'd be much too hot inside." 

"Even then I doubt whether a cloud of this mass could 
produce very much radiation," remarked Marlowe. "Its 
mass is much too small. According to the mass-luminosity 



11 



relation it'd be down as compared with the Sun by a 
fantastic amount. No, you're barking up a wrong tree 
there, Bill." 

"I've a question that I'd like to ask," said Parkinson. 
"Why do you always refer to your beast in the singular? 
Why shouldn't there be lots of little beasts in the 
Cloud?" 

"I have a reason for that, but it'll take quite a while to 
explain." 

"Well, it looks as if we're not going to get much sleep 
tonight, so you'd better carry on." 

"Then let's start by supposing that the Cloud contains 
lots of little beasts instead of one big beast. I think you'll 
grant me that communication must have developed be- 
tween the difiEerent individuals." 

"Certainly." 

"Then what form will the communication take?" 

"You're supposed to be telling us, Chris." 

"My question was purely rhetorical. I suggest that com- 
munication would be impossible by our methods. We com- 
municate acoustically." 

"You mean by talking. That's certainly your method all 
right, Chris," said Ann Halsey. 

But the point was lost on Kingsley. He went on. 

"Any attempt to use sound would be drowned by the 
enormous amount of background noise that must exist in- 
side the Cloud. It would be far worse than trying to talk in 
a roaring gale. I think we can be pretty sure that communi- 
cation would have to take place electrically." 

"That seems fair enough," 

"Good. Well, the next point is that by our standards the 
distances between the individuals would be very large, since 
the Cloud by our standards is enormously large. It would 
obviously be intolerable to rely on essentially D.C. methods 
over such distances." 

"D.C. methods? Chris, will you please try to avoid jar- 
gon." 

"Direct current." 

"That explains it, I supposel" 

"Oh, the sort of thing we get on the telephone. Roughly 
speaking the difference between D.C. communication and 
A.C. communication is the difference between the tele- 
phone and radio." 

Marlowe grinned at Ann Halsey. 

"What Chris is trying to say in his inimitable manner is 
that communication must occur by radiative propaga- 
tion." 

"If you think that makes it clearer. . . ." 

"Of course it's clear. Stop being obstructive, Ann. Radi- 
ative propagation occurs when we emit a light signal or a 
radio signal. It travels across space through a vacuum at a 
speed of 186,000 miles per second. Even at this speed it 
would still take about ten minutes for a signal to travel 
across the Cloud. 



12 



Close Reasoning 



"My next point is that the volume of information that 
can be transmitted radiatively is enormously greater than 
the amount that we can communicate by ordinary sound. 
We've seen that with our pulsed radio transmitters. So if 
this Cloud contains separate individuals, the individuals 
must be able to communicate on a vastly more detailed 
scale than we can. What we can get across in an hour of 
talk they might get across in a hundredth of a second." 

"Ah, I begin to see light," broke in McNeil. "If com- 
munication occurs on such a scale then it becomes some- 
what doubtful whether we should talk any more of separate 
individuals!" 

"You're home, John I" 
"But I'm not home," said Parkinson. 

"In vulgar parlance," said McNeil amiably, "what 
Chris is saying is that individuals in the Cloud, if there are 
any, must be highly telepathic, so telepathic that it becomes 
rather meaningless to regard them as being really separate 
from each other." 

"Then why didn't he say so in the first place?" — from 
Ann Halsey. 

"Because like most vulgar parlance, the word 'telepa- 
thy' doesn't really mean very much." 

"Well, it certainly means a great deal more to me." 
"And what does it mean to you, Ann?" 
"It means conveying one's thoughts without talking, or 
of course without writing or winking or anything like 
that." 

"In other words it means — if it means anything at all 
— communication by a non-acoustic medimn." 

"And that means using radiative propagation," 
chipped in Leicester. 

"And radiative propagation means the use of alter- 
nating currents, not the direct currents and voltages we use 
in our brains." 

"But I thought we were capable of some degree of 
telepathy," suggested Parkinson. 

"Rubbish. Our brains simply don't work the right way 
for telepathy. Everything is based on D.C. voltages, and 
radiative transmission is impossible that way." 

"I know this is rather a red herring, but I thought these 
extrasensory people had established some rather remarkable 
correlations," Parkinson |>ersisted. 

"Bloody bad science," growled Alexandrov. "Correla- 
tions obtained after experiments done is bloody bad. Only 
prediction in science." 
"I don't follow." 

"What Alexis means is that only predictions really count 
in science," explained Weichart. "That's the way Kings- 
ley downed me an hour or two ago. It's no good doing a 
lot of experiments first and then discovering a lot of correla- 
tions afterwards, not unless the correlations can be used for 



13 



making new predictions. Otherwise it's like betting on a 
race after it's been run." 

"Kingsley's ideas have many very interesting neurologi- 
cal implications," McNeil remarked. "Communication 
for us is a matter of extreme difficulty. We ourselves have to 
make a translation of the electrical activity— essentially D.C. 
activity — in our brains. To do this quite a bit of the brain is 
given over to the control of the lip muscles and of the vocal 
cords. Even so our translatioi. is very incomplete. We 
don't do too badly perhaps in conveying simple ideas, but 
the conveying of emotions is very difficult. Kingsley's little 
beasts could, I suppose, convey emotions too, and that's 
another reason why it's rather meaningless to talk of sepa- 
rate individuals. It's rather terrifying to realise that every- 
thing we've been talking about tonight and conveying so 
inadequately from one to another could be communicated 
with vastly greater precision and understanding among 
Kingsley's little beasts in about a hundredth of a second." 

"I'd like to follow the idea of separate individuals a 
little further," said Barnett, turning to Kingsley. "Would 
you think of each individual in the Cloud as building a 
radiative transmitter of some sort?" 

"Not as building a transmitter. Let me describe how I 
see biological evolution taking place within the Cloud. At 
an early stage I think there would be a whole lot of more 
or less separate disconnected individuals. Then communica- 
tion would develop, not by a deliberate inorganic building 
of a means of radiative transmission, but through a slow 
biological development. The individuals would develop a 
means of radiative transmission as a biological organ, rather 
as we have develojied a mouth, tongue, lips, and vocal 
cords. Communication would improve to a degree that we 
can scarcely contemplate. A thought would no sooner be 
thought than it would be communicated. An emotion 
would no sooner be experienced than it would be shared. 
With this would come a submergence of the individual and 
an evolution into a coherent whole. The beast, as I visual- 
ise it, need not be located in a particular place in the 
Cloud. Its different parts may be spread through the 
Cloud, but I regard it as a neurological unity, interlocked 
by a communication system in which signals are transmitted 
back and forth at a speed of 186,000 miles a second." 

"We ought to get down to considering those signals 
more closely. I suppose they'd have to have a longish 
wave-length. Ordinary light presumably would be useless 
since the Cloud is opaque to it," said Leicester. 

"My guess is that the signals are radio waves," went on 
Kingsley. "There's a good reason why it should be so. To 
be really efficient one must have complete phase control in a 
communication system. This can be done with radio waves, 
but not so far as we know with shorter wave-lengths." 



14 



Close Reasoning 



McNeil was excited. 

"Ovir radio transmissions!" he exclaimed. "They'd have 
interfered with the beast's neurological control." 

"They would if they'd been allowed to." 

"What d'you mean, Chris?" 

"Well, the beast hasn't only to contend with our tranv 
missions, but with the whole welter of cosmic radio waves. 
From all Quarters of the Universe there'd be radio waves 
interfering with its neurological activity unless it had devel- 
oped some form of protection." 

"What sort of protection have you in mind?" 

"Electrical discharges in the outer part of the Cloud 
causing sufficient ionisation to prevent the entry of external 
radio waves. Such a protection would be as essential as the 
skull is to the human brain." 

Aniseed smoke was rapidly filling the room. Marlowe sud- 
denly found his pipe too hot to hold and put it down 
gingerly. 

"My God, you think this explains the rise of ionisation 
in the atmosphere, when we switch on our transmitters?" 

"That's the general idea. We were talking earlier on 
about a feedback mechanism. That I imagine is just what 
the beast has got. If any external waves get in too deeply, 
then up go the voltages and away go the discharges until 
the waves can get in no farther." 

"But the ionisation takes place in our own atmos- 
phere." 

"For this purpose I think we can regard our atmosphere 
as a part of the Cloud. We know from the shimmering of 
the night sky that gas extends all the way from the Earth to 
the denser parts of the Cloud, the disk-like parts. In short 
we're inside the Cloud, electronically speaking. That, I 
think, explains our communication troubles. At an earlier 
stage, when we were outside the Cloud, the beast didn't 
protect itself by ionising our atmosphere, but through its 
outer electronic shield. But once we got inside the shield 
the discharges began to occur in our own atmosphere. The 
beast has been boxing-in our transmissions." 

"Very fine reasoning, Chris," said Marlowe. 

"Hellish fine," nodded Alexandrov. 

"How about the one centimetre transmissions? They 
went through all right," Weichart objected. 

"Although the chain of reasoning is getting rather long 
there's a suggestion that one can make on that. I think it's 
worth making because it suggests the next action we might 
take. It seems to me most unlikely that this Cloud is 
unique. Nature doesn't work in unique examples. So let's 
suppose there are lots of these beasts inhabiting the Galaxy. 
Then I would expect communication to occur between one 
cloud and another. This would imply that some wave- 
lengths would be required for external communication pur- 



15 



poses, wave-lengths that could penetrate into the Cloud and 
would do no neurological harm." 

"And you think, one centimetre may be such a wave- 
length?" 

"That's the general idea." 

"But then why was there no reply to our one centimetre 
transmission?" asked Parkinson. 

"Perhaps because we sent no message. There'd be no 
point in replying to a perfectly blank transmission." 

"Then we ought to start sending pulsed messages on the 
one centimetre," exclaimed Leicester. "But how can we 
expect the Cloud to decipher them?" 

"That's not an urgent problem to begin with. It will be 
obvious that our transmissions contain information — that 
will be clear from the frequent repetition of various pat- 
terns. As soon as the Cloud realises that our transmissions 
have intelligent control behind them I think we can expect 
some sort of reply. How long will it take to get started, 
Harry? You're not in a position to modulate the one centi- 
metre yet, are you." 

"No, but we can be in a couple of days, if we work 
night shifts. I had a sort of presentiment that I wasn't 
going to see my bed tonight. Come on, chaps, let's get 
started." 

Leicester stood up, stretched himself, and ambled out. 
The meeting broke up. Kingsley took Parkinson on one 
side. 

"Look, Parkinson," he said, "there's no need to go 
gabbling about this until we know more about it." 

"Of course not. The Prime Minister suspects I'm ofiE 
my head as it is." 

"There is one thing that you might pass on, though. If 
London, Washington, and the rest of the political circus 
could get ten centimetre transmitters working, it's just pos- 
sible that they might avoid the fade-out trouble." 

When Kingsley and Ann Halsey were alone later that 
night, Ann remarked: 

"How on earth did you come on such an idea, Chris?" 

"Well, it's pretty obvious really. The trouble is that 
we're all inhibited against such thinking. The idea that the 
Earth is the only possible abode of life runs pretty deep in 
spite of all the science fiction and kid's comics. If we had 
been able to look at the business with an impartial eye we 
should have spotted it long ago. Right from the first, things 
have gone wrong and they've gone wrong according to a 
systematic sort of pattern. Once I overcame the psychologi- 
cal block, I saw all the difficulties could be removed by one 
simple and entirely plausible step. One by one the bits of 
the puzzle fitted into place. I think Alexandrov probably 
had the same idea, only his English is a bit on the terse 
side." 



16 



Close Reasoning 



"On the bloody terse side, you mean. But seriously, do 
you think this communication business will work?" 

"I very much hoj>e so. It's quite crucial that it 
should." 

"Why do you say that?" 

"Think of the disasters the Earth has suffered so far, 
without the Cloud taking any purposive steps against us. A 
bit of reflection from its surface nearly roasted us. A short 
obscuration of the Sun nearly froze us. If the merest tiny 
fraction of the energy controlled by the Cloud should be 
directed against us we should be wiped out, every plant and 
animal." 

"But why should that happen?" 

"How can one tell? Do you think of the tiny beetle or 
the ant that you crush under your foot on an afternoon's 
walk? One of those gas bullets that hit the Moon three 
months ago would finish us. Sooner or later the Cloud will 
probably let fly with some more of 'em. Or we might be 
electrocuted in some monstrous discharge." 

"Could the Cloud really do that?" 

"Easily. The energy that it controls is simply monstrous. 
If we can get some sort of a message across, then perhaps 
the Cloud will take the trouble to avoid crushing us under 
its foot." 

"But why should it bother?" 

"Well, if a beetle were to say to you, 'Please, Miss 
Halsey, will you avoid treading here, otherwise I shall be 
crushed,' wouldn't you be willing to move your foot a 
trifle?" 



17 



Scientists often stress that there Is no single scientific 
method. Bridgman emphasizes this freedom to choose 
between many procedures, a freedom essential to sci- 
ence. 



On Scientific Method 

Percy W. Bridgman 

An excerpt from his book Reflections of a Pfiysicist, 1955. 



It seems to me that there is a good deal of ballyhoo 
about scientific method. I venture to think that the 
people who talk most about it are the people who 
do least about it. Scientific method is what working 
scientists do, not what other people or even they 
themselves may say about it. No working scientist, 
when he plans an experiment in the laboratory, asks 
himself whether he is being properly scientific, nor 
is he interested in whatever method he may be using 
as method. When the scientist ventures to criticize 
the work of his fellow scientist, as is not inicommon, 
he does not base his criticism on such glittering 
generalities as failure to follow the "scientific 
method," but his criticism is specific, based on some 
feature characteristic of the particular situation. The 
working scientist is always too much concerned with 
getting down to brass tacks to be willing to spend 
his time on generalities. 

Scientific method is something talked about by 
people standing on the outside and wondering how 
the scientist manages to do it. These people have 
been able to uncover various generalities applicable 
to at least most of what the scientist does, but it 
seems to me that these generalities are not very pro- 
found, and could have been anticipated by anyone 
who knew enough about scientists to know what is 
their primary objective. I think that the objectives 
18 of all scientists have this in common — that they are 



On Scientific Method 



all trying to get the correct answer to the particular 
problem in hand. This may be expressed in more 
pretentious language as the pursuit of truth. Now if 
the answer to the problem is correct there must be 
some way of knowing and proving that it is correct 
— the very meaning of truth implies the possibility 
of checking or verification. Hence the necessity for 
checking his results always inheres in what the 
scientist does. Furthermore, this checking must be 
exhaustive, for the truth of a general proposition 
may be disproved by a single exceptional case. A 
long experience has shown the scientist that various 
things are inimical to getting the correct answer. He 
has found that it is not sufficient to trust the word 
of his neighbor, but that if he wants to be sure, he 
must be able to check a result for himself. Hence 
the scientist is the enemy of all authoritarianism. 
Furthermore, he finds that he often makes mistakes 
himself and he must learn how to guard against 
them. He cannot permit himself any preconception 
as to what sort of results he will get, nor must he 
allow himself to be influenced by wishful thinking 
or any personal bias. All these things together give 
that "objectivity" to science which is often thought 
to be the essence of the scientific method. 

But to the working scientist himself all this ap- 
pears obvious and trite. What appears to him as 
the essence of the situation is that he is not con- 
sciously following any prescribed course of action, 
but feels complete freedom to utilize any method or 
device whatever which in the particular situation 
before him seems likely to yield the correct answer. 
In his attack on his specific problem he suffers no 
inhibitions of precedent or authority, but is com- 
pletely free to adopt any course that his ingenuity is 
capable of suggesting to him. No one standing on 
the outside can predict what the individual scien- 
tist will do or what method he will follow. In short, 
science is what scientists do, and there are as many 
scientific methods as there are individual scientists. 



19 



This is Polya's one-page summary of his book in which 
he discusses strategies and techniques for solving prob- 
lems. Polya's examples are from mathematics, but his 
ideas are useful in solving physics problems also. 



How to Solve It 

George Polya 

An excerpt from his book How To Solve It, 1945. 



UNDERSTANDING THE PROBLEM 

What is the unknown? What are the data? What is the condition? 
Is it possible to satisfy the condition? Is the condition sufficient to 
determine the unknown? Or is it insufiBcient? Or redundant? Or 
contradictory? 

Draw a figure. Introduce suitable notation. 

Separate the various parts of the condition. Can you write them down? 



First. 

You have to understand 
the problem. 



Second. 

Find the connection between 

the data and the unknown. 

You may be obliged 

to consider auxiliary problems 

if an immediate connection 

cannot be found. 

You should obtain eventually 

a plan of the solution. 



DEVISING A PLAN 

Have you seen it before? Or Jiave you seen the same problem in a 
slightly different form? 

Do you know a related problem? Do you know a theorem that could 
be useful? 

Look at the unknown! And try to think of a familiar problem having 
the same or a similar unknown. 

Here is a problem related to yours and solved before. Could you use it? 
Could you use its result? Could you use its method? Should you intro- 
duce some auxiliary element in order to make its use possible? 
Could you restate the problem? Could you restate it still differently? 
Go back to definitions. 

If you cannot solve the proposed problem try to solve first some related 
problem. Could you imagine a more accessible related problem? A 
more general problem? A more special problem? An analogous problem? 
Could you solve a part of the problem? Keep only a part of the condi- 
tion, drop the other part; how far is the unknown then determined, 
how can it vary? Could you derive something useful from the data? 
Could you think of other data appropriate to determine the unknown? 
Could you change the unknown or the data, or both if necessary, so 
that the new unknown and the new data are nearer to each other? 
Did you use all the data? Did you use the whole condition? Have you 
taken into account all essential notions involved in the problem? 



Third. 

Carry out your plan. 



CARRYING OUT THE PLAN 

Carrying out your plan of the solution, check each step. Can you see 
dearly that the step is correct? Can you prove that it is correct? 



Fourth. 
Examine the solution obtained. 



LOOKING BACK 
Can you check the result? Can you check the argument? 
Can you derive the result differently? Can you see it at a glance? 
Can you use the result, or the method, for some other problem? 



20 



The advice is directed primarily to the student planning 
a career in the'sciences, but it should be of interest to 
a wider group. 



5 Four Pieces of Advice to Young People 

Warren Weaver 

Part of a talk given in Seattle during the Arches of Science Award Seminars, 1966. 

One of the great prerogatives of age is the right to give advice to the young. 
Of course, the other side of the coin is that one of the prerogatives of youth 
is to disregard this advice. But. . . I am going to give you four pieces of ad- 
vice, and you may do with all four of them precisely what you see fit. 

The first one is this: I urge each one of you not to decide prematurely what 
field of science, what specialty of science you are going to make your own. 
Science moves very rapidly. Five years from now or ten years from now there 
will be opportunities in science which are almost not discernible at the pres- 
ent time. And, I think there are also, of course, fads in science. Science 
goes all out at any one moment for work in one certain direction and the 
other fields are thought of as being rather old-fashioned. But, don't let that 
fool you. Sometimes some of these very old problems turn out to be extremely 
significant. 

May I just remind you that there is no physical entity that the mind of man has 
thought about longer than \he phenomenon of light. One would ordinarily say 
that it would be simply impossible at the present day for someone to sit down 
and get a brand new idea about light, because think of the thousands of 
scientists that have worked on that subject. And yet, you see this is what 
two scientists did only just a few years ago when the laser was invented. They 
got a brand new idea about light and it has turned out to be a phenomenally 
important idea. 

So, I urge you not to make up your minds too narrowly, too soon. Of course, 
that means that what you ought to do is to be certain that you get a very solid 
basic foundation in science so that you can then adjust yourselves to the 
opportunities of the future when they arise. What is that basic foundation? 



21 



Well, of course, you don't expect me to say much more than mathematics, do 
you? Because I was originally trained as a mathematician and mathematics is 
certainly at the bottom of all this. But 1 also mean the fundamentals of physics 
and the fundamentals of chemistry. These two. Incidentally, are almost 
indistinguishable nowadays from the fundamentals of biology. 

The secondpieceof advice that I will just mention to you because maybe some 
of you are thinking too exclusively in terms of a career in research. In my 
judgment there is no life that is possible to be lived on this planet that is more 
pleasant and more rewarding than the combined activity of teaching and 
research. 

I hope very much that many of you look forward to becoming teachers. It is a 
wonderful life. I don't know of any better one myself, any more pleasant one, 
or any more rewarding one. And the almost incredible fact is that they even 
pay you for it. And, nowadays, they don't pay you too badly. Of course, 
when I started, they did. But, nowadays, the pay is pretty good. 

My third piece of advice — may I urge every single one of you to prepare your- 
self not only to be a scientist, but to be a scientist-citizen. You have to 
accept the responsibilities of citizenship in a free democracy. And those are 
great responsibilities and because of the role which science plays in our modern 
world, we need more and more people who understand science but who are also 
sensitive to and aware of the responsibilities of citizenship. 

And the final piece of advice is — and maybe this will surprise you: Do not 
overestimate science, do not think that science is all that there is, do not 
concentrate so completely on science that you end up by living a warped sort 
of life. Science is not all that there is, and science is not capable of solving 
all of life's problems. There are also many more very Important problems that 
science cannot solve. 

And so I hope very much there's nobody in this room who Is going to spend the 
next seven days without reading some poetry. I hope that there's nobody In 
this room that's going to spend the next seven days without listening to some 
music, some good music, some modern music, some music. I hope very much 
that there's nobody here who is not Interested in the creative arts, Interested in 
drama, interested in the dance. I hope that you interest yourselves seriously in 
religion, because if you do not open your minds and open your activities to this 
range of things, you are going to lead too narrow a life. 



22 



The size of an animal is related to such physical factors 
as gravity and temperature. For most animals there ap- 
pears to be an optimum size. 

On Being the Right Size 

J. B. S. Haldane 

An excerpt from his book Possible Worlds, 1928. 



From what has already been demonstrated, you can plainly see the impossi- 
bility of increasing the size of structures to vast dimensions either in art 
or in nature; likewise the impossibility of building ships, palaces, or temples 
of enormous size in such a way that their oars, yards, beams, iron bolts, 
and, in short, all their other parts will hold together; nor can nature pro- 
duce trees of extraordinary size because the branches would break down 
under their own weight, so also it would be impossible to build up the bony 
structures of men, horses, or other animals so as to hold together and per- 
form their normal functions if these animals were to be increased enor- 
mously in height; for this increase in height can be accomplished only by 
employing a material which is harder and stronger than usual, or by en- 
larging the size of the bones, thus changing their shape until the form and 
appearance of the animals suggest a monstrosity. This is perhaps what our 
wise Poet had in mind, when he says, in describing a huge giant: 
"Impossible it is to reckon his height 
So beyond measure is his size." — Galileo Galilei 

THE most obvious diflferences between different animals are differences 
of size, but for some reason the zoologists have paid singularly little atten- 
tion to them. In a large textbook of zoology before me I find no indication 
that the eagle is larger than the sparrow, or the hippopotamus bigger than 
the hare, though some grudging admissions are made in the case of the 
mouse and the whale. But yet it is easy to show that a hare could not 
be as large as a hippopotamus, or a whale as small as a herring. For 
every type of animal there is a most convenient size, and a large change 
in size inevitably carries with it a change of form. 

Let us take the most obvious of possible cases, and consider a giant man 
sixty feet high — about the height of Giant Pope and Giant Pagan in the 
illustrated Pilgrim's Progress of my childhood. These monsters were not 
only ten times as high as Christian, but ten times as wide and ten times as 
thick, so that their total weight was a thousand times his, or about eighty 
to ninety tons. Unfortunately the cross sections of their bones were only 
a hundred times those of Christian, so that every square inch of giant bone 
had to support ten times the weight borne by a square inch of human 
bone. As the human thigh-bone breaks under about ten times the human 
weight, Pope and Pagan would have broken their thighs every time they 
took a step. This was doubtless why they were sitting down in the picture 
I remember. But it lessens one's respect for Christian and Jack the Giant 
Killer. 



23 



To turn to zoology, suppose that a gazelle, a graceful little creature with 
long thin legs, is to become large, it will break its bones unless it does one 
of two things. It may make its legs short and thick, like the rhinoceros, 
so that every pound of weight has still about the same area of bone to 
support it. Or it can compress its body and stretch out its legs obliquely to 
gain stability, like the giraffe. I mention these two beasts because they 
happen to belong to the same order as the gazelle, and both are quite suc- 
cessful mechanically, being remarkably fast runners. 

Gravity, a mere nuisance to Christian, was a terror to Pope, Pagan, 
and Despair. To the mouse and any smaller animal it presents practically 
no dangers. You can drop a mouse down a thousand-yard mine shaft; 
and, on arriving at the bottom, it gets a slight shock and walks away. A 
rat would probably be killed, though it can fall safely from the eleventh 
story of a building; a man is killed, a horse splashes. For the resistance 
presented to movement by the air is proportional to the surface of the 
moving object. Divide an animal's length, breadth, and height each by 
ten; its weight is reduced to a thousandth, but its surface only to a hun- 
dredth. So the resistance to falling in the case of the small animal is 
relatively ten times greater than the driving force. 

An insect, therefore, is not afraid of gravity; it can fall without danger, 
and can cling to the ceiling with remarkably little trouble. It can go in for 
elegant and fantastic forms of support like that of the daddy-long-legs. But 
there is a force which is as formidable to an insect as gravitation to a 
mammal. This is surface tension. A man coming out of a bath carries with 
him a film of water of about one-fiftieth of an inch in thickness. This 
weighs roughly a pound. A wet mouse has to carry about its own weight 
of water. A wet fly has to lift many times its own weight and, as every 
one knows, a fly once wetted by water or any other liquid is in a very 
serious position indeed. An insect going for a drink is in as great danger 
as a man leaning out over a precipice in search of food. If it once falls 
into the grip of the surface tension of the water — that is to say, gets wet — 
it is likely to remain so until it drowns. A few insects, such as water- 
beetles, contrive to be unwettable, the majority keep well away from their 
drink by means of a long proboscis. 

Of course tall land animals have other difficulties. They have to pump 
their blood to greater heights than a man and, therefore, require a larger 
blood pressure and tougher blood-vessels. A great many men die from 
burst arteries, especially in the brain, and this danger is presumably still 
greater for an elephant or a giraffe. But animals of all kinds find difficul- 
ties in size for the following reason. A typical small animal, say a micro- 
scopic worm or rotifer, has a smooth skin through which all the oxygen 
it requires can soak in, a straight gut with sufficient surface to absorb its 
food, and a simple kidney. Increase its dimensions tenfold in every direc- 
tion, and its weight is increased a thousand times, so that if it is to use 
its muscles as efficiently as its miniature counterpart, it will need a thou- 
sand times as much food and oxygen per day and will excrete a thousand 
times as much of waste products. 

Now if its shape is unaltered its surface will be increased only a hun- 
dredfold, and ten times as much oxygen must enter per minute through 



24 



On Being the Right Size 



each square millimetre of skin, ten times as much food through each 
square miUimetre of intestine. When a Umit is reached to their absorptive 
powers their surface has to be increased by some special device. For ex- 
ample, a part of the skin may be drawn out into tufts to make gills or 
pushed in to make lungs, thus increasing the oxygen-absorbing surface in 
proportion to the animal's bulk. A man, for example, has a hundred 
square yards of lung. Similarly, the gut, instead of being smooth and 
straight, becomes coiled and develops a velvety surface, and other organs 
increase in complication. The higher animals are not larger than the lower 
because they are more complicated. They are more complicated because 
they are larger. Just the same is true of plants. The simplest plants, such 
as the green algae growing in stagnant water or on the bark of trees, are 
mere round cells. The higher plants increase their surface by putting out 
leaves and roots. Comparative anatomy is largely the story of the struggle 
to increase surface in proportion to volume. 

Some of the methods of increasing the surface are useful up to a point, 
but not capable of a very wide adaptation. For example, while vertebrates 
carry the oxygen from the gills or lungs all over the body in the blood, 
insects take air directly to every part of their body by tiny blind tubes 
called tracheae which open to the surface at many different points. Now, 
although by their breathing movements they can renew the air in the 
outer part of the tracheal system, the oxygen has to penetrate the finer 
branches by means of diffusion. Gases can diffuse easily through very 
small distances, not many times larger than the average length travelled 
by a gas molecule between collisions with other molecules. But wnen such 
vast journeys — from the point of view of a molecule — as a quarter of an 
inch have to be made, the process becomes slow. So the portions of an 
insect's body more than a quarter of an inch from the air would always 
be short of oxygen. In consequence hardly any insects are much more 
than half an inch thick. Land crabs are built on the same general plan as 
insects, but are much clumsier. Yet like ourselves they carry oxygen 
around in their blood, and are therefore able to grow far larger than any 
insects. If the insects had hit on a plan for driving air through their 
tissues instead of letting it soak in, they might well have become as large 
as lobsters, though other considerations would have prevented them from 
becoming as large as man. 

Exactly the same difficulties attach to flying. It is an elementary prin- 
ciple of aeronautics that the minimum speed needed to keep an aeroplane 
of a given shape in the air varies as the square root of its length. If its 
linear dimensions are increased four times, it must fly twice as fast. Now 
the power needed for the minimum speed increases more rapidly than the 
weight of the machine. So the larger aeroplane, which weighs sixty-four 
times as much as the smaller, needs one hundred and twenty-eight times 
its horsepower to keep up. Applying the same principles to the birds, we 
find that the limit to their size is soon reached. An angel whose muscles 
developed no more power weight for weight than those of an eagle or a 
pigeon would require a breast projecting for about four feet to house the 
muscles engaged in working its wings, while to economize in weight, its 
legs would have to be reduced to mere stilts. Actually a large bird such as 



25 



an eagle or kite does not keep in the air mainly by moving its wings. It 
is generally to be seen soaring, that is to say balanced on a rising column 
of air. And even soaring becomes more and more difficult with increasing 
size. Were this not the case eagles might be as large as tigers and as 
formidable to man as hostile aeroplanes. 

But it is time that we passed to some of the advantages of size. One 
of the most obvious is that it enables one to keep warm. All warm-blooded 
animals at rest lose the same amount of heat from a unit area of skin, for 
which purpose they need a food-supply proportional to their surface and 
not to their weight. Five thousand mice weigh as much as a man. Their 
combined surface and food or oxygen consumption are about seventeen 
times a man's. In fact a mouse eats about one quarter its own weight of 
food every day, which is mainly used in keeping it warm. For the same 
reason small animals cannot live in cold countries. In the arctic regions 
there are no reptiles or amphibians, and no small mammals. The smallest 
mammal in Spitzbergen is the fox. The small birds fly away in the winter, 
while the insects die, though their eggs can survive six months or more 
of frost. The most successful mammals are bears, seak, and walruses. 

Similarly, the eye is a rather inefficient organ until it reaches a large 
size. The back of the human eye on which an image of the outside world 
is thrown, and which corresponds to the film of a camera, is composed 
of a mosaic of 'rods and cones' whose diameter is little more than a length 
of an average light wave. Each eye has about half a million, and for two 
objects to be distinguishable their images must fall on separate rods or 
cones. It is obvious that with fewer but larger rods and cones we should 
see less distinctly. If they were twice as broad two points would have to be 
twice as far apart before we could distinguish them at a given distance. 
But if their size were diminished and their number increased we should 
see no better. For it is impossible to form a definite image smaller than a 
wave-length of light. Hence a mouse's eye is not a small-scale model of a 
human eye. Its rods and cones are not much smaller than ours, and there- 
fore there are far fewer of them. A mouse could not distinguish one 
human face from another six feet away. In order that they should be of 
any use at all the eyes of small animals have to be much larger in pro- 
portion to their bodies than our own. Large animals on the other hand 
only require relatively small eyes, and those of the whale and elephant 
are little larger than our own. 

For rather more recondite reasons the same general principle holds 
true of the brain. If we compare the brain-weights of a set of very similar 
animals such as the cat, cheetah, leopard, and tiger, we find that as we 
quadruple the body-weight the brain-weight is only doubled. The larger 
animal with proportionately larger bones can economize on brain, eyes, 
and certain other organs. 

Such are a very few of the considerations which show that for every 
type of animal there is an optimum size. Yet although Galileo demon- 
strated the contrary more than three hundred years ago, people still 
believe that if a fllea were as large as a man it could jump a thousand feet 
into the air. As a matter of fact the height to which an animal can jump 
is more nearly independent of its size than proportional to it. A flea can 



26 



On Being the Right Size 



jump about two feet, a man about five. To jump a given height, if we 
neglect the resistance of the air, requires an expenditure of energy pro- 
portional to the jumper's weight. But if the jumping muscles form a 
constant fraction of the animal's body, the energy developed per ounce of 
muscle is independent of the size, provided it can be developed quickly 
enough in the small animal. As a matter of fact an insect's muscles, al- 
though they can contract more quickly than our own, appear to be 
less efficient; as otherwise a flea or grasshopper could rise six feet into 
the air. 

And just as there is a best size for every animal, so the same is true 
for every human institution. In the Greek type of democracy all the citi- 
zens could listen to a series of orators and vote directly on questions of 
legislation. Hence their philosophers held that a small city was the largest 
possible democratic state. The English invention of representative gov- 
ernment made a democratic nation possible, and the possibility was first 
realized in the United States, and later elsewhere. With the development 
of broadcasting it has once more become possible for every citizen to 
listen to the political views of representative orators, and the future may 
perhaps see the return of the national state to the Greek form of democ- 
racy. Even the referendum has been made possible only by the institution 
of daily newspapers. 

To the biologist the problem of socialism appears largely as a problem 
of size. The extreme socialists desire to run every nation as a single busi- 
ness concern. I do not suppose that Henry Ford would find much diffi- 
culty in running Andorra or Luxembourg on a socialistic basis. He has 
already more men on his pay-roll than their population. It is conceivable 
that a syndicate of Fords, if we could find them, would make Belgium Ltd. 
or Denmark Inc. pay their way. But while nationalization of certain in- 
dustries is an obvious possibility in the largest of states, I find it no easier 
to picture a completely socialized British Empire or United States than 
an elephant turning somersaults or a hippopotamus jumping a hedge. 



27 



Not only the scientist Is Interested In motion. This ar- 
ticle comments briefly on references to motion In poetry, 



Motion in Words 



James B. Gerhart and Rudi H. Nussbaum 



An excerpt from their monograph, Motion, 1966. 



Man began describing movement 
with words long before there were 
physicists to reduce motion to laws. 
Our age-old fascination with moving 
things is attested to by the astonish- 
ing number of words we have for motion. 
We have all kinds of words for all 
kinds of movement : special words for 
going up, others for coming down; words 
for fast motion, words for slow motion. 
A thing going up may rise, ascend, 
climb, or spring. Going down again, it 
may fall or descend; sink, subside, or 
settle; dive or drop; plunge or plop; 
topple, totter, or merely droop. It 
may twirl, whirl, turn and circle; 
rotate, gyrate; twist or spin; roll, 
revolve and wheel. It may oscillate, 
vibrate, tremble and shake; tumble or 
toss, pitch or sway; flutter, jiggle, 
quiver, quake; or lurch, or wobble, 
or even palpitate. All these words 
tell some motion, yet every one has 
its own character. Some of them you 
use over and over in a single day. 
Others you may merely recognize. And 
still they are but a few of our words 
for motion. There are special words 
for the motions of particular things. 
Horses, for example, trot and gallop 
and canter while men run, or stride, 
or saunter. Babies crawl and creep. 
Tides ebb and flow,, balls bounce, arm- 
ies march . Other words tell the qual- 
ity of motion, words like swift or 
fleet, like calm and slow. 

Writers draw vivid mental pictures 
for the reader with words alone. Here 
is a poet's description of air flowing 
across a field on a hot day: 



There came a wind like a bugle: 
It quivered through the grass, 
and a green chill upon the heat 
so ominous did pass. 

Emily Dickinson 

Or again, the motion of the sea caused 
by the gravitational attraction of the 
moon : 

The western tide crept up along 

the sand, 
and o'er and o'er the sand, 
and round and round the sand, 
as far as the eye could see. 

Charles Kingsley, 
The Sands of Dee 

Or, swans starting into flight: 

I saw . . . all suddenly mount 
and scatter wheeling in great 

broken rings 
upon their clamorous wings. 

W. B. Yeats, 
The Wild Swans at Coole 

Sometimes just a single sentence will 
convey the whole idea of motion: 
Lightly stepped a yellow star 
to its lofty place 

Emily Dickinson 

Or, this description of a ship sailing: 

She walks the water like a thing 
of life 

Byron, The Corsair 



28 



Motion in Words 



How is it that these poets de- 
scribe motion? They recall to us what 
we have seen; they compare different 
things through simile and metaphor; 
they rely on the reader to share their 
own emotions, and they invite him to 
recreate an image of motion in his own 
mind. The poet has his own precision 
which is not the scientist's precision 
Emily Dickinson well knew it was the 
grass, not the wind, that quivered, 
and that stars don't step. Byron never 
saw a walking boat. But this is irrel- 
evant . All of us can appreciate and 
enjoy their rich images and see that 
they are true to the nature of man's 
perception, if not to the nature of 
motion itself . 

From time to time a physicist 
reading poetry will find a poem which 
describes something that he has 
learned to be of significance to his, 
the physicist's description. Here is 
an example : 

A ball will bounce, but less and 
less. It's not a light-hearted 
thing, resents its own resilience. 
Falling is what it loves, . . . 

Richard Wilbur, Juggler 

Relativity is implicit in this next 
example : 

The earth revolves with me, yet 

makes no motion. 
The stars pale silently in a coral 

sky . 
In a whistling void I stand before 

my mirror unconcerned, and tie 

my tie. 

Conrad Aiken, 

Morning Song of Senlin 



The poet's description of motion 
is a rich, whole vision, filled with 
both his perceptions and his responses 
Yet complete as it is, the poetic de- 
scription is far from the scientific 
one. Indeed, when we compare them, it 
is easy to forget they deal with the 
same things. Just how does the scien- 
tific view of motion differ? And to 
what purpose? Let's try to answer 
these questions by shifting slowly 
from the poet's description to the 
scientist's. As a first step, read 
this excerpt from a biography of a 




Fig. 1.10 Multiple- flash photograph show- 
ing the precession of a top. 



physicist of the last century. Lord 
Kelvin. The biographer is trying to 
convey the electric quality of Kelvin's 
lectures to his University classes." He 
describes a lecture on tops (referred 
to as gyrostats here) : 

The vivacity and enthusiasm of the 
Professor at that time was very 
great. The animation of his coun- 
tenance as he looked at a gyrostat 
spinning, standing on a knife edge 
on a glass plate in front of him, 
and leaning over so that its center 
of gravity was on one side of the 
point of. support; the delight with 
which he showed that hurrying of 
the precessional motion caused the 
gyrostat to rise, and retarding the 
precessional motion caused the gy- 
rostat to fall, so that the freedom 
to precess was the secret of its 
not falling; the immediate applica- 
tion of the study of the gyrostat 
to the explanation of the preces- 
sion of the equinoxes, and illustra- 
tion by a model ... - all these 
delighted his hearers, and made the 
lecture memorable. 

Andrew Gray, Lord Kelvin, An 

Account of his Scientific 

Life and Work 

This paragraph by Gray deals with 
motion, but still it is more concerned 



29 



with human responses - Kelvin's obvi- 
ous pleasure in watching the top, and 
his student's evident delight in watch- 
ing both Kelvin and Kelvin's top. At 
the same time it says much about the 
top's movement, hints at the reasons 
behind it, and mentions how under- 
standing the top has led to under- 
standing the precession of the earth's 
axis in space. 

Gray used some of the everyday 
words for motion: rise, fall, spin, 
hurry, retard. But he used other words 
and other phrases, too - more techni- 
cal, less familiar: precess, center 
of gravity, equinoxes. Technical words 
are important for a scientific descrip- 
tion of motion. When the scientist has 
dissected a motion and laid out its 
components, the need for new terms 
enters, the need for words with more 
precise meanings, words not clothed 
with connotations of emotional re- 
sponse. Still, the scientist never can 
(and never really wants to) , lose the 
connotations of common words entirely. 
For example, here is Lord Kelvin's at- 
tempt to define precession (see Fig. 
1.10), in the sense that Gray used it: 

This we call positive precessional 
rotation. It is the case of a com- 
mon spinning-top (peery), spinning 
on a very fine point which remains 
at rest in a hollow or hole bored 
by itself; not sleeping upright, 
nor nodding, but sweeping its axis 
round in a circular cone whose 
axis is vert ical . 

William Thomson (Lord Kelvin) 

and P. G. Tait, Treatise 

on Natural Philosophy 

This definition is interesting in 
several ways. For one thing, it seems 
strange today that Kelvin, a Scot, 
should feel the need to explain "spin- 
ning-top" by adding "peery," an ob- 
scure word to most of us, but one that 
Kelvin evidently thought more collo- 
quial. Think for a moment of how 
Kelvin went about his definition. He 



used the words of boys spinning tops 
for fun, who then, and still today, 
say a top sleeps when its axis is 
nearly straight up, and that it nods 
as it slows and finally falls. He re- 
minded his readers of something they 
all had seen and of the everyday words 
for it. (He obviously assumed that 
most of his readers once played with 
tops.) In fact, this is the best way 
to define new words - to remind the 
reader of something he knows already 
and with words he might use himself. 
Having once given this definition 
Kelvin never returns to the picture he 
employed. It is clear, though, that 
when he wrote, "positive precessional 
rotation," he brought this image to 
his own mind, and that he expected his 
readers to do the same. 

Of course, it is not necessary to 
use as many words as Kelvin did to de- 
fine precession. Another, more austere, 
and to some, more scientific defini- 
tion is this : 

When the axis of the top travels 
round the vertical making a con- 
stant angle i with it, the motion 
is called steady or precessional. 

E. J. Routh, Treatise on the 

Dynamics of a System of 

Riffid Bodies 



All that refers to direct, human ex- 
perience is missing here. The top is 
now just something with an axis, no 
longer a bright-painted toy spinning 
on the ground. And it is not the top 
that moves, but its axis, an imagined 
line in space, and this line moves 
about another imagined line, the ver- 
tical. There is no poetry here, only 
geometry. This is an exact, precise, 
and economical definition, but it is 
abstract, and incomplete. It does not 
describe what anyone watching a real 
top sees. In fact, it is only a few 
abstractions from the real top's mo- 
tion on which the physicist-def iner 
has concentrated his attention. 



30 



The treatment of speed and acceleration demonstrates the 
value of simple calculus in analyzing and describing motion. 



8 Motion 



Richard P. Feynman, Robert B. Leighton and Matthew Sands 
A chapter from The Feynman Lectures on Physics — Volume 1, 1963. 



8-1 Description of motion 

In order to find the laws governing the various changes tnat take place in 
bodies as time goes on, we must be able to describe the changes and have some way 
to record them. The simplest change to observe in a body is the apparent change 
in its position with time, which we call motion. Let us consider some solid object 
with a permanent mark, which we shall call a point, that we can observe. We 
shall discuss the motion of the little marker, which might be the radiator cap of an 
automobile or the center of a falling ball, and shall try to describe the fact that it 
moves and how it moves. 

These examples may sound trivial, but many subtleties enter into the descrip- 
tion of change. Some changes are more difficult to describe than the motion of 
a point on a solid object, for example the speed of drift of a cloud that is drifting 
very slowly, but rapidly forming or evaporating, or the change of a woman's 
mind. We do not know a simple way to analyze a change of mind, but since the 
cloud can be represented or described by many molecules, perhaps we can describe 
the motion of the cloud in principle by describing the motion of all its individual 
molecules. Likewise, perhaps even the changes in the mind may have a parallel 
in changes of the atoms inside the brain, but we have no such knowledge yet. 

At any rate, that is why we begin with the motion of points; perhaps we should 
think of them as atoms, but it is probably better to be more rough in the begin- 
ning and simply to think of some kind of small objects — small, that is, compared 
with the distance moved. For instance, in describing the motion of a car that is 
going a hundred miles, we do not have to disdnguish between the front and the 
back of the car. To be sure, there are slight differences, but for rough purposes we 
say "the car," and likewise it does not matter that our points are not absolute 
points; for our present purposes it is not necessary to be extremely precise. Also, 
while we take a first look at this subject we are going to forget about the three 
dimensions of the world. We shall just concentrate on moving in one direction, 
as in a car on one road. We shall return to three dimensions after we see how to 
describe motion in one dimension. Now, you may say, "This is all some kind of 
trivia," and indeed it is. How can we describe such a one-dimensional motion — 
let us say, of a car? Nothing could be simpler. Among many possible ways, one 
would be the following. To determine the position of the car at diff'erent times, 
we measure its distance from the starting point and record all the observations. 



31 



Table 8-1 



/ (min) 


sift) 








1 


1200 


2 


4000 


3 


9000 


4 


9500 


5 


9600 


6 


13000 


7 


18000 


8 


23500 


9 


24000 



Fig. 8-1. Graph of distance versus 
time for the car. 



S 15000 • 




2 4 6 

TIME IN MINUTES 



In Table 8-1, s represents the distance of the car, in feet, from the starting point, 
and / represents the time in minutes. The first line in the table represents zero 
distance and zero time — the car has not started yet. After one minute it has started 
and has gone 1200 feet. Then in two minutes, it goes farther — notice that it picked 
up more distance in the second minute — it has accelerated; but something hap- 
pened between 3 and 4 and even more so at 5 — it stopped at a light perhaps? Then 
it speeds up again and goes 13,000 feet by the end of 6 minutes, 18,000 feet at the 
end of 7 minutes, and 23,500 feet in 8 minutes; at 9 minutes it has advanced to 
only 24,000 feet, because in the last minute it was stopped by a cop. 

That is one way to describe the motion. Another way is by means of a graph. 
If we plot the time horizontally and the distance vertically, we obtain a curve some- 
thing like that shown in Fig. 8-1. As the time increases, the distance increases, 
at first very slowly and then more rapidly, and very slowly again for a little while 
at 4 minutes; then it increases again for a few minutes and finally, at 9 minutes, 
appears to have stopped increasing. These observations can be made from the 
graph, without a table. Obviously, for a complete description one would have to 
know where the car is at the half-minute marks, too, but we suppose that the graph 
means something, that the car has some position at all the intermediate times. 

The motion of a car is complicated. For another example we take something 
that moves in a simpler manner, following more simple laws: a falling ball. 
Table 8-2 gives the time in seconds and the distance in feet for a falling body. 
At zero seconds the ball starts out at zero feet, and at the end of 1 second it has 
fallen 16 feet. At the end of 2 seconds, it has fallen 64 feet, at the end of 3 
seconds, 144 feet, and so on; if the tabulated numbers are plotted, we get the 
nice parabolic curve shown in Fig. 8-2. The formula for this curve can be written 
as 

s = 16/2. (g j^ 

This formula enables us to calculate the distances at any time. You might say 
there ought to be a formula for the first graph too. Actually, one may write such 
a formula abstractly, as 

s = fit), (8.2) 

meaning that s is some quantity depending on / or, in mathematical phraseology, 



32 



Motion 



Table 8-2 



/ (sec) 


^(ft) 








1 


16 


2 


64 


3 


144 


4 


256 


5 


400 


6 


576 



Fig. 8-2. Graph of distance versus 
time for a falling body. 



400- 




2 3 4 

TIME IN SECONDS 



5 is a function of /. Since we do not know what the function is, there is no way we 
can write it in definite algebraic form. 

We have now seen two examples of motion, adequately described with very 
simple ideas, no subtleties. However, there are subtleties — several of them. In 
the first place, what do we mean by time and space ? It turns out that these deep 
philosophical questions have to be analyzed very carefully in physics, and this 
is not so easy to do. The theory of relativity shows that our ideas of space and 
time are not as simple as one might think at first sight. However, for our present 
purposes, for the accuracy that we need at first, we need not be very careful about 
defining things precisely. Perhaps you say, "That's a terrible thing — I learned that 
in science we have to define everything precisely." We cannot define anything 
precisely! If we attempt to, we get into that paralysis of thought that comes to 
philosophers, who sit opposite each other, one saying to the other, "You don't 
know what you are talking about!" The second one says, "What do you mean 
by know ? What do you mean by talking ? What do you mean by you ?,'" and so on. 
In order to be able to talk constructively, we just have to -agree that we are talking 
about roughly the same thing. You know as much about time as we need for the 
present, but remember that there are some subtleties that have to be discussed; 
we shall discuss them later. 

Another subtlety involved, and already mentioned, is that it should be possible 
to imagine that the moving point we are observing is always located somewhere. 
(Of course when we are looking at it, there it is, but maybe when we look away 
it isn't there.) It turns out that in the motion of atoms, that idea also is false — 
we cannot find a marker on an atom and watch it move. That subtlety we shall 
have to get around in quantum mechanics. But we are first going to learn what the 
problems are before introducing the complications, and then we shall be in a better 
position to make corrections, in the light of the more recent knowledge of the 
subject. We shall, therefore, take a simple point of view about time and space. 
We know what these concepts are in a rough way, and those who have driven a 
car know what speed means. 

8-2 Speed 

Even though we know roughly what "speed" means, there are still some 
rather deep subtleties; consider that the learned Greeks were never able to adequately 
describe problems involving velocity. The subtlety comes when we try to compre- 



33 



hend exactly what is meant by "speed." The Greeks got very confused about this, 
and a new branch of mathematics had to be discovered beyond the geometry and 
algebra of the Greeks, Arabs, and Babylonians. As an illustration of the diffi- 
culty, try to solve this problem by sheer algebra: A balloon is being inflated so 
that the volume of the balloon is increasing at the rate of 100 cm ^ per second; 
at what speed is the radius increasing when the volume is 1000 cm^? The Greeks 
were somewhat confused by such problems, being helped, of course, by some very 
confusing Greeks. To show that there were difficulties in reasoning about speed 
at the time, Zeno produced a large number of paradoxes, of which we shall men- 
tion one to illustrate his point that there are obvious difficulties in thinking about 
motion. "Listen," he says, "to the following argument: Achilles runs 10 times as 
fast as a tortoise, nevertheless he can never catch the tortoise. For, suppose that 
they start in a race where the tortoise is 100 meters ahead of Achilles; then when 
Achilles has run the 100 meters to the place where the tortoise was, the tortoise has 
proceeded 10 meters, having run one-tenth as fast. Now, Achilles has to run 
another 10 meters to catch up with the tortoise, but on arriving at the end of that 
run, he finds that the tortoise is still 1 meter ahead of him; running another meter, 
he finds the tortoise 10 centimeters ahead, and so on, ad infinitum. Therefore, at 
any moment the tortoise is always ahead of Achilles and Achilles can never catch 
up with the tortoise." What is wrong with that? It is that a finite amount of time 
can be divided into an infinite number of pieces, just as a length of line can be 
divided into an infinite number of pieces by dividing repeatedly by two. And so, 
although there are an infinite number of steps (in the argument) to the point at 
which Achilles reaches the tortoise, it doesn't mean that there is an infinite amount 
of time. We can see from this example that there are indeed some subtleties in 
reasoning about speed. 

In order to get to the subtleties in a clearer fashion, we remind you of a joke 
which you surely must have heard. At the point where the lady in the car is caught 
by a cop, the cop comes up to her and says, "Lady, you were going 60 miles an 
hour!" She says, "That's impossible, sir, I was travelling for only seven minutes. 
It is ridiculous — how can I go 60 miles an hour when I wasn't going an hour?" 
How would you answer her if you were the cop ? Of course, if you were really the 
cop, then no subtleties are involved; it is very simple: you say, "Tell that to the 
judge!" But let us suppose that we do not have that escape and we make a more 
honest, intellectual attack on the problem, and try to explain to this lady what 
we mean by the idea that she was going 60 miles an hour. Just what do we mean? 
We say, "What we mean, lady, is this: if you kept on going the same way as you 
are going now, in the next hour you would go 60 miles." She could say, "Well, 
my foot was off the accelerator and the car was slowing down, so if I kept on going 
that way it would not go 60 miles." Or consider the falling ball and suppose we 
want to know its speed at the time three seconds if the ball kept on going the way 
it is going. What does that mean — kept on accelerating, going faster? No — kept 
on going with the same velocity. But that is what we are trying to define! For if 
the ball keeps on going the way it is going, it will just keep on going the way it is 
going. Thus we need to define the velocity better. What has to be kept the same? 
The lady can also argue this way: "If I kept on going the way I'm going for one 
more hour, I would run into that wall at the end of the street!" It is not so easy to 
say what we mean. 

Many physicists think that measurement is the only definition of anything. 
Obviously, then, we should use the instrument that measures the speed — the 



34 



Motion 



speedometer — and say, "Look, lady, your speedometer reads 60." So she says, 
"My speedometer is broken and didn't read at all." Does that mean the car is 
standing still? We believe that there is something to measure before we build 
the speedometer. Only then can we say, for example, "The speedometer isn't 
working right," or "the speedometer is broken." That would be a meaningless 
sentence if the velocity had no meaning independent of the speedometer. So we 
have in our minds, obviously, an idea that is independent of the speedometer, 
and the speedometer is meant only to measure this idea. So let us see if we can get 
a better definition of the idea. We say, "Yes, of course, before you went an hour, 
you would hit that wall, but if you went one second, you would go 88 feet; lady, 
you were going 88 feet per second, and if you kept on going, the next second it 
would be 88 feet, and the wall down there is farther away than that." She says, 
"Yes, but there's no law against going 88 feet per second! There is only a law 
against going 60 miles an hour." "But," we reply, "it's the same thing." If it is 
the same thing, it should not be necessary to go into this circumlocution about 
88 feet per second. In fact, the falling ball could not keep going the same way 
even one second because it would be changing speed, and we shall have to define 
speed somehow. 

Now we seem to be getting on the right track; it goes something like this: 
If the lady kept on going for another 1/1000 of an hour, she would go 1/1000 of 
60 miles. In other words, she does not have to keep on going for the whole hour; 
the point is that /or a moment she is going at that speed. Now what that means 
is that if she went just a little bit more in time, the extra distance she goes would 
be the same as that of a car that goes at a steady speed of 60 miles an hour. Per- 
haps the idea of the 88 feet per second is right; we see how far she went in the last 
second, divide by 88 feet, and if it comes out 1 the speed was 60 miles an hour. 
In other words, we can find the speed in this way: We ask, how far do we go in a 
very short time? We divide that distance by the time, and that gives the speed. 
But the time should be made as short as possible, the shorter the better, because 
some change could take place during that time. If we take the time of a falling 
body as an hour, the idea is ridiculous. If we take it as a second, the result is 
pretty good for a car, because there is not much cl^ange in speed, but not for a 
falling body; so in order to get the speed more and more accurately, we should 
take a smaller and smaller time interval. What we should do is take a millionth 
of a second, and divide that distance by a millionth of a second. The result gives 
the distance per second, which is what we mean by the velocity, so we can define 
it that way. That is a successful answer for the lady, or rather, that is the definition 
that we are going to use. 

The foregoing definition involves a new idea, an idea that was not available 
to the Greeks in a general form. That idea was to take an infinitesimal distance 
and the corresponding infinitesimal time, form the ratio, and watch what happens 
to that ratio as the time that we use gets smaller and smaller and smaller. In other 
words, take a limit of the distance travelled divided by the time required, as the 
time taken gets smaller and smaller, ad infinitum. This idea was invented by 
Newton and by Leibnitz, independently, and is the beginning of a new branch 
of mathematics, called the differential calculus. Calculus was invented in order to 
describe motion, and its first application was to the problem of defining what is 
meant by going "60 miles an hour." 

Let us try to define velocity a little better. Suppose that in a short time, 
e, the car or other body goes a short distance x; then the velocity, v, is defined as 



35 



V = x/e, 

an approximation that becomes better and better as the e is taken smaller and 
smaller. If a mathematical expression is desired, we can say that the velocity 
equals the limit as the € is made to go smaller and smaller in the expression x/e, or 

V = lim - . (8.3) 

We cannot do the same thing with the lady in the car, because the table is in- 
complete. We know only where she was at intervals of one minute; we can get 
a rough idea that she was going 5000 ft/min during the 7th minute, but we do not 
know, at exactly the moment 7 minutes, whether she had been speeding up and the 
speed was 4900 ft/min at the beginning of the 6th minute, and is now 5100 ft/min, 
or something else, because we do not have the exact details in between. So only 
if the table were completed with an infinite number of entries could we really 
calculate the velocity from such a table. On the other hand, when we have a com- 
plete mathematical formula, as in the case of a falling body (Eq. 8.1), then it is 
possible to calculate the velocity, because we can calculate the position at any time 
whatsoever. 

Let us take as an example the problem of determining the velocity of the 
falling ball at the particular time 5 seconds. One way to do this is to see from 
Table 8-2 what it did in the 5th second; it went 400 — 256 = 144 ft, so it is going 
144 ft/sec; however, that is wrong, because the speed is changing; on the average 
it is 144 ft/sec during this interval, but the ball is speeding up and is really going 
faster than 144 ft/sec. We want to find out exactly how fast. The technique in- 
volved in this process is the following: We know where the ball was at 5 sec. 
At 5.1 sec, the distance that it has gone all together is 16(5.1)^ = 416.16 ft (see 
Eq. 8.1). At 5 sec it had already fallen 400 ft; in the last tenth of a second it fell 
416.16 - 400 = 16.16 ft. Since 16.16 ft in 0.1 sec is the same as 161.6 ft/sec, 
that is the speed more or less, but it is not exactly correct. Is that the speed at 
5, or at 5.1, or halfway between at 5.05 sec, or when is that the speed? Never mind 
— the problem was to find the speed at 5 seconds, and we do not have exactly 
that; we have to do a better job. So, we take one-thousandth of a second more than 
5 sec, or 5.001 sec, and calculate the total fall as 

s = 16(5.001)2 = 16(25.010001) = 400.160016 ft. 

In the last 0.001 sec the ball fell 0.160016 ft, and if we divide this number by 0.001 
sec we obtain the speed as 160.016 ft/sec. That is closer, very close, but it is 
still not exact. It should now be evident what we must do to find the speed exactly. 
To perform the mathematics we state the problem a little more abstractly: to 
find the velocity at a special time, to, which in the original problem was 5 sec. 
Now the distance at to, which we call ^o, is 16/o, or 400 ft in this case. In order 
to find the velocity, we ask, "At the time /q + (a little bit), or to + €, where is 
the body?" The new position is 16(/o + e)^ = 16/o + 32/oe + 166^. So it is 
farther along than it was before, because before it was only 16/o. This distance 
we shall call sq + (a little bit more), or .Sq + ^ (i^ ^ is the extra bit). Now if we 
subtract the distance at to from the distance at to + e, we get x, the extra distance 
gone, as X = 32/o • e + \(>e^- Our first approximation to the velocity is 

V = - = 32/0 + 16c. (8.4) 

e 

36 



Motion 



The true velocity is the value of this ratio, x/e, when € becomes vanishingly small. 
In other words, after forming the ratio, we take the limit as e gets smaller and 
smaller, that is, approaches 0. The equation reduces to, 

V (at time to) = 32/o. 

In our problem, to = 5 sec, so the solution is y = 32 X 5 = 160 ft/sec. A few 
lines above, where we took e as 0.1 and 0.01 sec successively, the value we got for 
V was a little more than this, but now we see that the actual velocity is precisely 
160 ft/sec. 

8-3 Speed as a derivative 

The procedure we have just carried out is performed so often in mathematics 
that for convenience special notations have been assigned to our quantities e and x. 
In this notation, the e used above becomes At and x becomes As. This At means 
"an extra bit of /," and carries an implication that it can be made smaller. The 
prefix A is not a multiplier, any more than sin 6 means s • i • n • 6 — it simply 
defines a time increment, and reminds us of its special character. As has an 
analogous meaning for the distance s. Since A is not a factor, it cannot be can- 
celled in the ratio As/At to give s/t, any more than the ratio sin ^/sin 26 can be 
reduced to 1/2 by cancellation. In this notation, velocity is equal to the limit of 
As/ At when At gets smaller, or 

V = lim ^. (8.5) 

This is really the same as our previous expression (8.3) with e and x, but it has the 
advantage of showing that something is changing, and it keeps track of what is 
changing. 

Incidentally, to a good approximation we have another law, which says that 
the change in distance of a moving point is the velocity times the time interval, 
or As = V At. This statement is true only if the velocity is not changing during 
that time interval, and this condition is true only in the limit as At goes to 0. 
Physicists like to write it ds = v dt, because by dt they mean At in circumstances 
in which it is very small; with this understanding, the expression is valid to a close 
approximation. If At is too long, the velocity might change during the interval, 
and the approximation would become less accurate. For a time dt, approaching 
zero, ds = v dt precisely. In this notation we can write (8.5) as 

,= lim ^ = ^. 

^<-*o A/ dt 

The quantity ds/dt which we found above is called the "derivative of s with 
respect to ?" (this language helps to keep track of what was changed), and the com- 
plicated process of finding it is called finding a derivative, or differentiating. 
The ds's and dt's which appear separately are called differentials. To familiarize 
you with the words, we say we found the derivative of the function 16/^, or the 
derivative (with respect to /) of 16/^ is 32/. When we get used to the words, the 
ideas are more easily understood. For practice, let us find the derivative of a more 
complicated function. We shall consider the formula s = At^ -\- Bt + C, which 



37 



might describe the motion of a point. The letters A, B, and C represent constant 
numbers, as in the famihar general form of a quadratic equation. Starting from 
the formula for the motion, we wish to find the velocity at any time. To find 
the velocity in the more elegant manner, we change / to / + A/ and note that 
s is then changed to s + some A^; then we find the As in terms of At. That is to 
say, 



s + As = A(t -\- Atf + B(t + AO + C 

but since 
we find that 



= At^ -\- Bt + C -j- 3At^At -\- BAt -\- 3At(At)^ + A(At)\ 

s =■ At^ + 5/ + C, 
A5 = 2>At^ At -\- BAt + 3At(At)^ + A(At)^. 



But we do not want As — we want As divided by At. We divide the preceding equa- 
tion by At, getting 

^ = 3^/2 + 5 + 3At(At) + A(At)^. 

As At goes toward the limit of As/ At is ds/dt and is equal to 

3At^ + B. 



ds^ 
dt 



This is the fundamental process of calculus, diff'erentiating functions. The process 
is even more simple than it appears. Observe that when these expansions con- 
tain any term with a square or a cube or any higher power of A/, such terms may be 
dropped at once, since they will go to when the limit is taken. After a little prac- 
tice the process gets easier because one knows what to leave out. There are many 
rules or formulas for diff'erentiating various types of functions. These can be 
memorized, or can be found in tables. A short list is found in Table 8-3. 

Table 8-3. A Short Table of Derivatives 

s, u, V, w are arbitrary functions of / ; a, b, c, and n are arbitrary constants 



Function 


Derivative 


s =r'* 


dt 




ds du 


s = cu 


dt ~ ^ ~dt 


s = u -{- V -'r w -\- • • ■ 


di ~ 'dt '^ dt '^ dt '^ 


s = c 


dt 




ds /^ ^" 1 * '^ 1 ^ ^^ 1 . . . ] 

di- ~ ^y'i^'di ^ V dt'^ w dt~^ / 


s = U V w . . . 



38 



Motion 

Table ^-4 
Velocity of a Falling Ball 



r(sec) 


V (ft/sec) 








1 


32 


2 


64 


3 


96 


4 


128 



8-4 Distance as an integral 

Now we have to discuss the inverse problem. Suppose that instead of a table of 
distances, we have a table of speeds at different times, starting from zero. For the 
falling ball, such speeds and times are shown in Table 8-4. A similar table could 
be constructed for the velocity of the car, by recording the speedometer reading 
every minute or half-minute. If we know how fast the car is going at any time, can 
we determine how far it goes? This problem is just the inverse of the one solved 
above ; we are given the velocity and asked to find the distance. How can we find 
the distance if we know the speed? If the speed of the car is not constant, and the 
lady goes sixty miles an hour for a moment, then slows down, speeds up, and so 
on, how can we determine how far she has gone? That is easy. We use the same 
idea, and express the distance in terms of infinitesimals. Let us say, "In the first 
second her speed was such and such, and from the formula As = v At we can 
calculate how far the car went the first second at that speed." Now in the next 
second her speed is nearly the same, but slightly different; we can calculate how 
far she went in the next second by taking the new speed times the time. We pro- 
ceed similarly for each second, to the end of the run. We now have a number 
of little distances, and the total distance will be the sum of all these little pieces. 
That is, the distance will be the sum of the velocities times the times, or s = 
'^v At, where the Greek letter ^ (sigma) is used to denote addition. To be more 
precise, it is the sum of the velocity at a certain time, let us say the /-th time, 
multiplied by At. 

s = Zv(ti)At. (8.6) 

The rule for the times is that ti+i = ?i + At. However, the distance we obtain 
by this method will not be correct, because the velocity changes during the time 
interval At. If we take the times short enough, the sum is precise, so we take them 
smaller and smaller until we obtain the desired accuracy. The true s is 

s = lim Ev(ti)At. (8.7) 

At-*0 i 

The mathematicians have invented a symbol for this limit, analogous to the symbol 
for the differential. The A turns into a. d to remind us that the time is as small as 
it can be; the velocity is then called v at the time /, and the addition is written 
as a sum with a great "5," J (from the Latin summd), which has become distorted 
and is now unfortunately just called an integral sign. Thus we write 



5 = 



= j v(t) dt. (8.8) 



39 



This process of adding all these terms together is called integration, and it is the 
opposite process to differentiation. The derivative of this integral is v, so one 
operator (d) undoes the other (J). One can get formulas for integrals by taking 
the formulas for derivatives and running them backwards, because they are re- 
lated to each other inversely. Thus one can work out his own table of integrals 
by differentiating all sorts of functions. For every formula with a differential, 
we get an integral formula if we turn it around. 

Every function can be differentiated analytically, i.e., the process can be carried 
out algebraically, and leads to a definite function. But it is not possible in a simple 
manner to write an analytical value for any integral at will. You can calculate it, 
for instance, by doing the above sum, and then doing it again with a finer interval 
A/ and again with a finer interval until you have it nearly right. In general, given 
some particular function, it is not possible to find, analytically, what the integral 
is. One may always try to find a function which, when differentiated, gives some 
desired function ; but one may not find it, and it may not exist, in the sense of being 
expressible in terms of functions that have already been given names. 

8-5 Acceleration 

The next step in developing the equations of motion is to introduce another 
idea which goes beyond the concept of velocity to that of change of velocity, 
and we now ask, "How does the velocity change?'' In previous chapters we have 
discussed cases in which forces produce changes in velocity. You may have heard 
with great excitement about some car that can get from rest to 60 miles an hour 
in ten seconds flat. From such a performance we can see how fast the speed 
changes, but only on the average. What we shall now discuss is the next level of 
complexity, which is how fast the velocity is changing. In other words, by how 
many feet per second does the velocity change in a second, that is, how many feet 
per second, per second? We previously derived the formula for the velocity of 
a falling body as y = 32/, which is charted in Table 8-4, and now we want to 
find out how much the velocity changes per second; this quantity is called the 
acceleration. 

Acceleration is defined as the time rate of change of velocity. From the 
preceding discussion we know enough already to write the acceleration as the 
derivative dv/dt, in the same way that the velocity is the derivative of the distance. 
If we now differentiate the formula /' = 32/ we obtain, for a falling body, 

a = ~ = 32. (8.9) 

at 

[To differentiate the term 32/ we can utilize the result obtained in a previous 
problem, where we found that the derivative of Bt is simply B (a constant). So 
by letting B = 32, we have at once that the derivative of 32/ is 32.] This means 
that the velocity of a falling body is changing by 32 feet per second, per second 
always. We also see from Table 8-4 that the velocity increases by 32 ft/sec in 
each second. This is a very simple case, for accelerations are usually not constant. 
The reason the acceleration is constant here is that the force on the falling body 
is constant, and Newton's law says that the acceleration is proportional to the force. 
As a further example, let us find the acceleration in the problem we have 
already solved for the velocity. Starting with 



40 



Motion 



s ^ At^ + Bt -^ C 
we obtained, for v = ds/dt, 

V = 3/4/2 ^ ^ 

Since acceleration is the derivative of the velocity with respect to the time, we need 
to differentiate the last expression above. Recall the rule that the derivative of the 
two terms on the right equals the sum of the derivatives of the individual terms. 
To differentiate the first of these terms, instead of going through the fundamental 
process again we note that we have already differentiated a quadratic term when 
we differentiated I6t^, and the effect was to double the numerical coefficient and 
change the /" to /; let us assume that the same thing will happen this time, and you 
can check the result yourself. The derivative of 3 At^ will then be 6 At. Next we 
differentiate B, a constant term; but by a rule stated previously, the derivative of 
B is zero; hence this term contributes nothing to the acceleration. The final 
result, therefore, is a = dv/dt = 6 At. 

For reference, we state two very useful formulas, which can be obtained by 
integration. If a body starts from rest and moves with a constant acceleration, 
g, its velocity v at any time / is given by 

V = gt. 

The distance it covers in the same time is 

s = igt^ 

Various mathematical notations are used in writing derivatives. Since velocity 
is ds/dt and acceleration is the time derivative of the velocity, we can also write 



e) 



d's 



" = 7\7,) - W-' (S'O) 

which are common ways of writing a second derivative. 

We have another law that the velocity is equal to the integral of the accelera- 
tion. This is just the opposite oi a = dv/dt\ we have already seen that distance is 
the integral of the velocity, so distance can be found by twice integrating the ac- 
celeration. 

in the foregoing discussion the motion was in only one dimension, and space 
permits only a brief discussion of motion in three dimensions. Consider a particle 
P which moves in three dimensions in any manner whatsoever. At the beginning 
of this chapter, we opened our discussion of the one-dimensional case of a moving 
car by observing the distance of the car from its starting point at various times. 
We then discussed velocity in terms of changes of these distances with time, and 
acceleration in terms of changes in velocity. We can treat three-dimensional motion 
analogously. It will be simpler to illustrate the motion on a two-dimensional 
diagram, and then extend the ideas to three dimensions. We establish a pair of 
axes at right angles to each other, and determine ihe position of the particle at any 
moment by measuring how far it is from each of the two axes. Thus each position 
is given in terms of an ;c-distance and a >'-distance, and the motion can be described 
by constructing a table in which both these distances are given as functions of time. 



41 



(Extension of this process to tiiree dimensions requires only another axis, at right 
angles to the first two, and measuring a third distance, the z-distance. The dis- 
tances are now measured from coordinate planes instead of lines.) Having con- 
structed a table with x- and >'-distances, how can we determine the velocity? 
We first find the components of velocity in each direction. The horizontal part of 
the velocity, or x-component, is the derivative of the x-distance with respect to 
the time, or 

iv, = dx/dt. (8.11) 



Similarly, the vertical part of the velocity, or >'-component, is 

Vy = dy/dt. 

v^ = dz/dt. 



In the third dimension, 



(8.12) 
(8.13) 



Now. given the components of velocity, how can we find the velocity along the 
actual path of motion? In the two-dimensional case, consider two successive 
positions of the particle, separated by a short distance A5 and a short time in- 
terval t2 — ti = ^t. In the time A/ the particle moves horizontally a distance 
Ax ~ Tj- A/, and vertically a distance ^y ~ Vy^t. (The symbol "~" is read 
"is approximately.") The actual distance moved is approximately 



^s ~ v'(Aa:)2 ^ (^^y)'^ 



(8.14) 



as shown in Fig. 8-3. The approximate velocity during this interval can be obtained 
by dividing by A/ and by letting A/ go to 0, as at the beginning of the chapter. 

We then get the velocity as 



ds 



V = j^ = V{dx/dty + (dy/dt^) = Vvfhvl 



(8.15) 



For three dimensions the result is 



= VV^ + ^,2 + y 



(8.16) 



Asw^Ax)* + (Ay)' 





Fig. 8-3. Description of the motion 
of a body in two dimensions and the 
computation of its velocity. 



Fig. 8-4. The parabola described by 
a falling body with an initial horizontal 
velocity. 



42 



Motion 



In the same way as we defined velocities, we can define accelerations: we have 
an A:-component of acceleration a^, which is the derivative of Vx, the jc-component 
of the velocity (that is, Ox = d'^x/dt^, the second derivative of x with respect to 
/), and so on. 

Let us consider one nice example of compound motion in a plane. We shall 
take a motion in which a ball moves horizontally with a constant velocity u, and 
at the same time goes vertically downward with a constant acceleration —g', 
what is the motion ? We can say dx/dt = Vx = u. Since the velocity Vx is constant, 

X = ut, (8.17) 

and since the downward acceleration — ^ is constant, the distance y the object 
falls can be written as 

y = -hgt'- (8.18) 

What is the curve of its path, i.e., what is the relation between y and x? We can 
eliminate / from Eq. (8.18), since t = x/u. When we make this substitution we 
find that 

y= -^2^'- (819) 

This relation between ;; and x may be considered as the equation of the path of 
the moving ball. When this equation is plotted we obtain a curve that is called a 
parabola; any freely falling body that is shot out in any direction will travel in 
a parabola, as shown in Fig. 8-4. 



43 



The twentieth century artist has been able to exploit 
his interest in motion in various ways in works of art. 



9 Representation of Movement 

Gyorgy Kepes 

A chapter from his book Language of Vision, 1944. 

Matter, the physical basis of all spatial experience and thus the source 
material of representation, is kinetic in its very essence. From atomic hap- 
penings to cosmic actions, all elements in nature are in perpetual interac- 
tion — in a flux complete. We are living a mobile existence. The earth 
b rotating; the sun is moving; trees are growing; flowers are opening 
and closing; clouds are merging, dissolving, coming and going; light and 
shadow are hunting each other in an indefatigable play; forms are appear- 
ing and disappearing; and man, who is experiencing all this, is himself 
subject to all kinetic change. The perception of physical reality cannot 
escape the quality of movement. The very understanding of spatial facts, 
the meaning of extension or distances, involves the notion of time — a 
fusion of space-time which is movement. "Nobody has ever noticed a 
place except at a time or a time except at a place,*' said Minkowsky in his 
Principles of Relativity. 

The sources of movement perception 

As in a wild jungle one cuts new paths in order to progress further, man 
builds roads of perception on which he is able to approach the mobile 
world, to discover order in its relationships. To build these avenues of 
perceptual grasp he relies on certain natural factors. One is the nature 
of the retina, the sensitive surface on which the mobile panorama is pro- 
jected. The second is the sense of movement of his body — the kinesthetic 
sensations of his eye muscles, limbs, head, which have a direct correspond- 
ence with the happenings around him. The third is the memory association 
of past experience, visual and non-visual; his knowledge about the laws 
of the physical nature of the surrounding object-world. 

The shift of the retinal image 

We perceive any successive stimulation of the retinal receptors as move- 
ment, because such progressive stimulations are in dynamic interaction 
with fixed stimulations, and therefore the two different types of stimulation 
can be perceived in a unified whole only as a dynamic process, movement. 
II the retina is stimulated with stationary impacts that follow one another 



44 



Representation of Movement 



in rapid succession, the same sensation of optical movement is induced. 
Advertising displays with their rapidly flashing electric bulbs are per- 
ceived in continuity through the persistence of vision and therefore pro- 
duce the sensation of movement, although the spatial position of the light 
bulbs is stationary. The movement in the motion picture is based upon 
the same source of the visual perception. 

The changes of any optical data indicating spatial relationships, such as 
size, shape, direction, interval, brightness, clearness, color, imply motion. 
If the retinal image of any of these signs undergoes continuous regular 
change, expansion or contraction, progression or graduation, one per- 
ceives an approaching or receding, expanding or contracting movement. If 
one sees a growing or disappearing distance between these signs, he 
perceives a horizontal or vertical movement. 

"Suppose for instance, that a person is standing still in a thick woods, 
where it is impossible for him to distinguish, except vaguely and roughly 
in a mass of foliage and branches all around him, what belongs to one 
tree and what to another, and how far the trees are separated. The moment 
he begins to move forward, however, everything disentangles itself and 
immediately he gets an apperception of the content of the woods and the 
relationships of objects to each other in space."* 

From a moving train, the closer the object the faster it seems to move. A 
far-away object moves slowly and one very remote appears to be station- 
ary. The same phenomenon, with a lower relative velocity, may be noticed 
in walking, and with a still higher velocity in a landing aeroplane or in a 
moving elevator. 

The role of relative velocity 

The velocity of motion has an important conditioning effect. Motion 
can be too fast or too slow to be perceived as such by our limited sensory 
receiving set. The growth of trees or of man, the opening of flowers, the 
evaporation of water are movements beyond the threshold of ordinary 
visual grasp. One does not see the movement of the hand of a watch, of 
a ship on a distant horizon. An aeroplane in the highest sky seems to 
hang motionless. No one can see the traveling of light as such. In certain 
less rapid motions beyond the visual grasp, one is able, however, to 
observe the optical transformation of movement into the illusion of a 
solid. A rapidly whirled torch loses its characteristic physical extension, 
but it submerges into another three-dimensional-appearing solid — into the 
virtual volume of a cone or a sphere. Our inability to distinguish sharply 
beyond a certain interval of optical impacts makes the visual impressions 
a blur which serves as a bridge to a new optical form. The degree of 
velocity of its movement will determine the apparent density of that new 
form. The optical density of the visible world is in a great degree con- 
ditioned by our visual ability, which has its particular limitations. 



• Helmhollz, Physiological Optics 45 



The kinesthetic gensation 

When a moving object comes into the visual field, one pursues it by a 
corresponding movement of his eyes, keeping it in a stationary or nearly 
stationary position on the retina. Retinal stimulation, then, cannot alone 
account for the sensation of movement. Movement-experience, which is 
undeniably present in such a case, is induced by the sensation of muscle 
movements. Each individual muscle-fibre contains a nerve end, which 
registers every movement the muscle makes. That we are able to sense 
space in the dark, evaluate direction-distancess in the absence of contacted 
bodies, is due to this muscular sensation — the kinesthetic sensation. 







E. G. Lukacs. A<:lion 

from Herbert Bayer De<isn Class 



H. L. C.iirpeiiler. yiuremenl • 

• fTork done for the author's course in Visual Fundamentals. 



PhiiI Riiiid. Cover Design 




46 



Memory sources 



Experience teaches man to distinguish things and to evaluate their physi- 
cal properties. He knows that bodies have weight: unsupported they 
will of necessity fall. When, therefore, he sees in midair a body he 
knows to be heavy, he automatically associates the direction and velocity 
of its downward course. One is also accustomed to seeing small objects 
as more mobile than large ones. A man is more mobile than a mountain; 
a bird is more frequently in motion than a tree, the sky. or other visible 
units in its background. Everything that one experiences is perceived in 
a polar unity in which one pole is accepted as a stationary background 
and the other as a mobile, changing figure. 

Through all history painters have tried to suggest movement on ihe 
^tationary picture surface, to translate some of the optical signs of move- 
ment-experience into terms of the picture-image. Their efforts, however, 
have been isolated attempts in which one or the other sources of move- 
ment-experience were drawn upon; the shift of the retinal image, the 



Representation of Movement 




kinesthetic experience, or the memory of past experiences were suggested 
in two-dimensional terms. 

These attempts were conditioned mainly by the habit of using things 
as the basic measuring unit for every event in nature. The constant 
characteristics of the things and objects, first of all the human body, 
animals, sun, moon, clouds, or trees, were used as the first fixed points of 
reference in seeking relationships in the optical turmoil of happenings. 

Therefore, painters tried first to represent motion by suggesting the 
visible modifications of objects in movement. They knew the visual 
characteristics of stationary objects and therefore every observable change 
served to suggest movement. The prehistoric artist knew his animals, 
knew, for example, how many legs they had. But when he saw an 
animal in really speedy movement, he could not escape seeing the visual 
modification of the known spatial characteristics. The painter of the 
Altamiro caves who pictures a running reindeer with numerous legs, or 
the twentieth century cartoonist picturing a moving face with many 
superimposed profiles, is stating a relationship between what he knows and 
what he sees. 

Other painters, seeking to indicate movement, utilized the expressive dis- ch. d. Gibson. 

tortion of the moving bodies. Michaelangelo, Goya, and also Tintoretto, ^'"' ^''•""— «"'» W''""""" -• i^oo 

by elongating and stretching the figure, showed distortion of the face 

under the expression of strains of action and mobilized numerous other 

psychological references to suggest action. 

The smallest movement is more possessive of the attention than the 
greatest wealth of relatively stationary objects. Painters of many different 
periods observed this well and explored it creatively. The optical vitality 
of the moving units they emphasized by dynamic outlines, by a vehement 
interplav of vigorous contrast of light and dark, and by extreme contrast 
of colors. In various paintings of Tintoretto, Maffei, Veronese, and Goya, 
the optical wealth and intensity of the moving figures are juxtaposed 
against the submissive, neutral, visual pattern of the stationary back- 
ground. 
The creative exploitation of the successive stimulations of the retinal 

receptors in terms of the picture surface was another device many painters 
found useful. Linear continuance arrests the attention and forces the eye 
into a pursuit movement. The eye, following the line, acts as if it were 
on the path of a moving thing and attributes to the line the quality of 
movement. When the Greek sculptors organized the drapery of their 
figures which they represented in motion, the lines were conceived as 
optical forces making the eye pursue their direction. 

We know that a heavy object in a background that does not offer sub- 
stantial resistance will fall. Seeing such an object we interpret it as action. 



47 



Haruiiobu. Windy Day Under Willow j 

Courtesy of The Art Institute oj Chicago 





MafTei. Painting 



We make a kind of psychological qualification. Every object seen and in- 
terpreted in a frame of reference of gravitation is endowed with potential 
action and could appear as falling, rolling, moving. Because we custom- 
arily assume an identity between the horizontal and vertical directions on 
the picture surface and the main directions of space as we perceive them 
in our everyday experiences, every placing of an object representation on 
the picture surface which contradicts the center of gravity, the main direc- 
tion of space — the horizontal or vertical axis — causes that object to appear 
to be in action. Top and bottom of the picture surface have a significance 
in this respect. 

Whereas the visual representation of depth had found various complete 
systems, such as linear perspective, modelling by shading, a parallel devel- 
opment had never taken place in the visual representation of motion. 
Possibly this has been because the tempo of life was comparatively slow; 
therefore, the ordering and representation of events could be compressed 
without serious repercussions in static formulations. Events were meas- 
ured by things, static forms identical with themselves, in a perpetual 
fixity. But this static point of view lost all semblance of validity when 
daily experiences bombarded man with a velocity of visual impacts in 
which the fixity of the things, their self-identity, seemed to melt away. 



48 



Representation of Movement 



G. McVicker. Study of Linear Movenieiil 

Work done lor the author's course 
in Visual Fundamentals 

Sponsored by The Art Director s Club 
ol Chicago. 1938 



Lee King. Study of Movement Represenlnlion 

Work done for the author's course 
in Visual Fundamentals 

School ol Design in Chicago 





49 



The more complex life became, the more dynamic relationships confronted 
man, in general and in particular, as visual experiences, the more neces- 
sary it became to revaluate the old relative conceptions about the fixity of 
things and to look for a new way of seeing that could interpret man's 
surroundings in their change. It was no accident that our age made the 
first serious search for a reformulation of the events in nature into 
dynamic terms. This reformulation of our ideas about the world included 
almost all the aspects one perceives. The interpretation of the objective 
world in the terms of physics, the understanding of the living organism, 
the reading of the inner movement of social processes, and the visual 
interpretation of events were, and still are, struggling for a new gauge 
elastic enough to expand and contract in following the dynamic changes 
of events. 

The influence of the technological conditions 

The environment of the man living today has a complexity which cannot 
be compared with any environment of any previous age. The skyscrapers, 
the street with its kaleidoscopic vibration of colors, the window-displays 
with their multiple mirroring images, the street cars and motor cars, 
produce a dynamic simultaneity of visual impression which cannot be 
perceived in the terms of inherited visual habits. In this optical turmoil 
the fixed objects appear utterly insufficient as the measuring tape of the 
events. The artificial light, the flashing of electric bulbs, and the mobile 
game of the many new types of light-sources bombard man with kinetic 
color sensations having a keyboard never before experienced. Man, 
the spectator, is himself more mobile than ever before. He rides in street- 
cars, motorcars and aeroplanes and his own motion gives to optical impacts 
a tempo far beyond the threshold of a clear object-perception. The ma- 
chine man operates adds its own demand for a new way of seeing. The 
complicated interactions of its mechanical parts cannot be conceived in a 
static way; they must be perceived by understanding of their movements. 
The motion picture, television, and, in a great degree, the radio, require 
a new thinking, i.e., seeing, that takes into account qualities of change, 
interpenetration and simultaneity. 

Man can face with success this intricate pattern of the optical events only 
as he can develop a speed in his perception to match the speed of his 
environment. He can act with confidence only as he learns to orient 
himself in the new mobile landscape. He needs to be quicker than the 
event he intends to master. The origin of the word "speed" has a revealing 
meaning. In original form in most languages, speed is intimately con- 
nected with success. Space and speed are, moreover, in some early forms 
of languages, interchangeable in meaning. Orientation, which is the basis 
of survival, is guaranteed by the speed of grasping the relationships of 
the events with which man is confronted. 



50 



Representation of Movement 

Social and pgychological motivations 

Significantly, the contemporary attempts to represent movement were made 
in the countries where the vitality of living was most handicapped by 
outworn social conditions. In Italy, technological advances and their eco- 
nomical-social consequences, were tied with the relics of past ideas, institu- 
tions. The advocates of change could see no clear, positive direction. 
Change as they conceived it meant expansion, imperialist power policy. 
The advance guard of the expanding imperialism identified the past with 
the monuments of the past, and with the keej>ers of these monuments; 
and they tried to brejik, with an uninhibited vandalism, everything which 
seemed to them to fetter the progress toward their goals. "We want to 
free our country from the fetid gangrene of professors, archaeologists, 
guides and antique shops," proclaimed the futurist manifesto of 1909. 
The violence of imperialist expansion was identified with vitality; with 
the flux of life itself. Everything which stood in the way of this desire 
of the beast to reach his prey was to be destroyed. Movement, speed, 
velocity became their idols. Destructive mechanical implements, the 
armoured train, machine gun, a blasting bomb, the aeroplane, the motor 
car', boxing, were adored symbols of the new virility they sought. 
In Russia, where the present was also tied to the past and the people 
were struggling for the fresh air of action, interest also focused on the 
dynamic qualities of experience. The basic motivation of reorientation 
toward a kinetic expression there was quite similar to that of the Italian 
futurists. It was utter disgust with a present held captive by the past. 
Russia's painters, writers, like Russia's masses, longed to escape into a 
future free from the ties of outworn institutions and habits. Museums, 
grammar, authority, were conceived of as enemies; force, moving masses, 
moving machines were friends. But this revolt against stagnant traditions, 
this savage ridiculing of all outworn forms, opened the way for the 
building of a broader world. The old language, which as Mayakovsky said 
"was too feeble to catch up with life," was reorganized into kinetic 
idioms of revolutionary propaganda. The visual language of the past, 
from whose masters Mayakovsky asked with just scorn, "Painters will 
you try to capture speedy cavalry with the tiny net of contours?" was 

infused with new living blood of motion picture vision. 

In their search to find an optical projection which conformed to the 

dynamic reality as they sensed and comprehended it, painters uncon- 
sciously repeated the path traced by advancing physical science. 
Their first step was to represent on the same picture-plane a sequence of 
positions of a moving body. This was basically nothing but a cataloging of 
stationary spatial locations. The idea corresponded to the concept of 
classical physics, which describes objects existing in three-dimensional 
space and changing locations in sequence of absolute time. The concept 
of the object was kept. The sequence of events frozen on the picture- 

51 



plane only amplified the contradiction between the dynamic reality and the 
fixity of the three-dimensional object-concept. 

Their second step was to fuse the different positions of the object by 
filling out the pathway of their movement. Objects were no longer con- 
sidered as isolated, fixed units. Potential and kinetic energies were 
included as optical characteristics. The object was regarded to be either 
in active motion, indicating its direction by "lines of force," or in potential 
motion, pregnant with lines of force, which pointed the direction in which 
the object would go if freed. The painters thus sought to picture the 
mechanical point of view of nature, devising optical equivalents for mass, 
force, and gravitation. This innovation signified important progress, 
because the indicated lines of forces could function as the plastic forces 
of two-dimensional picture-plane. 

The third step was guided by desire to integrate the increasingly compli- 
cated maze of movement-directions. The chaotic jumble of centrifugal 
line of forces needed to be unified. Simultaneous representation of the 
numerous visible aspects composing an event was the new representational 
technique here introduced. The cubist space analysis was synchronized 
with the line of forces. The body of the moving object, the path of its 
movement and its background were portrayed in the same picture by 
fusing all these elements in a kinetic pattern. The romantic language of 
the futurist manifestos describes the method thus: "The sinmltaneosity 
of soul in a work of art; such is the exciting aim of our art. In painting 
a figure on a balcony, seen from within doors, we shall not confine the 
view to what can be seen through the frame of the window; we shall give 
the sum total of the visual sensation of the street, the double row of 
houses extending right and left the flowered balconies, etc. ... in other 
words, a simultaneity of environment and therefore a dismemberment 
and dislocation of objects, a scattering and confusion of details inde- 
pendent of one and another and without reference to accepted logic," said 
Marinetti. This concept shows a great similarity to the idea expressed by 
Einstein, expounding as a physicist the space-time interpretation of the 
general theory of relativity. "The world of events can be described by a 
static picture thrown onto the background of the four dimensional time- 
space continuum. In the past science described motion as happenings in 
lime, general theory of relativity interprets events existing in space-time." 
The closest approximation to representation of motion in the genuine 
terms of the picture-plane was achieved by the utilization of color planes 
as the organizing factor. The origin of color is light, and colors on the 
picture surface have an intrinsic tendency to return to their origin. Motion, 
therefore, is inherent in color. Painters intent on realizing the full motion 
potentialities of color believed that the image becomes a form only in the 
progressive interrelationships of opposing colors. Adjacent color-surfaces 
exhibit contrast effects. They reinforce each other in hue, saturation, and 
intensity. 



S2 



Representation of Movement 




Ciacotno Balla. Dog on Leath 1912. Courtesy oj The Museum of Modern Art 




Giacomo Balla. Automobile and ISohe. Courtesy ol An ol This Century 



53 




Marcrl Durhainp. 

Made Descending the Sluirt I91'i 

Keproduclion Courtesy 
The An Insiiiiife ol Chirne« 




Marcel Duchamp. Sad Young Man in a Trai 

Courtesy ol Art ol This Century 



54 



Representation of Movement 



W^- 



Gyorgy Kepes. Advertising Design 1938 

Courtesy of Container Corporation of America 





^ CONTAINER CORPORATION OF AMERICA 



Herbert Mailer. Advertising Design 

Courtesy oj Container Corporation of America 



CONTAINER CORPORATION OF AMERICA 



55 



Representation of Movement 




Harold E. Edgerton. Golfer 



Soviet Poster 




56 



Representation of Movement 




E. McKnight KaufTer. The Early Bird 1919 

Courtesy of The Museum ol Modern Art 



57 



Representation of Movement 




Driauney. Circular Rhythm Courtesy of The Guggenheim \1iiseum ol ,\onObjective Art 



58 



Representation of Movement 



The greater the intensity of the color-surfaces achieved by a carefully 
organized use of simultaneous and successive contrast, the greater their 
spatial movement color in regard to picture-plane. Their advancing, 
receding, contracting and circulating movement on the surface creates a 
rich variety, circular, spiral, pendular, etc., in the process of moulding 
them into one form which is light or, in practical terms, grey. "Form 
is movement," declared Delaunay. The classical continuous outline of the 
objects was therefore eliminated and a rhythmic discontinuity created by 
grouping colors in the greatest possible contrast. The picture-plane, 
divided into a number of contrasting color-surfaces of different hue, satu- 
ration, and intensity, could be perceived only as a form, as a unified 
whole in the dynamic sequence of visual perception. The animation of the 
image they achieved is based upon the progressive steps in bringing oppos- 
ing colors into balance. 

The centrifugal and centripetal forces of the contrasting color-planes 
move forward and backward, up and down, left and right, compelling the 
spectator to a kinetic participation as he follows the intrinsic spatial- 
direction of colors. The dynamic quality is based upon the genuine 
movement of plastic forces in their tendency toward balance. Like a spin- 
ning top or the running wheel of a bicycle, which can find its balance 
only in movement, the plastic image achieves unity in movement, in per- 
petual relations of contrasting colors. 




A. M. (!«•■>» iiilrr. Poster 



59 



In his witty and provocative book. About Vectors, from which 
this opening chapter is taken, Banesh HofFmann confesses that 
he seeks here "to instruct primarily by being disturbing and 
annoying." 



10 Introducing Vectors 

Banesh Hoffmann 

A chapter from his book About Vectors, 1966. 

Making good definitions is not easy. The story goes that when the philos- 
opher Plato defined Man as "a two-legged animal without feathers," Diogenes 
produced a plucked cock and said "Here is Plato's man." Because of this, the 
definition was patched up by adding the phrase "and having broad nails"; 
and there, unfortunately, the story ends. But what if Diogenes had countered 
by presenting Plato with the feathers he had plucked? 

Exercise 1 .1 What? [Note that Plato would now have feathers.] 

Exercise 1 .2 Under what circumstances could an elephant qualify as 
a man according to the above definition? 

A vector is often defined as an entity having both magnitude and direction. 
But that is not a good definition. For example, an arrow-headed line segment 
like this 



has both magnitude (its length) and direction, and it is often used as a draw- 
ing of a vector; yet it is not a vector. Nor is an archer's arrow a vector, though 
it, too, has both magnitude and direction. 

To define a vector we have to add to the above definition something 
analogous to "and having broad nails," and even then we shall find ourselves 
not wholly satisfied with the definition. But it will let us start, and we can try 
patching up the definition further as we proceed — and we may even find our- 
selves replacing it by a quite different sort of definition later on. If, in the end, 
we have the uneasy feeling that we have still not found a completely satisfac- 
tory definition of a vector, we need not be dismayed, for it is the nature of 
definitions not to be completely satisfactory, and we shall have learned pretty 
well what a vector is anyway, just as we know, without being able to give a 
satisfactory definition, what a man is — well enough to be able to criticize 
Plato's definition. 

Exercise 1 .3 Define a door. 

Exercise 1 .4 Pick holes in your definition of a door. 

Exercise 1 .5 According to your definition, is a movable partition 
between two rooms a door? 



60 



Introducing Vectors 
2. THE PARALLELOGRAM LAW 

The main thing we have to add to the magnitude-and-direction definition 
of a vector is the following: 




*^P 



Figure 2.1 

Let us think of vectors as having definite locations. And let the arrow-headed 
line segments OP and OQ in Figure 2.1 represent the magnitudes, directions, 
and locations of two vectors starting at a common point O. Complete the 
parallelogram formed by OP and OQ, and draw the diagonal OR. Then, when 
taken together, the two vectors represented by OP and OQ are equivalent to 
a single vector represented by the arrow-headed line segment OR. This vector 
is called the resultant of the vectors represented by OP and OQ, and the above 
crucial property of vectors is called the parallelogram law of combination of 
vectors. 

Exercise 2.1 Find (a) by drawing and measurement, and (b) by 

calculation using Pythagoras' theorem, the magnitude and direction of 
the resultant of two vectors OP and OQ if each has magnitude 3, and OP 
points thus — > while OQ points perpendicularly, thus ] .[Ans. The 
magnitude is 3v^, or approximately 4.2, and the direction bisects the 

right angle between OP and OQ.] 

Exercise 2.2 Show that the resultant of two vectors OP and OQ 
that point in the same direction is a vector pointing in the same direction 
and having a magnitude equal to the sum of the magnitudes of OP and 
OQ. [Imagine the parallelogram in Figure 2.1 squashed flat into a line.] 

Exercise 2.3 Taking a hint from Exercise 2.2, describe the resultant 
of two vectors OP and OQ that point in opposite directions. 

Exercise 2.4 In Exercise 2.3, what would be the resultant if OP and 
OQ had equal magnitudes? [Do you notice anything queer when you 
coropare this resultant vector with the definition of a vector?] 
Exercise 2.5 Observe that the resultant of OP and OQ is the same 
as the resultant of OQ and OP. [This is trivially obvious, but keep it in 
mind nevertheless. We shall return to it later.] 

In practice, all we need to draw is half the parallelogram in Figure 2.1 — 
either triangle OPR or triangle OQR. When we do this it looks as if we had 
combined two vectors OP and PR (or OQ and QR) end-to-end like this, even 




^P 




Figure 2.2 (For clarity, the arrow heads meeting 
at R have been slightly displaced. We shall occa- 
sionally displace other arrow heads under similar 
circumstances.) "' 



though they do not have the same starting point. Actually, though, we have 
merely combined OP and OQ by the parallelogram law.* But suppose we 
were dealing with what are called free vectors — vectors having the freedom to 
move from one location to another, so that OP and QR in Figure 2.2, for 
example, which have the same magnitude and the same direction, are officially 
counted not as distinct vectors but as the same free vector. Then we could indeed 
combine free vectors that were quite far apart by bringing them end-to-end, 
like OPand PR in Figure 2.2. But since we could also combine them accord- 
ing to the parallelogram law by moving them so that they have a common 
starting point, like OP and OQ m Figure 2.1, the parallelogram law is the 
basic one. Note that when we speak of the same direction we mean just that, 
and not opposite directions — north and south are not the same direction. 

♦Have you noticed that we have been careless in sometimes speaking of "the vector 
represented by OP," at other times calling it simply "the vector OP," and now calling it 
just "OP''? This is deliberate — and standard practice among mathematicians. Using 
meticulous wording is sometimes too much of an effort once the crucial point has been 
made. 

Exercise 2.6 Find the resultant of the three vectors OA, OB, and 
OC in the diagram. 




Solution We first form the resultant, OR, of OA and OB like this : 




and then we form the resultant, OS, of OR and OC like this : 




This figure looks complicated. We can simplify it by drawing only half of 
each parallelogram, and then even omitting the line OR, like this: 




From this we see that the resultant OS can be found quickly by thinking 
of the vectors as free vectors and combining them by placing them end- 
to-end; /I/?, which has the same magnitude and direction as 05, starts 
where OA ends; and then RS, which has the same magnitude and direction 
as OC, starts where AR ends. 



62 



Introducing Vectors 



Exercise 2.7 Find, by both methods, the resultant of the vectors in 
Exercise 2.6, but by combining OB and OC first, and then combining 
their resultant with OA. Prove geometrically that the resultant is the 
same as before. 

Exercise 2.8 




The above diagram looks like a drawing of a box. Show that if we drew 
only the lines OA, AR, RS, and OS we would have essentially the last 
figure in Exercise 2.6; that if we drew only the lines OB, BT, TS, and OS 
we would have a corresponding figure for Exercise 2.7; and that if we 
drew only OA, AU, US, and OS we would have a figure corresponding to 
our having first combined OA with OC and then their resultant with OB. 
Exercise 2.9 In Exercises 2.6, 2.7, and 2.8, is it essential that the 
three vectors OA, OB, and OC lie in a plane? Give a rule for finding the 
resultant of three noncoplanar vectors OA, OB, and OC that is analogous 
to the parallelogram law, and that might well be called the parallelepiped 
law. Prove that their resultant is the same regardless of the order in 
which one combines them. 

Exercise 2.10 Find the resultant of the three vectors 0^4, 05, and 
OC below by combining them in three different orders, given that vectors 
OA and OC have equal magnitudes and opposite directions. Draw both 
the end-to-end diagrams and the full parallelogram diagrams for each 
case. 



C-*- 



■^A 



3. JOURNEYS ARE NOT VECTORS 

It is all very well to start with a definition. But it is not very enlightening. 
Why should scientists and mathematicians be interested in objects that have 
magnitude and direction and combine according to the parallelogram law? 
Why did they even think of such objects? Indeed, do such objects exist at all 
— outside of the imaginations of mathematicians? 

There are, of course, many objects that have both magnitude and direc- 
tion. And there are, unfortunately, many books about vectors that give the 
reader the impression that such objects obviously and inevitably obey the 
parallelogram law. It is therefore worthwhile to explain carefully why most 
such objects do not obey this law, and then, by a process of abstraction, to 
find objects that do. 



63 



Suppose that I live at A and my friend lives 10 miles away at B. I start 
from A and walk steadily at 4 m.p.h. for 2| hours. Obviously, I walk 10 miles. 
But do I reach 5? 

You may say that this depends on the direction I take. But what reason is 
there to suppose that I keep to a fixed direction? The chances are overwhelm- 
ing that I do not — unless I am preceded by a bulldozer or a heavy tank. 
Most likely I walk in all sorts of directions; and almost certainly, I do not 
arrive at B. I may even end up at home. 

Exercise 3.1 Where are all the possible places at which I can end, 
under the circumstances? 

Now suppose that I start again from A and this time end up at B. I may 
take four or five hours, or I may go by bus or train and get there quickly. 
Never mind how I travel or how long I take. Never mind how many times I 
change my direction, or how tired I get, or how dirty my shoes get, or whether 
it rained. Ignore all such items, important though they be, and consider the 
abstraction that results when one concentrates solely on the fact that I start at 
A and end at B. Let us give this abstraction a name. What shall we call it? 
Not a "journey." That word reminds us too much of everyday life — of rain, 
and umbrellas, and vexations, and lovers meeting, and all other such items 
that we are ignoring here; besides, we want to preserve the word "journey" 
for just such an everyday concept. For our abstraction we need a neutral, 
colorless word. Let us call it a shift. 

Here are routes of four journeys from A to B: 




Figure 3.1 

All four journeys are different — with the possible but highly improbable 
exception of (b) and (c). 

Exercise 3.2 Why "highly improbable"? 

But though the four journeys are not all the same, they yield the same 
shift. We can represent this shift by the arrow-headed line segment AB. It has 
both magnitude and direction. Indeed, it seems to have little else. Is it a 
vector? Let us see. 

Consider three places A, B, and C as in Figure 3.2. If I walk in a straight 




Figure 3.2 

line from A to B and then in a straight line from B to C,l make a journey 
from A to C, but it is not the same as if I walked directly in a straight line 
from A to C: the scenery is different, and so is the amount of shoe leather 
consumed, most likely, and we can easily think of several other differences. 



64 



Introducing Vectors 



Exercise 3.3 Why "most likely"? 

Thus, though we could say that the walks from A io B and from 5 to C 
combine to give a "resultant" journey from A to C, it is not a journey in a 
straight line from ^ to C: the walks do not combine in a way reminiscent of 
the way in which vectors combine; they combine more in the tautological 
sense that 2+1=2+1 than 2+1=3. 

Journeys, then, are not vectors. But when we deal with shifts we ignore 
such things as the scenery and the amount of shoe leather consumed. A shift 
from A to B followed by a shift from 5 to C is indeed equivalent to a shift 
from A to C. And this reminds us so strongly of the vectorial situation in 
Figure 2.2 that we are tempted to conclude that shifts are vectors. But there 
is a crucial difference between the two situations. We cannot combine the 
above shifts in the reverse order (compare Exercise 2.5). There is no single 
equivalent to the shift from 5 to C followed by the shift from A to B. We can 
combine two shifts only when the second begins where the first ends. Indeed, 
in Figure 2.1, just as with journeys, we cannot combine a shift from O to P 
with one from O to g in either order. Thus shifts are not vectors. 

4. DISPLACEMENTS ARE VECTORS 

Now that we have discovered why shifts are not vectors, we can easily see 
what further abstraction to make to obtain entities that are. From the already 
abstract idea of a shift, we remove the actual starting point and end point and 
retain only the relation between them : that B lies such and such a distance from 
A and in such and such a direction.* Shifts were things we invented in order 
to bring out certain distinctions. But this new abstraction is an accepted ma- 
thematical concept with a technical name : it is called a displacement. And it is 
a vector, as we shall now show. 

In Figure 4.1, the arrow-headed line segments AB and LM are parallel and 




Figure 4.1 

of equal length. Any journey from y4 to 5 is bound to be different from a 
journey from L to M. Also, the shift from A to B is different from that from 
L to M because the starting points are different, as are the end points. But the 
two shifts, and thus also the various journeys, yield the same displacement: 
if, for example, 5 is 5 miles north-northeast of A, so too is M 5 miles north- 
northeast of L, and the displacement is one of 5 miles in the direction north- 
northeast. 

Exercise 4.1 Starting from a point A, a man bicycles 10 miles due 

east to point B, stops for lunch, sells his bicycle, and then walks 10 miles 
due north to point C. Another man starts from B, walks 4 miles due north 
and 12 miles due east and then, feeling tired, and having brought along 

*We retain, too, the recollection that we are still linked, however tenuously, with 
journeying, for we want to retain the idea that a movement has occurred, even though we 
do not care at all how or under what circumstances it occurred. 



65 



a surplus of travellers' checks, buys a car and drives 6 miles due north 
and 2 miles due west, ending at point D in the pouring rain. What dis- 
placement does each man undergo? [Ans. lOV^ miles to the northeast.] 

Now look at Figure 2.1. The shift from O to P followed by the shift from 
P to R is equivalent to the shift from O to R. The shift from P to R gives a 
displacement PR that is the same as the displacement OQ. Therefore the 
displacement OP followed by the displacement OQ is equivalent to the dis- 
placement OR. 

Exercise 4.2 Prove, similarly, that the displacement OQ followed 
by the displacement OP is also equivalent to the displacement OR. 

Thus, displacements have magnitude and direction and combine according 
to the parallelogram law. According to our definition, they are therefore 
vectors. Since displacements such as AB and LM in Figure 4.1 are counted as 
identical, displacements are free vectors, and thus are somewhat special. In 
general, vectors such as AB and LM are not counted as identical. 



5. WHY VECTORS ARE IMPORTANT 

From the idea of a journey we have at last come, by a process of succes- 
sive abstraction, to a specimen of a vector. The question now is whether we 
have come to anything worthwhile. At first sight it would seem that we have 
come to so pale a ghost of a journey that it could have little mathematical signifi- 
cance. But we must not underestimate the potency of the mathematical process 
of abstraction. Vectors happen to be extremely important in science and 
mathematics. A surprising variety of things happen to have both magnitude 
and direction and to combine according to the parallelogram law; and many 
of them are not at all reminiscent of journeys. 

This should not surprise us. The process of abstraction is a powerful one. 
It is, indeed, a basic tool of the mathematician. Take whole numbers, for 
instance. Like vectors, they are abstractions. We could say that whole numbers 
are what is left of the idea of apples when we ignore not only the apple trees, 
the wind and the rain, the profits of cider makers, and other such items that 
would appear in an encyclopedia article, but also ignore even the apples them- 
selves, and concentrate solely on how many there are. After we have extracted 
from the idea of apples the idea of whole numbers, we find that whole numbers 
apply to all sorts of situations that have nothing to do with apples. Much the 
same is true of vectors. They are more complicated than whole numbers — so 
are fractions, for example — but they happen to embody an important type of 
mathematical behavior that is widely encountered in the world around us. 

To give a single example here: forces behave like vectors. This is not 
something obvious. A force has both magnitude and direction, of course. But 
this does not mean that forces necessarily combine according to the parallelo- 
gram law. That they do combine in this way is inferred from exp>eriment. 

It is worthwhile to explain what is meant when we say that forces combine 
according to the parallelogram law. Forces are not something visible, though 
their effects may be visible. They are certainly not arrow-headed line segments, 
though after one has worked with them mathematically for a while, one almost 



66 



Introducing Vectors 



comes to think they are. A force can be represented by an arrow-headed line 
segment OP that starts at the point of application O of the force, points in the 
direction of the force, and has a length proportional to the magnitude of the 
force — for example, a length of x inches might represent a magnitude of x 
pounds. When a force is represented in this way, we usually avoid wordiness 
by talking of "the force OP." But let us be more meticulous in our wording 
just here. To verify experimentally that forces combine according to the paral- 
lelogram law, we can make the following experiment. We arrange stationary 
weights and strings, and pulleys A and B, as shown, the weight W being the 




Wz£^ 



Figure 5.1 



sum of the weights W^ and W^. Then along OA we mark off a length OP of W^ 
inches, where W^ is the number of pounds in the weight on the left and, thus, 
a measure of the force with which the string attached to it pulls on the point 
O where the three pieces of string meet. Similarly, we mark off on OB a length 
OQ of W2 inches. We then bring a vertical piece of paper up to the point O, 
and on it complete the parallelogram defined by OP and OQ. We find that 
the diagonal OR is vertical and that its length in inches is W, the number of 
pounds in the weight in the middle. We conclude that the resultant of the 
forces W^ and W^ in the strings would just balance the weight W. Since the 
forces W^ and W2 also just balance the weight W, we say that the resultant is 
equivalent to the two forces. We then do the experiment over again, with 
different weights, and reach a similar conclusion. After that, we do it yet 
again; and we keep at it till our lack of patience overcomes our skepticism, 
upon which we say that we have proved experimentally that forces combine 
according to the parallelogram law. And we bolster our assertion by pointing 
to other experiments, of the same and different types, that indicate the same 
thing. 

We all know that it is much easier to get through a revolving door by 
pushing near the outer edge than by pushing near the central axis. The effect 
of a force depends on its location. Home runs are scarce when the bat fails to 
make contact with the ball. Thus forces do not behave like free vectors. 
Unlike displacements, vectors representing forces such as AB and LM in Figure 
4.1, though they have the same magnitude and the same direction, are not 
counted as equivalent. Such vectors are called bound vectors. 

Perhaps it worries us a little that there are different kinds of vectors. Yet 
we have all, in our time, survived similar complications. Take numbers, for 
example. There are whole numbers and there are fractions. Perhaps you feel 
that there is not much difference between the two. Yet if we listed the prop- 
erties of whole numbers and the properties of fractions we would find con- 
siderable differences. For instance, if we divide fractions by fractions the results 
are always fractions, but this statement does not remain true if we replace the 
word "fractions" by "whole numbers." Worse, every whole number has a 



67 



next higher one, but no fraction has a next higher fraction, for between any 
two fractions we can always slip infinitely many others. Even so, when trying 
to define number we might be inclined to insist that, given any two different 
numbers, one of them will always be the smaller and the other the larger. Yet 
when we become more sophisticated and expand our horizons to include 
complex numbers like 2 + 3V— 1, we have to give up even this property of 
being greater or smaller, which at first seemed an absolutely essential part of 
the idea of number. With vectors too, not only are there various tyf)es, but 
we shall learn that not every one of their attributes that seems at this stage to 
be essential is in fact so. One of the things that gives mathematics its power 
is the shedding of attributes that turn out not to be essential, for this, after 
all, is just the process of abstraction. 

Exercise 5.1 Find the resultants of the following displacements: 

(a) 3 ft. due east and 3 ft. due north. [Ans. 3yfY ft. to the northeast.] 
jn (b) 5 ft. due north and 5 ft. due east. 

(c) 9 cm. to the right and 9V^cm. vertically upwards. [Ans. 18 cm. in 
an upward direction making 60° with the horizontal towards the 
right.] 

(d) 9 cm. to the left and 9a/T cm. vertically downward. 

(e) the resultants in parts (c) and (d). 

(f ) X units positively along the x-axis and y units positively along the y- 
axis. [Ans, Vx" + y'^ units in the direction making an angle 
ian' y/x with the positive x-axis.] 

Exercise 5.2 Like Exercise 5.1 for the following: 

(a) 8 km. to the left and 3 km. to the left. 

(b) 5 fathoms vertically downward and 2 fathoms vertically upward. 

(c) a units to the right and /9 units to the left. [There are three different 
cases. What are they? Show how they can be summed up in one 
statement.] 

(d) h miles 60° north of east and h miles 60^ south of east. 

Exercise 5.3 What single force is equivalent to the following three 
horizontal forces acting on a particle at a point O? (1) magnitude 1 lb. 
pulling to the north; (2) magnitude 1 lb. pulling to the east; (3) magnitude 
V 2 lb. pulling to the northwest. [Ans. 2 lbs. acting at point O and 
pulling to the north.] 

Exercise 5.4 What force combined with a force at a point of 1 lb. 
pulling to the east will yield a resultant force of 2 lbs. pulling in a direc- 
tion 60° north of east? 

Exercise 5.5 Vector OP has magnitude 2a and points to the right 
in a direction 30° above the horizontal. What vector combined with it 
will yield a vertical resultant, OR, of magnitude 2v^a? 

Exercise 5.6 Find two forces at a point O, one vertical and one 
horizontal, that have a resultant of magnitude h, making 45° with the 
horizontal force. [Ans. The forces have magnitude h/\/^.] 

Exercise 5.7 Find two forces at a point O, one vertical and one 
horizontal, that have a resultant of magnitude h that makes an angle of 
30° with the horizontal force. 



68 



Introducing Vectors 



Exercise 5.8 Find two displacements, one parallel to the x-axis and 
the other to the ^'-axis, that yield a resultant displacement of magnitude 
h ft. making a positive acute angle a with the positive x-direction. 

Exercise 5.9 What is the resultant of n vectors, each starting at the 
point O, each having magnitude h, and each pointing to the pole star? 
[We could have shortened this by asking for the resultant of n equal 
vectors. But we have not yet defined "equal" vectors — even though we 
have already spoken of the equality of free vectors! You may find it 
instructive to try to do so here; but be warned that it is not as easy as it 
seems, and that there is something lacking in the wording of the ques- 
tion.] 

Exercise 5.10 A particle is acted on by two forces, one of them to 
the west and of magnitude 1 dyne, and the other in the direction 60° 
north of east and of magnitude 2 dynes. What third force acting on the 
particle would keep it in equilibrium (i. e., what third force would make 
the resultant of all three forces have zero magnitude)? [Ans. Magnitude 
V~3 dynes pointing due south.] 

6. THE SINGULAR INCIDENT OF THE VECTORIAL TRIBE 

It is rumored that there was once a tribe of Indians who believed that 
arrows are vectors. To shoot a deer due northeast, they did not aim an arrow 
in the northeasterly direction; they sent two arrows simultaneously, one due 
north and the other due east, relying on the powerful resultant of the two 
arrows to kill the deer. 

Skeptical scientists have doubted the truth of this rumor, pointing out that 
not the slightest trace of the tribe has ever been found. But the complete 
disappearance of the tribe through starvation is precisely what one would 
expect under the circumstances; and since the theory that the tribe existed 
confirms two such diverse things as the Nonvectorial Behavior of Arrows 
and the Darvv'inian Principle of Natural Selection, it is surely not a 
theory to be dismissed lightly. 

Exercise 6. 1 Arrow-headed line segments have magnitude and direc- 
tion and are used to represent vectors. Why are they nevertheless not 
vectors? 

Exercise 6.2 Given the three vectors represented by OP, OQ, and 

OR in Figure 2.1, form three new entities having the same respective 
directions, but having magnitudes equal to five times the magnitudes of 
the respective vectors. Prove geometrically that these new entities are so 
related that the third is a diagonal of the parallelogram having the other 
two as adjacent sides. 

Exercise 6.3 If in Exercise 6.2 the new entities had the same 
respective directions as the vectors represented by OP, OQ, and OR, but 
had magnitudes that were one unit greater than the magnitudes of the 
corresponding vectors, show that the new entities would not be such that 
the third was a diagonal of the parallelogram having the other two as 
adjacent sides. 



69 



Exercise 6.4 Suppose we represented vectors by arrow-headed line 
segments that had the same starting points and directions as the vectors, 
but had lengths proportional to the squares of the magnitudes of the 
vectors, so that, for example, if a force of 1 lb. were represented by a seg- 
ment of length 1 inch, then a force of 2 lbs. would be represented by one of 
4 inches. Show that, in general, these representations of vectors would not 
obey the parallelogram law. Note that the statement of the parallelogram 
law in Section 2 therefore needs amending, and amend it accordingly. [If 
you think carefully, you will realize that this is a topsy-turvy question 
since, in proving the required result, you will assume that the vectors, 
when "properly" represented, obey the parallelogram law; and thus, in a 
sense, you will assume the very amendment you are seeking. But since 
you have probably been assuming the amendment all this while, you will 
be able to think your way through. The purpose of this exercise is to 
draw your attention to this rarely mentioned, usually assumed amend- 
ment.] 

7. SOME AWKWARD QUESTIONS 

When are two vectors equal? The answer depends on what we choose to 
mean by the word "equal" — we are the masters, not the word. But we do 
not want to use the word in an outrageous sense: for example, we would not 
want to say that two vectors are equal if they are mentioned in the same 
sentence. 

Choosing a meaning for the word "equal" here is not as easy as one might 
imagine. For example, we could reasonably say that two vectors having the 
same magnitudes, identical directions, and a common starting point are equal 
vectors. And if one of the vectors were somehow pink and the other green, 
we would probably be inclined to ignore the colors and say that the vectors 
were still equal. But what if one of the vectors represented a force and the 
other a displacement? There would then be two difficulties. 

The first difficulty is that the vector representing a displacement would be 
a free vector, but the one representing the force would not. If, in Figure 4.1, 
we counted free vectors represented by AB and LM as equal, we might find 
ourselves implying that forces represented by AB and LM were also equal, 
though actually they have different effects. [Even so, it is extremely convenient 
to say such things as "a force acts at A and an equal force acts at L." We shall 
not do so in this book. But one can get by with saying such things once one 
has explained what is awkward about them, just as, in trigonometry one gets 
by with writing sin^ 6 after one has explained that this does not stand for 
sin(sin B) but (sin Of.] 

As for the second difficulty about the idea of the equality of vectors, it 
takes us back to the definition of a vector. For if, in Figure 2.1, OP represents 
a force and OQ a displacement, the two vectors will not combine by the paral- 
lelogram law at all. We know this from experiments with forces. But we can 
appreciate the awkwardness of the situation by merely asking ourselves what 
the resultant would be if they did combine in this way. A "disforcement"?* 
[Compare Exercise 5.9.] 

♦Actually, of course, lack of a name proves no more than that if the resultant exists, 
it has not hitherto been deemed important enough to warrant a name. 



70 



Introducing Vectors 



If two vectors are to be called equal, it seems reasonable to require that 
they be able to combine with each other. The situation is not the same as it is 
with numbers. Although 3 apples and 3 colors are different things, we can say 
that the numbers 3 are equal in the sense that, if we assign a pebble to each 
of the apples, these pebbles will exactly suffice for doing the same with the 
colors. And in this sense we can indeed combine 3 apples and 3 colors — not 
to yield 6 apples, or 6 colors, or 6 colored apples [it would surely be only 3 
colored apples], but 6 items. There does not seem to be a corresponding sense 
in which we could reasonably combine a vector representing a force with one 
representing a displacement, quite apart from the question of bound versus 
free vectors: there does not seem to be a vectorial analogue of the numerical 
concept of a countable item.* 

Though OP and O^ do not combine according to the parallelogram law 
if, for example, OP represents a force and OQ a. displacement, they never- 
theless represent vectors. Evidently our definition of a vector needs even 
further amendment. We might seek to avoid trouble by retreating to the 
definition of a vector as "an entity having both magnitude and direction," 
without mentioning the parallelogram law. But once we start retreating, where 
do we stop? Why not be content to define a vector as "an entity having 
direction," or as "an entity having magnitude," or, with Olympian simplicity, 
as just "an entity"? Alternatively, we could make the important distinction 
between the abstract mathematical concept of a vector and entities, such as 
forces, that behave like these abstract vectors and are called vector quantities. 
This helps, but it does not solve the present problem so much as sweep it 
under the rug. We might amend our definition of a vector by saying that 
vectors combine according to the parallelogram law only with vectors of the 
same kind : forces with forces, displacements with displacements, accelerations 
(which are vectors) with accelerations, and so on. But even that is tricky since, 
for example, in dynamics we learn that force equals mass times acceleration. 
So we would have to allow for the fact that though a force does not combine 
with an acceleration, it does combine with a vector of the type mass-times- 
acceleration in dynamics. 

We shall return to this matter. (See Section 8 of Chapter 2.) But enough of 
such questions here. If we continue to fuss with the definition we shall never 
get started. Even if we succeeded in patching up the definition to meet this 
particular emegency, other emergencies would arise later. The best thing to do 
is to keep an open mind and learn to live with a flexible situation, and even 
to relish it as something akin to the true habitat of the best research. 



*Even with numbers there are complications. For example, 3 ft. and 3 inches can be 
said to yield 6 items ; yet in another sense they yield 39 inches, 3^ ft., and so on — and 
each of these can also be regarded as a number of items, though the 3^ involves a further 
subtlety. Consider also 3 ft. and 3 lbs., and then 2.38477 ft. and 2.38477 lbs. 



71 



Galileo uses a thought experiment in discussing projec- 
tile motion, a typical device of the scientist to this day. 
Galileo's book was originally published in 1632. 



11 Galileo's Discussion of Projectile Motion 



Gerald Helton and Duane H. D. Roller 

An excerpt from their book Foundations of Modern Physical Science, 1958. 

3.1 Galileo's discussion of projectile motion. To this point we have 
been solely concerned with the motion of objects as characterized by their 
speed; we have not given much consideration to the direction of motion, or 
to changes in direction of motion. Turning now to the more general prob- 
lem of projectile motion, we leave the relatively simple case of bodies 
moving in a straight line only and expand our methods to deal with pro- 
jectiles moving along curved paths. Our understanding of this field will 
hinge largely on a far-reaching idea: the observed motion of a projectile 
may be thought of as the result of two separate motions, combined and 
occurring simultaneously; one component of motion is in a horizontal 
direction and without acceleration, whereas the other is in a vertical direc- 
tion and has a constant acceleration downward in accordance with the 
laws of free fall. Furthermore, these two components do not interfere with 
each other; each component may be studied as if the other were not present. 
Thus the whole motion of the projectile at every moment is simply the 
result of the two individual actions. 

This principle of the independency of the horizontal and vertical com- 
ponents of projectile motion was set forth by Galileo in his Dialogue on the 
great world systems (1632). Although in this work he was principally con- 
cerned with astronomy, Galileo already knew that terrestrial mechanics 
offered the clue to a better understanding of planetary motions. Like the 
Two new sciences, this earlier work is cast in the form of a discussion among 
the same three characters, and also uses the Socratic method of the Platonic 
dialogues. Indeed, the portion of interest to us here begins with Salviati 
reiterating one of Socrates' most famous phrases, as he tells the AristoteUan 
Simplicio that he, Simplicio, knows far more about mechanics than he is 
aware:* 

Salviati: . . . Yet I am so good a midwife of minds that I will make you con- 
fess the same whether you will or no. But Sagredus stands very quiet, and yet, 
if I mistake not, I saw him make some move as if to speak. 

Sagredo: I had intended to speak a fleeting something; but my curiosity 



*These extracts from Galileo's Dialogue on the great world systems, as well as 
those appearing in later chapters, are taken from the translation of T. Salusbury, 
edited and corrected by Giorgio de Santillana (University of Chicago Press, 
72 1953). 



Galileo's Discussion of Projectile Motion 



aroused by your promising that you would force Simplicius to uncover the 
knowledge which he conceals from us has made me depose all other thoughts. 
Therefore I pray you to make good your vaunt. 

Salviati: Provided that Simplicius consents to reply to what I shall ask him, 
I will not fail to do it. 

Simplicio: I will answer what I know, assured that I shall not be much put 
to it, for, of those things which I hold to be false, I think nothing can be 
known, since Science concerns truths, not falsehoods. 

Salviati: I do not desire that you should say that you know anything, save 
that which you most assuredly know. Therefore, tell me; if you had here a 
flat surface as polished as a mirror and of a substance as hard as steel that 
was not horizontal but somewhat inclining, and you put upon it a perfectly 
spherical ball, say, of bronze, what do you think it would do when released? 
Do you not believe (as for my part I do) that it would lie still? 

Simplicio: If the surface were inclining? 

Salviati: Yes, as I have already stated. 

Simplicio: I cannot conceive how it should lie still. I am confident that it 
would move towards the declivity with much propenseness. 

Salviati: Take good heed what you say, Simplicius, for I am confident that 
it would lie still in whatever place you should lay it. 

Simplicio: So long as you make use of such suppositions, Salviatus, I shall 
cease to wonder if you conclude most absurd conclusions. 

Salviati: Are you assured, then, that it would freely move towards the 
declivity? 

Simplicio: Who doubts it? 

Salviati: And this you verily believe, not because I told you so (for I 
endeavored to persuade you to think the contrary), but of yourself, and upon 
your natural judgment? 

Simplicio: Now I see your game; you did not say this really believing it, but 
to try me, and to wrest words out of my mouth with which to condemn me. 

Salviati: You are right. And how long and with what velocity would that 
ball move? But take notice that I gave as the example a ball exactly round, 
and a plane exquisitely polished, so that all external and accidental impedi- 
ments might be taken away. Also I would have you remove all obstructions 
caused by the air's resistance and any other causal obstacles, if any other 
there can be. 

Simplicio: I understand your meaning very well and answer that the ball 
would continue to move in infinitum if the inclination of the plane should last 
so long, accelerating continually. Such is the nature of ponderous bodies that 
they acquire strength in going, and, the greater the declivity, the greater 
the velocity will be. 



Simplicio is next led to express his belief that if he observed the ball 
rolling up the inclined plane he would know that it had been pushed or 
thrown, since it is moving contrary to its natural tendencies. Then Sal- 
viati turns to the intermediate case: 



Salviati: It seems, then, that hitherto you have well explained to me the 
accidents of a body on two different planes. Now tell me, what would befall 
the same body upon a surface that had neither acclivity nor declivity? 

Simplicio: Here you must give me a little time to consider my answer. There 



73 



being no declivity, there can be no natural inclination to motion; and there 
being no acclivity, there can be no resistance to being moved. There would 
then arise an indifference between propulsion and resistance; therefore, I think 
it ought naturally stand still. But I had forgoi myself; it was not long ago 
that Sagredus gave me to understand that it would do so. 

Salviati: So I think, provided one did lay it down gently; but, if it had an 
impetus directing it towards any part, what would follow? 

Simplicio: That it should move towards that part. 

Salviati: But with what kind of motion? Continually accelerated, as in 
declining planes; or successively retarded, as in those ascending? 

Simplicio: I cannot tell how to discover any cause of acceleration or re- 
tardation, there being no declivity or acclivity. 

Salviati: Well, if there be no cause of retardation, even less should there be 
any cause of rest. How long therefore would you have the body move? 

Simplicio: As long as that surface, neither inclined nor declined, shall last. 

Salviati: Therefore if such a space were interminate, the motion upon it 
would likewise have no termination, that is, would be perpetual. 

Simplicio: I think so, if the body is of a durable matter. 

Salviati: That has been already supposed when it was said that all external 
and accidental impediments were removed, and the brittleness of the body in 
this case is one of those accidental impediments. Tell me now, what do you 
think is the cause that that same ball moves spontaneously upon the inclining 
plane, and does not, except with violence, upon the plane sloping upwards? 

Simplicio: Because the tendency of heavy bodies is to move towards the 
center of the Earth and only by violence upwards towards the circumference. 
[This is the kernel of the Scholastic viewpoint on falling bodies (see Section 
2.3). Salviati does not refute it, but turns it to Galileo's purposes.] 

Salviati: Therefore a surface which should be neither declining nor ascending 
ought in all its parts to be equally distant from the center. But is there any 
such surface in the world? 

Simplicio: There is no want of it, such is our terrestrial globe, for example, 
if it were not rough and mountainous. But you have that of the water, at 
such time as it is calm and still. 



Here is the genesis of one of the fundamental principles of the new 
mechanics: if all "accidental" interferences with an object's motion are 
removed, the motion will endure. The "accidents" are eliminated in this 
thought experiment by: (1) proposing the use of a perfectly round, per- 
fectly hard ball on a perfectly smooth surface, and (2) by imagining the 
surface to be a globe whose surface is everywhere equidistant from the 
center of the earth, so that the ball's "natural tendency" to go downward is 
balanced by the upward thrust of the surface. (We shall return to this 
latter point in our discussion of isolated systems in Chapter 16.) Note 
carefully the drastic change from the Scholastic view: instead of asking 
"What makes the ball move?" Galileo asks "What might change its 
motion?" 

Having turned the conversation to smooth water, Galileo brings in the 
motion of a stone dropping from the mast of a moving ship. Since the 
stone is moving horizontally with the ship before it is dropped, it should 
continue to move horizontally while it falls. 



74 



Galileo's Discussion of Projectile Motion 




Sagredo: If it be true that the 
impetus with which the ship moves 
remains indeUbly impressed in the 
stone after it is let fall from the 
mast; and if it be further true that 
this motion brings no impediment 
or retardment to the motion directly 
downwards natural to the stone, 
then there ought to ensue an effect 
of a very wonderful nature. Suppose 
a ship stands still, and the time of 
the falling of a stone from the mast's 
round top to the deck is two beats 
of the pulse. Then afterwards have 
the ship under sail and let the same 
stone depart from the same place. 
According to what has been prem- 
ised, it shall still take up the time of 
two pulses in its fall, in which time 
the ship will have gone, say, twenty 
yards. The true motion of the stone 
then will be a transverse line [i.e., a 
curved line in the vertical plane, see 
Fig. 3.1], considerably longer than 
the first straight and perpendicular 
line, the height of the mast, and 
yet nevertheless the stone will have 
passed it in the same time. Increase 
the ship's velocity as much as you 
will, the falling stone shall describe 
its transverse lines still longer and 
longer and yet shall pass them all in 
those selfsame two pulses. In this 
same fashion, if a cannon were lev- 
eled on the top of a tower, and fired point-blank, that is, horizontally, and 
whether the charge were small or large with the ball falling sometimes a 
thousand yards distant, sometimes four thousand, sometimes ten, etc., all 
these shots shall come to ground in times equal to each other. And every 
one equal to the time that the ball would take to pass from the mouth of the 
piece to the ground, if, without other impulse, it falls simply downwards in 
a perpendicular line. Now it seems a very admirable thing that, in the 
same short time of its falling perpendicularly down to the ground from the 
height of, say, a hundred yards, equal balls, fired violently out of the piece, 
should be able to pass four hundred, a thousand, even ten thousand yards. 
All the balls in all the shots made horizontally remain in the air an equal 
time [Fig. 3.2]. 

Salviati: The consideration is very elegant for its novelty and, if the effect 
be true, very admirable. Of its truth I make no question, and, were it not for 
the accidental impediment of the air, I verily believe that, if at the time of the 
ball's going out of the piece another were let fall from the same height directly 
downwards, they would both come to the ground at the same instant, though 
one should have traveled ten thousand yaids in its range, and another only a 
hundred, presupposing the surface of the Earth to be level. As for the impedi- 



FiG. 3.1. A stone dropped from the 
mast of a ship in uniform motion. From 
the shore the trajectory of the stone is 
seen to be a curved line (parabola). 



75 














Fig. 3.2. For cannon balls fired horizontally with different initial forward 
speeds, "all the balls in all the shots made horizontally remain in the air an 
equal time." 

ment which might come from the air, it would consist in retarding the extreme 
swift motion of the shot. 



76 



This chapter from a beginning college physics text is not 
simple, but the reward of this numerical approach to 
Newtonian mechanics is a more powerful understanding 
of how the laws of motion work. 



12 Newton's Laws of Dynamics 

Richard P. Feynman, Robert B. Leighton and Matthew Sands 

A chapter from their textbook The Feynman Lectures on Physics, Volume 1, 1963. 



9-1 Momentum and force 

The discovery of the laws of dynamics, or the laws of motion, was a dramatic 
moment in the history of science. Before Newton's time, the motions of things 
like the planets were a mystery, but after Newton there was complete under- 
standing. Even the slight deviations from Kepler's laws, due to the perturbations 
of the planets, were computable. The motions of pendulums, oscillators with 
springs and weights in them, and so on, could all be analyzed completely after 
Newton's laws were enunciated. So it is with this chapter: before this chapter we 
could not calculate how a mass on a spring would move; much less could we 
calculate the perturbations on the planet Uranus due to Jupiter and Saturn. After 
this chapter we will be able to compute not only the motion of the oscillating mass, 
but also the perturbations on the planet Uranus produced by Jupiter and Saturn! 

Galileo made a great advance in the understanding of motion when he 
discovered the principle of inertia: if an object is left alone, is not disturbed, it 
continues to move with a constant velocity in a straight line if it was originally 
moving, or it continues to stand still if it was just standing still. Of course this 
never appears to be the case in nature, for if we slide a block across a table it stops, 
but that is because it is not left to itself — it is rubbing against the table. It required 
a certain imagination to find the right rule, and that imagination was supplied 
by Galileo. 

Of course, the next thing which is needed is a rule for finding how an object 
changes its speed if something is affecting it. That is the contribution of Newton. 
Newton wrote down three laws: The First Law was a mere restatement of the 
Galilean principle of inertia just described. The Second Law gave a specific way 
of determining how the velocity changes under difl'erent influences called forces. 
The Third Law describes the forces to some extent, and we shall discuss that at 



77 



another time. Here we shall discuss only the Second Law, which asserts that the 
motion of an object is changed by forces in this way: the time-rate-of-change of a 
quantity called momentum is proportional to the force. We shall state this mathe- 
matically shortly, but let us first explain the idea. 

Momentum is not the same as velocity. A lot of words are used in physics, 
and they all have precise meanings in physics, although they may not have such 
precise meanings in everyday language. Momentum is an example, and we must 
define it precisely. If we exert a certain push with our arms on an object that is 
light, it moves easily; if we push just as hard on another object that is much heavier 
in the usual sense, then it moves much less rapidly. Actually, we must change the 
words from "light" and "heavy" to less massive and more massive, because there 
is a diff'erence to be understood between the weight of an object and its inertia. 
(How hard it is to get it going is one thing, and how much it weighs is something 
else.) Weight and inertia are proportional, and on the earth's surface are often 
taken to be numerically equal, which causes a certain confusion to the student. 
On Mars, weights would be different but the amount offeree needed to overcome 
inertia would be the same. 

We use the term mass as a quantitative measure of inertia, and we may 
measure mass, for example, by swinging an object in a circle at a certain speed and 
measuring how much force we need to keep it in the circle. In this way we find a 
certain quantity of mass for every object. Now the momentum of an object is a 
product of two parts: its mass and its velocity. Thus Newton's Second Law may 
be written mathematically this way: 

F=~{mv). (9.1) 

Now there are several points to be considered. In writing down any law such as 
this, we use many intuitive ideas, implications, and assumptions which are at 
first combined approximately into our "law." Later we may have to come back 
and study in greater detail exactly what each term means, but if we try to do this 
too soon we shall get confused. Thus at the beginning we take several things for 
granted. First, that the mass of an object is constant; it isn't really, but we shall 
start out with the Newtonian approximation that mass is constant, the same all 
the time, and that, further, when we put two objects together, their masses add. 
These ideas were of course implied by Newton when he wrote his equation, for 
otherwise it is meaningless. For example, suppose the mass varied inversely as the 
velocity; then the momentum would never change in any circumstance, so the law 
means nothing unless you know how the mass changes with velocity. At first 
we say, // does not change. 

Then there are some implications concerning force. As a rough approximation 
we think of force as a kind of push or pull that we make with our muscles, but 
we can define it more accurately now that we have this law of motion. The most 
important thing to realize is that this relationship involves not only changes in 
the magnitude of the momentum or of the velocity but also in their direction. 



78 



Newton's Laws of Dynamics 



If the mass is constant, then Eq. (9.1) can also be written as 

F = m -J = ma. (9.2) 

The acceleration a is the rate of change of the velocity, and Newton's Second 
Law says more than that the effect of a given force varies inversely as the mass; 
it says also that the direction of the change in the velocity and the direction of the 
force are the same. Thus we must understand that a change in a velocity, or an 
acceleration, has a wider meaning than in common language: The velocity of a 
moving object can change by its speeding up, slowing down (when it slows down, 
we say it accelerates with a negative acceleration), or changing its direction of 
motion. An acceleration at right angles to the velocity was discussed in Chapter 7. 
There we saw that an object moving in a circle of radius R with a certain speed v 
along the circle falls away from a straightline path by a distance equal to ^(v^/R)t^ 
if / is very small. Thus the formula for acceleration at right angles to the motion is 

a = v^R, (9.3) 

and a force at right angles to the velocity will cause an object to move in a curved 
path whose radius of curvature can be found by dividing the force by the mass to 
get the acceleration, and then using (9.3). 



iZ 









Fig. 9-1. A small displacement of an object. 



9-2 Speed and velocity 



In order to make our language more precise, we shall make one further 
definition in our use of the words speed and velocity. Ordinarily we think of speed 
and velocity as being the same, and in ordinary language they are the same. But in 
physics we have taken advantage of the fact that there are two words and have 
chosen to use them to distinguish two ideas. We carefully distinguish velocity, 
which has both magnitude and direction, from speed, which we choose to mean 
the magnitude of the velocity, but which does not include the direction. We can 
formulate this more precisely by describing how the x-, y-, and z-coordinates of 
an object change with time. Suppose, for example, that at a certain instant an 
object is moving as shown in Fig. 9-1. In a given small interval of time At it 



79 



will move a certain distance Ax in the A:-direction, Ay in the >'-direction, and Az in 
the z-direction. The total effect of these three coordinate changes is a displacement 
As along the diagonal of a parallelepiped whose sides are Ax, Ay, and Az. In terms 
of the velocity, the displacement Ax is the x-component of the velocity times At, 
and similarly for Ay and Az: 



Ax 



VxAt, 



Ay = Vy At, Az = Vz At. 



(9.4) 



9-3 Components of velocity, acceleration, and force 

In Eq. (9.4) we have resolved the velocity into components by telling how fast the 
object is moving in the x-direction, the >'-direction, and the z-direction. The 
velocity is completely specified, both as to magnitude and direction, if we give the 
numerical values of its three rectangular components: 



Vj = dx/dt, Vy = dy/dt, v^ = dz/dt. 



(9.5) 



On the other hand, the speed of the object is 



ds/dt = \v\ 



= v^ 



' + 1-; + vi 



(9.6) 



Next, suppose that, because of the action of a force, the velocity changes to 
some other direction and a different magnitude, as shown in Fig. 9-2. We can 
analyze this apparently complex situation rather simply if we evaluate the changes 
in the x-, y-, and z-components of velocity. The change in the component of the 
velocity in the A-direction in a time At is Ar^ = Oj. At, where Uj- is what we call the 
.v-component of the acceleration. Similarly, we see that Avy = Oy At and Av^ = 
Qz At. In these terms, we see that Newton's Second Law, in saying that the force 
is in the same direction as the acceleration, is really three laws, in the sense that 
the component of the force in the x-. r-, or z-direction is equal to the mass times 



/ 




Fig. 9-2. A change in velocity in 
which both the magnitude and direction 
change. 



80 



Newton's Laws of Dynamics 

the rate of change of the corresponding component of velocity: 

F^ = m{dvjdt) = m{d'^x/dt-) = ma^, 

Fy = m{dvy/dt) = m(d^y/dt^.) = may, (9.7) 

F, = m{dvjdt) = m{d'h/dt^) = ma^. 

Just as the velocity and acceleration have been resolved into components by 
projecting a line segment representing the quantity and its direction onto three 
coordinate axes, so, in the same way, a force in a given direction is represented 
by certain components in the x-, y-, and z-directions: 

Fj, = F cos {x, F), 

Fy = F COS (y,F), (9.8) 

Fz = F cos (z, F). 

where F is the magnitude of the force and {x, F) represents the angle between the 
jc-axis and the direction of F, etc. 

Newton's Second Law is given in complete form in Eq. (9.7). If we know the 
forces on an object and resolve them into x-, y-, and z-components, then we can 
find the motion of the object from these equations. Let us consider a simple 
example. Suppose there are no forces in the y- and z-directions, the only force 
being in the x-direction, say vertically. Equation (9.7) tells us that there would be 
changes in the velocity in the vertical direction, but no changes in the horizontal 
direction. This was demonstrated with a special apparatus in Chapter 7 (see 
Fig. 7-3). A falling body moves horizontally without any change in horizontal 
motion, while it moves vertically the same way as it would move if the horizontal 
motion were zero. In other words, motions in the ;c-, y-, and z-directions are 
independent if Xht forces are not connected. 

9^ What is the force? 

In order to use Newton's laws, we have to have some formula for the force; 
these laws %2iy pay attention to the forces. If an object is accelerating, some agency 
is at work; find it. Our program for the future of dynamics must be Xo find the 
laws for the force. Newton himself went on to give some examples. In the case 
of gravity he gave a specific formula for the force. In the case of other forces he 
gave some part of the information in his Third Law, which we will study in the 
next chapter, having to do with the equality of action and reaction. 

Extending our previous example, what are the forces on objects near the 
earth's surface? Near the earth's surface, the force in the vertical direction due 
to gravity is proportional to the mass of the object and is nearly independent of 
height for heights small compared with the earth's radius i?: F = GmM/R"^ = mg, 
where g = GM/R^ is called the acceleration of gravity. Thus the law of gravity 
tells us that weight is proportional to mass; the force is in the vertical direction 
and is the mass times g. Again we find that the motion in the horizontal direction 



81 



v/////\ 



-EQUILIBRIUM 
X POSITION 

,-i Fig. 9-3. A mass on a spring. 

is at constant velocity. The interesting motion is in the vertical direction, and 
Newton's Second Law tells us 

mg = m{d'^xldt\ (9.9) 

Cancelling the w's, we find that the acceleration in the ;c-direction is constant and 
equal to g. This is of course the well known law of free fall under gravity, which 
leads to the equations 

Vx = ^0 + ^t, 

X = xo + 1^0/ + \gt'^- (9.10) 

As another example, let us suppose that we have been able to build a gadget 
(Fig. 9-3) which applies a force proportional to the distance and directed oppositely 
— a spring. If we forget about gravity, which is of course balanced out by the 
initial stretch of the spring, and talk only about excess forces, we see that if we 
pull the mass down, the spring pulls up, while if we push it up the spring pulls 
down. This machine has been designed carefully so that the force is greater, the 
more we pull it up, in exact proportion to the displacement from the balanced 
condition, and the force upward is similarly proportional to how far we pull down. 
If we watch the dynamics of this machine, we see a rather beautiful motion — up, 
down, up, down, . . . The question is, will Newton's equations correctly describe 
this motion? Let us see whether we can exactly calculate how it moves with this 
periodic oscillation, by applying Newton's law (9.7). In the present instance, 
the equation is 

-kx = m{dvjdt). (9.11) 

Here we have a situation where the velocity in the x-direction changes at a rate 
proportional to x. Nothing will be gained by retaining numerous constants, so 
we shall imagine either that the scale of time has changed or that there is an 
accident in the units, so that we happen to have kim = 1. Thus we shall try to 
solve the equation 

dvjdt = -X. (9.12) 

To proceed, we must know what Vj, is, but of course we know that the velocity is 
the rate of change of the position. 

9-5 Meaning of the dynamical equations 

Now let us try to analyze just what Eq. (9.12) means. Suppose that at a 
given time / the object has a certain velocity r^ and position x. What is the velocity 



82 



Newton's Laws of Dynamics 



and what is the position at a sUghtly later time / + 6? If we can answer this 
question our problem is solved, for then we can start with the given condition and 
compute how it changes for the first instant, the next instant, the next instant, and 
so on, and in this way we gradually evolve the motion. To be specific, let us suppose 
that at the time / = we are given that x = 1 and Vx = 0. Why does the object 
move at all? Because there is a force on it when it is at any position except x = 0. 
If x > 0, that force is upward. Therefore the velocity which is zero starts to 
change, because of the law of motion. Once it starts to build up some velocity 
the object starts to move up, and so on. Now at any time /, if e is very small, 
we may express the position at time / + e in terms of the position at time / and 
the velocity at time / to a very good approximation as 

x(t + e) = x(t) + €v,(t). (9.13) 

The smaller the e, the more accurate this expression is, but it is still usefully accurate 
even if e is not vanishingly small. Now what about the velocity? In order to get 
the velocity later, the velocity at the time / + €, we need to know how the velocity 
changes, the acceleration. And how are we going to find the acceleration? That 
is where the law of dynamics comes in. The law of dynamics tells us what the 
acceleration is. It says the acceleration is —x. 

v,(t + e) = v,0) + eax(t) (9.14) 

= vAO - €x(t). (9.15) 

Equation (9.14) is merely kinematics; it says that a velocity changes because of 
the presence of acceleration. But Eq. (9.15) is dynamics, because it relates the 
acceleration to the force; it says that at this particular time for this particular 
problem, you can replace the acceleration by —x(t). Therefore, if we know both 
the X and y at a given time, we know the acceleration, which tells us the new 
velocity, and we know the new position — this is how the machinery works. The 
velocity changes a little bit because of the force, and the position changes a little 
bit because of the velocity. 



9-6 Numerical solution of the equations 

Now let us really solve the problem. Suppose that we take e = 0.100 sec. 
After we do all the work if we find that this is not small enough we may have to 
go back and do it again with e = 0.010 sec. Starting with our initial value x(0) = 
1.00, what is a:(O.I)? It is the old position x(0) plus the velocity (which is zero) 
times 0.10 sec. Thus x(0.\) is still 1.00 because it has not yet started to move. 
But the new velocity at 0.10 sec will be the old velocity i'(O) = plus e times the 
acceleration. The acceleration is —x(0) = —1.00. Thus 

/•(O.l) = 0.00 - 0.10 X 1. 00 = -0.10. 



83 



Now at 0.20 sec 

x(0.2) = x(O.l) + €^0.1) 

= 1.00 - 0.10 X 0.10 = 0.99 
and 

r(0.2) = KO.l) + 6a(0.1) 

= -0.10 - 0.10 X 1.00 = -0.20. 

And so, on and on and on, we can calculate the rest of the motion, and that is 
just what we shall do. However, for practical purposes there are some little tricks 
by which we can increase the accuracy. If we continued this calculation as we have 
started it, we would find the motion only rather crudely because e = 0.100 sec 
is rather crude, and we would have to go to a very small interval, say e = 0.01. 
Then to go through a reasonable total time interval would take a lot of cycles of 
computation. So we shall organize the work in a way that will increase the pre- 
cision of our calculations, using the same coarse interval e = 0.10 sec. This can 
be done if we make a subtle improvement in the technique of the analysis. 

Notice that the new position is the old position plus the time interval e times 
the velocity. But the velocity when? The velocity at the beginning of the time 
interval is one velocity and the velocity at the end of the time interval is another 
velocity. Our improvement is to use the velocity halfway between. If we know 
the speed now, but the speed is changing, then we are not going to get the right 
answer by going at the same speed as now. We should use some speed between 
the "now" speed and the "then" speed at the end of the interval. The same 
considerations also apply to the velocity: to compute the velocity changes, we 
should use the acceleration midway between the two times at which the velocity 
is to be found. Thus the equations that we shall actually use will be something 
like this: the position later is equal to the position before plus e times the velocity 
at the time in the middle of the interval. Similarly, the velocity at this halfway point 
is the velocity at a time e before (which is in the middle of the previous interval) 
plus e times the acceleration at the time /. That is, we use the equations 

x{t + €) = x{t) + ev{i + €/2), 
v{t + €/2) = lit - 6/2) + ea{t\ (9.16) 

a{t) = -x{t). 

There remains only one slight problem: what is t'(e/2)? At the start, we are given 
t'(0), not i'(— e/2). To get our calculation' started, we shall use a special equation, 
namely, v{e/2) = r(0) + (€/2)a(0). 

Now we are ready to carry through our calculation. For convenience, we 
may arrange the work in the form of a table, with columns for the time, the position, 
the velocity, and the acceleration, and the in-between lines for the velocity, as 
shown in Table 9-1 . Such a table is, of course, just a convenient way of representing 
the numerical values obtained from the set of equations (9.16), and in fact the 
equations themselves need never be written. We just fill in the various spaces in 



84 



Newton's Laws of Dynamics 



Table 9-1 

Solution of c^yi/J/ = —x 
Interval: e = 0.10 sec 



/ 


X 


Vx 


Ox 


0.0 


1.000 


0.000 
-0.050 


-1.000 


0.1 


0.995 


-0.150 


-0.995 


0.2 


0.980 


-0.248 


-0.980 


0.3 


0.955 


-0.343 


-0.955 


0.4 


0.921 


-0.435 


-0.921 


0.5 


0.877 


A CT5 


-0.877 






0.523 — 




0.6 


0.825 


-0.605 


-0.825 


0.7 


0.764 


-0.682 


-0.764 


0.8 


0.696 


-0.751 


-0.696 


0.9 


0.621 


-0.814 


-0.621 


1.0 


0.540 


f\ O/'O 


-0.540 


1.1 


0.453 


U.ODO 

-0.913 


-0.453 


1.2 


0.362 


-0.949 


-0.362 


1.3 


0.267 


-0.976 


-0.267 


1.4 


0.169 


-0.993 


-0.169 


1.5 


0.070 


1.000- 


-0.070 








1.6 


-0.030 




+0.030 



the table one by one. This table now gives us a very good idea of the motion : 
it starts from rest, first picks up a little upward (negative) velocity and it loses 
some of its distance. The acceleration is then a little bit less but it is still gaining 
speed. But as it goes on it gains speed more and more slowly, until as it passes 
a: = at about t = 1.50 sec we can confidently predict that it will keep going, 
but now it will be on the other side; the position x will become negative, the ac- 



85 




0.5- 







r. 




PLANE r(x,y) 








^ 


y^ 

' ' 




^^ 


F 




'i 




SUN 








( 



Fig. 9-4. Graph of the motion of a 
moss on a spring. 



Fig. 9-5. The force of gravity on a 
planet. 



celeration therefore positive. Thus the speed decreases. It is interesting to compare 
these numbers with the function x = cos /, which is done in Fig. 9-4. The agree- 
ment is within the three significant figure accuracy of our calculation! We shall 
see later that x = cos / is the exact mathematical solution of our equation of 
motion, but it is an impressive illustration of the power of numerical analysis that 
such an easy calculation should give such precise results. 



9-7 Planetary motions 

The above analysis is very nice for the motion of an oscillating spring, but 
can we analyze the motion of a planet around the sun? Let us see whether we 
can arrive at an approximation to an ellipse for the orbit. We shall suppose that 
the sun is infinitely heavy, in the sense that we shall not include its motion. Suppose 
a planet starts at a certain place and is moving with a certain velocity; it goes 
around the sun in some curve, and we shall try to analyze, by Newton's laws of 
motion and his law of gravitation, what the curve is. How? At a given moment 
it is at some position in space. If the radial distance from the sun to this position 
is called r, then we know that there is a force directed inward which, according to 
the law of gravity, is equal to a constant times the product of the sun's mass and 
the planet's mass divided by the square of the distance. To analyze this further 
we must find out what acceleration will be produced by this force. We shall need 
the components of the acceleration along two directions, which we call x and y. 
Thus if we specify the position of the planet at a given moment by giving x and y 
(we shall suppose that z is always zero because there is no force in the z-direction 
and, if there is no initial velocity v^, there will be nothing to make z other than 
zero), the force is directed along the line joining the planet to the sun, as shown 
in Fig. 9-5. 

From this figure we see that the horizontal component of the force is related 
to the complete force in the same manner as the horizontal distance x is to the 
complete hypotenuse r, because the two triangles are similar. Also, i( x is positive. 



86 



Newton's Laws of Dynamics 



F^ is negative. That is, Fj./\F\ = —x/r,orF^ = —\F\x/r = —GMmx/r\ Now 
we use the dynamical law to find that this force component is equal to the mass of 
the planet times the rate of change of its velocity in the x-direction. Thus we find 
the following laws: 

m{dvjdt) = -GMmx/r\ 

m{dvy/dt) = -GMmy/r\ (9.17) 

r = Vx^ + y'. 

This, then, is the set of equations we must solve. Again, in order to simplify the 
numerical work, we shall suppose that the unit of time, or the mass of the sun, has 
been so adjusted (or luck is with us) that GM = I. For our specific example we 
shall suppose that the initial position of the planet is at x = 0.500 and y = 0.000, 
and that the velocity is all in the j^-direction at the start, and is of magnitude 
1.6300. Now how do we make the calculation? We again make a table with 
columns for the time, the x-position, the x-velocity r^, and the x-acceleration a^; 
then, separated by a double line, three columns for position, velocity, and accelera- 
tion in the >^-direction. In order to get the accelerations we are going to need 
Eq. (9.17); it tells us that the acceleration in the x-direction is —x/r'\ and the 
acceleration in the >^-direction is —y/r'\ and that r is the square root of jc^ + y^. 
Thus, given x and y, we must do a little calculating on the side, taking the square 
root of the sum of the squares to find r and then, to get ready to calculate the two 
accelerations, it is useful also to evaluate l/r'\ This work can be done rather 
easily by using a table of squares, cubes, and reciprocals: then we need only 
multiply X by l/r^, which we do on a slide rule. 







y 








• = '°^. . . . . 


:/' 


«0.5 




t = l.5— y^ . 




•0.5 






• 










• 










1 = 20-1,. 

1 1 J 1 1 


1 1 1 1 1 1 1 ^ 


. 1 1 


1 1 


• t=o 


-10 


-05 


SUN 




as X 



,=0 Fig- 9-6. The calculated motion of a 

planet around the sun. 



Our calculation thus proceeds by the following steps, using time intervals 
€ = 0.100: Initial values at / = 0: 



jc(0) = 0.500 
vAO) = 0.000 



>;(0) = 0.000 
Vy(0) = +1.630 



From these we find: 



/-(O) = 0.500 lA'^(O) = 8.000 
a, = -4.000 a„ = 0.000 



87 



Thus we may calculate the velocities y;c(0.05) and Vy(0.05): 

r^(0.05) = 0.000 - 4.000 X 0.050 = -0.200; 
i'j,(0.05) = 1.630 + 0.000 X 0.100 = 1.630. 

Now our main calculations begin: 

a:(0.1) = 0.500 - 0.20 X 0.1 = 0.480 

y(0.l) = 0.0 + 1.63 X 0.1 = Q.163 

r = VOASO^ + 0.163^ = 0.507 

1/r'' = 7.67 

aAO.l) = 0.480 X 7.67 = -3.68 

^^(0.1) = -0.163 X 7.70 = -1.256 

t';,(0.15) = -0.200 - 3.68 X 0.1 = -0.568 

/',X0.15) = 1.630 - 1.26 X 0.1 = 1.505 

a:(0.2) - 0.480 - 0.568 X 0.1 = 0.423 

y{0.2) = 0.163 + 1.50 X 0.1 = 0.313 
etc. 

In this way we obtain the values given in Table 9-2, and in 20 steps or so we have 
chased the planet halfway around the sun! In Fig. 9-6 are plotted the x- and 
>'-coordinates given in Table 9-2. The dots represent the positions at the succession 
of times a tenth of a unit apart; we see that at the start the planet moves rapidly 
and at the end it moves slowly, and so the shape of the curve is determined. Thus 
we see that we really do know how to calculate the motion of planets! 

Now let us see how we can calculate the motion of Neptune, Jupiter, Uranus, 
or any other planet. If we have a great many planets, and let the sun move too, 
can we do the same thing? Of course we can. We calculate the force on a particular 
planet, let us say planet number /, which has a position x„ yi, Zi (i = 1 may repre- 
sent the sun,/ = 2 Mercury, / = 3 Venus, and so on). We must know the positions 
of all the planets. The force acting on one is due to all the other bodies which 
are located, let us say, at positions Xj, yj, Zj. Therefore the equations are 

dvix v^ Gnji/nXxi — Xj) 
nii — ;— = > — — ' 

,„, ^ = t- ^'"•'"'^l' - ■^'> ■ (9.18) 

dvj^ _ v^ _ Gmjmjizi - z,) 

'"' "dt ~ ^ ;:3^ * 



88 



Newton's Laws of Dynamics 



Table 9-2 

Solution of (/vxM = -x/r^,dvy/dt = -y/r^. 
Interval: < = 0.100 
Orbit Vy =1.63 v, = x = 0.5 y = 



r = Vx' + y' 
at 1=0 



1 


X 


fx 


o. 


y 


"v 


ay 


r 


l/r' 


0.0 


0.500 


-0.200 


-4.00 


0.000 


1.630 


0.00 


0.500 


8.000 


0.1 


0.480 


-0.568 


-3.68 


0.163 


1.505 


-1.25 


0.507 


7.675 


0.2 


0.423 


-0.859 


-2.91 


0.313 


1.290 


-2.15 


0.526 


6.873 


0.3 


0.337 


-1.055 


-1.96 


0.442 


1.033 


-2.57 


0.556 


5.824 


0.4 


0.232 


-1.166 


-1.11 


0.545 


0.771 


-2.62 


0.592 


4.81 


0.5 


0.115 


-1.211- 


-0.453 


0.622 


- 0.526- 


-2 AS 


0.633 


3.942 


0.6 


-0.006 


4-0.020 


0.675 


-2.20 


0.675 


3.252 






-1.209 






0.306 








0.7 


-0.127 


-1.175 


+0.344 


0.706 


0.115 


-1.91 


0.717 


2.712 


0.8 


-0.245 


-1.119 


+0.562 


0.718 


-0.049 


-1.64 


0.758 


2.2% 


0.9 


-0.357 


-1.048 


+0.705 


0.713 


-0.190 


-1.41 


0.797 


1.975 


1.0 


-0.462 


-0.968- 


+0.796 


0.694 


-0.310- 


-1.20 


0.834 


1.723 


1.1 


-0.559 


+0.858 


0.663 


-1.02 


0.867 


1.535 






-0.882 






-0.412 








1.2 


-0.647 


-0.792 


+0.90 


0.622 


-0.499 


-0.86 


0.897 


1.385 


1.3 


-0.726 


-0.700 


+0.92 


0.572 


-0.570 


-0.72 


0.924 


1.267 


1.4 


-0.796 


-0.607 


+0.93 


0.515 


-0.630 


-0.60 


0.948 


1.173 


1.5 


-0.857 


- -0.513- 


+0.94 


0.452 


- -0.680- 


-0.50 


0.969 


1.099 


1.6 


-0.908 


+0.95 


0.384 


-0.40 


0.986 


1.043 






-0.418 






-0.720 








1.7 


-0.950 


-0.323 


+0.95 


0.312 


-0.751 


-0.31 


1.000 


1.000 


1.8 


-0.982 


-0.228 


+0.95 


0.237 


-0.773 


-0.23 


1.010 


0.970 


1.9 


-1.005 


-0.113 


+0.95 


0.160 


-0.778 


-0.15 


1.018 


0.948 


2.0 


-1.018 


- -0.037- 


+0.96 


0.081 


- -0.796- 


-0.08 


1.021 


0.939 


2.1 


-1.022 


+0.95 


0.001 


0.00 


1.022 


0.936 






+0.058 






-0.796 








2.2 


-1.016 




+0.96 


-0.079 


-0.789 


+0.07 


1.019 


0.945 


2.3 



















Crossed x-axis at 2.101 sec, . period = 4.20 sec. 
Vx = at 2.086 sec. 

1.022 + 0.500 
Cross X at 1.022, . . semimajor axis = 

f„ = 0.796. 



= 0.761. 



Predicted time 7r(0.761)^' 2 = ir(0.663) = 2.082. 



89 



Further, we define rij as the distance between the two planets / andy; this is equal to 



r,, = V{^- x,y^ + {y, - y,y + (z, - z,)^. (9.19) 

Also, X! means a sum over all values of y — all other bodies — except, of course, 
fory = /. Thus all we have to do is to make more columns, lots more columns. 
We need nine columns for the motions of Jupiter, nine for the motions of Saturn, 
and so on. Then when we have all initial positions and velocities we can calculate 
all the accelerations from Eq. (9.18) by first calculating all the distances, using 
Eq. (9.19). Hov, long will it take to do it? If you do it at home, it will take a 
very long time! But in modern times we have machines which do arithmetic very 
rapidly; a very good computing machine may take 1 microsecond, that is, a 
millionth of a second, to do an addition. To do a multiplication takes longer, 
say 10 microseconds. It may be that in one cycle of calculation, depending on 
the problem, we may have 30 multiplications, or something like that, so one cycle 
will take 300 microseconds. That means that we can do 3000 cycles of computation 
per second. In order to get an accuracy, of, say, one part in a billion, we would 
need 4 X 10' cycles to correspond to one revolution of a planet around the sun. 
That corresponds to a computation time of 130 seconds or about two minutes. 
Thus it take only two minutes to follow Jupiter around the sun, with all the 
perturbations of all the planets correct to one part in a billion, by this method! 
(It turns out that the error varies about as the square of the interval e. If we make 
the interval a thousand times smaller, it is a million times more accurate. So, let 
us make the interval 10,000 times smaller.) 

So, as we said, we began this chapter not knowing how to calculate even the 
motion of a mass on a spring. Now, armed with the tremendous power of Newton's 
laws, we can not only calculate such simple motions but also, given only a machine 
to handle the arithmetic, even the tremendously complex motions of the planets, 
to as high a degree of precision as we wish! 



90 



An experimental study of a complex m.otion, that of a 
golf club, is outlined. If you do not have a slow-motion 
movie camera, similar measurements can be made using 
the stroboscopic picture. 



13 The Dynamics of a Golf Club 



C. L Stong 



An article from Scientific American, 1964. 




91 



With the aid of a slow-motion 
movie camera and a co-opera- 
tive friend any golf player can 
easily explore the dynamics of his club 
head during the split second of the drive 
that separates the sheep from the goats 
of golfdom. The procedure, as applied 
by Louis A. Graham, a consulting en- 
gineer in Naples, Fla., analyzes the travel 
of the club head throughout the swing, 
including its velocity and acceleration 
at the critical moment of impact— factors 
that determine whether a squarely struck 
ball will merely topple off the tee or go 
a history-making 445 yards to match the 
performance of E. C. Bliss in August, 
1913. 

"The procedure is essentially simple," 
writes Graham, "but the reliability of 
the results will reflect the care with 
which certain measurements are made. 
I pick a sunny day for the experiment 
and, having arrived at the golf course 
with my co-operative friend and acces- 
sories, tee my ball. Then I place a tee 
marker precisely four feet in front of 
the ball and another four feet behind it 
to make a line that points toward the 
first green. My friend stations the tripod- 
mounted camera for a medium close-up 
shot on a line that intersects the ball at 
right angles to the tee markers. I address 



the bail, facing the camera. My friend 
photographs the complete drive from 
address to follow-through at the rate of 
48 frames per second. The known dis- 
tance between the tee markers and their 
position in relation to the club head 
scales the pictures with respect to dis- 
tance. The exposure rate— the number of 
frames per second— of the camera pro- 
vides the time dimension. (If the expo- 
sure rate is not known accurately, it can 
be calibrated by photographing a phono- 
graph turntable marked with a chalk line 
and turning at 45 or 78 revolutions per 
minute. ) 

"The film is developed and analyzed. 
One can use either a film-editing device 
that projects an enlarged image of each 
frame or a set of enlarged prints of each 
frame, mounted serially and numbered 
for identification. 

"The next step is to plot the position 
of the club head during the course of the 
swing. Since a point in a plane is deter- 
mined by its distance from two other 
known points, the position of the club 
head can be plotted in relation to that of 
the two tee markers [see illustration be- 
low]. First, I draw a base line near the 
bottom of a sheet of graph paper ruled 
with rectangular co-ordinates and on it 
locate three equally spaced points: the 



tee marker P, the ball (O) and the 
tee marker Q. 1 usually space these 
points four inches apart, thus establish- 
ing a scale of 12 inches of club head 
travel per inch of graph paper. 

"The location of the club head (C) 
with respect to that of the tee markers 
can be transferred to the graph by one 
of three methods. Proportional dividers 
are handy for transferring the scaled 
distance from P to C and from C to Q. 
Alternatively, the angles CPQ and CQP 
can be measured with a protractor and 
reconstructed on the graph, point C be- 
ing located at the intersection of lines 
projected from P and Q. If no protractor 
is at hand, the vertical and horizontal 
distances between C, P and Q can be 
measured with a square and ruler and 
similarly transferred to the graph. 

"Plot enough points to establish a rea- 
sonably smooth track, skipping several 
frames during slow portions of the 
swing. The resulting graph is of course 
not extremely accurate. The plane in 
which the club head swings, for example, 
is inclined to the plane of the film. The 
track plotted from the image therefore 
diff^ers slightly from the true excursion 
of the club head, but the error is not 
large and can be ignored. By the same 
token, the travel of the club head from 




Graph of successive club head positions 



92 



The Dynamics of a Golf Club 





50 52 5d 

Selected frames from slow-motion film of a golf swing 



point to point is subsequently measured 
along stiaiglit lines, whereas the club 
head actually follows a cursed path. 
Error introduced by this source can be 
minimized by speeding up the camera. 
My camera, an inc.xpensi\e one, is limit- 
ed to a maximum speed of 48 frames per 
second, a rate that records the event 
adefjuately for the objectives of this ex- 
periment. 

"The total distance traveled by the 
club head and its velocity and accelera- 
tion are derived from a second set of 
graphs prepared from the grapli of club 
head position. On a second sheet of 
graph paper ruled with rectangular co- 
ordinates di\ide the abscissa into a series 
of uniform increments equal to the total 
number of frames occupied by the su ing 
and note the corresponding time inter- 
vals in seconds as well as the frame num- 
bers. The ordinate will carry two scales: 
club head travel in feet and club head 
speed in miles per hour. The scales of the 
ordinate should provide for a total club 
head travel of 36 feet and a maximum 
velocity of about 80 miles per hour. 
Graphs of convenient proportion result 
when the length of the ordinate repre- 
senting 36 feet equals the length repre- 
senting one second on the abscissa. The 
maximum velocity of 80 miles per hour 
need not occupy more than half of the 
ordinate scale, as shown in the accom- 
panying graph [tipper illustration on 

page 94]. 

"Data for plotting club head travel 
against time are derived by measuring 
the graph of club head position. Make 
a table of three columns, for frame num- 
ber, time and distance. Beginning with 
the point on the graph of club head 
travel that shows the head addressing the 
ball, scale the distance to the next point 
and convert to equivalent feet by refer- 
ring the measurement to the base line 
that includes P, O and Q. Measure and 
tabulate the remaining position points 
in the same way. When the table is com- 
plete, add the distance increments pro- 
gressively, plot distance against time and 
draw a smooth curve through the points. 

"The speed of the club head at any 
point is found from this graph by the 
familiar graphical method of slopes. To 
find the speed of the club head at about 
tiie point of impact (frame No. 43), 
draw a tangent LKM of arbitrary length 
through K. The sides MN and LN are 
found by referring to the scale to equal 
11.2 feet and .11 second respectively. 
The speed of the club head at this instant 
is equal to the ratio 11.2/. 11, or 102 
feet per second. The result can be 
expressed in miles per hour by multi- 



93 



plying it by the number of seconds 
per hour and dividing the product by 
the number of feet per mile: 102 X 
3,600/5,280 = 70 miles per hour. Re- 
pe.it the procedure for each of the 
frames, tabulate the results, plot speed 
versus time and draw a smooth cur\e 
through the points. 

"Club head acceleration can be 
graphed in the same way or merely com- 
puted from the graph of club head speed 
at frames of particular interest, such as 
the frame showing the moment of im- 
pact. For example, to determine the ac- 
celeration of the club head depicted by 
frame No. 38, draw a tangent to the 
graph at this point. Then, at some arbi- 
trary point above, say at the point cor- 
responding to a velocity of .56 miles per 
hour, drop a perpendicular MN from the 
tangent. At anothei arbitrary point be- 
low, say at the point corresponding to a 
velocity of 12 miles per hour, draw a 
line LN parallel to the abscissa and in- 
tersecting both the tangent and MS. 
Inspection of the abscissa discloses that 
the length LN is analogous to a time 
interval of .1 second. Acceleration is 
defined as the rate of change of velocity 
and is equal to the difference between 
the final velocity and initial velocity 
divided by the time interval between 
the two. In this example the velocity 
difference is 56 miles per hour minus 
12 miles per hour, or, expressed in feet 
per second: (.56 - 12) X 5,280/3,600 
= 64 feet per second. The acceleration 
is 64/. 1 = 640 feet per second per sec- 
ond. The acceleration of gravity (g) 
amounts to 32 feet per second per sec- 
ond. The acceleration of the club head at 
frame No. 38 in terms of g is accord- 
ingly 640/32, or 20 g! 

"Having performed this rainy-after- 
noon portion of the procedure, what 
reward awaits the dufler? For one thing, 
he can see at a glance why his drives do 
not match those of a professional golfer. 
The graphs discussed so far show the 
performance of golf professional Dick 
Bull using an iron. His swing from ad- 
dress to follow-through required 1.17 
seconds. The club head traveled 31 feet. 
His backswing occupied .6 second. He 
paused at the top about .1 second. More 
interesting than these figures, in my 
opinion, are those of the club head speed 
and acceleration Bull achieved: the in- 
crease in club head speed during the 
.1 second before impact from 15 miles 
per hour to an amazing 70 miles per 
hour, representing an acceleration of 
slightly over 20 g. Graphs of Bull's per- 
formance with a driver, although differ- 
ent in many respects from tJiose of his 



irons, show exactly the same figure foi 
speed, 70 miles per hour, and an accel- 
eration of 22 g, a remarkably uniform 
performance. Similar analysis of the per- 
formance of a fairly good amateur using 
a driver shows precisely half the veloc- 
ity of Bull's club, 35 miles per hour, and 
an acceleration at impact of only seven g 
[see lower illuntration helow]. 

"Although these methods of analyzing 



motion are routine in engineering cir- 
cles, I am not familiar with their prior 
application to the game of golf. As with 
many procedures, they are easier to ap- 
ply than to describe. I find them interest- 
ing because they clearly reveal why Bull 
and other professionals achieve their 
long drives. Duffers with movie cameras 
may well begin asking each other, 
'How's your v and g?' " 



.15 .50 .75 

time in seconds 



J.25 




24 30 36 

frames 

Speed and acceleriition graph for u prolessioniil's swing 



.25 .50 .75 

time in seconds 



J.25 




Similar graph for an amateur's performance 



94 



Athletic events involve measurements of distance and 
time, and so bring In the same error considerations that 
one also meets in the laboratory. 



14 Bad Physics in Athletic Measurements 



P. Klrkpatrick 



An article from The American Journal of Physics, 1944. 



THE physics teacher has been accustomed to 
find in athletic activities excellent problems 
involving velocities, accelerations, projectiles and 
impacts. He has at the same time overlooked a 
rich source of illustrations of fictitious precision 
and bad metrology. When the student is told 
that the height of a tree should not be expressed 
as 144.632 ft if the length of its shadow has been 
measured only to the nearest foot, the student 
may see the point at once and yet ask, "What 
difference does it make?" But when shown that 
common procedures in measuring the achieve- 
ments of a discus thrower could easily award a 
world's record to the wrong man, the student 
agrees that good technic in measurement is 
something more than an academic ideal. The 
present discussion^ has been prepared partly to 
give the physics teacher something to talk about, 
but also to start a chain of publicity which may 
ultimately make athletic administrators better 
physicists and so make their awards more just. 

If physicists were given charge of the measure- 
ments of sport, one may feel sure that they 
would frown upon the practice of announcing the 



' Some of the material in this article appeared in a pap)er 
by the author in Scientific American, April 1937, and is 
incorporated here by permission of the editors. 



speed of a racing automobile in six or seven 
digits — see, for example, the World Almanac for 
any year — when neither the length of the course 
nor the elapsed time is known one-tenth so 
precisely. They could and would point out such 
inconsistencies as that observed in some of the 
events of the 1932 Olympic games when races 
were electrically and photographically timed to 
0.01 sec, but with the starting gun fired from 
such a position that its report could not reach 
the ears of the waiting runners until perhaps 
0.03 to 0.04 sec after the official start of the race. 
In this case, electric timing was used only as an 
unofficial or semi-official supplement to 0.1-sec 
hand timing; but it is easy to see that a sys- 
tematic error of a few hundredths of a second will 
frequently cause stopwatch timers to catch the 
wrong tenth. 

Scientific counsel on the field would immedi- 
ately advise judges of the high jump and pole 
vault to measure heights from the point of take- 
off instead of from an irrelevant point directly 
below the bar which should be at the same level 
but sometimes isn't. Physicists would suggest 
equipping field judges with surveying instru- 
ments for determining after each throw, not only 
how far the weight traveled but also the relative 



95 



elevation of the landing point and the throwing 
circle. Certainly it is meaningless if not deceptive 
to record weight throws to a small fraction of an 
inch when surface irregularities may be falsifying 
by inches the true merit of the performance. 

In shot-putting, for example, a measured 
length will be in error by practically the same 
amount as the discrepancy between initial and 
final elevations, since the flight of the shot at its 
terminus is inclined at about 45° to the hori- 
zontal. For the discus the effect is some three 
times as serious because of the flatter trajectory 
employed with this missile, while broad jumpers 
under usual conditions must be prepared to give 
or take as much as 0.5 ft, according to the luck 
of the pit. Meanwhile, the achievements in these 
events go down in the books with the last eighth 
or even the last sixteenth of an inch recorded. 

At the 1932 Olympic Games an effective device 
was used to grade the broad-jumping pit to the 
level of the take-off board before each leap, but 
the practice has not become general. Athletic 
regulations, indeed, recognize the desirability of 
proper leveling in nearly all the field events, but 
in actual usage not enough is done about it. 
Since sprinters are not credited with records 
achieved when blown along before the wind, 
there is no obvious reason why weight hurlers 
should be permitted to throw things down hill. 

The rule books make no specification as to the 
hardness of the surface upon which weights shall 
be thrown, but this property has a significant 
effect upon the measured ranges of the shot and 
hammer, since it is prescribed that measurement 
shall be made to the near side of the impression 
produced by the landing weight. In a soft surface 
this impression may be enlarged in the backward 
direction enough to diminish the throw by several 
times the ostensible precision of the measurement. 

A physicist would never check the identity of 
three or four iron balls as to mass by the aid of 
grocers' scales or the equivalent and then pretend 
that there was any significance in the fact that 
one of them was thrown a quarter of an inch 
farther than the others. In measuring the length 
of a javelin throw, no physicist who wanted to 
be right to | in. would be content to establish his 
perpendicular from the point of fall to the 
scratchline by a process of guesswork, but this 



is the way it is always done by field judges, even 
in the best competition. 

Among the numerous errors afflicting measure- 
ments in the field sports, there is none which is 
more systematically committed, or which could 
be more easily rectified, than that pertaining to 
the variation of the force of gravity. The range 
of a projectile dispatched at any particular angle 
of elevation and with a given initial speed is a 
simple function of g. Only in case the end of the 
trajectory is at the same level as its beginning 
does this function become an inverse proportion- 
ality; but in any case the relationship is readily 
expressed, and no physicist will doubt that a 
given heave of the shot will yield a longer put in 
equatorial latitudes than it would in zones where 
the gravitational force is stronger. Before saying 
that the 55-ft put achieved by A in Mexico City 
is a better performance than one of 54 ft, 11 in. 
which B accomplished in Boston, we should 
surely inquire about the values of g which the 
respective athletes were up against, but it is 
never done. As a matter of record, the value of g 
in Boston exceeds that in Mexico City by j per- 
cent, so the shorter put was really the better. 
To ignore the handicap of a larger value of g is 
like measuring the throw with a stretched tape. 
The latter practice would never be countenanced 
under AAU or Olympic regulations, but the 
former is standard procedure. 

Rendering justice to an athlete who has had to 
compete against a high value of g Involves ques- 
tions that are not simple. It will be agreed that 
he is entitled to some compensation and that in 
comparing two throws made under conditions 
similar except as to g, the proper procedure would 
be to compare not the actual ranges achieved, 
but the ranges which would have been achieved 
had some "standard" value of g — say 980 cm/ 
sec^ — prevailed in both cases. The calculation of 
exactly what would have happened is probably 
impossible to physics. Although it is a simple 
matter to discuss the behavior of the implement 
after it leaves the thrower's hand and to state 
how this behavior depends upon g, the depend- 
ence of the initial velocity of projection upon g 
depends upon the thrower's form and upon char- 
acteristics of body mechanics to which but little 
attention has so far been devoted. 



96 



Bad Physics in Athletic Measurements 



The work done by the thrower bestows upon 
the projectile both potential and kinetic energy. 
In a strong gravitational field, the imparted 
potential energy is large and one must therefore 
suppose the kinetic energy to be reduced, since 
the thrower's propelling energy must be dis- 
tributed to both. We have no proof, however, 
that the total useful work is constant despite 
variation of g, nor do we know the manner of its 
inconstancy, if any. The muscular catapult is not 
a spring, subject to Hooke's law, but a far more 
complicated system with many unknown charac- 
teristics. The maximum external work which one 
may do in a single energetic shove by arms, legs 
or both obviously depends partly upon the re- 
sisting force encountered. Only a little outside 
work can be done in putting a ping-pong ball 
because the maximum possible acceleration, 
limited by the masses and other characteristics 
of the bodily mechanism itself, is too slight to 
call out substantial inertial forces in so small a 
mass. The resisting force encountered when a 
massive body is pushed in a direction that has an 
upward component, as in shot-putting, does of 
course depend upon g; and until we know from 
experiment how external work in such an effort 
varies with resisting force, we shall not be able 
to treat the interior ballistics of the shot-putter 
with anything approaching rigor. 

Several alternative assumptions may be con- 
sidered. If we suppose that the velocity of de- 
livery, or "muzzle velocity," v, of the missile is 
unaffected by variations of g, we have only the 
external effect to deal with. Adopting the ap- 
proximate range formula R = v'^/g (which neg- 
lects the fact that the two ends of the trajectory 
are at different levels and which assumes the 
optimum angle of elevation) we find that the 
increment of range dR resulting from an in- 
crement dg is simply —Rdg/g. On the more 
plausible assumption that the total work done on 
the projectile is independent of g, this total to 
include both the potential and kinetic energies 
imparted, one obtains as a correction formula, 



dR 



2h\ dg 



/ ^n\ ag 



(1) 



where h is the. vertical lift which the projectile 
gets while in the hand of the thrower. A third 



assumption, perhaps the most credible of all, 
would hold constant and independent of g the 
total work done upon the projectile and upon a 
portion of the mass of the thrower's person. It is 
not necessary to decide how much of the thrower's 
mass goes into this latter term; it drops out and 
we have again Eq. (1), provided only that the 
work done on the thrower's body can be taken 
into account by an addition to the mass of the 
projectile. 

These considerations show that a variation of g 
affects the range in the same sense before and 
after delivery, an increase in g reducing the 
delivery velocity and also pulling the projectile 
down more forcibly after its flight begins. They 
indicate also that the latter effect is the more 
important since, in Eq. (1), l>2h/R by a factor 
of perhaps five in the shot-put and more in the 
other weight-throwing events. 

One concludes that the least which should be 
done to make amends to a competitor striving 
against a large value of g is to give him credit 
for the range which his projectile would have 
attained, for the same initial velocity, at a 
location where g is "standard." This is not quite 
justice, but it is a major step in the right direc- 
tion. The competitor who has been favored by a 
small value of g should of course have his achieve- 
ment treated in the same way. 

The corrections so calculated will not be 
negligible magnitudes, as Fig. 1 shows. They are 
extremely small percentages of the real ranges, 
but definitely exceed the ostensible probable 
errors of measurement. It is not customary to 
state probable errors explicitly in connection 
with athletic measurements, but when a throw 
is recorded as 57 ft, 1^ in., one naturally con- 
cludes that the last thirty-second inch, if not 
completely reliable, must have been regarded as 
having some significance. 

ROTATION OF THE EARTH 

It is customary to take account of the effects 
of terrestrial rotation when aiming long-range 
guns, but athletes and administrators of sport 
have given little or no attention to such effects 
in relation to their projectiles. As a matter of 
fact they should, for at low latitudes the range of 
a discus or shot thrown in an eastward direction 



97 



977 




Fig. 1 . Graphs for normalizing 
shot-put ranges to the common 
value g = 980 cm/sec'. Ranges 
achieved where g = 980 cm/sec* 
are not in need of adjustment, 
but a range of 50 ft (see inclined 
line marked 50') achieved at 
Glasgow, whereg = 981 .6cm/sec?, 
is entitled to a premium of 1 J in. 
which should be added before 
comparing the put with one 
achieved elsewhere. Distances 
accomplished where g<980 
cm/sec' should be subjected to 
the deductions indicated by 
graphs in the third quadrant. 



exceeds that of a westward throw by more than 
the ostensible precision of such measurements. 
The difference between the range of a projectile 
thrown from the surface of the real earth and 
the range of one thrown from a nonrotating earth 
possessing the same local value of g is given by^ 

IV sin 2a 4co Fo« 
Range = 1 

Xsin a[4 cos' a— 1] cos X sin fi, (2) 

where g is the ordinary acceleration due to 
weight, Vo is the initial speed of the projectile, 
a is the angle of elevation of initial motion 
(measured upward from the horizontal in the 
direction of projection), co(rad/sec) is the angular 
speed of rotation of the earth, X is the geographic 
latitude of the point of departure of the pro- 
jectile, and n is the azimuth of the plane of the 
trajectory, measured clockwise from the north 
point. 

A derivation of this equation (though not the 
first) is given in reference 2, along with a dis- 
cussion of its application to real cases. The 
approximations accepted in the derivation are 
such as might possibly be criticized where long- 



' P. Kirkpatrick, Am. J. Phys. 11, 303 (1943). 



range guns are considered, but they introduce no 
measurable errors into the treatment of athletic 
projectiles. 

The first term of the right-hand member of 
Eq. (2) is the ordinary elementary range ex- 
pression, and naturally it expresses almost the 
whole of the actual range. The second term is a 
small correction which is of positive sign for 
eastbound projectiles (0</x<180°) and negative 
for westbound. The correction term, being pro- 
portional to Fo', increases with Vo at a greater 
rate than does the range as a whole. Hence the 
percentage increase or decrease of range, because 
of earth rotation, varies in proportion to \'\ or to 
the square root of the range itself. Evidently this 
effect is a maximum at the equator and zero at 
the poles. Inspection of the role of a shows that 
the correction term is a maximum for a 30" 
angle of elevation and that it vanishes when the 
angle of elevation is 60°. 

By the appropriate numerical substitiTtions in 
Eq. (2), one may show that a well-thrown discus 
in tropic latitudes will go an inch farther east- 
ward than westward. This is many times the 
apparent precision of measurement for this event, 
and records have changed hands on slimmer 
margins. Significant effects of the same kind, 
though of lesser magnitude, appear in the cases 



98 



Bad Physics in Athletic Measurements 



of the javelin, hammer, shot and even the broad 
jump, where the east-west differential exceeds the 
commonly recorded sixteenth of an inch. 

Figures 1 and 2 are types of correction charts 
that might be used to normalize the performances 
of weight throwers to a uniform value of g and 
a common direction of projection. Figure 1 has 
been prepared with the shot-put in mind, but is 
not restricted to implements of any particular 
mass. The inclined straight lines of this figure are 
graphs of —dR versus dg from Eq. (1). Values of 
the parameter R are indicated on the graphs. 
The uniform value 100 cm has been adopted for 
h, an arbitrary procedure but a harmless one in 
view of the insensitivity of dR to h. 

Figure 2, particularly applicable to the hammer 
throw, furnishes means for equalizing the effect 
of earth spin upon athletes competing with the 
same implement but directing their throws vari- 
ously as may be necessitated by the lay-out of 
their respective fields. An angle of elevation of 
45** has been assumed in the construction of these 
curves, a somewhat restrictive procedure which 
finds justification in the fact that no hammer 
thrown at an angle significantly different from 
45° is likely to achieve a range worth correcting. 
These curves are plotted from Eq. (2); their 



application to particular cases is described in the 
figure legend. 

Upon noticing that some of these corrections 
are quite small fractions of an inch, the reader 
may ask whether the trouble is worth while. 
This is a question that is in great need of 
clarification and one that may not be answered 
with positiveness until the concept of the prob- 
able error of a measurement shall have become 
established among the metrologists of sport. 
Physicists will agree that to every measurement 
worth conserving for the attention of Record 
Committees should be attached a statement of 
its probable error; without such a statement 
there will always be the danger of proclaiming a 
new record on the basis of a new performance 
that is apparently, though not really, better than 
the old. If the corrections of Fig. 2 exceed the 
probable error to be claimed for a measurement, 
then those corrections must be applied. 

The aim of the American Athletic Union in 
these matters is hard to determine. Watches 
must be "examined," "regulated" and "tested" 
by a reputable jeweler or watchmaker, but one 
finds no definition of what constitutes an accept- 
able job of regulation. Distances must be meas- 
ured with "a steel tape." The Inspector of 









^ 




— 




*i>V^ 










(80' /x 




\ 


_ 




l«0' / /.X ^ 




^ 


— 




I40^__///0^ 




\ 


N 


E 


^f 


w 


\ 




1 1 1 


J 


T 1 


I ■■n 




^=^ 


/ 






^_ 


- 









(/) 



Fig. 2. Curves for rendering throws in various directions comparable. The 
assumed latitude is 30°, either north or south, and the assumed angle of elevation 
is 45°. Since the range has a maximum for about this angle of elevation, the 
curves also apply well to angles several degrees on either side. The curves show, 
for example (circled point), that a missile thrown 200 ft in a direction 30° south 
of east should have ^ in. subtracted from its range in order to bring it into fair 
comparison with unadjusted northward or southward throws or with throws in 
any other direction which have been adjusted by reference to curves of this type 
appropriately constructed for their respective latitudes. 



99 



Implements must find the weights of the imple- 
ments "correct." Such ideals of perfection are not 
realistic, and the only alternative is to recognize 
the existence of error and state its magnitude. 
The minimum permissible weight for each im- 
plement is prescribed both in pounds and in 
kilograms by AAU rules, but in no instance are 
the prescriptions exactly equivalent. A discus 
thrower whose implement just satisfies the metric 
specification will use a discus 4 gm, or \ percent, 
lighter than that of a competitor whose discus 
just passes as judged by an inspector using per- 
fect scales calibrated in British units. Those 4 gm 
will give the former athlete two or three extra 
inches of distance, an advantage that might be 
decisive. 

Similar comments could be made about the 
rules of competition of the ICAAAA, where one 
reads that the javelin throw is measured from 
the point at which the point of the javelin first 
strikes the ground. This is a mark that cannot in 
general be determined to the often implied i in. 
since it is obliterated by the subsequent penetra- 
tion of the implement. Any javelin throw as 
correctly measured by ICAAAA rules will show 
a greater distance than if measured by AAU 
rules, but few field judges know this nor could 
they do much about it if they did. It is probable 
that the rules do not say what was meant in 
these cases. It is interesting that whereas the 
hammer, shot and discus must be thrown upon a 
level surface, there is no such requirement in the 
case of the javelin. 

Any serious attempt to put the measurements 
of sport upon a scientific basis would be met with 
vast inertia if not positive hostility. The training 
of athletes is still very largely an art, and there 
is no reason to suppose that those who are at 
present practicing this art with success will be 
predisposed to changes involving ways of thought 
which, however commonplace in other disciplines, 
are novel in athletic competition. One eminent 
track and field coach, a producer of national, 
Olympic and world champions, told the writer 
that he had no interest in hairsplitting; that 
leveling the ground accurately would be too 



much trouble; that common sense is better than 
a wind gage for estimating the effect of wind 
conditions on sprinters; that a man can't put the 
shot by theory— it's all in the feeling; that the 
exact angle of elevation is unimportant as long 
as he gets it in the groove. 

A few years ago, the writer published some 
criticisms along the lines of the present article 
and sent reprints to each of the several hundred 
National Committeemen of the AAU. One ac- 
knowledgment was received, but no reactions to 
the subject matter. In a sense, this indifference 
was only just recompense for the writer's habit 
of ignoring communications from nonphysicists 
proposing novel theories of the atom, or other- 
wise instructing the physicist as to the founda- 
tions of his science. 

There probably exists a general feeling that 
part of the charm of sport resides in accident and 
uncertainty. Any discussion of the possibility of 
replacing the balls-and-strikes umpire in base- 
ball by a robot will bring out the opinion that 
the fallibilities of the umpire are part of the 
entertainment for which the public pays. An 
optical instrument for determining from the side- 
lines whether or not a football has been ad- 
vanced to first down was tried out in California 
a few years ago. It was technically successful, 
but a popular failure. The crowd was suspicious 
of a measurement that it did not understand 
and could not watch; the players begrudged the 
elimination of the breather which a chain meas- 
urement affords; and even the linemen protested 
the loss of their dramatic moment. 

Though entertained by such attitudes, the 
physicist will hardly be able to dismiss a feeling 
that in any field of popular importance or in- 
terest, it is improper to keep up the appearances 
of accurate and comparable measurement with- 
out doing what might be done to gain the reality. 
In the matter of athletic records, he and very 
few others know what to do about it.' 



'The author will be pleased to furnish reprints of this 
article to readers who would find interest in bringing it to 
the attention of athletic authorities. 



100 



Observation of nature by Renaissance artists and crafts- 
men was a precursor of the new scientific outlook. This 
in turn accelerated technology, leading to the industrial 
revolution. 



15 The Scientific Revolution 

Herbert Butterfield 

An article from Scientific American, 1960. 



The preceding article leaves Homo 
sapiens in about 2500 B.C., after 
his invention of the city-state. 
Our story does not really get under way 
until some 4,000 years later. Thus, in 
turning to the next major revolution in 
man's impact on his environment, we 
seem to pass over almost all of recorded 
human history. No revolution is without 
its antecedents, however. Although the 



scientific-industrial age is a recent and 
original achievement of Western man, 
it has deep historical roots. 

Western civilization is unique in its 
historical-mindedness as well as in its 
scientific character. Behind it on the one 
hand are the ancient Jews, whose re- 
ligious literature was largely historical, 
who preached a God of history, and 
taught that history was moving to a 



mighty end, not merely revolving in 
cycles of growth and decay. On the 
other hand are the ancient Greeks. Their 
literature has provided a training in 
logic, a stimulus to the exercise of the 
critical faculties and a wonderful 
grounding in mathematics and the phys- 
ical sciences. 

In western Europe civilization had a 
comparatively late start. For thousands 




ANATOMY, studied by Renaissance artists, wag the first of the 
sciences to be placed on a modern footing. This drawing is from a 



copy of Albrecht Diirer's work De Symmetria Partium Humanorum 
Corporum in the Metropolitan Museum of Art in New York. 



101 



of years the lands at the eastern end of 
the Mediterranean had held the leader- 
ship in that whole section of the globe. 
It was in the West, furthermore, that the 
Roman Empire really collapsed, and was 
overrun by "barbarian invaders." Here 
much of the ancient culture was lost, and 
society reverted to comparatively primi- 
tive forms. In the meantime a high By- 
zantine civilization had its center in Con- 
stantinople, and a brilliant Arabian one 
in Baghdad. It would be interesting to 
know why Western man, though he 
started late, soon proved himself to be 
so much more dynamic than the peoples 
farther to the east. 

In the formative period of a civiliza- 
tion religion plays a more important part 
than we today can easily understand. 
After the fall of the Roman Empire the 
comparatively primitive peoples in much 
of Europe were Christianized by con- 
quest or through royal command; in the 
beginning it was a case of pagans mere- 
Iv changing the names of their gods. But 
in the succeeding centuries of the Mid- 
dle Ages the Church deepened spiritual 
life and moral earnestness. It became the 
great educator, recovering ancient schol- 
arship and acting as the patron of the 
arts. By the 13th century there had de- 
veloped a lofty culture, very much 
under the presidency of religion, but a 
religion that nourished the inner life, 
stimulated heart-searchings and exam- 
inations of conscience and set an eternal 
value upon each individual soul. The 
Western tradition acquired a high doc- 
trine of personality. 

By the year 1500, when the Renais- 
sance was at its height, the West had 
begun to take the command of world 
history. The expansion of Islam had been 
contained. The terrible Asian hordes, 
culminating in the Mongols and the 
Turks, that had overrun the eastern 
Mediterranean lands had been stopped 
in central Europe. One of the reasons 
first for survival and then for progress in 
the West was its consolidation into some- 
thing like nation-states, a form of polity 
more firm and more closely knit than the 
sprawling Asiatic empires. 

Yet the Renaissance belongs perhaps 
to the old (that is, the medieval) 
world rather than to the new; it was 
still greatly preoccupied with the re- 
covery of the lost learning of ancient 
Greece and Rome. Its primary interest 
was not in scientific studies, but now, 
after something like a thousand years 
of effort, the West had recaptured virtu- 
ally all it ever was to recover of ancient 
Greek scholarship and science. Only 
after this stage had been reached could 

102 



the really original developments in the 
study of the physical universe begin. 
The Western mind was certainly becom- 
ing less other-worldly. In the later Mid- 
dle Ages there was much thought about 
the nature of man as well as about the 
nature of God, so that a form of Christian 
humanism had already been develop- 
ing. The Renaissance was essentially 
humanistic, stressing man as the image 
of God rather than as the doomed sinner, 
and it installed in western Europe the 




GOTHIC CLOCK, dating from ihe early 
16th century, was photographed at the 
Smithsonian Institution. Stone at bottom 
is the driving weight; arm at top is part 
of escapement. Clockworks were among 
earliest examples of well-ordered machines. 



form of classical education that was to 
endure for centuries. The philosophy of 
the time dwelt much on the dignity of 
man. Oiu" modern Western values there- 
fore have deep historic roots. 

And the men of the Renaissance were 
still looking backward, knowing that the 
peak of civilization had been reached 
in remote antiquity, and then lost. It 
was easy for them to see the natural 
process of history as a process of decline. 

Signs of something more modern had 
begun to appear, but they belong cliieflv 
to the realm of action rather than to 
that of thought. Theories about the uni- 
verse (about the movements of the plan- 
ets, for example) had still to be taken 
over bodily from the great teachers of 
the ancient world. On the other hand, in 
action Western man was already proving 
remarkably free and adventurous: in his 
voyages of discovery, in the develop- 
ment of mining and metallurgy and in 
the creative work of the Renaissance art- 
ists. Under these conditions scientific 
thought might make little progress, but 
technology had been able to advance. 
And perhaps it was the artist rather than 
the writer of books who, at the Renais- 
sance, was the precursor of the modem 
scientist. 

The artists had emancipated them- 
selves from clerical influence to a great 
degree. The Florentine painters, seek- 
ing the faithful reproduction of nature, 
sharpened observation and prepared the 
way for science. The first of the sciences 
to be placed on a modern footing— that of 
anatom\— was one which the artists cul- 
tivated and which was governed by di- 
rect observation. It was the artists who 
even set up the cr\ that one must not be 
satisfied to learn from the ancients or to 
take everything from books; one must 
examine nature for oneself. The artists 
were often the engineers, the designers 
of fortifications, the inventors of gadgets, 
they were nearer to the artisan than 
were the scholars, and their studios often 
had the features of a laboratory or work- 
shop. It is not surprising to find among 
them Leonardo da N'inci— a precursor of 
modern science, but onlv a precursor, in 
spite of his brilliance, because the mod- 
ern scientific method had not yet 
emerged. 

Records show that in the 1.5tli centur\ 
a Byzantine scholar drew the attention 
of his fellow-countr) men to the techno- 
logical superioritN' of the West. He men- 
tioned progress in machine saws, ship- 
building, textile and glass manufacture 
and the production of cast iron. Three 
other items should be added to the list: 
the compass, gunpowder and the print- 
ing press. Although they might not have 



The Scientific Revolution 



originated in Christendom, they had not 
been handed down from classical antiq- 
uity. They came to be the first concrete 
evidence generally adduced to show that 
the moderns might even excel the an- 
cients. Before 1500, artillery had assisted 
the consolidation of government on 
something like the scale of the nation- 
state. Printing was to speed up intellectu- 
al communication, making possible the 
wider spread of a more advanced kind 
of education and facilitating the rise of 
a lay intelligentsia. 

Tn setting the stage for modern develop- 
-*- ments the economic situation is of 
fundamental importance. By this time a 
high degree of financial organization had 
been attained. The countryside might 
look much as it had done for a thousand 
yeajs, but the Renaissance flourished 
primarily in the city-states of Italy, the 
Netherlands and southern .Germany, 
where commerce and industry had made 
great advances. The forms of economic 
life were calculated to bring out indi- 
vidual enterprise; and in the cities the 
influence of priests declined— the lay 
intelligentsia now took the lead. There 
had existed greater cities and even an 
essentially urban civilization in ancient 
times. What was nev^ was the form of 
the economic life, which, by the oppor- 
tunities it gave to countless individuals, 
possessed dynamic potentialities. 

It was a Western world already 
steeped in humanism that entered upon 
a great scientific and technological de- 
velopment. But if Western man decided 
now to take a hand in shaping his own 
destiny, he did it, as on so many other 
occasions, only because he had been 
goaded by problems that had reduced 
him to desperation. The decisive prob- 
lems were not material ones, however. 
They were baffling riddles presented to 
the intellect. 

The authority of ancient scholarship 
was shaken when it came to be realized 
that the great Greek physician Galen 
had been wrong in some of his observa- 
tions, primarily in those on the heart. 
In the 16th century successive discover- 
ies about the heart and the blood vessels 
were made in Padua, culminating a little 
later in William Harvey's demonstration 
in England of the circulation of the 
blood. The whole subject was now set on 
a right footing, so that a flood of further 
discoveries was bound to follow very 
quickly. Harvey's work was of the 
greatest importance, moreover, because 
it provided a pattern of what could be 
achieved bv observation and methodical 
experiment. 



The older kind of science came to 
shipwreck, however, over two problems 
connected with motion. Aristotle, having 
in mind a horse drawing a cart, had 
imagined tliat an object could not be kept 
moving unless something was pulling or 
pushing it all the time. On this view it 
was difficult to see why projectiles stayed 
in motion after they had become sepa- 
rated from the original prppulsive force. 
It was conjectured that a flying arrow 
must be pushed along by the rush of air 
that its previous motion had created, but 
this theory' had been recognized to be 
unsatisfactory. In the 16th century, 
when artillery had become familiar, the 
student of motion naturally tended to 
think of the projectile first of all. Great 
minds had been defeated by this prob- 
lem for centuries before Galileo altered 
the whole approach and saw motion as 
something that continued until some- 
thing intervened to check it. 

A great astronomical problem still re- 
mained, and Copernicus did not solve 
it alone. Accepting the recognized data, 
he had shown chiefly that the neatest 
explanation of the old facts was the 
hypothesis of a rotating earth. Toward 
the end of the century new appearances 
in the sky showed that the traditional 



astronomy was obsolete. They demon- 
strated that the planets, for example, in- 
stead of being fixed to crystalline spheres 
that kept them in their proper courses, 
must be floating in empty space. There 
was now no doubt that comets belonged 
to the upper regions of the sky and cut a 
path through what had been regarded as 
the hard, though transparent, spheres. It 
was now not easy to see how the planets 
were held on a regular path. Those who 
followed Copernicus in the view that 
the earth itself moved had to face the 
fact that the science of physics, as it then 
existed, could not possibly explain how 
the motion was produced. 

In the face of such problems it began 
to be realized that science as a whole 
needed renovation. Even in the 16th 
century people were beginning to ex- 
amine the question of method. In this 
case a great historic change was willed 
in advance and consciously attempted. 
Men called for a scientific revolution be- 
fore the change had occurred, and be- 
fore they knew exactly what the situa- 
tion demanded. Francis Bacon, who 
tried to establish the basis for a new 
scientific method, even predicted the 
magnitude of its possible consequences 
—the power that man was going to ac- 




COMPASS ROSE is reproduced from The Art of !\avigntion, published in France in 1666. 
The invention of the compass, wliich was not an achievement of classical antiquity, en- 
couraged the men of the Renaissance to believe that they might come to excel the ancients. 



103 




MOVABLE TYPE CAST FROM MATRICES was contribution of Johann Gutenberg to 
art of printing. Sample of his type, enlarged about four diameters, is from his Bible, printed 
about 1456. Bible in which this type appears is in Pierpont Morgnn Library in New York. 



quire over nature. It was realized, fur- 
thermore, that the authority of tlje 
ancient world, as well as that of the Mid- 
dle Ages, was in question. The French 
philosopher Rene Descartes insisted 
that thinking should be started over 
again on a clean slate. 

rphe impulse for a scientific revolution 
-^ came from the pressure of high intel- 
lectual needs, but the tools of civilization 
helped to give the new movement its di- 
rection. In the later Middle Ages men 
had become more conscious of the ex- 
istence of the machine, particularly 
through mechanical clocks. This may 
have prepared them to change the for- 
mulation of their problems. Instead of 
seeking the "essence" of a thing, they 
were now more prepared to ask, even of 
nature, simply: How does it work? 

The student of the physical universe, 
like the artists before him, became more 
familiar with the workshop, learning 
manipulation from the artisan. He in- 
terested himself in problems of the prac- 
tical world: artillery, pumps, the deter- 
mination of longitude. Experimentation 
had long existed, but it now became 
more organized and methodical as the 
investigator became more conscious of 
what he was trying to do. In the 1 7th 
century, moreover, scientific instruments 
such as the telescope and the microscope 
came into use. 

But theory mattered too. If Galileo 
corrected a fallacious view of motion, it 
was because his mind was able to change 
the formulation of the whole problem. At 
least as important as his experimentation 
was his mathematical attack on the prob- 
lem, which illustrated the potential role 



of mathematics in the transformation of 
science. 



Another momentous factor in devel- 
oping the new outlook was the revival 
of an ancient view: that matter is com- 
posed of infinitesimally small particles. 
This view was now at last presented in a 
form that seemed consistent with Chris- 
tianity (because the combinations of the 
particles which produced the varied 
world of physical things were no longer 
regarded as the mere product of chance) , 
so that the atomic theory was able to ac- 
quire a wide currency. It led to a better 
appreciation of the intricate texture of 
matter, and it proved to be the source of 
innumerable new hypotheses. The the- 
ory seemed to open the way to a purely 
mechanical explanation of the universe, 
which should account for everything by 
the shape, the combination and the mo- 
tion of the particles. Long before such 
an explanation had been achieved, men 
were aspiring to it. Even religious men 
were arguing that Creation itself would 
have been imperfect if God had not 
made a universe that was a perfectly 
regular machine. 



fiv gram frW 



i2L'»>. 




NEW COSMOLOGY OF COPERNICUS placed a fixed sun (Sol) at the renter of the 
universe. The sphere of the fixed stars (/.) and the spheres of the six known planets re- 
volved around the sun. Circle inscribed around the earth (Terra) is the lunar sphere. This 
woodcut appears in Copernicus's On the Revolution of the Celestinl Spheres (1543). 



104 



The Scientific Revolution 



The civilization that Iiad begun its 
westward shift in the later Middle 
Ages was moving north and west. At the 
Renaissance Italy still held the primacy, 
but with the Reformation the balance 
shifted more definitely to the north. By 
the closing decades of the 17th century 
economic, technological and scientific 
progress centered on the English Chan- 
nel. The leadership now belonged to 
England, France and the Netherlands, 
the countries that had been galvanized 
by the commerce arising from the over- 
seas discoveries of the 15th century. And 
the pace was quickening. Technique was 
developing apace, economic life was ex- 
panding and society was moving for- 
ward generally in an exhilarating way. 

The solution of the main problems of 
motion, particularly the motion of the 
earth and the heavenly bodies, and the 
establishment of a new notion of scien- 
tific method, took a hundred years of 
effort after the crisis in the later decades 
of the 16th century. A great number of 
thinkers settled single points, or made 
attempts that misfired. In the period 
after 1660 a host of workers in Paris and 
London were making science fashion- 
able and bringing the scientific revolu- 
tion to its culmination. Isaac Newton's 
Principia in 1687 synthesized the results 
of what can now be seen to have been a 
century of collaborative effort, and 
serves to signalize a new era. Newton 
crowned the long endeavor to see the 
heavenly bodies as parts of a wonderful 
piece of clockwork. 

The achievements of ancient Greece 
in the field of science had now been un- 
mistakably transcended and outmoded. 
The authority of both the ancient and 
the medieval worlds was overthrown, 
and Western man was fully persuaded 
that he must rely on his own resources 
in the future. Religion had come to a low 
ebb after generations of fanaticism, 
persecution and war; now it was in a 
weak position for meeting the challenge 
of the new thought. The end of the 18th 
century sees in any case the decisive mo- 
ment in the secularization of European 
society and culture. The apostles of the 
new movement had long been claiming 
that there was a scientific method which 
could be adapted to all realms of inquiry, 
including human studies— history, poli- 
tics and comparative religion, for ex- 
ample. The foundations of what has 
been called the age of reason had now 
been laid. 

At the same time society itself was 
changing rapidly, and man could see it 
changing, see it as no longer static but 
dynamic. There began to emerge a dif- 
ferent picture of the process of things in 




TRAJECTORIES OF PROJECTILES were calculated with aid of protractor device (right) 
invented by Niccolo Tartaglia, an Italian engineer and mathematician who died in 1377. 
Ballistics problems drew attention to the inadequacy of the Aristotelian ideas about motion. 



time, a picture of history as the em- 
bodiment of progress rather than of de- 
cline. The future now appeared to offer 
opening vistas and widening horizons. 
Man was coming to feel more capable 
of taking charge over his own destiny. 

It was not merely man's tools, and not 
merely natural science, that had carried 
the story forward. The whole complex 
condition of society was involved, and 
movement was taking place on a wide 
front. The age of Newton sees the foun- 
dation of the Bank of England and the 
national debt, as well as the develop- 
ment of speculation that was to culmi- 



nate in the South Sea Bubble. An eco- 
nomic order congenial to individualism 
meant that life was sprouting from mul- 
titudinous centers, initiatives were being 
taken at a thousand points and ingenuity 
was in constant exercise through the 
pressure of need or the assurance that it 
would have its reward. The case is illus- 
trated in 17th-century England by the 
famous "projectors"— financial promoters 
busy devising schemes for making mon- 
ey. They slide easily into reformers mak- 
ing plans for female education or a so- 
cialistic order or a better form of gov- 
ernment. 




STRENGTH OF A BEAM was one of the problems in which Galileo demonstrated the pow- 
er of mathematical methods in science. Illustration is taken from his Discorsi e dimostra- 
zioni matematiche, in which he described the "new sciences" of mechanics and motion. 

105 



The whole of Western society was in 
movement, science and technology, 
industry and agriculture, all helping to 
carry one another along. But one of the 
operations of society— war— had probably 
influenced the general course of things 
more than is usually recognized. War 
above all had made it impossible for a 
king to "live of his own," enabling his 
subjects to develop constitutional ma- 
chinery, to insist on terms in return for 
a grant of money. Because of wars, 
kings were allied with advanced cap- 
italistic developments from the closing 
centuries of the Middle Ages. The 
growing demands of governments in 
the extreme case of war tightened up 
the whole development of the state and 
produced the intensification of the idea 
of the state. The Bank of England and 
the national debt emerge during a con- 
flict between England and France, 
which almost turned into a financial war 
and brought finance into the very struc- 
ture of government. In the 17th century 
armies had been mounting in size, and 
the need for artillery and for vast num- 
bers of uniforms had an important effect 
on the size of economic enterprises. 
The popularity in England of the nat- 
ural sciences was paralleled to a degree 
by an enthusiasm for anti(juarian pur- 
suits. In the later decades of the 17th 
century the scientific method began to 
affect the development of historical 
study. In turn, the preoccupation with 
the process of things in time seems to 
have had an influence upon scientists 
themselves. Perhaps the presiding sci- 
entific achievement in the next hundred 
years was the application of biology, 
geology and allied studies to the con- 
struction of a history of the physical uni- 
verse. By the end of the period this 
branch of science had come almost to 
the edge of the Darwinian theory of 
evolution. For the rest, if there was fur- 
ther scientific "revolution" in the 18th 
century, it was in the field of chemistry. 
At the beginning of the period it had not 
been possible to isolate a gas or even 
to recognize clearly that different gases 
existed. In the last quarter of the century 
Lavoisier reshaped this whole branch of 
science; water, which had been regarded 
for thousands of years as an element, was 
now seen to be a compound of owgen 
and hydrogen. 

By this time England— the nation of 
shopkeepers— was surprising the world 
with developments in the industrial field. 
A class of men had emerged who were 
agile in intellect, capable of self-help and 
eager for novel enterprises. They often 
lacked the classical education of the 
time, and were in a sense cut off from 



JtlZ- ^^— ^— v^'^- 




r^A^'T. 










DETAILS OF STEAM ENGINE are reproduced from J:inies Wall's patenl of 1769. The 
rliange from water to steam power in textile factories intensified llie industrial revolution. 



their cultural inheritance; and they no 
longer had the passion to intervene in 
theological controversy. Science and 
craftsmanship, combined \s ith the state 
of the market, enabled them, howe\er, 
to indulge their zeal for gadgets, me- 
chanical improvements and inventions. 
A considerable minor literature of the 
time gives evidence of the widespread 
passion for the production of technical 
devices, a passion encouraged sometimes 
by the policy of the government. Betw eon 
1760 and 1785 more patents were taken 
out than in the preceding 60 years; and 
of the estimated total of 26,000 patent> 



for the whole century, about half wen- 
crowded into the 15 years after 1785. In 
1761 the Society for the Encourage- 
ment of the Arts, Manufactures and 
Commerce, established a few years 
earlier, offered a prize for an invention 
that would enable six threads to be spun 
by a single pair of hands. A few years 
later Hargreave's spinning jenny ani 
Arkwright's water frame appeared. Tlu 
first steam engine had emerged at tht 
beginning of the century, but textile fac- 
tories began by using water power. The 
change to steam both here and in the 
production of iron greatly intensified the 



106 



The Scientific Revolution 



industrial revolution that was to alter 
the landscape so profoundly in the 19th 
century. 

'T'he country was able to meet the needs 
^ of a rapidly expanding population, 
especially as industrial development 
was accompanied by an agrarian revolu- 
tion—the birth of something like modern 
farming. Possibly as a result of a change 
in the prevalent type of rat, England 
ceased to suffer from the plague that had 
ravaged it for centuries. Advances in 
public-health techniques helped reduce 
the death rate, especially among infants. 
During the 18th century the English 
population rose from 5.5 to nine million. 
And people flocked to swell the growing 
industrial towns, as though assured that 
they were fleeing from something worse 
to something better. 

Even in 1700 most Englishmen were 
still engaged in occupations of a primary 
nature, connected with farming, fishing, 
mining and so on. London had perhaps 
half a million inhabitants, but Bristol, 
which came next, may have had only 
20,000. Very few towns had a population 
exceeding 10,000. Each country town 
had its miller, its brewer, its tanner and 
so on; each village had its baker, its 
blacksmith and its cobbler. Man\' of the 
people who were employed in industry 
—in the making of textiles, for example 
—carried on the work in their own homes 
with hand looms and spinning wheels; 
they supplemented their income by 
farming. 

The coming of the factory system and 
the growth of towns represented an un- 
precedented transformation of life and 
of the human environment, besides 
speeding up the rate of all future change. 
This denser and more complicated world 
required more careful policing, more 
elaborate administration and a tremen- 
dous increase in the tasks of government. 
The mere growth and distribution of 
population, and the fresh disposition of 
forces that it produced within society, 
are fundamental factors in the history 
of the 19th century. 

With gathering momentum came 
railways, the use of electricity, the in- 
ternal-combustion engine and today the 
world of electronics and nuclear weap- 
ons. Science, so long an aid to the in- 
ventor, now seems itself to need the en- 
gineer and the industrial magnate. And 
all the elaborate apparatus of this techni- 
cal civilization is easily communicable to 
every quarter of the globe. Our scientif- 
ic-industrial revolution is a historical 
landmark for those peoples to whom 
Renaissance and Reformation have no 
relevance, since Christianity and Greek 



antiquity are not in their tradition. The 
material apparatus of our civilization is 
more communicable to other continents 
than are our more subtle and imponder- 
able ideas. 

"y/^et the humanism that has its roots so 
^ far back in our history has by no 
means lost its hold on the world. In the 
West, indeed, it now touches vastly wider 
classes of peoples than were able to read 
at all before the days of the industrial 
revolution. That revolution requires the 
spread of education, and at the same 
time provides the apparatus for it. The 
extraordinary speeding-up of communi- 
cations and the increased mobility of life 
have themselves had colossal educative 
results. It was under the ancient order 
that the peasantry were sometimes felt 
to be like cows; John Wesley\ although 
he held so firmly that the lowest classes 
were redeemable, himself described 
them with astonishing frequency as wild 
beasts. The new era has raised the 
stature of men, not lowered it, as some 
have imagined; and seems to require (or 
to produce) a more genuine kind of 
moral autonomy. 



Great literature is perhaps more wide- 
ly appreciated at the present day than 
ever in previous history. The rights 
and freedoms of man and the indepen- 
dence and self-respect of nations have 
never been more glorified than in our 
own century. And we have transmitted 
these ideals to other parts of the globe. 
The scientific-industrial revolution has 
operated to a great saving of life. At the 
same time it has provided a system 
which, where it has prevailed, has so 
far enabled the expanded population to 
live. 

The vastness of populations and the 
character of the technical revolution it- 
self have led, however, to certain dan- 
gers. The development of high-powered 
organization means that a colossal ma- 
chine can now be put at the service of 
a possible dictatorship. It is not yet clear 
that the character of the resulting civil- 
ization will necessarily undermine the 
dictatorship and produce the re-estab- 
lishment of what we call Western values. 
In this sense the elaborate nature of the 
system may come to undermine that 
wonderful individualism that gave it its 
start. At the same time, when nations 




SPINNING FRAME, patented by Richard Arkwright in 1769, produced superior yarn. In 
his application the inventor said the machine would be of "great utility" to manufacturers 
and to the public "by employing a great number of poor people in working said machinery." 



107 



are ranged against one another, each lution, but it may eventually prove a 
may feel forced to go on elaborating and necessary concomitant of that revolu- 
enlarging ever more terrible weapons, tion, wherever the revolution may 
though no nation wants them or ever in- spread. At this point we simply do not 
tends to use them. Weapons may then know. There are certain things we can- 
defeat their own ends, and man may find not achieve without tools. But the tools 
himself the slave of the machine. in themselves do not necessarily deter- 

The Western ideal of democracy is mine our destiny, 
older than the scientific-industrial revo- 



108 



The effect of the rise of physics in the age of Galileo 
and Newton, particularly on literature and religion, is 
discussed in this brief article. 



16 How the Scientific Revolution of the Seventeenth Century 
Affected Other Branches of Thought 

Basil Willey 



An article from A Short History of Science, Origins and Results 
of tfie Scientific Revolution, ^951. 

IN order to get a bird's-eye view of any century it is quite 
useful to imagine it as a stretch of cotintry, or a land- 
scape, which we are looking at from a great height, let us 
say from an aeroplane. If we view the seventeenth century 
in this way we shall be struck immediately by the great 
contrast between the scenery and even the climate of its 
earUer and that of its later years. At first we get movmtain 
ranges, torrents, and all the picturesque interplay of alter- 
nating storm and brightness; then, further on, the land 
slopes down to a richly cultivated plain, broken for a while 
by outlying heights and spurs, but finally becoming level 
coimtry, watered by broad rivers, adorned with parks and 
mansions, and fit up by steady sunshine. The mountains 
connect backwards with the central medieval Alps, and the 
plain leads forwards with Utde break into our own times. To 
drop the metaphor before it begins to be misleading, we 
may say that the seventeenth century was an age of transi- 
tion, and although every century can be so described, the 
seventeenth deserves this label better than most, becaxise it 
hes between the Middle Ages and the modem world. It 
witnessed one of the greatest changes which have ever 
taken place in men's ways of thinking about the world they 
five in. 

I happen to be interested in literature, amongst other 
things, and when I turn to this century I cannot help no- 
ticing that it begins with Shakespeare and Donne, leads on 
to Milton, and ends with Dryden and Swift: that is to say, 
it begins with a Uteratiu-e full of passion, paradox, imagina- 
tion, curiosity and complexity, and ends with one dis- 
tinguished rather by clarity, precision, good sense and 
definiteness of statement. The end of the century is the be- 
ginning of what has been called the Age of Prose and 
Reason, and we may say that by then the qtialities neces- 
sary for good prose had got the upper hand over those 
which produce the greatest kinds of poetry. But that is not 



109 



all: we find the same sort of thing going on elsewhere. Take 
architecture, for example; you all know the style of build- 
ing called Elizabethan or Jacobean— it is quaint and fanci- 
ful, sometimes rugged in outline, and richly ornamented 
with carving and decoration in which Gothic and classical 
ingredients are often mixed up together. Well, by the end 
of the century this has given place to the style of Christo- 
pher Wren and tlie so-called Queen Anne architects, which 
is plain, well proportioned, severe, and purely classical 
without Gothic trimmings. And here there is an important 
point to notice: it is true that the seventeenth centiiry begins 
with a blend of medieval and modem elements, and ends 
with the trivmiph of the modem; but observe that in those 
days to be 'modem' often meant to be 'classical', that is, 
to imitate the Greeks and Romans. We call the age of 
Dryden, Pope and Addison the 'Augustan' Age, and the 
men of that time really felt that they were living in an epoch 
like that of the Emperor Augustus— an age of enlighten- 
ment, learning and true civilisation— and congratulated 
themselves on having escaped from the errors and super- 
stitions of the dark and monkish Middle Ages. To write and 
build and think like the ancients meant that you were rea- 
sonable beings, cultivated and urbane— that you had aban- 
doned the shadow of the cloister for the cheerful light of 
the market place or the coflFee house. If you were a scientist 
(or 'natural philosopher') you had to begin, it is true, by 
rejecting many ancient theories, particiJarly those of Aris- 
totle, but you knew all the while that by thinking inde- 
pendently and taking nothing on trust you were following 
the ancients in spirit though not in letter. 

Or let us glance briefly at two other spheres of interest: 
politics and religion, beginning with politics. Here again 
you notice that the century begins with Cavalier and 
Roimdhead and ends with Tory and Whig— that is to say, 
it begins with a division arousing the deepest passions and 
prejudices, not to be settled without bloodshed, and ends 
with the mere opposition of two political parties, differing 
in principle of course, but socially at one, and more ready 
to alternate peaceably with each other. The Hanoverians 
succeed the Stuarts, and what more need be said? The 
divine right of kings is little more heard of, and the scene 
is set for prosaic but peaceful development. Similarly in re- 
ligion, the period opens with the long and bitter stmggle 
between Puritan and Anglican, continuing through civil 
war, and accompanied by fanaticism, persecution and exile, 
and by the multiplication of hostile sects; it ends with the 
Toleration Act, and with the comparatively mild dispute 
between the Deists and their opponents as to whether 



no 



How the Scientific Revolution of tfie Seventeentfi Century 
Affected Other Branches of Thought 



Nature was not after all a clearer evidence of God than 
Scripture, and the conscience a safer guide than the creeds. 
In short, wherever you turn you find the same tale repeated 
in varying forms: the ghosts of history are being laid; dark- 
ness and tempest are yielding to the hght of common day. 
Major issues have been settled or shelved, and men begin 
to think more about how to live together in concord and 
prosperity. 

Merely to glance at this historical landscape is enough 
to make one seek some explanation of these changes. If the 
developments had conflicted with each other we might 
have put them down to a nimiber of different caiises, but 
since they all seem to be setting in one direction it is natu- 
ral to suppose that they were all due to one common 
underlying cause. There are various ways of accounting for 
historical changes: some people believe, for instance, that 
economic causes are at the bottom of everything, and that 
the way men earn their hving, and the way in which wealth 
is produced and distributed, determine how men think and 
write and worship. Others believe that ideas, rather than 
material conditions, are what control history, and that the 
important question to ask about any period is what men 
then believed to be true, what their philosophy and religion 
were like. There is something to be said on both sides, but 
we are concerned with a simpler question. We know that 
the greatest intellectual change in modem history was com- 
pleted during the seventeenth centxuy: was that change of 
such a kind as to explain aU those parallel movements we 
have mentioned? Would it have helped or hindered that 
drift towards prose and reason, towards classicism, enlight- 
enment and toleration? The great intellectual change was 
that known as the Scientific Revolution, and I think the 
answer to these questions is— Yes. 

It is not for me to describe that revolution, or to discuss 
the great discoveries which produced it. My task is only 
to consider some of the effects it had upon men's thoughts, 
imaginations and feelings, and consequently upon their 
ways of expressing themselves. The discoveries— I am think- 
ing mainly of the Copemican astronomy and the laws of 
motion as explored by Galileo and fully formiJated by 
Newton— shocked men into realising that things were not 
as they had always seemed, and that the world they were 
living in was really quite different from what they had been 
taught to suppose. When the crystal spheres of the old 
world-picture were shattered, and the earth was shown to 
be one of many planets rolling through space, it was not 
everyone who greeted this revelation with enthusiasm as 
Giordano Bruno did. Many felt lost and confused, because 



111 



the old picture had not only seemed obviously true to com- 
mon sense, but was confirmed by Scripture and by Ar- 
istotle, and hallowed by the age-long approval of the 
Church. What Matthew Arnold said about the situation in 
the nineteenth century applies also to the seventeenth: re- 
ligion had attached its emotion to certain supposed facts, 
and now the facts were failing it. You can hear this note 
of loss in Donne's well-knovra hnes: 

And new philosophy calls all in doubt; 

The element of fire is quite put out; 

The sun is lost, and th' earth, and no man's wit 

Can well direct him where to look for it. 

Not only 'the element of fire', but the very distinction be- 
tween heaven and earth had vanished— the distinction, I 
mean, between the perfect and incorruptible celestial bod- 
ies from the moon upwards, and the imperfect and cor- 
ruptible terrestrial bodies below it. New stars had appeared, 
which showed that the heavens could change, and the tele- 
scope revealed irregularities in the moon's surface— that is, 
the moon was not a perfect sphere, as a celestial body 
should be. So Sir Thomas Browne could write: 

'While we look for incorruption in the heavens, we 
find they are but like the earth;— durable in their main 
bodies, alterable in their parts; whereof, besides comets 
and new stars, perspectives (i.e. telescopes) begin to tell 
tales, and the spots that wander about the sun, with 
Phaeton's favour, would make clear conviction.' 

Naturally it took a long time for these new ideas to sink 
in, and Milton still treats the old and the new astronomies 
as equally acceptable alternatives. The Copemican scheme, 
however, was generally accepted by the second half of the 
century. By that time the laws governing the motion of 
bodies on earth had also been discovered, and finally it was 
revealed by Newton that the law whereby an apple falls 
to the ground is the very same as that which keeps the 
planets in their courses. The realisation of this vast unify- 
ing idea meant a complete re-focusing of men's ideas about 
God, Nature and Man, and the relationships between them. 
The whole cosmic movement, in the heavens and on earth, 
must now be ascribed no longer to a divine pressure acting 
through the Primum Mobile, and angelic intelligences con- 
trolling the spheres, but to a gravitational pull which could 
be mathematically calculated. The universe turned out to 
be a Great Machine, made up of material parts which all 
moved through space and time according to the strictest 
rules of mechanical causation. That is to say, since every 



112 



How the Scientific Revolution of tlie Seventeentfi Century 
Affected Other Branches of Thought 



effect in nature had a physical cause, no room or need was 
left for supernatural agencies, whether divine or diabolical; 
every phenomenon was explicable in terms of matter and 
motion, and could be mathematically accounted for or pre- 
dicted. As Sir James Jeans has said: 'Only after much study 
did the great principle of causation emerge. In time it was 
foimd to dominate the whole of inanimate nature. . . . The 
final establishment of this law . . . was the triumph of the 
seventeenth century, the great century of Galileo and New- 
ton.' It is true that mathematical physics had not yet con- 
quered every field: even chemistry was not yet reduced to 
exactitude, and stiU less biology and psychology. But New- 
ton said: 'Would that the rest of the phenomena of natin-e 
could be deduced by a like kind of reasoning from me- 
chanical principles'— and he beheved that they could and 
would. 

I referred just now to some of the immediate effects of 
the 'New Philosophy' (as it was called); let me conclude 
by hinting at a few of its vdtimate effects. First, it produced 
a distrust of all tradition, a determination to accept nothing 
as true merely on authority, but only after experiment and 
verification. You find Bacon rejecting the philosophy of the 
medieval Schoolmen, Browne writing a long exposure of 
popular errors and superstitions (such as the behef that a 
toad had a jewel in its head, or that an elephant had no 
joints in its legs), Descartes resolving to doubt everything 
—even his own senses— until he can come upon something 
clear and certain, which he finally finds in the fact of his 
own existence as a thinking being. Thus the chief intellec- 
tual task of the seventeenth century became the winnowing 
of truth from error, fact from fiction or fable. Gradually a 
sense of confidence, and even exhilaration, set in; the uni- 
verse seemed no longer mysterious or frightening; every- 
thing in it was explicable and comprehensible. Comets and 
eclipses were no longer dreaded as portents of disaster; 
witchcraft was dismissed as an old wives' tale. This new 
feeling of security is expressed in Pope's epitaph on New- 
ton: 

Nature and Nature's laws lay hid in night; 
God said, Let Newton be! and all was light! 

How did all this affect men's rehgious beliefs? The effect 
was very different from that of Darwinism on nineteenth- 
century religion. In the seventeenth century it was felt that 
science had produced a conclusive demonstration of God, 
by showing the evidence of His wisdom and power in the 
Creation. True, God came to be thought of rather as an 
abstract First Cause than as the personal, ever-present God 



113 



of religion; the Great Machine impHed the Great Mechanic, 
but after making the machine and setting it in motion God 
had, as it were, retired from active superintendence, and 
left it to run by its ovvna laws without interference. But at a 
time when inherited religious sentiment was still very pow- 
erful, the idea that you could look up through Nature to 
Nature's God seemed to oflFer an escape from one of the 
worst legacies of the past— rehgious controversy and sec- 
tarian intolerance. ReUgion had been endangered by inner 
conflict; what could one believe, when the Churches were 
all at daggers drawn? Besides, the secular and rational tem- 
per brought in by the new science soon began to undermine 
the traditional foimdations of behef. If nothing had ever 
happened which could not be explained by natural, physi- 
cal causes, what about the supernatural and miraculous 
events recorded in the Bible? This was a disturbing thought, 
and even in the seventeenth century there were a few who 
began to doubt the literal truth of some of the biblical nar- 
ratives. But it was reserved for the eighteenth century to 
make an open attack upon the miraculous elements in 
Christianity, and to compare the Old Testament Jehovah 
disparagingly with the 'Supreme Being' or 'First Cause' of 
philosophy. For the time, it was possible to feel that science 
was pious, because it was simply engaged in studying 
God's own handiwork, and because whatever it disclosed 
seemed a further proof of His almighty skill as designer of 
the universe. Addison exactly expressed this feeling when 
he wrote: 

The spacious firmament on high. 

With all the blue ethereal sky, 

And spangled heavens, a shining frame. 

Their great Original proclaim. 

Th' unwearied Sim from day to day 

Does his Creator's power display; 

And publishes to every land 

The work of an Almighty hand. 

Science also gave direct access to God, whereas Church and 
creed involved you in endless uncertainties and difiBculties. 
However, some problems and doubts arose to disturb the 
prevailing optimism. If the universe was a material mecha- 
nism, how could Man be fitted into it?— Man, who had 
always been supposed to have a free will and an immortal 
soul? Could it be that those were illusions after all? Not 
many faced up to this, though Hobbes did say that the soul 
was only a function of the body, and denied the freedom of 
the will. What was more immediately serious, especially 
for poetry and religion, was the new tendency to discount 



114 



How the Scientific Revolution of tfie Seventeenth Century 
Affected Other Branches of Thought 



all the products of the imagination, and all spiritual insight, 
as false or fictitious. Everything that was real could be 
described by mathematical physics as matter in motion, and 
whatever could not be so described was either unreal or 
else had not yet been truly explained. Poets and priests had 
deceived us long enough with vain imaginings; it was now 
time for the scientists and philosophers to take over, and 
speak to us, as Sprat says the Royal Society required its 
members to do, in a 'naked, natural' style, bringing all 
things as close as possible to the 'mathematical plainness'. 
Poets might rave, and priests might try to mystify us, but 
sensible men would ignore them, preferring good sense, and 
sober, prosaic demonstration. It was said at the time that 
philosophy (which then included what we call science) 
had cut the throat of poetry. This does not mean that no 
more good poetry coxild then be produced: after all. Dry- 
den and Pope were both excellent poets. But when all has 
been said they do lack visionary power: their merits are 
those of their age— sense, wit, brilliance, incisiveness and 
point. It is worth noticing that when the Romantic move- 
ment began a himdred years later, several of the leading 
poets attacked science for having killed the universe and 
turned man into a reasoning machine. But no such thoughts 
worried the men of the Augustan Age; their prevailing feel- 
ing was satisfaction at Hving in a world that was rational 
through and through, a world that had been explained 
favourably, explained piously, and explained by an Eng- 
hshman. The modem beUef in progress takes its rise at this 
time; formerly it had been thought that perfection lay in 
antiquity, and that subsequent history was one long decUne. 
But now that Bacon, Boyle, Newton and Locke had arisen, 
who could deny that the ancients had been far surpassed? 
Man could now hope to control his environment as never 
before, and who could say what triumphs might not lie 
ahead? Even if we feel that the victory of science was then 
won at the expense of some of man's finer faculties, we can 
freely admit that it brought with it many good gifts as well 
—tolerance, reasonableness, release from fear and super- 
stition—and we can pardon, and even envy, that age for its 
temporary self-satisfaction. 



115 



Maxwell, the developer of electromagnetic theory (Unit 4), 
wrote light verse. The reference in the first line of the poem 
is to the members of the British Association for the Advance- 
ment of Science. 



17 Report on Tait's Lecture on Force, 
at British Association, 1876 



James Clerk Maxwell 



Verse written in 1876 and published in Life of James Clerk l^axwell, 1884. 



Ye British Asses, who expect to lie;a' 

Ever some new thiiii;, 
I've nothing new to tell, but wliat, I fear, 

May be a true thing. 
For Tait comes with his plummet and his line, 

Quick to detect your 
Old bosh new dressed in what you call a tine 

Poi)ular lecture. 

Whence comes that most peculiar smattering, 

Heard in our section ? 
Pure nonsense, to a scientific swing 

Drilled to j^erfection 1 
That small word "Force," they make' a barlier's l)louk, 

Ready to put on 
Meanings most strange and various, tit to shock 

Pupils of Newton. 

Ancient ;iud foreign ignorance they tlirow 

Into the bargain ; 
The shade of Leibnitz- mutters from lielow 

Horrible jargon. 
The phrases of last century in this 

Linger to play tricks — 
Vis Viva and Vis Mortua and Vis 

Acceleratrix : — 



Those long-nebbed words that to our te.xt books still 

Cling by their titles, 
And from them creep, as entozoa will. 

Into our vitals. 
But see ! Tait writes in lucid symbols clear 

One small equation ; 
And Force becomes of Energy a mere 

Space-variation. 
Force, then, is Force, but mark you ! not a thing, 

Only a Vector ; 
Thy barbM arrows now have lost their sting, 

Impotent spectre ! 



Thy reign, Force ! is over. Now no more 

Heed we thine action ; 
Repulsion leaves us where we were before, 

So does attraction. 

Both Action and Reaction now are gone. 

Just ere they vanished, 
Stress joined their hands in peace, and made tlicm one 

Then they were banished. 
The Universe is free from pole to pole. 

Free from all forces. 
Rejoice ! ye stars — like blessed gods ye roll 

On in your courses. 

No more the arrows of the Wrangler race, 

Piercing shall wound you. 
Forces no more, those symbols of disgrace, 

Dare to surround you : 
But those whose statements baffle all attacks, 

Safe by evasion, — 
Whose definitions, like a nose of wax, 

Suit each occasion, — 

Whose unreflected rainbow far sur]);vs.«i'd 

All our inventions. 
Whose very energy appears at last 

Sciint of dimensions : — 

Are tliesc the gods in whom ye put your trusi. 

Lordlings and ladies ? 
The hidden^ potency of cosmic dust 

Drives them to Hades. 

While you, brave Tait ! who know so well the way 

Forces to scatter, 
Calmly await the slow but sure decay, 

Even of Matter. 



116 



This after-dinner address to the American Physical Society 
attempts to point up in a simplified way the amusing, as well as 
some of the more serious, problems which arise in connection 
with flight into space, including the Impracticality of using the 
moon as a military base or of solving the population problem 
by colonizing the planets. 



18 Fun in Space 

Lee A. DuBridge 

An article from The American Journal of Physics, 1960. 



A WONDERFUL thing has happened during 
the past three years. A new subject has 
been opened up which even an old-fashioned 
physicist can understand. A new subject that 
involves no relativity corrections, no strange- 
particle theory — not even any Fermi statistics. 
Just good old-fashioned Newtonian mechanics! 

Space ! 

All you have to do is get an object a couple of 
hundred miles above the earth and give it a 
horizontal speed of 5 or 10 miles/sec, and from 
that time on you can tell exactly what's going to 
happen to it — maybe even for a billion years — by 
just using Newton's laws of motion and his law 
of gravitation. The mathematical details get a 
little rough now and then, but a good IBM 
machine will take care of that — if you can find 
someone who knows how to use it. But there is 
nothing in principle that any physicist can't 
understand. 

I personally prefer to talk about space to non- 
scientific audiences. In the first place, they can't 
check up on whether what you are saying is right 
or not. And, in the second place, they can't make 
head or tail out of what you are telling them 
anyway — so they just gasp with surprise and 
wonderment, and give you a big hand for being 
smart enough to say such incomprehensible 
things. And I never let on that all you have to do 
to work the whole thing out is to set the centri- 
fugal force equal to the gravitational force and 
solve for the velocity. That's all there is to it! 
Knowing v, you can find the period of motion, ot 
course, and that's practically all you need. 



* Text of remarks at the Banquet of the 1960 Spring 
Meeting of the American Physical Society, Sheraton Hall, 
Washington, D. C, .April 27, 1960. 



To show what I mean, let me give a simple 
example that I heard discussed at an IRE 
meeting a couple of years ago. 

Imagine two spacecraft buzzing along in the 
same circular orbit around the earth — say 400 
miles up — and one ship is 100 yards or so ahead 
of the other one. The fellow in the rear vehicle 
wants to throw a baseball or a monkey wrench 
or a ham sandwich, or something, to the fellow 
ahead of him. How does he do it? 

It sounds real easy. Since the two ships are in 
the same orbit, they must be going at the same 
speed — so the man in the rear could give the 
baseball a good throw forward and the fellow 
ahead should catch it. 

But wait! When you throw the ball out, its 
speed is added to the speed of the vehicle so now 
it is going too fast for that orbit. The centrifugal 
force is too great and the ball goes off on a tan- 
gent and rises to a higher orbit. But an object in 
a higher orbit must go slower. In fact, the faster 
he throws the ball, the higher it rises and the 
slower it goes. So our baseball pitcher stares in 
bewilderment as the ball rises ahead of him, then 
seems to stop, go back over his head, and recede 
slowly but surely to the rear, captured forever in 
a higher and slower and more elliptical orbit 
while the pitcher sails on his original course. 

You must make a correction, of course, if you 
assume the ball's mass is not negligible and you 
take account of the conservation of momentum. 
Then, as the ball is pitched forward, the vehicle 
is slowed down — whercuDon it falls into a lower 
orbit where, of course, it goes faster. So in this 
case the ball appears to rise higher and fall 
behind faster. 

But now our ball thrower decides to try again. 
This time he is going to be smart. If you can't 



117 



reach the guy ahead by throwing forward, the 
obvious thing to do is throw the ball to the rear. 
Now its speed is subtracted from that of the 
vehicle ; hence it is going too slow for its orbit ; 
hence it falls to a lower orbit and goes faster, 
passes underneath the rear vehicle, moves forward 
and passes underneath the forward vehicle, and 
then on into its orbit. It will be left as an exercise 
for the student to determine just how the baseball 
may be launched in order to hit the forward 
vehicle. One way, of course, is to first circle the 
earth and come back on the second lap, but there 
are other ways. 

Now, that's all very simple Newtonian me- 
chanics, of course. But you can see how, when 
you start to explain that to make an object go 
faster you slow it down and to make it go slower 
you speed it up, people begin to think you are 
either crazy or very smart. However, tonight I 
am talking to physicists and they are used to 
far crazier things than that — so they will have 
no trouble believing me at all. 

So let's get on with more serious problems. 

For example, last summer there appeared in 
a military journal an article on the use of the 
moon as a military base. This article is an inex- 
haustible source of fascinating problems for your 
students. 

The first point made by the writer is that 
military men have always cherished "high 
ground." First a hill or a mountain, then a 
balloon, then an airplane, then a higher airplane, 
then a ballistic missile, and now — what could be 
more logical — the moon. Next, of course (though 
the author fails to mention this), comes Venus, then 
Mars, then Mercury, then the sun\ Eventually, 
of course, we'd like to get out to Alpha Centauri 
(the nearest large star). But at the speeds of 
present space ships it would take 100 000 years 
or so to get to Alpha Centauri. And, who knows, 
the war might be over by then. 

But let's stick to the moon. Our article suggests 
it's a real interesting possibility to hit an enemy 
target from the moon. The author does not 
mention that it would be a lot quicker, cheaper, 
and easier to hit it from Iowa, or Alaska, or 
Maine. But the moon is higher — and so is less 
vulnerable. Besides— here is the clincher — the 
velocity of escape from the moon is only 1.5 
miles/sec, while the initial velocity of an ICBM 
is nearly 5 miles/sec. Think of all the fuel you 
save! Of course, there is a little matter of getting 
the rocket and fuel up to the moon in the first 
place. But that presumably will be charged to the 



Military Air Transp>ort Service and so can be 
neglected. 

Now you can easily prove that if you fired a 
rocket from the moon at just over 1.5 miles/sec, 
and did it just right, you could put it into an 
elliptical earth orbit which would intersect the 
earth's surface after a flight time of about five 
days. And, if you timed it just right and the earth 
kept spinning at just the right speed, your target 
might rotate into position under the point of 
entry just as the rocket came in. But if you made 
an error of a few percent in the velocity and the 
flight took only 4| days — then maybe New York 
would appear at the point of impact, or maybe 
the middle of the Pacific Ocean, or, more likely, 
the ellipse might miss the earth's surface entirely 
and the object return to its starting point. 
Except, the starting point, the moon — now, 10 
days later — won't be there anymore! The moon 
will be a third of the way around its orbit! 

It is, of course, very unimaginative of me not 
to recognize that you could shoot the rocket 
faster than 1.5 miles/sec and get the payload to 
the earth faster than five days. So you could. 
That takes more fuel of course — and soon you 
will wonder why you didn't stay home in the 
first place. But, the article says, you could reduce 
the flight time from moon to earth to a few 
minutes if you wished. Again, so you could. All 
you need to do is to accelerate to an average 
speed of a million miles per hour. That's 275 
milcs/scc. That's 55 times as fast as an earth- 
bound ICBM, or 3000 times as much kinetic 
energy. So, if the ICBM takes 100 000 pounds 
of fuel, to launch our rocket from the moon will 
take 5.5 million pounds. And that's quite a load 
to get off the earth and up to the moon in the 
first place. In fact, you'll burn up one billion 
pounds of fuel just lifting it off the earth. 

Well, you begin to see why space research is so 
much fun. And I think it's wonderful to have 
something turn up again that's fun. We ahva>s 
used to say that we went into physics just because 
it was fun. But then, with big machines and big 
crews and big budgets, physics research got 
deadly serious. I have a physicist friend who is 
thinking of going into biolog>' where all he needs 
is a microscope and some viruses — and he can 
have a lot of fun. But I think space may save him 
for physics because that's fun too — especially if 
you're a theoretical physicist, as he is. As long 
as you don't have to go up into space, but can 
just think about it, it is a lot of fun. 

There is another bundle of space problems that 



118 



Fun in Space 



can be a source of considerable amusement. Have 
you ever tried to explain to your wife why it is 
that if she were in a space capsule in an orbit 
around the earth she would have lost all her 
weight. Now the idea of losing a few pounds of 
weight might appeal to her, but I am sure the 
notion of weightlessness is something incompre- 
hensible to most people. If you ask most laymen 
why the condition of weightlessness exists, they 
would tell you that since you are above the 
earth's atmosphere there isn't any gravity and 
so, of course, you must be weightless. To such 
people one must carefully explain that the force 
of gravity 200 miles above the surface of the 
earth is only 10% less than it is on the earth's 
surface. Even at 4000 miles the gravity is reduced 
only to one-quarter of its value on the earth's 
surface; and at 8000 miles, to one-ninth. Since it 
is obviously gravity that holds a satellite in a 
circular orbit, and since the earth's gravity is 
even strong enough out at the distance of the 
moon — 240 000 miles — to hold the moon in its 
orbit, the weightlessness in an earth satellite is 
evidently not caused by the absence of gravity. 
Then what is it caused by? Of course, if you 
want to be a real coward, you will choose the 
easy way out and simply say that in a circular 
orbit the force of gravity is canceled by the 
centrifugal force, and the condition of weightless- 
ness results. You know very well, of course, that 
that isn't the proper explanation. The centrifugal 
force is the force that the satellite exerts on the 
earth and is not a force on the satellite. The force 
on the satellite is toward the earth and, indeed, 
it is the force of gravity which supplies the 
centripetal force which keeps the satellite in its 
orbit. In other words, gravity and centripetal 
force are in the same direction, not opi)ositc. So, 
when this is pointed out by some unkind person, 
you get more sophisticated and say simply, 
"Well, in any freely falling object the condition 
of weightlessness exists. It would exist, for 
example, for passengers in a freely falling 
elevator." But, since not many people have been 
passengers in a freely falling elevator, this explan- 
ation usually falls fairly flat also. At this point I 
recommend that the argument be abandoned and 
we retreat into technical jargon by saying, "Well, 
it's just one of Newton's laws of motion that 
whenever the inertial reaction and the acceler- 
ating force are equal, no tendency toward further 
acceleration can exist, and hence the system 
behaves as though no gravitational field were 
present." No one can quarrel with that state- 



ment. Even if nobody understands it, it's true. 
And it even holds for an elliptical orbit where 
centrifugal force and gravity are not always 
equal, but weightlessness exists anyway. 

By this time I suppose you will all be convinced 
that I am against space. However, that's not 
true. The Caltech Jet Propulsion Laboratory has 
a 50-million-dollar-a-year contract to do space 
research. I would not dare be against it! 

I seriously believe that when all the popular 
nonsense on space is swept away, we can soberly 
recognize that the achievement of getting man- 
made vehicles into space orbits and having them 
transmit scientific information back to earth is 
one of the great triumphs in the history of 
technology. And, as so often happens when a new 
technological development occurs, new types of 
scientific exploration become possible. 

I don't know much about the military value of 
space weapons. And the little I do know does not 
impress me. Nor do I know much about the 
psychological value of space ventures — how all 
the people in Asia and Africa think the greatest 
nation on earth is the one that puts up the 
heaviest satellite. That doesn't impress me 
either. But the possibilities of doing scientific 
experiments in space vehicles is something I can 
get really excited about. 

Look at the very first thing that happened — 
the discovery of the Van Allen layers of charged 
particles. Think of the many exciting experiments 
still ahead to unravel the mysteries which that 
discovery opened up. And it's only the start. 
Now at last we can explore the earth's gravita- 
tional, magnetic, and electric fields ; look down 
on its storm patterns; determine the nature of 
highly rarefied matter in the space through which 
the earth moves, the radiation fields present 
throughout space. We can now look, unimpeded, 
at the sun, the planets, and the stars — and a new 
era in astronomy is in the offing. We'll be able to 
examine the moon directly with instruments 
landed on its surface — and clear up many 
mysteries about the origin of the solar system. 
We'll discover somenew mysteries too, no doubt. 
Mars and Venus, and eventually other planets, 
will soon be in the range of direct examination, 
too. We may actually live to see the day when 
we will know for sure whether the green patches 
on Mars are living plants or not — and, if so, 
whether they consist of the same type of organic 
molecules with which we are familiar on earth. 

One of the most astonishing developments — to 
me at least — is that of the art of radio communi- 



119 



cation which makes it possible to transmit infor- 
mation over millions of miles of space. Pioneer V 
is being heard over 5 million miles away with 
only 5 w of power. Its 150-w transmitter should 
be heard out to 50 million miles — possibly to 
100 million if we get some sensitive new receivers 
going in time. Clearly, objects within a distance 
equal to the diameter of the earth's orbit can 
soon be listened to — out to a quarter of a billion 
miles perhaps. I wonder what we can do beyond 
that! The inverse-square law is a pretty imposing 
barrier. But the ingenuity of the electronic engi- 
neer is beyond calculation. (Incidentally, as an 
old-time worker in the field of photoelectricity, I 
take especial pleasure in watching the develop- 
ment of the solar cell. Without it we would be in 
real trouble. However, when Professor Hughes 
and I wrote our book on Photoelectric Phenomena, 
I regarded the photovoltaic cell as such a boring 
subject that I was glad to let him write that 
chapter. Solar cells flying in space did not occur 
to us as being an imminent necessity in 1931.) 

One of the most fascinating aspects of the 
space age is that it has given birth to a new 
science — space science. The only trouble is that 
no one is very clear about what space science is. 
Is it the study of the contents of space itself? If 
so, do we mean the space between the stars? The 
space between the planets? The space between 
the meteorites? The space between the hydrogen 
atoms? Or do we include everything? If we mean 
everything — then all the astronomers have been 
space scientists for 2000 years. And, if I judge 
correctly, many astronomers are a little disgusted 
with all the Johnny-come-latelys' who act as 
though they had discovered space — or even in- 
vented it. Or is space science the science you do 
with instruments that are in space? Thus, when 
you take pictures of the earth's clouds from a 
satellite, is that space science? Or is it still 
meteorology? When you are interested in the 
structure of the planet earth, you are a geologist. 
If you are interested in the moon, you are a 
selenologist (after Selene, the moon goddess). Is 
a selenologist a space scientist? Then why not a 
geologist too? If you are interested in Venus, then 
you have to look up the Greek word for Venus to 
find out what you are. And, since the Greek word 
for Venus is "Aphrodite," I still don't know what 



' At this point my secretary inserted the following note: 
"I suppose, it this is published, we should use 'Johnnies- 
come-lately,' although for oral delivery I much prefer the 
term you use — it has more style and zip and is more 
pleasing phonetically." 



to call a Venusian geologist. Maybe "space 
science" isn't such a bad term after all! 

All I hope is that we don't let the glamor of the 
term "space science" confuse us. There is a lot 
we can learn about the moon, for example, by 
just using lowly earthbound astronomical tele- 
scopes. Let's not be seduced into sending expedi- 
tions to the moon just to look for things we can 
see perfectly well from Palomar Mountain — or 
from Kitt Peak or Mt. Hamilton. 

Professor Bolton and Mr. Roberts and Mr. 
Radhakrishnan, of the Caltech Radio Astronomy 
Observatory, in just a few nights observing 
recently found that the radio radiation from 
Jupiter is partially polarized and that the 
polarized part appears to come from a belt which 
is separated from the planet's disk. In other 
words, they have probably observed synchrotron 
radiation from a Van Allen belt around Jupiter. 
That's space science for you — and achieved in a 
California desert at a cost far less than the cost 
of even a very small rocket ! 

On the other hand, the Pioneer V package has 
measured the earth's magnetic field out to nearly 
a million miles. Preliminar>' analysis shows that 
it appears to be a pretty good dipole field out to 
35 000 km, but beyond that shows small pertur- 
bations not yet analyzed. Here, clearly, is space 
science at its best — obtaining information avail- 
able in no other way. Pioneer V is also observ- 
ing charged-particle radiation far away from 
the earth's magnetic field — and has observed 
fluctuations which are correlated with distur- 
bances on the sun. And, of course. Pioneer V is 
at last obtaining data on the real primary cosmic 
radiation. We have heard some excellent papers 
on space physics at this very meeting of the 
American Physical Society. 

At last I believe the American people are 
beginning to realize that these are the real pur- 
poses of space research — to obtain scientific 
information. At last they are asking not just 
whether our satellites weigh more than the 
Russians', but whether they provide us with 
more information. We can be thankful that 
NASA did not yield to hysterical demands to 
perform useless stunts in space just to rival the 
Russians, but insisted on laying out a long-term 
program of space research. It's going to be a slow 
program and an expensive one. But, in the long 
run, solid scientific achievements will provide 
more national prestige than useless tricks. I 
believe even the Mercury man-in-space program, 
in spite of all the nauseating journalistic publicity 



120 



about the astronauts, has now been converted 
into a needed research program to study bio- 
logical problems which must be understood by 
the time sending men into space becomes a really 
useful scientific venture. 

Speaking of men in space, I am reminded of the 
recent television program on the population 
explosion in which a British economist calmly 
announced that rising population on earth would 
be no problem — we'll just ship the excess ofT 
into space! Now there is a concept to provide 
real merriment for your space discussions. I am 
told that excess population is piling up on earth 
at the rate of 45,000,000 people per year, or 
123,000 per day. What a passenger business 
that's going to be! The first colony will be on the 
moon, I suppose. But who is going to lay the 
pipeline to get oxygen up to them? And water? 
And what about food? And space suits? With a 
few million people on the moon, I wonder how 
many space suits will get punctured every day. 
(A punctured space suit in a perfect vacuum is 
a most unpleasant accident.) 



Fun in Space 

Every day! That reminds me — a day on the 
moon is 28 earth-days long. Sunshine for 336 
hours, then darkness for 336 hours. A sizzling 
temperature of 220°F by day and minus 220° at 
night. In view of all the trouble, I propose instead 
that we build a huge floating platform all over 
the Pacific Ocean and put our excess population 
there. It would have just as much area as the 
moon. And, if we include the rest of the oceans, 
it would have as much as Mars too. And it would 
be a lot cheaper. And at least the people would 
have air to breathe! 

Then we can save the moon for the people who 
ought to be there — physicists, chemists, biolo- 
gists, geologists, and astronomers. Then, I think 
the moon might be an interesting place to visit! 

Please forgive me for making jokes about a 
serious subject. My only hope is that by laughing 
at ourselves a little bit we may get back our sense 
of perspective. And a sense of perspective is 
important, no matter what problem we are 
dealing with. 



121 



In tracing the relation of science to other ports of 
modern life, Jacob Bronowski interviews on artist, 
Eduardo Paolozzi, an architect, Eero Saarinen, a 
physicist, Abdus Salam, and a writer, Lawrence 
Durrell . 



19 The Vision of Our Age 



J. Bronowski 



A chapter from his book, Insight-Ideas of Modern Science, 1964. 



This book began at the birth of a child, and traced 
its development until it enters 'the gateway to 
imagination and reason'. This is the stage when the 
child can manipulate objects in thought as well as 
with its hands: when it can make images of them. 
The child has little knowledge yet, in the ordinary 
sense of the word; but it has the mental equipment 
to learn and create knowledge. Once a child can make 
images, it can also reason, and build for itself a 
coherent picture of the world that is more than 
separate bundles of sense impressions. 

We have just seen that when a child enters 'the 
gateway to imagination', it leaves all animals be- 
hind. Before it learns to make images, a young 
human develops in much the same way as a young 
animal. Children and animals alike have to learn to 
co-ordinate their various senses and to recognise 
objects. But after that, animals fall behind. They have 
no power of imagination. That is, they cannot carry 
images in the mind; and without imagery, without 
an inner language, they cannot manipulate ideas. 

The theme of imagination runs through this book. 
We have examined some of the great achievements 
of science and seen that they are imaginative ideas. 
Science does not merely plod on like a surveyor, 
laboriously mapping a stretch of country, square 
mile by square mile. Of course nature must be sur- 
veyed, and very laborious that is at times; but the 
survey is not the end. The great moments in science 
come when men of imagination sit down and think 
about the findings — when they recreate the land- 
scape of nature under the survey. 

Science must be solidly grounded in fact and in 



experiment. But a blind search for experimental 
facts is not enough; it could never have discovered 
the theory of relativity. Science is a way of looking 
at things, an insight, a vision. And the theories of 
science are the underlying patterns that this way of 
looking at the world reveals. Many of the patterns 
are unexpected even at the simplest beginnings. (For 
example, common sense would not even have ex- 
pected to find that stars and human beings are put 
together from the same basic building bricks of 
matter.) And the more unexpected the pattern, the 
greater the feat of imagination that is needed to see 
it for the first time. 

What place have these imaginative ideas of science 
in our daily thoughts? Science and technology have 
transformed the physical world we live in; but have 
they yet had much effect on thought? Many people 
even dislike the ideas of science, and feel that they 
are abstract and mechanical. They reject science 
because they fear that it is in some way inhuman. 

This book shows that science is as much a creation 
of the human imagination as art is. Science and art 
are noc opposites; ihey spring from the same human 
impulses. In this last chapter, we shall examine their 
relations to one another, in the past and today. In 
particular, we shall see how both enter and combine 
into the way man in the twentieth century sees the 
world: the vision of our age. For this purpose, we 
shall include personal statements about their own 
work by an artist, an architect, a scientist, and a 
writer. 

The artist is the sculptor Eduardo Paolozzi. The ; 
group of pictures show him in his studio, then one 



122 



The Vision of Our Age 



of his sculptures being cast in the foundry, then" one 
of his finished sculptures called San Sebastian — with 
a jet engine standing in the background — and finally 
another recent work. 

This is what Eduardo Paolozzi had to say about 
his work and the world for which it is made. 

'I am a sculptor, which means that I make images. 
As a sculptor I was taught at the Slade the classical 
idea of being an artist. The best one could do would 
be to emulate Victorian ideals and to work in a 
studio executing portraits or monuments. 

'But there has been a rejection now of the class- 
ical idea of tracing art out of art, which is in a way 
a sort of death process leading to the provincial 
gallery, with the atmosphere of the death-watch 
beetle— a gilt-edged, sure-thing idea of art. 

'In this century we have found a new kind of 
freedom — an opening up of what is possible to the 
artist as well as to the scientist. So I don'^t make 
copies of conventional works of art. Tm not working 
for Aunt Maud; I'm trying to do things which have 
a meaning for us living today. So I work with 
objects which are casual and natural today, that is, 
mechanisms and throwaway objects. To me they are 
beautiful, as my children are beautiful, though in a 
different way. I think they are different definitions of 
beauty. 

'I haven't got any desire to make a sculpture of 
my children; but a wheel, a jet engine, a bit of a 
machine is beautiful, if one chooses to see it in that 
way. It's even more beautiful if one can improve it, 
by incorporating it in one's iconography. For in- 
stance, something like the jet engine is an exciting 
image if you're a sculptor. I think it can quite fairly 
sit in the mind as an art image as much as an 
Assyrian wine jar. I think it's a beautifully logical 
image, in the sense that anything in its delicate 
structure, with its high precision standards, has got 
a reason, almost in a way like human anatomy. 

'My San Sebastian was a sort of God I made out 
of my own necessity; a very beautiful young man 
being killed by arrows, which has a great deal of 
symbolism in it. I think this is a good thing for young 
artists to identify themselves with, in a way that 
doing the Madonna and Child may not be a thing 
they can identify themselves with. It has two legs, 
which are decorated columns, it has a rather open, 
symbolical square torso, with disguised, warped, 
twisted, mechanic elements. Then the final element 
is a sort of drum with a space cut in the middle. 

'What I feel about using the human diagram is 
that it points up in a more specific way the relation- 
ship between man and technology. There isn't any 
point in having a good idea in sculpture unless there 
is some kind of plastic or formal organisation. So 
I don't reproduce the jet engine, I transform it. And 
I use the wheel a lot in my sculpture as a symbol. 










123 




as a quickly read symbol, of the man-made object. 
This also refers back to my crude peasant idea of 
science, which is that the wheel gives the idea of man 
being able to get off the ground. The wheel to me is 
important, and the clock. I think this is very sig- 
nificant — I find the clock moving because I find 
modem science moving. I see it as a sort of heroic 
symbolism. 

'In the last fifty years, science seems to be the 
outstanding leading direction, the most considerable 
direction that man has taken. It is trying continually 
to go beyond what was possible till that very 
moment. I think there is a possibility in what I call, 
crudely, higher science, a tremendous possibility of 
man being free. And I think it can give me a certain 
kind of mora! strength, in the sense that art can 
move into a similar category of freedom. In my 
sculpture I am trying to speak for the way people 
are freeing themselves from traditional ideas. I'm a 
sculptor and so I put these ideas into images. If I 
do this well they'll be heroic images, ones that will 
survive and ones which other ages will recognise. 
Image making gives me the sense of freedom in a 
way that nothing else can.' 

A word to which Paolozzi returns several times is 
'free'. He feels that science frees man, from his 
conventions, from the restrictions of his environ- 
ment, from his own fears and self-doubts. If this is 
true, then man has gained this growing freedom by 
imagination: in science, by imagining things that have 
not yet happened. Paolozzi wants to communicate 
the same sense of growing freedom in the images of 
his sculpture. He wants people to feel that they are 
heroic images. 

Science and art are both imaginative activities, 
and they present two sides of the imagination. The 
two sides have often tried and often failed to come 
together, in the past and in recent time. This chapter 
itself, and this book, is an attempt to help bring 
them together. Paolozzi's work is also an attempt to 
bring them together, in a different language. He 
uses the everyday products of technology (the 
stamped shapes in the first picture, for example) as 
the raw material of his art, because they seem to him 
as natural and expressive in modern civilisation as 
the human body itself. 



It is interesting to look at the two sides of the 
human imagination in an earlier civilisation. We 
have evidence for them, long even before writing was 
invented. These paintings, in the caves of Lascaux in 
southern France, are at least twenty thousand years 
old. They are the most famous and the finest ex- 
amples of art from the Stone Age. The word 'art' 
is not out of place, and yet it is most unlikely that 
these pictures were created in the same spirit as 



124 



The Vision of Our Age 



classical art. The caves of Lascaux were not a Stone 
Age art gallery that people came to visit. Art of this 
kind was an integral part of the civilisation of 
Stone Age man. 

The Lascaux paintings are a product of one side 
of the imagination of the men who lived twenty 
thousand years ago. This picture shows a product of 
the other side of their imagination. It is a tool: a 
harpoon, cut from bone. It has barbs, like a modem 
fish hook, to stop it from being pulled out when it 
lodges in an animal. 




The next picture shows a tool again, and of a 
subtler kind. It does not look as impressive as the 
harpoon, yet it is in fact a more far-sighted invention. 
For it is a tool for making tools: it is a stone graver 
of the sort that must have been used to cut the barbs 




in the harpoon. The men who invented this were able 
to think beyond the immediate needs of the day — 
killing an animal, cutting it up, scraping its hide. 
When they invented a tool for making tools (today 
we should call that a machine-tool) they took a new 
step of the imagination. 

What is the link between paintings on the wall of 
a cave, and primitive tools made of bone and flint? 
Separated as we are by twenty thousand years from 
the men who created both, we can only speculate. 
But we are surely right in speculating that the paint- 





ings served some purpose other than mere decoration. 
Look at another Lascaux painting. It represents 
three bulls and (probably) a boar. A bull is being 
struck by a spear with barbs — a spear like the one 
that we have seen. This is plainly a hunting scene. 
Many of the other cave paintings show similar 
scenes. The painters were constantly preoccupied 
with hunting. This is why most authorities agree that 
the paintings were some kind of magic, and were 
intended to help the hunter to dominate the animal 
before the hunt started. 

Unhappily, 'magic' is one of those words 
('instinct' is another) that does not really explain 
anything. It merely says that we do not know the 
explanation. What kind of magic were the painters 
making? What did they feel they were doing for the 
hunters? How did they think that they were helping 
them to dominate the hunted animal? 

Here I will give my personal view. I think that the 
paintings helped the men who painted them, and 
the men who lived in the caves with them, to conquer 
their fear of the hunted animal. A bull was (and is) 
a dangerous beast, and out in the open there would 
not be much time to think about him. By drawing 
him you become familiar with him, get used to the 
idea of meeting and hunting him, and imagine ways 
in which he can be outwitted. The close-up makes 
the bull familiar to you; and the familiar is never 
as frightening as the unknown. 

It is not far-fetched here to draw an analogy with 
modem methods of training. Consider, for example, 
the training of spacemen. They have to face a 
frightening situation, in which what they fear is 
simply the unfamiliar and unknown. They will not 
survive if they panic; they will do the wrong thing. 
So a long and life-like training programme is de- 
signed to make them familiar in advance with every 
situation that they are likely to encounter. The 
spaceman's training is more than a matter of simply 
learning to press the right buttons. It is also a 
psychological preparation for the unknown. 

I believe that the Stone Age cave paintings were 
also a psychological preparation for the unknown. 



125 



They helped the Stone Age hunters to dominate 
their psychological environment, just as flint and 
bone tools helped them to dominate their physical 
environment. That is the connecting link between 
the two. Both are tools, that is, instruments which 
man uses to free himself and to overcome the 
limitations of nature. It was Benjamin Franklin who 
first defined man as 'the tool-making animal'. He 
was right, and the tools are mental as well as 
physical. 



We move forward now many thousand years, to a 
time and place where the two sides of the human 
imagination worked more closely together than ever 
before, and perhaps ever since. The pictures on the 
right come from Athens of the fifth century B.C. 
The men who built this city had suddenly burst out 
of the confines of the cave and come into the light 
of freedom. Their civilisation recognised that man's 
most powerful tool in the command of nature is the 
human mind. The Greeks named their city, and the 
great temple of the Parthenon in it, after the goddess 
of wisdom, Athene. Light and reason, logic and 
imagination together dominated their civilisation. 

Greek architecture, for example, has a strong 
mathematical basis, yet it never appears stiff and 
mechanical. Look at the Parthenon, as perfect a 
creation in architecture as man has made; and it is 
dominated by a precise sense of numbers. Numbers 
had a mystical significance for the Greeks (Pythag- 
oras made them almost into a religion) and this 
expressed itself in all they did. 

The Parthenon has 8 columns along the front and 
17 along each side. That to the Greeks was the ideal 
proportion. The number of columns along each side 
of a temple should be twice the number along the 
front, plus one more. No Greek architect would 
have built otherwise. 

Numbers that are perfect squares seemed to the 
Greeks equally fascinating and beautiful. The Par- 
thenon is 4 units wide and 9 units long; for 4 is the 
square of 2, and 9 is the square of 3 — the two 
smallest squares. The ratio of height to width along 
the front of the building is also 4 to 9; and so is the 
ratio of the thickness of the columns to the distance 
between them. 

Yet all this arithmetic is not a dead ritual. The 
Greeks found it exciting because they found it in 
natural objects. To them, it expressed the mystery 
of nature, her inner structure. Numbers were a key 
to the way the world is put together: this was the 
belief that inspired their science and their art 
together. 

So the Parthenon is nowhere merely a set of 
mathematical relations. The architect is guided by 
the numbers, but he is never hidebound by them. 





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126 



The Vision of Our Age 



His plan begins with arithmetic, but after that the 
architect himself has taken command of the building, 
and has given it freedom, lightness and rhythm. For 
example, the end columns are closer to their neigh- 
bours than are the other columns; and the end 
columns are also a little thicker. This is to make the 
building compact, to make it seem to look inwards 
at the corners. And all the columns lean slightly 
inwards, in order to give the eye (and therefore the 
building) a feeling of upward movement and of 
lightness. 

The pictures on the right are of the Erictheum. It 
stands close to the Parthenon, but is less famous. 
Perhaps that is because the Erictheum is less monu- 
mental, more slender, more delicate in its whole 
conception. Yet the mathematics is still there. The 
porch of the Erictheum, for instance, is designed on 
the 'golden section'. That is, the canopy has the same 
proportion to the base as the base has to the human 
figures which support the canopy. The golden section 
was a mathematical relation which was based on 
nature: on the proportions of the human body. 

The human figures which support the canopy are 
made to seem in movement; two rest on the right 
foot, two on the left. Everywhere in the Erictheum 
there is the feeling of movement. The different levels 
of the building are joined together with suppleness 
and rhythm. This is what the Erictheum expresses in 
architecture: an almost musical sense of rhythm. And 
this reminds us that Pythagoras prided himself, right- 
ly, on having discovered the mathematical structure 
of the musical scale. 

The fusion of the mathematical order with the 
human, of reason with imagination, was the triumph 
of Greek civilisation. The artists accepted the math- 
ematics, and the mathematicians did not resent the 
architects imposing their individuality on the math- 
ematical framework. It was a civilisation which 
expressed itself in the way things were put together — 
buildings, ideas, society itself. Greek architecture 
survives to illustrate this, perhaps better than any 
other record. 

All architecture must begin with technical effi- 
ciency. Walls have to stand up, roofs have to keep 
the rain out. So an architect can never be unpractical, 
as can a painter or a sculptor. He cannot be content 
with the mere look of the thing. The side of the 
human imagination which made the Stone Age tools 
cannot be left out. But a bad architect can play it 
down, and can take the practical for granted, as a 
painter takes his canvas for granted. 

The strength of the best architecture today is that 
it does not despise the practical purposes of build- 
ings. It does not hide the structure and function 
under merely elegant decoration. Structure and func- 
tion in modern buildings play the same fundamental 
part as numbers in Greek architecture. They form 








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a.r 



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127 



the framework on which the architect imposes his 
individual imagination. And he does not pretend 
that the framework is not there. 



Our next personal statement comes from a famous 
architect, Eero Saarinen. He was born in Finland but 
built most of his great buildings in America. The 
pictures below show the building that he did 
not live to finish, the TWA Air Terminal at Idlewild 
Airport in New York. The lines of the building are 
very dramatic, and the form is consciously mathe- 
matical and aerodynamic. The question is: Is the 
bold, flying shape necessary, or is it a romantic 
artifice without a true function? I discussed this with 
Eero Saarinen during the building, and this is how 
he replied. 

To really answer your question, I would have to 
go a little bit back, and talk philosophically about 
architecture. As you know, we all, in architecture, 
have been working in this modern style, and certain 
principles have grown up within it. The basic prin- 
ciples are really three. There is the functional part. 
There is the structural part, honestly expressing the 
structure of the building. And the third thing is that 
the building must be an expression of our time. In 




other words, the technology of our time must be 
expressed in a building. 

'Now those are the principles that we are all 
agreed on — the principles that one might have said 
ten years ago were the only principles. I think since 
that time more thought and maybe some more 
principles have grown up. I would say one of these 
additional principles, one which I believe in, is that 
where buildings have a truly significant purpose they 
should also express that purpose.' 

Function and purpose were not the same thing in 
Saarinen's mind. The TWA Air Terminal has a clear 
function: to handle passengers into and out of 
aeroplanes. But for Saarinen, it also had a deeper 
purpose: from here people were to fly, and he wanted 
to give them the sense of freedom and adventure 
which flying has for earth-bound men. The vaulted 
shapes of the building were well-conceived as struc- 
tures, but they were meant to be more: their aero- 
dynamic and birdlike look was to express what 
Saarinen called the purpose — the sense of going off" 
to fly. And the long spurs reaching out from the 
building show that it is not something self-contained, 
an end-point. They suggest entering the building and 
leaving it, which is of course what the passengers do. 

Eero Saarinen went on: 

'The last thing that I've become convinced of, and 
I'm not the only one, there are many others, is that 
once you've set the design, it must create an archi- 
tectural unity. The idea of the barrel vaults making 
the roof of the Air Terminal building is carried 
through in all the details, even the furnishings. 

'Basically architecture is an art, though it is half- 
way between an art and a science. In a way it 
straddles the two. I think to a large degree the 
motivating force in the designing of architecture 
comes from the arts side. If you ask. Are these curves 
and everything derived from mathematics? the an- 
swer is No. They are sympathetic with the forces 
within the vaults, which is mathematical, but there 
are so many choices which one has, and these really 
come from the aesthetic side. 

'To me architecture is terribly important because 
it is really an expression of the whole age. After 
we're dead and gone, we're going to be judged by 
our architecture, by the cities we leave behind us, 
just as other times have been. What man does with 
architecture in his own time gives him belief in him- 
self and in the whole period. Architecture is not just 
a servant of society, in a sense it's a leader of 
society.' 

Architecture straddles art and science. That state- 
ment is true of the Greek architecture of two thous- 
and years ago as well as of the architecture of today. 
In this, the Greek imagination is close to our own. 
The Greeks were preoccupied with the idea of struc- 
ture; and we have seen in this book that the idea of 



128 



The Vision of Our Age 



structure is also central to modern science. Like the 
Greeks, the modern scientist is always looking at the 
way things are put together, the bones beneath the 
skin. How often in this book have we used such 
phrases as 'the architecture of matter' ! 

For example, the Greeks invented the idea of the 
atom as the smallest unit of matter from which 
everything in the world is built. Plato thought there 
were five kinds of atom, and he pictured them as the 
five regular solids of geometry. The first four were 
the atoms of the four kinds of matter: earth, air, 
fire, and water; one of these is shown in the first 
picture below. The fifth was the universe itself, the 
unity of the other four— we still call it the quintessence; 
it is shown, as Plato imagined it, in the second pic- 
ture. 

This conception is fantastic, and the atoms it 
pictures have no relation to the facts. And yet the 
fanciful pictures are a first attempt to solve, imagina- 
tively, the same problems of structure and behaviour 
that the modern physicist faces. The Greek concep- 
tion and the modern theories about atoms are both 
attempts to explain the bewildering complexity of 
the observable world in terms of an underlying, 
unifying order. Greek scientific theories are now only 
of historical interest. Yet before the Greeks, no one 
had thought about the world in this way at all. 
Without them, there would have been no modern 
science. It was the Greeks who first formulated the 
problems that modern science tries to answer. 



Our third personal statement comes from a physi- 
cist: Professor Abdus Salam, of the Imperial College 
of Science in London. He describes some modern 
ideas about atoms. They are a long way from Plato's 
regular solids; yet, as Professor Salam points out, 
that is where they started. Here is what Salam said. 

'I am a theoretical physicist, and we theoretical 
physicists are engaged on the following problem. 
We would like to understand the entire complexity 
of inanimate matter in terms of as few fundamental 
concepts as possible. This is not a new quest. It's the 
quest which humanity has had from the beginning 
of time — the Greeks were engaged on it. They con- 
ceived of all matter as being made up of fire, water, 
earth and air. The Arabs had their ideas about it 
too. Scientists have been worried about this all 
through the centuries. The nearest man came to 
solving this problem was in 1931 when, through the 
work done in the Cavendish Laboratory in Cam- 
bridge, we believed that all matter consisted of just 
two particles — electrons and protons — and all forces 
of nature were essentially of two kinds, the gravita- 
tional force and the electrical force. 

'Now we know that this view of 1931 was erron- 



eous. Since that time the number of particles has 
increased to thirty, and the number of elementary 
forces to four. In addition to the electrical and 
gravitational forces, we now believe that there are 
two other types of force, both nuclear — one extremely 
strong, and the other extremely weak. And the task 
we are engaged on is to try to reduce this seeming 
complexity to something which is simple and 
elementary. 




'Now the type of magic which we use in order to 
solve our problem is first to rely on the language 
which we use throwing up ideas of its own. The 
language which we use in our subject is the language 
of mathematics, and the best example of the language 
throwing up ideas is the work of Dirac in 1928, He 
started with the idea that he would like to combine 
the theory of relativity and the theory of quantum 
mechanics. He proceeded to do this by writing a 
mathematical equation, which he solved. And to his 




astonishment, and to everyone's astonishment, it was 
found that this equation described not only the part- 
icles — electrons and protons — which Dirac had de- 
signed the equation for, but also particles of so-called 
anti-matter — anti-electrons, anti-protons. 

'So in one stroke Dirac had increased the number 
of particles to twice the number. There are the 
particles of matter, there are the particles of anti- 
matter. In a sense, of course, this produces simplicity 
too, because when I speak of thirty particles, really 



129 



fifteen of them are particles and fifteen of them are 
anti-particles. The power of mathematics as a lang- 
uage that suggests and leads you on to something, 
which we in theoretical physics are very familiar with, 
reminds me of the association of ideas which follows 
when possibly a great poet is composing poetry. He 
has a certain rhyme, and the rhyme itself suggests 
the next idea, and so on. That is one type of way in 
which invention comes about. 

'The second type of idea which we use to solve 
our problems is the idea of making a physical picture. 
A very good illustration is the work of the Japanese 
physicist Yukawa in 1935. Yukawa started to ponder 
on the problem of the attractive force between two 
protons, and he started with the following picture. 
Suppose there are two cricketers, who have a cricket 
ball, and they decide to exchange the ball. One 
throws the ball and the other catches it, perhaps. 
Suppose they want to go on exchanging the ball, 
to and fro, between them. Then the fact that they 
must go on exchanging the ball means that they 
must keep within a certain distance of each other. 

'The result is the following picture. If one proton 
emits something which is captured by the second 
one, and the second one emits something which is 
captured by the first one, then the fact that they have 
to capture, emit, re-absorb constantly means that 
they will remain within a certain distance of each 
other. And someone who cannot see this inter- 
mediate object, this ball, the object we call the meson, 
will think that these two protons have an attractive 
force between them. This was Yukawa's way of 
explaining the attractive force between two elemen- 
tary particles. 

'The result of Yukawa's work was that he pre- 
dicted that there do exist such particles which play 
the role of intermediate objects. And he predicted 
that such particles would have a mass about three 
hundred times that of electrons. Yukawa made this 
prediction in 1935. In 1938 these particles were 
discovered, and we now firmly believe that the forces 
of nature, all forces of nature, are transmitted by 
this type of exchange of intermediary particles. 

'Now so far I have been talking about our 
methods, but what is really important are our aims. 
Our aim in all this is to reduce the complexity of the 
thirty elementary particles and the four fundamental 
forces into something which is simple and beautiful. 
And to do this what we shall most certainly need is a 
break from the type of ideas which I have expressed 
— a complete break from the past, and a new and 
audacious idea of the type which Einstein had at the 
beginning of this century. An idea of this type comes 
perhaps once in a century, but that is the sort of 
thing which will be needed before this complexity is 
reduced to something simple.' 



The ideas put forward by Salam are vivid. But 
more than the specific ideas, we are interested here in 
his description of science itself. For him, science is 
the attempt to find in the complexity of nature some- 
thing which is simple and beautiful. This is quite 
different from the usual view that science collects 
facts and uses them to make machines and gadgets. 
Salam sees science as a truly imaginative activity, 
with a poetic language of its own. This is an arresting 
point that Salam made: that the mathematics in 
science is a poetic language, because it spontaneously 
throws up new images, new ideas. 



Science can learn from the language of poetry, and 
literature can learn from the language of science. 
Here we bring in our fourth contributor. He is 
Lawrence Durrell, who wrote the four famous books 
which make up The Alexandria Quartet. In this four- 
fold novel, space and time are treated in an unusual 
way, and Durrell began by talking about this 




T was hunting for a form which I thought might 
deliver us from the serial novel, and in playing 
around with the notions of relativity it seemed to me 
that if Einstein were right some very curious by- 
products of his idea would emerge. For example, that 
truth was no longer absolute, as it was to the 
Victorians, but was very provisional and very much 
subject to the observer's view. 

'And while I felt that many writers had been 
questing around to find a new form, I think they 
hadn't succeeded. I don't know of course, I've only 
read deeply in French. There may well be Russian 
or German novels which express this far better than 
I have. 

'But they hadn't expressed what I think Einstein 
would call the 'discontinuity' of our existence, in the 
sense that we no longer live (if his reality is right) 
serially, historically, from youth to middle age, to 
death; but in every second of our lives is packed, in 
capsule form, a sort of summation of the whole. 



130 



The Vision of Our Age 



That's one of the by-products of relativity that I got. 
'In questing around for a means of actually pre- 
senting this in such an unfamiliar form as a novel, 
1 borrowed a sort of analogy, perhaps falsely, from 
the movie camera. I'd been working with one, and 
it seemed to me that when the camera traverses 
across a field and does a pan shot, it's a historic shot 
in the sense that it goes from A to B to C to D. And 
if it starts with a fingernail and backtracks until you 
get a whole battlefield, that seemed to me a spa- 
tialisation. It was rooted in the time sequence that 
it was spatialising; it was still enlarging spatially. 

'I tried to mix these two elements together, and 
see what would happen to ordinary human charac- 
ters in what is after all a perfectly old-fashioned type 
of novel — an ordinary novel, only not serial. I found, 
somewhat to my own surprise, that I was getting a 
kind of stereoscopic narrative, and getting a kind of 
stereophonic notion of character. This excited me so 
much that I finished it and tried to add the dimension 
of time by moving the whole thing forward — you 
know, "read our next issue" — five years later. And 
there it is, ready for the critics to play with.' 

Here are Lawrence Durrell's answers to some 
questions about his work: 

Q. You said that you got from relativity the feeling 
that truth was provisional, or at least depended 
very much on the observer. 
A. Well, the analogy again is the observer's position 
in time and space. It's so to speak the fulcrum 
out of which his observation grows, and in that 
sense it is not an absolute view, it's provisional. 
The subject matter is conditioned by the ob- 
server's point of view. 
Q. You're really making the point that the most 
important thing that relativity says is that there 
are no absolutes? 
A. I was saying, most important for me. I think that 
any average person who's not a mathematician 
would assume that that was probably the most 
important part of it. 
Q. I want to recall another phrase that you've just 
used. You said of your novel that 'after all 
it's a perfectly old-fashioned novel'. Now I don't 
feel that. I feel that your novel could have been 
written at no time but in the twentieth century. 
A. Yes, in that sense certainly. But I was trying to 
distinguish between the form which, I believe, if 
it has come off at all, is original, and the content. 
When I was building the form I did something 
new. I said to myself, this is the shape: there are 
three sides of space, one of time. How do I shift 
this notion into such an unusual domain as the 
novel? And at the back of my mind I wondered 
whether we in the novel couldn't escape our 
obsession with time only. 



Q. Your dimensions, as it were, deepen out each 
character as a recession in space. You show how 
different he becomes when he is seen by someone 
else from another point. 

A. Stereoscopically, you see. 

Q. I want to ask you a crucial question. Do you feel 
that the kind of inspiration that you've drawn 
from the scientific idea of relativity here is valid 
for everyone? That we can all in some way make 
a culture which combines science and the arts? 

A. Surely a balanced culture must do that. And I 
think all the big cultures of the past have never 
made very rigid distinctions. Also I think that 
the very great artists, the sort of universal men, 
Goethe for example, are as much scientists as 
artists. When Goethe wasn't writing poetry he 
was nourishing himself on science. 

Q. We can't expect everybody to be a Goethe, so 
how are we going to unify what is obviously 
different — the sense of what the artist is doing 
and the sense of what the scientist is doing? 

A. I think by understanding that in every generation 
the creative part of the population feels called 
upon to try and attack this mysterious riddle of 
what we're doing, and to give some account of 
themselves. We're up against a dualism, because 
some people have more intelligence and less 
emotion, and vice versa. So the sort of account 
they give may suddenly come out in a big poem 
like Dante's, or it may come out in a Newtonian 
concept. In other words, the palm isn't equally 
given in each generation. But I feel that they're 
linked hand in hand in this attack on what the 
meaning of it all is. 

The meaning of it all: the meaning of the pattern 
of nature, and of man's place in nature. Durrell's 
quest is also Salam's quest, and Saarinen's, and 
Paolozzi's. It is the quest of every man, whether 
scientist or artist or man in the street. 

The driving force in man is the search for freedom 
from the limitations which nature has imposed. Man, 
unlike the animals, is able to free himself. The first 
crude attempts were already made by Stone Age 
man with his tools and paintings. Now, twenty 
thousand years later, we are still struggling for free- 
dom. We try to reach it by understanding the mean- 
ing of things. Our age tries to see things from the 
inside, and to find the structure, the architecture 
which underlies the surface appearance of things. 
We command nature by understanding her logic. 

Our age has found some unexpected turns in the 
logic of nature. How atoms evolve, much like living 
species. How living things code and pass on their 
pattern of life, much like a machine. How the 
rigorous laws of nature are averaged from the million 



131 



uncertainties of atoms and individuals. How time 
itself is an averaging and a disordering, a steady loss 
of the exceptional. 

How life opposes time by constantly re-creating 
the exceptional. And how profoundly our ideas of 
so safe and absolute a concept as time once seemed 
to be can be changed by the vision of one man, who 
saw and proved that time is relative. 

Above all, our age has shown how these ideas, and 
all human ideas, are created by one human gift: 
imagination. We leave the animals behind because 
they have no language of images. Imagination is the 
gift by which man creates a vision of the world. 

We in the twentieth century have a vision which 
unifies not only the physical world but the world of 



living things and the world of the mind. We have a 
much greater sense of person than any other age. 
We are more free than our ancestors from the 
limitations both of our physical and of our psycho- 
logical environments. 

We are persons in our own right as no-one was 
before us. It is not only that we can travel into space 
and under the oceans. Nor is it only that psychology 
has made us more at home with ourselves. It is a 
real sense of unity with nature. We see nature not 
as a thing but as a process, profound and beautiful; 
and we see it from the inside. We belong to it. This 
above all is what science has given us: the vision of 
our age. 



132 



In this chapter from her book. The Making of a Scientist, Anne 
Roe, on eminent clinical psychologist, reports on her interviews 
with several men who became physicists. As these scientists 
individually describe their family backgrounds, the interests 
and activities of their youth, and their education, it becomes 
clear that there is no single pattern. 

20 Becoming a Physicist 

Anne Roe 

An excerpt from her book The Making of a Scientist. 1952. 

Here are the stories of several of the men who became physi- 
cists. Since the theorists and experimentahsts are quite unhke 
in some ways, I shall include both. Again it is true that some 
of them knew quite early that the physical sciences were a 
vocational possibility, and others did not hear of them in such 
a connection until well along in school. You can know that 
there is a school subject called physics, and men who teach 
it, and you probably will have learned that there have been 
famous men called physicists, who found out certain things 
about the world, but this is very diflPerent from realizing that 
you can make a living at finding out things in this field. 

Martin was the son of a consulting engineer, who had had 
some college training. His mother had worked as a reporter 
for a while after she finished high school. He says, 

"I can't remember much about grade school except the fact 
that I got reasonably decent grades right along and that I was 
fairly interested in science and mathematics. I had a friend 
in 7th or 8th grade who was the son of a druggist and we got 
a chemistry set between us and played around with it and 
almost blew up the house. We spent our spare time memoriz- 
ing the table of elements. I never got along in languages, I 
couldn't see any sense in memorizing grammar. In history I 
read so much I had many more facts than the rest whether 
they were right or not. I think probably the interest in science 
was partly because of father. When he was home he liked to 
do shop work and I used to do some with him. He was rather 
meticulous and in some ways this was discouraging for a be- 
ginner." 

Several things about this statement are very characteristic 
of theoretical physical scientists. All of them liked school. 



133 



Most of them preferred mathematics and science to other sub- 
jects. A number of them spoke of dabbhng in chemistry, and 
of still being surprised that they had not blown up the house, 
and many of them did other sorts of things with their hands, 
such as the shop work mentioned by Martin. His mention of 
memorizing the table of elements reminds me of another of 
this group who became interested in mineralogy when he was 
a boy and who papered his room with sheets of paper on which 
he had copied tables and descriptions of minerals. 

Martin goes on to say, 

"I was rather sickly. I imagine it was more allergic than 
anything else, although it was not recognized at the time, and 
I was out sick two or three months each year. One term in 
high school I was only there for a month. It was always some- 
thing special; my brothers and sisters always had measles and 
things like that but those never bothered me. I had tonsils 
and adenoids, hay fever, a mastoid, and appendicitis. This 
meant that during most of the winter months I didn't get out 
and I got to reading fairly early. Since I was in the 8th grade 
I've been in the habit of reading 4 books or more a week. I 
read pretty much anything. If Tm working hard in physics 
I like to relax by reading history or almost anything but phys- 
ics. One spell in high school, when I was sick for three months, 
I decided I was going to go into history and I spent the time 
in drawing up a historical chart beginning with the Egyp- 
tians." 

His frequent illnesses, and his omnivorous reading are also 
characteristic of this group. There were only three who had 
had no serious physical problems during childhood, and all 
of them read intensely and almost anything they could get 
their hands on. Two of them remarked that they thought they 
got their first interest in science from reading science fiction. 
Reading, of course, is not a very social occupation, and the 
physicists, like the biologists, rather tended to be quite shy. 
Martin, however, is unlike the others in that he got over this 
rather suddenly, although not very early. 

"I did very little going out in high school. Mother was very 
worried about it. I felt very shy. I started in my junior year 
in college and all of a sudden found it interesting and easy 
and rather overdid it for a while. Let's see if I can remember 
how it happened. I just happened to get in with a group of 
fellows and girls who were interested in artistic things. I 



134 



Becoming a Physicist 



started going to the symphony concerts at that time and we 
got in the habit of going Saturdays to Little Italy and sitting 
around and drinking wine and talking. Since that time it's 
been a thing I could turn on or off at will. There were a num- 
ber of periods before my marriage that I did a lot of running 
around and other times I'd be too interested in something 
else. I've always been self-conscious at social functions and 
never cared very much for them. With a few people it's differ- 
ent." 

In high school one of the teachers had great influence on 
him, and this experience oriented him towards science at the 
same time that out of school experiences convinced him that 
he did not want to be a business man. Not all of these men 
had occasion to spend any time in commercial activities, but 
quite a few of them did, usually iij the course of making 
enough money to go to school. None of them liked business 
except one of the biologists who found it of interest but was 
glad to go back to science. The extreme competitiveness, the 
indifference to fact, the difficulty of doing things personally, 
all were distasteful to them. 

"The first few years in high school I don't remember any- 
thing special about, except that I managed to get fairly de- 
cent grades in mathematics. I took physics and didn't like it. 
I had taken chemistry before I got there, but there was an 
extra course that sounded interesting so I took it and it turned 
out there were only four students in the course and a very 
interesting teacher. He sort of took personal charge and let 
us do pretty much what we wanted except that he was ex- 
tremely insistent that we take care and do a good job. We 
worked through all of analytical chemistry there and I got a 
feeling for looking for small traces of elements, etc. This con- 
vinced me that I wanted to be a chemist. A little earlier I had 
gotten a job with the phone company which was with a fellow 
studying to be a chemist. I read Slosson, Creative Chemistry. 
This was the romantic thing to be. I think that teacher had 
more individual influence on me than any other." 

Some firm, apparently interested in increasing the supply 
of chemists, had sent Creative Chemistry around to a num- 
ber of high schools, and it seems to have been a very successful 
promotion. At least several others of my subjects mentioned 
having been influenced by it. 



135 



"when I was still in high school I took a job one summer 
at a Yacht Club. It was a navy camp and one of the instructors 
had been a radio operator. He got me interested in radio and 
we played around a certain amount. That winter he and two 
other radio amateurs decided to open a small radio equip- 
ment store in town and they asked me to go in. Perhaps they 
thought father might help. Dad did put up some money and 
we opened a small store and for a while I spent part time 
there. When the craze hit in 1922 or 1923 the place was about 
swamped, it was the only store in town. What was made on 
the store pretty much paid my way through college. While 
this episode was interesting I was pretty sure I didn't want 
to go into business. You always got essentially people fight- 
ing you. During part of this time in addition to working at the 
store I had been a part-time radio writer for one of the papers. 
While that was interesting, too, it didn't appeal as a life work 
either. By then I was convinced I wanted to go on in academic 
work. 

"College was actually pretty much taken for granted. My 
mother was convinced from the beginning that all her chil- 
dren were going to college. I just went to college expecting to 
be a chemist. I had no very special idea about it. Two things 
happened in my freshman year. I took the college chemistry 
course plus the lab course. The lab course threw me for a com- 
plete loss. I think it was taught by a poor teacher who was 
careless of the reagents and they weren't pure. I got traces of 
everything and reported it. I didn't like the way the course 
was taught because I was told everything I was supposed to 
do and it soured me on chemistry. 

"I got acquainted with a young man who had just come 
there as an astronomer and was teaching mathematics. He 
was perhaps the most inspiring teacher I had. He let you go 
if you wanted to go. I needed some money so I helped arrange 
the library and so I had a chance to look over the mathematics 
books. At the end of the year I decided the devil with chem- 
istry, I'm going into physics. 

"At that time the college had a course in physics which was 
not popular. My class had three students and this gave us per- 
sonal attention. I thought of going on with it. My father was 
very dubious about it. He wasn't sure that physics was a thing 
you could get along with but he didn't push it very hard. He 
talked to me about it once and said, 'You will have to go on 



136 



Becoming a Physicist 



in university work and won't make any money.' I said I knew 
that and he said If you reahze it, that's all right.' There was 
nothing special about the course except at the end of that 
year a prize examination was given. At that time physics was 
taught practically everywhere without the use of calculus 
and still is in many places. We didn't get calculus until our 
sophomore year in mathematics and I still can remember the 
annoyance and the feeling of being cheated out of an extra 
year or so of activity by not having had it earlier. At any rate 
the physics course was given with the calculus but didn't use 
it. So about the middle of the second term I got disgusted and 
decided I wanted to learn physics the right way and asked 
the teacher for a text. He smiled and gave me one and I studied 
that so when the exam came along I gave it all in calculus 
and got the prize. This confirmed me, of course, and the next 
two years were extremely pleasant. I divided my time pretty 
much between astronomy and physics. There were just three 
of us and we'd go to the professor and say we had finished up 
this and what should we do next and he would say, 'What 
do you want to do?' So we'd tell him and he would give us 
manuals and get the old apparatus out and usually it would 
have to be cleaned and fixed up, and he would tell us to work 
it up and we would have a fine time. 

"My teacher felt I should go on to do graduate work. This 
was kind of a surprise to the family and a little bit of a worry 
because my brothers and sisters were coming along and there 
wasn't too much money. But I applied for scholarships at 
three places and took the second oflFer. My main danger the 
first year was to keep from galloping oflF in 24 different direc- 
tions at once. I found it extremely interesting and exciting. I 
started work on an experimental problem, but then I would 
get an idea for a theoretical paper and work on that for a while, 
and then go back to the other. 

"I think my teacher in high school had given me a few 
nudges in the direction of research. Both the professors at 
college wdth whom I was in close personal contact and saw 
daily were active in research themselves and I just soaked 
that stuff up. I find it hard to think back to the time when the 
idea of research and just spending all the time I had available 
on trying to understand anything wasn't just there." 



137 



The story of George, who became an experimental physi- 
cist is quite a different one, but it is fairly characteristic (rf 
the experimentalists. He did some manual things as farm boys 
do, but was not particularly interested, and he did not have 
radio sets and gadgets of one sort or another. Farm boys didn't 
then. Nor did he do any particular amount of reading. So far 
as he knows none of his family had gone to college before 
him, although some have gone since; his father had had about 
a 6th grade education and his mother one year of high school. 
He started out in the usual 7 months country school, near 
home, but his going on was unusual. He says, 

"My father and mother were rather an exception in the 
community which can be pointed out in this way. We lived 
out in the country about 7 or 8 miles from a high school. The 
country school to which we went was very close but when I 
finished seventh grade the school was having its usual ups and 
downs and the high school was no good. So my father and 
mother decided to send me to another school and it required 
boarding me away from home, and that was quite the talk of 
the area, that they would waste money boarding me. 

"My recreations were the usual ones, physical activities. 
Whereas most parents in that neighborhood believed that 
children when not in school should work along with the hired 
help, both father and mother adopted the attitude that they 
expected me to do a certain amount of work but didn't care 
when I did it. They would lay out a certain amount per day 
and if I wanted to get up and work hard and be through with 
it that was up to me. That was always criticized because I 
was always enticing the other boys away when they were 
supposed to be at work. I earned the title of being one of the 
laziest boys. Father required only that I do my work and do 
it well. He did this with the other help as far as possible, too, 
like piece work. From that I learned how to make time on 
manual things and at the same time to do as well as required. 
But we had no tools and I did no carpentering. Up until I 
went to graduate school I never knew I had any ability in 
that respect at all. I didn't do a great deal of reading. In those 
days the books that were available were novels and I wasn't 
particularly interested. 

"I think I wanted to go to high school. At least I was per- 
fectly willing to go. It came rather suddenly. I don't think 



138 



Becoming a Physicist 



very much was said about it until possibly a few days before 
I went. I suspect my mother had more to do with it, she had 
thought it out very well, but I don't think she said much even 
to father. His reaction was that as long as I did well he'd help 
me go to school. If I failed I could come home and work. He 
always thought farming too hard for anyone and that anyone 
who had intelligence would get oflF it. The first year or so was 
pretty rugged. It was difficult to find a satisfactory place to 
stay. We had one little course in physics in high school, not 
a lab course, and the usual mathematics. I think I was proba- 
bly the top of the class in that. 

"There was an incident there that has always been amus- 
ing to me. The only time I had any trouble in school was with 
the physics teacher. About the middle of the year she was 
showing how the water level in the boilers was determined. 
She left the gauge open and I said all the water would go out. 
The argument got hotter and hotter and finally I volunteered 
to show her, at which time I got thrown out of class. There 
again it was what father always said, you have to think things 
out for yourself." 

This is the sort of incident that can happen when a teacher 
( or parent ) is so insecure as to be unable to tolerate the sug- 
gestion that she might be mistaken, or might lack some par- 
ticular piece of knowledge. 

The experimentalists are like the theorists in their early 
preference for mathematics and science classes, and their dis- 
interest in languages, and difiiculty with them is somewhat 
greater than that shown by the theorists. Very few of the ex- 
perimentalists were avid readers. The teachers at George's 
school were all college graduates, and the principal talked a 
good deal about going on to college. George was early deter- 
mined to go. He liked school work, he did not like farming, 
and he had some idea of going into medicine. He tells how 
he happened to think of this. 

"I started out for medicine. Along about the time I was 14, 
there was a young doctor came to the community and he 
boarded in my home. I used to drive a car for him and I got 
rather interested. My real interest got started from an inci- 
dent one afternoon when a colored child had gotten badly 
burnt. Neither parent could hold the child and a neighbor 
couldn't do it either so he came out to the car and asked me 
if I thought I could hold the child and give it ether. It was 



139 



badly burned. Apparently I succeeded because that night he 
told my mother she had a young surgeon in the family. Maybe 
that started it, but when I went to college I intended to go 
into medicine. 

"I went to the nearest college. The medicine idea shifted 
gradually. Two things happened, I think, that caused a shift. 
One was that by pure accident, in the first year mathematics 
course I was lucky to be in the section of an exceedingly good 
teacher. I always liked to be in the back of the room if I could. 
It seemed that during the first week this professor would start 
asking questions and begin at the front end, and by the time 
it came back to me I would have been able to get the answer, 
from the book or by working it out. Then he began another 
trick, if he didn't get the answer on the first three or four he 
would say, 'How about my old standby?' and call on me so 
I felt I had to know it. From that he began to take quite an 
interest in my work and before the year was out began talk- 
ing about my working up the second year for myself during 
the summer. So I promised I'd try and he said he'd give me 
an examination in the fall and then I could go into the third 
year which he taught. I never have known if I passed it or 
if he let me by, but I went on with him. He wanted me to spe- 
cialize in mathematics, and along with that there happened 
another incident. 

"I had become engaged to my wife and she wasn't keen 
about being a doctor's wife and undoubtedly that had an in- 
fluence on me. She wanted her husband at home a reasonable 
amount of the time. As it turned out, especially during the 
war, that isn't just what she got. So I gradually drifted in 
the direction of mathematics. The second summer I worked 
up some other courses and at the end of the third year had 
completed four years of mathematics. Along with it I took one 
course in physics but I wasn't particularly interested, and I 
had one year of chemistry. The last year I found all I lacked 
for a B.A. instead of a B.S. which wasn't considered as good a 
degree, was a year of Greek so I took that. It was a kind of 
training that to my mind is lacking today. I even wound up 
with the highest grade in the class. 

"The idea of going on to graduate school came from this 
math professor. When I started I only intended to go through 
for an M.A. I didn't see my way clear further. This professor 



140 



Becoming a Physicist 



helped me to get a fellowship and that plus my father plus 
my wife's working made it possible for me to go. I started out 
intending to spend a year and a half and get an M.A. and go 
out teaching in mathematics. 

"Then again one of these things happened. The first sum- 
mer I took two courses in mathematics and for some strange 
reason I was assigned a course in physics. The two courses in 
mathematics were taught by two foreigners and they were the 
two most discouraging courses I've ever had in my life. One 
in particular was taught by a famous English mathematician 
and he was teaching completely over our heads. I thought it 
was my own dumbness. I worked as hard as I ever worked in 
my life and accomplished as little. A few days before the exam 
I mentioned it to one of the other students and he was feel- 
ing the same way. So the next class he had the nerve to go in 
before the teacher came in and he went up front and asked 
and pretty soon he discovered most of us were in the same 
boat so when the professor came in we stopped him and told 
him this. He asked around the class and they mostly said the 
same. He had assumed we had had two years of mathematics 
that we hadn't had and so he gave an exam I could have passed 
in high school. I was thoroughly disgusted with mathematics. 
The only course that was half decent was the physics course 
but I wasn't prepared for that. 

"At the end of the summer I thought I wouldn't go on with 
graduate school and I decided to go down town and get a 
job. If I still felt the same way I'd just continue working in- 
stead of going back next term. I got a job as a salesman. That 
was another lucky stroke. I went down and started putting 
the same effort into that. I began selling boys' shirts and I'd 
never bought a shirt in my life, mother always did. So I went 
to the library and got out three books on cloth. I read two 
that night and by the second day I understood a little more. 
I thought that if you wanted to be helpful in selling and it 
would be your job to learn what you were selling and it paid 
off as far as sales were concerned. Of course then it was said 
I was a sales grabber so I was told to take my turn. I said 
that was all right and did take my turn but I still maintained 
the highest sales, but it was because by then I was selecting 
out the good quality. I got called down for that, and they 
said there would be a lot of returns, but I asked them to check 



141 



it and there were hardly any. Then I had a run-in with the 
buyer and was transferred upstairs to sports goods and the 
same thing happened there. It was the same old trouble. No 
one ever bothered to study their stuflF. At the end of the month 
I saw very clearly that in an industrial job you didn't get any- 
where by knowing more or doing more than anyone else. By 
that time I was convinced that that side of the world was a 
pretty sorry one. 

"By then I had also decided I didn't want to go on in mathe- 
matics. That one course convinced me that physics was what 
I wanted. I had my fellowship transferred and had a long fuss 
with the Dean who wanted to assign courses and I wanted to 
work up to them. So I started out from there and with essen- 
tially undergraduate courses. 

"I liked it very much better and I found I somehow had 
time on my hands and very soon I wanted to try my hand 
in the lab. I had never had any tools in my hand. Again I 
had a lucky break. I went down and told the professor and 
said I'd like to try and I'd be glad to begin by opening boxes 
or anything else. He laughed and said as it happened there 
were a lot of boxes to open and so he put me to work. Presuma- 
bly lying dormant in my fingers was an ability I didn't know 
I had. Within a month I challenged him that I could make an 
electroscope work better than he and I won. I've always won- 
dered if he let me do it; he never would admit it but I would 
not expect him to. 

"I found that almost anything in experimental work I had 
no difficulty in doing. Glass-blowing and so on just came to 
me overnight. I learned mainly just by doing it. Machine work 
was all pretty much the same way. Handling the tools just 
came naturally as if I had been doing it for years. So much 
so that when I came here and took over the shop I said I'd 
never ask them to do anything I couldn't do myself. At first 
they sometimes said they couldn't do things, but I always 
showed them and since then there hasn't been any question." 

It is rare to find any planning ahead in the early years. 
Mostly the men just go from one thing to another, as occasion 
off^ers. The next story is particularly interesting from this point 
of view. He had an early bent to mechanical things. He went 
to college, largely because of his mother's dreams for him, 



142 



Becoming a Physicist 



but even there and after he had courses in physics, it was some 
time before he found out about research. His story is a par- 
ticularly good illustration, too, of a sort of unconsciousness 
about many aspects of living that is not uncommon at the col- 
lege years, and not unheard of beyond them. Ernest described 
himself to me as an experimentalist but one of his colleagues 
once told me that his greatest contributions had been theoreti- 
cal. 

"I really can't say when I got interested in things mechani- 
cal but it's just about as early as I can remember. About 6 or 
so I was interested in pretty much anything electrical, the 
usual things that kids are interested in, autos and so on. 

"Father never got even through high school and started 
at practically hard labor at 13 and got from that to be a star 
salesman. I don't know when he found time for the things 
he did. He was quite athletic and at that time there were 
amateur athletic groups and he was stroke. I never realized 
how good he was at the time but later I found some old papers 
and found that his crew was the best anywhere around. All 
the training was done after a day's work. Then some time later 
some of the books I read when I was a kid were some Inter- 
national Correspondence School texts on engineering which 
he had studied. That's a lot of work when you are working 
hard too. Father was a better man than I was or ever will be. 
Even when I was young and strong, my father was much 
stronger and tougher than I was always." 

References to parents show marked differences in the at- 
titudes of the sons. Ernest's respect for his father was very 
great, and this is generally characteristic of the physical sci- 
entists. It is less characteristic for them to have any great feel- 
ing of closeness to their fathers, or great aflFection, but Ernest 
and his father seem to have been very close. 

"Father had a strong mechanical bent and I learned quite 
a bit from him without realizing it. From the age of ten or so 
I was entrusted with keeping his car serviced. By the time 
I was 12 there were several of us interested in radio and we 
made a set. I was sort of leader and I did most of the design- 
ing and construction, the others did the operating. This was 
a transmitting and receiving station. I was always sure I 
wanted to be something of the engineering sort. I had never 
heard the word physicist, of course, and neither had either 
of my parents. I had fairly large sets. Meccano and Erector, 



143 



at a rather early age. You can get a lot of action for a reasona- 
ble amount of money. The folks would buy motors for toys 
and when I got to be old enough to be a radio amateur I was 
more organized and then it was mainly a question of making 
up my mind what I needed. We had all kinds of complicated 
arrangements. For a while we formed a small company to 
manufacture transformers. It was sort of a joke. The power 
company was putting in a lot of new transformers, and so we 
got any amount of stuff given us by the uncle of one of the 
boys and then we cooked up a deal with another's uncle to 
dig a cellar for $20 or $30 worth of wire, and we made some 
transformers and sold them. I never worked so hard in my 
life. We sure found things out the hard way. We had consider- 
able instruction but it was practically all of it from books and 
we found out how to do it the wnrong way first always. It just 
happened there were no radio amateurs around who knew 
more than we did so they learned from us. 

"Father never helped me make anything. On the other hand 
if I asked him how to do something he always knew and he 
had tools around which he got for his own purposes and which 
I appropriated so it's hard to describe. He never gave me any 
formal instruction but I learned a lot. Not about electricity 
but about mechanical things he was very, very good. 

"In high school I took chemistry and physics, all there was 
of both, about a year of each, and then some odds and ends 
of surveying and such courses. I took all there was of math 
and some that didn't exist, i.e. the math teachers were very 
interested in me and awfully kind to me and gave me instruc- 
tion in things that weren't really on the books and I learned 
some on the side myself. 

"I got through high school quite young and my folks didn't 
think I ought to go to college quite so soon so they sent me 
for a year to the technical high school there, so I had perhaps 
better training than ordinary in that way. That was a well-run 
course. I spent most of my time in the machine shop. 

"Going to college wasn't taken for granted. My father was 
the son of immigrant parents and had his first job as a black- 
smith, so college tradition in the family wasn't strong. It was 
mother's idea. Her father was a minister and she was of a 
fairly well educated family. Among my boy friends none went 
to college. I always had had a good time in school and would 



144 



Becoming a Physicist 



just read anything. I wouldn't say I liked all my studies but I 
liked anything scientific or mathematical and was all in favor 
of more school. Father was all for it but it was mother's idea 
in the first place. 

"I got a scholarship and went to college intending to be- 
come an electrical engineer that being the nearest thing we 
knew of to what I was interested in. Then my money ran out 
and I went home and continued in the college there. About 
then I had to take sophomore courses in physics and the pro- 
fessor thought well of me and he said, 'Why don't you go into 
physics?' It seemed a lot of fun and he thought he could stir 
me up a job at another college and said there wasn't much dif- 
ference between the physics and the electrical engineering 
courses and I could change back if I wanted to. I guess he 
must have done some considerable wrangling but he got me 
a job as assistant when I was a junior, and I came up here and 
thought that was a lot of fun. 

"I was pretty young and I guess not any too noticing about 
some things. I didn't realize there was such a thing as research 
either at that time. One fine day I was downstairs and saw 
someone wandering down the hall with a soldering iron, some- 
thing I recognized. He was a graduate student and didn't 
look like he knew what he was going to do so I went with him 
to help and spent most of my junior year working on his re- 
search and had a high old time working on it. 

"This was a small place in those days. No one told me how 
things ran. I didn't know about any of the places where peo- 
ple gathered. I'd seen this fellow around the teaching labs 
but I'd never heard of the idea of research. I'd taken courses 
and I thought that teaching was what professors did. The 
fellow I assisted for was one of the few that did not do re- 
search and I just saw him in his teaching laboratory. I didn't 
have any idea of what the student I helped was trying to do. 
I could see he was building things that he didn't know how 
to do and I did so I helped him for the fun of it. 

"There was an International Research Fellow here. He's 
a smart guy but pretty excitable and not dependable. By the 
time I got to be a senior it got to be recognized that I was 
pretty useful in the lab so they gave me to him for research 
associate and by that time it got time for me to graduate and 
I began to wonder what to do. This research Fellow was of- 
fered a job elsewhere and he could bring along anyone he 



145 



wanted so he asked me if I wouldn't like to go and I said sure. 
The next day I ran into the department head and told him this 
and he didn't say anything about it, but after a couple of 
weeks passed I got an offer of an instructorship here and that 
surprised me and I accepted. So I stayed here to get a Ph.D. I 
was only 20 and just had hardly grown up yet. I took chem- 
istry too and got along well in it and had a good time. I'm sure 
I would have been happy as a chemist only I just had more 
experience of thinking mechanically that made me seem to fit 
into physics better. 

"As it happened I worked on several problems at once, but 
the one I did my thesis on was a joint paper with the head, 
so he really suggested the problem and I just worked with 
him. It's a very rare student that can tell a good problem when 
he sees one, can start it off and carry it through. I certainly 
couldn't have." 



146 



Attempts to predict when things will happen, and what will 
be available in the future, are as fascinating as they are risky. 
Arthur Clarke, a science-fiction writer and scientist, has had 
unusual success in predicting future technical advances. 



21 Chart of the Future 

Arthur C. Clarke 

From his book Profiles of tfie Future — An Inquiry into the Limits 
of ttie Possible, 1962. 



THE PAST 



148 







Communication 


Materials 


Biology 




Date 


Transportation 


Information 


Manufacturing 


Chemistry 


Physics 


1800 


Locomotive 


Camera 

Babbage calcu- 
lator 


Steam engines 


Inorganic chem- 
istry 

Urea synthesized 


Atomic theory 




Steamship 


Telegraph 


Machine tools 




Spectroscope 


1850 










Conservation of 








Electricity 


Organic chem- 
istry 


energy 






Telephone 












Phonograph 






Electromagnetism 






0£Bce machines 






Evolution 




Automobile 




Diesel engine 






1900 


Airplane 




Gasoline engine 


Dyes 


X-rays 
Electron 






Vacuum tube 


Mass production 
Nitrogen fixa- 


Genetics 
Vitamins 
PlasUcs 


Radioactivity 


1910 




Radio 


tion 


Chromosomes 


Isotopes 
Quantum theory 


1920 








Genes 


Relativity 
Atomic structure 


1930 




TV 




Language of bees 
Hormones 


Indeterminacy 
Wave mechanics 
Neutron 


1940 


Jet 

Rocket 
Helicopter 


Radar 

Tape recorders 












Electronic com- 


Magnesium 


Synthetics 


Uranium fission 






puters 


from sea 


Antibiotics 


Accelerators 






Cybernetics 


Atomic energy 


Silicones 


Radio astronomy 


1950 




Transistor 


Automation 








Satellite 


Maser 


Fusion bomb 


Tranquihzers 


IG.Y. 




GEM 


Laser 






Parity overthrown 



Chart of the Future 



NOW 







Communication 


Materials 


Biology 




Date 


Transportation 


Information 


Manufacturing 


Chemistry 


Physics 


1960 


Spaceship 


Communication 




Protein struc- 


Nucleon struc- 






satellite 




ture 


ture 


THE FUTURE 




Space lab 










1970 


Lunar landing 


Translating 










Nuclear rocket 


machines 


EfiBcient electric 
storage 


Cetacean lan- 
guages 




1980 


Planetary land- 
ings 








Gravity 






Personal radio 




Exobiology 


waves 


1990 




Artificial intel- 


Fusion power 


Cyborgs 




2000 


Colonizing 


hgence 


"Wireless" en- 








planets 




ergy 


Time, perception 


Sub-nuclear 






Global Lbrary 


Sea mining 


enhancement 


structure 


2010 


Earth probes 


Telesensory de- 
vices 


Weather control 






2020 




Logical lan- 






Nuclear cata- 




Interstellar 


guages 




Control of 


lysts 




probes 


Robots 




heredity 




2030 




Contact with 
extra-terrestri- 
als 


Space mining 


Bioengineering 




2040 






Transmutation 


Intelligent animals 




2050 


Gravity control 
"Space drive" 


Memory playback 


Planetary 


Suspended 
animation 




2060 




Mechanical edu- 
cator 

Coding of artifacts 


engineering 


Artificial Lf e 


Space, time 
distortion 


2070 


Near-Lght speeds 




Climate 
control 






2080 


Interstellar flight 


Machine inteUi- 
gence exceeds 
man's 








2090 


Matter transmitter 




Replicator 








Meeting with 


World brain 




Immortality 




2100 


extra-terres- 




Astronomical 








trials 




engineering 







149 



Authors and Artists 



PERCY WILLIAMS BRIDGMAN 

P. W. Bridgman was born in Cambridge, Massachu- 
setts in 1882, entered Harvard in 1900, received 
his Ph.D. in physics there in 1908, and in 1913 
became Professor. He retired in 1954, and died in 
1961. Bridgman's experimental work was in high- 
pressure physics, for which he received the Nobel 
Prize in 1946. He has made important contributions 
to philosophy of science; for example, we owe him 
first detailed articulation of the concept of opero- 
tional definition. 



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- control led ap- 
prooch project. He has won the Kolinga Prize, 
given by UNESCO for the popularization of science. 
The feosibility of many of the current space de- 
velopments was perceived and outlined by Clarke 
in the 1930's. His science fiction novels include 
Childhoods End and The City ond the Stars . 



JACOB BRONOWSKI 



LEE DuBRIDGE 



Jacob Bronowski, who received his Ph. D. from 
Cambridge University in 1933 is now a Fellow of 
the Salk Institute of Biological Studies in Califor- 
nia. He has served as Director of General Process 
Development for the Notional Coal Board of Eng- 
land, 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 Pro- 
fessor at the Massachusetts Institute of Technology. 



HERBERT BUTTERFIELD 



Lee DuBridge was born in Terre Haute, Indiana in 
1901, and educated at Cornell College (Iowa) and 
the University of Wisconsin. During World War II 
he served as Director of the Radiation Laboratory 
at the Massachusetts Institute of Technology, 
where Rador was perfected. In 1946 he became the 
president of the California Institute of Technology 
and served in that capacity until becoming the Ad- 
viser to President Nixon on Science and Technol- 
ogy. His special fields of interest include bio- 
physics, nuclear physics, and photoelectric and 
thermionic emission. 



Herbert Butterfield is Professor of Modern History 
at the University of Cambridge. He graduated from 
Cambridge ond was elected a Fellow of Peterhouse 
at the same institution in 1923. He become Master 
of Peterhouse in 1955 and vice chancellor of the 
University in 1959. His writings include books on 
the history of religion, international affairs, and 
the history of science. 



ALEXANDER CALDER 

Alexander Colder, the American sculptor and in- 
ventor of the mobile, was born in Pennsylvania in 
1898. Intending to become an engineer, Colder en- 
tered the Stevens Institute of Technology, gradu- 
oting in 1919. But by 1926 he had already pub- 
lished his first book ( Animol Sketches ) and pre- 
sented his first exhibition of paintings. A visit 
with the Dutch artist Piet Mondrion in 1930 
oriented him toward abstraction, and the next year 
he produced the first "stabiles," and in 1932, the 
first "mobiles." In these mobiles. Colder was 
able to incorporate motion into sculpture. 



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 has been Tolmon Professor of Physics 
at the California Institute of Technology. Professor 
Feynmon received the Albert Einstein Award in 
1954, and in 1965 was named o Foreign Member of 
the Royol Society. In 1966 he was awarded the 
Nobel Prize in Physics, which he shared with 
Shinichero Tomonago and Julian Schwinger, for 
work in quantum field theory. 



JAMES BASIL GERHART 

James Gerhart is Professor of Physics at the Uni- 
versity of Washington in Seattle. Before coming to 
Washington, he tought at Princeton, where he re- 
ceived his Ph.D. in 1954. Professor Gerhart's 
specialty is nuclear physics. 



150 



I 



J. B. S. HALDANE 



GYORGY KEPES 



J. B. S. Haldane was a British geneticist who 
served as Professor of Biometry at University 
College, London. He pioneered in the application 
of mathematics to the study of natural selection 
and to other aspects of evolutionary theory. His 
broad grounding in mathematics, physics, and 
biology has enabled him to moke insightful con- 
tributions in many different areas. 



BANESH HOFFMANN 

Banesh Hoffman, born in Richmond, England in 
1906, attended Oxford and Princeton. He has been 
a member of the Institute of Advanced Study, elec- 
trical engineer at the Federal Telephone and Radio 
Laboratories, researcher at King's College, London, 
and a consultant for Westinghouse Electric Corpora- 
tion's science talent search tests. He has won the 
distinguished teacher award at Queens College, 
where he is Professor of Mathematics. During the 
1966-1967 year he was on the staff of Harvard 
Project Physics. 



GERALD HOLTON 



Gyorgy Kepes was born in 1906 in Selyp, Hungary. 
From 1930 to 1936 he worked in Berlin and London 
on film, stage, and exhibition design. In 1937 he 
came to the United States to head the Light and 
Color Department at the Institute of Design in 
Chicago. Since 1946 he has been Professor of 
Visual Design at the Massachusetts Institute of 
Technology. He has written The New Londscope 
in Art and Science, Language of Vision , and edited 
several books, including those in the Vision + 
Value series. Professor Kepes is one of the major 
painters; his work is included in the permanent 
collections of many museums. 



PAUL KIRKPATRICK 

Born in South Dokoto, Paul Kirkpotrick received 
his doctorate in physics in 1923. Before reaching 
Stanford in 1931, he tought in China and Hawaii. 
At Stanford, he was named Professor of Physics in 
1937, and became Professor Emeritus in 1959. 
Professor Kirkpotrick has served as education ad- 
visor with the U.S. Overseas Mission to the Philip- 
pines, and with the UNESCO mission to India. 



Gerald Holton received his eorly education in 
Vienna, at Oxford, and at Wesleyan University, 
Connecticut. He has been at Harvard Univer- 
sity since receiving his Ph.D. degree in physics 
there in 1948; he is Professor of Physics, teach- 
ing courses in physics as well os in the history 
of science. He was the founding editor of the 
quarterly Daedolus . Professor Holton's experi- 
mental research is on the properties of matter 
under high pressure. He is a co-director of Har- 
vard Project Physics, the group that developed 
materials on which the Project Physics Course 
is based. 



FRED HOYLE 

Fred Hoyle is an English theoretical astronomer, 
born in Yorkshire in 1915. Now Professor of Astro- 
nomy at Cambridge University, he is perhaps best 
known for one of the major theories on the struc- 
ture of the universe, the steady-state theory. Hoyle 
is well known for his scientific writing, ond his 
success in elucidating recondite matters for the 
layman. 



JAMES CLERK MAXWELL 

James Clerk Maxwell was born in Edinburgh, of a 
prominent Scottish family, in 1831. He graduated 
second in his class in mathematics at Cambridge, 
ond was appointed to a professorship at Aberdeen 
in 1856. Shortly thereafter he demonstrated that 
Saturn's rings were composed of small particles.. 
Next, Moxwell considered the mechanics of gases, 
and helped develop the kinetic theory. Maxwell's 
crowning achievement was his mothematical for- 
mulation of the laws of electricity and magnetism. 
He showed that electricity and magnetism were re- 
lated, and proposed that light was one form of elec- 
tromagnetic radiation. In 1871, Maxwell was ap- 
pointed first Professor of Experimental Physics at 
Cambridge. He died eight years later, his life cut 
short by cancer. 



151 



Authors and Artists 



HERBERT MATTER 



DUANE H. D. ROLLER 



Herbert Matter was born in Engelberg, Switzerland, 
on April 25, 1907. After graduating from college, he 
studied painting at L'Ecole des Beaux Arts in 
Geneva, and under Fernand Leger in Paris. In 1936 
he came to the United States to work as a free- 
lance photographer for Harper's Bazaar, Vogue, 
and others. Presently he is Professor of Photo- 
graphy and Graphic Design at Yale University. 



RUDI HANS NUSSBAUM 

Rudi Nussbaum was born in Germany in 1922, he 
received his Ph.D. from the University of Amster- 
dam in experimental physics in 1954. Since then 
he has served os UNESCO research fellow ot the 
Nuclear Physics Laboratory in Liverpool, as a 
senior fellow at CERN in Geneva, and is now 
Professor of Physics at Portland State College. 



Duane H. D. Roller was educated at Columbia 
University, Purdue University ond at Harvard Uni- 
versity. Since 1954 Dr. Roller has been at the 
University of Oklahoma, where he is McCasland 
Professor of the History of Science. 



C. L. STONG 

C. L. Stong was born in 1902 in Douds, Iowa. He 
attended the University of Minnesota, the Armour 
Institute in Chicago, ond the University of Michi- 
gan (Detroit). For thirty years he was an engineer 
with Western Electric. Mr. Stong has also been ir^ 
volved in movie production, ond in the eorly 1920's 
he was a stunt flier. Since 1948 he hos been a con- 
tributor to Scientific American , where his column. 
The Amoteur Scientist, appears monthly. 



GEORGE POLYA 



WARREN WEAVER 



George Polyo was born in Budapest in 1887. He 
studied in Vienna, Gottingen, and Budapest, where 
he received his doctorate in mathematics in 1912. 
He taught in Zurich, and in this country at Brown 
University, Smith College, and Stanford University, 
where he served as Professor of Mathematics from 
1946 to 1953. He is now Professor Emeritus. 



JACOPO DA PONTORMO uACOPO CARRUCCI) 

Born at Pontormo, Italy, May 24, 1494, Jacopo 
Carrucci, later to be known as Jacopo do Pontormo, 
wos one of the first of the Florentine Mannerists. 
Apprenticed to Leonardo da Vinci ond later to Al- 
bertinelli and Piero di Cosimo, Pontormo broke 
away from the classical High Renaissance style. 
His altarpiece (still in the church of S. Michele 
Visdomini, Florence) exemplifies his intensely 
emotional style, in contrast to the traditional har- 
monically balanced style. Pontormo was buried in 
Florence on January 2, 1557. 



ANNE ROE 

Anne Roe, a psychologist and educator, born in 
Denver, Colorado, was educated at the University 
of Denver and Columbia University. From 1947 to 
1951 she was the director of a psychological study 
of scientists, that resulted in the book The Moking 
of a Scientist . She is the wife of biologist George 
Gaylord Simpson. 



Warren Weaver received his Ph.D. in mathematics 
and physics from the University of Wisconsin in 
1921, and remained at his alma mater, becoming 
Professor of Mathematics and Chairman of the De- 
partment in 1928. In 1932 he was appointed Direc- 
tor of Natural Sciences at the Rockefeller Foundo- 
tion, and in 1955 was named Vice-president. He 
later wos associated with the Sloon-Kettering In- 
stitute, and since 1959 with the Alfred P. Sloan 
Foundation. He is the recipient of the Arches of 
Science Award given by the Pacific Science Center 
of Seattle "for outstanding contributions to the im- 
proved public understanding of science." 



BASIL WILLEY 

Bosil Willey was born in 1897 and later attended 
Peterhouse College, Cambridge, where he read his- 
tory and English. From 1946 to 1964 he served as 
King Edward VII Professor of English Literature 
at Cambridge. In 1958 he was selected as Presi- 
dent of Pembroke College, Cambridge, and is now 
an Honorary Fellow. His published works include 
many studies in English and the history of ideas. 



152 



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